银河系中心区域近星摄动对相对论进动的影响(英文)（可编辑）

银河系中心区域近星摄动对相对论进动的影响(英文)（可编辑） 银河系中心区域近星摄动对相对论进动的影响(英文) 中 国 科 学 院 上 海 天 文 台 年 刊 2012 年 总第 33 期 ANNALS OF SHANGHAI OBSEIWAI ,0 1又 Y ACADEMIA SINICA No. 33, 2012 0 fl P er t u r b in g E ffe et s o f a C lose St a r 1ll R ela t iv ist ie P r e eession s of a St a r G a la et ie C en t er H A N 认7 匕n 一 biao Shanghai Astronomieal Observa 七 ory, Chinese Aeademy ofSeienees, Shanghai 200030 A bs t r a e t :A s ProPose d by Cliffo r d MW ill, observa t ions of the pr eeessing or bits of a ela s o f st ar s ar o u n d t h e su p erm a ssiv e b la ek h ole in t h e ga la et ie een ter w it h ve ry sh o rt p er io d O0.1 yr eould provide measurements ofthe spin and quadrupole moment of the blaek hole a nd the reby t est t he no 一 ha ir t heorem s of genera l re la tivit y, if the obser vat ion aee ura ey fr oni t he E ar t h e an be re aehed 10 L ta s a nd l 件a s in t he n ear fu t ureH owever , in t lle G ala et ie eent er m a ny r esear ehes show e d tha t the re 15 a ste llar densit y eusp , a r egion of diverging den ity ar ou nd t h e su p er m a ssiv e b laek h ole. T h e res 一 ilt 15 th a t th e o b se rve d st ar s a re Pe rt u rb ed b y o th er sta rs u n av o id ab lyI n o rd e r to inv estiga te th e in fl u en ees o f su eh p ert u r b in g effe et s, w e n u m er iea lly ea le ula te t he Per t urb at ion effe et s on preee ssions of p er ihelion and or bit al Pla ne eom ing fr om a ve ry elose per t urbing st ar a nd t he n eom pa re w it h t he pr eeessions eaused by th e Sehw ar zseh ild part mass, fr ame一 dragging effe et angular momentum and quadrupole moment. Our results show t llat t he P ert ur bing e ffe et 15 m ueh sm aller t han the Sehw a rzsehild Pr eeession , b ut p ossibly ea n a ffe et t he m e asur em e nt of t he fr a m e 一 d ra gging effe et a nd q uadr upole m om ent , esPee ia lly fo r t he latt er on e. K e y w or d s : bla ek hole Physies: G alaxy: e ent er : r elat ivity C h in e se L ib r a r y C la ssifl e a t io n : P 2 28 .4 1 I n t r o d u et io n It 15 widely believe d that there 15 a supermassive blaek hole SMBH assoeiated with th e rad io sou ree s g r A * in ou r G alaetie eenter b eyo n d an y reason ab le d o ub t ! 3. T he nzas s o f su p erm a sive b laek h ole w as est ilna t ed ab o ut 4 m illion sozar m ass 51. T h e dist an ee fr om tlze SM BH to th e Su n 15 ab ou t 8 k Pe, 15 100 t im es elos er than th e SM BH in A n d rom ed a , th e n earest large galax y. F O r th is reason , th e G alaetie b la ek h ole offe rs th e b e t lab o ratory fo r strong grav itationa一fi eld physie nd testing general relativity Se review 6?, fo r more details. 收 稿 日期: 2 0 1 2 一0 7 一0 6 : 修 回 日期: 2012一 08一15 资助项 目: 国家 自然科 学基 金 No.11243002总第 33 期 W .HAN : Perturbing E ffe ets on Relativistie Preeession s in G al aetie Center 13 U sin g star s n ear t h e Gal aet ie SM BH ean d eteet Post 一New t on ian effe ets a nd test gen er al relativity GR in thew e ak an d strong fi eld limits near the SMBH.Several sueeessfu l exper- im ent s h ave d on e in o ur solar system fo r deteet in g ve ry weak G R e 旋 ets [8一 ], b ut th e s MB H w ill g ive u s th e b est eh an ee t o test G R in stron g gr av itat ion al fl eld ne ar a su Per m as sive b lack h ole. T h e 52 star , on e of a gro u p of 5 stars at d istan ees ran gin g fr om 100 一102 m p e fr om th e Gal aetieeent er,with an orbital period ab out 15 ars,has a periapse adva nee ab out 0.20/yr b as ed on genera l relat iv ityB ut th e Len se一T hirr in g effe et of 5 2 15 to o sm al l to b e d eteeted n owA s su 既 ested b y w ill 2], if w e ea n fi n