Problem 4.10 & 4.11

PRECESSION OF THE PERIHELION OF MERCURY

1. Abstract

4.10 Calculate the precession of the perihelion of Mercury, following the approach described in this section.
4.11 Investigate how the precession of the perihelion of a planet's orbit due to general relativity varies as a function of the eccentricity of the orbit. Study the precession of different elliptical orbits with different eccentricities, but with the same value you can compare it with the results shown in this section.

2. Background

• In celestial mechanics, apsidal precession or orbital precession is the precession (rotation) of the orbit of a celestial body. More precisely, it is the gradual rotation of the line joining the apsides of an orbit, which are the points of closest and farthest approach. Perihelion is the closest point to the Sun. The apsidal precession is the first derivative of the argument of periapsis, one of the six primary orbital elements of an orbit.
  
• Newton derived an early theorem which attempted to explain apsidal precession. This theorem is historically notable, but it was never widely used and it proposed forces which have been found not to exist, making the theorem invalid. This theorem of revolving orbits remained largely unknown and undeveloped for over three centuries until 1995. Newton proposed that variations in the angular motion of a particle can be accounted for by the addition of a force that varies as the inverse cube of distance, without affecting the radial motion of a particle. Using a forerunner of the Taylor series, Newton generalized his theorem to all force laws provided that the deviations from circular orbits are small, which is valid for most planets in the Solar System. However, his theorem did not account for the apsidal precession of the Moon without giving up the inverse-square law of Newton's law of universal gravitation. Additionally, the rate of apsidal precession calculated via Newton's theorem of revolving orbits is not as accurate as it is for newer methods such as by perturbation theory.
  
• An apsidal precession of the planet Mercury was noted by Urbain Le Verrier in the mid-19th century and accounted for by Einstein's general theory of relativity.Einstein showed that for a planet, the major semi-axis of its orbit being $\alpha$,the eccentricity of the orbit e and the period of revolution T, then the apsidal precession due to relativistic effects, during one period of revolution in radians, is
  $ϵ=24π3α2T 2 c 2 ( 1 − e 2 ) {\displaystyle \epsilon =24\pi ^{3}{\frac {\alpha ^{2}}{T^{2}c^{2}(1-e^{2})}}} \epsilon = 24 \pi^3 \frac{\alpha^2}{T^2 c^2 (1-e^2)}$
  where c is the speed of light. In the case of Mercury, half of the greater axis is circa $5.79×10^{10}$ m, the eccentricity of its orbit is 0.206 and the period of revolution 87.97 days or $7.6×10^{6}$ s. From these and the speed of light (which is ~$3×10^{8}$ m/s), it can be calculated that the apsidial precession during one period of revolution is $\epsilon = 5.028×10^{−7}$ radians ($2.88×10^{−5}$ degrees or 0.104 arcseconds). In one hundred years, Mercury makes approximately 415 revolutions around the Sun, and thus in that time, the apsidal perihelion due to relativistic effects is approximately 43 arcseconds, which corresponds almost exactly to the previously unexplained part of the measured value.
  

3. Main Body

3.1 Cauculation

According to Newton's law of gravitation the magnitude of this force is given by
$F_{G}=\frac{GM_{s}M_{E}}{r^{2}}$
where $M_{E}$ and $M_{S}$ are the masses of the Sun and Earth, r the distance between them. From Newton's second law of motion we have
$\frac{d^{2}x}{dt^{2}} = \frac{F_{G,x}}{M_{E}}$
$\frac{d^{2}y}{dt^{2}} = \frac{F_{G,y}}{M_{E}}$
where $F_{G,x} = -\frac{GM_{s}M_{E}}{r^{2}}cos\theta = -\frac{GM_{s}M_{E}x}{r^{3}}$
with a similar result for $F_{G,y}$ .
For circular motion we know that the force must be equal to $M_{e}v^{2}/r$, which leads to $\frac{M_{E}v^{2}}{r} = F_{G} = \frac{GM_{S}M_{E}}{r^{2}}$
where v is the velocity of Earth. Rearranging we find
$GM_{S} = v^{2}r = 4\pi ^{2} AU^{3}/yr^{2}$
where we have used the fact that the velocity of Earth is $2\pi r/(1yr) = 2\pi (AU/yr)$.

3.2 Thus we have the Pseudocode for subroutine planet

At each time step i calculate the position (x,y) and the velocity ($v_{x},v_{y}$)for time step i+1 using the Euler-Cromer method.
calculate the distance $r_{i}$from the sun: $r_{i} = (x^{2}+y^{2})^{1/2}$
Compute $v_{x,i+1}=v_{x,i} - \frac{4\pi ^{2}x_{i}}{r_{i}^{3}}\Delta t$and $v_{y,i+1}=v_{x,i} - \frac{4\pi ^{2}y_{i}}{r_{i}^{3}}\Delta t$
The Euler-Cromer step: calculate $x_{i+1}$ and $y_{i+1}$ using $v_{x,i+1}$ and $v_{y,i+1}$: $x_{i+1} = x_{i} + v_{x,i+1}\Delta t$, $y_{i+1} = y_{i} + v_{y,i+1}\Delta t$
Record the new position or plot it as it becomes available.
Repead for desired number of time steps.

3.3 Results

For e = 0.206:

For e =0.2:

For e =0.19:

For e=0.21:

For e=0.22:

5. Acknowledgement

• Prof.Cai
• Wikipedia