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5453c993b6ccedda872b2b32e4fe29062bc7df60 | subsection | 19 | 20 | Conclusions | Visualisation is one of the most effective techniques to explore and present scientific data,
understanding at a glance its basic features and properties. Proper graphical solutions
must be developed and exploited in order to fully achieve such objective. Such solutions can
change according to the target users, dependi... | {
"cite_spans": []
} | 10.1088/1367-2630/10/12/125006 | 0807.1742 | Splotch: Visualizing Cosmological Simulations | [
"K. Dolag",
"M. Reinecke",
"C. Gheller",
"S. Imboden"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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c99f23fd6223cdbf1881bef2c21646dd8bc526fb | abstract | 0 | 27 | Abstract | We present the results of NICMOS imaging of a sample of 16 high mass
passively evolving galaxies with 1.3<z<2, taken primarily from the Gemini Deep
Deep Survey. Around 80% of galaxies in our sample have spectra dominated by
stars with ages >1 Gyr. Our rest-frame R-band images show that most of these
objects have compac... | {
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} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
"astro-ph"
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dc606d859ac767445f7822c0d265f945811a79f2 | subsection | 1 | 27 | Introduction | The formation mechanism of elliptical galaxies has long been controversial and
remains a key test of more general galaxy formation models. The original `nature' ( monolithic collapse) vs. `nurture' (formation through mergers, , ,
debate is still with us, but is now set in a \Lambda CDM cosmological
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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fbed59f76c9a99678e837e3784d1d85a0acc9bbd | subsection | 2 | 27 | Introduction | However
only a limited number of simulations of this process have been done and they
have not yet been incorporated into cosmological models in a detailed way
such that they can be compared with data (e.g., numbers, sizes and masses of galaxies).
Further it is not clear that a dry
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
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"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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ce9c9a71c58061627d2301ae25cb7dc652079a76 | subsection | 3 | 27 | Introduction | Our use of NICMOS allows us to more robustly show that the old components in the galaxies are truly compact.
Finally, we are able to unify the optical and infrared work by introducing a new `stellar mass Kormendy relation'
which we use to better quantify evolution in the sizes of early-type galaxies as a function of st... | {
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} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
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"R. G. Carlberg",
"Inger Jorgensen",
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56c84fcbaacffb3de2f3bcc6c302fe6bcf9897d6 | subsection | 4 | 27 | Sample definition | Our sample of galaxies was taken mainly from the GDDS, crafted to sample
the galaxy population in the critical 1<z<2 interval with an emphasis on red
galaxies abr04. While modest in area (120 square arcminutes), the survey is spread over
four independent and representative sightlines. Redshifts for \sim 300 galaxies br... | {
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"Ivana Damjanov",
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"Roberto G. Abraham",
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"Damien Le Borgne",
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"R. G. Carlberg",
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52cf67b4f307c1ce9efcf6352732fad80dcbc392 | subsection | 5 | 27 | Sample definition | As will be described below, this additional data provided us with a useful check of our methodology by allowing us to compare results from our analysis pipeline against those published
in .lccc
4
0pc
Properties of the ten galaxies in GDDS sample
ID z Massa Ageb[10^{11} M_{}] [Gyr]
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
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"Richard Murowinski",
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"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
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31ff3958738d9a5765cb534444576f0f2dfaa388 | subsection | 6 | 27 | NICMOS Observations | The ten GDDS galaxies were observed with Camera 3 on NICMOS using
the F160W filter. Each individual exposure was 896 seconds in duration with multiple samples using
the STEP64 read pattern. A single orbit contained three exposures and we observed
each target over four HST orbits for a total integration time of 10740 se... | {
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"Roberto G. Abraham",
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"Damien Le Borgne",
"Sandra Savaglio",
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"R. G. Carlberg",
"Inger Jorgensen",
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4e23dd0fa2035460d64a50be5ad1643a0819583a | subsection | 7 | 27 | Surface brightness profiles | Using the Galfit software package , we derived
two-dimensional (2D) surface brightness profiles by fitting synthetic galaxy images to
our data using a
range of surface brightness profiles, ellipticities and orientations. A series of models
were constructed using exponential surface brightness profiles, de Vaucouleurs R... | {
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
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"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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faa27b6745c1254ff94f10c0d5515e2603acb013 | subsection | 8 | 27 | Surface brightness profiles | On the other hand, the best fit R^{1\slash n} profile for S2F1_443 has index n=2.8, higher than n=1.9 reported by . For the rest of the MUNICS sample the difference in the goodness of fit for R^{1\slash 4} profile between our and analysis is \Delta (\chi ^2)\lesssim 0.2.
[Figure: Upper panels: major axis surface bright... | {
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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b8fb3be88160e90605b1f3d398d9661706baf53b | subsection | 9 | 27 | Surface brightness profiles | REF .lcccccccccc
7
0pc
Morphological parameters of the galaxies in the GDDS sample
ID F160W n r_e R_{e} \langle \mu \rangle _e^{160W} \langle \mu \rangle _e^{corr} b/a \chi ^2[mag] [arcsec] [kpc] [mag/arcsec^2] [mag/arcsec^2]
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} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
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"Ivana Damjanov",
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"Roberto G. Abraham",
"Karl Glazebrook",
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"Damien Le Borgne",
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"Richard Murowinski",
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"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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669489549bc12343cd02e1c3c792d5be87c62f69 | subsection | 10 | 27 | Surface brightness profiles | 79\pm 0.09 4 0.25 \pm 0.06 2.1 \pm 0.5 19.8 \pm 0.5 16.6 \pm 0.5 0.82 \pm 0.07 0.49520.78 \pm 0.06 4.1 \pm 0.9 0.25 \pm 0.04 2.1 \pm 0.3 19.8 \pm 0.3 16.7 \pm 0.3 0.82 \pm 0.07 0.49512-607222.30 \pm 0.08 4 0.04 \pm 0.04 0.3 \pm 0.3 17 \pm 3 14 \pm 3 0.97^{+0.03}_{-0.3} 0.17422.31 \pm 0.07 1.7 \pm 1.2 0.09 \pm 0.04 0.8 ... | {
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
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"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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b325e7f6cb44187885c4b2f1637a8d6a3c6edf2a | subsection | 11 | 27 | Surface brightness profiles | We constructed a set of galaxy images from
our best-fit model for each galaxy and convolved these with a range of PSFs (i.e., PSFs obtained
from different stars) and added these to the background images. We dithered the position image
about the central value to explore the importance of binning, and used RMS images to ... | {
"cite_spans": []
} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
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"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
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"Richard Murowinski",
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"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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a495af670b519041a4da013fe0dc8da66f12d158 | subsection | 12 | 27 | K-corrections and cosmological dimming | Our analysis requires comparison between the properties of our 1.2 < z < 2 samples
observed at 1.6 \mu m (H-band) to those of present-day galaxies observed at visible
wavelengths. In order to make a proper comparison, we need to transform
the various data sets to a common bandpass and apply a K-correction.
