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As the heading says: given an equation of an elliptical orbit, is it possible to find satellite´s speed at a certain point? I'm playing a bit with simulations of planets and satellites in the solar system, and I'd like to display satellite velocities at given pre-defined points on the orbit.

My equation is

$$ \frac{x^2}{a^2} + \frac{x^2}{b^2} = 0 $$

and I need to find the velocity at $[m, 0]$, $[m, n]$ and $[0, n]$ (where all these points $\epsilon$ orbit).

What I tried was to find $\frac{\textrm{dy}}{\textrm{dx}}$ which turned out to be

$$\frac{\textrm{dy}}{\textrm{dx}} = -\frac{x}{y}\frac{b^2}{a^2}$$

It looks good for $[m, n]$, but it doesn't give any meaningful answer for $[m, 0]$ and $[0, n]$.

Is my approach OK, or is there any other way to do this? One which came up to my mind was to convert the equation to polar form and use the fact that

$$ [m, 0] \equiv [m, \pi] $$ $$ [0, n] \equiv [n, \frac{\pi}{2}] $$

... but that seems to be too complicated.

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My equation is $$\frac{x^2}{a^2} + \frac{x^2}{b^2} = 0$$

That's not the equation you want for a satellite. That equation describes an ellipse with its center at the origin. You want an ellipse with the origin at one of the foci: $$r = \frac{a(1-e^2)}{1+e\cos\theta}$$ where

  • $r$ is the distance from the origin to a point on the ellipse.
  • $a$ is the semi-major axis length, half of the length of the ellipse's major axis.
  • $e$ is the eccentricity of the ellipse.
  • $\theta$ is the angle subtended between the line segments extending from the origin to the periapsis point (closest approach) and from the origin to the satellite.

While the above describes the path, it does not yet describe the speed. With a bit of work (not shown), differentiating $\vec r = r\hat r$ with respect to time yields $$\vec v \equiv \frac{d\vec r}{dt} = r\dot\theta\left(\frac {e\sin\theta}{1+e\cos\theta} \hat r + \hat \theta\right)$$ There are two problems with the above. One is that $\dot\theta$ is not constant. The other is that it doesn't say a thing about $\dot\theta$. You need some simple physics for that. What you need is Newton's law of gravitation, conservation of energy, and conservation of angular momentum. The result is, without derivation, the vis-viva equation: $$v^2 = G(m_1+m_2)\left(\frac 2 r - \frac 1 a\right)$$ where

  • $v$ is the magnitude of the velocity vector of one body with respect to the other,
  • $G$ is the Newtonian gravitational constant,
  • $m_1$ and $m_2$ are the masses of the two bodies,
  • $r$ is the distance between the centers of the two bodies, and
  • $a$ is the length of the semi-major axis of one body's orbit about the other.
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