Extragalactic Astronomy ASTR 5630
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Date Created: 09/21/15
Whittle EXTRAGALACTIC ASTRONOMV ml Toolbox 8 STELLAR DVNAMICSII 3D SVSTEMS Index Questions Images References Prim Next M 1 Introduction We have of oourse already begun our study of Stellar Dynamics neai in large pan these notes follow though smolify the treatment in am 39 s and Stellar Dynamics To set the sage lets firs compare stellar systems With atomic of molecular gases E 0 nt E a 39U 3 First some similarities Each an be described by distributions in space and velocity 7 eg Maxwellian yelooty distributions uniform density sphericallycofloefltfated etc Starsiamaiiiiiiiiiaiuiiid A quot momentum Now some cfuaal dillerences as atoms interao only With their neighbors during orief elastic repulsye oollisons gravlt eg uniform medium F ix 3 pt39Zdr l 2 ix der l 2 ix pdf a equal force from all distances The relative frequency of strong encounters is radiaclly different as for atoms enoounters arelrequent and all are strong ieAv V W for 381 i n nii nti dynamics gt ooncepts sucn asTemperature and Pressire can be appiied to seiiar systems gt we use analog to tne equations of uid dynamics and nydrostatios tnere are 2190 some interesting diiierenoes gt pressures in steiiar systmes can be anisotropic gt 3 Heat 39 b A Path Through the Subiect iiii iii i coveri Tne geometry of gravitational potentials is a good starting point a metnods to derive grayitationa potentiaisfrom mass distributions and vim yerm Potentiais define now stars move gt consider steHar orbit shapes and diyide tnem into orbit classes Tne grawtationai neid and steiiar motion are deepiy interconnected tne ai Theorem reiates tne giobai potentia energy and kinetic energy of tne system Tne Vina Tneorem can be used to investigate a tne masse of steiiar s ems a now energy isreieased during grayitationa coiiapse a now Seifrgravitating s ems naye negative specinc neat a now tne ratio of rotation to disperson sippon can define gaaxy attening A more detaied approacn requires usto Work witn a D t ut n Funct on DF a n n outtn For coiiisioniess systems tne DF is oonstrained by a continuity equation tne CBE Tnis can be recast in more obseryationai terms astneJeans Equation Tne Jeans iutionstotn Witn tnese DFs we can conarua seiicon ent models ofeguilibrium seiiar systems Some smpie systems are consdered in detaii wniie more compiex ones are toudied on We introduoe situations wnere tne potentia is changing in time UsJaHythis is untreatabie except wnen tne dianges are rapid and iarge V ient reiaxat n Tnis is important in desoibing gaiaxyformation and gaiaxy merging Finaiy we reiax tne coiiisioniess assimption and introduce starstar interactions SJ systems are desaibed by tne FokkerPlanck equation Tnis reyeais a number of siow processes wnicn ocwr in dense steiiar systems a 2rEod reiaxation amp equipartition 4 Core coiiapse amp tne gravothermai catastropne a Evaportion amp eiection Additionai importanttnemes are postponed to iater Topics a Effea of nudear biack noies on seiiar distributions 9 u Egg 0 sn mg as 0 era 85 5 ast impuisye enoounters is a Merging amp sateiiite aocretion is Nitl ml ml 2 Potential Theory a Preliminaries We initiaiiy charaaene mass dis39mbutions as smooth func uons u this is usuaiiy iegmmate for galaxies See 810 beiow The gravitationai potential energy is a scalar field 812 81b quotMGKT 8 2a 17rG r 8 2b 8 2c 8 2b is Poisson s equation for iocations within the mass diaribution e 2c is Lapiaoe s equation for iocations outside the mass dis mbution For a voiume v With surface A enciosng mass M We have Hang DivergenceGausSsTheorem inGM 7 ma 4011311 882 V 7V 1mm ear 125 V A ce the force fieid isthe gradient of a potentiai n is conservative is the energy required to move m sfrom r110 r is independent of the path the total Potential Energy istherefore well defined semng c 0 at r on We get EHZJQ p 59 7 l 3 7 1 2 3 H 7 EVprlt1gtrrlr 7 i Viv It 84 Note hag With this de nition potential energy is almys negative 8 8b Sn b Selected Examples of DensityPotential Pairs Often in 31w l l l39ll There are however a number of useful llluslratlve analytlc lr lt gt ltIgtr pars i Point Mass ltIgtr GM lr Hr VQrd IgtdrrGM l V020 GM lrrlt1gtr Vex2r2GM lrr2lt1gtr where vC amp veSC are the clrcular and escape veloatles respealvely Thls ls called a Keplerian Potential shoe lt pertainsto the solar syserh ii Uniform Spherical Shell QJISlde dgtr lhsloe 61gtr VGM lr Keplerlan ons t l Fr0 Homogeneous Sphere Sph re s Wlthproorl QJISlde qgtrrGM lr Keplerlan lhsloe chm 27era 7 ls l F 79 Mr l 2 r4 37erx r whlch glves SHM Wlth perlod p radlu s rlt a 37rlepquot anolreelall quotwt P epquot vC l4le7rep1quot x r sothat Um Oorls39L a solid body rotation note also that PC P iv Logarithmic Potentials from Flat Flotation cunles Many rotatlorl curves arellat at large raoh v0 vol so we have F V do 7 7 17 ltIgt7 Vnzlnl o nst 85 v Spherical Systems Power Laws pp0 raj39o have Mltr we a3 pa l 3 390 x rlaa and dgtr we 22 p0 I 3 7010172 x rla239 V02 l m2 o 3 rs a break point For 01 gt 3 Mltr a w for rgt0 we have lhhhlle rhass auhe orlglh Fora lt 3 Mltr a w for rgt on mass dlverges atlarge r However for2 ltolt3 the potentlal lelrllte as are VC and V690 at all raoh 01 2 lsspeoal lt lsthe singular isothermal sphere Wlth VC 7 4 7r 3 a2 1 ooha at all raoll ylelolhgtbh 4 7r 3 a2 F0 lhr l a See 8 sao for other lSothermal and related th9 spheres lt link gt Hernqulsi 1990 and Jaffe l 988 models have p X r4 at large r Wmch ms E 92 We L and stheoreuwHy grounded m mdem re axauon at we r Jaffe core ss teeper than Hemqms t core Ma GM Wm M ltIgtHr 7 7m 8 ea Ma GM T mm W mm 7 771nlt7agt 86b F39mmmer 19m Sphere sana yuc So uuon of hydrgaauc suppon for po ytropwc s1euar sys1em of mdex 5 see 8 80 lt lmk gt pm matdwes GO We L but 5100 seep at arge rfor Exhpucds p vx H5 GM I Pm r 2 e7 7 15 Pwummen sothermal daffe and Hemqms t denswty aws are shown here wage vi Axlsymmetrlc Thin Disks dxsks m 1 H u We have two condmons within a dsx of vdume densty go near the p ane above a dslt of surface densnyE Usng eduauon 8 3p We have TGIIn z wide 8 ea 8 8b 27er above UsJaHy caxcuxaung g obal D and F for dxsks s algebraca y dense Here are afew swmp e examp es Megs dslt HR 20 R0 R has constant VC V020 27rGEO R0 GMltR ms 5 unusual m that VCR doesn t depend on mass outswde R Exponermal dslt EREoexprRRd tmsmsme hght prome of sp al dxsks mudw better than Mesie dxsk and has arw ar Ve oaty V30 ki GEanZIQInKn 113K13 89 Where y R 2Rd and x Kn are Besse funawons or the 1 s and 2nd band see eqn 5 71 for an analyucapproxwmauon and rotanon curve Kumm 1955 dtsx nas a snnpte form for botn R and potenttat 2R7 M ltIgtR7 G M 7 R2a232 1t 2 7 