Gravitational Dynamics: Part II. Size and Density of a BH. Adiabatic Compression due to growing BH. Boundary of Star Cluster

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Lec: Gowth of a Black Hole y captuing ojects in Loss Cone Gavitational Dynamics: Pat II Non-Equiliium systems AS40, Pat II A small BH on oit with peicente p <R h is lost (as a whole) in the igge BH.

Newtonian theoy is not adequate (Nealy adial) oits with angula momentum J<J lc =*c*r h =4GM h /c entes `loss cone` (lc) When two BHs mege, the new BH has a mass somewhat less than the sum, due to

A BH has a density (/4Pi) M h /R h, hence smallest holes ae densest. Compae density of 0 8 Msun BH with Sun (o wate) and a giant sta (0Rsun).

1 Lec: Gowth of a Black Hole y captuing ojects in Loss Cone Gavitational Dynamics: Pat II Non-Equiliium systems AS40, Pat II A small BH on oit with peicente p <R h is lost (as a whole) in the igge BH. The final pocess is at elativistic speed. Newtonian theoy is not adequate (Nealy adial) oits with angula momentum J<J lc =*c*r h =4GM h /c entes `loss cone` (lc) When two BHs mege, the new BH has a mass somewhat less than the sum, due to gavitational adiation. AS40, Pat II Size and Density of a BH A lack hole has a finite (schwazschild) adius R h = G M h /c ~ au (M h /0 8 M sun ) veify this! What is the mass of cm BH? A BH has a density (/4Pi) M h /R h, hence smallest holes ae densest. Compae density of 0 8 Msun BH with Sun (o wate) and a giant sta (0Rsun). Adiaatic Compession due to gowing BH A sta ciculating a BH at adius has a velocity v=(gm h /) /, an angula momentum J = v =(GM h ) /, As BH gows, Potential and Oital Enegy E changes with time. But J conseved (no toque!), still cicula! So J i = (GM i i ) / =J f =(GM f f ) / Shink f / i = M i /M f <, oit compessed! AS40, Pat II AS40, Pat II 4 Bounday of Sta Cluste Limitted y tide of Dak-Matte-ich Milky Way Tidal Stipping TIDAL RADIUS: Radius within which a paticle is ound to the satellite athe than the host galaxy. Conside a satellite (mass m s ) moving in a spheical potential φ g (R) made fom a host galaxy (mass M). R M AS40, Pat II 5 AS40, Pat II 6

2 If satellite plunges in adially the condition fo a paticle to e ound to the satellite m s athe than the host galaxy M is: GM GM Gm! " s ( R! ) ( R + ) Diffeential (tidal) foce on the paticle due to the host galaxy Gms! GM U R!! " # " # 4 If << R then U = % - &! % + & $ +... ' R ( ' R ( R! GM Gm k s, k R " = Foce on paticle due to satellite AS40, Pat II 7 Instantaneous Tidal adius Geneally,! ms " t ( t) = R( t) # $ % km ( R) & fudge facto k vaies fom to 4 depending on definitions. t is smallest at peicente R p whee R is smallest. t shinks as a satellite losses mass m. AS40, Pat II 8 The meaning of tidal adius (k=) Paticle Bound to satellite if the mean densities ms ( ) M ( R) " 4 4!! R The less dense pat of the satellite is ton out of the system, into tidal tails. Shot question Recalculate the instantaneous Roche Loe fo satellite on adial oit, ut assume Host galaxy potential Φ(R)= V 0 ln(r) Satellite self-gavity potential φ()= v 0 ln(), whee v 0,V 0 ae constants. Show M= V 0 R/G, m = v 0 /G, Hence Show t /R = cst v 0 /V 0, cst =k / AS40, Pat II 9 AS40, Pat II 0 Shot questions Tun the Sun s velocity diection (keep amplitude) such that the Sun can fall into the BH at Galactic Cente. How accuate must the aiming e in tem of angles in acsec? Find input values fom speed of the Sun, BH mass and distances fom liteatue. Conside a giant sta (of 00sola adii, sola mass) on cicula oit of 0.pc aound the BH, how ig is its tidal adius in tems of sola adius? The sta will e dawn close to the BH as it gows. Say BH ecomes 000 as massive as now, what is the new tidal adius in sola adius? AS40, Pat II Lec : otating potential of satellite-host Conside a satellite oiting a host galaxy Usual enegy E and J NOT conseved. The fame (x,y,z), in which Φ is static, otates at angula velocity Ω = Ω e z Effective potential & EoM in otating fame: && = "#$ " (% & & ), $ = $ " % R eff eff Pove JACOBI S ENERGY conseved v EJ ' E " % ( J =! eff + & AS40, Pat II

