PostNewtonian dynamics for orbiting compact objects


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1 PostNewtonian dynamics for orbiting compact objects Achamveedu Gopakumar TheoretischPhysikalisches Institut FriedrichSchillerUniversität Jena Turku, Finland 10/08/2008
2 Aim & Outline AIM: To introduce NBody astrophysicists to postnewtonian (PN) approximation, useful to describe the orbiting dynamics of compact objects Outline PN approach to General Relativity; why it is interesting? & what can it provide.. Symbolic demonstration of a PNcomputation Solving PNaccurate dynamics: subtitles Going beyond PNaccurate dynamics & 3Body interactions
3 Motivations:I Coalescing blackhole binaries have become the focus of a large number of astrophysically & theoretically motivated analytic, semianalytic & numerical investigations Most promising sources of Gravitational Waves (GWs) for LIGO/VIRGO GW astronomy with LISA/ SKA will require GWs from Coalescing Massive Black holes. Recoiling black holes associated with binary black hole merger should have observational & cosmological consequences We need detailed analysis of the dynamics of compact binaries in GR
4 PN approximation:i In General Relativity (GR), zeroth order approximation gives Newtonian gravity npn order: corrections of order to the Newtonian gravity ( v c ) 2n ( Gm rc 2 m, v & r denote total mass, orbital velocity & separation ) n Black holes & neutron stars are modeled as point particles
5 PN approximation:ii In the case of nonspinning compact binaries, for LIGO/VIRGO applications, one needs to tackle two problems (usually analyzed separately) Problem of finding equations of motion ẍ Problem of computing gravitationalwave luminosity L, h,+ ẍ N 1PN 2PN 2.5PN 3PN 3.5PN 4PN 4.5PN 5PN 5.5PN 6PN L N 1PN 1.5PN 2PN 2.5PN 3PN 3.5PN h,+ N 0.5N 1PN 1.5PN 2PN 2.5PN 3PN
6 Motivations:II The response function of the laserinterferometric detector to gravitational waves from coalescing compact binary in circular orbits: h(t) L/L = C d ( φ(t) ) 2/3 sin ( 2φ(t) + α ), (1) d is the distance of the binary to the Earth; C and α are some constants, ( / ) φ(t) is the orbital phase of the binary & φ(t) dφ(t) dt. The secular orbital phase φ = φ(t) is computed from the balance equation de dt = L The expression for h(t) in terms of φ & φ requires h,+
7 Motivations:III GWs from ICBs are being searched by Matched Filtering theoretical template sets against the output of LIGO/VIRGO Theoretical templates should match the expected (& weak) inspiral signals to within a fraction of a GW cycle in the sensitive bandwidth This requires inclusion of higher order PN terms in the evolution of φ(t) appearing in h(t)
8 Motivations:IV The accumulated number of gravitationalwave cycles, N GW (f/ḟ)df, at a PN order in a LIGO/VIRGOtype detector Initial & final values of f are 10 Hz & 1/(6 3/2 πm) Hz It clearly shows 2PN is NOT sufficient Are we justified to use PN approximation when r/m 6? We can not treat ICBs as testparticle in a Sch. BH spacetime 2 1.4M 10M + 1.4M 2 10M Newtonian 16, First PN 439(104) 212(26) 59(14) Tail (1.5PN) Second PN 9(3) 10(2) 4(1)
9 PN quantities For comparable mass nonspinning compact binaries in circular orbits, following quantities are available to 3/3.5PN order they are sufficient to describe accurately the inspiral regime 3PN accurate dynamical (orbital) energy E(x) as a PN series in x = ( G m ω 3PN /c 3) 2/3 ω 3PN (t) the 3PN accurate orbital angular frequency Damour, Jaranowski & Schäfer (2001) 3.5PN accurate expression for GW energy luminosity L(x) Blanchet et.al (2002) & (2005) 3PN amplitude corrected expressions for h + (t) & h (t) in terms of the orbital phase φ(t) and x(t) Blanchet et.al (2008) Approximation techniques that describe inspiralling compact binaries usually require that (v/c) 2 ( G m ) [ slowmotion & weakfields ] c 2 r
10 Point particles Effacement property allows to reduce the problem of motion of centers of mass of N bodies to the problem of motion of N pointmasses [ Arguments due to T. Damour] Ellipticity due to tidal distortions arising from Gravitational interactions ǫ ( G M L ) R ( 3 G M L 2 ) ( ) 3 L (2) R Tidal quadrupole moments ǫ M L 2 & structure dependent interbody forces F St Gǫ M 2 L 2 /R 4 / F St G M 2 F St F N R 2 ǫ L 2 /R 2 (L/R) 5 (3) For compact objects, L G M/c 2 This F St F N ( G M c 2 R) 5 ; 5PN order...
