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[EQUATION] where [MATH] is the synthetic ratio of extinction in the Johnson V band to color excess in [MATH] color at each grid point. |
We use the inverse of the variances of the input colors as weights to find the weighted average for the extinction. From the previous equations and the definition of [MATH] the weights are |
[EQUATION] To remove negative [MATH] , only grid points with [MATH] were selected, where [MATH] is a weighted dispersion of [MATH] |
For each point in the grid we have theoretical magnitudes, colors, theoretical excess ratios, and an estimation of the extinction [MATH] . Now we calculate for each point a synthetic reddened color, |
[EQUATION] and remove points that are out of the [MATH] range from the input color of the star. Finally, we select points with [EQUATION] |
for each color [MATH] and have N grid points as possible solutions for the star. 4.2 The second part of the method: cluster analysis |
We selected N points that satisfy our requirement (Eq. 4.1.1 ), that are within the allowed extinction range ( [MATH] ), and that are inside the error box of the stellar colors. All these points are possible solutions for the input star must be grouped properly to form a set of possible solutions in the next step. |
The distribution of the grid points is uniform in the parameter space [MATH] , [M/H]). We selected all grid points that are inside the error box of stellar colors, so are now able to simulate a more realistic physical distribution of grid points inside the error box assuming a normal distribution of errors of the color... |
The weighted distribution of grid points inside the error box will be projected onto the axes in the parameter space to subdivide the subspace with selected grid points into the set of possible solutions and to assign a probability to each solution. |
4.2.1 Weighting of points Suppose that the “observed” star with colors [MATH] has a normal distribution of errors in each color ( [MATH] ) and all colors are independent variables. From Eq. ( 14 ) we can find the distance of each grid point from the center of the error box (from the observed color of star), |
[EQUATION] where c is one of the colors, [MATH] The probability that a point occupies the 4-dimensional color space from [MATH] to [MATH] [MATH] to [MATH] [MATH] |
[MATH] is [EQUATION] where [EQUATION] The same distribution for other colors is assumed. The real distribution of grid points is [MATH] This is a number of points with [MATH] between [MATH] |
and [MATH] [MATH] and [MATH] , and so on; i.e. [MATH] The one-dimensional weight for the color (B-V), for example, is [MATH] and for a point in the 4-dimensional color space |
[EQUATION] The optimal bin in the case of the one-dimensional distribution is [MATH] and, in the case of d dimensions, the optimal bin is [MATH] |
(see Scott et al. Scott ( 2004 ). After the weighting of the points we normalise the weight of all selected points to 1, so that |
[MATH] 4.2.2 The final set of solutions The distribution of the weighted grid points was projected on each axis in the parameter space [MATH] [MATH] , [M/H]) to group the grid points into clusters in the parameter space. These clusters are the final set of possible solutions for the input star. |
Let us suppose that the grid points are distributed between a minimum [MATH] and a maximum [MATH] . We used the original bin of the grid ( [MATH] dex) to construct the projection of the distribution of N selected grid points on the temperature axis: |
[EQUATION] where i runs through 1 to [MATH] , and [MATH] is the number of bins of the grid between [MATH] and [MATH] [MATH] Note that [MATH] [MATH] and [M/H] are discrete values of the grid. |
Let us suppose that we have found [MATH] local maxima in [MATH] space. We estimate the significance of each local maximum in accordance with the maximum likelihood criterion for the significance of the maximum (see Materne materne ( 1979 ), i.e. for the k-th local maximum the criteria for significance is |
[EQUATION] where [EQUATION] and [MATH] is the likelihood in the absence of the k-th local maximum. We use the 10% significance level ( [MATH] ). If [MATH] the maximum is neglected and joined with the neighboring, most significant maximum. This procedure decreases the number of clusters in [MATH] space to [MATH] In each... |
axis, we are looking for the clustering on the metallicity axis with the same procedure. For example, there are [MATH] clusters on the temperature axis, the i-th cluster on the temperature axis has [MATH] clusters on the gravity axis and the j-th cluster on the gravity axis has |
[MATH] clusters on the metallicity axis. The final number of clusters (and this is the final set of solutions) is [EQUATION] We have distributed N grid points selected over [MATH] clusters, the sum of weights of points inside the cluster gives us the weight of each cluster: |
[EQUATION] where [MATH] is the j-th cluster. To find average values of temperature, gravity, and metallicity for each cluster, we repeated the procedure of weighting described in Sect. 4.2.1 , this time inside each cluster, and compute the weighted average solutions for [MATH] [MATH] , [M/H], [MATH] , and distance r wi... |
4.3 The third part of the method: selection of the solution We forme a set covering all possible solutions for a star. The final and most complicated step is the selection of one solution from the set. |
As we see from Fig. there is no possibility of selecting a single “true” solution in general. Nevertheless we can select “the statistically most probable” solution by the appropriate criteria. We would like to underline that, by these criteria, we do not use guesses about the astrophysical parameters themselves, so the... |
There are a number of methods for this selection: we can take the solution with the largest weight or with minimal distance from the input “observed” color in color space or take one with an unbiased distribution of grid points around the center of the error box in color space. |
Let us show an example of the determination of parameters to point out the major troubles we have to face. We used the point from the synthetic grid with [MATH] K, [MATH] , and [M/H]=0.0, photometric errors equal to 0.05 mag in all colors were taken. The first step of the method returned 6000 points distributed between... |
and between -2.5 and 0.5 dex in metallicity. All points were weighted and grouped in 5 possible solutions (see Table ). The second part of the method (the cluster analysis) is illustrated by Fig. . The first maximum on the [MATH] axis is rejected by the significance level assumed. Finally Fig. presents the clusters in ... |
[MATH] and the [MATH] planes. We define the following two quantities that describe the error of the estimated parameter: 1. The offset (bias) of the selected solution from the true parameters [MATH] |
[MATH] [MATH] . These values are the differences between the estimated parameter (the center of the selected cluster) and the true parameter known only in simulations. |
2. The dispersion of the estimated parameter [MATH] [MATH] [MATH] From Fig. we see that it is possible to subdivide the first cluster (first solution in Table ). This possibility was ruled out by the significance level we assumed (see Fig. , the first peak on the distribution function). If we decrease the significance ... |
4.3.1 The most preferable regions in parameter space We have determined regions in the parameter space where we can best estimate astrophysical parameters; in other words, if the solution falls into this region, we can be sure that this estimation is a correct, unbiased solution for the star. |
To find the most preferable regions, we performed simulations using the synthetic grid points as input “observed” colors. We interpolated between the grid points to create from a grid point with { [MATH] [MATH] , [M/H]} new points with [MATH] |
[MATH] [MATH] }, [MATH] , and [MATH] [MATH] , and [MATH] are intervals between points of the initial grid in temperature, gravity, and metallicity. For each point of this enhanced grid, we calculated synthetic colors. The resulting test grid includes points with the largest scatter from the points of the initial grid i... |
We tested the method on two sets of errors in colors; in the first case, all colors had an error 0.01 mag and in the second all colors had an error 0.05 mag. Figure shows the solution selected by the maximum weight with offsets (biases) caused by the selection of the “false” cluster or the shift inside the selected “tr... |