R2alz 2 8 03 Note tnat because Potsson se ua on S hnear dinerenees between anytwo 7 pans 5 also alt1gt 7 pan and dinerentials of D or eg w rt shape constants are also D 7 pans for exampte a2 n1 S a Kumm 1 99 noo S a Gaussan dtsx 1 7171 1 n71 2120 2MB 720 K swab Unfonunate y n v Axisymmetric Flattened Systems ofoourse n tn was W n an Wu tnnndtsks We need potenttats wntdn are botn cornbtned e attened potentials MtyamotorNagat 1975 attened Syaem t nnages reducesto tne F39mmmer rnodet f go and tne Kumm dtsx tr b0 Batch attened systems are denved m srnnar manner to theToomre dtsks nnages 2 2 2 MAR 2 1 2 LR aSBrlB em 1 R2 01 B2d2Ba I lt1th 2 7 G B2 7 22 b2 R2aB2 sub sR 2 lt 1M I s R 2 39 e 110 Triaxial Ellipsoids more oomphcated see 3amp12 2 5 ix Mullipole Expansion mi Somttons mohe spherical harmonics Vtquot a Dlt PH W cos 9 eXpU wnere Pt m m are assoctated Legendre funwons a rnonopote 00 r r eacn wtn assooated amphtudes For ooordmates wrtn ongrn at center of rnass a rnonobote argeaterm assocrated wtn VcrGM r a drbote ro outsrde tne rnass because no rve rnass a quadrubote nrs stgm wntnonrmheanerm Nexll Prev Topl 3 Orbit Classes TEID Nexll Prev Topl 4 Numerical NBody Methods Often astrophysca y mteres tmg Syaems are algebraca y rntraoabte Computatrona rnetn ods orovrde a Way forward romca y ernptoyrng tne Newtonth rorce aw can be a dtmaer hard force aw X r 2 I3 tr aught brnaryrorrns tnrs can be a oornputatrona smk So so en theforoe aw XArA 2 note tnrs may be rnaooropnater r 62 or smat Syaems Wnere boursons are rrnoortant Several rnetnods are used See 3amp12 2 9 and Josh Harness mce wnteub for more deta s lt lInk gt Drect Summauon or parwrse rorces omy oosbte ror N 50000 0N2 operatrons perumeaep Drnde regron mto canesan ceHs poputatron m eadn ceH cnanges ArJ392 omy evauated once strmng done usng FFTS smoe Q 2 MJ 3 Amy N resernbtes a oonvotutron takes 2N21 4 ogg2N 12 steps oornbared to N4 so very erncrent for N gt15 Typtca y 32 x 32 x 32 cube 327w oeHs W 105 stars r ror oentraHy concentrated drsks dnoose ootar gnd spaced m mm and D Wbysenes r A carouate tota potentrat by summmg tnese over 2H pamctes resohmon naturaHy better near nucteus N oatcutatrons perumeaep so very ernoent EM To 5 The Virial Theorem snared between Kmeuc energyK and poterm a energy W Specmey we are rntereseo m tnerr ratto KMW note K rs atways we vv aways sve g case a Simple Illustralions i circular Orbit Consroer a sateHrte rnassrn m arcu ar orbrt about M gtgtm rn v2 Ir 3 rn M muhpybyr mV GmMr a 2Kevv 2KW0 KHW zandErK a Krnetrc energyrsnar tne we botentra energy a Tne tota energyE K W srve and e qua to mmus tne Kmeuc energy As we 912 see 6 v2 rsa enaractenstrc shared by a wroe range of systerns ii Tlme Averaged Keplerian Orbit tn generat f K WW changes aong a Kebtenan orbrt batn rrnage oornpare f at pencenter and ab oen er 6p fa ra rp 1 usng rp vp ra Vafrom AM conseryatron However takmg trrne averages over an orbwt we nno ltVWgtltGMrgt GM Hr GM x Hay and M lt1rr 122 gt vzeM x Ha a and we recover once agam lt6 v2 and E ltKgtlt 22gtG ltKgt v2 As e wru see however 6 v2 always notos wnen we average over an pamdes m a systern For our Kebtenan orbwt rn and M are tne wnote systern wrtn M havmg ero KE b The General Case Onoe agam we ask wnat Sf tne ratro of Krnetrcto potentra energres There are 3 equatrons of mouon for member at r representsx y take tne 151 moment m posmon mummy by xfquot and sum oyerot t represents x d an of to es me an equatron of energies tons se can be nea y wntten usng 3 x 3 rnatneest re tensors of order 2 thrs set of equatrons conartute the Tensor virial Theorem 1 d2 EWL J 2K1J Wm 2TiJ Hm Wm 818 where the ve tensors are mnmem of me ia m 1 7 137 I1I11JgtII3T 7 mm KB 814 ordered KE pltv1ltt1137 pai1137 1 r gGfmrwr a b c d e 7 1 Ti H i 7 rPrrFr39 PE r the Kmeuc an for examp e Consrdenngru T W Trace Trace so for the statrc case we get the Seal 2K W l Consrdenng th the total en ts yatue rs equ mmusthe hat the ath 2Tr W K For steady state systems 12 d t2 o and we get 2K1 W n 8152 d poterma energres are rexated loreach tensorelement they rs are rexated Separate y ong each ax e dwiglonat terms we also have age K totat kmeuc energy and total potermal e nergy ar virial Theorem 8 15b etotal energy E we nnd K 2 8150 ergyrsnegative the systern rs bound a to ether ve Kmeuc Energy or eye Potentra Energy Here is a very useful iittie diagram to iiiustrate tne situation image Briefly reviewing tne conditions necesmry to use tnese simbie equations tne system must be seii gravitatin tne system must be in steady state orbit timescale ltlt evoiution timeswle 39 v Note tnat tne system may be eitner ooiiisoniess steiiar or coiiisiona gaseous 1 Mass Determination Tne mos famous use ottne viria tneorem isto determinetne masses of seiiar systems V2 For a stem of total massM and mean squared veiooty ltv2gti K is smpiy v2 M lt gt tne Ilrlal tneorem tnen gives lt2gt 7W l M GM le wnicn in braoioe defines tne gravitational radius Knowing R9 and measuring ltv2gt allows usto determine Mi tne system mass Wnat to use for R9 isn t obviousfor mos seiiar systems witn no ciear edge or sie Howeveri we can make use of tne median radius Rm wnidi encioses nat tne mass For many steiiar systems itturns out tnat Rm 1 o 4 R9 note Rm is written rh in am we tnen nave 816 notice tnat for our circuiar Keplerian orbit we recover tne simbie reiation 2 e M IR d Binding Energy Energy Released During Collapse it tne system starts very spread out and at rest E s K Aiter settiing dowry we nave once again E s K a energy mus be released it tne system ooiiabses W0 7K a tne vaiue of tne binding energy is eoua to tne remaining KE a tne tota gravitational energy reieased lS39W of wnidi nait goes into KE and nat esc es tne system Here is anotner iittie diagram to iiiustrate tne stuation image Examples 0 Coiiaosng brotostars are iuminous a tney radiate nat tneir gravitationa potentia energy 0 Kelvin considered a gravitationa origin for tne Sun s energy via gradua oontraoion a ForagaiaxyiK VzM V02 105 E io ULQ e rs tnisisexm397 ofthe res mass ie V02 02 x M tnis is negiigibie in gaaxystarburs formation nudear burning is 7x10393 M02 e Stellar Systems Have Negative Specific Heat a Try to siow Earth s orbita motion by buiiing badlt l e remove orbita energy t faHs m to tower orbtt and speeds up a CoHapsng gas ctoud radtatee energy ooHapsesfunher and heats up a Add energyto a star cmaer e g by acoeterattngtne stare tne duster expands and boots Here are dtagrarne to Huarate tne stuatton Wage f Rotational Flattening Constder an axteymrnetnc systern rotaung about tne ext El Symmetry and W are 2 dtagonal x amp y eternente ofthese tensors are tne Ef e Tne tensor wnat tneorern gwes 21 Hxx Wxx o 2TZZ 1391ZZ WZZ We also nave T22 o rotatton about a no dnrt H to 2T 7 1Lx M002 ootstne rnasswetgnted dtsoerston 1391ZZ E LJJHXX 145mm 6 lt trneasuresantsotropy vvxxvvZZ m AEIJW we V2I ltV gt2 dar v2 M v02 v0 tstnernasswetgnted rotauon speed NE 5 ext ratto of tsodensty smaces Fma y sts ututmg aH tneee mto tne ratto of tne two tensor retattone above we get 8 17a T71 g 4 5 snowstnts retatton for t and EIamp Several 6 mdudmg projeaton oorreatons Wages for tsotroptc Vetoattes d 0 we get for sma e 8 17b tntntem T M r rm r OoServannaHy m Tobtc7 we found mm ng 4 or trnageet a Low turntnoety EHtpttcals and Em esfoHoW tne teotrobtcretatton a Lummous EHtpttcals often fa m tne antsotrobtc 60 regton t tm 6 Describing Collisionless Systems We ttrst oonstoer collisionless dynamtcs Cotttston x here meanssiarrs tar de ection not otrecttrnoao gataxtes tnts ts almost always a very good approxtrnatton tn 8 to we oonstoer wnen and now aarraar enoounters are retevant a The Distribulion Function DF fr v t fr v to3ro3tt nurnber of stars at r wttn v at ttrne t tn range dar and oat Knowteoge of tne DP tsa noty gratt stnoe tt wetos cornotete tnformatton about tne stern tn praatoe however we onty observe oenam projections ottne DP eg 1th VpR 17pR Reoovenngtne DP otrecttytrorn observattons ts esserma y trnposstbte To proceed we need to tntroouce further constraints on tne DP vt However h ere are other con aratnts b Collisionless Boltzmann Vlasov Equation CBE Look for a coniinuiiyequation stnoe n ars createddestroyed ow conserves stars stars do notjump across tne pnase space te no detlectitte encounters Vtew tne DP ovtng Hutd of stars tn 67D space r V tex ytgvx vy 2 stars move ow tnrougn tne regton astnetr oostttons and vetoottes onange Constoer 217D examote usng x and vx and reca f ts a nurnber density ocus on a sma eterne tonase soaoe at x and vx wttn ste dx by dvx tnts tntage wttt neto VtZUZHSe tne sttuatton tn tntervat dt net ow tnx ts wilt lvr u 150 7f1 1 150 715 df thud tne net ow due to tne vetooty graotent ts 11 1 1f fa 121 t 7 f1 1lw In 0 7 1171 1 818b tne surn of tnese equatstne net change to f tn tne regton te atx vx of ste dx dvx lvr if It 61 101 g BI 12 etec oh dtmdthg by dx dvx at We get if V Bf dzI 6f 7 a 1ra I a 7 l 8192 8 19b 8 190 8 19d Thts tsthe collisionless Boltzmann equation CHE The CEIE descnbes how the DF changes m Ume tt ts a dtrect consequence of 1 Conservator of stars 2 starstouow smooth orbts a ow of starsthrough r dehhes hhphatythe toeatton v Md 4 ow of stars through v ts gwen exphatty ter Smce space Howeveh constdertheLagrangian total or convective derwaIHe DfDtEdfdt orbit 1 But thts Lagrangtan derwaIHe ts hoththg more than the LHS ot the use tif39 39 rm at t d 81 m C ear y the phase space densty f along the stars orbtt is constant e the ow ts thcompressbte m phasespace for exampte it a region gets more dense o Wiii inoease it a region expands 0 WM decrease ex pie arat on raoe start n high Av high i end n OW Av iow The CHE appiies to an sub opuiations of stars eg eadn speorai dass even though no singie ciass determines the otentia in 8 7o We introduce a seifrconsaent f which itseii generateslt1gt lt link gt c The Jeans Equations As it sands the CHE isot rather iimited use a the constrants it provides are siii insuttiCientto find ttriiiit a the compiexty otttriiiit renders it observationa y inacoessioie What We observe are a mean yeiooties y a yeiocity dispersions 0 which is reiatedto lt 12 gt a seiiar densities h aso tor mass density ori for iuminosty denisty We need to recast the CHE in terms of these quantities ieariyi these ooseriaoie quantities are oontained Within the DE f nut they an be extraaed bytakirig appropriate averages or moments for exampie number density nri t Inn iii t day 0th moment in y mean yeiooty ltiri tgt in yimi iii t day is moment in y it We take moments otthe CHEi We transorm it into equations in these neW variaoies Lets iook in more detaii atthese first We moments in y see Hme we Using the TD x axis as exampiei smpiy integrate the CHE eds 190 over ai yx 0th moment in yx We obtain 821 This is a smpie continuity equation for the number of stars aiongthe x axis 2H yx on rearranging and using eds 21 above We obtain 1st moment in yx 64gt l 8070 1 w 61 8 22a where of isthe yeiooty dispersion about the mean yeiooty 2 gt it arises from ltyx2gt repeating this in SD requires a iittie oare Hme 4 e we obtan tneJeans Equation for ooordinatei 8 22b wnere tne summation convention appiies st oyer repeated indioes nerei 2 3 and isi 2 3 refer to x eg xQ E y and y2 E y ThisJeans equation isakm to Newtonss seoond iaw dyidt Fm witn LHS istne derivative of v RHS are foroe terms a it is instructive to compare this to Euler s Equation ioriluidilow e 23 in 8 22b n 0 2 is a stress tensor wnicn takesthe roie of an anisotropic pressure i 8 a g a i Q i m E 5 2 e e o a i Q in afiuidi pressure isa scalar and istnerefore aways isotropic for seiiar systems 0 J isatensor wnicn can be anisotropic 17W is symmetric i e axes eXiS wnere 71 1 172 2 173 3 are Semiraxes of a yeiocity eiiipsoid if 01 s 022 s a we naye isotropic disperson a Jeans and Euier equations are identica For ooiiisoniess systemstnere is no equation o1 state iinking presstre of to density Ustaiiy therefore we areforced to assume am on equiyaentiyi tne anisotropy parameter Reoentiy noweyeri tne LOSVD nas been used to constrain 13 see T 5 7a lt link gt d Applications of the Jeans Equalion Tne Jeans equation wnen combined witn observations nas a number of appiications a deriving ML profiies in spnerica gaaXies Hum 4 21d tatin p 4 2 i e y s of asymmetric drift Hum 4 2 ia a Emacs density and yoiume densty in tne gaiactic di9lt Hum 4 2 ib a anayss of tne iota yeiooty eiiipsoid in terms of Ooft s constants Hum 4 2 ic Here we iook briefiy at tnefirs and second i Spherically Symmelric Steady Slate Systems Tnis is of course an important specia case to consider For steady state tnefirs term in eq 22b is ero For spneritzi symmetry ltygt 3 o giving lt y2 gt 02 and lt yf gt of Aftertransorming to spnerica poiar ooordinates tnedeans Equation reads We 1 1 MI 1 11 e e2a37lta5aggte W 8242 n 17 7 7 17 ntroduang anisotropyparameters g 709202 and 1ro 2m2 and Wrmng 2 for g 13 and vm forltv gt ms becomes wmch S equwalent to the equanon of hydrostatwc support dpdr anwsotropwccorrewon oenmfuga correctwon Fgw Gomg a We when recaamg d dr as GM r 2 v02 r vC arw ar vexocny and rewrmng the ra term m eq 8 24b m oganthrmc gradwents we have GM lt7 szot 171 E 3LT 25 V22 822m 2d1nn d1nn3 T Tm para e sthe equauon for hydrostanc stport of an dea 92 Where p nkT the equwalences are 02 E T dUn n dUn r d n T d n r E np dp dr 2 and v r are amSotropy and rotauon correc39uon terms Ely measurmg bngmness promes and vexocny dwmerson amp rotanon promes we can derwe assuming 13 MU and hence ML r Tm 5 very mportant eg m we Seardw for nudear mack ho essee Topwc 14 2 hnk ii Rotational Flattenng Revisited Verlical Disk Structure TEID Next Prev Top 7 Steady State The DF as fE L Lz Takmg moments of the CEIE ost a most 2H deta ed nformanon from the DF Rather than Workmg wwth thequ DE the Jean equauon Works wwth just n v and lt2gt Gan we remtroduce memu DF and regam a more oomp ete descnpuon of a Syaem 7 The anewer weyesw by wntroducwng two new powerruw oonetrante a demand that the eyetern wewn steady state n equwhbrwum a p We H We oonedertheee wn turn a Integrals of Motion and he Jeans Theorem When a rn we wn etead etatew D and f are notexplicit func uons of twrn wn thwe Se we may wntroduce a powerruw new entwty Integrals ot mofon I An wntegra of motworw weamnctwon w x v whwch we constant awong a eare orbwtElampT71 3 w w e exampwee of poeebwe wntegrawe of rnotwon are E n v E van t pm energy per umt rnaee wn astatic potential L n v rx v totaw AM wn aephen39cal static potential LZ n v x2 y2 v component of AM wn an axisymmetric static potential Hwnce w x v we conetant awong an orbwt wt we aeo a eowutwon to the eteady etate CHE epecwrwcawy H W e 25 Swnce the CHE we a hnear equawon then funawone of eowutwone aretherneewvee sowutwone Thwe ywewdsthedeans Theorem ll Anyfunawon ofwntegraeof rnotwon f hr 2 war is also a solution at the steady state CBE g g rnotwon eg the DE ME L2 N0 E2 3L2 we a eowutwon to the CHE for an astymmemc potentwaw Gowng funher the strongJeans Theorem aatee ERIN 4 4 y m w w 442 that 17 a DEe of theform E wLw rnuet have an anisotropic vewoaty dweperewon a 09 17 e have theform E M a DEe of theformfE rnuet have an isotropic vewoaty dweperewon a e 09 us Rather than trywrwgto nd DE eowutwone to the CHE of therorrnm v wnetead choose DFs whrdn are expressed asfunarons of for examp e E LL LZ b SelfConsistency Both the CEIE and theueans Equauon tnc ude a potentra gradtent VltIgt n nether equatton howeven are these potermals hnked exphcmyto the DP reca Irr v dav nr r whrch oou d m prmcwp e denne As rt stands the DP on y desmbe tracer popu a ons C ear y an rrnportant step tsto require that the DP also yte dsthe potermal ltIgtr e rlrrGfrv113v 17er1 8252 VAN 8 25b where r here tsthe mass DP re We ve mumphed r by the mean stehar mass Takmgthe sphencdrorrnrorVthrsreads egroraDPortherorrnnE m l 1 2m 7 i r 7 drrG 7 2 17 17 butf Such a Sohmon now desmbes a Se froonstaent phystca y p ausb e aeHar dynamtcal system 2 1 r TXV MSV 827 many syslems 4 4 a relative potential W ltIgtOPltIgt a relative energy E Wevzv2 a note ooth W and E are rnore we for rnore oound stars deeper m the Syaem a dnoose be sothat r gt0 for E gt0 bound 0 at gHen W E spans range 0 to W as v spansthe range from t2W Vega to o c Spherical Isotropic Systems DF fE r r q dav47r2 dv e I21392 11 e 282 1 il erGn HE dB 8 28b nonrrotatmg They Wm be our s tamng pont m oonaruamg speoho ephenoa rnodeTe m 8 e d Deriving fE from Pr for NonRolating Spherical Systems n nram h r We can 7 The anewerteyee F rst evauate Wm r dgtr GMltr r from r and ehmmate rto hnd 1W We then nd E fromthe Eddington 1916 Formula EMT 8d mi 1 E 12 M 1 1p 39 n2g m2 raw 5 m1 H 829 E exarnptes are deVauoomeurs RV TaW amp Jaffe aw rnages ETETA g412 Tm rnethod can be extended to rotatmg sphentzt systems WW1 HE LT as We as ameymrnetnc systems WW1 HE L2 and HE LZ a e From fEd3r d3v to NEdE For Nepody srnmanons t soften usem to evauate NE e the tota number of stars as amncnon of energy E s not srnptythe DF E snoe ms descnbesthe of energy E at each point in phase spaoe r v m the range dar BETA Eq 4A49 BETA 452aand 458 Note that NE number of stare dav For examp e rm the MaxWeHrElotmann dtstnputton gwes NE convenuona y Wrmen as Nc c speed Conetder net the regton at r Wm r m 50 V 2W r Equot and dv W a Equot dE the vetogty phase space eternent day 4 7r So the number of stare at r of energy E m the hterva dE 5 ME r dE dar X E x er Equot dE d r Nouoe that at r E rangesfrom o pvesc to Wm v0 5 for examp e the Km form exp E 02 A see 8 so then r starts ate for Veer t then neee and then faHs agan to 0 when mesa sow ME 3 a ntegratTng over aH posmons WefmdElampT71 44 5 a 16172 fE 7 E IE 2II7 7 E 17 e 30 I v u I where rm argea radmsout to whch a aar wnh E can befound e v0 at Wmquot E WhHe E typwca yincreases exoohehhaw wnh E NE E o a Mos Stars are nea yunbound E 0 a Eew Ears are deepw bound E W0 EXamp es of ME dE forthe oeVaucomeurs ng and two Jaffe mode s am 9 415 Wages Nexll Prev Topl 8 Model Building Using DFs We oegh wnh the srhpes cases eomhorwrh nonrrotatmg sphe cal systems e DE E The process goes as foHoWs 1 Choose 2 DE whmh sa mhcuoh of energy E E W 7 M from Jean Theorem E 5 already a Sohmon to the Steady Sate CEIE so u 2 ntegrate the DE over v to ho W e evaluates 25a 3 Sche Eossoh s equahoh 8 28a to ho Wr 4 Corhohe WP and Wr to gwe the mass mahouuoh r Here are some examp es a Polytropic Sphere Power Law fE oohsoer a power law DE E E EM for E gt0 otherwse E o ntegrate E over Ve oatyto rho the densty h terms of W eoe 25a on m x p 117 fE1211 drrF Iliy yk gdv 83 n after suosmmhg 2WV cosa we nd mg an 1 W gt0 where c 5 a consiant depehohg oh h Subanute ms me the sphe cal Verson of Eossoh s equahoh eoh 8 28a 8 32 equ111bnurn of a se frgrawtatmg sonere of po1ytrop1c gas 1e equauon of sate p lt F7 Tnus nnuma1 1 W 1 b sars nun DE 1x EMS2 and gas nun bo1v1rob1c equauon of gate and 11 1 1n S1rnb1e so1unons on1v ex1afor n 5 W 55 1115 1sthe Plummer Sphere nun Hr 1x 1 rb2395 2 1t has mte rnass and1swe11benaved a r 1115 a good macn to e1obu1ar Ous Lers but 15100 Steep at 1arge rfor E111bnoas n gt5 systems are rn re extended and nave whmte rnass densty promesfor 1213145 are shown nere lt graph gt densty botennaL rotanon amp 1rnage for a P1urnrner sbnere are snown nere 1rnage 1 and b 1x I1 Wn1cn 1sthe isothermal eduanon of State reca1 p n k T b We nave na on n 00 so W for n w the above anavs1s breaks down u an aner we approa b Isolhermal Sphere Exponential fE Cons1der an exponential ao1trnann DF 8 as Rem rnore ve W amp E rneans rnore bound A so note HE gt0 for E lt0 were are unbound Siars We annobae brob1erns at 1arge radn 04 subsmuung W e 12112 for E and 1ntegrat1ngfE over v gwes F1 exp W102 P1ugg1ng ms mto F39o1sson sequat1on gves 7243 774 72 834 17 T111513 1n faa the eduanon for a nvdros1auc sonere of1sothermal gas nun 02 kTm n 1stn1s 7 At every pomL N X exppvQOon for both the s1e11ar svs1ern and a gas of atoms 111 W 1 2m 1h rf r Whether not atoms the rest Tradmonauv We cons1derthe so1unons to e 34 as 1 a spec1a1case and i Singular lsolhermal Sphere SIS d1sperson ve1ootv 402 evervwnere 1sotherma1 1 LD 502 1 and nas whmte