puzzle: satellite clock crossover

[John Denker <jsd@MONMOUTH.COM>]

I wrote:
> > Note that for GPS satellites in orbit, the relativistic contributions
> > are very significant, necessary corrections.

Then at 11:33 PM 1/27/01 -0800, Leigh Palmer wrote:
> ...and guess which effect is dominant, the special relativistic time
> dilation due to the satellite's speed relative to clocks on Earth's
> surface, or the general relativistic time contraction which may be
> viewed as a gravitational red shift due to the gravitational
> potential difference between the positions of the satellite and the
> Earthbound clock.

Even though I like to think I'm a good guesser, and even though I know the
right answer to Leigh's question, I claim it would be rather hard to guess
this one.  I particular, if you leave out the reference to specific "GPS"
satellites, then no amount of guessing will give a reliable answer, for the
following reason:

Consider a satellite in such a low orbit that it just barely clears the
roof of your lab.  Now, relative to a roof-top clock in the lab frame, the
satellite clock sees no GR shift, but it sees a substantial SR shift.  Now
consider other satellites in successively higher orbits:  The GR shift gets
bigger and bigger, while the SR shift gets smaller and smaller since the
orbital velocity is going down.

The two contributions are of opposite sign, so you cannot even guess the
sign of the total effect without knowing something about the size of the orbit.

This led me to ask myself the following question:  at what orbital radius
does the SR effect just cancel the GR effect?  Express the answer in terms
of the relevant variables such as planetary radius, planetary density, et
cetera.  Make whatever simplifying assumptions seem reasonable.

Small hint:  When I did it, I made the following simplifying assumptions:
 -- spherical airless nonrotating planet
 -- planet density low enough so that nonlinear GR effects can be ignored,
    i.e. planet not on verge of collapsing into black hole

Hint / warning:  Rote application of textbook formulas probably won't lead
to the right answer.  You need to know only the tiniest amount about
relativity, but you need some "physics sense".  Remember, a great deal of
real-world physics involves making controlled approximations.  This includes
 -- knowing what can be approximated to lowest order,
    to keep things manageably simple,
 -- knowing what you had better not approximate just yet,
    to preserve enough accuracy, and
 -- knowing what textbook formulas already contain sneaky hidden
    assumptions that need to be undone before you can proceed.
    For example:  Is "g" a fundamental constant, or is it an
    assumption-infested approximation to some more general expression?