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 reactionto the orbits in Kerr Spacetime
Norichika Sago (Osaka)
qr-qc/0506092 (accepted for publication in PTP)
NS, Tanaka, Hikida, Nakano
NS, Tanaka, Hikida, Nakano and Ganz
8th Capra meeting, 11-14 July 2005
at the Coesner’s House, Abingdon, UK
adiabatic approximation, radiative field
geometry, constants of motion, geodesics
evaluation of dE/dt, dL/dt, dQ/dt, consistency check
slightly eccentric, small inclination case
solar mass CO
We consider the motion of a particle orbiting the Kerr geometry.
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!
T : orbital period
: timescale of the orbital evolution
At the lowest order, we assume that a particle moves
along a geodesic.
Use the radiative field instead of the retarded field.
Take infinite time average.
Radiative field is a homogeneous solution.
It has no divergence at the location of the particle.
We do not need the regularization!!
Gal’tsov, J.Phys. A 15, 3737 (1982)
The results are consistent with the balance argument.
Mino, PRD 67,084027 (2003)
This estimation gives the correct long time average.
There are three constants of motion for Kerr geometry.
: proper time
In addition, we define another notation for the Carter constant:
r- andq - component
Assume that the radial andqmotion are bounded:
In this case, we can expand in the Fourier series.
divide into oscillation term and constant term
We can obtainf - motion in a similar way.
Here we define the above vector field as
: mode function of metric perturbation
We can replace –iwafter mode decomposition.
+ (up-field term)
: periodic function
: periodic function + linear term
After mode decomposition andl-integration, we can obtain:
In a similar manner, we can obtain dL/dt:
Using the vector field, , we rewrite this formula.
This term vanishes in this case.
reduce it by integrating by parts
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.
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.
We consider a slightly eccentric orbit with small inclination.
e << 1, y << 1
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 !!
We considered a scheme to compute the evolution of constants of
motion by using the adiabatic approximation method and the radiative
We found that the scheme for the Carter constant can be dramatically
We checked the consistency of our formula for circular orbits and
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.
: orbital parameter
Analytic evaluation for general orbits at higher PN order
(Ganz, Tanaka, Hikida, Nakano and NS)
Hughes, Drasco, Flanagan and Franklin (2005)
Fujita, Tagoshi, Nakano and NS: based on Mano’s method
Solve EOM for given constants of motion
Using Fujita-Tagoshi’s code
Parameter estimation accuracy with LISA
(Barack & Cutler (2004), Hikida et al. under consideration)