Orbits with the Same Semimajor Axes

The orbital period \(\tau\), or the time taken for an object to complete an orbit is given by $$ \tau = \frac{2\pi}{\sqrt{GM}}a^{3/2} $$ where G is the universal gravitational constant, M is the mass of the body that is being orbited, and a is the semi-major axis of the elliptical orbit.

The major axis is just the length of an ellipse. The semi major axis a is just half of that value.

semimajor a major axis

The equation says nothing at all about the eccentricity of the orbit (how oval it is). You can have orbits with the same semi major axis but different eccentricities, and they will all have the same orbital period!

The animation below shows three orbits with the same semi major axes but different eccentricities. The objects all line up at the same time, twice per orbit. If you measure with a ruler the lengths of the ellipses (and width of the circle) they are the same.

(Due to rounding errors in the code and the limitations of Euler's Method, there will be some drift after about a dozen orbits.)

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