diff --git "a/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt" "b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/b9FAT4oBgHgl3EQfXx3V/content/tmp_files/load_file.txt" @@ -0,0 +1,1251 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf,len=1250 +page_content='RASTI 000, 1–20 (2022) Preprint 23 January 2023 Compiled using RASTI LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 Overcoming Separation Between Counterparts Due to Unknown Proper Motions in Catalogue Cross-Matching Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson1★ ID 1School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' in original form ZZZ ABSTRACT To perform precise and accurate photometric catalogue cross-matches – assigning counterparts between two separate datasets – we need to describe all possible sources of uncertainty in object position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With ever-increasing time baselines between observations, like 2MASS in 2001 and the next generation of surveys, such as the Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Rubin Observatory’s LSST, Euclid, and the Nancy Grace Roman telescope, it is crucial that we can robustly describe and model the effects of stellar motions on source positions in photometric catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While Gaia has revolutionised astronomy with its high-precision astrometry, it will only provide motions for ≈10% of LSST sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' additionally, LSST itself will not be able to provide high-quality motion information for sources below its single-visit depth, and other surveys may measure no motions at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This leaves large numbers of objects with potentially significant positional drifts that may incorrectly lead matching algorithms to deem two detections too far separated on the sky to be counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To overcome this, in this paper we describe a model for the statistical distribution of on-sky motions of sources of given sky coordinates and brightness, allowing for the cross- match process to take into account this extra potential separation between Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We further detail how to fold these probabilistic proper motions into Bayesian cross-matching frameworks, such as those of Wilson & Naylor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will vastly improve the recovery of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' very red objects across optical-infrared matches, and decrease the false match rate of photometric catalogue counterpart assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Key words: Algorithms – methods: statistical – catalogues – astrometry – proper motions – Galaxy: kinematics and dynamics 1 INTRODUCTION Counterpart assignment, the merging of bandpass detections in two (or more) datasets, enables a wide range of value-added science, and is therefore a crucial aspect of many areas of astronomical re- search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Fundamentally, we require the ability to answer the question ‘are these two detections observations of two different objects, or two observations or the same astrophysical object?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, to pro- vide accurate and precise cross-matches between two photometric catalogues, we require a complete description of all sources of sep- aration between detections of a single astrophysical object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Unfortunately for astronomers, there are many reasons for the same source, detected by two different telescopes in different parts of the world at different times, to have recorded positions that are not perfectly aligned with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first, and frequently assumed only, contribution is that from the act of measuring the position of the source on the detector image as part of the catalogue creation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This ‘centroid’ uncertainty is related to the size ★ Email: t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='wilson@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' onoddil@pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='me of the telescope and the wavelength of the observation, as well as the atmospheric seeing, if applicable – all of which affect the telescope ‘point spread function’ (PSF), as well as the signal-to- noise ratio (SNR) of the detection, related to its brightness (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' King 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor (2017) highlighted an additional source of positional shift that can affect detections in crowded fields: sources, too close together on the sky to be resolved by the telescope, can appear as a single observation, leading the fainter source to influence the position of the (assumed singular) brighter object (sometimes referred to as ‘classical confusion’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' see also e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hogg 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we consider an extra source of apparent separation be- tween detections: that of the physical motion of the source across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the observations to be combined are sufficiently sepa- rated in time, the ‘proper motion’ of sources introduces a drift in the separation between consecutive measurements of the objects’ locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus for pairs of observations with significant baselines, the proper motion-induced separations can become significant for large enough numbers of objects that, if we failed to consider these motions, we would decide the objects were too far apart to be coun- terpart detections of one object, and fail to assign them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08536v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='SR] 20 Jan 2023 2 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson For probabilistic cross-matching algorithms this problem of object drift is further compounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is not only objects with sig- nificant proper motion that suffer, those few objects with motions large enough to render them completely incompatible with the hy- pothesis that the two detections are counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' All objects, even those with relatively small motions, are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Any motion on the same length scale as the astrometric precisions will impact the derived match confidence, and potentially render quoted match or non-match probabilities meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This issue of your chosen model completely encapsulating the information contained within your data (or not) is often referred to as ‘model (mis)specification’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Therefore, the effect of source motion must be accounted for, even if not so extreme as to completely move an object beyond its prior position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If it is not taken into account, users of any resulting cross- match tables may not be able to put trust in the quoted match likelihoods and be able to take reliable, high-confidence cuts of the merged datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect is particularly important for the upcoming Vera C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Rubin Observatory’s Legacy Survey of Space and Time (LSST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), for a few key reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, it will operate from ∼2025-2035, and thus have a two or three-decade baseline to the numerous surveys that operated during the 2000s and 2010s, such as 2MASS (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2006) or SDSS (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' And second, it will lack measured proper motions for almost all of its sources for a large fraction of its survey lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In part this is because the survey will require a multi-year baseline before reliable proper motions can be derived, but more simply because most ob- jects within the full LSST catalogue will be below the completeness limit of the single-visit images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Even in this specific case, with Ru- bin’s high-fidelity time-series capabilities, proper motions will only ever be available for objects that appear in multiple images, which sets the proper motion magnitude limit much higher than that of inclusion in the full coadd catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Worse still, the sheer number density of objects in the full LSST catalogue mean that up to 10 LSST sources will be potential counterparts to every single oppos- ing catalogue object, and the ‘re-shuffle’ of objects, even of order the precision of the measured positions, may lead to false matches being returned for a sizeable fraction of the catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, the survey will be especially susceptible to this match misspecification due to proper motion drift, primarily at faint magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Addition- ally, other upcoming missions such as Euclid (Laureijs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2011) and the Nancy Grace Roman Space Telescope (Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2012) will likely lack the multi-epoch capabilities that Rubin and LSST offer, but still suffer the effects of decade-long time baselines back to previous generations of deep surveys, such as SDSS or VISTA (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' VHS, McMahon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As time goes on, and we accumulate increasing numbers of surveys we wish to combine to maximum scientific return, we will increasingly no longer be able to ignore even relatively small lev- els of apparent on-sky motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' One obvious solution is to use the individual proper motions available through datasets such as the Gaia (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2016) mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These positions – combined with the rate-of-change of position from the proper mo- tions – can be ‘fast-forwarded’ through time, allowing for sources to be placed in the epoch of the opposing catalogue, removing on- sky drift as a factor in considering the separation between sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, this is impractical for surveys such as LSST for a couple of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, and simplest, is that proper motions are not available for the entire Gaia catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Something like 20% of sources in the early Data Release 3 (eDR3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lin- degren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021) do not have the five- or six-parameter solutions necessary to include proper motions, and a not insignificant frac- tion of those that have quoted proper motions have uncertainties that render the quoted values useless for any meaningful position projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Second, this would result in needing to run two Gaia- to-other-catalogue matches, merge the most likely of those matches in turn, and then run an internal Gaia-Gaia look up to get the in- ner join of the two catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This would significantly affect the quality of the resulting matched datasets, with probabilistic cross- matching processes not being able to provide the proper probability of sources in the two ‘other catalogue’ datasets matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Third, and most crucial, is the dynamic range consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia is, for all its superb data, a relatively bright survey – at least by LSST standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It also lacks coverage against longer wavelength surveys, where Galactic extinction is less oppressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' LSST will, with its ∼7 magnitude fainter completeness limit, include at least an order of magnitude more stars (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), and VISTA, as an example infrared (IR) catalogue of consideration as an ancillary dataset to extend LSST information with, will have little overlap with Gaia due to differing wavelength coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Other surveys with deeper completeness limits, such as Euclid, Roman, and SDSS, will also suffer significant numbers of matches beyond the Gaia com- pleteness limit, with neither offering reliable proper motions at 21st magnitude or fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, even if we did decide to peg Gaia as our gold standard, this would leave perhaps 9 out of every 10 LSST Galactic sources without a proper motion match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Those LSST ob- jects, and many others in other catalogues, would be in need of a separate, and worse, cross-match once we had handled those few objects with a Gaia proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If we cannot trust matches between observations with signifi- cant time between observations, and we cannot necessarily use an- cillary datasets with measured proper motions, how can we recover robust catalogue counterpart assignments through cross-matching?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Naively, we might think that we can ‘re-center’ the distribution of offsets, by subtracting some mean separation between all of our counterpart pairings to account for the drift of our objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' How- ever, different average proper motions across the dynamic range of the two catalogues would cause further systematics, affecting the distribution of counterpart separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' One may also think to use a Galactic model that calculates stellar velocities, and hence provides proper motions as viewed from Earth, for example the Besançon model (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, as discussed in more detail in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8, there are various reasons that these proper motions do not provide robust enough statistics for the determination of a statistical separation drift between two potential counterpart stars during a probabilistic cross-match process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, in this paper we put forward a model to build a statistical distribution of proper motions of sources, based on their Galactic motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Modelling all sources of motion a star orbiting the Galactic center is subject to – its bulk circular orbit, and any ‘random’ scatter of sources from e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' stellar cluster interactions – combined with the Sun’s motion, we are able to build a picture of the apparent motion of the given object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do so, we begin with the velocity of the star as it orbits around the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The conversion from velocity – in units like km s−1 – to proper motion – in units like arcsec yr−1 – is, roughly speaking, an inverse relation with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As we detail later, instead of using distance directly, we intend to use the brightness of an object as its more readily available surrogate, accepting that this is only an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, combining the apparent motions of a wide range of objects at the same sky position and brightness we obtain all proper motions such a source might have – faint M dwarfs close by would have larger apparent motions than very intrinsically bright super- RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 3 giants, but all ‘types’ of object contribute to the spread of motion drifts a source of this particular brightness could have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We are not particularly interested in a precise