Advance of Perihelion: Mercury's Formula & Approximations

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In summary, the formula for the advance of the perihelion of Mercury is derived from the Schwarzschild solution and involves the approximation that the mass of the sun is much smaller than the distance between Mercury and the sun. This approximation is made in order to drop an oscillating term from the result. The full derivation can be found in the textbook "The Mathematical Theory of Relativity" by Sir Arthur Eddington.
  • #1
NoobixCube
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The formula for the advance of the perihelion of Mercury is given as :
[tex]\Delta T = \frac{c^{2}a(1-\epsilon^{2})P}{3GM_{sun}}[/tex]
for the time taken to advance through [tex]2\pi[/tex]. I was wondering what approximations were made from the Schwarzschild solution to get to this result.
 
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  • #2
The full derivation takes a couple of pages and the only approximation is near the end, when a Newtonian value is substituted into the full EOM with the assumption that

[tex] \frac{M}{r} << 1[/tex]

An oscillating term is dropped from the result of this substitution. Please see any standard textbook of GR.
 
  • #3
"The Mathematical Theory of Relativity" written by Sir Arthur Eddington, in the 1920's and still available through Dover Books, devotes a chapter to the derivation of the "advance of periheliion".
 
  • #4
Thanks for the input guys
 

1. What is the advance of perihelion in relation to Mercury?

The advance of perihelion is a phenomenon observed in Mercury's orbit where the closest point to the sun (perihelion) shifts slightly with each revolution. This shift is caused by the gravitational pull of other planets, primarily Venus, and is an important factor in understanding Mercury's orbit and the theory of general relativity.

2. How is the advance of perihelion calculated?

The advance of perihelion can be calculated using a formula developed by the German astronomer Friedrich Bessel in the 19th century. It takes into account the mass of other planets, their distance from Mercury, and the eccentricity of Mercury's orbit. This formula has been refined over the years and is still used to calculate the advance of perihelion today.

3. What are the approximations used in the calculation of the advance of perihelion?

One of the main approximations used in the calculation of the advance of perihelion is the assumption that all other planets are in circular orbits around the sun. This simplifies the calculation and provides a good estimate, although it is not entirely accurate. Other approximations include neglecting the effects of non-gravitational forces and considering the orbits to be perfectly coplanar.

4. How accurate is the formula for the advance of perihelion?

The formula for the advance of perihelion is accurate to within a few seconds of arc per century. This means that over a period of 100 years, the predicted shift in Mercury's perihelion will only differ from the observed value by a few seconds of arc. This level of accuracy is impressive considering the complex nature of planetary orbits and the number of approximations used in the formula.

5. Why is the advance of perihelion important in the study of general relativity?

The advance of perihelion was one of the first pieces of evidence that supported Einstein's theory of general relativity. The observed shift in Mercury's perihelion could not be explained by classical Newtonian mechanics, but was accurately predicted by Einstein's theory. This provided strong evidence for the theory and helped to solidify its place in modern physics.

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