d so me stars aro un d th e G ala etie eent er at ve ry small semimaj ors a 龙1mPe and with high eeeentrieity, the orbitalplane Preeessions indueed by the sPin angular moment um J and quadruPole moment Q of the SMBH ean be larger than 10 淋s/yr observing fr om the Earth an d ean be deteeted 饰 some eoming projeets, fo r ex am p le, G R AV IT YF u rth er m ore , o bserve d Pr eeessions in or b it al Plan es of tw o su eh stars ean in PrineiPle fi x J an d Q of the SMBH then test general relativistie no一 hair,, theorems whieh demajn d that Q 一J Z/M where M 15the mass of the SMBH. T hese m eas u rem ents req u ire th at th e ob serve d sou rees are in ar ea wh ieh th e d istan ee less t han 1 mPe fr om the Galaet ie eenter. In sueh very eent ral region, one ean not just tak e th e ob serve d st ars as test m ass b u t need eon sid er P otent ial eom Plieat ion s. Nu m erou s th eo ret ieal m od el w it h diffe rent dy n am ieal in itial an d b ou nd ar y eon d ition s sh ow e d th at th e d istrib u tion of stars arou n d t h e SM BH fo rm s a stellar den sity eu sP , a region o f d ive rgin g density around th sMBH 3一7. Thi mean in 龙 1 mp region, ther r dens f tars inelu d in g stellar b lack h olesA n d extr ap olat in g th e ob serve d s tella r d en sities at dist an ees of 1 p e fr om th e SM BH sh ow th at of ord er 100 一 102 stars sh ou ld b e in th is region l 8 , Aeeordingly, the orbit of target star defi nitely ean be perturbed by other star s nearby. Thus p ert urb a tion fr om an oth er star m ay b e ea n p ollu te th e m eas u rem ent s of th ese relativ istie Preeess ion s if the p ertu rb in g st ar 15 m as sive an d elo ses eno u ghMerritt et al. 2010 did a N一 body simulat ion and diseussed the infl uenee of star distri- b u tio n o n th e m eas u rem en ts of gener az rela tivistie p reeession s81. A nd in 20 1 1, s ad egh ia n an d w ill d iseu ssed th e alm ost sa m e p rob lem by an al ytie orb ital p ertu rb a tion th eo ry l 9T h ey eo n elu d ed th at fo r a ran ge of p ossib le stellar d istrib u tion s, th e effe ets of st ellar elust er P ert ur bat ion s ean n ot o bs eu re th e relat iv istie sp in an d qu ad ru p o le effe et s fo r an o bs erve d s tar su ffi eiently elose to th e b laek h ole. T h es e r esu lts are eon sisten t w ith th ose fr om the nu merieal N一 body sim u lat ions by 181, H ow e ve r , b oth of th e ab ov e two res ear eh es fo eu sed on t h e p ert ur b ation effe et of N 一 s tar distribution in the surrounding eluster each star 15 fa r aw 盯 fr om the tar get one on the tar get sta r,5 orb it al pr eees sion s an d eo m p ar ing with th e rela tiv istie effe etsA s a diffe ren t on e fr om t h ese tw o resear eh es , in the Pres en t PaP er , w e eon sid er th e Per turb a tion effe et eom es fr o m on ly o ne p ertu rb in g st ar b u t 15 ver y elose to th e target star b ut n ot a m oo n of itw 七sim u lat e th is th ree一b od y sy stem n um erieal ly an d fi n d th at th e p er tur b ation fr oml 4 中 国 科 学 院 上 海 天 文 台 年 刊 2012 年 su eh elose st ar 15 p ossib le to Pollut e th e ob serva tion s of relativistie Pr eeession s es Peeially th e q u adr u p ole on eC on seq u ent ly w e mu st st ate h ere th at th e fo eu s of ou r work 15 d iffe rent fr om th e tw o p 即 ers[ 8, 9] th ou gh th ey look sim ilarA n d th e resu lt s ar e also d i旋 rent b eeau se w e th in k th at the p ertu r batio n effe et ean in terfe re t he measu rem ents of th e relat iv istie effe Cts. T he Pa per 15 orga nized as fo llow s. In See.11 we briefl y introduee the relativistie pre- eessions of a star near t he SM BH in the G