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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0182195ae7cbe1fa8c61e2f8c62667c916d2fb55 | subsection | 13 | 27 | Morphologies of Passive Galaxies at | All of the objects in our NICMOS F160W sample (shown in
Figures REF and REF ) have compact morphologies and none
show obvious evidence of interactions, such as double nuclei or disturbed isophotes
at bright levels. The star-forming massive galaxies drawn from the GDDS sample,
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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901f15b885c96051f8e40985ee541cb0e873fc16 | subsection | 14 | 27 | Surface Brightness Profiles & Sizes | Azimuthally averaged surface brightness profiles presented in
Figure REF confirm that six of our 10 GDDS galaxies are well-fit by R^{1\slash 4} profiles.
The effective radii for these six objects range from as small as 005 to as large as 042, or
from 0.4 to 3.6 kpc. The median effective radius is 026 or 2.2 kpc.
As Fig... | {
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Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
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"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
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"Hsiao-Wen Chen",
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28a4da93490d0b2221ebd020d3672699a3441310 | subsection | 15 | 27 | Surface Brightness Profiles & Sizes | The lower redshift samples
, , cover a broad range of the parent luminosity functions while
the higher redshift objects (1<z<3),
including the DRGs, the GDDS and MUNICS samples (, gla04, ), sample the
high mass end of the galaxy population and thus are biased to large values in their median sizes.
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
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b57f089943a3a410a2e2e880e16d60056ba3df0e | subsection | 16 | 27 | The Kormendy Relation to | In Figure REF we present the rest-frame r-band Kormendy
relation, \langle \mu \rangle _e vs. R_e, for the GDDS and MUNICS samples. As noted earlier,
our construction of this diagram is particularly robust because our
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
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"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
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0bfb87211209309ef460619b466325a4832d3032 | subsection | 17 | 27 | The Mass-Size Relation and the Stelar Mass Kormendy Relation | As the previous section illustrates, a proper comparison between galaxy samples at high
and low redshifts nearly always entails corrections for luminosity evolution. We can, however, improve on the standard procedure of
using simple models of luminosity evolution by using multi-color
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... | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
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1d039ccbd939aee84b87fb92db6a3acacb75d5c6 | subsection | 18 | 27 | The Mass-Size Relation and the Stelar Mass Kormendy Relation | The number density of compact galaxies with R_e<1 kpc (`red nuggets') in the redshift range 1.1<z<2
is 2\times 10^{-5} Mpc^{-3}. In contrast, number density of these objects in the SDSS sample
is 3\times 10^{-8} Mpc^{-3}, three orders of magnitude lower than that for the higher redshift objects. The
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Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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670c82bbfc4860c6f49106f70217427f2601ca9b | subsection | 19 | 27 | The Mass-Size Relation and the Stelar Mass Kormendy Relation | In the
lower redshift sub-sample 6/18 galaxies, or \sim 33% of the sample, fall within the range
of the local sample, while in the high redshift sample, only 4/25, or \sim 17% of the galaxies
fall within the locus of the local systems. Thus it appears that the strongest evolution in size
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} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
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b9eb4406f691894438e9900e7b526d96611637cb | subsection | 20 | 27 | Discussion | The key result of this paper are that the sizes and projected mass densities of early-type
passively evolving galaxies have changed very significantly since z \sim 2. A number of other
studies, noted above, have reached similar conclusions in samples with higher and overlaping
redshift intervals. Our analysis has remov... | {
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"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
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"R. G. Carlberg",
"Inger Jorgensen",
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61ad5bfa5d6d86fe4b6bb6fe6cb9e195d64debe4 | subsection | 21 | 27 | Discussion | If the
mass is lost more quickly than the dynamical timescale, then the
expansion of the system will be larger than this estimate. It is trivial to show that as the system loses
mass the dynamical timescale increases in proportion to 1\over {(1-\frac{\Delta m}{m})^2} while the escape
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Kormendy Relation | [
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"Erin Mentuch",
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"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
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7a23c8d368b2c4e4614301cc462a217d27187a64 | subsection | 22 | 27 | Discussion | Another constraint on the importance of adiabatic
expansion is that is does not explain the steady factor
of (at least) three
growth in the stellar mass density locked up in massive
galaxies over the redshift range 1<z<2
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"Roberto G. Abraham",
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3237702779d4aec65b4be9b9f2cc095976bb4a7f | subsection | 23 | 27 | Discussion | Limiting stellar mass densities for the 90th percentiles for the SDSS objects with stellar mass densities above and below the median value are given with upper and lower dashed lines, respectively. Following the discussion on the quality of CFRS imaging in section corresponding points are excluded from this fugure.]So... | {
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6e5ff56ae5fcc96919ce7e0835e6eb45a2bb65f7 | subsection | 24 | 27 | Discussion | The dispersion
on the size-mass plot in the 0.2<z<1 regime is large (upper panels in Figures REF and REF ), but there seems
to be some evidence for a systematic offest relative to the local
trends with the increasing redshift. The GOODS/DEIMOS data in Figures REF and REF spans both
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dd8aa45261b45865aed126f27143990c1e100b34 | subsection | 25 | 27 | Discussion | A successful model will have to simultaneously explain
the size change in the galaxies, the duty cycle
for this size change, and the epoch in a galaxy's
life history at which the change occurs.
And, as noted above, mass density growth over
the redshift interval being probed suggests
that the size growth
being witnessed... | {
"cite_spans": []
} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
"astro-ph"
] | 2,008 | en | Physics | [
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cc2e52c279923acac35534dfd5a9224100e825ee | subsection | 26 | 27 | Conclusions | The size-mass
relationship for early-type
galaxies evolves significantly from z=2 to z=1.
