The Solar System

Astronomy

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Abstract

Background

This is our solar system.

Motion equations

According to Newtonian second law and Newton's law of gravitation, we can get the motion equations of planets in solar system:

$$\frac{dv_x}{dt}=-\frac{GM_Sx}{r^3}$$
$$\frac{dx}{dt}=v_x$$
$$\frac{dv_y}{dt}=-\frac{GM_Sy}{r^3}$$
$$\frac{dy}{dt}=v_y$$
In astronomical units, $GM_S=2\pi$
Initial conditions:
$x_0=a(1+e)$
$y_0=0$
$v_{x0}=0$
$v_{y0}=2\pi\sqrt{\frac{1-e}{a(1+e)}}$

Planets orbits

The orbits of 9 planets in solar system acquired by calculation:

The 4 terrestrial planets:

corresponding program code
solar system.py

Different force law

If the force law deviates from an inverse-square dependence,it is of the form

$$F_G=\frac{GM_SM_E}{r^\beta}$$
Take the example of Mercury, its semimajor axis$a=0.39AU$, eccentricity$e=0.206$.
when $\beta=3$,its trajectory is shown in the following figure:

It is obivious that its motion is unstable when $\beta=3$.

When $\beta=2$,$\beta=2.01$,$\beta=2.1$ and $\beta=2.5$,its orbits are shown in the following figure:

corresponding program code
4.2 non2.py

The precession of perihelion of Mercury

The force law predicted by general relativity is approximately

$$F_G=\frac{GM_SM_M}{r^2}(1+\frac{\alpha}{r^2})$$
$\alpha\approx1.1\times10^{-8}AU^2$
Because $\alpha$ is so small, in order to research this question and obtain precession angle, we use the method of linear fit.
Change the value of $\alpha$, we can get the following figure:

These green points represent aphelions
corresponding program code
Mercury.py
and here is its output data
Mercury.txt

Linear fit

Firstly,use the linear fit of precession angle $\theta$ and time.

They have very well linear dependence.
Its slope of the fit straight line is precession rate $\frac{d\theta}{dt}$.
Similarly,we can obtain different precession rates when $\alpha$ has different values:

Secondly,make use of these data

Its slope is $206.5rad/yr.AU^{-2}$
For $\alpha\approx1.1\times10^{-8}AU^2$,
The precession rate of Mercury estimated is
$1.1\times10^{-8}\times206.5rad/yr\times180\div\pi\times3600\div100\approx46.9''/century$
The result approaches 43 arcseconds/century which can't be explained.There may exists calculation errors.

The precession and eccentricity

If we increase or decrease eccentricity of Mercury orbit,
When $e=0.4$,

When $e=0.3$,

When $e=0.1$,

When $e=0$,

Even though when $e=0$,its orbit predicted by general relativity still has precession! This may be errors caused by approximation about initial conditions.
In short, precession rate of its orbit will be larger with larger eccentricity.

Conclusions

• Plotted the orbits of 9 planets in solar system.
• If the force law deviates from an inverse-square dependence, when $\beta=3$, it is unstable;when $\beta$ approaches 2, its orbit will precess.
• Calculated the precession rate of perihelion of Mercury predicted by general relativity, which approaches observed value.
• Precession rate of perihelion of Mercury predicted by general relativity will be larger with larger eccentricity of its orbit.