alaet ie eenterA nd w e deseribe the dynam ieal equ at io n s of ou r t hree 一 b o d y sy stem , an d g ive a sim Ple an alysis b ased on th e orb it al Per- tu rb atio n th eo ry. T h en in See. IV , w e Presen t our n um eriea l resu lts. In th e last seetio n , eon elusio ns an d d iseu ssion s are m ade2 R e lat iv ist ie e ffe e t s in G a la e t ie e e nt e r a n d d y n a m ie a l e q u a- t io n s As ment ioned in See .I, a super massive blaek hole with M 4 x 106叮 loeates in our G alaetie eent erT h e Seh war zseh ild rad ii of th e SM BH 15 ab o ut 0 .08 A U an d 10 卜a s see fr om the Earth. If a star wit h semimajor ax is a orbits around the SMBH, the orbital Period in u n it of year 15: 2 开a3/ 2 尸 - 七 1.48 瓦3/ 2 1 v 勺五 了 un its of where 瓦means the semimaj or axis in mpe. R om Eq.1, w e ean see if a star with semimajor ax is 0.1 1mPe, the Period 15 about 0.1 1yr. The orbital periapse and plane preeessions per orbit are given as !121, 1了 尹 z r了 了汀 9JO 口 AS 一ZAJeos 乞 一去A 1一3eosZ乞 , 2 刀 A J 一 A Q co s 乞, w h ere 口 公 + eo s 艺 口 15 t he Preeession of Per ieent er relat iv e to th e fi xed refe renee d ireet io n , 乞, 公 an d 刀 are th e orb ital in elin ation , argu m en t of PeriaPse, an d lo ng itu d e o f as cen d ing n od e resp eetively. As , A 了 an d A Q rep resen t th e relat iv istie effe ets d u e to th e blaek hole5mass Sehw arzsehild part, angular momentum fr ame一 dragging and quadrupole moment respeetive ly g, in nit f remin/yr,they a b written as : 67r G 八 了 2 廿 尹 尹 .11. A s Z r 叮 了 .tl廿 了 4 勺 b1 力 七8 351l 一 2一&一5/2 eZ1 一eZa J一 鲁 荞鸽压 3/2 刘 一1 一2,一 3/一3, , 一 丫 万 练 2 二7979一 卜2,一 22& 一 7/2 wheree 15the veloeity oflight , 已the orbital eeeent rieity,and 义 J/M Zthe Kerr parameter. W 晚ea n easily fi n d th at th e Sehw arzseh ild Preeessio n 15 m ueh larger t h an th e ot her tw o .总 弟 33 期 WHAN : P erturbing E ffe et s on R elat ivist ie P reeessions in Galaet ie C enter 1 5 F O r th e sim Plieity an d fo eusing on th e m ain Pu rPo se, we h ere ad oPt a Newt on ian gr av i- ta tion al eq u ation s b u t n ot Post 一 N ew to nian on e t o ealeu lat e th e Pertu rb at ion effe ets. Beea use that the relativistie preeessions hav e been analytieal worke d out see Eqs.2一 6, and grav i- tatio n al fi eld of the p ertu rb in g star 15 w e ak en o ugh to u se N ew to n ia n grav ity. T h e om ission of th e Post 一 N ewto n ia n t erm s in ou r nu m er iea l sim u lation s d oes n ot eh a nge th e r esu lts in th is Pap er , b eeau se in o u r sim u lat ion s we w all 咬to fi gu re o ut th e th ird b o dy 5 Pertu rb atio nT he Pos t 一N ewt onian ter m s ad d ed or n o a dd ed in th e nu m er ieal eo des W i ll b e n o in fl u en ee on th e ex tra etio n of th e Pert urb atio n effe et fr o m th e n ear star on th e tar get o neC on sid erin g th e m ass o f th e SMB H 15 m u eh la rger th an th e orb itin g st ars, th e aeeeleration of target sta r ean b e w rit t en as , G 八 了R G m , r 几 一 7 尹3 R 3 w h ere m , 15 t he m ass of p er turb in g star , R is th e veetor fr om th e SM BH Pointin g to th e ta rg et star an d r t he ve etor fr om th e Per tur b in g st ar to th e targ et on e. r 15 ve ry sm all eo mp arin g to R , but 15large enough to avo id the tw o stars beeoming a binary. It 15 ve ry easy to estimate the Hill radius of the target star with eireular orbit, r a /3M /3 4.4 x 10一3a. F o r h igh eeeentrieity , th e H ill rad ius w ill b e sm allerT hu s r m u st b e larger th an r 万. T h ou gh th e d ist anee b etw e en th e target an d P ert urb in g st ars 15 very srn all, th e m as s of th e p ert ur bin g star 15 on ly 10 一6 o f t lie SMB H叭三st ill ea n eon sid er th e g rav it ation al fo ree of m , as a Pertu rb at ion on th e or b it of th e targ et starB as ed on a sim Ple Pertu rb ation an aly sis, we ean fi nd th at fo r th e target st ar, b eeause o f the Pert urbat ion, there 15no seeular va riation of the semimajor a, but t he inelination 乞and eeeent r ieity e h ave slow ly an d seeu larly eont rary eh a nges d u e to the K ozai m eeh an ism 201, an d a lso th e orb ital Periap se an d Plan e ex ist seeular preeess io n s. A s w e k n ow , ev en w ith th e nu m erieal sim u lat ion s , eom Pletely in vestig atin g t he Preeess ion b y th e t hird b o d y Pertu rb a-- tio n in qu al it at ive an d q uan titative 15 qu ite d iffi eu lt, b eeau se the Pertu rb ation m ag n it u de d ep end s on to o m any Param etersT he p a ram eters w e eh ose in o ur n um erieal m o d els in th e n ext eh ap ter tr y t o m ake t he p ertu rb ation relat ive ly larg e in ord er t o st ud y its in fl uen ee on th e Po ssib le m ea surem ent s of t h e r ela tiv istie effe et s in th e fu tu re3 N u m e r ie a l r e su lt s Based on the analysis in the last ehapter, the semimaj or of the target st ar 15 set fr om 0.1 mPe to about 1 mpe, and the di价renee of semimajors betw e en the target star and the Pert u rb in g on e 15 0 .0 1 m PeT he las t n u m b er 15 b ased on t h e ext rap o lat ion est im at ion w hieh su gg est s th at th er e are of ord er 10 0 一 102 sta rs in sid e th is r egionA n d a lso , th is sep aration m akes t h at t h e t wo st ar s ar e v ery h ard t o ent er t h e H ill sPh ere of eaeh ot h erIt 15 m ean s th at th ey are im p oss ib le to b eeom e a b in ary in th e b aekgro un d grav itation al fl eld o f t he SM BHT h e m as s of targ et star 15 on e solar m as s an d th e m as s of p er tur b in g on e ean b e set中 国 科 学 院 上 海 天 文 台 年 刊 2012 年 16 as o ne , two or th ree solar m ass. F o r sim Plifi eatio n , w e Pu t th e in it ial orb it of t he Pertu rb in g star on th e eq u atorial Plan e of th e SMBH , b ut th e t arget o n e h av e an orb ital in elin a tion 乞 the angle between the orbital angular moment um of the star and the spin of the SMBH. A nd we set th e K err sPin Par a meter as its m ax imu m va lu e 1 du rin g all ealeu lat io n s. F o r ealeu latin g th e va lu es of Preeession s n u m erieally, w e sim u late the o rb it al evolu tion s about a fe w ten thousand periods to make the preeession rates 刀 tend toeonstant values. Then we take these eonstant s as the ave rage va lues of preeession rates in unit aremin/yr. A Runge一 Kutta- F e hlberg 78 integrator is adopted fo r integrat ing thedynamieal equat ions. A nd we eheek t he eon serva tion s of en ergy an d an gu lar mo ment um to m ake sure t h e nu m erieal Preeision an d relia b ility. In th e toP 一 left Pan el o f F ig. 1, we Plot th e orb ital Preeession s o f a h igh eeeent r ieity star e 0.9 produeed by the spin and quadrupole of the SMBH, and the perturbation eaused b y a o ne solar 一m ass nearb y st ar. It ean b e elearly seen th at th e Pertu rb ation effe et 15 h ar d t o b e ob served by t he in s tru m ents w ith 10 幽 5 Preeision in the wh ole reg ion w e eo nsid eredAt the same time, the fr ame一 dragging effe et ean be observed if the semimajor a 龙0.7 mpe a n d th e qu ad ru Pole on e also ean b e ob served if th e star 15 eloser to th e G a laet ie eenter a 龙 0.17 mpe. But fo r a l 件 as observa tion aeeuraey, the perturbation effe et eompletely eove rs the quadrupole part fo r 0.2 mpe 龙瓦龙0.5 mpe. N ow , w e k eeP all Par am et ers exeePt t h e m ass o f Pert u rb in g st a r b eing eh an ged t o * 3 对乙, w e ean see the quadruPole effe et will be Polluted obviously. Even the fr ame- dragging e旋et will be eove red by the perturbation fo r 瓦之0.7 mpe with l 件as observation aeeuraey see the top一 right panel ofFig.l. T hen , in th e b o ttom 一left Panel of F igl , we p ut th e p ert urb in g sta r with on e so lar mass a little fa r fr om the target star, as &, 一& 0.1 mpe. 