Over the whole of this
redshift range early type galaxies tend to be a factor
of 2-3 smaller
than local counterparts of similar mass. Similarly
compact galaxies are seen at z>2 , and
we speculate that
the very compact galaxies stu... | {
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} | 10.1088/0004-637X/695/1/101 | 0807.1744 | Red Nuggets at z~1.5: Compact passive galaxies and the formation of the
Kormendy Relation | [
"Ivana Damjanov",
"Patrick J. McCarthy",
"Roberto G. Abraham",
"Karl Glazebrook",
"Haojing Yan",
"Erin Mentuch",
"Damien Le Borgne",
"Sandra Savaglio",
"David Crampton",
"Richard Murowinski",
"Stephanie Juneau",
"R. G. Carlberg",
"Inger Jorgensen",
"Kathy Roth",
"Hsiao-Wen Chen",
"Rona... | [
"astro-ph"
] | 2,008 | en | Physics | [
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57048c0fed8fa545206f7e8159f2ae86c8795d1d | abstract | 0 | 9 | Abstract | A ring of dense molecular gas extending 2-7 pc orbits the supermassive black
hole Sgr A* at the center of our Galaxy. Using the Green Bank Telescope, we
detected water maser lines and both narrow (0.35 km/s) and broad (30 - 50 km/s)
methanol emission from the molecular ring. Two of the strongest methanol lines
at 44 GH... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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61b296a35029480f2108d654053bb17f3f2b9335 | subsection | 1 | 9 | Introduction | The circumnuclear molecular ring (CMR) extends from 2 to 7 pc
from the massive black
hole at the center of the Galaxy (Güsten et al. 1987; Jackson et al. 1993;
Marshall, Lasenby & Harris 1995; Latvakoski et
al. 1999; Yusef-Zadeh et
al. 2006; Bradford et al. 2005; Herrnstein & Ho 2005;
Schödel et al. 2002; Ghez et al. 2... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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7a53c69efcb65deabdb46315d6044b87d725e709 | subsection | 2 | 9 | Observations | Using the Green Bank Telescope (GBT) of the National Radio Astronomy
ObservatoryThe National Radio Astronomy Observatory is a facility
of the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc., we searched for 7_0-6_1 A^+ methanol I
(44.0694 GHz) and 6_{16} - 5_{23} water ... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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a770fa5fc5c935b970111263ba055a99c1739077 | subsection | 3 | 9 | Results | Our search detected five methanol sources, three of which arise
from the CMR and two from sources SgrA-A and SgrA-D of the SgrA East HII regions.
Figure 1 shows the positions of detected methanol and water line emission from the
CMR; the water spectra are shown in panels around the HCN image of the molecular
ring. A to... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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22f23882922c4830a320a5286613ebbcf2d06f34 | subsection | 4 | 9 | Results | Our VLA observations had insufficient sensitivity (56 mJy per channel) and spectral resolution (1.2
km s^{-1}) to confirm the location within the GBT beam.
[Figure: (Bottom)Contours of HCN emission from clump V are shown to the left andthe methanol spectrum at the position indicated by the 15^{\prime \prime } circle (c... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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a67c336ac6cd7aa77c23de6d7a062aef020c7bfa | subsection | 5 | 9 | Results | The
spectrum (Fig. 3) shows three narrow lines between 52 and 55 km s^{-1} with peak flux density
of 2.38 Jy at 50.2 km s^{-1} and a
broad red-shifted wing extending up to 100 km s^{-1}, similar to the
velocity profile seen toward clump V. This is the only
one among the five detected methanol sources that shows a count... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
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fcf6894fb16d362caf4901f0204cf3b6235d7963 | subsection | 6 | 9 | Results | The narrow velocity width and the high brightness temperature confirms
the maser nature of the methanol emission detected with the GBT.Although methanol emission has been detected from three regions in the
CMR, we find additional broad, red-shifted HCN profiles, as in the region
in the southern lobe of the CMR at
\alph... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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7d3fb14a4d79169208fa0ce1ab4ef319fb6fd27b | subsection | 7 | 9 | Discussion and Conclusion | The presence of Class I methanol masers or thermal methanol
emission and the absence of ultra
compact HII regions in the immediate vicinity of clumps F and G are
consistent with the earliest stages of massive star formation, within
a few 10^4 \, \textrm {yr} of the onset of gravitational collapse of a
molecular cloud c... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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9e4a147c131693bd04eeec13370a242aa6c6a322 | subsection | 8 | 9 | Discussion and Conclusion | It is also consistent with the high densities recently inferred
from HCN measurements (CSSY), which imply the existence of
gravitationally bound
clumps, the precursors of collapse. Furthermore, the asymmetry in the northern and
southern halves of the CMR suggests that the southern half
of the CMR has not begun its star... | {
"cite_spans": []
} | 10.1086/591731 | 0807.1745 | Massive Star Formation in the Molecular Ring Orbiting the Black Hole at
the Galactic Center | [
"F. Yusef-Zadeh",
"J. Braatz",
"M. Wardle",
"D. Roberts"
] | [
"astro-ph"
] | 2,008 | en | Physics | [
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145c3e2c4d63054276de18757dd225706fc2aa1e | abstract | 0 | 7 | Abstract | New resonances with masses of order a few ${\rm TeV}$ might be discovered at
the LHC. We show that no resonance that couples to electrons only through
Standard Model interactions can decay to both $e^+e^-$and $\gamma \gamma$ with
significant branching ratios. This means that finding both electron-positron
and two-photo... | {
"cite_spans": []
} | 0807.1746 | Exotic Implications of Electron and Photon Final States | [
"Lisa Randall",
"Mark B. Wise"
] | [
"hep-ph"
] | 2,008 | en | Physics | [
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894fdc58c523100a012a2b0d3c4f6467decf229a | subsection | 1 | 7 | Body | Exotic Implications of Electron and Photon Final StatesLisa Randall
Jefferson Laboratory of Physics, Harvard University
Cambridge, Massachusetts 02138, USA.Mark B. Wise
California Institute of Technology, Pasadena, CA 91125New resonances with masses of order a few {\rm TeV} might be discovered at the LHC. We show that ... | {
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6ed319f11fbffd1bc8a5678edb5f7e816d6becdc | subsection | 2 | 7 | Body | The interactions that give rise to the bound-state are not shown.]However, as we now show, if the electron mass is neglected it is never possible for such a resonance to decay at tree level via Standard Model electroweak interactions to both e^+e^-Our conclusions apply to any charged lepton, in particular, \mu ^+ \mu ^... | {
"cite_spans": []
} | 0807.1746 | Exotic Implications of Electron and Photon Final States | [
"Lisa Randall",
"Mark B. Wise"
] | [
"hep-ph"
] | 2,008 | en | Physics | [
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26cf6fac4be69ee096e970852bf33a6d62527571 | subsection | 3 | 7 | Body | So although spin zero resonances can decay to photons, in the limit m_e \rightarrow 0, they cannot decay to e^+e^- when the new physics, beyond that in the minimal Standard Model, does not couple directly to the electrons.There is a simple example of beyond-the-minimal-Standard-Model physics that can have a spin zero r... | {
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"do... | 0807.1746 | Exotic Implications of Electron and Photon Final States | [
"Lisa Randall",
"Mark B. Wise"
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91ef063c6da7c9f08953e356b97f7769c536d021 | subsection | 4 | 7 | Body | It remains to consider spin-2 resonances and beyond. However, they cannot decay to e^+ e^- through a virtual gauge boson because for such resonances R the matrix element \langle 0|J_{\mu } |R(p,\epsilon )\rangle vanishes since their polarization vectors have more than one Lorentz index and the momentum p contracted wit... | {
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} | 0807.1746 | Exotic Implications of Electron and Photon Final States | [
"Lisa Randall",
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] | [
