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Added experimental values for precessions~~~~
*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 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.
:This created quite a problem—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. The primary resistance to this approach came from mathematicians unable to do the integration without an exponent of precisely 2, and they insisted, incorrectly, that was impossible for the exponent to be slightly different from 2. Due to this desire for mathematical elegance rather than objective observation-based science, Newcomb's approach was not pursued.
:This created quite a problem—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. The primary resistance to this approach came from mathematicians unable to do the integration without an exponent of precisely 2, and they insisted, incorrectly, that was impossible for the exponent to be slightly different from 2. Due to this desire for mathematical elegance rather than objective observation-based science, Newcomb's approach was not pursued. Furthermore as can be seen from the table below the measured values of the anomalous precessions of other planets agree well with the predictions of general relativity but poorly with those predicted by Newcomb and Hall.
: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.
!Anomalous precession, Newtonian with exponent of 2.000000157
!Anomalous precession, Newtonian with exponent of 2.000000157
!Anomalous precession, general relativity
!Anomalous precession, general relativity
!Measured anomalous precession (estimated uncertainty)
|-
|-
|Mercury
|Mercury
|43
|43
|43
|43
|43.5(5)
|-
|-
|Venus
|Venus
|16.6
|16.6
|9
|9
|8(5)
|-
|-
|Earth
|Earth
|10.3
|10.3
|4
|4
|5(1)
|-
|-
|Mars
|Mars
|5.5
|5.5
|1.4
|1.4
|
|-
|-
|Jupiter
|Jupiter
|0.87
|0.87
|0.07
|0.07
|
|-
|-
|Saturn
|Saturn
|0.35
|0.35
|0.014
|0.014
|
|-
|-
|Uranus
|Uranus
|0.12
|0.12
|0.002
|0.002
|
|-
|-
|Neptune
|Neptune
|0.063
|0.063
|0.0008
|0.0008
|
|}
|}