11 How to get Newtonian dynamics from GR For a slowly moving testparticle in quasistationary & weak gravitational field g µ ν = η µ ν + h µ ν, h µ ν 1 (4) The geodesic equation d 2 x µ dτ 2 = Γ µ α β dxα dτ dx β dτ becomes d 2 x i dτ 2 = Γi α β dx α dτ dx β dτ (5) In our case dx0 dτ 1 & dxi dτ 0 d 2 x i dt 2 ( ) 1 = Γi h 0 0,i h 0 i,0 (6) d 2 x i dt h 0 0,i ẍ = φ (7) This => g 0 0 = (1 + 2 φ)
12 Quantities derivable from PNaccurate dynamics
13 PNaccurate compact binary dynamics:i By iterating Einstein s field equations,in principle, it is possible to compute 3PNaccurate Lagrangian L harmonic L[y A (t), v A (t), a A (t)],a=1,2&i=1,2,3) (8) instantaneous positions y i A (t) y A(t) coordinate velocities v i A (t) v A(t) = dy A /dt, coordinate accelerations a i A (t) a A(t) = dv A /dt. The explicit derivation of 1PNaccurate Lagrangian will be demonstrated later.. It provides a number of useful quantities.. We neglect the effects of radiation reaction..
14 PNaccurate compact binary dynamics:ii To 1PN accuracy 1 2 L = Gm 1m 2 + m 1v1 2 2r Gm 1m 2 r c 2 { G2 m 2 1 m 2 2r m 1v ( 1 4 (n 12v 1 )(n 12 v 2 ) v (v 1v 2 )) } (9) n 12 = (y 1 y 2 )/r 12, and the scalar products are written e.g. (n 12 v 2 ) = n 12.v 2. E = m 1v Gm 1m 2 r 12 { G 2 m1 2m 2 2r m 1v1 4 8 Gm 1m r 12 c 2 ( 1 4 (n 12v 1 )(n 12 v 2 ) v (v 1v 2 )) } (10)
15 PNaccurate compact binary dynamics:iii 1PNaccurate linear momentum { P i = m 1 v1 i + 1 c 2 n12 i Gm 1 m 2 (n 12 v 1 ) 2r 12 + v i 1 ( 1PNaccurate angular momentum { J i Gm 1m 2 + m 1v 2 ) } 1 (11) 2r 12 2 = ε ijk m 1 y j 1 v k c 2ε ijk y j 1 v k 2 7Gm 1 m 2 2r 12 + y j 1 y k 2 ( y j 1 v 1 k 3Gm1 m 2 + m 1v1 2 r 12 2 } Gm 1 m 2 2r12 2 (n 12 v 1 ) ) (12) 1PNaccurate COM integral G i = P i t + K i G i = m 1 y1 i + 1 ( {y i c 2 1 Gm 1m 2 + m 1v 2 ) } 1 (13) 2r 12 2
16 PNaccurate compact binary dynamics:iv In the COM frame K P 0, y1 i = η y i + η δ 2 c 2 y2 i = η y i + η δ 2 c 2 η = µ/m; δ = (m 1 m 2 )/m. v v 1 v 2 { v 2 G m r = G m r 2 { v 2 G m r } y i } y i (14) n + G m {( ) ( ) } c 2... n+... v r For general orbits, we know v to O(1/c 7 ) & conserved quantities to O(1/c 6 ) order More details in Andrade, Blanchet & Faye, grqc/ (15)
17 Usable 3.5PN v, E&J i ẍ for nonspinning comparable mass compact binaries to 3.5PN order in Eqs. (2.7),(2.8) & (2.9) in T. Mora & C. M. Will, Phys. Rev. D 69, (2004) a d 2 x dt 2 m = r 3 «1 + A n + B v A = A 1 (r, ṙ, φ, m, η) + A 2 (r, ṙ, φ, m, η) + A 2.5 (r, ṙ, φ, m, η) + A 3 (r, ṙ, φ, m, η) + A 3.5 (r, ṙ, φ, m, η) Similar expressions for B (16) 3PNaccurate conserved orbital energy & angular momentum are given by Eq. (2.11) & (2.12) E = E 0 + E 1 + E 2 + E 3 ( v 2 E 0 = µ 2 m ), r J = J 0 + J 1 + J 2 + J 3 = µ r v (17) J 0
18 h,+ :I GW polarizations h,+ are defined h + = 1 ) (p i p j q i q j hij TT, (18) 2 hij TT, the transversetraceless (TT) part of the radiation field h TT ij = 2G { c 4 r I (2) ij + 1 [ 1 c 3 n ai (3) ija + 4 ] TT 3 ε ab(ij (2) j)a n b +...} (19) A ij TT = Alm (P il P jm 1 2 Pij P lm ), where P ij = δ ij n i n j ; n i : unit vector from source to observer I ij ; PNaccurate mass quadrupole moment & J ij ; current quadrupole moment hij TT is expressible in terms of STF multipoles of source densities; usually computed via BlanchetDamourIyer formalism
19 h,+ :II h TT ij analogues to A j appearing in electromagnetism; A j = 1 c r d T j ; d T j P jk d k L em r 2 ( Ȧ j Ȧ j ) dω(n ) 1 c 3 d T j d T j (20) However, in GR, there are NO mass dipole & mass magnetic dipole radiations as total P i & J i are conserved d j = A m A ẍ A j = A p A j 0 (21) mass magnetic dipole ( ) µ i ε ijk x j A m A va k = A A J A j 0 (22) Therefore, the lowestorder radiation in GR is quadrupolar...
