We have argued that a first order perturbation in the expansion of the potential in inverse powers of r causes a precession of the node of Mercury. Prior to this, Einstein and his followers have implicated the quadrupole moment as the cause of the precession. Danby has shown that insofar as the quadrupole is concerned, there is a small term in the solution of the orbital equation that "after a long enough time it is sure to have a perceptible effect."
How long of a time is not mentioned.
Rather, the first order effect has no such concerns, and, in addition, will cause a change in the semi-latus rectum from its unperturbed value, p=h^2/m, where h is the angular momentum per unit mass and m is the gravitational parameter of the sun, GM.
First consider Einstein's modified equation for u=1/r,
u" + u = 1/p + 3m/c^2 u^2 (1)
where the primes stand for differentiation with respect to the true anomaly. Now, the rhs of (1) is the force divided by h^2 u^2. If we expand the potential in inverse powers of r we get to lowest order
V= m/r(1+c_1 (a/r)+c_2 (a/r)^2+....) (2)
where the c_i are zonal harmonics, and a is a characteristic parameter. Taking the derivative of (2) with respect to r and dividing through by h^2 u^2 gives
c_2= h^2c^2 /a^2 (3)
on the condition that c_1 by assuming that the origin corresponds to the center of gravity.
So Einstein's correction in (1) is a quadrupole term. Through approximations and neglect of terms this leads to a precession of the ellipse given by
u= [1 + e cos k(f-f_0)]/p (4)
where e is the eccentricity and
k= [1-6(m/ch)^2]^{1/2}=1-3(m/ch)^2 (5)
The angular increase caused by the precession is
2 pi X 3(m/ch)^2 (6)
which is proportional to the square of the gravitational fine-structure constant.
Instead had be considered the first order term and neglected the second for reason of its smallness we would have gotten the orbital equation
u" + (1-2c_1 a/p )u =1/p (7)
On account of the smallness of the first order correction we can write (7) equivalently as
(1+2c_1 a/p)u" + u = 1/p(1+2c_1 a) (8)
It is clear from (8) that the true anomaly will be changed into f->[1-2c_1 a/p]^{1/2}f, and
the perihelion will become p->p[1-2c_1 a].
The second zonal coefficient in Einstein's case is
c_2 = (h/c)^2 /a^296
which resembles the term (C-A)/a^2 where C and A are the moments of inertia of a Maclaurin ellipsoid. Since the resulting shift (6) must be the same, this fixes the first zonal coefficient as
c_1= (3m/c^2)/a,
which has the form of a dipole moment with a moment given by (3/2) the Schwarzschild radius, 2m/c^2.
The key to distinguishing these two effects is a precise measurement of the semi-latus rectum of Mercury. If the quadrupole were operative, there would be no change from the value h^2/m, while if the dipole moment were operative there would be a shift of
6 p(m/hc)^2,
the order of the precession.