Generalized true- and eccentric-anomaly parametrizations for the perturbed Kepler motion

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by  László Á. Gergely

GENERALIZED TRUE- AND ECCENTRIC-ANOMALY
PARAMETRIZATIONS FOR THE PERTURBED KEPLER MOTION
LA´SZLO´ A´. GERGELY1,2, ZOLTA´NI. PERJE´S1 ANDMA´TYA´S VASU´TH1
1 KFKI Research Institute for Particle and Nuclear
Physics, Budapest 114, P.O.Box 49, H-1525 Hungary
7 2 Astronomical Observatory and Department of Experimental Physics,
0 University of Szeged, Szeged, D´om t´er 9, H-6720 Hungary
0
2 The true- and eccentric-anomaly parametrizations of the Kepler motion are gen-
eralized to quasiperiodic orbits by considering perturbations of the radial part of
n kineticenergyasaseriesinthenegativepowersoftheorbitalradius. Atoolboxof
a methods foraveraging observables interms of the energy E andangular momen-
J tumLisdeveloped. AbroadrangeofsystemsgovernedbythegenericBrumberg

7 force, as well as recent applications of the theory of gravitational radiation in-
2 volveintegralsoveraperiodofmotion. Theseintegralsareevaluatedbyusingthe
residuetheorem. Itisshownthatthepoleoftheintegrandislocatedintheorigin
1 andthatundercertaincircumstances anadditionalpoleemerges.
v
1
5 In our previous works 1,2 we have analyzed the motion of a compact binary
1 system with rotating components in order to provide realistic signal templates for
1 theinterferometricgravitational-waveobservatories. Wehavecalculatedthecumu-
0
lative radiation reaction effects on the motion of the spinning binary. In obtaining
7
secular changes one has to evaluate integrals of type
0
/
c T ω
q dt , (1)
r2+n
- Z0
r
g where T is the radialperiod, n aninteger andω a function with no manifest radial
v: dependence. For example T is determined by n = −2, while averaging instanta-
i neous radiative losses of quantities characterizing the orbit of a binary system one
X
calculates this type of integrals with n = 1,...,7. During the computations it is
r
convenient to use the generalized true- and eccentric-anomaly parameters defined
a
as follows.
The radial equation of a perturbed Kepler problem is considered as
2E 2m L2 p ϕ
r˙2 = + − + i , (2)
µ r µ2r2 µ2ri
i=0
X
wherethesmallcoefficientsϕ coupletheperturbationstothe system. The conser-
i
vationofenergyE andmagnitudeoftheorbitalangularmomentumLareassumed
to hold from symmetry considerations. A wide class of perturbed Kepler motions
3
accomplishtheseconditions,forinstancesystemsdescribedbytheBrumberg force
and the dynamics of spinning binaries.
The generalized true- and eccentric-anomaly parametrizations are introduced
by the relations
dr dr
=−γr2 , =−κ , (3)
d(cosχ) d(cosξ)
MG9: submitted to World Scientific on February 28, 2001 1
where γ and κ are positive constants. The scale of the parameters is adjusted
to the turning points r(0) = r , r(π) = r , which are solutions of the radial
min max
equation r˙ =0. To firstorder one has p−2 unphysicalroots (even if real, they are
muchsmallerthantheradiusr)andr andr ,whichdifferfromtheKeplerian
min max
turning points in linear terms in ϕ . With the above properties the familiar form
i
of the parametrizations is obtained :
2 1+cosχ 1−cosχ 1+cosξ 1−cosξ
= + , 2r= + . (4)
r r r r−1 r−1
min max min max
The most remarkable properties of these parametrizations are summarized in
the following theorems:
Theorem 1: The integral r−(2+n)dt , n ≥ 0 is given by the residue located
in the origin of the plane of the complex true-anomaly parameter z =eiχ.
Theorem 2: The integral R r−(2+n)dt with n < 0 is given by the sum of the
residuesintheoriginofthecomplexparameterplanew =eiξ and[forperturbative
R
terms with powers i≥2−n of r in the radial equation (2)] at
1/2
mµ2− −2µEL2
w = .
1
mµ2+p−2µEL2!
Theabovestatementsareprovedbyppassingtothe true-andeccentric-anomaly
parameters in the integral (1) and examining this expression in linear order of the
perturbations ϕ . In doing so one can specify the possible ω functions which will
i
not destroy the advantageous property of the integrand. It turns out, that ω can
be any polynomial in sine and cosine of the parameters. In particular, it can have
the form
ω =ω +ϕ(ψ,r˙) , (5)
0
whereω isconstantandtheperturbingtermϕ(ψ,r˙)isapolynomialinsinψ,cosψ
0
andr˙. Tofirstorderthe expressionsofthe azimuthalangleψ andr˙ intermsofthe
true- and eccentric-anomaly parameters are needed to Keplerian order.
The two types of generalized Keplerian parametrizations studied here are ca-
pable of describing a wide variety of systems (2). They do not give a complete
parametrization of the motion, but are suitable for handling radial effects. When
using them and the residue theorem the evaluation of various integrals (1) turns
out to be a simple task. They have been tested in computing the secular radia-
tive changes of a compact binary system1,2. In our view the solution of the radial
motion presentedhere is at least a viable candidate amongstother approaches4,5.
References
1. L.Gergely, Z.Perj´es and M.Vasu´th, Phys. Rev. D 58, 124001 (1998).
2. L.A´.Gergely, Phys. Rev. D 61, 024035 (2000).
3. Brumberg,V.A.: Essential Relativistic Celestial Mechanics, Bristol: Adam
Hilger, 1991.
4. R. Rieth and G. Scha¨fer, Class. Quantum Grav. 14, 2357 (1997).
5. Gopakumar,A. and Iyer,B.R., Phys. Rev. D 56, 7708 (1997).
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