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be6d4980d2995c2424b68f720af4dad8bc0cfbe7 | subsection | 5 | 7 | Body | To naturally suppress flavor-changing effects of new physics that couples directly to electrons requires a gauge principle. Known examples would involve strong gauge interactions that couple both to the new resonance and to electrons. Perhaps the best-motivated examples involve electron compositeness, with the new reso... | {
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8b2ba42472eb313eaf0917a293b026797c2293ad | subsection | 6 | 7 | Body | If only Standard Model final states are possible then the above results imply that {\rm Br}( h^{\rm KK}\rightarrow e^+e^-) \simeq 0.022 and {\rm Br}( h^{\rm KK}\rightarrow \gamma \gamma ) \simeq 0.044.In conclusion, measuring branching the fractions for resonant decay to e^+e^- and \gamma \gamma can tell us that the el... | {
"cite_spans": []
} | 0807.1746 | Exotic Implications of Electron and Photon Final States | [
"Lisa Randall",
"Mark B. Wise"
] | [
"hep-ph"
] | 2,008 | en | Physics | [
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40bef8bf46ecbd5b3038a3b33624d6459b257e9a | abstract | 0 | 74 | Abstract | We generalize the Newtonian n-body problem to spaces of curvature k=constant,
and study the motion in the 2-dimensional case. For k>0, the equations of
motion encounter non-collision singularities, which occur when two bodies are
antipodal. This phenomenon leads, on one hand, to hybrid solution singularities
for as few... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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9e775d9dd8103363cc633f309b99c4e7604de89d | subsection | 1 | 74 | Introduction | The goal of this paper is to extend the Newtonian n-body problem of celestial
mechanics to spaces of constant curvature. Though attempts of this kind existed
for two bodies in the 19th century, they faded away after the birth of special and general relativity, to be resurrected several decades later, but only in the ca... | {
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} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
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63e10a6abe03c5ece2b3b08cde5ec8c9b339ad18 | subsection | 2 | 74 | History of the problem | The first researcher who took the idea of gravitation beyond {\bf R}^3 was Nikolai Lobachevsky. In 1835, he proposed a Kepler problem in the 3-dimensional hyperbolic space, {\bf H}^3, by defining an attractive force proportional to the inverse area of the 2-dimensional sphere of the same radius as the distance between ... | {
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"raw": "N. I. Lobachevsky, The new foundations of geometry with full theory of parallels [in Russian], 1835-1838, In Collected Works, V. 2, GITTL, Moscow, 1949, p. 159.",
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b08b68340d065429861f7ea382251550d32d08b7 | subsection | 3 | 74 | History of the problem | A bibliography of works about mechanical problems in spaces
of constant curvature is given in .Several members of the Russian school of celestial mechanics, including Valeri Kozlov and Alexander Harin, , , Alexey Borisov, Ivan Mamaev, and Alexander Kilin, , , , , , Alexey Shchepetilov, , , , and Tatiana Vozmischeva, , ... | {
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4a5ee7d885db2303aea42c71fd1726aaa61dde9f | subsection | 4 | 74 | Relativistic | Before trying to approach this problem with contemporary tools, we were
compelled to ask why the direction of research proposed by Lobachevsky was
neglected after the birth of relativity. Perhaps this phenomenon occurred because relativity hoped not only to answer the questions this research direction had asked, but al... | {
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83e65782e495e13a8551a20e883554037bf3c702 | subsection | 5 | 74 | Our approach | The present paper overcomes the above-mentioned difficulties encountered in defining a meaningful n-body problem prone to the same mathematical depth achieved in the classical case, by replacing the differential-geometric approach
used for n=2 in the case of the cotangent potential with the variational method of constr... | {
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78764d0b4a6f7e2d0ce5feadc19d4fdddf59f235 | subsection | 6 | 74 | Equations of motion | In Section 3, we extend the Newtonian potential of the n-body problem
to spaces of constant curvature, \kappa , for any finite dimension. For
\kappa \ne 0, the potential turns out to be a homogeneous function of
degree zero. We also show the existence of an energy integral as well as
of the integrals of the angular mom... | {
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72a889729f3556385096063bfa87773b727a8d68 | subsection | 7 | 74 | Singularities | In Section 4 we focus on singularities, and distinguish between singularities
of the equations of motion and solution singularities. For any \kappa \ne 0,
the equations of motion become singular at collisions, the same as in the
Euclidean case. The case \kappa >0, however, introduces some new singularities,
which we ca... | {
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} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
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58ec5d24ebce7fe2b0170f8a0e2cb556e299a718 | subsection | 8 | 74 | Relative equilibria | The rest of this paper, except for the Appendix, focuses on the results we obtained
in {\bf S}^2 and {\bf H}^2, mainly because these two surfaces are representative for the cases \kappa >0 and \kappa <0, respectively. Indeed, the results we proved for these surfaces can be extended to different curvatures of the same s... | {
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} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
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5be82d4e69a8b33b50da6b0acfc4db7b309e8603 | subsection | 9 | 74 | Relative equilibria | Thus Lagrangian solutions with unequal masses are specific to the Euclidean case.Theorems REF and REF show that analogues to the collinear (Eulerian) orbits in the 3-body problem of the classical case exist in {\bf S}^2 and {\bf H}^2, respectively. While nothing surprising happens in {\bf H}^2, where
we prove the exist... | {
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} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
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3ba440f72dfc8c64690caa3f91f261f699a95720 | subsection | 10 | 74 | Saari's conjecture | Our extension of the Newtonian n-body problem to spaces of constant curvature
also reveals new aspects of Saari's conjecture. Proposed in 1970 by Don Saari in
the Euclidean case, Saari's conjecture claims that solutions with constant moment of inertia are relative equilibria. This problem generated a lot of interest fr... | {
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5360876546a16f46f651c5adb0edc5fcdddc383d | subsection | 11 | 74 | Some physical remarks | Does our gravitational model have any connection with the physical reality? Since there is no unique extension of the Newtonian n-body problem to spaces of constant curvature, is our generalization meaningful from the physical point of view or does it lead only to some interesting mathematical properties?We followed th... | {
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02b9162e4545d37d7309064b2d289b77bbe32b45 | subsection | 12 | 74 | Some physical remarks | They don't show up in the Euclidean case, and neither do they appear in this model within the observable universe. The properties we proved for relative equilibria are also in agreement with the classical n-body problem, the only exception being the Lagrangian solutions for \kappa \ne 0, which, unlike in the Euclidean ... | {
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5a5cbff04858e8aa487c655bdbb9aba11563f76f | subsection | 13 | 74 | Unified trigonometry | Let us first consider what, following , we will call trigonometric \kappa -functions, which unify elliptical and hyperbolic
trigonometry. We define the
\kappa -sine, {\rm sn}_\kappa , as{\rm sn}_{\kappa }(x):=\left\lbrace
\begin{array}{rl}
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6f7e107eddd619e0c0ca1f6527b7c3873798adca | subsection | 14 | 74 | Differential-geometric approach | In a 2-dimensional Riemann space, we can define geodesic polar
coordinates, (r,\phi ), by fixing an origin and an oriented geodesic through it.
