Finding orbital eccentricity

Question

I have this problem: They give me, from a satellite that is in orbit in earth, a value for the period, and the closest height to earth surface, the ask me what the eccentricty of the orbit is. I have no idea how to do this. I've tried using Binet's equation, and the equation that comes for these movements ($p/r=1+e\cos(\theta-\theta_0)$) or the conservation of angular momentum to try to get some relations between quantities, but I can't get anything.

For example, the angular momentum is conserved, so: $$l=mr^2\dot \phi$$ I can get from here: $$\int_0^T\frac{l}{mr^2}dt=2\pi$$ Being T the period, but I don't know $r(t)$ Or the other way: $$T\frac{l}{m}=\int_0^{2\pi}r(\phi)d\phi$$ Now I now the function $r(\phi)=\frac{p}{1+e\cos\phi}$, where $p=\frac{l^2}{\mu k}$, being $l$ the angular momentum and $k$ the constant of the gravity potential: $U=-k/r$. I don't know how to integrate that if it's possible or how to use the known value of the closest point. Some help?

Answer 1

You're trying all of the right things, but the problem is actually much simpler---you just need to use better relations for this problem. In particular, think about kepler's law of orbital periods; and some of the general ellipse equations, like:

$r(\theta) = \frac{ a \, (1 \, - \,e^2) }{1 \, - \, e \cos \theta}$ (which is the equation you have, just expressed a little differently), and

$e = \frac{ r_{max} \, - \, r_{min} }{r_{max} \, + \, r_{min}} = \frac{r_{max} \, - \, r_{min} }{2a}$

Here, $a$ is the 'semi-major axis', $e$ the eccentricity, $\theta$ the angle of the orbit, and $r_{max}$ and $r_{min}$ are the maximum and minimum separations respectively (also referred to as 'apocenter' and 'pericenter').
Does that help?