reconstruction of Galactic orbital dynamics – not being overly concerned with the details of spiral arm dynamics, or the specifics of the Milky Way Bar shape, for ex- ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The use of the proper motions here is to ‘spread’ the motion drift, the separation between potential counterparts, allowing for the recovery, and increasing the reliability of match probability, of these objects within a Bayesian cross-matching framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There- fore, the exact shape is less important than its central location and width;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' so long as these match reality to a fraction of the precision of the objects’ positions and the underlying distribution width to a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 − 2, the model has served its purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The probability of two stars being counterpart ranges over many orders of magnitude, and hence the resulting probability density functions (PDFs) we derived to model the statistical proper motions having widths, and overall PDF heights, correct to a factor two is sufficient to improve counterpart recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also desire computational simplicity, and thus speed, over an overly prescriptive or detailed exact model of the Galaxy, as this model must fit within a wider counterpart assign- ment framework and be able to be run on-the-fly for some arbitrary sets of sky positions and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once we have constructed our distribution of theoretical proper motions, we can then consider their effect on our potential cross- match pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we can, in a similar way to how we would han- dle known Gaia proper motions, translate one object’s position into the epoch of the second catalogue observations, and consider the additional separation caused by the motion of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We must then test all modelled proper motions, with appropriate weighting, ultimately accounting for these potential additional time-based sep- arations in answering the question of whether these two detections are two physical sky objects or one source viewed twice in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Paper Layout This paper is split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we detail the construction of a simple analytic model for the statistical distribution of potential proper motions of a source of a given magnitude and sky coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In Section 2 we detail the constituent parts necessary to build the model of proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We describe the process of building the distributions in Section 3, while Section 4 discusses the preci- sion and accuracy of the model at various Galactic sightlines and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we will return to Gaia, using its high-precision stellar proper motions across many Galactic sightlines to evaluate and corroborate our model distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The second part of this work describes how to include this unknown proper motion distribution – or any distribution of proper motions, theoretical or poorly constrained yet detected – in the cross-matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We discuss the mathematical framework necessary for including proper motion drift in the evaluation of the separation between potential counterpart detections in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also touch upon how to use these statistical distributions of proper motion drift as a discriminator between stars and galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Concluding remarks are given in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In Appendix A we outline the various coordinate systems and derive the transformation matrices used throughout this work, while in Appendix B we detail the inclusion of the proper motion PDFs within a probabilistic cross-match astrometric separation likelihood framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2 CONSTRUCTING THE PROPER MOTIONS We need to build a model to describe the observed motions of sources across the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There are, essentially, three components that matter: first, the ‘peculiar’ motion of the Sun itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' second, the expected velocity of a source moving with the Galactic rotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and third, the random velocities of the Galactic sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' these will be discussed individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we must consider how we will build this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As the model involves consideration of the Galactic rotation (and we will see later that source random motion is location depen- dent), we will require a description of the position of the source in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first two components are easy: sky coordinate in Galactic coordinates (converting Equatorial 𝛼 and 𝛿 by rotation to longitude 𝑙 and latitude 𝑏, if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The only other component we would need is a distance, or parallax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' however, if we have paral- lax we likely have a unique proper motion, as these are generally fit for simultaneously, and hence we use the next best proxy: magni- tude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will blend several ‘types’ of source together (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' dwarfs and giants of the same brightness are at different distances).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will see that while this might introduce extra scatter in the proper motion drifts, our models match Gaia proper motions at magnitude cuts well – and indeed account for the fact that we do not know the type of any individual source from its photometry a priori!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we do need some distance metric, and hence we turn to the TRILEGAL simulations (Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2005) to provide a theoretical magnitude-distance relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, while our catalogue sources have their proper motions built as a function of sky coordi- nates and photometric brightness, our model is coordinate/distance based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the following sections we describe how we formulate a description of the observed proper motion of a set of sources based on their given parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Solar Peculiar Motion The first component in the Galactic motion is the unique velocity of the Sun, relative to the local standard of rest (LSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Sun’s motion through the Galaxy will induce a ‘secular’ parallax effect (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' distance-dependent, albeit non-periodic, as a trigonometric par- allax would be) in the apparent movement of all other sources in the sky, with opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence we need to know the Sun’s mo- tion, relative to this ‘zero point’ motion, the LSR, defined in the Heliocentric Cartesian coordinate frame, (𝑈, 𝑉, 𝑊) – velocities corresponding to the (𝑥, 𝑦, 𝑧) coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we use the values of Schönrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010): 𝑈⊙ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 km s−1 𝑉⊙ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 km s−1 𝑊⊙ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactic Rotation The main component of our model that will dictate the Galaxy-wide observed motions of sources is that of the Galactic rotation, and the stellar streaming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Descriptions of this motion go back to Oort (1927), with the Oort constants describing the motion of stars on closed orbits around the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, through modern kine- matic surveys, obtaining the three-dimensional velocities and in- dependent distances to a host of well-characterized objects, it is possible to directly measure the rotation curve of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 4 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson Thus, we can derive the average tangential velocity of sources or- biting the Galactic center at a given radius, here following Model 3 of Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Obtaining the rotational velocity Θ at a given Galactocentric radius 𝑅𝑐, we can transform this Galactocentric Cylindrical az- imuthal velocity into a Galactocentric Cartesian coordinate frame and subtract the Solar peculiar and LSR motion, obtaining 𝑈1, 𝑉1, and 𝑊1 (Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019, equations 5-7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do so, we use the transformation T𝑡 = ����� � 𝑅2 𝑐+𝑅2 ⊙−𝑑2 ip 2 𝑅𝑐 𝑅⊙ 𝑑ip 𝑅𝑐 sin(𝑙) 0 − 𝑑ip 𝑅𝑐 sin(𝑙) 𝑅2 𝑐+𝑅2 ⊙−𝑑2 ip 2 𝑅𝑐 𝑅⊙ 0 0 0 1 ����� � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2) here 𝑅⊙ is the Solar Galactocentric Cylindrical radius, 𝑑ip is the in-plane distance from the Sun to the particular location, and 𝑙 is Galactic longitude – see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Deviating from Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019), we set (𝑈𝑠, 𝑉𝑠, 𝑊𝑠), the non-circular motion of the source, all to zero, and hence have 𝑈1 = Θ(𝑅𝑐) × 𝑑ip 𝑅𝑐 sin(𝑙) − 𝑈⊙ 𝑉1 = Θ(𝑅𝑐) × 𝑅2𝑐 + 𝑅2 ⊙ − 𝑑2 ip 2 𝑅𝑐 𝑅⊙ − 𝑉⊙ − Θ⊙ 𝑊1 = −𝑊⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (3) Once we have the Galactocentric Cartesian components of the rota- tional velocity, relative to the Sun, we can transpose into the in-plane Heliocentric radial and tangential velocities, 𝑣𝑑 = 𝑈1 cos(𝑙) + 𝑉1 sin(𝑙) 𝑣𝑙 = −𝑈1 sin(𝑙) + 𝑉1 cos(𝑙), (4) along with 𝑣𝑧 = 𝑊1, since the two axes are still in alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, once we have appropriate Heliocentric Cylindrical velocities, we can construct our latitudinal velocity as 𝑣𝑏 = 𝑣𝑧 cos(𝑏) − 𝑣𝑑 sin(𝑏) (5) and relate the Heliocentric longitudinal and latitudinal velocities to their proper motions through 𝜇𝑙∗ ≡ 𝜇𝑙 cos(𝑏) = 𝑘 × 𝜋𝑣𝑙, (6) 𝜇𝑏 = 𝑘 × 𝜋𝑣𝑏, (7) where 𝜋 is the parallax of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As our distances come from Galactic models, we assume they are not subject to any observational bias or uncertainty, and simply treat 𝜋−1 = 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The factor 𝑘 describes the translation from units of km s−1 kpc−1 to mas yr−1, and is given by 𝑘 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2108 mas yr−1 km−1 s kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In practice, the TRILEGAL simulations do not provide either distance or parallax, but provide its distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, to obtain a (three-dimensional) distance 𝑑 in kpc, we invert the absolute magnitude equation: 𝑑 = 10−3 × 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2(𝑚−𝑀)+1 (8) where 𝑚 − 𝑀 is the distance modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Asymmetric Drift Velocity The above equations describe the average Galactic rotation velocity around the center of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, objects have other sources of velocity that impact their observed proper motions, and this leads to a deviation away from the expected velocity, and thus proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is termed the asymmetric drift velocity, and essentially controls how much of the theoretical velocity a source should have is taken by other, random motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, we must include this component in the velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We model three Galactic components (see Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 for more details of their construction) in our simulations: the Galac- tic thin and thick discs, and a single Galactic (outer) halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Each of these is given their own asymmetric drift, as a measure of the levels to which their motions are different from ‘pure’ streaming motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We assume the thin disc of the Galaxy has an azimuthal asymmetric drift velocity of 10 km s−1 (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019) used Classical Cepheids in the derivation of their ro- tation curve, we also assume the rotation curve is valid for the thin disc and already folds in any drift velocity, and therefore just need to consider the relative drift velocities of the thick disc and the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use a thick disc drift velocity of 49 km s−1 (Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2012a), and give the halo a drift velocity of 240 km s−1 (Golubov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013), to essentially counteract the motion of the LSR, mod- elling the halo as stationary relative to the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore use 𝑣𝑎,𝜙 = {0 , 39 , 230} km s−1 for the relative thin disc, thick disc, and halo drift velocities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For a given location in the Galaxy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' our decomposition of the drift velocity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' from Galactocentric Cylindrical coordinates into He- liocentric Cylindrical coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' is given by the transformation 𝒗′drift = T𝑐 𝒗drift,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (9) with T𝑐 = ����� � 𝑅2 𝑐+𝑑2 ip−𝑅2 ⊙ 2𝑅𝑐𝑑ip 𝑅⊙ 𝑅𝑐 sin(𝑙) 0 𝑅⊙ 𝑅𝑐 sin(𝑙) − 𝑅2 𝑐+𝑑2 ip−𝑅2 ⊙ 2𝑅𝑐𝑑ip 0 0 0 1 ����� � (10) and 𝒗drift = �� � 0 𝑣𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='𝜙 0 �� � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (11) with 𝑣𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='𝜙 taking on any one of the three given drift velocities above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' depending on which component of the Galaxy is being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For details on the derivation of this transformation (rotation and mirror) matrix, see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Velocity Dispersion The asymmetric drift velocity suggests that some of the motion that ought to be used by a given source in its rotation around the Galaxy is being used otherwise, in a random component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, a collection of sources in a particular part of the Galaxy will have some spread of their velocities around some mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, to be able to model our collection of sources in the Galaxy we require a description of the dispersion of the velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Each component of the Galaxy modelled – thin and thick discs, and halo – have their own prescription of velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, when simulating sources, we do not initially know to which component to assign a given simulated object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Therefore we simply generate a set of proper motion realisations for each of the three components, weighted according to their a priori density at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, in the following sections we also describe the formulation of each component’s density profile, which are simply re-normalised by the sum of their densities to provide a prior probability, used as the weight for the proper motion distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 5 We currently do not consider the Bulge, a common component of Galactic simulation models such as TRILEGAL, in our proper motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We chose to ignore this additional component for this initial, exploratory model, focussing on relatively bright sources, with detected Gaia proper motions to compare and verify our model against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This should mean they are sufficiently far