Trajectory in gravitational field

[GhostLoveScore]

So, I'm attempting to programm orbit in Unity game engine. So I need equation that shows r and phi depending on time.

Equation for gravitational potential is

U(r)=-k/r+M^2/(2*μ*r^2). Force is -1*derivation of U(r) by r. So I get lots of stupid stuff.

If anybody could help me to integrate this equation to get r(t) I would be grateful.

 

[ShayanJ]

It doesn't work that way. You should integrate the differential equations of motion numerically.

 

[GhostLoveScore]

Can you please explain, what differential equations? Is it m*d^2(x)/dt^2=gravitational force + centrifugal force?

 

[ShayanJ]

If you use polar coordinates $(\rho,\varphi)$, then the equations are:

$\rho^3\ddot \rho+GM \rho=R^4 \omega^2$
and
$\rho^2 \dot\varphi=R^2 \omega$

where $R$ is the initial distance from the centre of force and $\omega$ is the initial angular speed.

 

[GhostLoveScore]

Actually I'm using spherical coordinate system

 

[ShayanJ]

Actually I'm using spherical coordinate system

Well, there is absolutely no difference. Because the orbits of an inverse-square field lie on a plane and so the problem is two dimensional and spherical coordinates reduce to polar coordinates in two dimensions.

 

[vanhees71]

Don't forget that
$$M=\mu r^2 \dot{\varphi}=\text{const}.$$
Then you can use the energy-conservation Law
$$\frac{\mu}{2} \dot{r}^2 +U(r)=E=\text{const}$$
substitute
$$\dot{r}=\frac{\mathrm{d}r}{\mathrm{d} \varphi} \dot \varphi=\frac{\mathrm{d}r}{\mathrm{d} \varphi} \frac{M}{\mu r^2}.$$
With this you get a differential equation for the orbit in terms of polar coordinates $r=r(\varphi)$.

To integrate it, just substitute
$$s=\frac{1}{r}$$
into this differential equation.

You find the calculation in my mechanics FAQ (which is, however, in German, but with a lot of equations, so that you should be able to follow the arguments with the above given summary).

http://theory.gsi.de/~vanhees/faq/mech/node42.html

 

[GhostLoveScore]

Thanks guys, that will help.

vanhees71 - in the end I get r=r(phi), but I need r=r(t) and phi=phi(t).

From shyan's equation I get

d$\varphi$/dt=R^2*$\omega$/ρ^2. I don't know how to get dρ/dt

 

[ShayanJ]

The equations I gave are already proper for being integrated numerically. You don't need to change them. You only need to learn about some methods of numerically solving differential equations.

 

[GhostLoveScore]

OK, I will look into that. Few years ago we were solving it in Sage (python).