# Geostationary orbit and rotation speed of a planet

Assume for planets similar to Earth but with different rotation speeds (including zero), and we put a satellites at altitude of 35 786 km/hr (similar to the Geostationary orbit or earth), with speed 11 068 km/hr, similar to the speed of communication satellites. these satellites, can rotate around the center point of the planets at any direction regardless of it's rotation axis, as long as it's perpendicular to the line from it's center and the planet center.

For any planet, if we consider it as the reference point (we will exclude the rotation speed), it has multiple satellites at same altitude but with different speed but they don't fall. How is that possible? or we should consider another reference point, with absolute zero of speed and rotation, which does not exist? or there is another factor missing?

rephrasing: I put an assumption that the planets are similar to earth to ease calculation

The satellite at orbital speed of 11 068 km/hr, can rotate around the planet in any direction as in the image . it doesn't have to be around equator. only the one around the equator, will look as fixed in the space.

• Sorry, but right now I find it impossible to understand what you're asking. Could you try to rephrase the question more clearly? – Danu May 11 '14 at 8:53
• this case has two frames of reference, Inertial frame of reference (en.wikipedia.org/wiki/Inertial_reference_frame) & Rotating frame of reference (en.wikipedia.org/wiki/Rotating_reference_frame) Observer on earth uses rotating frame of reference, he can see satellites at same altitude but with different speed and direction. for another observer outside the planet, he uses inertial frame of reference, he sees satellites traveling at the same speed. – user46273 May 12 '14 at 0:47

My question is about the orbital speed, why at the same altitude, we can have multiple satellite, at different speed relative to earth.

If satellites are at the same orbital speed and the same altitude, they cannot have different speeds relative to earth. If the equatorial orbit takes 8 hours, all the orbits take 8 hours. The non-equatorial orbits see different regions under them as they orbit. If you map a non-equatorial orbit on a flat map, it will look like a sinusoidal wave. The orbit looks longer than an equatorial orbit (but it's not) because of the way the map is drawn. Horizontal distances are stretched (more and more) away from the equator in order to get the map to be flat.

Also, satellites miss each other because of their position in their orbits. The satellites are positioned so that they don't collide.

• they can have different speed, relative to earth. ex. one satellite east to south, and the other north to east – user46273 May 11 '14 at 22:01
• @user46273 That's a different direction, not speed. – LDC3 May 11 '14 at 22:04

A satellite goes around the Earth in an elliptical orbit. It is nice to arrange the orbit so that it matches to rotation of the earth, and the satellite stays above the same spot of the Earth's surface. It is possible to choose a special orbit that does this.

First a circle is a special case of an ellipse. If we choose a circular orbit, the satellite will always stay at the same altitude.

The orbit is in a plane. For a circular orbit, the center of the orbit is the center of the Earth.

Next the orbit must in the plane of the equator. Otherwise, half of the orbit will be north of the equator and the other half south. This orbit does not stay above the same point of the Earth's surface.

Last, the orbit must be chosen so that it completes one rotation is exactly one day. The Earth also completes one rotation in a day.

It is possible to do this by choosing the altitude of the orbit. Lower altitudes have a shorter orbital period. Most satellites are 100 miles or so above the surface. Just high enough to avoid the last whisps of atmosphere. These satellites orbit every hour or so.

The moon is much higher - about 239,000 miles up. Its orbital period is almost a month.

The altitude for a 1 day orbit is 22,236 miles.

For other planets, you can make the same choices.

It won't always work. If the planet doesn't rotate, the satellite would have to be infinitely high to stay over the same point. If the planet rotates too fast, the orbit would have to be so low that it would be underground. But for many planets it should be possible.

--- EDIT

For a circular orbit, the speed and altitude are constant. Speed can be calculated from the altitude.

But for elliptical orbits, speed and altitude change along the orbit. Speed is fastest at the lowest point. You could choose different elliptical orbits where the speeds are different at the same altitude. I am not sure if that is what you are asking about.

• If a planet rotates to slow it could also mean that the satellite would have to orbit outside the Hill sphere, which would mean an unstable orbit. And if the planet is tidally locked to its sun you could put it at a Lagrange point. – fibonatic May 11 '14 at 10:22
• My question is about the orbital speed, why at the same altitude (not only the orbit over the equator), we can have multiple satellite, at different speed relative to earth. – user46273 May 11 '14 at 13:31

multiple satellites at same altitude but with different speed but they don't fall. How is that possible?

This is not possible. For a planet with mass $M$, the orbital speed of a satellite in circular orbit at distance $r$ from the center of the planet is $v = \sqrt{GM/r}$. If you want to increase the speed of a satellite, you must bring it in closer to the planet. If you want to slow it down, you must push it farther away.

Note that the above puts no restrictions on the direction of the orbit. All satellites with an orbital period equal to the planet's period of rotation will be geosynchronous. So if you modify the planet's period of rotation, you will need to change the altitude of your satellites to keep their orbits geosynchronous. But also note that these geosynchronous orbits are not necessarily geostationary; only those orbit directly over the equator will be geostationary.