from the Galactic center to not be influenced by the Bulge, avoiding the complexities that the inner region of the Galaxy impose on velocities of orbiting stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, with the upcoming LSST survey and the need to model much fainter, more distant objects in the next few years, we will investigate a robust Galactic Bulge/Bar model for inclusion within this proper motion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We simply conclude, for now, that it would be easy to add additional components, and we could model a simple Bulge component after e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Jackson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once a given Galactic component has its dispersion vector – its covariance matrix – in the Heliocentric Cylindrical coordinate system, then a realisation is drawn from a multivariate normal, given by 𝒗noisy,𝑖 ∼ N (𝒗 − 𝒗′drift,𝑖, 𝚺′𝑖) (12) where 𝑖 ∈ {thin, thick, halo} and 𝚺′𝑖 is the Cylindrical frame rotated covariance matrix of the 𝑖th component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Thin Disc The thin disc is modelled as an exponentially decaying density profile with given radial and vertical scale heights, as per Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008): 𝜌(𝑅𝑐, 𝑧) = Γ exp � − 𝑅𝑐 − 𝑅⊙ 𝑙thin − 𝑧 + 𝑧⊙ ℎthin � , (13) with 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='09 kpc (Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), 𝑧⊙ = 25 pc (Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Γ is an irrelevant normalisation constant, used simply to explicitly cancel in the re-normalisation from density to weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The radial and vertical scale lengths we use are the bias-corrected values calculated by Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' : 𝑙thin = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc, and ℎthin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The dispersion vector for the thin disc is based on observations of RAVE stars from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, the nature of the observations limit their calculation of covariances to approximately 1 kpc from the Sun, and we need to extrapolate these dispersions out to perhaps five times that distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we turn to Amendt & Cuddeford (1991) for relations between the various (co-)variances in the dispersion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we assume that the variance in the vertical direction, 𝜎2𝑧𝑧, scales with radial distance in the mid-plane of the Galaxy: 𝜎2 𝑧𝑧 (𝑅𝑐, 𝑧 = 0) = 𝜎2 𝑧𝑧(0, 0) exp �� − 𝑅𝑐 𝑙thin � = 𝜎2 𝑧𝑧(𝑅′⊙, 0) exp � − 𝑅𝑐 − 𝑅′⊙ 𝑙thin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (14) As discussed by Amendt & Cuddeford, this scaling relation is also sometimes assumed for 𝜎2 𝑅𝑐𝑅𝑐, but a second, valid formalism can be used where the rotation curve of the Galaxy is flat – which it should be safe to assume given the very small gradient from Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019) for most of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This formalism is based on a constant Toomre (1964) local stability parameter, and gives 𝜎2 𝑟𝑟 ≡ 𝜎2 𝑅𝑐𝑅𝑐 ∝ 𝑅2 𝑐 exp � −2 𝑅𝑐 ℎthin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (15) Hence we use1 𝜎2 𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 𝑟𝑟 (𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthin � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (16) For the vertical extrapolation, we assume a local Taylor expan- sion to first order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we extrapolate linearly to above and below the plane, from 𝑧 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We limit this extrapolation to the inner one kiloparsec of the plane, and assume a constant dispersion beyond that, and hence: 𝜎2 𝑧𝑧(𝑅𝑐, 𝑧) ≃ 𝜎2 𝑧𝑧(𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑧𝑧 (𝑅𝑐, 0) 𝜕|𝑧| (17) 𝜎2 𝑟𝑟 (𝑅𝑐, 𝑧) ≃ 𝜎2 𝑟𝑟 (𝑅𝑐, 0) + min (1 kpc, |𝑧|) 𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) 𝜕|𝑧| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (18) Using the Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' data in the range 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 kpc ≤ 𝑅𝑐 ≤ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 kpc, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc ≤ 𝑧 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc, we find 𝜎2 𝑧𝑧(𝑅′⊙, 0) = 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='71 km2 s−2, 𝜕𝜎2𝑧𝑧(𝑅𝑐, 0) 𝜕|𝑧| = 306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='84 km2 s−2 kpc−1, 𝜎2 𝑟𝑟 (𝑅′⊙, 0) = 715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='93 km2 s−2, 𝜕𝜎2𝑟𝑟 (𝑅𝑐, 0) 𝜕|𝑧| = 1236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='97 km2 s−2 kpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (19) Additionally, to be consistent with the data as derived by Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', we use 𝑅′⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc – note that 𝑅′⊙ ≠ 𝑅⊙ – for extrapolating the dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we have assumed their quoted location of the Sun ‘in the range 𝑅 ∈ ]8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6] kpc’ implies2 an assumed default location in the middle of the bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We assume, following Amendt & Cuddeford (1991) and Val- lenari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2006), that 𝜎2 𝑟𝑧 (𝑅𝑐, 𝑧) ≃ 𝜎2 𝑟𝑧(𝑅𝑐, 0) + 𝑧 𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) 𝜕𝑧 (20) where the first term on the right hand side vanishes by symmetry at 𝑧 = 0, and the derivative is given by 𝜕𝜎2𝑟𝑧 (𝑅𝑐, 0) 𝜕𝑧 = 𝜆(𝑅) 𝜎2𝑟𝑟 (𝑅𝑐, 0) − 𝜎2𝑧𝑧(𝑅𝑐, 0) 𝑅𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (21) Given no information on the radial dependence of 𝜆, we fix it to the local value of 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 (Amendt & Cuddeford 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In cases where the linear extrapolation would result in a correlation � 𝜌𝑟𝑧 ≡ 𝜎2 𝑟𝑧 𝜎𝑟𝑟 𝜎𝑧𝑧 � larger in absolute value than one, we force the correlation back to either +1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Following Vallenari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2006), this is the only off-diagonal term we consider for the thin disc covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, again following the prescription of Amendt & Cuddeford, we assume that the azimuthal and radial dispersions are related by a constant, and hence use 𝜎2 𝜙𝜙 = −𝐵 𝐴 − 𝐵 𝜎2 𝑟𝑟, (22) with 𝐴 and 𝐵 the Oort (1927) constants, for all 𝑅𝑐 and 𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use the Olling & Dehnen (2003) Oort constant values (their table 5, figure 1 Using 𝜎2𝑟𝑟 for the Galactocentric Cylindrical frame radial dispersion component, to avoid the slightly clunky notation 𝜎2 𝑅𝑐 𝑅𝑐 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2 Inclusive of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc but exclusive of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 kpc, equivalent to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 6 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson 6), as a function of intrinsic colour 𝐵 − 𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we interpolate 𝐴 and 𝐵 as a linear function of (𝐵 − 𝑉)0, fitting: 𝐴 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='94553 × (𝐵 − 𝑉)0 + 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='33138 𝐵 = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='63360 × (𝐵 − 𝑉)0 − 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='60611 (23) as shown in Figure 1 (left-hand panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As we are using TRILEGAL simulations, we require a conversion from available TRILEGAL parameters to (𝐵 − 𝑉)0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' for this we use the dwarf colour sequence of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We fit a two-step function to the intrinsic B-V colour as a function of effective temperature (with 𝑇 in units of Kelvin), (𝐵 − 𝑉)0 = ����� ����� − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='40739 + 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='07836 × exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='27083 × 𝑇/1000K) 𝑇 < 10000 K − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='35093 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='69012 × exp(−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08179 × 𝑇/1000K) 𝑇 ≥ 10000 K (24) as shown in Figure 1, right-hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once we have all of the terms, we rotate the covariance ma- trix from its Galactocentric cylindrical coordinate frame into the Heliocentric coordinate system by 𝚺′ = T𝑐 𝚺 T𝑇 𝑐 = T𝑐 ��� � 𝜎2𝑟𝑟 0 𝜎2𝑟𝑧 0 𝜎2 𝜙𝜙 0 𝜎2𝑟𝑧 0 𝜎2𝑧𝑧 ��� � T𝑇 𝑐 (25) using the rotation matrix as defined in equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Thick Disc The formalism for the thick disc is very similar to that of the thin disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also use the exponential decay model for the density profile, albeit with different scale lengths: 𝜌(𝑅𝑐, 𝑧) = Γ 𝑓thick exp � − 𝑅𝑐 − 𝑅⊙ 𝑙thick − 𝑧 + 𝑧⊙ ℎthick � , (26) again following the Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) formalism, with 𝑅⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='09 kpc and 𝑧⊙ = 25 pc again, and 𝑙thick = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 kpc, and ℎthick = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='9 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, the parameter 𝑓thick = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='13 sets the relative densities of the thin and thick discs, and Γ again is an arbitrary normalisation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The thick disc dispersion vector uses the data from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012a), again following the same radial scaling relations for the thin disc: 𝜎2 𝑟𝑟 (𝑅𝑐, 0) = 𝜎2 𝑟𝑟 (𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthick � (27) 𝜎2 𝜙𝜙(𝑅𝑐, 0) = 𝜎2 𝜙𝜙(𝑅′⊙, 0) � 𝑅𝑐 𝑅′⊙ �2 exp � −2 𝑅𝑐 − 𝑅′⊙ ℎthick � (28) 𝜎2 𝑧𝑧(𝑅𝑐, 0) = 𝜎2 𝑧𝑧(𝑅′⊙, 0) exp � − 𝑅𝑐 − 𝑅′⊙ ℎthick � (29) where, once again, we use 𝑅′⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 kpc from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012b), assuming the two papers were jointly analysed and hence have the same 𝑅′⊙, although neither paper in the series quote a specific value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This time, we do not describe any vertical dependency of the dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, we take the diagonal terms as presented by Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2012a) within or without the Solar circle as the values approximately at (𝑅′⊙, 0), as given by their tables 3 and 4 respectively, but ignore the off-diagonal terms, which are all within ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5𝜎 of zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Exactly the same as with the thin disc, we rotate the Galac- tocentric Cylindrical reference frame into Heliocentric Cylindrical coordinates by 𝚺′ = T𝑐 𝚺 T𝑇 𝑐 = T𝑐 ��� � 𝜎2𝑟𝑟 0 0 0 𝜎2 𝜙𝜙 0 0 0 𝜎2𝑧𝑧 ��� � T𝑇 𝑐 (30) again using the cylindrical rotation matrix of equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Halo The halo Galactic component follows the density profile 𝜌(𝑅𝑐, 𝑧) = Γ 𝑓ℎ ����� � 𝑅⊙ √︂ 𝑅2𝑐 + � 𝑧 𝑞 �2 ����� � 𝑛 , (31) again using the Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) and Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2008) formalism, with 𝑓ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0051, 𝑞 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='64, 𝑛 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Γ is a normalising constant once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This parameterization of an inverse power law leads, at 𝑅𝑐 = 0, 𝑧 = 0, to an infinite halo density, and hence unphysical normalising weighting in the Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore truncate the halo density within the solar circle, 𝑅⊙, fixing it at its value at 𝑅𝑐 = 𝑅⊙ at smaller radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This ought to be possible because the old Galactic halo should be negligible in relative density by the solar circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The dispersion vector for the halo is derived from King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2015), given in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We take the full covariance matrix from the closest radial bin from the ‘Equally Populated Bins’ in their table 3 for a given set of (𝑅𝑠, 𝜙, 𝜃) parameters for a given source, with the exception of their 𝑅𝑠 = 12 kpc bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This bin gives a covariance matrix that is not positive semi-definite, and hence we ignore the off-diagonal terms for that individual bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These have no scaling applied to them and are taken exactly as quoted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To rotate into the Heliocentric Cylindrical reference frame, we use 𝚺′ = R𝑠𝑐 𝚺 R𝑇 𝑠𝑐 (32) where3 𝚺 = ��� � 𝜎2𝑟𝑟 Σ𝑟 𝜙 Σ𝑟 𝜃 Σ𝑟 𝜙 𝜎2 𝜙𝜙 Σ𝜙𝜃 Σ𝑟 𝜃 Σ𝜙𝜃 𝜎2 𝜃 𝜃 ��� � , (33) R𝑠𝑐 = T𝑐R𝑠, (34) R𝑠 = �� � cos(𝛽) 0 −𝑑/𝑅𝑠 sin(𝑏) 0 1 0 𝑑/𝑅𝑠 sin(𝑏) 0 cos(𝛽) �� � , (35) and where Σ𝑟 𝜃 ≡ 𝜎2 𝑟 𝜃, following the King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' R𝑠 describes the rotation from Galactocentric Spherical coordinates to Galactocentric Cylindrical coordinates, with T𝑐, as before, defining the rotation from Galactocentric Cylindrical to Heliocentric Cylin- drical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝛽 is defined as the angle between the spherical radial vector and the Galactic plane (𝑏 = 0◦), with 𝑑 the three- dimensional distance to the source in question, and 𝑅𝑠 the three- dimensional Galactocentric distance to the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For more details on the derivation of this transformation matrix, see Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 3 Once again, 𝑟 has been used instead of 𝑅𝑠 for notation’s sake, analogous to the thin and thick disc notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 7 0 10 20 30 40 Teff / 103 K 0 1 2 (B - V)0 / mag 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 (B - V)0 / mag −10 0 10 20 Oort Constant / km s−1 kpc−1 A B Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Relationships used to derive the dependencies of 𝐴 and 𝐵 on intrinsic colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Left: linear relationships between intrinsic B-V and Oort constants, using the Oort constants as derived by Olling & Dehnen (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right: a two-piece fit between effective temperature and intrinsic B-V, using the empirical colour sequence of Pecaut & Mamajek (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 3 CREATING A PROPER MOTION DISTRIBUTION Now that we have described the model for simulating the velocity of a source at a given position in the Galaxy, we can create a theoretical distribution of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In a small sky coordinate window (in our tests limiting ourselves to a few square degrees in the Galactic plane, and relatively small polar cap latitude windows), we run a TRILEGAL simulation in the center of the defined region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We simulate either 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 million sources down to Gaia 𝐺 = 25, or as many as we are allowed within 10 square degrees, the maximum limit of the public simulation API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distances for these simulated sources are derived from their absolute distance modulus, and – with no positional information in the simulated dataset – we randomly place the sources within the rectangle defining the coordinate window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For a given small magnitude range of sources, each source then has its proper motions calculated as though it were from each of the three Galactic components in turn, with some number of realisations (𝑁 ≈ 1000) of the multivariate dispersion drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We then calculate a weighted histogram of proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For each source, 𝑗 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 𝑀, where 𝑀 is the number of simulated stars (and thus distances), the three Galactic components at the given Galactic longitude, latitude, and distance have their respective weights 𝑤𝑖 𝑗 (𝑖 ∈ {1, 2, 3}, or 𝑖 ∈ {thin, thick, halo}) calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The weighted histogram is therefore built with each derived proper motion being given weight 𝑤𝑖 𝑗/𝑁 (𝑁 the number of derived Galactic velocities for the 𝑗th source, in each of the three components), for each of the 3 × 𝑁 × 𝑀 derived proper motions, across all 𝑀 objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The histogram (which will contain 𝑀 weighted counts across all bins) is then converted to a PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the purposes of visualisation and testing, we extract all of the proper motions of Gaia eDR3 sources with flux SNRs greater than five in the same coordinate window and magnitude range de- fined for the simulated proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, one additional step is taken, solely for the purposes of distribution comparison: we convolve the model with the median uncertainty of the Gaia proper motions in the dataset for this magnitude cut and sightline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This allows for the inclusion of non-negligible Gaussian uncertainties in our comparison of our generated model to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This step was purely for visualisation purposes, and is not part of the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4 ASSESSING THE ACCURACY AND PRECISION OF THE PROPER MOTION MODEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Overall Model Shape The simulated proper motions are good across all sightlines and brightnesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' some examples are shown in Figures 2-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We get good agreement in the mean proper motion, and shape of the dis- tributions, of bulk source motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For most sightline-brightness combinations the agreement is quantitative, while sometimes the shapes are merely broadly in agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Disagreement in modal proper motion drift is likely largely caused by our Galactic rotation model not capturing the fine detail of Galactic potentials or inac- curacies in our asymmetric drift velocity, while distribution width issues can mostly be explained by the extrapolation of the velocity dispersion vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we stress that the model’s simplicity is one of its strengths in the context of inclusion within a larger cross-match process, and that these minor differences are more than acceptable for the purpose of improving Bayesian match likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These distributions are intended to reflect a wide range of potential positional shifts through time, rather than model any one specific proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence so long as the rough widths – to within some- thing like a factor two, which we achieve – and mean offsets – good to high accuracy using the Galactic rotation curve – are modelled to reasonable accuracy, our distributions are good enough for our work, and as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 5 shows some reduced statistics for the entire set of sightline-brightness combinations we tested in the Galactic plane – 𝐺 = {12, 15, 18, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5}, 𝑙 in the range from 0◦ to 345◦ in 15 degree intervals, and 𝑏 = {−50◦, −30◦, 0◦, 25◦, 40◦}, as well as the Galactic north and south poles at −90◦ ≤ 𝑏 ≤ −80◦, −80◦ ≤ 𝑏 ≤ −70◦, 70◦ ≤ 𝑏 ≤ 80◦, and 80◦ ≤ 𝑏 ≤ 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, we find that the widths of the Gaia data are approximately 80% that of our model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' our model is too wide by 25%) across all positions and brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' There is a roughly 10% spread in relative widths – middle column, Figure 5, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2, bottom right panel, where our red model has a slightly wider wing than the histogram of the black Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also see evidence for overly narrow simulated proper motion distributions (Gaia-to-model ratios larger than one) along various sightlines, in approximately 8% of cases – but, again, get extremely good agreement along others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These slightly-too-wide RASTI 000, 1–20 (2022) 8 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 ∆l / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions (Galactic longitude, left-hand columns, and latitude, right-hand columns) for sources at 𝑙 = 90◦, 𝑏 = 0◦, for 𝐺 = 12, 𝐺 = 15, 𝐺 = 18, and 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 (each respective row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Proper motions have been converted from a per-year drift to decadal positional change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Gaia proper motions are shown in the black histogram, with sim- ulated distributions of proper motions in the red solid lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Errorbars in the corner of each subplot show the typical uncertainty of each individual Gaia proper motion, while the plot labels show the Galactic longitude and latitude, and 𝐺 magnitude, of the subset of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' distributions are likely related to our modelled radial and vertical dependencies of the thin disc dispersion vector, as the thin disc is the dominant term at the distances our Gaia data probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, as can be seen in the top row of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2, low-number statistics of brighter Gaia stars could be interpreted as lower standard deviations, as the ‘real’ distributions fail to probe the wings of the simulated distributions to high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Relative to the standard deviation of the distributions, our bi- ases – the mean motion offsets – are within ∼ 10% two-thirds of the time, and almost always within 15 − 20%, as shown by right-hand column, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Absolutely, on a decade baseline, we find most of our mean motion offsets are within approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 arcseconds (or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 arcseconds on a 25-year baseline, as maybe be important for LSST), well within the ‘centroid’ precision of most photometric catalogues – left-hand column, Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These results – even where qualitative (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 4, bottom-left panel) as opposed to quantita- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 ∆l / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 ∆l / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 180◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' tively (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 4, top-left panel) good fits – are satisfactory, and we therefore have chosen not to over-explore the residuals, as the subtleties of the spiral arm structure of the Milky Way are outside of the scope of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactic Poles All previous examples shown (Figures 2-4) were limited in Galac- tic latitude to |𝑏|≤ 50◦, exploring primarily the proper motions of sources roughly in the Galactic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we must also verify that our model is good at high absolute Galactic latitudes, where we are viewing sources orbiting around the Galactic cen- ter ‘above’ us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As shown in Figure 6, we get good agreement for the Galactic longitudinal and latitudinal proper motions, af- ter removing objects with parallax 𝜋 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 mas (𝑑 ≳ 20 kpc) or 𝜋/𝜎𝜋 < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, close to 𝑏 = 90◦, our equations for the average rotational velocities (equations 3-7) simplify somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, look- ing straight up out of the Galactic plane, we have 𝑑ip ≈ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we also have 𝑅𝑐 ≈ 𝑅⊙, and hence (𝑅2𝑐 + 𝑅2 ⊙ − 𝑑2 ip)/(2 𝑅𝑐 𝑅⊙) ≈ 1, and can assume Θ(𝑅𝑐) = Θ⊙, giving 𝑈1 = −𝑈⊙, 𝑉1 = −𝑉⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This further gives 𝑣𝑑 = −𝑈⊙ cos(𝑙) −𝑉⊙ sin(𝑙), 𝑣𝑙 = 𝑈⊙ sin(𝑙) −𝑉⊙ cos(𝑙), and RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 2 4 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 2 4 6 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 ∆l / arcsecond [10yr baseline] 0 2 4 6 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 2 4 6 8 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 PDF / arcsecond−1 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 255◦, 𝑏 = 25◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' hence, along with a simplified 𝑣𝑏 = −𝑣𝑑, 𝜇𝑙∗ = 𝑘 𝑑 [𝑈⊙ sin(𝑙) − 𝑉⊙ cos(𝑙)] , (36) 𝜇𝑏 = 𝑘 𝑑 [𝑈⊙ cos(𝑙) + 𝑉⊙ sin(𝑙)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (37) This renders the orbital motions effectively just those of the Sun, with our dispersion vector giving good shape agreement to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, we see good agreement in the shape of our model and the Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Widths of Distributions vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Position and Proper Motion Precisions At this point it is worth briefly considering if, or at what bright- nesses, this additional information is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figures 2-6 show a representative sample of Galactic sightlines and the various widths of the distributions of potential proper motions in each sightline- magnitude combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, the widths of these distributions are approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 arcsecond drifts over a 10 yr baseline (20 − 50 mas yr−1) at the bright end of our tests (𝐺 = 12), reduc- ing to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 arcsecond drifts in 10 years (10 − 30 mas yr−1) at 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The first parameter we should compare the proper motions to is the precision on an individual position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If one or both of the positions in a given cross-match were highly uncertain, factors 10 or higher than the proper motion drift, this would dominate over the extra positional spread caused by the potential proper motion of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Its inclusion would then not contribute to the determination of potential counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, for a decade-long baseline, the spread of separations induced by unknown proper motion is at least a factor two or three higher than typical astrometric precisions, with even small motions over long enough baselines moving objects several astrometric precisions apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For Gaia astrometric precisions are vastly higher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' while 80% of its sources will also have incredibly high precision proper motions even the remaining sources will have coordinate positions significantly higher than the unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A more typical ground-based survey, LSST should have at worse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='07 arcsecond precision on each individual visit at 𝑟 = 24 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While this is a factor ≈ 3 times smaller than the widths of our proper motion distributions on decade baselines, the real power of LSST lies in its repeated observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Depending on whether the object is in the full ‘Wide-Fast-Deep’ (WFD) survey or in the Galactic Plane footprint, it will either be observed approximately 800 or 150 times across LSST’s full survey lifetime (Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence the statistical precision on a co- added detection at 𝑟 = 24 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='003−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='006 arcseconds depending on the exact number of visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Even including ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 arcsec systematic precision (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) this is far below the widths of our models for proper motion drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While those objects will likely also have proper motions after LSST DR3-4, 𝑟 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 coadded detections will have statistical astrometric precisions a factor √ 10 higher, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='008 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 arcseconds, still a factor 10 or more below our 10-year baseline drift spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It will be therefore important to take these long-baseline drifts into account for faint LSST objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For current-generation surveys such as SDSS, its very faintest sources have statistical positional uncertainties comparable to the tightest of our proper motion distribution widths (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 arcsec) so 𝑟 = 24 objects in SDSS may only see limited gains matching across a 10-year timespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Of course, the drifts increase linearly with time, and so a 15-year baseline (2015-2030, for example, in the case of SDSS-LSST) increases the potential proper motion drifts to a larger impact than astrometric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, in the context of crowded field Bayesian cross-matching, even a ‘one- sigma’ positional movement will be enough to significantly disrupt match likelihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is also useful to ask if proper motion precisions are ever com- parable to the width of potential unknown proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Again, for Gaia this is not the case due to its extremely high precision and repeated observations of all objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At 𝐺 = 20 the median precision on its proper motions are of order 1 mas yr−1 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For LSST, quoted proper motion uncertainties are also of order 1 mas yr−1 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) – but these assume ≈ 800 visits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence the stellar proper motion precisions for Galactic Plane objects will be a factor ≈ 3 higher due to the reduced number of observations within the same timeframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, even 5 mas yr−1 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 arcsec over a 10-year timeframe and therefore a smaller, but sizeable, fraction than the unknown proper motion distribution widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right at the detection limit of proper motions with LSST it may be the case that it is preferable to not use the detected-but- unconstrained proper motions, though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' SDSS has typical limiting proper motion precisions of 5 mas yr−1 by 𝑟 ≈ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, like LSST, where constrained individual SDSS proper motions brighter than about 20th magnitude are likely preferable to unknown proper motions, but below this limit and the single-visit detection limit of RASTI 000, 1–20 (2022) 10 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 Gaia − Model Average / arcsecond [10yr baseline] 0 25 50 75 100 125 150 N Mean Median Mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Ratio of Gaia-to-Model Distribution Widths 0 20 40 60 80 100 N StDev 90th−10th 84th−16th 75th−25th −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Gaia − Model Median / Gaia StDev 0 10 20 30 40 50 60 70 N −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='03 Gaia − Model Average / arcsecond [10yr baseline] 0 25 50 75 100 125 150 N Mean Median Mode 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Ratio of Gaia-to-Model Distribution Widths 0 20 40 60 80 N StDev 90th−10th 84th−16th 75th−25th −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Gaia − Model Median / Gaia StDev 0 10 20 30 40 50 60 N Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Comparison between Gaia proper motions and those of our model across all sightlines and brightnesses in the Galactic plane, in Galactic longitude (top row) and latitude (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Left: Data-to-model average proper motion drift offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Middle: Ratio of data and model proper motion drift distribution widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Right: Ratio of the median proper motion drift offset between Gaia data and the model distribution, normalised by the standard deviation of the Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑟 = 22 unknown proper motions may be the more precise constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the IR, CatWISE (Eisenhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020) has proper motion un- certainties of 20 mas yr−1 at 𝑊1 ≈ 15 (Marocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2021) and the VVV survey (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2018) cites uncertainties of 10 mas yr−1 around 𝐾𝑠 ≈ 16;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' below these brightnesses the precisions on in- dividual proper motions become comparable to or larger than the widths of typical unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4 Missing Galactic Components As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4, we do not currently include a full prescription for the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In particular, we do not model the Galactic Bulge (or Bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This may have an effect at very low Galactic longitudes and latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Gaia data show a broader, almost flat distribution of longitudinal proper motions, where our simpler Galactic model, using the thin disc as the dominant term, has a bi- modal distribution of two narrower peaks, as shown in Figure 7, left hand panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It can also be seen in the data (Figure 7, right hand panel) that there is a slightly too narrow distribution of latitudinal proper motions, as compared to the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This likely comes back to the minor effects of radial dependencies of the 𝜎2𝑟𝑟 term, either following a Gaussian- or Rayleigh-like distribution (as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, it could also be the case that our radial and vertical dispersion scalings are failing at these smaller Galactic radii, as a significant fraction of Gaia sources ought to be sufficiently far removed from the Galactic center to be Bar or Bulge objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once again, we deem these minor issues beyond the scope of this preliminary work – the combined bi-modal longitudinal distribution almost entirely covers the distribution of Gaia motion drifts, to within better than a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 or so, which is our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we highlight the issue that the very inner few degrees of the Galactic center may suffer systematic proper motion effects due to the nature of the Galactic Bulge and Bar complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also have not modelled the Magellanic Clouds, and in- deed during testing found that several of our test fields are heavily ‘contaminated’ by sitting on the Small Magellanic Cloud (SMC) and Large Magellanic Cloud (LMC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These extra terms, as with the Bulge, would be easy to implement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' provided a relative num- ber density of sources, with some positional distribution, and bulk and dispersal proper motion – assuming the Magellanic Clouds are orbiting internally, and around the Milky way – the proper motions can be modelled in much the same way with the Galactic discs and halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For now, we also simply urge the reader to take care when sim- ulating sources centered on the Magellanic Clouds (SMC 𝑙 ∼ 300◦, ��� ∼ −45◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' LMC 𝑙 ∼ 280◦, 𝑏 ∼ −35◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 Missing Binarity Perturbation Our model for motions of objects in the plane of the sky assumes all sources are single stars, subject solely to the Galactic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, half of objects are in some form of higher-order system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Raghavan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2010) and should therefore be subject to ad- ditional on-sky motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is therefore reasonable to ask whether the non-inclusion of this effect, of unresolved binary objects, would have any impact on our derived proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the binary were equal mass, any orbital motion of the two sources around their common barycentre would completely cancel by symmetry, and show no impact on the photocentre and proper motion of the blended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, if the objects were very unequal in mass then both the barycentre and photo- centre of the pair will be dominated by the larger, brighter main source, and effectively reduce to a singular object for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆l / arcsecond [10yr baseline] 0 1 2 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = -75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆b / arcsecond [10yr baseline] 0 1 2 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆l / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 ∆b / arcsecond [10yr baseline] 0 1 2 3 PDF / arcsecond−1 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 180◦, 𝑏 = −75◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, Gaia objects were filtered for parallaxes consistent with zero to remove extragalactic contamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 ∆b / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 0◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' An object of approximately half the mass of the primary, however, contributes very little in luminosity but significantly in astrometric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Placing such an object on a worse-case orbit of approxi- mately 10 AU would give an orbital period around 25 years, and a half-phase orbit on our key decade-long time interval between pho- tometric catalogue ‘generations’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In that time, the primary object would travel halfway around the orbit, appearing to move a total of ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5AU, twice its orbital distance from the barycentre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At a typical distance of roughly 1 kpc this is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 mas or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='54 mas yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Such a perturbation is well below the of order 10 mas yr−1 widths to the proper motion distributions observed for faint objects in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the object were significantly closer – say 100 pc instead – the motion effects would be 10 times higher, and comparable to the model widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In those cases the object would be much brighter, and likely have an individually measured proper motion or be known to be a multiple system through other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore believe the non-inclusion of higher-order systems is justifiable at the resolution we are aiming to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='6 Random Positions of Sources As noted in Section 3, the TRILEGAL simulations we use to con- struct our models of proper motions do not provide individual posi- tions for simulated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To overcome this, we simply uniformly distributed sources within the rectangular area we sampled our Gaia proper motions in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect may explain some small disagree- ments between our simulated and Gaia proper motion distributions, as we are therefore not properly modelling any clustering, extinction effects, or other non-uniformity and correlations in the distances and positions of Galactic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, the effect on each individual proper motion should be relatively small, as the regions in question were mostly limited to several degrees in extent, and cos(𝑥 + 5◦) − cos(𝑥) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08, sin(𝑥 + 5◦) − sin(𝑥) ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='08 over the entire Galactic longitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus our values for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the decomposition of 𝑈⊙, or Θ, within our proper motion equations, are of order 8% wrong at their most extreme, in the case of a simulated patch of sky five degrees wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As an ensemble, however, this assumption should be a reasonable one, and the ‘incorrectness’ should average out, with uniformity of source distribution acceptable for small enough patches of sky, providing a statistical distribution of variations of velocity decomposition across the whole region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='7 Gaia Proper Motion Uncertainty To compare our ensemble proper motion distribution with the distri- bution of Gaia proper motions, we included the Gaia measurement uncertainty in our theoretical distribution of drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Gaia data have individual uncertainties but to smooth the model with the uncertainty we had to select a single average value (the error bar included in the corners of sub-plots in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A small part of the discrepancies between model and data in our analysis could therefore stem from this simplifying assumption, with no bearing on the model itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If the Gaia data have a particularly broad dis- tribution of measurement uncertainties, as they tend to at fainter magnitudes, our single uncertainty value would not produce an uncertainty-convolved motion drift distribution that reflected that of the Gaia data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Testing more complex treatments of Gaia uncer- tainty distributions in the comparison between model and data, we found that more fully describing the non-singular value of mea- surement precision did produce theoretical drift distributions that RASTI 000, 1–20 (2022) 12 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='50 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 5 10 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆b / arcsecond [10yr baseline] 0 10 20 30 40 PDF / arcsecond−1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 270◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In addition, the dashed blue line shows simulated Besançon proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' better matched the expected data, but not completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore still find a few sightlines with slight differences in central proper motion or distribution width, but perhaps 30% of these tensions are explainable by the different measurement precisions of faint Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ultimately, however, as mentioned in Section 3, we do not include this measurement uncertainty in the final model, just performing the convolution to compare to the Gaia distributions more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8 Comparison with the Besançon Model Throughout this work we have used the TRILEGAL simulations to provide a set of theoretical distances for sources of a particular Galactic sightline and magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We could, of course, use any model of the Milky Way to achieve this, such as the Besançon model (Robin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2003, 2012, 2014, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Czekaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Bienaymé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With the Besançon models, however, we receive simulated proper motions for the objects returned in our query, unlike with TRILEGAL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We can use these simulated proper motions to further verify the robustness of our proper motion model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but this then raises the question of why we simply do not use these simulated proper motions for use in our cross-matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will address both of these issues in the next two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Verifying the Accuracy of Our Model with Besançon With simulated Besançon proper motions, we can compare our model’s statistical distribution of proper motions with those of the Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Shown in Figure 8 are distributions of proper mo- tions at 𝑙 = 270◦, 𝑏 = 0◦ for Gaia, our simple model for stellar velocities, and Besançon proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall, at fainter mag- nitudes (bottom row), both models are in agreement with the Gaia data, with our distribution a slightly better match in Galactic latitude than the Besançon model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, there are some sightlines within the Galaxy where our model has some mismatches to the Gaia data – an example sightline demonstrating this effect is shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, at faint magnitudes (𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5), the Besançon model better reproduces the Galactic longitude proper motion distribution seen with Gaia, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆l / arcsecond [10yr baseline] 0 5 10 15 20 PDF / arcsecond−1 G = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 l = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='0 ∆b / arcsecond [10yr baseline] 0 10 20 30 PDF / arcsecond−1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='00 ∆l / arcsecond [10yr baseline] 0 10 20 PDF / arcsecond−1 G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='025 ∆b / arcsecond [10yr baseline] 0 20 40 PDF / arcsecond−1 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Distributions of proper motions for 𝑙 = 60◦, 𝑏 = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Lines and symbols have the same meaning as in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' where our model shows a slight bias, and a distribution slightly too broad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Neither model can reproduce the Galactic latitude Gaia proper motions, and both look very similar in their over-broad dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At bright magnitudes (𝐺 = 12), however, we can see that our distribution (red solid lines) much better matches the Gaia data points than the Besançon simulation (blue dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In almost all cases, 𝑙 = 60◦ and 𝑙 = 270◦ in Figures 8 and 9, but more generally across multiple sightlines, the Besançon models are too sharp in distribution, and fail to match the Gaia data as well as our model for proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We discuss this magnitude-dependence of the Besançon model fits further in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we conclude that our model matches the Besançon models very well, as it does the Gaia data, and see cases where both our model and the Besançon model fail to match the Gaia data perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Why Not Just Use the Besançon Proper Motions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We have used TRILEGAL simulations to construct our Galactic model throughout this work, but we could have used any Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If we had used the Besançon model, we would also have been provided with simulated proper motions for the objects we use for their distances in constructing our proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is therefore reasonable to ask why we would go to the effort of using another model, if we already had a set of proper motions from which to construct a PDF of unknown proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, as our model is broken up into separate smaller sub- models, as opposed to being wrapped in a full Galaxy model, our magnitude-to-distance relation is flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As mentioned, we have been using TRILEGAL simulations to get our potential distribution of distances of sources of a given magnitude, but we could use any Galactic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Indeed, we do not need to use a model at all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' if we instead had a known distribution of tip of the red-giant branch stars, or some other class of standard candle, we would immediately know the distance of our sources from their brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore do not necessarily need to rely on fully resolved Galactic models to provide proper motions or distances with our simple model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, our options become slightly more limited if we wish to use a full Galactic model to obtain simulated proper motions in one pass, as opposed to generating more ‘static’ distributions of RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 13 object brightnesses (or distances), and using other functionality to continue on to create our final proper motion distributions, as we do here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The second consideration is that of dimensionality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' each Be- sançon source is provided with a simulated proper motion – but only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Our model uses the simulated distance for each source once, but draws 𝑁 simulated velocities – and hence 𝑁 simulated proper motions – for each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore much more com- pletely sample the 3-D velocity space than any one simulation from the Besançon Galactic model will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This effect can be seen in the 𝐺 = 12 panels of Figures 8 and 9, where our model (red solid lines) much better agrees with the Gaia proper motions, where limited sample size means the Besançon model is not fully populating the velocity dispersion dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At 𝐺 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='5, number counts have increased by a factor 100, and the velocity dispersion, having an inverse-distance component (and fainter stars being further away, on average), has reduced in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This reduces the effect of the lack of realisations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Besançon models therefore agree much better at these fainter magnitudes than they do at bright ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' By using each source only for its distance, as opposed to using it to sample the 4-D distance-velocity dimensionality, we much more accurately sample from the full potential proper motion distribution at bright magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is crucial for bright sources, being closer to the Sun on average, which have larger proper motions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the 𝑥 axis ranges on the top and bottom rows of Figures 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' It is here where our constructed model has the edge on the proper motions constructed from large-scale Galactic simulations, although brighter objects are, of course, more likely to have a robustly detected proper motion from other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5 INCLUDING PROPER MOTIONS IN PROBABILISTIC CATALOGUE CROSS-MATCHING No matter how you construct your proper motion distributions, it is still important to consider them in a match between two photo- metric catalogues of differing epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Astrometric Uncertainty Function (AUF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson & Naylor 2018b) is the description of the belief as to a true position of the source, given its measured position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is typically assumed to be a Gaussian, which describes the most obvious term affecting the measured posi- tions of sources in photometric catalogues, and hence the separation between two potential counterparts: the noise-based centroiding of the individual objects during the catalogue creation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The function representing the likelihood of two sources having a given separation under the assumption that they are counterparts to one another – two detections of the same physical object – is given by 𝐺(Δ𝑥, Δ𝑦) = (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥, Δ𝑦) (38) (Wilson & Naylor 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here Δ𝑥, Δ𝑦 are the two-dimensional sky offsets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' right ascension and declination, or Galactic longitude and latitude), ℎ𝛾 and ℎ𝜙 the AUFs of the sources from the two catalogues respectively, and ( 𝑓 ∗ 𝑔)(𝑥, 𝑦) denotes the convolution of two arbitrary functions 𝑓 and 𝑔 evaluated at 𝑥 and 𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As discussed by Wilson & Naylor (2018b), the AUF ℎ can be extended with any additional terms, ℎ𝛾 = ℎ𝛾,1∗ℎ𝛾,2∗ℎ𝛾,3 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here the additional ℎ𝛾,𝑖 components, after the first noise-based centroid term, describe extra potential movement away from the ‘true’ sky position of the source in the limit of infinite precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These could include, for example, stochastic processes such as the perturbation of objects due to hidden contaminants affecting the center-of-light of sources, or systematic effects like offsets of the coordinate frame of the catalogue from a common reference frame, such as the ICRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Whatever the effects, the point is that each source is considered individually, and has all of its ℎ𝛾,𝑖 components applied to it on an isolated, per-source basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Proper motion, however, does not work like this;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the effect of proper motion drift works on offsets between two positions, as opposed to affecting the absolute position of one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus the proper motion drift must be applied to 𝐺, giving, effectively 𝐺′ = 𝐺 ∗ ℎ′pm (see Appendix B1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' ℎ′pm should be calculated in the sense of mapping from oldest to youngest epoch, in units of distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' thus for Δ𝑡 > 0 we have, crudely, Δ𝛿 = 𝜇𝛿 × Δ𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Mapping from most recent to older data would have a negative Δ𝑡, but the proper motion would have to be of the opposite sign as well (being a ‘rewind’ of the motion), and thus the sign of Δ𝛿 would be the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' If a source has a purely positive proper motion distribution, such that all 𝜇𝑙∗ > 0 for this simulated source, then we would expect a source observed in the year J2000 to have a smaller Galactic longitude than a source observed at J2015, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This convolution can be performed as any other convolution done to calculate 𝐺 by the convolution of all ℎ components – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' either numerically, or through expression as a mixture of analytically convolvable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We also highlight here that while Sections 2-4 detail a method for the construction of a distribution of unknown proper motions, ℎ′pm can be constructed through any available means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For exam- ple, Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010) construct sets of data-driven proper mo- tion distributions for the purpose of improving cross-matches, using available proper motions to construct priors for weighting the search for unknown proper motions between potential source counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, the faint end of a photometric catalogue will sys- tematically have worse precision on its measurements (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3), and at some point will have detected the proper motion of an object but be unable to constrain it with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, ℎ′pm could very well be constructed as a Gaussian PDF with mean and covariance matrix that of the best-fit and uncertainty of the proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Star-Galaxy Separation Our model for proper motions assumes the source in question is a star – objects orbiting the Galactic center in some fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, for an all-sky catalogue cross-match we will also, at high Galactic latitudes, be matching a considerable number of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We there- fore need to model the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, that the sources being matched are stars, and hence have the statistically modelled un- known proper motion distribution, with which we wish to ‘blur’ out our potential match separations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Second, they are galaxies, which have zero proper motion, being altogether too distant to have visibly moved anywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore have a slightly different probability of match (sources being ‘counterparts’, under hypothesis 𝑐) given separation 𝑑, now also conditioned on the ‘type of source’ hypoth- esis, which we will denote as 𝑝(𝑐|𝑑, S) and 𝑝(𝑐|𝑑, G) for a ‘star’ and ‘galaxy’ pairing respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' When matching, we are generally only concerned with the overall probability of the two sources having a given sky separation under the hypothesis of their being matched, 𝑝(𝑑|𝑐) – this term is denoted 𝐺 by Wilson & Naylor (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Note that this differs from 𝑝(𝑐|𝑑), the probability of the two sources being counterparts given their sky separation, Wilson & Naylor (2018a)’s 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑝(𝑑|𝑐) we can RASTI 000, 1–20 (2022) 14 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson obtain by the marginalisation over the two hypotheses: 𝑝(𝑑|𝑐) = 𝑝(𝑑, S|𝑐) + 𝑝(𝑑, G|𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) + 𝑝(𝑑|𝑐, G)𝑃(G|𝑐) (39) where 𝑃(S|𝑐) is the prior probability that these counterparts (with given sky positions, brightnesses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=') are stars (or galaxies, in the opposite case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We work under the assumption there is no third type of object – crudely labelling objects as ‘in the Milky Way’ or ‘outside the Milky Way’ – and thus 𝑃(S|𝑐) + 𝑃(G|𝑐) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, we can also ask a related but separate question: ‘what is the probability that these two detections are of a star, given that they are counterparts with a given separation?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', which looks like 𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S|𝑐) 𝑝(𝑑|𝑐) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (40) Here we have the likelihood of the separation given the hypoth- esis that the sources are counterparts and stars, multiplied by the prior chance of the sources being stars given they are counterparts, normalised by the overall chance of either a galaxy or star pair hav- ing this particular detection offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To calculate both 𝑃(S|𝑑, 𝑐) and 𝑝(𝑑|𝑐) we therefore need both prior and likelihood terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Calculating the likelihood terms 𝑝(𝑑|𝑐, S) and 𝑝(𝑑|𝑐, G) is relatively straightforward, simply being the convolution of the re- spective AUFs (containing all relevant AUF components for the two catalogues) of the sources in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For 𝑝(𝑑|𝑐, G) this does not include any proper motion terms, as the ‘proper motion model’ for galaxies is a static one – mathematically, this is equivalent to the convolution of 𝐺 and a delta function at zero proper motion, with 𝑓 ∗ 𝛿 = 𝑓 – and hence 𝑝(𝑑|𝑐, G) = 𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For 𝑝(𝑑|𝑐, S), how- ever, we wish to include the motion of Galactic sources, and hence subsequently convolve by the ℎ′pm PDF, describing the potential additional on-sky movement due to the epoch difference between the two sets of observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑝(𝑑|𝑐, S) = 𝐺′ ≡ 𝐺 ∗ ℎ′pm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, the likelihood for our new question is easy to calculate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' we are therefore left with the derivation of the prior, 𝑃(S|𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The ‘conditioned on the fact that the sources are counterparts’ aspect of the prior is tricky to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We wish to know, analogous to Wilson & Naylor (2018a)’s derivation of photomet- ric likelihoods, the distribution of stars and galaxies as a function of the two bandpasses in question – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑟 and 𝐽, for a match be- tween optical and infrared data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, 𝑃(S|𝑐) is really ‘what is the probability that these two sources are stars given that they are counterparts with magnitude limits (or dynamic ranges) in their re- spective bandpasses?