# Prove that orbits are conic sections [duplicate]

How does one prove that an orbit of a satellite around a planet of significantly greater mass than the satellite is a conic section (i.e. an ellipse, circle, hyperbola, or parabola)?

Also, there is also the case of a degenerate orbit, which we have for example when the satellite starts at rest with respect to the planet, and the satellite moves along a straight line towards the center of the planet.

By the way, I realize this may be an extremely tough question, because I have seen once before the derivation with vector calculus that bounded orbits are ellipses, and it is definitely quite involved. As such, you may wish to provide an outline of how might this be done (does one have to consider bounded and open orbits separately, for example?) and omit the math.

Also note that I assume the satellite is small compared to the planet because I don't want to add the complexity of a two body-system orbiting around the center of mass just yet!

## marked as duplicate by Qmechanic♦Jun 13 '16 at 19:09

First, you have to prove that the equations of motion of the satellite with mass $m$ relative to the planet with mass $M$ is given by $${\bf\ddot{r}} = -{\frac{\mu}{r^3}} {\bf r}$$ where ${\bf r}$ (with magnitude $r=\|{\bf r}\|$) is the position vector of the satellite relative to the center of mass of the planet, and where $\mu=G(M+m)\approx GM$ is the standard gravitational parameter ($389\,600.4418$ km$^3$s$^{-2}$ for Earth).
Then you solve the equations of motion to obtain the orbit equation $$r = \frac{h^2/\mu}{1+e\sin\nu}$$ where $h>0$ (a constant) is the magnitude of the specific angular momentum of the satellite relative to the planet, $e\geq 0$ is the eccentricity of the orbit, and $\nu$ is the true anomaly. The orbit equation describes a circle if $e=0$, an ellipse if $0\leq e<1$, a parabola if $e=1$, and a hyperbola if $e>1$. Circles and ellipses are bounded orbits while parabolas en hyperbolas are unbounded orbits. If the specific angular momentum is zero, we have a degenerate orbit which cannot be described by the orbit equation.