We define a region in the parameter space as suitable for the unbiased determination of astrophysical parameters, if we can select a solution for the input point inside the region, and this final solution has biases for temperature [MATH] for gravity [MATH] rms-errors [MATH] [MATH] |
in the temperature-gravity plane and [MATH] [MATH] rms-errors [MATH] [MATH] in the temperature-metallicity plane. This is done in Figs. 13 and 14 . According to these figures we define the following regions in the [MATH] plane |
[MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] [MATH] as the most suitable for the determination of [MATH] and [MATH] and on the [MATH] plane |
[MATH] as the most suitable in the search for metallicity. We note that the definition of these regions depends on the assumed errors of photometry, i.e. observations; and in the case of real stars with different observational errors in different bands, the definition of the most suitable regions can vary. |
4.3.2 Setting bounds on extinction Estimating the most preferable regions for our method made in the previous section is based on the assumption that photometric errors are the same in all bands. In the case of real observations, we would have to recalculate the estimation of errors and biases at any point on the synth... |
For the actual application of the method to observations, we propose the following procedure: we estimate the extinction per photometric distance. We assume a maximum extinction in the galactic plane to be 2.0 mag/kpc in the Johnson V band and a minimum extinction in the direction of the galactic poles as 0.1 mag/kpc (... |
Finally there are three criteria for selecting a single solution for the star: 1. selection of the solution with the largest weight, |
2. selection based on the most preferable region in the parameter space, 3. selection based on setting a bound on the extinction per photometric distance. |
The method used in practice and problems of the method We used the 2096 stars of our compiled catalog (see Sect. for details) with Johnson [MATH] [MATH] , and 2MASS [MATH] [MATH] [MATH] photometry to test the method in practice. |
Table shows the statistics for the different combinations of the criteria described in the previous section: criterion 1, criterion 1 combined with criterion 2 for selected stars (we select a single solution with the largest weight for each star and select stars within the most preferable regions), and criterion 1 comb... |
[MATH] ). The results for the calibration subset, based on the different selection criteria are shown in Figs. 15 16 17 (left panel for criterion 1, middle panel for criterion 1 & 2, right panel for criterion 1 & 3). |
The results for all stars in the compiled catalogue are shown in Figures 18 19 , and 20 We selected stars that have solutions with weight [MATH] uncertainties of the input photometry better that 0.05 mag, rms-errors of the estimated parameters (dispersion of the points around the weighted average parameter for the solu... |
5.1 The problem of the selection of the synthetic model grid There are a number of libraries of synthetic models of atmospheres is available today. The most popular among them are the BaSeL library based on Kurucz models with an addition of synthetic and semi-empirical models for cool stars (Lejeune et al., BaSeL ( 199... |
We do not discuss the different choices of theoretical models leaving this subject to the user of the method, but we note that some of the discrepancies in the parameters estimated via our method in comparison with the parameters from spectroscopic estimations can be due to the different model sets used for photometric... |
The direct comparison of Kurucz models with the SEDs of real stars (see, for example, Straižys et al., Straizys-Kurucz ( 1997 shows problems with the ultraviolet part of spectra for main-sequence stars, as well as an inconsistency in the synthetic SEDs with real SEDs for late type stars (K7-M stars). This will not infl... |
Both photometric and spectroscopic determinations of astrophysical parameters based on synthetic models suffer from the uncertainties and systematic errors of the selected synthetic grid. We urge the users of this method to always make a reference to the synthetic models used and to accept the resulting parameters as p... |