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 𝑃(S|𝑐, 𝑚lim,r, 𝑚lim,J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Due to the nature of the simulated objects – being derived from one-sided distributions, a function of just a single magnitude in one bandpass in one of the two catalogues – we are unable to create two-dimensional relation- ships between stars and galaxies in the construction of these priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In fact, our likelihoods, 𝑝(𝑑|𝑐, S), should implicitly assume coun- terparts for sources, but are built from the full distribution of sources of just a single magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we could have, for example, a case where sources of 𝐽 = 17 either have optical brightnesses 𝑟 = 18 or 𝑟 = 25 (being two classes of objects at differing distances, say);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' this distance distribution is blurred into a bimodal proper motion distri- bution in the IR, but one class of object is rejected if we consider the dynamic range of the optical data for an example 𝑚lim,𝑟 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' At present, the explicit dependency on the two-sided, magnitude-magnitude relationship between sources in our two cat- alogues is beyond the scope of this work, due to the nature of the outputs available from most Galactic simulations being limited to a particular set of bandpasses for a specific catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore simply note here that for now, the construction of these models is one-sided – in contrast to the cross-matching algorithms of Wilson & Naylor (2018a), taking into account both catalogues symmet- rically, in both AUF-based astrometry and photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We thus sidestep this dependency by constructing our priors on star and galaxy counts on single magnitude source counts, effectively cre- ating 𝑃(S) and 𝑃(G), removing the dependency on 𝑐 within the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We can still, however, account for the dynamic range of each bandpass within its given catalogue on a per-filter basis, and hence implicitly use 𝑃(S|𝑚lim) in a practical implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With this minor practical dependency removed, we conclude that with the inclusion of a distribution of unknown proper motions for Galaxy-based stars, it is possible to discriminate between stars and galaxies in photometric catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The equations 𝑃(S|𝑑, 𝑐) = 𝑝(𝑑|𝑐, S)𝑃(S) 𝑝(𝑑|𝑐) (41) and 𝑃(G|𝑑, 𝑐) = 𝑝(𝑑|𝑐, G)𝑃(G) 𝑝(𝑑|𝑐) (42) allow for the drift of Galactic sources with time, recovering them as non-static sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is the most certain question that can be answered;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' stars, as shown in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Figure 3, can have a very high probability of small proper motions in certain sightlines in the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, zero proper motion does not necessarily mean galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but a combination of delta-function likelihood for Galactic proper motion and imbalanced priors at high Galactic latitudes mean that zero proper motion objects will bias towards being extragalactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' On the other hand, if a source has a proper motion distribution which is significantly non-zero, as is the case for Galactic longitude proper motions at 𝑙 = 270◦, 𝑏 = 0◦ (Figure 8), then we should see a breaking of this degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The offset between the two sources being considered as potential counterparts should now be able to tell whether the sources are further apart than their respective AUFs would suggest – at which point they are very likely detections of a star – or if they have an offset compatible with their AUFs – at which point they are very likely a galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Inclusion of Proper Motions in the Non-Match Hypothesis We also note that we should also consider the proper motions within the context of non-matches, but it is easy to see that this results in a trivial case, effectively ignoring the proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the counter hypothesis of ‘these sources are unrelated to one another, and separate detections of two physical sky objects’, each source can have its own proper motion, based on its own statistical distribution of potential motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, we need to compare to the hypothesis that these sources are not related to one another given the separation between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This involves the double, but separate, marginalisation over all possible unknown locations and proper motions, for both objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ultimately, as the integrals are separate the proper motions do not affect the end result – see Appendix B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This is obvious intuitively: the distribution of separations of unrelated, randomly placed objects is independent of the unknown motion history of those objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 6 WHEN ARE UNKNOWN PROPER MOTION DISTRIBUTIONS NEEDED?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The theoretical framework for accounting for unknown proper mo- tions presented here is relatively indifferent to the type of surveys RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 15 being matched and brightnesses at which it is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, prac- tically it is more useful in some situations than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, we summarise here some key surveys, magnitude ranges, and science cases for which the inclusion of statistical proper motions may be most crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The main criterion for considering whether the inclusion of unknown proper motion distributions is important or not is the surveys being matched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The model is most useful outside of Gaia dynamic ranges, as such high-precision individual proper motions may be too impor- tant to ignore at brighter magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The main downside here is that Gaia is quite a bright survey relative to the next generation of photometric catalogues, and therefore large numbers of objects won’t be detected in Gaia at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the case of LSST, it will also offer proper motions down to perhaps 𝑟 = 24 (Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019) but cannot offer proper motions for objects not detectable in its single-visit images, and thus those objects will have to rely solely on statistical proper motions to avoid risking underestimating match probability or unnecessary false match rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' More generally, any science done in the Northern Hemisphere, where LSST has no coverage, will be unable to take advantage of the dataset – for its increased proper motion coverage or otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' These cases will more likely require the falling back on unknown proper motions when outside of Gaia or SDSS proper motion dy- namic ranges (𝐺 ≈ 20 and 𝑟 ≈ 21 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Where proper motions are not important to the science case, and any motion drift is a nuisance parameter, it may also be prefer- able to avoid relying on matching to an intermediate catalogue that contains proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' When trying to match catalogue 𝐴 to cata- logue 𝐵, we may not want to perform separate LSST-𝐴 and LSST-𝐵 (or Gaia-𝐴 and Gaia-𝐵, depending on your source of individual proper motions) matches, then join across common LSST (Gaia) objects to obtain a final cross-match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, where the abil- ity to select high-quality matches using the added-value information from a probabilistic cross-match algorithm is important, reliance on proper motion distributions may suffice to gain in other areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' With the key exceptions of CatWISE, albeit with order-of- magnitude larger uncertainties than LSST or Gaia, and, but with much less sky coverage, VVV, most IR surveys are single-epoch, and matching longer wavelength surveys to one another therefore relies far more than optical catalogues on unknown proper motion distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In terms of science cases, the main areas that benefit from in- cluding unknown proper motions are those in which proper motions are crucial but lacking by other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Nearby faint objects, which LSST especially will find signifi- cant numbers of, will have appreciable on-sky motions that may not be derived as part of the survey’s dataset construction due to their faint fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Red objects will suffer a bias in current- and future-generation surveys such as LSST, Euclid, and Roman where they will system- atically be less likely to have measured proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Very faint transient progenitors will also suffer from a lack of known proper motions, and potentially may require matching back to a number of long-time-baseline surveys to probe progenitor characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For LSST, Galactic Plane science will systematically be af- fected due to the much lower number of visits currently planned than in the main WFD survey (Bianco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Current simu- lated LSST precisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Ivezić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019, table 3) assume WFD cadences and hence numbers of observations, but reduced visit count will lead to worse proper motion accuracies and precisions by factors of a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, care should be taken when attempting to extrapolate reasonably uncertain, but ‘detected’ proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases it may be more advantageous to not use the best-fit value, but marginalise over all potential proper motions based on the likely more robustly determined position and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Alternatively, the best-fit proper motion can be used, but ‘blurred’ out with the detection’s precision, representing the proper motion offset PDF ℎ′pm as a Gaussian with given mean proper motion and one-sigma uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 7 CONCLUSION We described a model of the bulk motion of a random set of sources through the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The model uses the rotation curve of the Galaxy, the Solar motion, and a prescription for the random motion of sources due to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' their interaction history to create a statistical distribution of potential proper motions of a source at a particular set of sky coordinates and brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We compared this model to Gaia sources in various sightlines across the Galactic plane – in the mid-plane and out of plane – in different magnitude regimes, and to the proper motions provided by the Besançon Galactic model, to verify its robustness and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Overall we find that our model matches the observed proper motions with a high degree of both accuracy and precision, and hence believe that our model is an acceptable description of the statistical proper motions of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will be invaluable when matching the next generation of deep photometric surveys to other datasets, in the regime where Gaia cannot provide individual proper motions for sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Without the inclusion of unknown proper mo- tions we could be subject to a source separation bias that will impact the number of cross-matches reported between two such catalogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This will be particularly crucial in the coming years in light of the revolution in Galactic studies that the Rubin Observatory’s LSST will bring, where – with its long time baseline back to previous brighter infrared surveys – this effect has the potential to dominate a systematic search for classes of sources such as faint, red objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We have made a Python version of the model described in this paper available through the macauff GitHub codebase4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' ACKNOWLEDGEMENTS TJW thanks the reviewers for their useful comments and sugges- tions, which much improved the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' TJW would also like to thank Sergey Koposov for useful conversations and suggestions that improved the accuracy of the model, and Tim Naylor for his helpful discussions and proofreading assistance throughout this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has been supported by STFC funding for UK participation in LSST, through grant ST/S 006117/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has made use of Python (Van Rossum & Drake 2009), and the SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020), NumPy (Harris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2020), Astropy (Astropy Collab- oration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2018), astroquery 4 At this URL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022) 16 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson (Ginsburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2019), Matplotlib (Hunter 2007), and F2PY (Pe- terson 2009) Python modules, as well as NASA’s Astrophysics Data System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/ gaia), processed by the Gaia Data Processing and Analy- sis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/web/ gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Funding for the DPAC has been pro- vided by national institutions, in particular the institutions partici- pating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' DATA AVAILABILITY The datasets used in this manuscript were derived from sources in the public domain, from the Gaia archive (https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='int/archive/), TRILEGAL (http://stev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='oapd.' metadata={'source': 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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', 2000, AJ, 120, 1579 APPENDIX A: COORDINATE SYSTEMS Here we define the coordinate systems we use in this paper, and how to transform from one to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We use several coordinate systems: Heliocentric Cartesian space (𝑥, 𝑦, 𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the observable, He- liocentric Spherical coordinate space (𝑑, 𝑙, 𝑏);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Heliocentric Cylin- drical coordinates (ˆ𝑣𝑑, ˆ𝑣𝑙, ˆ𝑣𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Galactocentric Cylindrical co- ordinate system (𝑅𝑐, 𝜙, 𝑧);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' the Galactocentric Spherical coordinate system (𝑅𝑠, 𝜙, 𝜃);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and the Galactocentric Cartesian coordinate sys- tem (𝑋, 𝑌, 𝑍).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Heliocentric Cylindrical coordinate system is defined as the radial and tangential velocity components of the in-plane stellar motions, as measured from the Sun in (and orthogonal to) the direc- tion towards the source, as well as the orthogonal, vertical compo- nent of the motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Its transformation from Heliocentric Spherical coordinates is a simple rotation from (𝑑, 𝑏) through the angle 𝑏 to (ˆ𝑣𝑑, ˆ𝑣𝑧), albeit with the caveat that the direction of rotation varies with the sign of 𝑏;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' its transformation from Galactocentric Cartesian coordinates is a rotation through longitudinal angle 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Heliocentric Cartesian coordinate system can be obtained from the Heliocentric Spherical coordinates, the observables, dis- tance 𝑑, and Galactic coordinates 𝑙 and 𝑏, with 𝑥 = 𝑑 cos(𝑙) cos(𝑏) (A1) 𝑦 = 𝑑 sin(𝑙) cos(𝑏) (A2) 𝑧 = 𝑑 sin(𝑏), (A3) where we have used the ‘right-handed’ system that defines 𝑥 as pointing towards the Galactic center from the Sun, to 𝑙 = 0◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 𝑦 towards 𝑙 = 90◦;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' and 𝑧 towards 𝑏 = +90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The Galactocentric Cartesian coordinates are a simple shift of zero-point, relative to the Heliocentric coordinates: 𝑋 = 𝑥 − 𝑅⊙ (A4) 𝑌 = 𝑦 (A5) 𝑍 = 𝑧 + 𝑧⊙ (A6) with a shift of the origin up by ≃ 8kpc and down ≃ 25pc in the 𝑋 and 𝑍 directions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Jurić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' For the Galactocentric non-Cartesian coordinate systems, we have to define new angles, as well as two additional radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The radii are fairly straightforward, being based simply on the Galactocentric Cartesian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, the Galactocentric Cylindrical radius, being defined as the in-plane radius, is given by 𝑅𝑐 = √︁ 𝑋2 + 𝑌2 (A7) RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 17 and the Galactocentric Spherical radius by 𝑅𝑠 = √︁ 𝑋2 + 𝑌2 + 𝑍2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A8) The angle 𝜙, used in both Galactocentric Cylindrical and Spherical coordinate systems, is defined as the angle around the Galaxy – if viewed top-down, from the Galactic North Pole – from the line running from the Sun through the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This an- gle, however, is defined as clockwise for the Galactocentric Cylin- drical coordinates (and during the derivation of the