rnassas r oo 1 but tne rnode1nas1nf1n1te densty at r Density potential rotation amp image for SIS are shown here image ii General Isothermal Sphere Choose as central boundary conditions at r 0 rquot0 P0 finite central density dl339dr r20 0 flat central density profile from eq 834 as r 0 we have LHS 22339 d dr and RHS 0 Integration of 834 yields lr39r images BampT1 figs 47 48 We find a constant nearnuclear density f539r 33390 within a radius r0 3 7 47139 G 4539012 this is a core and r0 is called the King or core radius lr0 05013 l0 so r0 is appropriately defined r0 is also the scale length of the r392 envelope see below big cores are in big galaxies circular velocity VC 2 0 d In l3 d In r12 When plotted as log 9310 vs log r to there is only one isothermal profile 0 At small radii eg rltfew r0 N N the density law resembles the Hubble density law l39r 1 r r023932 l339Hr IR fits OK to the centers of many Elliptical galaxies 0 At large radii eg r215 re the system resembles the SIS f339r 0 r r0392 and VC 2 Ix2 this is different from the Hubble density Law projected light profile does not fit Ellipticals well in the outer parts too flat The scale length and central density together define the dispersion 0 2 0 30 r02 for a given central density hotter galaxies are larger for a given core radius hotter galaxies are denser gt basically stars are bound and must not escape Quantitativer 0 2 4 9 7r G PO r02 To simplify calculations use G 45 X 10398 in units of pc kms and M13 Eg for a 100 kms r0 100 pc we have PO 2 159 Meg pc398 A good isothermal core match to the centers of Ellipticals can be used to estimate central ML obtain r0 and l0 from isothermal fits to IR and measure 0 express l0 in units of L 3 pc392 to allow simplified calculations with G 45 X 10393 j0 05 0 rO 210 9 02 4rr G r02 ML ri390 j0 This method is called quotcore fittingquot or quotKing39s methodquot typical values for ellipticals cores are 2 1020 h Mfgl Lg suggesting minimalno dark matter There is a problem with all isothermal models they have infinite total mass It is easy to see why the system is at least infinite in extent at any given radius stars have isotropic dispersion 0 at this radius at easi some aafs are therefore moving cummrd butfurther out the dispersion is mill 0 and stars are rnoying outward gt the system rnust have infinite extent Uitifnateiy negative E i e w i c Lowered Isothermal King Truncated Exponential fE Suppress stars at iarge radius ie as E aoi we wantfE a0 rnodifythe exponentiai DF fltEgt 7 8 85 27mg where 00 is a dispersion We pararneter Repeating the same anaysis as before we get for Poisson seqfi i1 d l 2 q 1 r m 7w 7 i imam r erf e i 1 e i 8 dr 42 mi Mfr 375i Sche this by integration choosing boundary oonditions at r o VI10 o 702 o gtoi iarge o deep oentrai potentiai d W dr o as before inner regions We isothermai with oore King radius rO defined as before Outer regions Wm decreases amp approacheso at r feca yeiooty range 211 iso a 2 so density f day 0 at r tidal or truncation radius edge of sphere iarger VIZo iarger o a iarger r amp Mm Aiternatiye pararneterto VIZo or o is concentration c ogm r r0 irnagesi ElampTfig4 9410411 singie sequence of King rnodeis by varying eouiyaentiy Wm or c images 3amp7 fig4 9 Empirica y we find 00 75 71752 3 a 7 fit GCsvefyweH 2 2 gt10 fit sorne EHiptiwis quite weii fits Hubbie iaw weii w isthe isotherrnai sphere King rnodeis are not isotherrna 02 EltV2gtZOO2 within r0 but drops at iarger radii irnagesi EampTfig411 Howeyeri as with isotherrna modes for eadn c or d we have a range of King rnodeis each of different 00 subiectto 002 ix F0 r02 eg for given r0 high 5390 has high 00 cl Spherical Models with Rotation DF fE L TBD e Axisymmetric Systems From the strong Jeans Theorem in general we expect DFs of the form fE LZ IS Finding such DFs is usually very difficult We therefore first consider the subset of simpler cases with fE LZ cases i amp ii below Note that since DF ifE we expect anisotropic velocities see 87b lt link gt i Thin Disks with DF fE Lz For thin disks the problem is simpler DF 2 fE LZ For given 2R virial theorem sets total KE since 2KE W 0 but this KE is shared between ordered rotational and random dispersion components Example consider DF 2 fE L2 F exp E 72qu Applying the earlier methods we derive a Mestel disk ER 20 r R0 with VC2 211139 G 20 R0 The parameter q 2 VC 0 2 1 measures the degree of centrifugal support we find 02 ltVR2gt the radial dispersion while ltVjigt 0 X P12q I 121q is the streaming rotation for 39q 1 the disk doesn39t rotate but is held up by dispersion for q 03 we have pure circular orbits with ltV39gt 2 VC Note this illustrates the phenomenon of asymmetric drift see T 54d lt link gt ii Thick Axisymmetric Systems with DF fE Lz In general axisymmetric systems have DFs with three integrals fE LZ I3 These are difficult see below so the subset with fE LZ has received more attention So far most chosen DFs have not yielded very realistic models BampT 452 LZ gt 0 otherwise f 0 What about reversing the process and starting with a density distribution As for spherical systems 87d above methods exist to convert any irquotRz into fE LZ unfortunatly they are highly unstable when trying to use observational data There is one nice result for systems with f fE LZ BampT p250 The radial and z velocity dispersions are equal ltVR2gt ltVZ2gt this is not true when a third integral is present iii Thick Axisymmetric Systems with DF fE Lz l3 Asjust mentioned whenever ltVR2gtEltVZ2gt we know a third integral IS must exist Bible M In the solar neighborhood we measure ltVR2gt 2 X ltVZ2gt the DF for the MW disk must involve some I3 in addition to E and LZ This quite shocked Jeans in 1915 who concluded the MW was not in steady state now we suspect MW is in steady state but don39t yet know the form of IS Exarnpte 2 some attened ethpttoat gaaxtes don t rotate rnuch see Toptc 7 we oonctude they are antsotroptc wtth ltR2gt gt 92gt These etttpttcas rnust have DFs whtch requtre t3 tn generat constructton of DFs wtth 3 tntegrats ts very dtf ou t 3r v ts usuaHy not anayttoatty srnpte F ts now unctton otthree other tunotons addtng to the dtf ou ttes More progress ts rnade oy ernp oytng action variables J J Ja Jt am 45 a Entt re conferences have been devoted to The Thtrd tntegra f Triaxial Systems eg Bars amp Some Elleticals Very dttttcutt stttt rnuch to be done Some tnsght from oldtheory of rotattng tnoornpresstote utds Newton 1680 Mactaunn 1742 Jacobt 1884 Retrnann 1890 Chandrasekhar 1950 Sohmons aso extst tn whtdn there tsttutd ow Retrnann For a rotattng nonaxisymmetric Syaem E L L are not conserved for orotttng stars Wever EJ E a V2 x r t2 Jacobt s tntegrat is conserved where I rotatton vector