Can a Quadrupole Moment Affect Elliptical Orbits?
It has always remained a curiousity of why Gerber's (1895) velocity-dependent potential gives the same numerical prediction as Einstein's (1916) relativity theory for the precession of orbital perihelia--to first order in the field strength.
In terms of the inverse radial coordinate, u=1/r, Gerber's equation to first order in the field reads
A(u) u" + u = 1/p (1)
where p is the semi-latus rectum and A is
A(u) = 1+6 m/c^2 u.
Only if r is set equal to the first Bohr radius, h^2/m, will the precession per orbit,
6 pi m/p c^2 = 24 pi^3 a^2/c^2 T^2 (1-e^2),
where a is the semi-major axis of the ellipse with eccentricity e, and T is the period which follows from the application of Kepler's III law, m=(2pi/T)^2a^3.
In contrast to (1), Einstein's modification of the equation of an ellipse is
u" + u = 1/p + B u^2, (2)
where B=3m/c^2. The seeminly diverse forms of (1) and (2) make it, indeed, surprising that both should result in the same expression for the precession of an elliptical orbit. But, it is not equation (2) that is used, but rather its linearized form
u" + (1-6m/c^2 u_0)u = 1/p. (3)
Like Gerber's equation, we must fix u_0, about which the equation (2) has been linearized. Setting u_0=m/h^2, the inverse of the first Bohr orbit gives identical results for the precession because (3) is equivalent to
A u" + u =A 1/p (4)
However, equation (4) makes an addition prediction that the semi-latus rectum will be shortened by an amount
p(1-6(m/hc)^2)= p - 6 (m/c^2) (5)
which is the same order as the magnitude of the precession per orbit. The correction to the semi-latus rectum on the lhs is the square of the gravitational fine-structure constant, which is equal to the square of the relative velocity, (v/c)^2, since v= m/h, the velocity of the first Bohr orbit.
This is indeed enligtening since whereas equation (2) invokes a quadrupole moment, the corresponding linearized equation (3) calls into action the dipole moment. Whereas the former is quadratic in the expansion of the potential, the latter is linear in the expansion. It is common to invoke the coincidence of the origin with the center of mass to eliminate this term leaving only the quadrupole as the next order correction. However, the quadrupole will not couple to a symmetric field of force, like that of Newton's inverse-square law. In such a field, all torques and forces will cancel out.
Thus, a quadrupole, such as that of the sun, should have no effect on the precession of Mercury's perihelion. One needs a dipole moment to create the precession of an elliptical orbit. It is only the linearization of (2) that reduces the quadrupole moment to a dipole moment. So dipole moments are coherent with conic sections, while quadruole, and higher order moments, with tori. Ellipses, hyperbolae, and parabola are obtained by slicing a conic section whereas, slicing a torus give Casini ovals. Whereas an ellipse is described as the locus of points where the arithmetic mean of the distances from the foci to a point on the ellipse remains constant, it is the geometric mean that remains constant for Cassini ovals [Seeing Gravity, p. 177]. An example would be three masses orbiting in a lemniscate--an inverted 8. Another example would be the Roche limit of stability which is the minimum distance that a body can be from a central mass and still be held to by gravitational forces only. Tidal forces come into play, being third order in the inverse radial coordinate, and the associated torques. The lowering of symmetry would be conducive to the creation of quadrupole moments.
Harmonic motion and the inverse-square law belong to the same class of dual laws [The Physics of Gravitation, pp. 43ff.] A quadrupole moment would indicate deviations from a spherical distribution of mass. And since an object with a spherical mass distribution looks just like a point mass gravitationally enabled Newton to derive his "shell" theorem attesting to the fact that a gravitational field outside a spherical shell is th same as if the entire mass were concentrated at the center of mass. This a priori excludes quadrupole forces which arise from non-spherical distribution of masses. Quadrupole moments, like moments of inertia are second order moments. Einstein used the the spinning rotor to derive the energy loss due to gravitational radiation. This was a classical derivation which had nothing whatsoever to do with his "general" theory of relativity, which was not relative at all.
In this context, we may question why plane-wave solutions of Maxwell's equation should be applicable to the plane-wave solutions of Einstein's vacuum field equations, since the putative source of the latter lies in the quadrupole interaction of masses? But, Einstein's vacuum field equations do not admit spherical waves for that would localize the source of the waves.