Ellipses and the Two-Body Problem

This is the third post on my summer research about three-body problems. Today we are discussing ellipses and the kepler problem.

This is my previous post about symplectic integrators if you have missed it.

Ellipses

Semimajor and Semiminor Axes

I have discussed the parametrization of ellipses here.

Recall that an ellipse can be written as:

\[\frac{X^2}{a^2}+\frac{Y^2}{b^2}=1\]

We can express:

\[b^2=a^2(1-e^2)\]

Where \(e\) is the eccentricity.

Radius

We can express the radius \(R\) (measured from center):

\[R= \frac{a \sqrt{1-e^2}}{[1-e^2 \cos^2{\theta_{\text{center}}}]^{\frac{1}{2}}}\]

Measured from the focus, it is expressed as:

\[r=\frac{a(1-e^2)}{1+e \cos{\theta_{\text{foci}}}}\]

or simply

\[r=\frac{p}{1+e \cos{\theta_{\text{foci}}}}\]

where \(p\) is the semi-latus rectum. It was introduced briefly in my other post.

Two-Body Problem

I have written a post on the Two-Body Problem Here. I will focus more about the results here.

Energy of the system

Consider:

\[\frac{1}{2}v^2=E+\frac{GM}{r}.\]

Now set \(E=- \alpha^2\).

Let \(a=\frac{GM}{2 \alpha^2}\), we obtain:

\[r \leq 2a.\]

Hence negative energy implies bounded orbit.

Angular Momentum and the Runge-Lenz Vector

As I discussed in this post angular momentum \(\mathbf{C}\) is conserved.

Now we also notice that:

\[\begin{aligned} \frac{\mathrm{d}}{\mathrm{d} t}\left(\mathbf{e}_r\right)=\frac{\mathrm{d}}{\mathrm{d} t}\left(\frac{\mathbf{r}}{r}\right) & =\frac{r \dot{\mathbf{r}}-\dot{\boldsymbol{r}} \mathbf{r}}{r^2}=\frac{r^2 \dot{\mathbf{r}}-r \dot{r} \mathbf{r}}{r^3} \\ & =\frac{(\mathbf{r} \cdot \mathbf{r}) \dot{\mathbf{r}}-(\mathbf{r} \cdot \dot{\mathbf{r}}) \mathbf{r}}{r^3}=\frac{\mathbf{r} \times(\dot{\mathbf{r}} \times \mathbf{r})}{r^3}=-\frac{\mathbf{r} \times \mathbf{C}}{r^3}=\frac{\mathbf{C} \times \mathbf{r}}{r^3} \end{aligned}\]

If \(\mathbf{C}=0\), then the two particles collide. If \(\mathbf{C} \neq 0\), we have \(r \cdot \mathbf{C}=0\), which implies \(\mathbf{C}\) is perpendicular to the motion of the particles.

\[\frac{\mathrm{d}}{\mathrm{d} t}\left(\mathbf{e}_r\right)=\frac{\mathbf{C} \times \mathbf{r}}{r^3}=-\frac{\mathbf{C} \times \ddot{\mathbf{r}}}{G M}=-\frac{1}{G M} \frac{\mathrm{d}}{\mathrm{d} t}(\mathbf{C} \times \dot{\mathbf{r}})=\frac{1}{G M} \frac{\mathrm{d}}{\mathrm{d} t}(\mathbf{v} \times \mathbf{C}) .\]

Integrating both sides,

\[\mu\left(\mathbf{e}_r+\mathbf{e}\right)=\mathbf{v} \times \mathbf{C}=\dot{\mathbf{r}} \times(\mathbf{r} \times \dot{\mathbf{r}})=(\dot{\mathbf{r}} \cdot \dot{\mathbf{r}}) \mathbf{r}-(\mathbf{r} \cdot \dot{\mathbf{r}}) \dot{\mathbf{r}}=v^2 \mathbf{r}-r \dot{r} \dot{\mathbf{r}} .\]

where \(\mathbf{e}\) is the constant vector upon integration.

Now as \(\mathbf{C}\) is normal to both \(\mathbf{r}\) and \(\mathbf{\dot{r}}\).

By considering \(\mathbf{e_r} \cdot \mathbf{C}=0\), we can easily see that \(\mathbf{e} \cdot \mathbf{C}=0\). The reader can show that this implies \(\mathbf{e}\) lies in the plane of the motion.

In fact \(\mathbf{e}\) is the Runge-Lenz vector, which is a constant of motion. Any reader having read my post of Noether’s Theorem can immediately see this implies a “hidden” conservation in the system. However in this case it is rather complicated: The reason behind this symmetry is that the two-body problem is mathematically equivalent to a particle moving freely on the surface of a four-dimensional (hyper-)sphere, so that the whole problem is symmetric under certain rotations of the four-dimensional space. This will be a focus in a later part of my research project.

Here contains more information about the Runge-Lenz Vector.

Kepler’s Laws

• First Law: Explained above.
• Second Law and Third Law: Explained in my other post.

Thoughts on the Project

This project is tougher than I thought. I expected 5-6 hours per day, and I ended up spending 8-9 (even 10!) hours per day. Part of it is because I want to work as hard as I can. Reading the literature is quite boring sometimes, but organising my thoughts about it really helps my understand the materials.

Credits

The whole post is based on the book Integrable Systems in Celestial Mechanics. The part about four-dimensional (hyper-)sphere is based on wikipedia.

I would also like to thank Dr Jenni Smillie for her guidance and support duing this project.