Question about the Gravity Probe B Schiff equation

jimholt

I've been reading about Gravity Probe B and the tests to verify the frame-dragging and geodetic precessions as described by the Schiff equation. (See e.g, here.) I am curious about what the equivalent equation might have looked like under pre-relativistic (Newtonian?) physics.

In order words, if the data from Gravity Probe B provide strong support for GTR, it must be that they are inconsistent with whatever would have been predicted without GTR. What does this alternative hypothesis look like? Are both components on the RHS of the Schiff equation simply equal to zero under Newtonian physics? If not, I really want to be able to compare the two equations side by side to see where the relativistic parts come in.