# Path of Orbital Bodies

I am trying to figure out how to parametrize the path of a body under the influence of gravity from another body, but I am stuck.

I have looked at the Wikipedia page on Kepler orbits, but it is rather hard to follow, and doesn't seem to ever actually give the final formula for the position WRT time. (It should somewhere near the end of the "Mathematical solution of the differential equation (1) above" section, right before the "Some additional formulae" section, right?)

I have a number of equations I've collected that describe the path of an orbital body, but I have been unable to figure out how to get $\overset{\rightharpoonup}{r}(t)$ directly in terms of $t$. I use$\theta(t)$ is to mean the polar angle of $\overset{\rightharpoonup}{r}(t)$, and $k$ to mean the whole "product of masses and gravitational constant" that keeps showing up everywhere ($k = m_1 \times m_2 \times g$).

$$\begin{matrix} \text{Newtonian Gravity:} && \frac{\mathrm{d}^2 \overset{\rightharpoonup}{r}(t)}{\mathrm{d}t^2} = \frac{\mathrm{d} \overset{\rightharpoonup}{v}(t)}{\mathrm{d}t} = \frac{k \cdot \hat{r}(t)}{ \overset{\rightharpoonup}{r}(t){}^2} \; \\ \text{Kepler's 1st Law (conics):} && \overset{\rightharpoonup}{r}(t)=\frac{\hat{r}(t) \cdot l}{1 + e\cos{\theta(t)}} \text{(} l \text{ and } e \text{ are scalar constants)} \\ \text{Kepler's 2nd Law (sweep):} && \int_{\theta(t_0)}^{\theta(t_0+\Delta t)} \overset{\rightharpoonup}{r}(t){}^2 \mathrm{d}\theta(t) = c \ \text{ constant WRT } \Delta t \\ \text{Potential Energy:} && \mathrm{d} \left \| \overset{\rightharpoonup}{v}(t) \right \| = \mathrm{d}\sqrt{\frac{k}{ \left \| \overset{\rightharpoonup}{r}(t) \right\|}} \text{(did I get this one right?)} \end{matrix}$$

I am somewhat lost here, could someone please tell me what I need to do, where to look online, etc, to find the information I need?

What is the correct parametrization for $\overset{\rightharpoonup}{r}(t)$?

## 2 Answers

There is no analytical solution for position in relation to time to date (I can not find a source to confirm this).

I asked a somewhat related question some time ago, in which I made an analytical solution for time as a function of the position, so the equation below. However this function does not seem to have an inverse, so which seems to confirm that there is no analytical solution for position in relation to time. $$t_{\theta_0,\theta_1}=\sqrt{\frac{a^3}{\mu}}\left[2\tan^{-1}\left(\frac{\sqrt{1-e^2}\tan{\frac{\theta}{2}}}{1+e}\right)-\frac{e\sqrt{1-e^2}\sin{\theta}}{1+e\cos{\theta}}\right]^{\theta_1}_{\theta_0}$$

• is there a formula like the one you wrote that allows one to compute the time between two poinst in a hyperbolic orbit? If I am correct, $a < 0$ for a hyperbolic orbit, so $\sqrt{\frac{a^3}{\mu}}$ is imaginary for a hyperbola... so is it only possible to do something like what you describe for elliptical orbits? – feralin Nov 11 '13 at 16:00
• No, this equation should still return correct values, due to the fact that $e$ will be bigger than 1 and therefore $\sqrt{1-e^2}$ will also be complex, which makes the final answer real again. – fibonatic Nov 11 '13 at 23:18

First, calculate the mean anomaly from the mean motion and the time.

Next, solve Kepler's equation to find the eccentric anomaly $\theta$.

Finally:

$\vec{r}(\theta) = a \left[\cos(\theta) , \ \ \sqrt{1-e^2} \sin(\theta) \right]$