If the space has constant curvature \kappa , the range of r depends on \kappa ; namely r\in [0,{\pi /(2\kappa ^{1/2})}] for \kappa >0 and r\in [0,\infty ) for
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b20ad14411487fa2959c1662ee24e754fe157fc8 | subsection | 15 | 74 | The potential | To generalize the above setting of the Kepler problem to the n-body problem
on surfaces of constant curvature, let us start with some notations. Consider
n bodies of masses m_1,\dots ,m_n moving on a surface of constant
curvature \kappa . When \kappa >0, the surfaces are spheres of radii
{\kappa }^{-1/2} given by the e... | {
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00f3dd26d2cac8ed43f9d4c7a066fe6f24287517 | subsection | 16 | 74 | The potential | We also define {\bf a}\otimes {\bf b} as either of the cross products{\bf a}\times {\bf b}:=(a_yb_z-a_zb_y, a_zb_x-a_xb_z, a_xb_y-a_yb_x) \ \ {\rm for}\ \ \kappa \ge 0,{\bf a}\boxtimes {\bf b}:=(a_yb_z-a_zb_y, a_zb_x-a_xb_z, a_yb_x-a_xb_y)\ \ {\rm for}\ \ \kappa <0.The distance between \bf a and \bf b on the surface of... | {
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b18671af1d9bc1fa078bd3de1f67ea5d0217b5a2 | subsection | 17 | 74 | The potential | \ \ \kappa >0\cr |{\bf a}-{\bf b}|, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \! \ \hspace{23.33147pt} \kappa =0\cr ({-\kappa })^{-1/2}\cosh ^{-1}{\kappa {\bf a}\boxdot {\bf b}\over \sqrt{\kappa {\bf a}\boxdot {\bf a}}\sqrt{\kappa {\bf b}\boxdot {\bf b}}},\hspace{4.5pt} \kappa <0.\cr \end{array}\right.}Notice that th... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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-... |
87a5b95fee4b94ecdf7b2f476bb52091ec5082ea | subsection | 18 | 74 | The potential | A straightforward computation shows thatU_\kappa ({\bf q})={1\over 2}\sum _{i=1}^n\sum _{j=1,j\ne i}^n{m_im_j
(\sigma \kappa )^{1/2}{\kappa {\bf q}_i\odot {\bf q}_j\over \sqrt{\kappa {\bf q}_i
\odot {\bf q}_i}\sqrt{\kappa {\bf q}_j\odot {\bf q}_j}}\over \sqrt{\sigma -\sigma \Big ({\kappa {\bf q}_i\odot {\bf q}_j\over \... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.01... |
b60c01c09d1a3412d7881e08949610b5a9eaeb90 | subsection | 19 | 74 | Euler's formula | Notice that U_{\kappa }(\eta {\bf q})=U_\kappa ({\bf q})=\eta ^0U_{\kappa }({\bf q})
for any \eta \ne 0, which means that the potential is a homogeneous function of degree zero. But for {\bf q}
in {\bf R}^{3n}, homogeneous functions F:{\bf R}^{3n}\rightarrow {\bf R} of degree \alpha satisfy Euler's formula, {\bf q}\cdo... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
0.0007402152405120432,
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-0.... |
d0fc885d9cc71dd91156e62d4adadf9051ee2388 | subsection | 20 | 74 | Derivation of the equations of motion | To obtain the equations of motion for \kappa \ne 0, we will use a variational
method applied to the force function (REF ). The Lagrangian of the n-body system has the formL_\kappa ({\bf q}, \dot{\bf q})=T_\kappa ({\bf q}, \dot{\bf q})+U_\kappa ({\bf q}),where T_\kappa ({\bf q},\dot{\bf q}):={1\over 2}\sum _{i=1}^nm_i(\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1162,
"openalex_id": "",
"raw": "I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall, Englewood Cliffs, New Jersey, 1963.",
"source_ref_id": "ff6f943de3ae2c9dac86714cbf10ba66cc690739",
"start": 574
}... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0... |
905709c0d22fbd2fe335d8b5cea5f00daf534ad0 | subsection | 21 | 74 | Derivation of the equations of motion | Since {\bf q}_i\odot {\bf q}_i=\kappa ^{-1} implies that
\dot{\bf q}_i\odot {\bf q}_i=0, it follows that{d\over dt}\Bigg ({\partial L_\kappa \over \partial \dot{\bf q}_i}\Bigg )=
m_i\ddot{\bf q}_i(\kappa {\bf q}_i\odot {\bf q}_i)+2 m_i(\kappa \dot{\bf q}_i\odot {\bf q}_i)=
m_i\ddot{\bf q}_i.This relation, together with... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.01564355194568634,
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... |
18c1bdbafef7f6d8d4bdb27a30d4cc3e7921b8ab | subsection | 22 | 74 | Derivation of the equations of motion | Substituting these values of the Lagrange multipliers into equations (REF ),
the equations of motion and their constraints becomem_i \ddot{\bf q}_i=\widetilde{\nabla }_{{\bf q}_i} U_{\kappa }({\bf q})-m_i\kappa (\dot{\bf q}_i\odot \dot{\bf q}_i){\bf q}_i, \ \ {\bf q}_i\odot {\bf q}_i=\kappa ^{-1}, \ \ \kappa \ne 0,\\
\... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.0020354848820716143,
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0.025196518748998642,
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... |
3da9d0c75083004bd77e0a3dbf89123c8865645a | subsection | 23 | 74 | First integrals | The equations of motion have the energy integralT_\kappa ({\bf q},{\bf p})-U_\kappa ({\bf q})=h,where, recall, T_\kappa ({\bf q}, {\bf p}):={1\over 2}\sum _{i=1}^nm_i^{-1}({\bf p}_i\odot {\bf p}_i)(\kappa {\bf q}_i\odot {\bf q}_i) is the kinetic energy, {\bf p}:=({\bf p}_1,\dots ,{\bf p}_n) denotes the momentum of the ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
0.010368824936449528,
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-0.0051767826080322266,
-0.016068244352936745,... |
199302f0458fc2ec480bfa6311cb2480ddc5c8a5 | subsection | 24 | 74 | First integrals | The last of the above identities follows from the skew-symmetry of
\otimes and the fact that {\bf q}_i\otimes {\bf q}_i={\bf 0}, \ i=1,\dots ,n. | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
0.0171628650277853,
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0.00... |
80ad47606167c9332f12e7a2c1429be3a2358c6c | subsection | 25 | 74 | Motion of a free body | A consequence of the integrals of motion is the analogue of the well known result from the Euclidean space related to the motion of a single body in the absence of any gravitational interactions. Though simple, the proof of this property is not as trivial as in the classical case.Proposition 1 A free body on a surface ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.022173630073666573,