20 h,+ :III The radiate energy loss (ve of the GW luminosity) can be computed (ḣtt ) dω(n ) ( de dt ) FZ = c3 r 2 32πG kmḣtt = G { 1 c 5 5 I(3) ij I (3) ij + 1 [ 1 c I(4) ijk I(4) ijk J(3) ij J (3) ij } + O(c 3 ] (23) For circular inspiral, the above quantity is computed to 3.5PN order [ neglected terms are O(c 8 order in the above Eq. ] 3PN accurate orbital energy, 3.5PN accurate L & 3PN accurate expressions for h,+ are the crucial quantities to do astrophysics with eventual GW observations of inspiralling compact binaries
21 GW search templates
22 PN quantities Blanchet, Damour, Schäfer & their collaborators, after many years of computations, provided FOUR valuable expressions for compact binaries in PN accurate circular orbits 3PN accurate dynamical (orbital) energy E(x) as a PN series in x = ( G m ω 3PN /c 3) 2/3 ω 3PN (t) the 3PN accurate orbital angular frequency Damour, Jaranowski & Schäfer (2001) 3.5PN accurate expression for GW energy luminosity L(x) Blanchet et.al (2002) & (2005) 3PN amplitude corrected expressions for h + (t) & h (t) in t erms of the orbital phase φ(t) and x(t) Blanchet et.al (2008)
23 LAL Routines The LSC Algorithms Library (LAL) employs these inputs to construct various types of search templates TaylorT1 Damour, Iyer & Sathyaprakash (2001) ( ) 2/3 G m ω(t) h(t) cos 2 φ(t), (24) c 3 PNaccurate LAL templates require that inspiral is along exact circular orbits!!
24 To compute TaylorT1 3.5PN h(t), one needs 3.5PN accurate GW luminosity L(ω) & 3PN orbital energy E LAL Routines The LSC Algorithms Library (LAL) employs these inputs to construct various types of search templates TaylorT1 Damour, Iyer & Sathyaprakash (2001) ( ) 2/3 G m ω(t) h(t) cos 2 φ(t), (24) c 3 PNaccurate LAL templates require that inspiral is along exact circular orbits!! TaylorT1 h(t) dφ(t) dt = ω(t); d ω(t) dt / de = L(ω) dω, (25)
25 LAL Routines The LSC Algorithms Library (LAL) employs these inputs to construct various types of search templates TaylorT1 Damour, Iyer & Sathyaprakash (2001) ( ) 2/3 G m ω(t) h(t) cos 2 φ(t), (24) c 3 PNaccurate LAL templates require that inspiral is along exact circular orbits!! TaylorT4 3.5PN h(t):very close to NR inspiral h(t), but not in LAL dφ(t) dt ω(t); d ω(t) dt = 96 5 { ( G M ω ) 5/3 c ω O(ν) + O(ν 3/2 ) 3 }, (25) +O(ν 2 ) + O(ν 5/2 ) + O(ν 3 ) + O(ν 7/2 )
26 GW phase evolution: PN Vs NR 1 ΦPN ΦNumerical Et T1 T PN Order
27 TaylorEt h(t) The restricted 3.5PN accurate TaylorEt h(t) is given by h(ˆt) Ẽ(ˆt) cos 2 φ(ˆt)
28 TaylorEt h(t) The restricted 3.5PN accurate TaylorEt h(t) is given by h(ˆt) Ẽ(ˆt) cos 2 φ(ˆt) dφ dˆt dẽ dˆt = Ẽ 3/2 {1 + Ẽ = 64 5 η Ẽ5 {1 + Ẽ [ ] [ ]}.. + Ẽ 2.., (26) [ ] [ ]} Ẽ 7/2.., Ẽ/2 is the dimensionless nonrelativistic energy per unit reduced mass GW driven inspiral is along PNaccurate circular orbit...