Roche Loe of Satellite A test paticle with Jakoi enegy E J is ound in a egion whee φ eff (x)<e J since v >0 always. Lagange points of satellite!

s ( ) " # R GM Gms = " v u " v " # R + R AS40, Pat II AS40, Pat II 4 If cicula oit Rotation angula fequency Ω =G(M+m)/R L point: Saddle point

" # m $ % m & t = ± R # $ ' ± R m ( ) # % & M M + $ * + # ( ), * M + $ - Roche Loe shapes to help Diffeentiate Newtonian, DM, o MOND Roche Loe:

than the giant BH, is tidally disupted fist.

3 Roche Loe of Satellite A test paticle with Jakoi enegy E J is ound in a egion whee φ eff (x)<e J since v >0 always. Lagange points of satellite!" eff!" = =! x! y eff 0, and 0 In satellite s oital plane ( pependicula to Ω) v u v v! eff ( ) =! g ( R + ) +! s ( ) " # R GM Gms = " v u " v " # R + R AS40, Pat II AS40, Pat II 4 If cicula oit Rotation angula fequency Ω =G(M+m)/R L point: Saddle point satisfies (afte Taylo Expand Φ eff at =R):! " # m $ % m & t = ± R # $ ' ± R m ( ) # % & M M + $ * + # ( ), * M + $ - Roche Loe shapes to help Diffeentiate Newtonian, DM, o MOND Roche Loe: equal effective potential contou going though saddle point AS40, Pat II 5 AS40, Pat II 6 Tidal disuption nea giant BH A giant sta has low density than the giant BH, is tidally disupted fist. Disuption happens at adius dis > R h, whee M h / dis ~ M * /R * Show a giant sta is sheded efoe eaching a million sola mass BH. Pat of the tidal tail feeds into the BH, pat goes out. AS40, Pat II 7 What have we leaned? Citeia to fall into a BH as a whole piece size, loss cone Adiaatic contaction Tidal disuption citeia Mean density Whee ae we heading? Fom -ody to N-ody system AS40, Pat II 8

4 Lec 4: Encounte a sta occasionally Oit deflected evaluate deflection of a paticle when encounteing a sta of mass m at distance : Gm g" = 0 g pep Gm ( x + ) X=vt cos = / & θ AS40, Pat II 9 Gm (. vt + % +, ) # &' - * # $ v! Stella Velocity Change Δv pep sum up the impulses dt g pep use s = vt / $ Gm % " v# = & g# dt = &( + s ) ds Gm " v# = g# " t =! v O using impulse appoximation: v %$ Gm = v whee g pep is the foce at closest appoach and the duation of the inteaction can e estimated as : Δt = / v AS40, Pat II 0 Cossing a system of N stas plus Dak Matte elementay paticles let system diamete e: R Ague Cossing time t coss =R/v Sta nume density pe aea ~ N/(R π) Total mass M =N*m* + N dm m dm > N*m* Typically m dm ~ Gev << m* = m N dm > 0 0 > N* = N AS40, Pat II Nume of encountes with impact paamete - + Δ # of stas on the way pe cossing $ % N ( ) " R " & N N * = & " R R ( )! N = " + & '" & v = 0 each encounte is andomly oiented sum is zeo: # AS40, Pat II +Δ Sum up the heating in kinetic enegy sum ove gain in (Δv pep )/ in one-cossing max " Gm # N d " Gm # v! + % N v & R + ' ( % Rv & min ' ( $ = $ = " m # max v! / v 8N ln, whee ~ N $ ) % * * = ' M & ( min conside encountes ove all < max ~ R ~ GM/v [M= total mass of system] > min ~ R/N AS40, Pat II Relaxation time Oit Relaxed afte n elax times acoss the system so that oit deflected y Δv /v ~ v N ' nelax!! whee N'=(M/m) / N " N # v 8ln $ thus the elaxation time is: N ' telax = nelaxtcoss! tcoss 8ln $ Ague two-ody scatteing etween sta-sta, sta- DM, lump-sta, lump-dm ae significant, ut not etween Gev paticles. AS40, Pat II 4 4

