# Calculate the orbital radius of a synchronous satellite

• james111
In summary, a geosynchronous satellite needs to be in the plane of the equator in order to send and receive signals. This is because the satellite's central acceleration is more constant in this plane than any other.
james111
I'v been trying to work out this question for ages, but nothings quite working for me, here goes..

q. a) calculate the orbital radius of a synchronous satellite (one period of 24hr, so appears stationary above anyone point). Approximately how many radii of Earth is this orbital radius? Why does the satellite have to be in the plane of the Equator?

b) draw a scale diagram to estimate the angle above the horizontal that a receiving aerial in latitude 45 deg must point in order to receive signals from the satellite.

For a) I figured that I could use the eqn a = (v^2)/r for circular motion substituting v = 2rPi / (24x60x60) ... but this gives r as 1853078.528km I think... which would be wrong. I don't know wot to do.

thanks for any help

IIRC, geosynchronous satellites are something like 24,000 miles up (dunno if that's above the Earth's surface or measured from the center of the Earth, though). Because the acceleration due to gravity falls off in space, you need to figure out where the central acceleration due to that lower gravitational acceleration is correct for the orbital velocity. Do you know the equation that gives the gravitational acceleration in terms of two masses m and M and the gravitational constant G and the radius r?

Hey, I just checked wikipedia, and I was pretty close with my guess. James, *after* you work out this problem and get an answer close to my guess, check out the page at wikipedia.org about "geosynchronous".

GMm/R ?? I still can't figure how we can apply this. Sorry, this is quite a new area for me, and I still haven't fully figured it out.

james111 said:
GMm/R ?? I still can't figure how we can apply this. Sorry, this is quite a new area for me, and I still haven't fully figured it out.
Um, no. Close, but not correct. Do you have a textbook for this class? It should definitely be giving you this information before asking you this question.

I went back to the wikipedia page about geosynchronous orbits, and followed a link at the bottom of that page to a web page about orbital mechanics. It's a nice write-up, and it has the correct equation (similar to yours but with one term changed) part-way down the page. Try reading through this link to see if it helps this question make more sense.

http://www.braeunig.us/space/orbmech.htm

## 1. What is a synchronous satellite?

A synchronous satellite is a type of satellite that orbits the Earth at the same rate that the Earth rotates, resulting in the satellite always being above the same location on the Earth's surface. This type of orbit is also known as a geosynchronous orbit.

## 2. Why is calculating the orbital radius important for a synchronous satellite?

The orbital radius determines the distance between the satellite and the Earth's surface. For a synchronous satellite, it is important to calculate the correct orbital radius in order for the satellite to maintain its synchronized orbit and stay above the same location on Earth.

## 3. How is the orbital radius of a synchronous satellite calculated?

The orbital radius can be calculated using the equation: R = (GMT^2 / 4π^2)^(1/3), where R is the orbital radius, G is the gravitational constant, M is the mass of the Earth, and T is the period of the satellite's orbit. This equation is derived from the law of gravitation and centripetal force.

## 4. What factors can affect the orbital radius of a synchronous satellite?

The orbital radius is mainly determined by the period of the satellite's orbit, which is affected by the mass of the Earth. However, other factors such as atmospheric drag, solar radiation pressure, and gravitational pull from other celestial bodies can also have a slight influence on the orbital radius.

## 5. Can the orbital radius of a synchronous satellite change over time?

Yes, the orbital radius can change over time due to various factors such as atmospheric drag and gravitational pull from other celestial bodies. However, for a synchronous satellite, special measures are taken to maintain its orbital radius and keep it synchronized with the Earth's rotation.

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