I'm quite stuck with this problem.

I know that I have an object in orbit. I know the eccentricity of that orbit, as well as the semi-major axis of the orbit.

Giving a true anomaly, how do I find the speed and altitude of that object? The true anomaly is the angle between the line made with the focus of the ellipse and the position of the object.

Thank you!

P.S. I'm more looking for a general help, more than a specific answer. That's why I didn't give any numbers.

  • $\begingroup$ What is an anomaly? $\endgroup$
    – FrankH
    Oct 23, 2011 at 21:29
  • $\begingroup$ @FrankH Added a link and a little description of the true anomaly. $\endgroup$
    – gcamp
    Oct 23, 2011 at 21:37

1 Answer 1


As far as I can tell the true anomaly is the same type of angle used in the standard solution of the Kepler problem since there we assume the sun is at a foci. When solving the equations of motion for a Keplerian orbit we obtain $r\left(\theta\right)=\frac{a\left(1-e^{2}\right)}{1\pm e\cos\theta}$ (- if $r\left(0\right)$ is through the origin and + if it is away from the origin) and we can express the velocity as $v=\sqrt{\mu\left(\frac{2}{r}-\frac{1}{a}\right)}$ where $\mu$ is the standard gravitational parameter which in the two body case is just $G\left(m_{1}+m_{2}\right)$.

The derivation of these formulae can be found in many mechanics textbooks. For example, Taylor - Classical Mechanics - chapter 8.

  • $\begingroup$ How to find the velocity direction? $\endgroup$
    – Robotex
    Nov 8, 2021 at 8:27

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