叭2 匕 ean fi nd that fo r the 10 淋 5 ob serva tion , we ean n ot see th e Pertu rb ation effe et. F O r t he l 件as aeeu raey, th e p ertu rb ation h as a sm all in fl u en ee o n th e q u ad ru Pole effe et ob serva tion , b u t alm ost n oth in g on t he fr am e一d ragg ing o bserva tionF u rth erm ore , th e m agn itu de of Pertu rb ation also dePen ds o n the eeeentr ieity e an d orb it al inelin at ion 乞. B ased on o ur n um erou s sim ulat io n s, u su ally, a h ig h er eeeent r ieit y an d sm aller in elin at io n w ill pro d u ee a lar ger p ert u rb at ion Preeess io nW 七u se e 0 .95 , 乞 0 .3 to replaee the val ues of eeeent rieity and inelination in the mentioned panels 0.9 and 0.1 respeetively. The results are show n in the bottom一 right panel. Though hav ing a higher eeeentrieit 叭beeause of thebigger orbital inelinat ion 乞 0.3,theperturbation effe et almost has no infl u en ee on th e fr am e一dr aggin g an d qu ad ru p ole ob serva tion s w ith b o th th e 10 协as and l 件as Preeisio nIn ad d it io n , with th e fu ture 10 件as eve n l 件as ob serva tion t eeh n o lo gy , PeoPle ean m ea-- su re t he Seh warzsehild Preeession s of th e s 一sta rs wh ieh h av e b een observ ed , fo r ex am p le, th e总第 33 期 W .H A N : P ertu rbing Effe et s on R ela tivistie P reeessions in G a laet ie Cent er 17 n,胜 nnll. 且 O C 01 10 0 0 \一 三 日 舅 补 \毛 口 舅 补 0010, 共 浪 卜 长 之 全 一 J. 几 冲 心 勺 q 七 PJ 0. 0一 众卜 二 冬0 .0 1 o01一 一,1 0 .10 0 .15 0 .2 0 0.30 0.50 0.70 1.00 1.50 0 10 0.15 0.20 0.30 0.50 0.70 1.00 50 0 /m pe a 加 pe 1000 000! ; 氢 100 一 层 \写 卜 日 口 已 U O J 月 工 00 一二 10 卜. , 臼 ~ l 0 泛, 二 泣 全 口 弓 \ 加 olr. , , , 0. 01 卜二 l 一, 0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.50 0.10 0.15 0.20 0.30 0.50 0.70 1.00 1.50 a 加 pe a /m pe Com P ar ison of t he t hir d body Per tur ba tion effe et w it h t he SM B H 5 sPin a nd ld qu ad u P o le e ffe et s fo r di价 r ent orbit al P ar am et ers. T he das he d eur ve , t he dot 一 das hed an d t he solid one reP resent t h e fr a m e- dr a gging, qu a dr uPole and p ert urb at ion effe et s resPe et ive ly; T he thick solid an d t hiek das hed eurve s ar e fr om the Earth respeetive ly hereaft er the sam e. The the 10 林 as /yr and l 冬 切5/yr observation leve ls orb it a l P ar a m et er s: , m 1 M 三, 刀, 口, , , e, e 0.9, 乞 0.1 and 云, 一 反 0.01mpe the *,, labels the eorresponding var iables of the perturbing starto卜left ; m, 3二 3 M , 二 刀 刀 口*** 刀 刀 叭 叭 e*e* 叭 叭 c 0 .9 乞 0.1 and a一 瓦 0.01 mpe top一 right ; 1 八 几 e 0 .9 乞 0.1 and 反* 一瓦 0.1 mpe bottom left ; , 二 I M台 口, , , e, 0.95, 乞 0.3 and 瓦, 一厄! 0.01mpe bottom right . 52 star with the orbital period about 15 years the shortest period in all observed s一 star s. A p ro b lem 15 th at it needs ten s years ob servation s to d et erm in e the va lu es of p reeessio ns b eeau se of th e lo ng orb ital p eriodJ u st fo r a d em on strat io n , w e Plot h ere th e P ert urb a tion effe et on the Sehw arzsehild preeession indueed by themas s part of the SMBH.W七ean see elea rly th at th e p ertu rb ation eann ot in fl uen ee on th e ob serva tion s fo r th e t arget stars w ith Semimajor fr om 5 to 50 mpe see Fig.2. 4 C o n e lu s io n s a n d d ise u ss io n s It 15 q u ite d iffi eu lt to eo