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[Phys-L] Re: Help on a problem from Goldstein



Not again! I just discovered an accidental sign error in one term
copied from my notes to a previous PHYS-L post concerning the exact
bound orbital apsidal precession of general relativity in a
Schwarzschild geometry. Earlier I had written:

It probably ought to be mentioned that not only can the orbital
precession be solved for exactly in the special relativistic version
of the problem, but the precession can *also* be solved for exactly
in the *general* relativistic version as well (I know this because I
was a bit surprised to be able to work it out when I tried my hand
at it). The exact result requires, however, the evaluation of an
elliptic integral function. If we use the previous parameter
definitions the *exact* *general* relativistic apsidal precession
per orbit is:

[delta-phi'] = 4*K(k)/sqrt(2*R*sin([theta]+ [pi]/3)/sqrt(3)) -2*[pi]

where K(k) is the complete elliptic integral of the first kind of
modulus k = sqrt(sin([theta])/sin([theta]+ [pi]/3)) and R is given
by R == sqrt(1 - 12*S^2) and [theta] is given by
[theta] == (1/3)*arccos((1 - S^2*(18 + 108*Q*(1 - Q/2)))/R^3) where
Q == E'/(m*c^2) and S == G*M/(h'*c).

If the above formula for [delta-phi'] is expanded to leading order
in 1/c^2 (for the case where the motion is quasi-Newtonian) the
result is:

[delta-phi'] = 6*[pi]*S^2

which is in agreement with the usual expression for the leading
order general relativistic apsidal precession per orbit.

There is a sign error in a term in the defining equation for the
quantity [theta] above. The proper defining equation for [theta] is

[theta] == (1/3)*arccos((1 - S^2*(18 + 108*Q*(1 + Q/2)))/R^3) .

Thus the Q/2 term above is supposed to be + Q/2 not - Q/2 in the
expression inside the arccos() function. Of course since Q is a
negative number for bound orbits (remember Q is a dimensionless
measure of the bound energy deficit relative to escape energy) a
value of +Q/2 is the same as -|Q|/2 in that expression. This
sign error has no effect on the leading order in 1/c^2
contribution, (i.e. [delta-phi'] = 6*[pi]*S^2) to the apsidal
precession however, and it only would have shown up for orbits
involving strong GR effects such as for a bound object orbiting a
neutron star just above its surface or for an object orbiting a
black hole whose orbital periapsis is a not a whole lot greater than
the hole's Schwarzschild radius.

In case anyone is interested, the exact expressions for the periapsis
value r_p and the apapsis value r_a are:

r_p = (3/2)*r_s/(1 - 2*R*cos([theta]+[pi]/3))


r_a = (3/2)*r_s/(1 - 2*R*cos([theta]-[pi]/3)) .

Here r_s == 2*M*G/c^2 is the Schwarzschild radius and the values of
r_p and r_a are given as Schwarzschild standard radial parameter
values.

David Bowman
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