### Description:

The motions of the planets in the Solar system are simulated. Each planet is simulated as a two body problem, planet - Sun. The Newtonian gravitational acceleration is replaced by the post-Newtonian approximation of the acceleration predicted by The General Theory of Relativity [GR].

For each planet, the advance of the perihelion is measured to show what GR predicts the rate of perihelion advance would be if the only existing bodies in the Universe were the planet and the Sun.

You can run the simulation as an Application, see how here

### The post-Newtonian approximation of the gravitational acceleration predicted by GR

The gravitational acceleration predicted by GR is:

$\frac{\mathrm d \vec{v}}{\mathrm dt} = -\frac{GM}{r^2}\left(1 - 4\frac{GM}{rc^2}+\frac{v^2}{c^2} \right)\hat{r}+ \frac{4GMv^2\left( \hat{r}\cdot \hat{v}\right)}{r^2c^2}\cdot \hat{v}$

Where: $\vec{r}$ = distance vector sun - planet [m] $r$ = |$\vec{r}$|, magnitude of $\vec{r}$ [m] $\hat{r}$ = unity vector parallel to $\vec{r}$ . $\vec{v}$ = distance vector sun - planet [m/s] $v$ = |$\vec{v}$|, magnitude of $\vec{v}$ [m/s] $\hat{v}$ = unity vector parallel to $\vec{v}$ . $c$ = speed of light [m/s] $G$ = Gravitational constant $[m^3 \cdot kg^{-1} · s^{-2}]$ $M$ = solar mass [kg]