offrame tn whtch ltIgt ts stattonary E V2V2 D tsnorrna KE F39E tn tnerttat frame v2 t n x r t2 v2 02 R2 ts PE assooated wtth centnmga force EJ V2V2 thequot where thequot ts an effeatve potenttat tn the rotattng frame Constder DFS of formf ttEJ n rotattng frame V o amp a 06 te 0 ts dtagonat tn thts use hedeans Equatton E hydrostattc support tn rotattng frame a sotuttons are Jacobt etttpsotds Proceed by choosng form of DE eg power taw f a potytrope retevant to rotattng stars so tong htsory of anaysts sotuttons onty gtve rnttd central oonoentratton a not good for Es a better for oars Mos progress so far usesnumertcal rnethods a orbtt tntegratorsftnd orbtt dtstrtbuttons oonststent wtth Fm e dewarjht d Memtt Statter o NrElody codesfoHoW coHapSe unttt trtaxta eqtrn estaottshed eg Aarseth e Ehnney m 9 olent Relaxation Varying potentials are usually intractable and require a numerical approach There are however a few situations which can be treated analytically Paradoxically one of these is when the potential is maximally fluctuating This is the case of violent relaxation which we now describe briefly For galaxies 2body encounters are negligible and evolution is determined by the CBE For a static potential energy E of a star is conserved and the DF doesn39t change Isolated galaxies in steady state do not therefore evolve dynamically we39re ignoring gas amp 2body processes here For a galaxy to change there needs to be a changing potential For each star dE dt 2 MW at at the star The DF evolves and the structure of the galaxy changes This occurs during i inhomogeneous collapse and ii encounters Topic 12 These are brief traumatic times quotgalaxy changes are by revolution rather than by evolutionquot nice quote from Binney39s EAA article In collapse of large cold system 1 changes rapidly stars gain and lose energy which broadens fE energy is redistributed via collective interactions this acts like a relaxation process Note the total energy remains constant this is a nondissipational process energy is not radiated away as with dissipational gaseous collapse If the total energy is initially zero eg diffuse system at rest then following collapse some stars will be strongly bound but some must also have been ejected Note scattering is independent of the star39s mass fundamentally different from 2body relaxation no segregation by mass eg heavy stars don39t sink to center Phase mixing helps smooth out lumps fairly quickly distribution is smooth after few collapse times violent relaxation timescale is few gtlt dynamical collapse timescale lf relaxation is complete then fully random scattering occurs results in isotropic velocity field and Boltzmannlike fE Usually however the density distribution becomes smooth before scattering is complete relaxation ceases and is incomplete residual anisotropies amp phasespace substructures lt viewgraph gt NBody example van Albada 1982 BampT 473 start with homogeneous sphere with low 0 1st infall dense center settles into Rm law qdrops with radius F anisotropy is 0 at nucleus 1 at edge most scattering occurred at small ron 1st infall most stars have low AM NE dE spreads out most stars have E 0 few are deeply bound images BampT figs 41923 If the initial distribution is hotter less concentrated If the initial distribution is rotating slowly less concentrated amp rotating oblate figure If the initial distribution is rotating faster even less concentrated amp prolatebar figure If the initial distribution is ellipsoidal rotating ellipsoid anisotropic everywhere 10 Introducing StarStar Encounters So ran We nave eonsldered star motlon ln a perlealy smooth potentlal However ln reallty lrldlvldual stars render tnls potentlal oumpy on nne scale 0 Hi i m a urnpn n valld7 a Estimating Encounter and Relaxa n Timescales As usual mean free lnA l ebetween encounters ll AV n for enoounter crossectlon A to2 o lmpact parameter star densty n and mean yelogty v We oonsldertnree reglmes i Direct collision or tidal capture For o few x Rm strongtldes dlsspate orbltal energy dependlng on clrcumstanoes tne stars may ultlm a tnls ls exceedingly rare ln present day galaxles ii Strong Deflection Derlned asAv v leadlng to tldal capture oal esoe oocumng Wnen o E r5 2 sror strong from mna tneorem e m2 l lssurnclently smal 2 r5 mV mV nere m ls star mass a tnls ls l AU rortne surl usng v 20 r so kms d vegror earth s orolt mar ln solar nelgnoomood Tne lnterval 5 between colllslonsl ln r52 v a tnlslsver r refer v3 l 3an2 n lo 5 yearstr sun most aellar Syaems may be relevant ln dense so nuclel galacth nudel and galaxy clusters Weak Deflection DerlnedasAv v S estlmate deflealon velodtytowardstarget aar tlme ln vlarllty of target star At m2o l v c leratlon m Gm l o so Av LH2Gm l ov Angle of deflealon A mAVi l v H2Gm l ov2 m2 arcsec ln solar nelgnoomood b gtgtr After many enoounters Av o 2 Av snoe tne direction of pull lsrandom Avm executes a random Walk tne amplltude squared ortnls resultant Veloclty after tlme t nglVerl oy 9pm 2 AV 2 Aw rV2llb lml D l WM W V I where A omax l own The Syaem has relaxed when the veloclty changes by100 le when Avm v lntegratlrlg over a Maxwelllan changes the numean constant sllghtly am s 4 Changlrlg v to 0 and ertlng m h as the stellar denslty we get am eq 8771 03 I 391 0347 I szp In A a sea 18Xloln 3 7 71 eeeo 1quot quot3 where am has unlts lo kms m has unlts of Mg and F3 has unlts 103 MG l p05 e e a surprlsnglyslmple alternatlve expresslorl torthe relaxatlorl tlme lt ls less preose but lsadequate ln many orwmstanoes We start as before Wlth equatlorl e 37o for a system of Sle R contamlrlg N stars h 8N l 47rR3 from the vlrlal theorem v2 GM l R GNm l R the system relaxes when Avm v take omasz bmmzrs Gm l v so omax l ommA N choose unlts oftlme tetrmS HRV Suoaltulng we get a as c Surprlsngly thls only depends on N the total number of stars tn the Syaem Notlcethattelaxgttcmss for N230 to good approxlmatlorl stars usually orblt tn the overall potentlal Equal loganthmlc lrltervalslrl o oorltrlbute equallyto long term deflectlon eg the ranges Rto szr m to VAR 2 to own all contrlbute equally However Slrlce the de ealon drops rapldly Wlth o asAV l v M l o a for h Example galaxy Wlth R10 kpc ow 1AU1M aars so ln A 20 half de ec39tlorl from encounters outslde o1 where ln Rlo1 10 34 deflection from encounters outside b2 Where In Rb2 15 b1 05 pc for which Av v bmmb1m10395 h2 0003 pc for which Av v bmm b2 3 015 Need care with NBody simulations when N ltlt Nstars I is more grainy than reality and trelaxsimulation ltlt trelaxreality avoid by softening star potentials to increase bmm care lose structure on scales R lt bmm Finally note that this treatment of 2body relaxation is almost identical to the derivation