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0.004032609518617392,
-0.00138299050740897... |
92bacf1f4dfaa75067c5f719c6bfa0a1384d6f6a | subsection | 26 | 74 | Hamiltonian form | The equations of motion (REF ) are Hamiltonian. Indeed, the Hamiltonian function
H_\kappa is given by{\left\lbrace \begin{array}{ll}
H_\kappa ({\bf q},{\bf p})=
{1\over 2}\sum _{i=1}^nm_i^{-1}({\bf p}_i\odot {\bf p}_i)
(\kappa {\bf q}_i\odot {\bf q}_i)-U_\kappa ({\bf q}),\cr {\bf q}_i\odot {\bf q}_i={\kappa }^{-1}, \ \... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.0038175894878804684,
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0.0036307144910097122,
-0.02532347105443477... |
9adedc166b18636d899951fa5b3a6689a2544117 | subsection | 27 | 74 | Hamiltonian form | The representative non-zero-curvature cases, however, are \kappa =1 and \kappa =-1, which characterize the motion for \kappa >0 and \kappa <0, respectively. Therefore we will further focus on the n-body problem in {\bf S}^2
and {\bf H}^2. | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.030565502122044563,
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0.013... |
f409981393ad20476e31c79b4a1be319c967c00b | subsection | 28 | 74 | Equations of motion in | In this case, the force function (REF ) takes the formU_1({\bf q})={1\over 2}\sum _{i=1}^n\sum _{j=1,j\ne i}^n\frac{m_im_j~ \frac{{\bf q}_i\cdot {\bf q}_j}{\sqrt{{\bf q}_i\cdot {\bf q}_i}\sqrt{{\bf q}_j\cdot {\bf q}_j}}}{
\sqrt{1-\left(\frac{{\bf q}_i\cdot {\bf q}_j}{\sqrt{{\bf q}_i\cdot {\bf q}_i}\sqrt{{\bf q}_j\cdot ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.01070378627628088,
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0.... |
bbf42bdd6422d9db1c22b0a17f53da11f567791f | subsection | 29 | 74 | Equations of motion in | \end{array}\right.}Since we will neither need the homogeneity of the gradient,
nor we will we differentiate it, we can use the constraints to write the
above system as{\left\lbrace \begin{array}{ll}
\ddot{x}_i=\sum _{j=1,j\ne i}^n{m_j[x_j-(x_ix_j+y_iy_j+z_iz_j)x_i]\over [1-(x_ix_j+y_iy_j+z_iz_j)^2]^{3/2}}-(\dot{x}_i^2+... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.005470738280564547,
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0.014405514113605022,
-0.007111196871846914,
... |
a851ba4ba4897d906bdb4f76283bfacd030ecbd5 | subsection | 30 | 74 | Equations of motion in | \end{array}\right.}Consequently the integral of energy has the form\sum _{i=1}^nm_i^{-1}({\bf p}_i\cdot {\bf p}_i)-\sum _{i=1}^n\sum _{j=1,j\ne i}^n\frac{m_im_j~ \frac{{\bf q}_i\cdot {\bf q}_j}{\sqrt{{\bf q}_i\cdot {\bf q}_i}\sqrt{{\bf q}_j\cdot {\bf q}_j}}}{
\sqrt{1-\left(\frac{{\bf q}_i\cdot {\bf q}_j}{\sqrt{{\bf q}_... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
0.0010774012189358473,
0.03060200996696949,
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0.015682386234402657,
-0.009908277541399002,... |
451c1ed8b7b6948799ba745cf3e32fcd2a31cdfe | subsection | 31 | 74 | Equations of motion in | \end{array}\right.}For the same reasons described in the previous subsection, we can use the constraints to write from now on the above system as{\left\lbrace \begin{array}{ll}
\ddot{x}_i=\sum _{j=1,j\ne i}^n{m_j[x_j+(x_ix_j+y_iy_j-z_iz_j)x_i]\over [(x_ix_j+y_iy_j-z_iz_j)^2-1]^{3/2}}+(\dot{x}_i^2+\dot{y}_i^2-\dot{z}_i^... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.004489149432629347,
0.020290803164243698,
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0.02131297066807747,
0.01292203739285469,
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-0.010625973343849182,
... |
3d2a0174ff091c877f89360253b1a550bcb2049d | subsection | 32 | 74 | Equations of motion in | \end{array}\right.}Consequently the integral of energy takes the form\sum _{i=1}^nm_i^{-1}({\bf p}_i\boxdot {\bf p}_i)+\sum _{i=1}^n\sum _{j=1,j\ne i}^n\frac{m_im_j~\frac{{\bf q}_i\boxdot {\bf q}_j}{\sqrt{-{\bf q}_i\boxdot {\bf q}_i}\sqrt{-{\bf q}_j\boxdot {\bf q}_j}}}{\sqrt{\left(\frac{{\bf q}_i\boxdot {\bf q}_j}{\sqr... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.004348426125943661,
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0.017454735934734344,
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0.002641478320583701,
... |
63410100a1fd802f22ffe55e8bfc8b48467c607f | subsection | 33 | 74 | Equations of motion in | But this extension
can be done as in general relativity with the help of the exterior product. However,
we will not get into higher dimensions in this paper. Our further goal is to study the
2-dimensional case. | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.0607965849339962,
0.05640165135264397,
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0.029497941955924034,
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0.007698764558881521,
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0.02304288186132908,
-0.0033934840466827154,
0.... |
d102fe9671d8b2a9979dfe31d1caf6bcf2c66a42 | subsection | 34 | 74 | Singularities of the equations | The equations of motion (REF ) have restrictions. First, the variables are
constrained to a surface of constant curvature, i.e. ({\bf q},{\bf p})\in {\bf T}^*({\bf M}_\kappa ^2)^n, where {\bf M}^2_\kappa is the surface of curvature \kappa \ne 0 (in particular, {\bf M}^2_1={\bf S}^2 and {\bf M}^2_{-1}={\bf H}^2),
{\bf T... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
-0.03432655707001686,
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0.008963045664131641,
-0.007490817923098803,
-0.026103440672159195,
-0.0015122756594792008,
-0.016964947804808617,... |
847718e3f90ab2e02a59850bff0125feb1e2d4c5 | subsection | 35 | 74 | Singularities of the equations | For \kappa <0, such singularities do not
exist because \kappa {\bf q}_i\odot {\bf q}_j\ge 1.In conclusion, the equations of motion are undefined for configurations that involve
collisions on spheres or hyperboloids, as well as for configurations with antipodal
