Adiabatic evolution of the constants of motion in resonance (I)

Transcription

1 Adiabatic evolution of the constants of motion in esonance (I) BH Gavitational 重 力力波 waves Takahio Tanaka (YITP, Kyoto univesity) R. Fujita, S. Isoyama, H. Nakano, N. Sago PTEP 013 (013) 6, 063E01 e-pint: axiv:

2 Extime Mass Ratio Inspial(EMRI) Inspial of 1~100M sol BH of NS into the supe massive BH at galactic cente (typically 10 6 M sol ) Vey elativistic wave fom can be calculated using BH petubation Many cycles befoe the coalescence ~O(M/µ) allow us to detemine the obit pecisely. Clean system The best place to test GR. BH

3 Leading ode wave fom Enegy balance agument is sufficient. de GW df Wave fom df de obit de obit fo quasi-cicula obits, fo example. de df obit Hee E is enegy divided by mass. de obit 0 + O µ + O µ de obit geodesic + O µ + O µ df leading ode ( ) ( ) ( ) ( ) ( ) 3

4 Evolution of geneal obits If we know fou velocity u µ at each time accuately, we can solve the obital evolution. On Ke backgound thee ae fou constants of motion Nomalization of fou velocity: Enegy: ( t ) E u µ ξµ Angula momentum: Q u L z u µ ξ µ 1 u µ u ( φ ) µ µ Killing vecto fo time tanslation sym. u ν constant in case of no adiation eaction Killing vecto fo otational sym. Cate constant: Killing tenso µν Quadatic and un-elated to Killing vecto K One-to-one coespondence E, L z, Q u Secula evolution of E,L z,q is necessay. µ de obit + de obit df O ( µ ) O( µ ) ( geodesic ) + O ( µ ) O( µ ) 4

5 The issue of adiation eaction to Cate constant " E, L z Killing vecto Conseved cuent fo the field coesponding to Killing vecto exists. µ ( GW ) E E GW d Σ t ξ ν µν E GW Howeve, Q Killing vecto As a sum consevation law holds. " We need to diectly evaluate the self-foce acting on the paticle. 5

6 3 Adiabatic appoximation fo Q T << τ RR T: obital peiod τ RR : timescale of adiation eaction " The tajectoy of a paticle is assumed to be given by a geodesic specified by E,L z,q. " We evaluate the adiative field ( ad ) ( et ) ( adv) h µν hµν hµν instead of the etaded field. " Self-foce is computed fom the adiative field, and it detemines the change ates of E,L z,q. dq d which is diffeent fom enegy balance agument. Appoximation pocedue 1 1 T Q lim dτ τ µ T T T u α F α [ ( )] ad h µν [ ] 6

7 Why does this appoximation wok? " Fo E and L z the esults ae equivalent to the balance agument. (shown by Gal tsov 8) " Fo Q, the estimate using the adiative field is shown to give the coect long time aveage. (shown by Mino 03) " Key point: Unde the tansfomation ( ) ( ) t,, θ, φ t,, θ, φ a geodesic is tansfomed back into itself. Radiative field is fee fom divegence at the location of the paticle and easy to evaluate. " Divegent pat is common fo both etaded and advanced fields. 7

8 Outstanding popety of Ke geodesic d λ dτ dλ + a cos θ dθ R( ) Θ( θ ) dλ - and θ -oscillations can be solved independently. Intoducing a new time paamete λ by d dλ ( - dependent pat) + ( θ - dependent pat) t ( ) ( θ ) ( λ) t + t + dλ Only discete Fouie components aise in an obit 1 m / dφ simila dλ ( m dφ dλ + n Ω + n ) n, nθ / dλ Ω λ Peiodic functions with fequencies θ Ω,Ωθ θ 8

9 dq Final expession fo dq/ in adiabatic appoximation The esulting fomula is so simple. (Sago, TT, Hikida, Nakano PTP 115 (005) 873) de dlz f ( ) g( ) + n, nθ l, m ( mode function ) ( souce ) nω 4 Al, tem d x amplitude of the patial wave A This expession is as easy to evaluate as de/ and dl/. de dl m A l, A l, Analytic fomula up to.5pn ode: Ganz, Hikida, Nakano, Sago, TT, PTP 117 (007) 1041 Numeical calculation: Fujita, Hikida, Tagoshi, PTP 11 (009) 843

10 " Key point: Unde Mino s tansfomation t,, θ, φ t,, θ, φ ( ) ( ) a geodesic is tansfomed back into the same geodesic. Howeve, fo esonant case: jθ Ω jω θ with intege j & j θ Δλ (sepaation fom θ max to max ) has physical meaning. max θ max Δλ Resonant obit θ Δλ λ (Mino time) Unde Mino s tansfomation, a esonant geodesic with Δλ tansfoms into a esonant geodesics with -Δλ. 10

11 G ( et ) ( ) ( ad ) ( ) ( sym x, xʹ G x, xʹ + G ) ( x, xʹ ) Fo the adiative pat (etaded-advaneced)/, a fomula simila to the non-esonant case can be obtained: dq Ω A de dl n ( ) ( ) + Ω f g B N N N, Al, n nθ nω + nθ Ωθ NΩ dq/ at esonance n, n θ A n B Ω Ω N n Al, n n Ω, θ n Ω + n Ω NΩ j θ θ Sum fo the same fequency is to be taken fist. A m N, (Flanagan, Hughes, Ruangsi, ), n θ j θ We ecently developed a method to evaluate the symmetic pat contibution. The next Soichio s talk 11

12 Impact of the esonance on the phase evolution d Δ λ ΔΩ d ΔΩ µ Ω Ω M Δ O µ t es ΔΩes O µ dδω If 0 fo Δλ Δλ c, d( Δ Δλc ) ΔΩ O Ω dδω µ O Ω( Δλ Δλc ) M ( 1 M Ω ): duation staying aound esonance ( M Ω) ( M µ ) Δ ϕ O es O((µ/Μ ) 0 ) : fequency shift caused by passing esonance : oveall phase eo due to esonance λ d ( λ Δλ ) Oscillation peiod is much shote than the adiation eaction time Δ c µ β Ω M ( Δλ Δλ ) If β stays negative, esonance may pesist fo a long time. 1 c

13 Conclusion Intoduction to the next talk. Soy fo containing nothing new. Adiabatic adiation eaction fo the Cate constant is as easy to compute as those fo enegy and angula momentum. leading ode second ode de obit + de obit df O ( µ ) O( µ ) ( geodesic ) + O ( µ ) O( µ ) Hence, the leading ode wavefom whose phase is coect at O(M/µ) has aleady been eady to compute. The obital evolution may coss esonance, which induces O((M/µ) 1/ ) coection to the phase. Fo the change ate of the Cate constant, we need to evaluate the symmetic pat in the esonance case. 13