29 Spin effects: a primer Kidder, PRD,bf 52, 821, 1995; Blanchet, Buonanno & Faye, grqc/ , grqc/ & recent papers from Jena
30 Including spin effects:i The dominant spin effect is that due to the relativistic spinorbit coupling Its contribution to reduced Hamiltonian H = H/µ H SO L S 1 c 2 r 3 = µ (r v) S 1 c 2 r 3 (27) This is formerly at 1PN order; BUT S 1 m co r co v spin & for compact objects r co G mco H c 2 SO at 2PN order, if v spin < c. However, if v spin = c, H SO contributions stand at 1.5PN order Relativistic spinorbit coupling provides corrections to ẍ at 2PN/1.5PN order order along with expressions for S 1, S 2, & L Its contributions to farzone GW luminosity is also available
31 Including spin effects:ii Recently, next to leading order corrections to ẍ, S 1, S 2, & L and farzone GW luminosity due to spinning pointparticles were obtained Black hole absorption occurs at this PN order for the first time... If both compact objects are spinning, spinspin interactions are important & its dominant contributions H SS S 1 S 2 + (n S 1 + n S 1 ) c 2 r 3 (28) S 1 G mco c 2 the dominant H SS appear at 2PN/3PN order.. Explicit contributions to ẍ, S 1, S 2, & L and farzone GW luminosity due to spinning pointparticles were obtained. Nexttoleading order contribution to H SS are conceptually & computationally difficult to compute.. ongoing efforts..
32 Recoil: a primer Blanchet, Qusailah& Will, astroph/ &...
33 GW induced recoil:i If gravitational radiation field created by a compact binary is asymmetric, GW induced recoil occurs m 1 m 2 &S 1 S 2 = 0; m 1 = m 2 ; S 1 &S 2 0 P ejected m 1 &m 2 0; S 1 &S 2 0 m 2 center of mass X v 1 m 1 Net recoil is due to GW induced damping v 2 P recoil To get lowest order asymmetric radiation field, we need to beat the mass quadrupole with mass octupole and current quadrupole moments. Figure from Wiseman s paper (1992) Higher order moments a tiny effect
34 r GW induced recoil:ii QNMs matching (BQW) F i P 8e05 6e05 4e05 2e v com t Various aspects of GW induced recoil during BBH coalescence based on EOB approach & for nonspinning BHs Maximum recoil during the merger phase Recoil estimates based on numerical relativity is consistent with these observations
35 v (km/s) Baker, et al Campanelli Damour and Gopakumar Herrmann, et al Sopuerta, et al Blanchet, et al η
36 GW induced recoil:iv In PN approximation, Linear momentum flux associated with the anisotropic emission of GWs FP i = G { 2 c 7 63 I(4) ijk I(3) jk + 16 } 45 ε ijk I (3) jl J (3) kl + O(c 2 ) J kl & I ijk are (1 4 η) 1/2 (29) To compute the GW induced recoil, one invokes momentum balance argument dp i = FP i dt (t), Pi (t) = t dt F P i (t) V i = P i / m 2 + P 2 V i P i /m (30) V i = 464 δm 105 η2 m x 4 n i For circular inspiral, FP i is known up to 2PN order... Numerical Relativity recoil estimates follows a similar procedure..
