Max and min orbital speed

An earlier post needed to calculate how much the speed of a planet varies in orbit. The planet moves fastest as perihelion, the point in its orbit closes to the sun, and it moves slowest at aphelion, when it is furthest from the sun.

The ratio of the maximum to minimum speed turns out to be a simple expression in terms of the eccentricity e of the orbit:

(1 + e)/(1 − e).

You can derive this fairly quickly from the vis-viva equation, which in turn is derived from conservation of energy.

There are several things I find interesting about this. First, that the expression is so simple. Second, it can be simplified even more for small e:

(1 + e)/(1 − e) ≈ 1 + 2e.

This comes from expanding the ratio as a series:

(1 + e)/(1 − e) = 1 + 2e + 2e² + 2e³ …

This explains two things from the previous post. First, that the variation in orbital speed, for both Earth and Mars, worked out to be about 2e. The eccentricity of Earth’s orbit is 0.0167 and orbital speed varies by about 3%. Mars’ orbit has eccentricity 0.0934 and its orbital speed varies by about 19%. Since the eccentricity of Mars orbit, while fairly small, is larger than that of Earth, the quadratic term matters more for Mars.

Finally, “I keep running into the function f(z) = (1 − z)/(1 + z),” as I first wrote four years ago, and wrote on again a few months ago. It comes up, for example, in computing impedance, in mental calculation tricks, and in efficient calculation of the perimeter of an ellipse. Now you can add to that list calculating the variation in orbital speed in a two body problem.