The estimation of the systematics in the photometry due to the difference between synthetic models (Martins & Coelho, martins ( 2007 ) shows that, for Johnson B, V, J, H, and K bands, they are within 0.05 mag, which can be accepted as an upper limit for this type of uncertainties in photometry. The rough estimation of ... |
5.2 The problem of stars outside the grid As we can see from the previous section and Figs. 15 16 17 we reached quite good agreement between spectroscopic and photometric estimation of astrophysical parameters, however there are some cases of a large differences between our estimated parameters and parameters from spec... |
The two stars with the greatest differences in the estimation of temperature are stars with peculiarities in the spectra. One of the cases where we failed to determine gravity is a Cepheid. For estimating the metallicity, we failed in the case of two close binaries, an RS CVn-like variable and a Cepheid (HD 101602, bot... |
There is only one star for which the reason for the failure is unclear: BD +30 2611. This star is a suspected Hipparcos binary star, although the radial velocity measurements did not prove the binarity (Sperauskas & Bartkevicius BD ( 2002 ). |
In all cases of failure, the stars in question did not satisfy the assumptions underlying our classification procedure. Binaries and variables cannot be estimated using the synthetic grid of models for “normal” single stars. Exotic stars (from the point of view of chemical abundance) like carbon stars were rejected at ... |
Results and conclusions We developed and tested a new method determining the astrophysical parameters from broad-band photometry. This method does not rely on a predefined model for the stellar population, but requires assuming the law for the extinction in the direction of the estimated star. In principle, the extinct... |
We used an interval-cluster analysis as the mathematical core for the method. For a given input photometric system (here Johnson B,V, and 2MASS J, H, and [MATH] ), this method allows us to select the regions in the parameter space where a (unique) solution is possible and to identify regions in the parameter space wher... |
We have compiled a catalog of stars with astrophysical parameters known from spectroscopic observations to calibrate the Kurucz models and the estimated parameters. |
In the Johnson B,V, and 2MASS J, H, and [MATH] photometric system and with mean errors of input colors better than 0.01 mag the best results are achieved for main sequence G-K stars (most preferable region, G6 to K9). In this case the parameters are accurate to [MATH] [MATH] , and [MATH] If the accuracy of the color me... |
We find that the biggest problem in determining astrophysical parameters is not to determine a solution but to distinguish between possible solutions that correspond to the input colors. We propose and tested a simple criterium for the selection of a solution based on the estimated extinction. Final tests with the comp... |
The proposed method was initially designed to search for sparse stellar groups in deep sky surveys, but it also suits the study of populations in the Galaxy without a predefined model for the distribution of stars. It is possible to use the method to determine an extinction law. |
Acknowledgements. This research was partially performed when ANB was at the Astronomisches Rechen-Institut, Heidelberg, where he gratefully acknowledges the DFG grant RO 528/9-1. ANB is grateful to Dr. Castelli for help with ATLAS9 program. This research made use of the SIMBAD database, operated at the CDS, Strasbourg,... |
# Source: arxiv 0808.1918 # Title: The Stellar Populations of M33's Outer Regions IV: Inflow History and Chemical Evolution # Sections: all # Downloaded: 2026-03-02T07:58:41.112920+00:00 |
The Stellar Populations of M33’s Outer Regions IV: Inflow History and Chemical Evolution (Accepted 2008 August 11. Received 2008 July 13; in original form 2008 May 1) |
Abstract We have modelled the observed color-magnitude diagram (CMD) at one location in M33’s outskirts under the framework of a simple chemical evolution scenario which adopts instantaneous and delayed recycling for the nucleosynthetic products of Type II and Ia supernovae. In this scenario, interstellar gas forms sta... |
keywords: galaxies: abundances, galaxies: evolution, galaxies: individual: Messier Number: M33, galaxies: stellar content, galaxies: spiral, galaxies: Local Group |