bodies on spheres of any curvature \kappa >0. In both cases... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1563,
"openalex_id": "",
"raw": "R. McGehee, Triple collision in the collinear three-body problem, Invent. Math. 27 (1974), 191-227.",
"source_ref_id": "feb7471646c0e1b6374a89a97bba5c6061d2b0dc",
"start": 1417
}
... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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... |
4ed3cef8b5e07eec7236de853b4999cc447a9f1a | subsection | 36 | 74 | Solution singularities | The set \bf \Delta is related to singularities which arise from the question
of existence and uniqueness of initial value problems. For initial conditions
({\bf q},{\bf p})(0)\in {\bf T}^*({\bf M}_\kappa ^2)^n with {\bf q}(0)\notin \bf \Delta , standard results of the theory of differential equations ensure
local exist... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1167,
"openalex_id": "",
"raw": "P. Painlevé, Leçons sur la théorie analytique des équations différentielles, Hermann, Paris, 1897.",
"source_ref_id": "29c6547de79f87d5ddf6f41dcf8c7614ae866372",
"start": 940
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... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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292bec33f81e9940c007a6b665ead847b4d6dcfb | subsection | 37 | 74 | Solution singularities | Then, if(a) \ddot{x}(t_0)> 0 and \ddot{y}(t_0)< 0, the force pulls
the body along the circle toward the point (x,y)=(1,0).(b) \ddot{x}(t_0)< 0 and \ddot{y}(t_0)> 0, the force pulls
the body along the circle toward the point (x,y)=(0,1).(c) \ddot{x}(t_0)\le 0 and \ddot{y}(t_0)\le 0, the force pulls the body toward the p... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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c2203771193a9fcf8dfd66f009c6d5d12a6d616e | subsection | 38 | 74 | Solution singularities | Case (iii) includes the case when the
acceleration is zero.In case (i), the acceleration vector lies on a line whose slope is larger
than y(t_0)/x(t_0), i.e. \ddot{y}(t_0)/\ddot{x}(t_0)>y(t_0)/x(t_0), so
the force pulls m toward (1,0). In case (ii), the acceleration vector
lies on a line of slope that is smaller than y... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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... |
4053f01b920577ef7e0c84de92dbf50e8e50fadd | subsection | 39 | 74 | Solution singularities | Then(i) there are values of m and M, as well as initial conditions, for which the solutions end in finite time in a collision-antipodal singularity;(ii) other choices of masses and initial conditions lead to solutions that are repelled from a collision-antipodal singularity;(iii) and yet other choices of masses and ini... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.0... |
3c4faf169ac6cf5de78c712f81e320e35c1dbe34 | subsection | 40 | 74 | Solution singularities | The computations lead
us to the system{\left\lbrace \begin{array}{ll}
\ddot{x}=-{M\over 4x^2y}+{my\over x^2}-(\dot{x}^2+\dot{y}^2)x\cr \ddot{y}={M\over 4xy^2}-{m\over x}-(\dot{x}^2+\dot{y}^2)y\cr \end{array}\right.}and the energy integral\dot{x}^2+\dot{y}^2={h\over M}-{2my\over x}+{M(2y^2-1)\over 2xy}.Substituting this... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.0010870418045669794,
0.013738682493567467,... |
309af9c8ccc903c5dbe2cebf71d49c1f27207202 | subsection | 41 | 74 | Solution singularities | Then equations
(REF ) have the form{\left\lbrace \begin{array}{ll}
\ddot{x}={m\over 2x^2y}-{h\over 2m}x\cr \ddot{y}={m\over 2xy^2}-{h\over 2m}y,\cr \end{array}\right.}with the integral of energy\dot{x}^2+\dot{y}^2+{m\over xy}={h\over 2m},which implies that h>0. Obviously, as x\rightarrow 0 and y\rightarrow 1, the force... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.005901... |
7f06afd19de96bd67d2b31bbe1233d9b8607108d | subsection | 42 | 74 | Solution singularities | Then 2x(0)/y(0)<y(0)/x(0) for any x(0) and y(0) with 0<x(0)<1/\sqrt{3} and the corresponding choice of y(0)>0 given by the constraint x^2(0)+y^2(0)=1. But the inequality 2x(0)/y(0)<y(0)/x(0) is equivalent to the condition \ddot{y}(t_0)/\ddot{x}(t_0)<y(t_0)/x(t_0) in Lemma REF (c), according to which the force pulls m_2... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.0007... |
b408b8fcd7616eb8f0d73136d64ce3922f60175f | subsection | 43 | 74 | Relative equilibria in | In this section we will prove a few results related to fixed points and elliptic relative equilibria in {\bf S}^2. Since, by Euler's theorem (see Appendix), every element of the group SO(3) can be written, in an orthonormal basis, as a rotation about the z axis, we can define elliptic relative equilibria as follows.Def... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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d3300b008b31f862f9aa8e87f15c7ae481083e9e | subsection | 44 | 74 | Fixed points | The simplest solutions of the equations of motion are fixed points. They
can be seen as trivial relative equilibria that correspond to \omega =0. In
terms of the equations of motion, we can define them as follows.Definition 2 A solution of system (REF ) is called a fixed point if\nabla _{{\bf q}_i}U_1({\bf q})(t)={\bf ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.01... |
29e3439a9d44a06daac2d42d034becbc66a34f88 | subsection | 45 | 74 | Fixed points | Since
antipodal bodies introduce singularities into the equations of motion, only the n-gons with an odd number of vertices are fixed points of equations (REF ).The proof that the regular tetrahedron is a fixed point can be merely done by
computing that 4 bodies of equal masses with initial coordinates given by
{\bf q}... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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... |
4aea6ee3b876216842cc4c3ce67bb76eda472d8f | subsection | 46 | 74 | Fixed points | Since all initial velocities are zero, only the mutual forces between
bodies act on m_1. Then, according to the equations of motion (REF ),
m_1\ddot{z}_1(0)={\partial \over \partial z_1}U_1({\bf q}(0)).
