In our last blog, we argued that a dipole interaction gives rise to both a precession of the nodes of Mercury as well as a perturbation of the semi-latus rectum of the ellipse. This is reasonable since the semi-latus rectum is the radius of curvature of the osculating circles at the apsides of the orbit.
We are going to show that whereas GR gives the precession rate it is moot on the change in the semi-latus rectum. Whereas the derivation of the effects of the dipole interaction is clear-cut, that of the quadrupole requires a whole host of approximations which would make it equivalent to the dipole interaction were it not for the fact that it predicts no change in the semi-latus rectum.
The spinning of an otherwise spherical body about its axis makes it oblate. This results in a non-spherically symmetric gravitational field, and, consequently, the force is not exactly inverse-square. To see this, split a mass in two and place the parts at a and b on the x-axis. A mass, also located on the x-axis will feel an acceleration
F/m= -GM/2{ 1/(r+a)^2+1/(r-b)^2}.
Expanding in inverse powers of the radial coordinate, r, we get to lowest order
F/m=-GM/r^2{1-(a-b)/r+3(a^2+b^2)/r^2+....
The first term is the monopole, the second the dipole and the third the quadrupole. From Newton's theorem of revolving orbits we know that the dipole term will cause a precession of the elliptic orbit turning it into a rosette. It will vanish only where a=b: the masses are equidistantly spaced from the origin. In that case there results
d^2r/dt^2-h^2/r^2=-GM/r^2- 3GM J2/r^4
where J2 is an unknown, referred to as the second zonal harmonic. Transforming to u=1/r, results in
u" + u= Q + Pu^2,
where Q=GM/h^2 and P=3GM J^2/h^2. We are following Bacon's mathpages notes "Newtonian precession of Mercury's perihelion.
Viewing the last equation as a quadratic equation for u,
Pu^2 -u + (Q-u") =0,
only the smaller root is seen to make sense, which is
u=[1-{1-4P(Q-u")]^{1/2}]/2P.
Making the unjustified assumption that the second term under the square root is small (we don't know what u" is) he gets after rearrangement
(1+2PQ)u" +u = Q(1+PQ) + Pu"^2.
Assuming the last term to be small, and hence can be neglected, we obtain the solution
u=[1+e cos([{1-2PQ}^{1/2}(theta)/1/Q(1+PQ).
The true anomaly and the semi-latus rectum are modified, but in a disproportionate way.
If the dipole interaction were pertinent, the equation of the orbit would read
u" + (1-2PQ) u= Q
since the dipole interaction is inverse cubic just like the centripetal acceleration, h/r^3. To this Bacon remarks:
...if J2=h^2 (corresponding to a term equal to h^2/r^3 in the potential [sic, force] this gives the same precession as general relativity for a perfectly spherical gravitating body.
He is impervious to the fact that a term comparable to h^2/r^3 is a dipole interaction, and that the Einstein modification is the result of an assumed quadrupole moment, so the body cannot be spherical. The remark about J2 being a fixed characteristic of the gravitating body "so it is not possible to replicate the relativistic precessions of more than one planet by hypothesizing solar oblateness" is absurd since the semi-latus rectum p=h^2/GM connect the angular momentum and the angular momentum. Also the statement that "In order to duplicate the relativistic prediction it would be necessary to hypothesize a gravitational potential that is dependent on the angular velocity of the test particle, not just on its position." This clearly shows that he has absolutely no idea of how J1 and J2 come about through an expansion of the potential in inverse powers of r. There are no angular velocities involved!
A "cleaner" derivation of the precession is given in Danby, "Fundamentals of Celestial Mechanics. And it clearly shows that GR does NOT predict any modification of the semi-latus rectum. What he does is to plug the solution to the unperturbed equation
u=[1+e cos(theta)]/p
back into the perturbed equation
u" + u = 1/p + A u^2,
where A=3GM/c^2. He comes out with a complete solution
u=[1+e cos(theta)]/p +(A/p^2){(1+e^2/2 +e(theta)sin(theta)-(1/6)e^2 cos2(theta)}. (*)
Neglecting the first and third terms under the curly brackets he finds
u=[1+e cos(theta)]/p +Ae/p^2 (theta)X sin(theta). (**)
The latter he writes as a double angle formula on the strength that terms squared in (A/p) can be neglected. Thus the solution he finds is
u=(1/p){1+e cos[(1-A/p)(theta)].
However, if we differentiate (**) twice we obtain
u" +u =1/p +2Ae/p^2 cos(theta).
Using the solution (**) to eliminate the cosine term, gives
u" + (1-2A/p) u = (1/p)(1-2A/p) -2A^2e/p^3 (theta) sin(theta).
With the neglect of the last term in the above equation, we come out with
u"/[1-2A/p] +u = 1/p.
This clear shows a precession in the orbit of 2 pi 3A/p, by expanding the square root of 1-2A/p, but absolutely no change in the semi-latus rectum.
Thus, the determining factor that will distinguish dipole and quadrupole interaction is whether the semi-latus rectum is modified, or not, respectively.
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