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===Precession of the orbit of Mercury===

===Precession of the orbit of Mercury===

The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury.  There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun.  These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century.  This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.

The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury.  There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun.  These are hard to calculate accurately, but, by 1860 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century.  This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.  This was based on observations going all the way back to about 1700.<ref>http://www.math.toronto.edu/~colliand/426_03/Papers03/C_Pollock.pdf</ref>

::One might wonder how astronomical observations accurate enough to measure such a tiny effect could have been made with the technology of that era.  A clever trick was used:  Solar transits of Mercury were observed and timed.  Some of these transits were timed with an accuracy of about 15 seconds.

: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.

: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.

: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 of the Schwartzschild metric, 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.