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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

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  • $\begingroup$ Is solving Newton's equation and finding that the solution is a polar conic not proof enough for you? $\endgroup$ – Ryan Unger Feb 12 '15 at 3:54
  • $\begingroup$ That is exactly what I am asking for. But you must consider both bounded and closed and degenerate orbits. If you can do this all in one shot I will be impressed. $\endgroup$ – Joshua Benabou Feb 12 '15 at 4:10
  • $\begingroup$ Every upper division and graduate text on classical mechanics has some version of these proofs, but they take a few pages and are moderately involved. $\endgroup$ – dmckee Feb 12 '15 at 4:35
  • $\begingroup$ BTW-after you make the canonical transformation to separation and reduced-mass there is no added complexity for solving the more general case. $\endgroup$ – dmckee Feb 12 '15 at 4:37
  • $\begingroup$ I thought it would be quite involved. Too be honest, I'm not interested in following a long proof, but I just want to know the method, and if you have to consider bounded and closed orbits separately. Also for future reference where can I find a full proof of this result online? $\endgroup$ – Joshua Benabou Feb 12 '15 at 5:02

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.


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