Changes

:This created quite a problem&mdash;physicists by then were accustomed to having their theories check out very accurately.  One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2.  He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>.  By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year.  Whatever value is chosen for <math>\delta\,</math>, it gives the same precession, per revolution, for all orbiting bodies, but gravitational effects from other planets diminish that effect the further the planet is from the sun.

:This created quite a problem&mdash;physicists by then were accustomed to having their theories check out very accurately.  One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2.  He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>.  By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year.  Whatever value is chosen for <math>\delta\,</math>, it gives the same precession, per revolution, for all orbiting bodies, but gravitational effects from other planets diminish that effect the further the planet is from the sun.

:A good approximation for the precession under general relativity is <math>\frac{3GM}{c^2 a(1-e^2)}</math> revolutions per planet's "year", where a is the semi-major axis and e is the eccentricity.  A simpler but less accurate one is <math>3{}v^2/c^2</math>, where <math>v</math> is the planet's average orbital speed.

:The approximation given at the end of Einstein's 1916 paper <ref>http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_GRelativity_1916.pdf "The Foundation of the General Theory of Relativity"</ref> is <math>24 \pi^3\frac{a^2}{T^2c^2 (1-e^2)}</math> revolutions per planet's "year", where a is the semi-major axis, T is the length of the planet's year, and e is the eccentricity.  A much simpler, but less accurate, approximation, designed to show how the precession relates to the planet's speed, is <math>3{}v^2/c^2</math> revolutions per planet's "year", where <math>v</math> is the planet's average orbital speed.  These are just approximations.  Getting an accurate value requires integrating the geodesic equation where Ricci's tensor is zero.

:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.

:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.