37 Symbolic demonstration : How to get 1PNaccurate L(y, v) Textbooks by MTW, N. Straumann, Will
38 Demo:I Aim is to solve R µ ν = T µ ν 1 2 g µ ν T ξ ξ χ ( ) G m 1/2 c 2 r v c We are only interested to χ 2 order & in a perturbative manner & let g 0 0 = 1 + (2) g 00 + (4) g 00, g 0 i = (3) g 0i, g ij = δ ij + (2) g ij (31) From g µ ξ g ν ξ = δ ν µ, we infer (2) g 00 = (2) g 00, (2) g ij = (2) g ij, (3) g 0i = (3) g 0i, (32) Compute Christoffel symbols & the components of the Ricci tensor R 0 0 = (2) R 00 + (4) R 00, R 0 i = (3) R 0i, R ij = (2) R ij (33) It is straightforward to obtain above quantities in terms of (n) g µ ν, their spatial & time derivatives
39 Demo:II In order to make RHS of (2) R 00, (4) R 00, (3) R 0i, & (2) R ij less complicated, we apply certain gaugecondition This leads to (2) R 00 (2) g 00, (2) R ij (2) g ij ( ) (4) R 00 (4) g 00, (2) g ij, (2) g 00,ij,... ( ) (3) R 0i (3) g 0i, (2) g ij,0j,... (34) Apply similar PN ansatz to S µ ν = T µ ν 1 2 g µ ν T ξ ξ ; T µ ν & g µ ν have similar PN expansions & this leads to (0) S 00 (2) T 00, (0) S ij δ (2) ij T 00, (2) S 00 (...), (1) S 0i (...) (35)
40 Demo:III (2) g 00 (2) T 00, (2) g ij δ (2) ij T 00, (36) (2) g 00 φ, (2) g ij δ ij φ, φ d 3 x (2) T 00 (t, x) x x (37) Using above expressions, somewhat complicated partial differential Eqs for (4) g 00 and (3) g 0i can simplified.. ( (4) g 00 φ 2 + ψ, ψ (2) T 00 + (2) T ii) (38) To solve the Eq for (3) g 0i, introduce two more potentials ξ i d 3 x (1) T 0i (t, x) x x Finally (3) g 0i ξ i + χ i0, χ d 3 x x x (2) T 00 (t, x) χ = φ (39)
41 Demo:IV The gauge condition introduced implies that, in general, T α β ;β 0 However, at this order, the potentials introduced do satisfy the above relation... To get 1PNaccurate Lagrangian, we compute the EOM of a particle in an external gravitational field, defined by (φ, ψ, χ, ξ i ) δ ( ) ( ) 2 dτ dτ dt dt = 0, dt = g µ ν dxµ dx ν dt dt ( ) 2 dτ = 1 v 2 (2) g 00 (4) g 00 2 (3) g 0i v i (2) g ij v i v j (40) dt PNaccurate Lagrangian L := 1 dτ dt & hence expressible in terms of (φ, ψ, χ, ξ i ), their spatial & temporal derivatives..
42 Demo:IV We are interested in pointparticles & therefore heuristically introduce T µν (x, t) = NX A=1 m A dy µ A dt dy ν A dt 1 dt g dτ δ(x y A(t)), (41) This makes the evaluation of potentials fast & easy For example, χ a m a x x a, & ξ i m a v i a a x x... a The Lagrangian L a of a particle a in the gravitational field of other particles & represented by (φ, ψ, χ, ξ i ) follows... The total Lagrangian should satisfy Limit ma 0 L m a 1 2 L = + = L a L = P a ma La Gm 1 m 2 + m 1 v ( 2r 12 2 c 2 G2 m 1 2 m 2 2r m 1 v4 1 8 Gm 1 m 2 1 (n 12 v 1 )(n 12 v 2 ) + 3 v1 2 7 (v 1 v 2 )«) (42) r
43 Solving PNaccurate orbital dynamics Papers from Jena during
44 PNaccurate solving:i Numerical solution of PNaccurate ({ ẍ} requires { } ) dx/dt v; dv/dt ẍ = A r, ṙ, φ x + r, ṙ, φ v Reactive contributions to ẍ is known only to relative 1PN order; but farzone fluxes are known to higher PN orders.. Numerical solution of ẍ need not be PNaccurate & this may lead to undesirable effects If ẍ is employed in an NBody code, isolating physical effects due to PNaccurate ẍ for isolated binaries from those due to manybody effects may become demanding.. It is desirable to have a semianalytic prescription to solve ẍ It is also required to construct h,+ (t) associated with compact binaries in inspiralling eccentric orbits..
45 PNaccurate Solving:II Let the relative acceleration of the compact binary be ẍ A = A 0 + A. A 0 is the conservative (integrable) part & A is the reactive perturbative part. The method first constructs the solution to the unperturbed system, whose dynamics is governed by A 0. The solution to the binary dynamics, governed by A, is obtained by varying the constants in the generic solutions of the unperturbed system.
