Problem 4 – Hint 3

Here are a few useful facts and tricks related to the Runge-Lenz vector.

As mentioned, it is a sum of two vectors, and one of these has a constant modulus and is pointing from the centre of attraction towards the orbiting body. The ratio of the magnitudes of the Runge-Lenz vector and of the term of constant modulus equals to the ellipticity of the orbit. Often, the R-L vector is normalized so that it is equal to the ellipticity by modulus, in which case it is also referred to as the ellipticity vector.

The other component of the R-L vector is proportional to the cross product of the angular momentum and the velocity of the orbiting body. Thus, if we divide the R-L vector with a suitably chosen constant, that term becomes equal to the cross product of a-perpendicular-to-the-orbital-plane unit vector with the velocity vector. Hence, this cross product of the velocity vector can be expressed as a sum of a re-scaled ellipticity vector, and a vector of constant magnitude.

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