Introduction The distribution of stars in a color-magnitude diagram (CMD) is a sensitive function of the past and present star formation rate and chemical composition. This fact allows the extraction of the star formation history (SFH) and chemical enrichment history (CEH) of a galaxy by fitting its observed CMD with a... |
(Freeman & Bland-Hawthorn, 2002 Another, equally important method, is chemical modelling, whose primary goal is to reproduce the elemental abundance distribution in a galaxy’s stars. This method involves tracking the effects of gas flows, star formation, and stellar nucleosynthesis on the abundances and abundance ratio... |
[MATH] Gyr (Greggio, 2005 The methods of CMD fitting and chemical modelling are complementary, but they have evolved mostly independently. Some examples in the recent literature which combine the two methods generally use the results of CMD fitting as external inputs to chemical evolution models. For example, |
Aparicio et al. ( 1997 derived the first SFH of the Local Group (LG) dwarf galaxy, LGS 3, from CMD fitting and used the chemical evolution equations to see how much gas inflow and outflow was required to match their results. |
Carigi et al. ( 2002 computed the chemical evolution of four dwarf spheroidal (dSph) satellites of the Milky Way (MW). As external constraints, these authors used the SFHs of these galaxies derived by |
Hernandez et al. ( 2000 from a CMD fitting method. Dolphin ( 2002 calculated the SFHs of 6 dSph MW satellites using synthetic CMD fitting, and then |
Lanfranchi & Matteucci ( 2003 2004 adopted his SFHs as inputs to their chemical evolution models for the same galaxies. Similarly, Fenner et al. ( 2006 modelled the chemical evolution of the Sculptor dSph using the SFH derived by |
Dolphin et al. ( 2005 from a CMD analysis. Lastly, after adopting the SFH derived by Wyder ( 2001 2003 Carigi et al. ( 2006 modelled the evolution of NGC 6822 in a cosmological framework. |
Conversely, the results of chemical modelling can be used as external inputs to CMD fitting. For example, Pagel & Tautvaišienė ( 1998 , hereafter PT98) modelled the chemical evolution of the Large Magellanic Cloud (LMC) and Small Magellanic Cloud (SMC), adopting gas inflow and non-selective galactic winds (i.e., the wi... |
to extract its SFH from the CMD and luminosity function (e.g., Holtzman et al., 1999 ; Smecker-Hane et al., 2002 All the aforementioned studies combined CMD fitting and chemical modelling in a two-step process, in which the first step was done independently of the second. However, the two steps are inextricably linked ... |
The recent works of Ikuta & Arimoto ( 2002 and Yuk & Lee ( 2007 represent a significant improvement over the studies mentioned above because they incorporate CMD fitting and chemical modelling simultaneously. |
Ikuta & Arimoto ( 2002 computed a few closed box evolution models for several dSph satellites of the MW. They performed a qualitative comparison of their model CMD and [Mg/Fe] vs. [Fe/H] relation with what was observed and found a reasonable agreement, but they had to invoke gas stripping via ram pressure or tidal shoc... |
Yuk & Lee ( 2007 improve upon the work of Ikuta & Arimoto ( 2002 by quantitatively fitting a closed box chemical model to the CMD of IC 1613, a relatively isolated LG dwarf irregular galaxy. Their model SFH and AMR are in good agreement with previous independent determinations based on the canonical CMD fitting method,... |
There are many lines of evidence that suggest not all galaxies evolve as closed boxes. As originally hypothesized by Larson ( 1974 and exemplified by the Ikuta & Arimoto ( 2002 results, gas outflows could explain the lack of gas in dSphs despite their low metallicities (see also Lanfranchi & Matteucci ( 2004 ). Other e... |
(e.g., Garnett, 2002 ; Tremonti et al., 2004 ; Pilyugin et al., 2004 ; Erb et al., 2006 extended extra-planar HI gas in spirals (Fraternali et al., 2004 ; Oosterloo et al., 2007 high velocity clouds (HVCs) around the MW with near solar metallicity |
(van Woerden & Wakker, 2004 ; Wakker, 2001 ; Richter et al., 2001 extra-planar optical and X-ray emission around starburst galaxies |
(Heckman et al., 1990 ; Lehnert & Heckman, 1996 ; Martin et al., 2002 ; Strickland et al., 2004 , and velocity shifts of high-ion absorption lines in damped Lyman- [MATH] systems |