of Dynamical Friction in Topic 123a b Timescales for Real Stellar Systems Here are rough timescales in years for a number of stellar systems System N R pc V kms tcross treiax tage agerelax Open Cluster 102 2 05 106 107 108 10 Globular Cluster 105 4 10 5 gtlt105 4 gtlt108 1010 20 Dwarf Galaxy 109 108 50 2 gtlt107 1014 1010 10394 Elliptical 1011 1045 250 108 4 x1016 1010 10397 Spiral Disk 1011 1045 20 15 x109 6 x1017 1010 10398 MW Nucleus 106 1 150 104 108 1010 100 Luminous Nucleus 108 10 500 2 x104 1010 1010 1 Galaxy Cluster 102 5 x105 500 109 3 X109 1010 3 The presence of dark matter complicates the situation in clusters see Topic 134c In practice 2body relaxation is not as fast as our simple analysis suggests 2Body relaxation may be relevant for star clusters and galaxy nuclei For most galaxies 2body relaxation is utterly negligible Because this course deals specifically with galaxies and not star clusters we will only briefly consider the ramifications of relaxation Dont forget relaxation times can vary greatly within a given system for example a GC core can be relaxing while the halo is not c Analytic Treatment The FokkerPlanck Equation You may be wondering when Max Planck amp Adriaan Fokker worked on stellar dynamics They didn39t much of this work has its roots in plasma physics Unlike neutral gases charges in plasmas have long range Coulomb interactions The early work on plasmas has been appropriated and applied to stellar systems we continue or a smooth potential the DF obeys the CEIE df i dt o with ehoouhters stars stztter into and out of the phase space traieoorv df l dt I tf I tf is a collision term and itself depends on f let w r V be the SD Sw AWJdawAt phase spaoe veaof and probability of scattering by Aw in time At ElampT use svmbol W for S giving the master equation off W rm SwaAw AW fwaAw eSwAwlfw WW 8 89 Now the maioritv of Swtterlrlg oomes from distant ehoouhters with small Aw so expand thefirs term in L scattering in asTavlor series upto 2nd order shoe 0th order term is2hd term in L these both dlmppear oh subtraction CEIE d f it 0 ii a iii 7 DAiai 7 DAiA W l 20mlle m OWXWIWWJ ml W where DAwi ahd DtAwnij are dillusion coellicients giving the ratethat stars diffuse through phase space due to ehcouhters 3 DAwi I Awi SwAwd w ohoe DAwi ahd DtAwn AwJ are know the PP eon isa F39DE which is in pflflap e solvable Much Work has been invested in solihhg the PP equation for stellar svsems Several good approximation help further 1 Orbit Averaging for beta gtgttmss Aw issmall during a shgle orbit ing achieved by Working in a ese Coordinate orbit average is smply integration over PP eqfl is how collapsed to a F39DE in 470 mime symmeth we lose another degree of freedom further reduced to a F39DE in SD up L s time or i i best ctionangle variables Jr 0 instead of r ii n th d3 for spherical a PP eqfl 2 Local Approximation shoe equal logarithmic intervals in b oohtribute equallvto Sw erlng D ii a oohsequehtly a scattering oocurs ohlvih V and not in r a diffusion terms DAx Axi AxJ DtAxiiAvj 20 they all vanish a ehoouhters oocur as Kepleriah hvperbolae a isotropic Field Star Velocity Distribution The symmetry of the scattering population eliminates many of the diffusoh terms ohlvthree are le expressed relative to the direction of thetes star DtAvlli DtAvlFi DtAvf Togetn err tn ese aHoW analyttc express ons for ottmston coetnctents am App 8 A mm iluwumn lsnwmmr emmnA 7 2 1quot u e 412 DAt ZnYGYm lut quotn m s Mb t 1 2a 21 A v so 0Anj Mfr lzntftwtdm e 410 t r wnere subsnpt a referstoftetd stars Autumn e 2 Combmmg these one may took at tne rate of change of KE of a star 7 2 v2 0 r r r 7 t t DAb 1m 6 mmuluA nuV A f LquL m tuwt J 8 42 r u U tne 1512nd terrns represent heatmg byof tne neto stars ote tnese cancet wnen may ran equipanition Often a Maxweutan forfaUa ts used to evaluate tne ottmson coetnctents expnotty from here one can now sche tne FokkerrF39tantx eon Solutions to the FokkerPlanck Equation Several rnetnoos nave peen exptoreo a Moment Equatons foHong earher 1s moment of CEIE a Jeans E oatton srnnany 151 rno ent of PP a usem moment equattons as before one neeosto assume aform tortne DFto proceed a Maxweutan tsotten qmte sJoces u eg Larson LyndeanleH and Eggteton a Monte Carto Numertca y calcutate a ssrnpte 1000 of orpts Sijea eaon to appropnatety onosen random perturbattons foHow tne response of tne orptts and therefore tne system a to m g 29 a Dtrect tntegratton tne FokkerrF39tanck eon s a F39DE m 3 vanaptes up L t sotuttons are now posspte by numertw tntegratton eg Conn d Results The Effects of Encounters i Relaxation seuar o For selfgravitating systems this can be a rather interesting process recall from 85e that such systems have negative specific heat if you remove energy heat stars fall deeper in the gravitational well they therefore speed up and that part of the system gets hotter In its simplest form this relaxation renders clusters more centrally concentrated stellar encounters in the core pass energy to envelope stars gt the core contracts and heats the envelope expands and cools after some time the envelope developes a density profile l339r 0 r39s395 radial anisotropy increases with time and radius stars have been kicked out from encounters in the core and carry little AM a successful DF is due to Michie and is fEL 0C expL2L02 gtlt expE m2 1 After about 15 trelax the process takes off in a runaway gravothermal catastrophe TBD explain why in intuitive terms This quoteventquot is called core collapse and leaves a density law 9539r 0 F223 infinite at r0 since G0 are about 20 X trelax old at least some have probably undergone core collapse In practice core collapse is not as dramatic as its name suggests e her o the core quotruns out of starsquot before densities become exotic o a hard binary forms which a scatters core stars heating the core and halting core collapse b ejects stars from the system accelerating evaporation the binary acts like a source of nuclear burning in a star see below 0 Similar behaviour is found in stars contracting cores heat while expanding envelopes cool LyndenBell and Eggleton 1980 derive powerlaw of 94 225 for a conducting gas sphere obviously this had no nuclear energy source so could undergo gravothermal collapse ii Equipartition Violent relaxation during formation leaves all stars the same velocity distribution consequently heavier stars have more kinetic energy this is unlike a gas where molecules have the same kinetic energy heavier ones move slower 2body encounters mimic molecular interactions energy passed from high mass to low mass stars in the limit of complete interaction energy is shared equally hence equipartition o More massive stars begin to sink deeper mass