Comment on "From geodesics of the multipole solutions to the perturbed Kepler problem"

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by  L. Fernández-Jambrina

Comment on “From geodesics of the multipole solutions to the perturbed Kepler
problem”
L. Ferna´ndez-Jambrina1,a
1
Matema´tica Aplicada, E.T.S.I. Navales, Universidad Polit´ecnica de Madrid,
Arco de la Victoria s/n,
E-28040 Madrid, Spain
(Dated: January 5, 2011)
Inthiscommentweexplainthediscrepanciesmentionedbytheauthorsbetweentheirresultsand
oursabout theinfluenceof thegravitational quadrupolemoment in theperturbativecalculation of
correctionstotheprecessionoftheperiastronofquasiellipticalKeplerianequatorialorbitsarounda
pointmass. Thediscrepancyappears tobeconsequenceof twodifferentcalculations oftheangular
momentum of the orbits.
1
PACSnumbers: 95.10.Eg,04.80.Cc,04.20.-q,96.30.Dz
1
0
2 In[1]the authorsmakeuse oftheir static andaxisym- where t and φ are the coordinates associated with the

n metric solution of Einstein’s vacuum equations with a isometriesofthe spacetimeandthe functions f, Aandγ
a finite number of multipole moments [2–4] in a system of dependonlyonthecoordinatesrandθ,aBinetequation
J
coordinates adapted to multipole symmetry to derive a is written for U = 1/r along timelike geodesics in the