Heliocentric Spherical proper motions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3), but counter-clockwise for the Galactocentric Spherical coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Equivalently, this counter-clockwise angle can be considered as being measured in the 𝑋 𝑌 plane, from the 𝑋 axis towards the 𝑌 axis (or −𝑌 axis, for a clockwise defined 𝜙), analagous to how 𝑙 is defined as the angle from the 𝑥 axis towards the 𝑦 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, when in Galactocentric Spherical coordinates, we could calculate 𝜃 by 𝜃 = arccos(𝑍/𝑅𝑠), (A9) where 𝜃 is the co-latitude, the angle as measured from the Cartesian 𝑍-axis, which differs from the system defining the Galactic latitude 𝑏, measured from the (𝑥, 𝑦) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We will find that we never need to consider 𝜙 or 𝜃 themselves, as they will entirely be used to de- fine rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' The rotation matrices to convert from Galactocentric Spherical to Galactocentric Cylindrical coordinates, or Galactocen- tric Cylindrical to Heliocentric Cylindrical coordinates, along with the conversion from Galactocentric Cylindrical to Galactocentric Cartesian coordinates, are derived in Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A1 Rotating Covariance Matrices In this section we briefly outline the transformation, reflection, and rotation matrices used to convert between four coordinate sys- tems: the Galactocentric and Heliocentric Cylindrical, and Galac- tocentric Cartesian and Spherical frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' First, we need to convert from Galactocentric Cylindrical to Galactocentric Cartesian coor- dinates, following the rotation curve-based methodology of Mróz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Additionally, to work entirely in Sun-based radial, tangential, and vertical velocity space, we need to rotate the co- variance matrices calculated from Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', and King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=', in Galactocentric Cylindrical and Spherical coordinates respectively, to ˆ𝑣𝑑 − ˆ𝑣𝑙 − ˆ𝑣𝑧 space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 Galactocentric Cylindrical to Galactocentric Cartesian Rotation First we need to calculate the rotation matrix describing the change from Galactocentric Cylindrical to Galactocentric Cartesian coordi- nates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Starting with Figure A1, we first consider the case of 𝑙 ≤ 180◦ (left-hand schematics), a counter-clockwise rotation from 𝑅 through angle 𝜔 to 𝑈.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As these are left-handed cartesian coordinate systems, this is a negative rotation, and hence the rotation matrix is TCCW = � cos(𝜔) sin(𝜔) − sin(𝜔) cos(𝜔) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A10) We therefore need to calculate sin(𝜔) and cos(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' sin(𝜔) can be derived using the law of sines, and is given by sin(𝜔) = 𝑑 𝑅 sin(𝑙), (A11) while the cosine can be calculated from its corresponding law, cos(𝜔) = 𝑅2 + 𝑅2 0 − 𝑑2 2 𝑅 𝑅0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A12) Sun GC R R0 d l ω Sun GC R R0 d l ω 360∘ − l V U ϕ R ω V U ϕ R ω Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- tric Cylindrical coordinates to Galactocentric Cartesian 𝑈 − 𝑉 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In the 𝑙 ≥ 180◦ case, right-hand side of Figure A1, we now have a clockwise rotation, which in our left-handed coordinate system is a positive rotation, TCW = �cos(𝜔) − sin(𝜔) sin(𝜔) cos(𝜔) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A13) We can, as before, calculate the sine and cosine of 𝜔: sin(𝜔) = 𝑑 𝑅 sin(360◦ − 𝑙) = − 𝑑 𝑅 sin(𝑙), (A14) cos(𝜔) = 𝑅2 + 𝑅2 0 − 𝑑2 2 𝑅 𝑅0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A15) We can therefore now see that the changing from positive to negative rotation in T, which changes the sign of sin(𝜔) in the rotation matrix, is correlated with a change of sign of sin(𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we can simplify our matrices, giving us T𝑡 = �� � 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 𝑑 𝑅 sin(𝑙) − 𝑑 𝑅 sin(𝑙) 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A16) Expanding to the full three-dimensions of our problem, we note that the third axis is unchanged by the rotation within the plane of the Galaxy, and therefore the final axis has a trivial transformation, giving T𝑡 = ���� � 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 𝑑 𝑅 sin(𝑙) 0 − 𝑑 𝑅 sin(𝑙) 𝑅2+𝑅2 0−𝑑2 2 𝑅 𝑅0 0 0 0 1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A17) A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 Galactocentric Cylindrical to Heliocentric Cylindrical Rotation Here we calculate the Pasetto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' rotation from the Galactic center- based cylindrical frame on to one centered on the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Consider the RASTI 000, 1–20 (2022) 18 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson Sun GC R R0 d l θ ̂vl R ϕ ̂vd α Sun GC R R0 d l θ ̂vl R ϕ ̂vd α 360∘ − l Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the transformation from 𝑅 − 𝜙 Galactocen- tric coordinates to a Heliocentric ˆ𝑣𝑑 − ˆ𝑣𝑙 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' left-hand panel of Figure A2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' to rotate from 𝑅 − 𝜙 coordinates to ˆ𝑣𝑑 − ˆ𝑣𝑙 is a negative (clockwise) rotation – working in the more traditional right-handed coordinate system – through 𝛼, as well as a mirroring around the 𝑅 axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' a flip of the 𝜙 axis on to the 𝑣𝑙 axis, after rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence a rotation-then-mirror transformation matrix would look like TCW = �1 0 0 −1 � � cos(𝛼) sin(𝛼) − sin(𝛼) cos(𝛼) � = �cos(𝛼) sin(𝛼) sin(𝛼) − cos(𝛼) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A18) As can be seen in Figure A2, 𝛼 = 𝜃, and hence cos(𝛼) = cos(𝜃) = 𝑅2 + 𝑑2 − 𝑅2 0 2𝑅𝑑 , (A19) sin(𝛼) = sin(𝜃) = 𝑅0 𝑅 sin(𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A20) In the right-hand case of Figure A2, where 𝑙 ≥ 180◦, we now have a counter-clockwise, positive rotation from 𝑅 through ˆ𝑣𝑑, but still have a mirror reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This simply changes the sign of sin(𝛼) in the rotation matrix, and hence TCCW = �1 0 0 −1 � �cos(𝛼) − sin(𝛼) sin(𝛼) cos(𝛼) � = � cos(𝛼) − sin(𝛼) − sin(𝛼) − cos(𝛼) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A21) Again, we can consider the inner triangle of 𝑅0−𝑑−𝑅 and calculate angles for 𝛼 using 𝜃: cos(𝛼) = cos(𝜃) = 𝑅2 + 𝑑2 − 𝑅2 0 2𝑅𝑑 , (A22) sin(𝛼) = sin(𝜃) = 𝑅0 𝑅 sin(360◦ − 𝑙) = − 𝑅0 𝑅 sin(𝑙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A23) Similar to Appendix A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1, we can see that no matter the di- rection of the rotation – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' if 𝑙 ≤ 180◦ or 𝑙 ≥ 180◦ – the sign R z θ ρ β R z θ ρ β Sun GC Sun GC ρ R0 d R0 b β d ρ b β Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Schematic showing the rotation from 𝜌−𝜃 Galactocentric Spher- ical coordinates to a Galactocentric Cylindrical 𝑅 − 𝑧 coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' of sin(𝛼) cancels with the sign within the sin(𝛼) elements of the transformation matrix, and hence TCCW = TCW = �� � 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 𝑅0 𝑅 sin(𝑙) 𝑅0 𝑅 sin(𝑙) − 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A24) We can also now explicitly include the third axis, the vertical coordinate in our three-dimensional cylindrical reference frame, a trivial continued alignment of the 𝑧 axis with our 𝑣𝑧 axis, giving the final transformation matrix as T𝑐 = ���� � 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 𝑅0 𝑅 sin(𝑙) 0 𝑅0 𝑅 sin(𝑙) − 𝑅2+𝑑2−𝑅2 0 2𝑅𝑑 0 0 0 1 ���� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A25) A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='3 Galactocentric Spherical to Galactocentric Cylindrical Rotation Finally, we consider the King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' rotation from a spherical refer- ence frame into a cylindrical one, albeit still centered on the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' To do this, we consider the frame goes from 𝜌 − 𝜙 − 𝜃 to 𝑟 −𝜙−𝑧;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' here, (𝜌, 𝜃) and (𝑅, 𝑧) are both in a right-handed cartesian coordinate systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore define our rotation matrices in the opposite sense to Section A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1, TCW = � cos(𝛽) sin(𝛽) − sin(𝛽) cos(𝛽) � , TCCW = �cos(𝛽) − sin(𝛽) sin(𝛽) cos(𝛽) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A26) The upper case of Figure A3 shows the rotation necessary for 𝑏 ≥ 0◦, with the left hand side showing the clockwise rotation through 𝛽, and the right hand side showing a schematic of the various known distances and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here, considering a negative – clockwise in a right-handed frame – rotation through 𝛽, we can calculate sin(𝛽) RASTI 000, 1–20 (2022) Overcoming Separation Between Counterparts Due to Unknown Proper Motions 19 and cos(𝛽) as sin(𝛽) = 𝑑 𝜌 sin(𝑏), (A27) where 𝜌2 = 𝑅2 0 + 𝑑2 − 2𝑅0𝑑 cos(𝑏), and cos(𝛽) = 𝑅2 0 + 𝜌2 − 𝑑2 2𝑅0𝜌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A28) For the lower case of Figure A3, 𝑏 < 0◦, with a positive rotation, cos(𝛽) = (𝑅2 0 + 𝜌2 − 𝑑2)/(2𝑅0𝜌), as previously, as the triangle is unchanged, just mirrored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' sin(𝛽) is a little more complicated to derive, however, as the triangle in Figure A3 uses 𝑏 as its modulus value, but it is negative in value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Using |𝑏| explicitly, the law of sines gives sin(𝛽) = 𝑑 𝜌 sin(|𝑏|), (A29) as previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, if we use, as we will in practice, 𝑏′ = −|𝑏|, we get sin(𝑏′) = − sin(|𝑏|), and hence sin(𝛽) = −𝑑/𝜌 sin(𝑏′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Once again, we find – as with Appendices A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='1 and A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content='2 – that the sign of sin(𝛽) cancels with the sign of the term within the rotation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus, for either orientation – positive and negative Galactic latitude – the rotation matrix from Galactocentric spherical to Galactocentric cylindrical coordinates (from the (𝜌, 𝜃) to (𝑟, 𝑧) plane) is given by R𝑠 = � cos(𝛽) 𝑑/𝜌 sin(𝑏) −𝑑/𝜌 sin(𝑏) cos(𝛽) � , (A30) with cos(𝛽) still defined consistently as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' While we are using 𝜙 to represent the two azimuthal angles, they are defined in the opposite sense (see Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' We therefore need to reflect the 𝜙 axis through the (𝑟, 𝑧) plane, after the rotation has occurred, given by R𝜙,reflect = �� � 1 0 0 0 −1 0 0 0 1 �� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (A31) Thus, our full three-dimensional transformation matrix is given by R𝑠 = �� � cos(𝛽) 0 𝑑/𝜌 sin(𝑏) 0 −1 0 −𝑑/𝜌 sin(𝑏) 0 cos(𝛽) �� � , (A32) APPENDIX B: CONVOLUTION MATHEMATICS FOR COUNTERPART AND NON-COUNTERPART HYPOTHESES B1 Counterpart Likelihood Including Proper Motion In this Appendix we detail the derivation of the inclusion of the proper motion PDF in the hypothesis that two objects are one astro- physical object given their separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Starting from a similar place to Wilson & Naylor (2018a)’s equation 14, we have 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝛾, 𝑦0 − 𝑦𝛾)× ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B1) Here we have the simultaneous marginalisation over an unknown common position – dropping the prior, 𝑝(𝑥0, 𝑦0) for being uniform and independent of unknown position (and proper motion), as per Wilson & Naylor (2018a) – and a marginalisation over the PDF of all unknown proper motions drifts 𝑝 (here representing proper motions in the two orthogonal sky directions with 𝑢 and 𝑣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Substituting Δ𝑥 = 𝑥𝜙 − 𝑥𝛾 and Δ𝑦 = 𝑦𝜙 − 𝑦𝛾 into ℎ𝛾 we get 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝜙 + Δ𝑥, 𝑦0 − 𝑦𝜙 + Δ𝑦)× ℎ𝜙(𝑥𝜙 − 𝑥0 − Δ𝑢, 𝑦𝜙 − 𝑦0 − Δ𝑣) d𝑥0 d𝑦0 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B2) Now we change variables from 𝑥0 and 𝑦0 to 𝑥 and 𝑦 via 𝑥 = 𝑥𝜙−𝑥0− Δ𝑢, 𝑦 = 𝑦𝜙 − 𝑦0 −Δ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This rearranges such that 𝑥0 −𝑥𝜙 = −Δ𝑢 −𝑥, 𝑦0 − 𝑦𝜙 = −Δ𝑣 − 𝑦, and thus 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) +∞ ∬ −∞ ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)× ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦 dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B3) As per Wilson & Naylor (2018a), we note that the inner integral is the definition of a convolution, and thus setting 𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) ≡ (ℎ𝛾 ∗ ℎ𝜙)(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) = +∞ ∬ −∞ ℎ𝛾(Δ𝑥 − Δ𝑢 − 𝑥, Δ𝑦 − Δ𝑣 − 𝑦)ℎ𝜙(𝑥, 𝑦) d𝑥 d𝑦, (B4) we have 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B5) Now it is clear that this is itself a convolution, of 𝑝 and 𝐺, and hence we can now write 𝐺′(Δ𝑥, Δ𝑦) ≡ (𝑝 ∗ 𝐺)(Δ𝑥, Δ𝑦) = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣)𝐺(Δ𝑥 − Δ𝑢, Δ𝑦 − Δ𝑣) dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B6) We note that our equation B1 is of similar form to equation 5 of Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (2010), with the interchange of integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Here we have chosen to construct a semi-analytic simulated model for the construction of the distribution of unknown proper motions, while Kerekes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' built theirs from survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' However, as discussed in Section 5, we can substitute such a data-driven distribution of proper motions within our matches, using any valid distribution as 𝑝(Δ𝑢, Δ𝑣) (or ℎ′pm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Finally, consistent with Wilson & Naylor (2018a)’s original derivation, we explicitly remind the reader that the AUFs ℎ must be defined such that ℎ(𝑥, 𝑦) = ℎ(−𝑥, −𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' B2 Unrelated Object Likelihood Including Proper Motion For the case where the sources are unrelated to one another, we have a slightly different equation to that of equation B1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' something more RASTI 000, 1–20 (2022) 20 Tom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Wilson like equation 10 of Budavári & Szalay (2008), 𝐺′ = +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) � +∞ ∬ −∞ ℎ𝛾(𝑥0 − 𝑥𝛾 − Δ𝑢, 𝑦0 − 𝑦𝛾 − Δ𝑣) d𝑥0 d𝑦0 � dΔ𝑢 dΔ𝑣 × +∞ ∬ −∞ 𝑝(Δ𝑢, Δ𝑣) � +∞ ∬ −∞ ℎ𝜙(𝑥0 − 𝑥𝜙 − Δ𝑢, 𝑦0 − 𝑦𝜙 − Δ𝑣) d𝑥0 d𝑦0 � dΔ𝑢 dΔ𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' (B7) Here, as with equation B1, we have explicitly assumed that 𝑝(𝑥0, 𝑦0) is independent of both unknown position and proper motion, and thus can be removed as a factor from the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' As ℎ𝛾 and ℎ𝜙 are normalised PDFs, the inner integral is trivially integrable to unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' but with 𝑝, the PDF of unknown proper motions, also normalised, the outer integral then also evaluates to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Thus we have the trivial case, for unrelated objects, that 𝐺 = 1 and 𝐺′ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' In these cases, as with the counterpart hypothesis having a prior 𝑝(𝑥0, 𝑦0) = 𝑁𝑐 as per Wilson & Naylor (2018a), we can say that the equivalent priors in the ‘unrelated’ hypothesis case are 𝑝(𝑥0, 𝑦0) = 𝑁 𝑓 , Wilson & Naylor (2018a)’s ‘field’ source density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' Hence, for the hypothesis of two sources being unrelated to one another and two detections of different sky objects, the PDF describing the likelihood of the objects having some separation is independent of proper motion, just as it is independent of the respective sources’ AUFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'} +page_content=' RASTI 000, 1–20 (2022)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf'}