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

L. Fernández-Jambrina* Matemática Aplicada, E.T.S.I. Navales, Universidad Politécnica de Madrid, Arco de la Victoria s/n, E-28040 Madrid, Spain

In this Comment we explain the discrepancies mentioned by the authors between their results and ours about the influence of the gravitational quadrupole moment in the perturbative calculation of corrections to the precession of the periastron of quasielliptical Keplerian equatorial orbits around a point mass. The discrepancy appears to be a consequence of two different calculations of the angular momentum of the orbits.

In [1] the authors make use of their static and axisym- written for U l/r along timelike geodesies in the metric solution of Einstein's vacuum equations with a finite spacetime: number of multipole moments [2-4] in a system of coor­ E2-f dinates adapted to multipole symmetry to derive a Binet U2 -2y Uz (4) equation for orbits in the equatorial plane and relate it to [f2(J - EA)2 the classical Keplerian problem. This allows them to write where E and /, respectively, energy and angular momen­ down the relativistic corrections to Newtonian elliptical tum per unit of mass, are the conserved quantities of orbits in terms of multipole moments of the source of the geodesic motion associated to the isometries of the space- gravitational field: time: d u M + u + 3Mu2 J2 hi/r-M. (i) E = f(i- A4>), J = fA(i -A&+J , (5) where u is the inverse of the radial coordinate, M is the and the dot means derivation with respect to proper time central mass, / is the angular momentum of the orbit, and along the geodesic. VpMM{u) is a generalized gravitational potential which This Binet equation arises from the normalization con­ encloses the perturbations due to the gravitational quadru­ dition of the velocity v = (i, r, 8, ) + (6) periastron in a revolution around the quasielliptical orbit: f and imposing the existence of conserved quantities in order (2) to remove the derivatives

2 and thereby eliminating the de­ where a is the semimajor axis of the unperturbed orbital pendence on proper time. ellipse of eccentricity e and, hence, of angular momentum In order to compare our results with [1], we take 2 A(r, 8) = 0 in order to consider only static spacetimes. / = ±Va(l - e )M and energy E = -M/2a. A dimen- 2 2 This equation is solved perturbatively [5] in powers of a sionless small parameter ^ = M /J has been introduced. In [5] a different approach was followed to a similar small dimensionless parameter e = M/J and a change of purpose. Starting with the general metric for a stationary variable if/ = cod), which allows us to get rid of secular axially symmetric vacuum spacetime, terms in the perturbation scheme. This frequence co is responsible for the precessed angle, ds2 = -f(dt - Add))2 A 2 32\ e2y(dr2 + r2dd2) + r2sin28d(f> 1-1 6e + + 15 E0 (7) (3) 77 CO (f whereas the energy is also expanded where t and are the coordinates associated with the isometries of the spacetime and the functions /, A, and y 2 E- 1 + E0e + -6-lQEo-^y, (8) depend only on the coordinates r and 8, a Binet equation is

in powers of e. The term E0 is related therefore to the leonardo.fernandez @upm.es; http://dcain.etsin.upm.es/ilfj .htm classical orbit. The discrepancy mentioned in [1] arises on identifying M Q the small parameters in both expansions by means of Veff(r) JL 2 r I = e2. The term 3M/a in (2) appears to be missing 2r in (7). and so their radius is a solution of

The explanation is simple and is due precisely to the 1 2 previous identification. The authors assign the classical Ma J a + 3Q = 0. 2 2 Keplerian values Ec = —M/2a, J = a(\ — e )M to the For J4 > 12MQ there are two circular orbits, but we are energy and momentum of the elliptical orbit. However, in interested in the exterior one, their calculations E and / are also the conserved quantities of geodesic motion, and hence / has the same meaning in 2 4 J + Vi - Í2MQ both notations, though the value J2 — a(\ — e2)M should 32 2 enclose the contribution of the quadrupolar moment. 2M M J ' Since we are interested in the first correction to the since it is the one that appears as a perturbation of the angular momentum, we perform just the classical classical orbit for small Q. calculation. The conserved quantities for these circular orbits, In classical mechanics the orbit of a particle moving under a central force of potential V(r) has two conserved M 32 JL -® J2 Ma + quantities, the angular momentum / and the energy per 2a2 a unit of mass E: are seen to have simple classical corrections due to the ¿2 2 2 presence of the quadrupole moment. J = r J ('-iM'-iM r r a counterterm appears that cancels out the last term in (2). but considering just orbits in the equatorial plane, it acts as Hence we have shown that the apparent discrepancy if it were central with V(r) = —M/r — Q/r3. between the formulas for the precession of the periastron For simplicity we consider circular orbits of radius of an equatorial orbit around a mass endowed with quad­ r = a. It is clear that they cannot be used for measuring rupole moment calculated in [1,5] is solved by including a precessed angles, but still they provide an easy computa­ first-order classical contribution to the Keplerian angular tion of (2) and (7). They are located at extrema of the momentum due to the gravitational quadrupole moment effective potential in [1].

[1] J. L. Hernández-Pastora and J. Ospino, Phys. Rev. D 82, [4] C. Hoenselaers and Z. Perjes, Classical Quantum Gravity 104001 (2010). 7, 1819 (1990). [2] R. Geroch, J. Math. Phys. (N.Y.) 11, 2580 (1970). [5] L. Fernández-Jambrina and C. Hoenselaers, J. Math. Phys. [3] R. O. Hansen, J. Math. Phys. (N.Y.) 15, 46 (1974). (N.Y.) 42, 839 (2001).