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Adiabatic radiation reaction to the orbits in Kerr Spacetime

Adiabatic radiation reaction to the orbits in Kerr Spacetime. Norichika Sago (Osaka). qr-qc/0506092 (accepted for publication in PTP) NS, Tanaka, Hikida, Nakano and preparing paper NS, Tanaka, Hikida, Nakano and Ganz. 8th Capra meeting, 11-14 July 2005 at the Coesner’s House, Abingdon, UK.

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Adiabatic radiation reaction to the orbits in Kerr Spacetime

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  1. Adiabatic radiation reactionto the orbits in Kerr Spacetime Norichika Sago (Osaka) qr-qc/0506092 (accepted for publication in PTP) NS, Tanaka, Hikida, Nakano and preparing paper NS, Tanaka, Hikida, Nakano and Ganz 8th Capra meeting, 11-14 July 2005 at the Coesner’s House, Abingdon, UK

  2. Talk Plan • Introduction • Strategy • Background • Calculation of dE/dt, dL/dt and dQ/dt • Easy case • Summary and Future works adiabatic approximation, radiative field geometry, constants of motion, geodesics evaluation of dE/dt, dL/dt, dQ/dt, consistency check slightly eccentric, small inclination case

  3. SMBH solar mass CO GW 1. Introduction We consider the motion of a particle orbiting the Kerr geometry. ] [ • test particle case A particle moves along a geodesic characterized byE, L, Q. In the next order, the particle no longer moves along the geodesic because of the self-force! Orbital evolution Evolution ofE,L,Q

  4. 2. Strategy • Adiabatic approximation T : orbital period : timescale of the orbital evolution At the lowest order, we assume that a particle moves along a geodesic. • Evolution of constants of motion Use the radiative field instead of the retarded field. Take infinite time average.

  5. Key point Radiative field is a homogeneous solution. It has no divergence at the location of the particle. We do not need the regularization!! Justification • ForEandL Gal’tsov, J.Phys. A 15, 3737 (1982) The results are consistent with the balance argument. • ForQ Mino, PRD 67,084027 (2003) This estimation gives the correct long time average.

  6. 3. Background • Kerr geometry (Boyer-Lindquest coordinate) • Killing vector : • Killing tensor : where

  7. Constants of motion for geodesics There are three constants of motion for Kerr geometry. • Particle’s orbit and 4-velocity : proper time • Constants of motion In addition, we define another notation for the Carter constant:

  8. Geodesic equations • New parametrization (Mino ’03) • Equations of motion r- andq - component are decoupled.

  9. r- and q - components of EOM Assume that the radial andqmotion are bounded: In this case, we can expand in the Fourier series. where

  10. t- and f - components of EOM • t - component of EOM divide into oscillation term and constant term linear term periodic functions with period, with period, We can obtainf - motion in a similar way.

  11. Here we define the above vector field as : mode function of metric perturbation 4. Evaluation of dE/dt, dL/dt and dQ/dt • Formula ofdE/dt (Gal’tsov ’82) We can replace –iwafter mode decomposition. where + (up-field term)

  12. : periodic function : periodic function + linear term Fourier transformation After mode decomposition andl-integration, we can obtain: In a similar manner, we can obtain dL/dt:

  13. Simplified dQ/dt formula • Formula ofdQ/dt Using the vector field, , we rewrite this formula. This term vanishes in this case. reduce it by integrating by parts

  14. Further reduction In general for a double-periodic function, Total derivative with respect tolr Using this relations, we can obtain: This expression is as easy to evaluate asdE/dtanddL/dt.

  15. Consistency check circular orbit case: This agrees with the circular orbit condition. (Kennefick and Ori, PRD 53, 4319 (1996)) equatorial orbit case: RewritingdQ/dtformula in terms ofC=Q-(aE-L)2 : An equatorial orbit does not leave the equatorial plane.

  16. 5. Easy case We consider a slightly eccentric orbit with small inclination. e << 1, y << 1 Inclinationy Eccentricitye In this case, we obtain the geodesic motion: If we truncate at the finite order of e and y, it is sufficient to calculate up to the finite Fourier modes. The calculation have not been done yet but is coming soon !!

  17. 6. Summary and Future works We considered a scheme to compute the evolution of constants of motion by using the adiabatic approximation method and the radiative field. We found that the scheme for the Carter constant can be dramatically simplified. We checked the consistency of our formula for circular orbits and equatorial orbits. As a demonstration, we are calculating dQ/dt for slightly eccentric orbit with small inclination, up to O(e2), O(y2), 1PN order from leading term.

  18. : orbital parameter Future works • (more) Accurate estimate ofdE/dt, dL/dtanddQ/dt Analytic evaluation for general orbits at higher PN order (Ganz, Tanaka, Hikida, Nakano and NS) Numerical evaluation Hughes, Drasco, Flanagan and Franklin (2005) Fujita, Tagoshi, Nakano and NS: based on Mano’s method • Evaluation of secular evolution of orbit Solve EOM for given constants of motion • Calculation of waveform Using Fujita-Tagoshi’s code Parameter estimation accuracy with LISA (Barack & Cutler (2004), Hikida et al. under consideration)

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