46 PNaccurate Solving:III Recall that for nonspinning compact binaries, conservative part is available to 3PN order For 3PNaccurate dynamics, in the COM frame, there are 4 first integrals. The 2PN accurate energy and angular momentum of the binary, denoted by c 1 & c i 2 : c 1 = E(x 1, x 2, v 1, v 2 ) 2PN CM, c i 2 = J i (x 1, x 2, v 1, v 2 ) 2PN CM, The vectorial structure of c2 i, indicates that the unperturbed motion takes place in a plane. Close inspection of 3.5PN accurate A similar picture even when radiation reaction is present We can introduce polar coordinates in the plane of the orbit
47 PNaccurate Solving:IV The functional form for the solution to the unperturbed (3PN accurate) equations of motion r = S(l; c 1, c 2 ) ; ṙ = n S l (l; c 1, c 2 ), W φ = λ + W(l; c 1, c 2 ) ; φ = (1 + k)n + n (l; c 1, c 2 ), l The basic angles l and λ are given by l = n(t t 0 ) + c l, λ = (1 + k)n(t t 0 ) + c λ l, λ & S(l), W(l), W l (l) are periodic in l with a period of 2 π. The radial period n = 2 π/t r & periastron advance parameter k are gauge invariant functions of c 1 & c 2 = c i 2. t 0 is some initial instant and the constants c l & c λ, the corresponding values for l & λ.
48 PNaccurate Solving:V We construct the solution of the perturbed system, defined by A in the following way. We keep the same the functional form for r, ṙ, φ & φ, as functions of l & λ, but allow temporal variation in c 1 = c 1 (t) & c 2 = c 2 (t). Also, we have following definitions for l & λ l t t 0 n dt + c l (t) λ t t 0 (1 + k) n dt + c λ (t). Note evolving quantities c l (t), & c λ (t). The four variables {c 1, c 2, c l, c λ } replace the original four dynamical variables r, ṙ, φ & φ and {c α } satisfies first order evolution equations.
49 PNaccurate Solving:VI The explicit expressions for {dc α /dt} read dc 1 = c 1(x, v) dt v i A i, dc 2 = c 2(x, v) dt v j A j, ( ) 1 { dc l S S dc 1 = + S } dc 2, dt l c 1 dt c 2 dt dc λ = W dc l W dc 1 W dc 2. dt l dt c 1 dt c 2 dt The evolution of Eqs. for c l & c λ follow from the fact that we have same functional form for ṙ & φ in unperturbed & perturbed cases.
50 PNaccurate Solving:VII c α (l) = c α (l) + c α (l); c α (l) a slow secular drift & c α (l) periodic fast oscillations It can be demonstrated that to all PN orders dc λ dt 0 dc l dt Therefore, the reactive secular evolution of a PNaccurate eccentric orbit can be obtained with d c 1 dt & d c 2 dt While pursuing GW phasing for eccentric binaries, it is advisable to use ξ 2 E (µ c 2 ) and e t as appropriate variables to describe PNaccurate eccentric orbit Idea is to impose ξ(t) and e t (t) on 3PN accurate quasikeplerian representation of PNaccurate eccentric orbit that provides explicit PNaccurate semianalytic solution to conservative 3PNaccurate ẍ
51 GQKP:I 3PNaccurate conservative orbital dynamics of nonspinning compact binaries, either in Lagrangian or Hamiltonian approach, allows Keplerian type parametric solution: Keplerian parametric solution for Newtonianaccurate orbital motion r = r(cosϕ, sinϕ) R = a(1 e cos u), [( ) 1/2 1 + e φ φ 0 = v 2 arctan tan u ], (43) 1 e 2 l n(t t 0 ) = u e sin u, (44) The orbital elements a, e, n are orbital energy E, & angular momentum L
52 GQKP:II For 3PNaccurate orbital motion, the radial motion is also parametrized by r = a r (1 e r cos u) a r, e r 3PN accurate semimajor axis & radial eccentricity. These are expressible in terms of orbital energy E, angular momentum L and m 1 & m 2. u is the eccentric anomaly.
