# Geostationary satellites

1. Aug 6, 2004

### kurious

How accurate does the orbiting velocity of a geostationary satellite have to be so that it keeps in synch with the same geographical part of the Earth over a period of ten years.

2. Aug 6, 2004

### marlon

ah, start calculation, kurious. Just apply perturbationtheory on the sattelite's orbit. The earth is not sferical so the potential has to be changed. The formula in function of Legendre functions can be found in any site of astronomy. You can make the following approximatiob though : the inclination of the earth's orbit and the sattelite's orbit is constant with respect to the ecliptica

greets
marlon.

3. Aug 6, 2004

### enigma

Staff Emeritus
Actually, they do oscillate a bit depending on the longitude they're orbiting over.

There are 2 nodes (I seem to recall over the midwest US and 180 degrees around the planet, but I could easily be mistaken) where they more-or-less stay put. Those nodes are caused by a slightly non-uniform gravitational field.

Any other place needs stationkeeping on the order of 10s of meters per second per year to maintain position.

4. Aug 7, 2004

### BobG

If the satellite is located at 105 degrees West or 75 degrees East (stable equilibrium points), it's altitude has to within 20 km or so (rough estimate) of a true geosynchronous orbit 42,164 km. That's a difference of maybe 2 m/s at the most.

Any other locations, the satellite has to be actively kept there (as Enigma said). Left unattended, the satellite will oscillate back and forth across the nearest stable equilibrium point, however big a range in longitude that may be.

The inclination doesn't stay constant, either, unless the satellite operator actively maintains the inclination. Because of the pull from the Sun and Moon, the inclination will try to match the plane of the ecliptic (the Earth's orbit around the Sun). Except the orbit plane is also precessing at the same time (imagine the orbit plane as a plate rolling along the floor, slowing, and the way it slowly settles to the floor through precession). The satellite's inclination increases for awhile and then decreases until the satellite is right back in the equatorial plane (it takes about 27 years). The cycle just continues forever (unless the satellite operator has pushed the satellite out of the geosynchronous belt right before the satellite died).

5. Aug 14, 2004

### Jenab

I haven't learned much yet about perturbations due to deviations of an extended mass from spherical symmetry. Are the perturbations modelled as though the mass distribution of a rotating planet were the integration of cylinders: rod connecting the poles, disk at the equator, and intermediate cylinders in between? That would explain why zonal harmonics are the only spherical harmonics used in the series expansion for the gravitational potential.

Jerry Abbott

6. Aug 15, 2004

### pervect

Staff Emeritus
I beleve that Legendre polynomials are usually used.

mi
^
|
|ri
|
|
c.of.m---------------r------------->M

If we have an extended body on the left, and a large point mass M on the right, we can form the total potential energy as the sum

V = -GMmi / rtot

where rtot is the distance from mi to M.

But we can re-write this sum in terms of r (the distance from the center of mass c.of.m) and ri, the distance from the c.of.m to mi as follows.

$$V = {GMm_{i}} / {r \sqrt{1+(\frac{r_{i}}{r})^2-2 \frac {r_{i}} {r} cos (\theta)}$$

where $$\theta$$ is the angle between the vectors r and ri.

This expression can be series expanded in terms of Legender polynomials Pn for r >> ri

$$V = \frac {GM}{r} \sum m_{i} (r_{i}/r)^i P_{i} (cos(\theta))$$

7. Aug 16, 2004

### BobG

True in general, but not specifically for geosynchronous satellites.

The sectorial harmonics are extremely small. A satellite would have to stay in the same region for a long time in order for the effects of sectorial harmonics to build up enough to be noticeable. Since geosynchronous satellites maintain a more or less constant location relative to the Earth's longitudes, these also have to be taken into account (for geos or other resonant orbits, only).

8. Aug 17, 2004

### remcook

ehmm...no they're not. It's just that the flattening of the Earth (due to rotation, expressed by the term of order 2, degree 0) is the largest effect in general.

PS: also note that the acceleration due to the Moon and Sun is larger than the acceleration due to the J22 (sectorial) term.
edit-for a geostationary satellite that is.

Last edited: Aug 17, 2004
9. Aug 17, 2004

### Jenab

Ah well, live and learn. One of these days, I'm going to know enough about spherical harmonics in a gravity potential field to write a hillbilly tutorial about it, but there's the winter wood supply to finish getting, etc.

Jerry Abbott

10. Aug 18, 2004

### remcook

looking forward to that :)

11. Sep 3, 2004

### tony873004

I've heard that geosync satellites that lose their ability to correct their orbits tend to gather over the Indian Ocean. I guess that's the 180 from the U.S.A. that Enigma was talking about. I've never heard of another node over the U.S, though. That would be useful :tongue2:
Not sure if my source is right though

12. Sep 4, 2004

### BobG

The ACTS satellite took advantage of this to extend its life. It sits at about 105 degrees West.

But, geo satellites that lose their ability to control their orbit won't gather at one of the 'gravity valleys' (unless you're talking really long term - the 'bulges' don't quite balance out). Instead, they cross back and forth across the valley, roughly moving however far away from the valley they started on both sides. For example, if a geo satellite died at 90 degrees West, it would drift from 90 degrees West to 120 degrees West. Plus, the inclination changes in response to the Sun and Moon. Otherwise, it would get very crowded at the 'gravity valleys'.

Rather than 'gathering' at the 'valleys', I think a more accurate description would be that the 'valleys' are busy intersections.

Last edited: Sep 4, 2004