Mercury's Orbital Precession Advance of its Perihelion
"Science without religion is lame, religion without science is blind"  Science, Philosophy and Religion: a Symposium (1941), chap. 13  Albert Einstein ( 1879  1955 )
Since 2008 when Messenger spacecraft first passed Mercury and subsequently going into orbit in 2011, this colorized video by NASA is a product of 160,000 stitched mosaic images from its Wide Angle Camera ( WAC ) as well as other observations taken as part of the Mercury Messenger mission. source: NASA/Johns Hopkins University Applied Physics Laboratory/Carnegie Institution of Washington, published June 11, 2013.
Short History of the Problem
After having taken into account Newton's inverse  square Law of Universal Gravity for the various other gravitational perturbative effects due to the other 4 inner planets of venus, earth and mars including the effects of the sun, which takes its largest value in the case of mercury's orbit determined with observational certainty, only the last 43 arcsecs per century remainder in the residual precession rate in the longitude of mercury's perihelion, not explainable alone by Newtonian gravity, is adequately accounted for, well within the bounds of experimental error, by Einstein's Theory of General Relativity mathematics first presented to the Prussian Academy of Science in November 1915 and subsequently successfully applied to "Mercury's Problem" throughout the period of 19151916.
That is, only the last 43 arcsecs per century remainder ... is adequately accounted for ... by Einstein's Theory of General Relativity mathematics!!
This then became the first successful test for Einstein's General Relativity mathematical physics theory for gravity.
In fact
demonstrates that gravitational interactions occur at the speed of light!
This mysteriously anomalous, yet unexplained, residual precession rate in mercury's equatorial orbit plane had been voluminously documented in the observational data gathered of mercury's transit across the face of the sun first by U.J. Le Verrier, Ann. Obs. Paris 5 ( 1859 ) and later confirmed by S. Newcomb, US Navy Astronomical Observatory Papers, Washington ( 1882 & 1898 ); and its significant definitive explanation by general relativity mathematics heralded one of the most important early fundamental direct tests of spacetime ( gravitational metric ) theory.
U.J. ( Urbain Jean Joseph ) Le Verrier ( 1811  1877 ) French mathematician, astronomer of celestial mechanics Predictor of Neptune and its position by mathematical calculation alone by employing a laborious computational method to assess the effects of gravity, known as "gravitational perturbation theory", Le Verrier successfully argued in favor of Neptune's actual existence Taught astronomy at the École Polytechnique in 1837 Discoverer of mercury's perihelion advance in 1855 Appointed director of the Paris Observatory in 1857 Authored major 1859 paper on mercury's perihelion precession contradicting Newton's Law of Gravity 
Simon Newcomb ( born Nova Scotia, Canada, 1835  1909, buried Arlington National Cemetery ) Brilliant Canadian  American astronomer, mathematician Also writer on economics, statistics and science fiction novelist 
The recent confirming results of the Gravity Probe B ( GPB ) experiment for the "frame  dragging" general relativistic effect upon an orbiting gyroscope axis under earth's gravitational influence ( directly analogous to mercury and the sun where mercury's effect upon the metric of the surrounding spacetime fabric is negligible compared to the sun ), finally further demonstrated by observational experiment the general relativity mathematics for the residual precession of a gyroscope's axis by an overwhelming gravity field of force.
Solar and Heliospheric Observatory  Realtime Nov. 8th, 2006 Mercury transit taken by the SOHO spacecraft directly passing in front of the sun
source: NASA's SOHO Project
[ note: look for a tiny black speck going from left to right in the lower 1/3rd of the video and appearing about 1/2 way across ]
mercury transit, Hawaii, Nov. 8th, 2006
CaK timelapse movie with Coronado CaK 70mm
source:The summit of Haleakala, the Visitor Information Station (VIS) and summit of Mauna Kea.
http://www.ifa.hawaii.edu/hilo/MercTransit06.html
[ note: look for a tiny black speck beginning from about 2 o'clock and going down towards 7 o'clock ]
The Geometry of Elliptical Orbits
All conic sections arise from this quadratic equation in the Cartesian coordinate system with proper choices for the A, B, C, D, E, and F constants
producing these geometric configurations
Newton's theory of planetary motion states that a smaller planet's mass moving under the gravitational influence of a greater planet's mass will be one of the conic sections. In fact, only the total energy available for any given planetary orbit will determine the type of planetary conic section being formed and, indeed, all the planetary orbits are ellipses with the sun at one of the foci.