53 GQKP:III However, the 3PNaccurate angular motion is given by ( f4ϕ ϕ ϕ 0 = (1 + k) v + c 4 + f ) 6ϕ c 6 sin 2v + ( g4ϕ + i 6ϕ c 6 sin 4v + h 6ϕ sin 5v, c6 [ ( ) ] 1/2 1+eϕ where v = 2 arctan 1 e ϕ tan u 2 is the true anomaly. k measures the advance of the periastron & e ϕ is the angular eccentricity c 4 + g 6ϕ c 6 ) sin 3v f 4ϕ, f 6ϕ, g 4ϕ, g 6ϕ, i 6ϕ, and h 6ϕ are 2PN & 3PN order orbital functions expressible in terms of E, L, m 1 & m 2
54 GQKP:IV The 3PN accurate Kepler equation, which connects the eccentric anomaly to the coordinate time reads ( g4t l n (t t 0 ) = u e t sin u + c 4 + g ) 6t c 6 (v u) + ( f4t c 4 + f 6t c 6 ) sin v + i 6t c 6 sin 2v + h 6t sin 3v c6 l is the mean anomaly, n the mean motion & e t the time eccentricity g 4t, g 6t, f 4t, f 6t, i 6t & h 6t are 2PN & 3PN order orbital functions expressible in terms of E, L, m 1 & m 2 The most accurate & efficient way of solving PNaccurate KE is via adapting Mikkola s method
55 How to derive GQKP:I Obtain 3PN accurate expressions ṙ & r 2 φ in terms of 2 E, J, m, η ṙ = n H r 2 r φ ˆp = H ˆp Let s = 1/r & this leads to ds dt & dφ dt being 7 th degree polynomial in s 3PNaccurate ds dt admits two positive real roots, s + & s s + & s periastron & apastron Factorize these roots from ds dt & dφ ds leads to t t 0 = s s A 0 + A 1 s + A 2 s 2 + A 3 s 3 + A 4 s 4 + A 5 s 5 (s s)( s s + ) s 2 d s. (45) φ φ 0 = s s B 0 + B 1 s + B 2 s 2 + B 3 s 3 + B 4 s 4 + B 5 s 5 d s, (46) (s s)( s s + )
56 How to derive GQKP:II Expressions for t t 0 & φ φ 0 can be [ integrated using the ansatz ( ) ] 1/2 r = a r (1 e r cos u) &ṽ = 2 arctan 1+er 1 e r tan u 2 We also introduce e φ = e r [ ] &e t = e r [ ] to arrive at 3PNaccurate GQKP It follows that a r = 1 s + s +, e r = s s +. (47) 2 s s + s + s + The integrals for t t 0 & φ φ 0 when evaluated between the limits s + & s lead to the radial orbital period & the periastron advance rate These two are gaugeinvariant quantities if expressed in terms of E & J.
57 The? of chaos Numerical solution to PNaccurate spinning compact binary dynamics is to be pursued carefully It was argued by J. Levin & her coworkers that 2PN accurate spinning compact binary dynamics can be chaotic even when only one of the object spins The 2PNaccurate conservative dynamics only contains the leadingorder spinorbit interactions Therefore, the dimension of phasespace is 8( ) & hence degrees of freedom is 4 There are 4 conserved quantities E, L, S, L S It should be integrable & hence can not be chaotic!!
58 GQKP with spins:i A parametric solution to PN accurate orbital dynamics that leading order relativistic spinorbit interactions exists for two specific configurations i) m 1 m 2, S 1 0 or S 2 0 (Single spin case ) ii) S 1 and S 2 are arbitrary but m 1 = m 2. For these cases, we have Keplerian type parametrization that describes not only the precessional motion of the orbit inside the orbital plane, but also the precessional motions of the orbital plane and the spins themselves
59 GQKP with spins:ii k = L/L J = Je Z e Z N Line of sight Θ θ j Invariable plane Υ φ ϕ n = r/r e Y e X, p i
60 GQKP with spins:iii r(t) = r(t) cos ϕ(t) i(t) + r(t) sin ϕ(t) j(t), L(t) = Lk(t), S(t) = Je Z Lk(t), The time evolution of the basic vectors (i, j, k) are given by i(t) = cosυ(t)e X + sin Υ(t)e Y, j(t) = cosθ sinυ(t)e X + cosθcosυ(t)e Y + sin Θ e Z, k(t) = sin Θ sinυ(t)e X sin Θ cosυ(t)e Y + cosθe Z Θ, the precessional angle of L, = S sin α J J = (L 2 + S 2 + 2LS cosα) 1/2 α is the angle between L & S.
