Adiabatic radiation reaction to the orbits in kerr spacetime
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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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Adiabatic radiation reaction to the orbits in kerr spacetime

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


Talk plan

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


1 introduction

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


2 strategy

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.


Adiabatic radiation reaction to the orbits in kerr spacetime

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.


3 background

3. Background

  • Kerr geometry (Boyer-Lindquest coordinate)

  • Killing vector :

  • Killing tensor :

where


Constants of motion for geodesics

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:


Geodesic equations

Geodesic equations

  • New parametrization

(Mino ’03)

  • Equations of motion

r- andq - component

are decoupled.


R and q components of eom

r- and q - components of EOM

Assume that the radial andqmotion are bounded:

In this case, we can expand in the Fourier series.

where


T and f components of eom

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.


4 evaluation of de dt dl dt and dq dt

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)


Adiabatic radiation reaction to the orbits in kerr spacetime

: 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:


Simplified dq dt formula

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


Further reduction

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.


Consistency check

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.


5 easy case

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 !!


6 summary and future works

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.


Adiabatic radiation reaction to the orbits in kerr spacetime

: 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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