(Fox et al., 2007 and Lyman break galaxies at [MATH] (e.g., Pettini et al., 2001 ; Adelberger et al., 2003 . For a more detailed review, we refer the interested reader to Veilleux et al. ( 2005 |
Evidence for gas inflows includes the so-called G-dwarf problem, which is the fact that the metallicity distribution function of low mass, long lived stars observed in the Solar vicinity (SV) and in many other galaxies is too narrow and contains too few metal poor stars compared to the closed box model |
(e.g., Tinsley, 1975 ; Rocha-Pinto & Maciel, 1996 ; Seth et al., 2005 ; Jørgensen, 2000 ; Wyse & Gilmore, 1995 ; Harris & Harris, 2002 ; Koch et al., 2006 ; Sarajedini & Jablonka, 2005 ; Mouhcine et al., 2005 ; Worthey et al., 2005 Additionally, gas inflow over several Gyr is required to explain many characteristics of... |
(e.g., Chiappini et al., 2001 ; Portinari et al., 1998 . Other evidence for inflows includes the kinematics of extra-planar HI gas in some spirals |
(Fraternali & Binney, 2008 , the low metallicities and large distances of some MW HVCs (Wakker et al., 1999 ; Richter et al., 2001 high velocity OVI absorption along various sight-lines through the MW’s halo (Sembach et al., 2003 and the prevalence of warps in HI discs (Bosma, 1991 We refer the reader to Sancisi et al.... |
There are theoretical expectations for gas inflows, as well. Cosmological simulations of galaxy formation in the [MATH] CDM framework predict disc galaxies to form from the accretion of gas after the last major merger (Abadi et al., 2003 ; Sommer-Larsen et al., 2003 ; Governato et al., 2004 2007 For galaxies with viria... |
[MATH] , the accreted gas is expected to be mostly cold ( [MATH] ). On the other hand, upon entering the haloes of more massive galaxies, most of the accreted gas experiences shock-heating to the virial temperature ( [MATH] ), creating surrounding reservoirs of hot gas (Kereš et al., 2005 ; Dekel & Birnboim, 2006 Therm... |
(Maller & Bullock, 2004 ; Sommer-Larsen, 2006 ; Kaufmann et al., 2006 In some of the most recent N-body simulations, these clouds have properties similar to the HVCs mentioned above (Peek et al., 2008 Furthermore, this idea is supported by the recent discovery of a hot [MATH] ) gaseous halo around the quiescent massive... |
(Pedersen et al., 2006 . This gas is too hot to be heated by SNe in the disc and the disc SNe rate is too low to have created the reservoir through the outflow of disc gas. Finally, theoretical simulations suggest the need for extended accretion of dilute gas to keep discs from being destroyed after a succession of min... |
Despite all this evidence, the precise nature and importance of gas flows in the evolution of galaxies, particularly spirals, is still uncertain. To help improve the situation, in the present paper, we develop a chemophotometric CMD fitting method as an extension to the canonical method used in many other studies, but ... |
Ikuta & Arimoto ( 2002 , we solve the chemical evolution equations to obtain a self-consistent SFH and AMR. We improve upon their work by allowing for gas inflow and outflow and by efficiently searching the full volume of parameter space to make a detailed and quantitative fit to the observed CMD. |
The photometric data we use in this paper and the reduction procedure were presented in Barker et al. ( 2007a , hereafter Paper II In summary, three co-linear fields located in projection |
[MATH] southeast of M33’s nucleus were observed with the Advanced Camera for Surveys on board the Hubble Space Telescope In Barker et al. ( 2007b , hereafter Paper III , we computed the SFHs for these fields using the canonical synthetic CMD fitting method with age and metallicity as free parameters. Because the outer ... |
Paper III for a discussion), we restrict ourselves to analyzing the innermost field in the current study. This field has a projected galactic area of [MATH] |
and it lies at a deprojected radius of [MATH] disk scale lengths (Ferguson et al., 2007 or [MATH] kpc, assuming a distance of 867 kpc (Tiede et al., 2004 , inclination of [MATH] and position angle of [MATH] |
(Corbelli & Schneider, 1997 At this radius in M33, the azimuthally averaged HI and stellar surface densities are, respectively, [MATH] and [MATH] |
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