segregation Probably occurred in GCs though difficult to check since visible giants all have similar mass May have played role in galaxy clusters but other effects dynamical friction mergers confuse interpretation iii Escape Ejection and Evaporation Encounters can result in stars with V gt VeSC this can occur in two ways a single encounter gives the star sufficient energy to escape ejection a star slowly wanders into unbound phase space due to many distant encounters From the analysis above 10aiii the second is much more important Using the fact that Vesc2 2 I r it is easy to show BampT p 490 that ltVesc2gt 4 ltV2gt so the rms escape velocity is just twice the rms velocity for a Maxwellian the fraction with V gt 2Vrms 7 gtlt10398 so this fraction is lost every trelax tevapxmo trelax Selection of Homework Questions Topic 4 Luminosity Functions 1 TheSchechter Function L 39 39 39 39 39 quotM Luminosity fundion of galaxies First the fundion reads ltIgtLdL n Laez39p L a 1 Considering just the low luminosity galaxies L ltlt Lw show that for 2 lt alt total number of galaxies is infinite but the total light is not 2 Derivean expression for the quotmidrankquot galaxyluminosity Lmid such that half the light comes from galaxies with L gt Lmid and half comes from galaxies with Llt Lmid What is Lmid for or 1 3 Transform the Scheduler luminosity fundion expressed in Ltoan equivalent fundion expressed in M absolute magnitude dontjust copy the formula given in BampM but show how it comesabout 4 Using whatever computing environment you prefer generate plots of the following related LFs a LogltIgtL vs Log LLw b LogdgtM vs M M 0 Log NgtL vs Log LLw d Log Nlt M vs M Mn brighter galaxies Take the normalization nto be unity take the range in LLto be from 1039210 10 and overplot lines with three values Ola 15 10 05 dotted solid dashed Be careful toacoount for the fact that graph a expressesvbper unit interval of luminosity dL while graph b expresses lt1gtper magnitude dM which isan interval in Log L Also note that graphs Wm mum mwm Aswanan x2272nn512 36 m Whime EXTRAGALAC39HC ASTRONOMV m Toolbox E 18 GALAXV FORMA39HON AND HIGHZ GALAXIES Index Images Quesuons Refs Prim Tm Tops m not yet been pvepaed Maybe next nme Spung 2mm 1 lnlmduclian 2 Nanlinearity Turnaraund amp Collapse 3 PressSchechher Formalism 4 Heirarchical Merging Next Prev Top m HA5wldmwmKnaSxsvaSKmdax mm mme Selection of Homework Questions Topic 7 Ellipticals 1 General a To first order elliptical galaxies appear to be an extremely homogeneous group Describe some f e parameter correlations Which support this claim explaining What physical principals lie behind the correlations b On closer scrutiny ellipticals seem to divide into two distind classes depending on the 2D shape of their isophotes How is this shape difference defined and in What other ways are these two types different from eachother7 c It has been suggested that While probably oversimplified each type has experienced a different formation scenario Outline these two possible scenarios and state Which one goes with Which type of elliptical galaxy 2 The Sersic Brighlness Profile The radial surface brightness distribution in flux units for a wide range of systems can be described by the Sersic Law 102 1Rcerpib RR1 e 1 071 Where Re encloses half the total light a Give the equivalent expression using surface brightness measured in magnitudes u b Express the central surface brightness l0 andu0 in terms of the surface brightness at the effective radius IRe anduRe Hence find equivalent expressions for IR anduR using l0 andu0 c The special case n1 yields an exponential profile which is usually expressed in terms of the scale length Rd IR l0 expRRd so that IRd 1e x l0 Show that Ltot 2 39FI39RdZ l0 and that Re 167 Rd Hence show that b 168 for this exponential disk Please don39t be confused by the subscripts here Re encloses half the light e for effective while Rd marks the radius at which the surface brightness has fallen to 1e of the central surface brightness d for disk since disks are close to exponentials d Show that the definition of Re namely looo 21TH IR dR 2 x l oRe 27rR IR dR can be recast as Jmooo x2 1 e39bx dx 2 x lo1 x2 1 e39bx dx Hence using numerical methods show that b 1999n 0327 for n 1 8 giving b in terms of n Hint you will need to evaluate the integrals numerically as well as hunt for the appropriate b that satisfies the equation for each n Having done that plot b vs n and confirm that the line b 1999n 0327 goes through the points e Using your previous results how does l0 IRe or equivalently 10 p Re depend on n Are ellipticals n 4 more or less concentrated than disks n 1 h For M87 and M32 look up BT and Ae in RC3 and assuming they both fit the R14 law at all radii calculate the factor by which their centers outshine the moonless sky which has 13 227 Also estimate D25 and compare it with the value given in RC3 3 Galaxy Shapes Of course we only ever see galaxies in projection on the sky And yet we feel we have significant knowledge of their 3D shapes Describe the observations and line of reasoning that has been taken to ascertain statistically the 3D shapes of a ellipticals b spiral disks Whmia mum WWW Aswaan l2272nnsn51w Whlme EXTRAGALAC39HC ASTRONOMV m Toolbox E 11 as o osmooov 7 EI11p11caIs 12 1me1ac11ons 17 LS Suuctuve 13 Goupsamp0usevs 1E Gaaxy Fo1ma11or1 9 ex amp 051 14 Nuc1e1 amp EH 1a Femmzanon amp18M a swan 1n Popu1a11ons 15 11611512131an 2n Dak Mane1 7 ELLIP39I10AL GALAXIES 1 lnlmduclian a The Myths 011 mew o1 E111p11ca gaax1es m dwawged gvea y 111 me 197D E111p11cas weve mougm 10 be Dames mges wm devwcouieuvs Rm pmmes awa oonsiawi densuy K1119 coves Ouale sphevmds 1a11er1ea by 1o1a11or1 Vow 019 a1 coman asng1e awem popu1anor1 0151a Fbiaxed dynam1caiyqu1esoemsys1ems To a1age extent all 011m above as new mougm 10 be Wong b 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anumbev m pvacnca dmwwmes Vamng wouva e sunaee photomeuy anon y amca Typcanyone am an R y yym man 000 men may not extend Va enoug ec oe ya ve ng mem yym me PSF typea y Gaman yym PL wngs a R tum we we a al Cole my lt1 0 empncuy deae es sgnmcanuy an R lt A a a a4 5 ected even M ev om CaIIhIanon y onen mme min typca aocuvacy 0ny6 e Nexll Prevl 2 Radial Light Pra les Filling Funclians Heve ae some exmwp e hgm pvomes nrnages numbev m anaync expvessons nave been used to m y and weaknesses Ovev me yea a R m Emma gaaxwes Tney eam nave mew s1vengms a deVaucouleurs Hm and Sersic H m Laws deVaJcomeuvs nausea ma 390 many empncas Ea R an my me Wm B ynages The awws usJaHy wnnen 1R IE 61 7767 RRQVAA 73 u nx me VoHowmg pmpemes 7 22 T Re we nun we Lm m mama n Hmmmmwmmesxmssznnaex mm
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