4 Binet equation for orbits in the equatorial plane and re- spacetime,

late it to the classical Keplerian problem. This allows
]
c them to write down the relativistic corrections to New- E2−f
-q ttohneiasonuerlcleipotfictahleogrbraitvsitiantitoenrmalsfioefldm,ultipole moments of Uφ2 =e−2γ(cid:26)f2(J −EA)2 −U2(cid:27), (4)
r
g whereE andJ,respectivelyenergyandangularmomen-
1 [ ddϕ2u2 +u= MJ2 +3Mu2− J12dduVpRMM(u), (1) tguemodepseicrmunotitionofamsoacsias,teadretotthheecisoonmseertvreiedsqoufathnteitsipeascoe-f
v where u is the inverse of the radial coordinate, M is time,
0
the central mass, J is the angular momentum of the or- r2
0 bit and VRMM(u) is a generalized gravitational poten- E =f(t˙−Aφ˙), J =fA(t˙−Aφ˙)+ φ˙, (5)
7 p f
tial which encloses the perturbations due to the grav-
0
. itational quadrupole moment Q after substracting the andthedotmeansderivationwithrespecttopropertime
1 Schwarzschild and centrifugal terms. We have taken the along the geodesic.
0
gravitationalconstantGandthemassoftheorbitingtest This Binet equation arises from the normalization
1 particle m equal to one. condition of the velocity, v = (t˙,r˙,θ˙,φ˙), of geodesics
1
: The authors obtain a result for the angle precessed by parametrised by proper time, v·v =−1,
v
i the periastron in a revolution around the quasielliptical e2γr˙2+r2φ˙2
X orbit, −1=−f(t˙−Aφ˙)2+ , (6)
f
r
a ∆φ=6π ζ+ζ2 Q −1 +3M , (2) andimposing the existenceofconservedquantities inor-
(cid:26) M3 (cid:18) 2 a (cid:19)(cid:27) der to remove the derivatives φ˙, t˙. Binet’s equation (4)
where a is the semi-major axis of the unperturbed or- is obtained dividing by φ˙2 and thereby eliminating the
bital ellipse of eccentricity e and, hence, of angular mo- dependence on proper time.
mentum J = ± a(1−e2)M and energy E = −M/2a. In order to compare our results with [1], we take
A dimensionlesspsmall parameter ζ = M2/J2 has been A(r,θ)=0 in order to consider only static spacetimes.
introduced. This equation is solved perturbatively [5] in powers of
In [5] a different approach was followed to a similar a small dimensionless parameter ǫ=M/J and a change
purpose. Startingwiththegeneralmetricforastationary of variable ψ =ωφ, which allows us to get rid of secular
axially symmetric vacuum spacetime, terms in the perturbation scheme. This frequence ω is
responsible for the precessed angle,
e2γ(dr2+r2dθ2)+r2sin2θdφ2
ds2 =−f(dt−Adφ)2+ , ∆φ 1 105 3Q
f = −1≃6ǫ2+ +15E − ǫ4, (7)
π ω (cid:18) 2 0 M3(cid:19)
(3)
whereas the energy is also expanded
E2
a [email protected];http://dcain.etsin.upm.es/ilfj.htm E ≃1+E0ǫ2+(cid:18)−6−10E0− 20(cid:19)ǫ4, (8)
2
in powers of ǫ. The term E is related therefore to the of the effective potential,
0
classical orbit.
The discrepancy mentioned in [1] arises on identifying J2 M Q
V (r)= − − ,
the small parameters in both expansions by means of eff 2r2 r r3
ζ = ǫ2. The term 3M/a in (2) appears to be missing in
(7). and so their radius is a solution of
The explanation is simple and is due precisely to the
Ma2−J2a+3Q=0.
previous identification. The authors assign the classical
Keplerian values E = −M/2a, J2 = a(1−e2)M to the
c c ForJ4 >12MQtherearetwocircularorbits,butweare
energy and momentum of the elliptical orbit. However,
interested in the exterior one,
intheircalculationsE andJ arealsotheconservedquan-
titiesofgeodesicmotionandhenceJ hasthesamemean-
inginbothnotations,thoughthe valueJ2 ≃a(1−e2)M a= J2+ J4−12MQ ≃ J2 − 3Q,
should enclose the contribution of the quadrupolar mo- p2M M J2
ment.
since it is the one that appears as a perturbation of the
Since we are interested in the first correction to the
classical orbit for small Q.
angular momentum, we perform just the classical calcu-
The conserved quantities por these circular orbits,
lation.
In classical mechanics the orbit of a particle moving
J2 M Q 3Q
underacentralforceofpotentialV(r)hastwoconserved E = − − , J2 =Ma+ ,
2a2 a a3 a
quantities, the angular momentum J and the energyper
are seen to have simple classical corrections due to the
unit of mass E,
presence of the quadrupole moment.
Infact, if in (2) we include the quadrupolarcorrection
J =r2φ˙, E = r˙2 + r2φ˙2 +V(r)= r˙2 + J2 +V(r), to the Keplerian angular momentum, J2 = Jc2 +3Q/a,
2 2 2 2r2 the lowest order, the Schwarzschildterm,
in spherical coordinates (r,θ,φ).
6πM2 6πM2 3Q 3Q
The gravitational potential due to a central mass M ≃ 1− =6π ζ− ζ2 ,
J2 J2 (cid:18) J2a(cid:19) (cid:18) M2a (cid:19)
and a quadrupole M is not central, c c
M Q(3cos2θ−1) a counterterm appears that cancels out the last term in
V(r,θ)=− + , (2).
r r3
Hence we have shown that the apparent discrepancy
butconsideringjustorbitsintheequatorialplane,itacts between the formulae for the precession of the perias-
as if it were central with V(r)=−M/r−Q/r3. tron of an equatorial orbit around a mass endowed with
For simplicity we consider circular orbits of radius quadrupole moment calculated in [1] and [5] is solved
r = a. It is clear that they cannot be used for mea- by including a first order classical contribution to the
suring precessed angles, but still they provide an easy Keplerian angular momentum due to the gravitational
computationof(2)and(7). Theyarelocatedatextrema quadrupole moment in [1].
[1] J.L. Hern´andez-Pastora, J. Ospino, Phys. Rev. D 82, [4] C. Hoenselaers, Z. Perjes, Class. Quantum Grav. 7 1819
104001 (2010). (1990)
[2] R. Geroch, Journ. Math. Phys. 11 2580 (1970). [5] L. Ferna´ndez-Jambrina, C. Hoenselaers, Journ. Math.
[3] R.O. Hansen,Journ. Math. Phys. 15 46 (1974). Phys. 42, 839 (2001).