But as no body has its z coordinate smaller than z_1,
the terms contained in the expression of {\partial \over \parti... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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9de13c31e2455edd953ab34ea93b6821911c7a0e | subsection | 47 | 74 | Polygonal solutions | We will further show that fixed points lying on geodesics of spheres can generate
relative equilibria.Theorem 4 Consider a fixed point given by the masses m_1, m_2,\dots , m_n that
lie on a great circle of {\bf S}^2. Then for every nonzero angular velocity,
this configuration generates a relative equilibrium along the ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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9aa7e14f6f96f775720ce5a7d667c1d1464b451e | subsection | 48 | 74 | Polygonal solutions | Therefore \bf q given by (REF ) satisfies equations (REF ).
Then, by Definition REF , \bf q is an elliptic relative equilibrium.The following result shows that relative equilibria generated by fixed points
obtained from regular n-gons on a great circle of {\bf S}^2 can occur only
when the bodies rotate along the great ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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7836eba3dfc4c6020aec475828ffb0d5677e59da | subsection | 49 | 74 | Polygonal solutions | This solution has the derivatives\dot{x}_i=-r_i\omega \sin (\omega t+\alpha _i), \ \dot{y}_i=r_i\omega \cos (\omega t+\alpha _i), \ \dot{z}_i=0,\ i=1,\dots ,n,\ddot{x}_i=-r_i\omega ^2\cos (\omega t+\alpha _i), \ \ddot{y}_i=-r_i\omega ^2\sin (\omega t+\alpha _i), \ \ddot{z}_i=0,\ i=1,\dots ,n.Then\dot{x}_i^2+\dot{y}_i^2... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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... |
c9975cdab1e848ee6be9e3b1b8ad06cc46b1d104 | subsection | 50 | 74 | Polygonal solutions | If n is even, this
property is still true if we exclude the case z=0.There are two cases to discuss: (i) n odd and (ii) n even.(i) To simplify the presentation, we further denote the bodies by m_i,
i=-s, -s+1,\dots , -1, 0, 1,\dots , s-1,s, where s is a positive integer,
and assume that they all have mass m.
Without lo... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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... |
ed79701be4b741fa0e227f88f0a254b1066fdfe5 | subsection | 51 | 74 | Polygonal solutions | Writing the denominator of
equation (REF ) explicitly, we are led to\sum _{j=1}^s{2\over (1-\cos \alpha _j)^{1/2}(1-z^2)^{3/2}[2-(1-\cos \alpha _j)(1-z^2)]^{3/2}}
={\omega ^2\over m}.The left hand side is always positive, so for any m>0 and z\in (-1,1) fixed,
there are a positive and a negative \omega that satisfy the ... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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0.... |
2c87ce94fb448b2b05ef1e38cb62de3b2741df8b | subsection | 52 | 74 | Lagrangian solutions | The case n=3 presents particular interest in the Euclidean case because the equilateral triangle is an elliptic relative equilibrium for any values of the masses, not only when the masses are equal. But before we check whether this fact holds in {\bf S}^2, let us consider the case of three equal masses in more
detail.C... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
"math.MP"
] | 2,008 | en | Mathematics | [
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952d485b729f2d10169ea3449f783022d93464f0 | subsection | 53 | 74 | Lagrangian solutions | The following result shows that, unlike in the Euclidean case, the answer is negative when the bodies move on the sphere in the same Euclidean plane.Proposition 2 In the 3-body problem in {\bf S}^2, if the bodies m_1, m_2, m_3
are initially at the vertices of an equilateral triangle in the plane z=
constant for some z\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1543,
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"raw": "A. Wintner, The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, N.J., 1947.",
"source_ref_id": "dd0033ef9f29c9e229900acd4e8bed70293a1dff",
"start":... | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
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c52294fd20e5b42d9024579b93a6fc639110f707 | subsection | 54 | 74 | Lagrangian solutions | In other words, we assume that this equilateral forms a constant angle with the rotation axis, z, such that each body describes its own circle on {\bf S}^2.
But for such a solution to exist, it is necessary that the total angular momentum is
either zero or is given by a vector parallel with the z axis. Otherwise this v... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
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35928c71c14ffbf102017b4d6698f93eef82ec6d | subsection | 55 | 74 | Eulerian solutions | It is now natural to ask whether such elliptic relative equilibria exist, since—as Theorem REF shows—they cannot be generated from regular n-gons. The answer in the case n=3 of equal masses is given by the following result.Theorem 8 Consider the 3-body problem in {\bf S}^2 with equal masses, m:=m_1=m_2=m_3.
Fix the bod... | {
"cite_spans": []
} | 10.1007/s00332-011-9116-z; 10.1007/s00332-011-9117-y | 0807.1747 | The n-body problem in spaces of constant curvature | [
"Florin Diacu",
"Ernesto Perez-Chavela",
"Manuele Santoprete"
] | [
"math.DS",
"math-ph",
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