Secular dynamics around a supermassive black hole via multipole expansion

In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is...
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Autor*in:	Fouvry, Jean-Baptiste [verfasserIn] Dehnen, Walter [verfasserIn] Tremaine, Scott [verfasserIn] Bar-Or, Ben [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022 May 18

Anmerkung:	Gesehen am 28.06.2022
Umfang:	17

Übergeordnetes Werk:	Enthalten in: The astrophysical journal - London : Institute of Physics Publ., 1995, 931(2022), 1, Artikel-ID 8, Seite 1-17
Übergeordnetes Werk:	volume:931 ; year:2022 ; number:1 ; elocationid:8 ; pages:1-17 ; extent:17

Links:	Volltext

DOI / URN:	10.3847/1538-4357/ac602e

Katalog-ID:	1808027116

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allfields	10.3847/1538-4357/ac602e doi (DE-627)1808027116 (DE-599)KXP1808027116 (OCoLC)1341461778 DE-627 ger DE-627 rda eng Fouvry, Jean-Baptiste verfasserin (DE-588)1260514374 (DE-627)1807365956 aut Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or 2022 May 18 17 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Gesehen am 28.06.2022 In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Dehnen, Walter verfasserin (DE-588)1194488994 (DE-627)167650396X aut Tremaine, Scott verfasserin aut Bar-Or, Ben verfasserin aut Enthalten in The astrophysical journal London : Institute of Physics Publ., 1995 931(2022), 1, Artikel-ID 8, Seite 1-17 Online-Ressource (DE-627)269019219 (DE-600)1473835-1 (DE-576)077662733 1538-4357 nnns volume:931 year:2022 number:1 elocationid:8 pages:1-17 extent:17 https://doi.org/10.3847/1538-4357/ac602e Verlag Resolving-System lizenzpflichtig Volltext GBV_USEFLAG_U GBV_ILN_2013 ISIL_DE-16-250 SYSFLAG_1 GBV_KXP GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 931 2022 1 8 1-17 17 2013 01 DE-16-250 4157189183 00 --%%-- --%%-- --%%-- --%%-- l01 28-06-22 2013 01 DE-16-250 00 s hd2022 2013 01 DE-16-250 01 s (DE-627)1410508463 wissenschaftlicher Artikel (Zeitschrift) 2013 01 DE-16-250 02 s per_4 2013 01 DE-16-250 03 s s_17 2013 01 DE-16-250 04 p (DE-627)1758741295 Dehnen, Walter 2013 01 DE-16-250 04 k (DE-627)1416733728 ZAH Astronomisches Recheninstitut 2013 01 DE-16-250 04 s (DE-627)1410501914 Verfasser 2013 01 DE-16-250 04 s pos_2
spelling	10.3847/1538-4357/ac602e doi (DE-627)1808027116 (DE-599)KXP1808027116 (OCoLC)1341461778 DE-627 ger DE-627 rda eng Fouvry, Jean-Baptiste verfasserin (DE-588)1260514374 (DE-627)1807365956 aut Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or 2022 May 18 17 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Gesehen am 28.06.2022 In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Dehnen, Walter verfasserin (DE-588)1194488994 (DE-627)167650396X aut Tremaine, Scott verfasserin aut Bar-Or, Ben verfasserin aut Enthalten in The astrophysical journal London : Institute of Physics Publ., 1995 931(2022), 1, Artikel-ID 8, Seite 1-17 Online-Ressource (DE-627)269019219 (DE-600)1473835-1 (DE-576)077662733 1538-4357 nnns volume:931 year:2022 number:1 elocationid:8 pages:1-17 extent:17 https://doi.org/10.3847/1538-4357/ac602e Verlag Resolving-System lizenzpflichtig Volltext GBV_USEFLAG_U GBV_ILN_2013 ISIL_DE-16-250 SYSFLAG_1 GBV_KXP GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 931 2022 1 8 1-17 17 2013 01 DE-16-250 4157189183 00 --%%-- --%%-- --%%-- --%%-- l01 28-06-22 2013 01 DE-16-250 00 s hd2022 2013 01 DE-16-250 01 s (DE-627)1410508463 wissenschaftlicher Artikel (Zeitschrift) 2013 01 DE-16-250 02 s per_4 2013 01 DE-16-250 03 s s_17 2013 01 DE-16-250 04 p (DE-627)1758741295 Dehnen, Walter 2013 01 DE-16-250 04 k (DE-627)1416733728 ZAH Astronomisches Recheninstitut 2013 01 DE-16-250 04 s (DE-627)1410501914 Verfasser 2013 01 DE-16-250 04 s pos_2
allfields_unstemmed	10.3847/1538-4357/ac602e doi (DE-627)1808027116 (DE-599)KXP1808027116 (OCoLC)1341461778 DE-627 ger DE-627 rda eng Fouvry, Jean-Baptiste verfasserin (DE-588)1260514374 (DE-627)1807365956 aut Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or 2022 May 18 17 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Gesehen am 28.06.2022 In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Dehnen, Walter verfasserin (DE-588)1194488994 (DE-627)167650396X aut Tremaine, Scott verfasserin aut Bar-Or, Ben verfasserin aut Enthalten in The astrophysical journal London : Institute of Physics Publ., 1995 931(2022), 1, Artikel-ID 8, Seite 1-17 Online-Ressource (DE-627)269019219 (DE-600)1473835-1 (DE-576)077662733 1538-4357 nnns volume:931 year:2022 number:1 elocationid:8 pages:1-17 extent:17 https://doi.org/10.3847/1538-4357/ac602e Verlag Resolving-System lizenzpflichtig Volltext GBV_USEFLAG_U GBV_ILN_2013 ISIL_DE-16-250 SYSFLAG_1 GBV_KXP GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 931 2022 1 8 1-17 17 2013 01 DE-16-250 4157189183 00 --%%-- --%%-- --%%-- --%%-- l01 28-06-22 2013 01 DE-16-250 00 s hd2022 2013 01 DE-16-250 01 s (DE-627)1410508463 wissenschaftlicher Artikel (Zeitschrift) 2013 01 DE-16-250 02 s per_4 2013 01 DE-16-250 03 s s_17 2013 01 DE-16-250 04 p (DE-627)1758741295 Dehnen, Walter 2013 01 DE-16-250 04 k (DE-627)1416733728 ZAH Astronomisches Recheninstitut 2013 01 DE-16-250 04 s (DE-627)1410501914 Verfasser 2013 01 DE-16-250 04 s pos_2
