Adiabatic evolution of resonant orbits on Kerr space time Kyoto - - PowerPoint PPT Presentation

Adiabatic evolution of resonant

• rbits on Kerr space time

Soichiro Isoyama

(Yukawa Institute for theoretical Physics)

Kyoto University

Extreme Mass Ratio Inspiral

A compact object inpirals into a massive black hole: Promising sources of gravity waves (Tests GR etc...) Required: understand the evolution of general orbits on a Kerr black hole

• 1. Three constants of motion

Kerr Compact object

Property of Kerr geodesics

• 2. Bi-periodic:

Carter constant

Inspirals in “resonance”

The two orbital frequencies are no longer independent

How to understand the radiation reaction effects ?

There is a special geodesic: resonant orbit

The dependence appears in the difference between the

• rigin of r and motion:

:Non resonance

Adiabatic approximation

(Orbital period) (Timescale of radiation reaction)

Oscillatory: not accumulate Accumulate for long time

Leading orbital evolution with radiation reaction: Long time averaged change: Expand in the Fourier series:

For resonant orbits:

Accumulate for long time Oscillatory

Resonant orbits evolve quite differently. The term accumulating for long time is different

Calls for better understandings of resonance

There :is only one common frequency

Evolution of consts. of motion

・Conservation laws with global Killing vectors Easy to compute via GWs flux. ・There is no “Carter constant GW flux” thus…

Needs Gravitational self-forces. (Singular at particle location)

Impossible to practical calculation for Kerr orbit, But…

Averaged value is still calculable in simple manner if the orbit is NOT in resonance

Asymptotic amplitudes of GWs. at the infinity and at the horizon

Our goal: Deduce the similar formula to compute

the long time averaged evolution of the Carter constant even when the orbit is in resonance.

Toy model: scalar casae

Charge (mass)

Gravitational case

Filed

(Linear perturbation)

Scalar model

Charge (scalar)

(Scalar field)

Filed

Simplify the discussion, use the scalar toy model as the first attempt.

Mode decomposition

Deduce the retarded solution via discrete mode sum ・Scalar field equation (separable ):

Amplitude depends on an orbit

A bound orbit only excites discretized frequencies.

Mode function

Evolution of the Carter constant

・ We return to the definition of the Carter constant

With geodesic equations

Note: the retarded field is singular at the particle location

Radiative and symmetric field

With the advanced field, decompose the retarded field into two pieces: radiative and symmetric field Singular structure at particle location is common both in retarded and advanced field.: The radiative field is regular everywhere

• 1. Radiation field part (Regular piece)

Radiation reaction folmula

The part from radiation field is essentially the same

expression as the non resonant case, if .

Some terms are modified fitting to the resonant case Calculable with asymptotic amplitude of scalar waves

Divergence in symmetric part

In resonant case, the contribution from symmetric field

exists. Diverges at particle location.

• 2. Symmetric field part (Singular piece)

Only makes sense in the resonance case.

Avoid divergence, introduced point splitting regularization

We can factorize the regularization terms. Rewrite symmetric part as the double Fourier series.

Mode sum Regularizations

Read out physical information, we also need to

subtract the singular portion from the symmetric field.

We can subtract mode by mode, which are regular The singular portion can be “smeared”

by inverse above Fourier transformation

Summary

・In the resonant case, the symmetric field also contributes the evolution of the Carter constant. ・We derive the formula for the long time averaged evolution of the Carter constant, applicable to a resonant orbit. ・Despite the divergence in the symmetric field, we can control it via mode sum regularization.

糸冬

お疲れ様でした。

Evolution of consts. of motion

・Conservation laws with global Killing vectors must always balance to GWs flux. ・For the Carter constant, there is no Killing vector…

Needs Gravitational self-forces. (Singular at particle location)

Impossible to practical calculation for Kerr orbit, But…

Retarded force and radiative force give the same long time average of the change of the Carter constant

Asymptotic amplitudes of GWs.

Numerically calculable formula [Drasco+(2005), Fujita+(2009)]

Radiation reaction formula

if and only if the orbit is off resonance.

Regular at particle location

since the orbit is mapped to itself: