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(Editing because I left out the actual question part. Oops!)

Given a situation like a comet going round the sun in an elliptical orbit, what would parametric equations for the comet's path look like? It'd be extra rad if it was expressed in terms of polar coordinates with the sun(one of the foci) at the origin. Even just pointers on a better way to approach the problem would be cool.

Here's how I approached it but it didn't get me to a solution:

The acceleration at any point is $$\begin{eqnarray*} F &=& \frac{Gm_{sun}m_{comet}}{r^2} \\ r &=& \|\vec{x}\|\\ \vec{a} &=& -\frac{F}{m_{comet}}\frac{\vec{x}}{\|\vec{x}\|}\\ &=& \frac{Gm_{sun}}{\|\vec{x}\|^2}\frac{\vec{x}}{\|\vec{x}\|} \\ &=& \frac{Gm_{sun}\vec{x}}{\|\vec{x}\|^3}\\ \end{eqnarray*}$$ and of course acceleration is the second derivative of the position, so $$\begin{eqnarray*} \vec{x} &=& \iint \frac{Gm_{sun}\vec{x}}{\|\vec{x}\|^3} dt\,dt\\ &=& Gm_{sun}\iint \frac{\vec{x}}{\|\vec{x}\|^3} dt\,dt \end{eqnarray*}$$ but I have no idea what do with integrating a norm like that, so breaking the vector down: $$\begin{eqnarray*} \vec{x} &=& \left \langle x,y \right \rangle \\ \vec{a} &=& \left \langle x'',y'' \right \rangle \\ \|\vec{x}\| &=& \left ( x^2+y^2 \right )^\frac{1}{2} \\ \|\vec{x}\|^3 &=& \left ( x^2+y^2 \right )^\frac{3}{2} \\ \frac{1}{\|\vec{x}\|^3} &=& \left ( x^2+y^2 \right )^{-\frac{3}{2}} \\ \left \langle x'',y'' \right \rangle&=& Gm_{sun} \left \langle \frac{x}{\|\vec{x}\|^3}, \frac{y}{\|\vec{x}\|^3} \right \rangle\\ \left \langle x'',y'' \right \rangle&=& Gm_{sun} \left \langle x\left ( x^2+y^2 \right )^{-\frac{3}{2}}, y\left ( x^2+y^2 \right )^{-\frac{3}{2}} \right \rangle \end{eqnarray*}$$ which gives me a system of differential equations I could write like $$\begin{eqnarray*} {\partial^2\over\partial t^2} x(t) &=& Gm_{sun}x(t)\left ( x(t)^2+y(t)^2 \right )^{-\frac{3}{2}} \\ {\partial^2\over\partial t^2} y(t) &=& Gm_{sun}y(t)\left ( x(t)^2+y(t)^2 \right )^{-\frac{3}{2}} \end{eqnarray*}$$ And I really don't know what to do with differential equations like those except take guesses at the form x(t) and y(t) take. Kepler's laws of planetary motion and the fact that it is known to be an ellipse and might be able to add useful constraints, but so far I haven't been able to really make progress using them myself.

Anyway, I'm sure two-body gravity problems like this are extremely well-understood by now, so I'm sure somebody worked this out already if I could just find the right words to type into Google :)

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Ah ha! It turns out this problem is indeed well studied! By none other than Johannes Kepler himself! Seems to explained pretty well here:

Between that and I think I've learned that there simply isn't any closed-form solution to this, which explains why I'm having so much trouble finding one :)

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