# How to compute the perihelion precession of Earth's orbit?

I want to simulate the perihelion precession of Earth. I tried searching other posts in stack exchange as well relating this question but may be it's my fault that I couldn't understand their answers. I read the history about mathematicians and physicists trying to find a good theory accounting the precession. I read that there are many planetary equation by Newton,Laplace,Hill,Poincare etc. But I am not sure which one gives the more accurate result. I am not even sure if I going on a right track searching the correct theory. I'd really appreciate if anyone could help me with the theory which gives fast and more accurate result?

• This page details the calculation required to model the precession of the orbits of the planets. The section following discusses the relativistic correction for Mercury which is a small correction. Apr 25, 2018 at 21:42
• This question is rather confused. The link points to the wrong place, the title of the question asks about the Earth's perihelion precession, and the body asks about the contributions of the Sun and Moon the this precession. Those two bodies contribute almost nothing to the Earth's perihelion precession. On the other hand, they are the primary drivers of the Earth's axial precession. Axial precession and perihelion precession are completely different beasts. Until the question is clarified, I am voting to close as "unclear what you are asking". Apr 26, 2018 at 1:11
• I don't know about the link, but I do believe that all the sources lead to exactly the same thing; there might be some discussion about correction which is negligible, especially for simulation. So, make sure that you understand these topics: Keplerian motion, conservation, virial theorem, open/close orbit. The most direct way for the simulation is to construct time grids and update parameters in the equations of those metioned topics. The updated parameters include the radial distance, and the total velocity, and the constants include the total angular momentum and total energy. Apr 26, 2018 at 10:54
• First of all,I am trying to understand the theory and mathematical part. I have problem with it.
– Rima
Apr 26, 2018 at 10:57

For simulation only, you should be able to show it by simple code as following: $$\vec{a}_t = - \frac{GM}{r_t^2} \hat{r} \\ \vec{v}_{t+1} = \vec{v}_t + \vec{a}_t \Delta t \\ \vec{r}_{t+1} = \vec{r}_t + \vec{v}_t \Delta t$$ where $t$ is time step, $\vec{a}$ is acceleration, $\vec{v}$ is velocity, $\vec{r}$ is position, $\hat{r}$ is a unit vector in radial direction from the origin (set the Sun at the origin), $G$ is gravitational constant, $M$ is Sun mass in this case, and $\Delta t$ is size of time step. You will have to play around with the initial values and step size to get certain orbit representing the case. To make it more real, you derive the initial values from the real parameters, if known.

• I want to simulate it and as well as plot the graph to show the real plot. I am not sure which equation to use. There are many post newtonian theories from laplace,lagrange etc. I want to show a real approximation solution. I hope you understand what I mean.
– Rima
Apr 26, 2018 at 16:55
• I found an equation of force predicted by GR and it gives best approximation for Mercury and I am not sure if I can apply that for other planets or moon.
– Rima
Apr 26, 2018 at 17:01
• My experience has been that a first-order numerical approach like this one is numerically unstable; you have to have more cleverness than this to get closed orbits. And in a purely Newtonian approach like this one, once you ironed out the numerical details, you'd get zero precession of perihelion; the precession of the perihelion in Newtonian mechanics comes from interactions with other orbiting bodies.
– rob
Apr 26, 2018 at 20:43
• @rob So do something like Verlet instead of Euler integration, include the other major planets, and make sure the timestep is small enough that total energy is conserved. Wouldn't the simulation reproduce some classical perturbation then? Apr 29, 2018 at 17:14
• @MikeG Yes. You might expand that comment into a useful answer.
– rob
Apr 29, 2018 at 18:39