## Note 404(4): Final Version of Note 404(2)

Agreed, the angular momentum of the satellite was used in the Newtonian approximation. Eq. (4) is OK because the angular part of the velocity squared is in the second term (Marion and Thornton chapter 7). The value of dr / dt is given by Eq. (30), which is the usual Newtonian expression. Then the spin connection and precession are calculated as in Note 404(2). The angular momentum is defined by Eq. (28), as in Marion and Thornton chapter seven, and the hamiltonian from Eq. (18), again as in Marion and Thornton chapter seven. So everything is standard Newton for the ellipse. The transformation (37) was used because wikipedia gives the orbital parameters with respect to the surface of the earth. They are needed with respect to the centre of the earth. The transformed alpha, b and a were used with the approximation (36).

In eq.(4) there should stand v^2 instead of (dr/dt)^2 but this has no relevance to the following.
The angular momentum (21) is that of the earth but I think it should be the orbital momentum of the satellite. This seems to be used in (26) ff.
How did you compute the values of dr/dt and Delta_phi? Did you compute a and b from (33) with the small value of alpha from (6) ? This does not give parameters of an ellipse. I changed

alpha –> alpha + r_E

and then computed a and b from (33) with this alpha. Inserting the results into (36) however gives

dr/dt=0.

Using (30) with r=a gives

dr/dt = 10 m/s.

Perhaps the perigee and apogee values should be used to computed a and b:

a = apogee + r[E]
b = perigee + r[E]

Horst

Am 04.04.2018 um 14:55 schrieb Myron Evans:

This derives dr / dt in the Newtonian approximation (30) and proceeds to the approximation (36). Eq. (37) is used to find the parameters with respect to the centre of the earth as required. The orbital parameters (6) to (10) given in a Wikipedia article seem to relate to the surface of the earth for some obscure reason. The result of this check for dr / dt is 751 metres per second. This compares with the initial guess made in Note 404(2) of 460 meters per second, the velocity of a point on the surface of the earth. Proceeding using eq. (40) it is found that the precession due to the rotation of the earth from the apsidal method is

delta phi (apsidal) = 0.86 ten power minus twelve radians per year

compared with the dubious experimental claim of

delta phi (Lense Thirring) = 1.02 ten power twelve radians per year

There is a typo in Eq (41) of the note, it should be ten power minus twelve. The key equation is the Newtonian Eq. (30) for the orbit of Gravity Probe B above the centre of the earth, as required by Newtonian dynamics, because the gravitational attraction of the earth regarded as a point mass M is at its centre, not its surface. So what is needed are a and b of Gravity Probe B with respect to the centre, and a point r such as the perihelion. The eccentricity of the Gravity Probe B orbit according to wikipedia is epsilon = 0.0014, so it is nearly circular. Therefore the apsidal method is an excellent approximation. This is an application of ECE2 gravitomagnetism.