61 GQKP with spins:iv Time evolution for r, ϕ & Υ are given by r = a r (1 e r cos u), l n (t t 0 ) = u e t sin u + ( f4t + c 4 + f 6t c 6 ϕ ϕ 0 = (1 + k)v + ( g4t c 4 ) sin v + i 6t ( f4ϕ c 4 + f 6ϕ c 6 + g 6t c 6 ) (v u) ( g4ϕ + c 4 + g ) 6ϕ c 6 sin 3v + i 6ϕ Υ Υ 0 = χ soj c 2 (v + e sin v) L3 [ ( ) 1/2 1+eϕ The true anomaly v = 2 arctan 1 e ϕ tan u 2 c 6 sin 2v + h 6t sin 3v, c6 ) sin 2v The orbital elements & functions are expressible in terms of E, L, S, m 1, m 2 & α. c 6 sin 4v + h 6ϕ sin 5v, c6 ].
62 Solving PNaccurate dynamics: Summary It is worthwhile to pursue a semianalytic approach to solve PNaccurate orbital dynamics to substantiate purely Numerical results It also allows one to specify orbital elements to PNaccurate order & to make sure that numerical results are physically justifiable
63 Effective onebody approach Ref. T. Damour;
64 EOB approach:i Real twobody problem (two masses m 1, m 2 orbiting around each other) Effective onebody problem (one test particle of mass m 0 moving in some background metric gαβ effective ) The mapping is performed with th help of HamiltonJacobi formalism Ŝ = Ê t + j φ + dr R(r; Ê, j), (48) Demand that Ŝ, j & I r coincide for the real & effective descriptions E effective = f(e real ); f is determined in the process of matching.
65 EOB approach:ii For a testparticle m 2 in BH of mass m 1 E t = m 1 p 1 p 2 m 1, Solving for E t leads to E t = m µ m This manner one develops H imp (r, p r, j) E eff µ = (E t) 2 m 2 1 m2 2 2m 1 m 2. (49) E eff µ. (50) µ dˆr dˆt = dˆp r dˆt Ĥimp, dφ = Ĥimp ˆω partialˆp r dˆt ĵ = Ĥimp ˆr, dĵ dˆt = F j (51a) (51b) F j is the farzone angular momentum flux & one usually employs Padé approximant version of PNaccurate angular momentum flux for circular inspiral
66 Hierarchical triplets Ref. Ford, Kozinsky & Rasio; 2000
67 HT in PN approach?:i Discussions about Hierarchical triplets appear in dense stellar cluster simulations Hierarchical triplets, consisting of 3 objects (0,1,2), usually modeled to consist of an inner (0,1) & an outer binary (2,3) [ 3 stands for the COM of (1,2) binary] If the mutual inclination angle between the two orbital planes is large enough, inner binary may experience oscillations in its eccentricity (Kozai resonances) m 1 Φ r 1 m 0 r 2 m 2 It is due to the time averaged tidal force on the inner binary
68 HT in PN approach?:ii Periastron advance, appearing at 1PN order, should destroy the Kozai resonances.. The inclusion of the above effect is done in an adhoc manner Therefore, evolution of Hierarchical triplets using fully 1PNaccurate 3Body Hamiltonian should be of some interest! Is it worth pursuing it?
69 HT in PN approach?:iii For HT, H sum of two terms representing the two decoupled motions and an infinite series representing the coupling of the orbits. H m 0m 1 2a 1 + (m 0 + m 1 )m 2 2a a 2 ( ) j ( ) j+1 (a 1 /a 2 ) j r1 a2 P j (cosφ), a 1 r 2 j=2 (52) It is customary to introduce a set of canonical variables (l 1, l 2 ), (g 1, g 2 ), (h 1, h 2 ) and their conjugate momenta (L 1, L 2 ), (G 1, G 2 ), (H 1, H 2 ) L i a i With the help of a specific canonical transformation, it is possible to introduce new canonical coordinates & momenta such the the Hamiltonian is independent of (l 1, l 2 ) & hence a i are conserved..
70 HT in PN approach?:iv The resulting Hamiltonian is available to (a 1 /a 2 ) 3 order & depends only on (e 1, e 2, g 1, g 2 ) and i, the mutual inclination angle It is straightforward, but tedious, to obtain dynamical equations for de i /dt and dg i /dt To probe the effect of GR, it is customary to add 1PN corrections to dg 1 /dt and 2.5PN (Newtonian) radiation reaction contributions to de 1 /dt We need to assume that the inner binary is isolated It should be possible to derive de i /dt and dg i /dt at least 1PN order & at quadrupolar order [ corrections are O((a 1 /a 2 ) 2 ) ] It is doable with 1PNaccurate 3Body Hamiltonian derived by G. Schäfer in 1987! & spin effects can be, in principle, added!!
71 Conclusions PNaccurate dynamics involving compact objects should be useful to the practitioners of NBody astrophysics, especially while trying to model realistically scenarios leading to potential GW sources
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