# Can an elliptical orbit take the same time as a circular obit?

In the picture below you can see two orbits of potential pbjects. The main aspect of the orbits is that they have a collision point at the maximum of the elliptical orbit. My question is, could the object travelling elliptical orbit take the same amount of time as the object taking the circular orbit?

The reason behind this question is that I am doing an analysis on the movie called Gravity. The only information provided in the movie is that the debris cloud will collide every 90 minutes (with the Hubble telescope) and that it is going at 50000 miles per hour (which is much faster that the Hubble telescope). Obviously, since they are going at different speeds, they have to be in different orbits. So this is the only possibility I have come up with where they could possibly collide every 90 minutes, or that the orbit time of the elliptical orbit would take 45 minutes, or 22.5 minutes (any factor of 90).

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It is a movie, don't try to rationalize it. Filmmakers have that thing called artistic freedom. –  this Jul 24 '14 at 12:17
That moment when you look at the numbers and realize how inaccurately slow the debris was moving in the collision scenes. –  ioSamurai Jul 24 '14 at 13:21
Simper if the debris is merely in the opposite orbit, both orbit having a period of 90 minutes. Why would some one do that you ask? Because nothing can go wrong obviously; except it did. –  Pieter Geerkens Jul 24 '14 at 14:28
@PieterGeerkens, As I recall, the debris field in the movie was in fact in a retrograde orbit. Despite the fact that the ex-satellite(s) shouldn't have been in reality. Just one more reason why the movie's science was terribad. –  Brian S Jul 24 '14 at 15:11
@BrianS: There is nothing scientifically wrong with a retrograde orbit - it's just unwise when all other orbital material isn't. –  Pieter Geerkens Jul 24 '14 at 16:03

## 1 Answer

The period of an elliptical orbit is given by:

$$T = 2\pi\sqrt{\frac{a^3}{GM}}$$

where $a$ is the semi-major axis. For a circular orbit of radius $r$ we have $a = r$.

The two orbits you show do not have the same semi-major axis, so they do not have the same period. However if the elliptical orbit had $a^3 = 4r^3$ then the period of the elliptical orbit would be 180 minutes so objects in the two orbits could collide every 180 minutes.

According to this article the collision every 90 minutes is one of the things the film got wrong. This could only happen if the cloud of debris were stationary, in which case it would simply fall towards the Earth.

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Google NdGT's massive, hilarious takedown of the many violations of orbital mechanics in this movie. (Neil deGrasse Tyson) –  Carl Witthoft Jul 24 '14 at 11:49
It's a debris cloud. It doesn't have to be stationary. The fact that it's a cloud means that there is debris along all parts of the orbit at a given time. That means the collision would happen every time the one object in the 90 minute orbit intersected the always occupied cloud in the larger orbit. The period of the other orbit is irrelevant, what is the problem with that? –  Jims Bond Jul 24 '14 at 14:31
@Jim the problem is that the volume that would necessarily be occupied by such a cloud in order for these kinds of collisions to be probable is huge. Even if they were several, non contiguous clouds, that seems even more improbable. –  KidElephant Jul 24 '14 at 15:32
@Jim - The two were presumably in similar orbits prior to the debris cloud formation. The 90 minute thing would work if the orbits of the bodies were inclined relative to each other. If they are in the same plane, however, it should take a different amount of time (which ignores the fact that as your speed changes, so does your altitude). So if the cloud were orbiting with an inclination of 90 degrees, then 90 minutes would be okay. –  honeste_vivere Jan 19 at 18:20