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 $\overset{\rightharpoonup}{r}(t)$?
No comments:
Post a Comment