Also, but just theoretically, if a planetary body had just enough energy to escape its bonds to infinity but without enough energy to maintain a finite constant outward velocity, the trajectory at infinity would be parabolic; if, however, there were enough kinetic energy beyond its initial escape for an indefinite amount of constant velocity, the trajectory path would be hyperbolic. [ source: Relativity Science Calculator Glossary: Ellipse; Elliptic Paths ]
The Mathematical Physics Tools Employed
The Four Equations of Geodesic Motion
The Mathematical Computations
Full  on view of Mercury taken by Mariner 10 spacecraft in 1974, surveying only one hemisphere, was replaced by Messenger ( MErcury Surface, Space ENvironment GEochemistry, and Ranging ), launched Aug. 3, 2003 for a full insertion into Mercury orbit in 2011 after having gone past Earth once and Venus twice using these planets gravitational pull. source: NASA
Let's now simplify the physics by assuming the equatorial plane of the spherical coordinates containing the initial velocity vector and the initial position vector for the central mass of the of orbiting planet mercury around the sun, possesses spherical angle phi and hence as well as . Therefore we will only be "mathematically observing" the perihelic angle theta :
STEREO ( Solar TErrestrial RElations Observatory ) Captures Huge Solar Eruption with superimposed spherical coordinates and an imagined planet mercury traversing the equatorial plane. source: NASA's Solar Terrestrial Probes Program ( STP ), April 12  13, 2010
Thus spherical angle is held constant whereby planet mercury's orbit is permanently confined to the equatorial plane and henceforth any orbital precession of the perihelion for mercury can be mathematically compared to any actual experimental observations!
Hence, the above remaining geodesic equations of motion become
and the final reduction for the remaining geodesic equations of motion is as follows:
or,
Now integrating equations and , we get:
However not to forget the '  equation' of above, a simpler equation to derive is the Schwarzschild metric equation for gravity's force field by substituting for and as follows:
Newton's Orbital Physics vs. Einstein's Orbital Physics
Now in the non  relativistic, classical physics of Newton and Kepler, the planetary orbital linear equation
and without the non  linear, general relativistic perturbation term
has the straight  forward solution
In fact, it can be more explicitly shown by means of proofs for Kepler's 1st Law ( Planetary Law of Ellipses: Sun  centered model ) and Kepler's 2nd Law ( Equal Areas in Equal Times: Variable  speed model ) that
In the meantime, observe the following proof:
Perturbation Method: Final Derivation of Mercury's Perihelion Advance
JPL Calculated and Observed Arcsec Values of per Julian century of the Four Inner Planets  

Planet . 
semimajor axis^{[1]} . 
eccentricity^{[1]} . 
orbital period^{[1, 2]} .( earth years ) 
. ( Myles Standish^{[3, 4]}  JPL calculation, 2000 ) 
astronomical perihelia observation^{[5]} .( arcsecs per Julian century ) 
arcsecs per revolution  
Mercury  5.791 x 10^{10}m  0.20563069  0.240848533^{∗∗∗∗}  42.980^{"}± 0.001^{"}  42.9764^{"}± 0.0050^{"}  0.1035^{"[6]}  
Venus  10.821 x 10^{10}m  0.00677323  0.615194857  8.618^{"}± 0.041^{"}  9.148^{"}± 0.30^{"}  0.0530^{"}  
Earth  14.960 x 10^{10}m  0.01671022  1.000000000  3.846^{"}± 0.012^{"}  3.8458^{"}± 0.0004^{"}  0.0038^{"}  
Mars  22.792 x 10^{10}m  0.09341233  1.880870765  1.351^{"}± 0.001^{"}  1.3511^{"}± 0.0005^{"}  0.0254^{"} 
^{1} source: http://nssdc.gsfc.nasa.gov/planetary/factsheet/
^{2} Tropical orbit period was used in the alignment of earth's axis of rotation with that of the individual planet axis as it better takes account for the precession
of the equinoxes than does the Sidereal orbit period for our purposes
^{3} Principal Member of Technical Staff at JPL calculated the 4 inner planets from A.D. 1800 to A.D. 2200 by comparing Mercury's perhelia every 400 days
using JPL's Solar System Data Processing System ( SSDPS ) numerical integration program
^{4} "Advance of Mercury's Perihelion", by Edmund Bertschinger & Edwin F. Taylor, 2010
^{5} "Relativistic Effects and Solar Oblateness from Radar Observations of Planets and Spacecraft", Copyright © by E. V. Pitjeva, Institute of Applied
Astronomy, Russian Academy of Sciences, published in Astronomy Letters, Vol. 31, No. 5, 2005, pgs. 340  349.
See: Table 3  Secular motions of the planetary perihelia
^{6}
Calculating Mercury's Perihelion Advance
§ References:
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