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# Solar Oblateness and the Gravitational Dipole Moment

Dicke attempted to relate the quadrupole moment of the sun as a partial cause of Mercury's node of 43" per century on the equatorial plane of the sun. He, and others, concluded that it contributes at a rate of 3".4 century. Realizing that observations are made in the ecliptic plane, it turned out to be only 0.27 per century that was inferior to the experimental error of 1" per century. So the matter was dropped.

If even a finite contribution could be found, it would have a resounding effect on the 43" per century found by GR. That would make it a fluke, and leave open the question of what the rate of precession is caused by.

Going beyond the orbital equation of an ellipse, one (everyone!) expands the potential in inverse powers of r, i. e.

V(r)= GM/r(1+ J_1(a/r)+ J_2(a/r) +....) (1)

where the J's are zonal coefficients, assumed to be decreasing in magnitude, and a is a characteristic distance of the orbit. And everyone eliminates odd order J's because a star is supposed to be symmetrical about its axis of rotation. So Dicke's analysis consisted in evaluating the quadrupole term J_2, assuming that J_1 is zero.

However, we are not only dealing with a central mass, but also one orbiting about it. In the old days before wave mechanics was born, Max Born wrote a book "The Mechanics of the Atom" in which he attempted to take into account the precession of an electron about the nucleus. For him, the J_1 term was not zero, and, in fact, it led to precession. So it would indicate that the dipole interaction is crucial, then one cannot merely transform the odd zonal harmonics out by moving the origin to the center of gravity of the star.

The orbital equation for the true anomaly (the angle measured from one of the foci) is

df/dr= h/{2E+2V(r)-h^2/r^2}^{1/2} (2)

or

df/dr= C^(1/2)/{-A +2B/r -C^2/r^2}^{1/2},

where E is the total energy, E<0 for a closed orbit (A>0), and h is the angular momentum per unit mass. If we retain only the lowest order term in (1) the integral of (2) is the |Keplerian orbit

r= (C/B)/{1+[1-AC/B^2]^{1/2} cos(f-f_0)}, (3)

where f_0 is a constant of integration. the term [1-AC/B^2]^{1/2}=e is the ellipticity.

Now, the key is to realize that if we retain, the dipole term, J_1, it will be comparable to the centrifugal term h^2/r^2, having the same power of r. Thus, if C is not only h^2, but

C= h^2-2GM a J_1

then the orbit will be modified to read

r = p/1+e cos[j(f-f_0)] (4)

where

j= {1- 2a/p J_1}^{1/2} (5)

with p=h^2/GM, the semi-latus rectum.

This was known to Newton that 1/r^3 terms in the force produced precessions of the node. It is this dipole interaction which causes the orbit to process in a form of a rosette--not the quadrupole term, J_2, which Dicke implicated.

And since it is lower order, it will be of a greater magnitude than the J_2 term. As a check, we can compare this to what Einstein found (and before him, Gerber (1895)),

j={1-6GM/c^2r}^{1/2}. (6)

Equating (5) and (6) gives

J_1=3(h/c)^2/ar. (7)

If we identify a with the classical radius, GM/c^2, called the (1/2) Schwarzschild radius,

J_1= 3h^2/GM/r,

the numerator being the gravitational analog of the first Bohr orbit. Alternatively, if we identify a with the first Bohr orbit, a=h^2/GM, then

J_1= (3/2)R_s/r (8)

where R_s is the Schwarzschild radius. All relativistic corrections are given to lowest order by J_1 given by (8).

This is not say that the relativistic correction (8) is the cause of the rate of precession of the node of Mercury. Rather, we should look for an effect due to the oblateness of the sun in the form of a dipole interaction. It is known that both the sun and Mercury possess magnetic dipole moments. And even if this were to give a finite contribution to the precession of Mercury, it would make the 43" per century obtained by GR as a pure anomaly in itself. In any event, the cancellation of the first order term in (1) was uncalled for since it leads to physical effects. This is irrespective of whether we are dealing with charges or neutral masses.