allfieldsGer	10.3847/1538-4357/ac602e doi (DE-627)1808027116 (DE-599)KXP1808027116 (OCoLC)1341461778 DE-627 ger DE-627 rda eng Fouvry, Jean-Baptiste verfasserin (DE-588)1260514374 (DE-627)1807365956 aut Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or 2022 May 18 17 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Gesehen am 28.06.2022 In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Dehnen, Walter verfasserin (DE-588)1194488994 (DE-627)167650396X aut Tremaine, Scott verfasserin aut Bar-Or, Ben verfasserin aut Enthalten in The astrophysical journal London : Institute of Physics Publ., 1995 931(2022), 1, Artikel-ID 8, Seite 1-17 Online-Ressource (DE-627)269019219 (DE-600)1473835-1 (DE-576)077662733 1538-4357 nnns volume:931 year:2022 number:1 elocationid:8 pages:1-17 extent:17 https://doi.org/10.3847/1538-4357/ac602e Verlag Resolving-System lizenzpflichtig Volltext GBV_USEFLAG_U GBV_ILN_2013 ISIL_DE-16-250 SYSFLAG_1 GBV_KXP GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 931 2022 1 8 1-17 17 2013 01 DE-16-250 4157189183 00 --%%-- --%%-- --%%-- --%%-- l01 28-06-22 2013 01 DE-16-250 00 s hd2022 2013 01 DE-16-250 01 s (DE-627)1410508463 wissenschaftlicher Artikel (Zeitschrift) 2013 01 DE-16-250 02 s per_4 2013 01 DE-16-250 03 s s_17 2013 01 DE-16-250 04 p (DE-627)1758741295 Dehnen, Walter 2013 01 DE-16-250 04 k (DE-627)1416733728 ZAH Astronomisches Recheninstitut 2013 01 DE-16-250 04 s (DE-627)1410501914 Verfasser 2013 01 DE-16-250 04 s pos_2
allfieldsSound	10.3847/1538-4357/ac602e doi (DE-627)1808027116 (DE-599)KXP1808027116 (OCoLC)1341461778 DE-627 ger DE-627 rda eng Fouvry, Jean-Baptiste verfasserin (DE-588)1260514374 (DE-627)1807365956 aut Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or 2022 May 18 17 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Gesehen am 28.06.2022 In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Dehnen, Walter verfasserin (DE-588)1194488994 (DE-627)167650396X aut Tremaine, Scott verfasserin aut Bar-Or, Ben verfasserin aut Enthalten in The astrophysical journal London : Institute of Physics Publ., 1995 931(2022), 1, Artikel-ID 8, Seite 1-17 Online-Ressource (DE-627)269019219 (DE-600)1473835-1 (DE-576)077662733 1538-4357 nnns volume:931 year:2022 number:1 elocationid:8 pages:1-17 extent:17 https://doi.org/10.3847/1538-4357/ac602e Verlag Resolving-System lizenzpflichtig Volltext GBV_USEFLAG_U GBV_ILN_2013 ISIL_DE-16-250 SYSFLAG_1 GBV_KXP GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_2522 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 931 2022 1 8 1-17 17 2013 01 DE-16-250 4157189183 00 --%%-- --%%-- --%%-- --%%-- l01 28-06-22 2013 01 DE-16-250 00 s hd2022 2013 01 DE-16-250 01 s (DE-627)1410508463 wissenschaftlicher Artikel (Zeitschrift) 2013 01 DE-16-250 02 s per_4 2013 01 DE-16-250 03 s s_17 2013 01 DE-16-250 04 p (DE-627)1758741295 Dehnen, Walter 2013 01 DE-16-250 04 k (DE-627)1416733728 ZAH Astronomisches Recheninstitut 2013 01 DE-16-250 04 s (DE-627)1410501914 Verfasser 2013 01 DE-16-250 04 s pos_2
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title_full	Secular dynamics around a supermassive black hole via multipole expansion Jean-Baptiste Fouvry, Walter Dehnen, Scott Tremaine, and Ben Bar-Or
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title_sort	secular dynamics around a supermassive black hole via multipole expansion
title_auth	Secular dynamics around a supermassive black hole via multipole expansion
abstract	In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Gesehen am 28.06.2022
abstractGer	In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Gesehen am 28.06.2022
abstract_unstemmed	In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauß, is to use the secular approximation of smearing out N stars over their Keplerian orbits, using K nodes along each orbit. In this study, we propose three novel improvements to this method. First, we reformulate the discretization of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order , reducing the overall computational cost from O(N 2 K 2) to . Finally, we show that the averaged dynamical system is equivalent to 2N interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme, and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics. Gesehen am 28.06.2022
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title_short	Secular dynamics around a supermassive black hole via multipole expansion
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Secular dynamics around a supermassive black hole via multipole expansion

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