5 How long does it take fo eal systems to elax? gloula cluste, N=0 5, R=0 pc t coss ~ R / v ~ 0 5 yeas t elax ~ 0 8 yeas << age of cluste: elaxed galaxy, N=0, R=5 kpc t coss ~ 0 8 yeas t elax ~ 0 5 yeas >> age of galaxy: collisionless cluste of galaxies: t elax ~ age AS40, Pat II 5 Self-heating/Expansion/Segegation of an isolated sta cluste: Relax! Coe of the cluste contacts, fom a tight inay with vey negative enegy Oute envelope of cluste eceives enegy, ecomes igge and igge. Size inceases y ode /N pe cossing time. Ague a typical gloula cluste has size-douled Low-mass stas segegate and gadually diffuse out/escape AS40, Pat II 6 Lec 5: Dynamical Fiction As the satellite moves though a sea of ackgound paticles, (e.g. stas and dak matte in the paent galaxy) the satellites gavity altes the tajectoy of the ackgound stas, uilding up a slight density enhancement of stas ehind the satellite The gavity fom the wake pulls ackwads on the satellites motion, slowing it down a little This effect is efeed to as dynamical fiction ecause it acts like a fictional o viscous foce, ut it s pue gavity. It ceates density wakes at low speed, & cone-shaped wakes if satellite tavels with high speed. AS40, Pat II 7 AS40, Pat II 8 Chandasekha Dynamical Fiction Fomula The dynamical fiction acting on a satellite of mass M moving at v s kms- in a sea of paticles of density m*n() with Gaussian velocity distiution v v # v $ s n( ) m f (, ) = f (,0)exp & %, (,0) ' f = (! ) ( "! ) Only stas moving slowe than M contiute to the foce. dvm dt = % 6" ln $ vm # f ( vm) vmdvm 0 G ( M + m) V M! M AS40, Pat II 9 Dependence on satellite speed Fo a sufficiently lage v M, the integal conveges to a definite limit and the fictional foce theefoe falls like v M -. Fo sufficiently small v M we may eplace f(v M ) y f(0), hence foce goes up with v M : dvm vm #! 6 $ ln " G f (0)( M + m) vm =! dt t This defines a typical fiction timescale t fic fic AS40, Pat II 0 5

6 Depends on M, n*()m* & n dm ()m dm Moe massive satellites feel a geate fiction since they can alte tajectoies moe and uild up a moe massive wake ehind them. Dynamical fiction is stonge in highe density egions since thee ae moe stas to contiute to the wake so the wake is moe massive. Note: oth stas (m*~msun) and dak matte paticles (m dm ~Gev) contiute to dynamical fiction. AS40, Pat II Fiction & tide: effects on satellite oit The dag foce dissipates oital enegy E(t) and J(t) The decay is faste at peicente staicase-like decline of E(t), J(t). As the satellite moves inwad the tidal ecomes geate so the tidal adius deceases and the mass m(t) will decay. AS40, Pat II Oital decay of Lage Magellanic Cloud: a poof of dak matte? Dynamical fiction to dag LMC s oit at R=50-00 kpc: density of stas fom Milky Way at 50 kpc vey low No dag fom odinay stas dak matte density is high at 50 kpc Dag can only come fom dak matte paticles in Milky Way Enegy (fom futue velocity data fom GAIA) diffeence ealie/late deis on the steam may eveal evidences fo oital decay AS40, Pat II Summay Relaxation is a measue of ganulaity in potential of N-paticles of diffeent masses Relaxation cause enegy diffusion fom coe to envelope of a system, expansion of the system, evapoation (~escape) of stas Massive lumps leaves wakes, tanspot enegy/momentum to ackgound. Cause oit decay, galaxies mege AS40, Pat II 4 Tutoial session AS40, Pat II 5 6

Physics 312 Introduction to Astrophysics Lecture 7

Physics 312 Introduction to Astrophysics Lecture 7 Physics 312 Intoduction to Astophysics Lectue 7 James Buckley buckley@wuphys.wustl.edu Lectue 7 Eath/Moon System Tidal Foces Tides M= mass of moon o sun F 1 = GMm 2 F 2 = GMm ( + ) 2 Diffeence in gavitational