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Re: puzzle: satellite clock crossover



From dimensional considerations alone the answer would appear to
be (a number of order unity)*R_e. I actually get (3/2) R_e from
an (overwhelmingly?) simpleminded approach. My answer at least
enjoys the twin virtues of being greater than R_e and less than
the orbital radius of GPS satellites.

John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm

On Mon, 29 Jan 2001, John Denker wrote:

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?


John Mallinckrodt mailto:ajm@csupomona.edu
Cal Poly Pomona http://www.csupomona.edu/~ajm