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How to Change a Coplanar Orbit

It would be difficult to think of a part of human life that is not affected by the information that satellites carry. Satellites watch the weather, carry telephone signals and provide navigation information for land, air and sea traffic. A satellite's orbit should match its task. A long-term weather observation satellite should be in a high, geosynchronous orbit so it can continuously monitor a face of the Earth, while navigation satellites might find lower orbits more efficient. The task of adjusting a satellite's orbit is an orbital mechanics problem, and one of the most common orbital mechanics problems is changing a coplanar orbit.

A satellite's orbit is determined by its location and its velocity. So two satellites that go through the exact same point can have completely different orbits if their velocities are different. That's the trick for changing coplanar orbits. At one point in a satellite's orbit, change its velocity to put it into a different orbit. Then let it go for a while until it gets where you want it to end up and change its velocity again to put it into its final orbit. The details are not all that complicated, given a few key equations.

Instructions

- 1
  Calculate the initial velocity of the satellite. The velocity is given by the square root of Newton's gravitational constant times the mass of the Earth divided by the satellite's orbital radius.
  
  For example, a satellite in a circular orbit 250 kilometers above the surface of the Earth has a radius equal to the radius of the Earth plus its altitude; that is
  
  6.378 x 10^6 + 250 x 10^3 meters = 6.628 x 10^6 meters.
  
  G x M for the Earth is 3.968 x 10^14 m^3/s^2 so the satellite's velocity is given by
  
  sqrt(G x M/r1) = sqrt (3.968 x 10^14/6.628 x 10^6) = 7755 meters per second (more than 17,000 miles per hour).
- 2
  Determine the velocity of the final orbit. The velocity is given by the same equation as in Step 1, just with the different radius.
  
  For example, say you wanted to move your satellite to a circular orbit 4,000 km above the surface of the Earth. The final velocity would be
  
  sqrt (3.968 x 10^14/10.378 x 10^6) = 6197 meters per second.
- 3
  Calculate the starting velocity of the transfer orbit to get from the initial to the final orbit. That is, the satellite doesn't just jump from one orbit to the next; it transfers by way of an elliptical orbit. The starting velocity of the elliptical orbit is given by
  
  sqrt ((G x M) x (2/r_initial - 2/(r_initial + r_final)).
  
  For the example problem this is
  
  sqrt (3.968 x 10^14 x (2/6.628 x 10^6 - 2/(6.628 x 10^6 + 10.378 x 10^6)) = 8569 meters per second.
- 4
  Operate the satellite's thrusters long enough to change the velocity of the satellite, a maneuver known in the industry as a "delta-V." The amount of delta-V is the difference between the velocity of the initial orbit and the velocity of the transfer orbit at that same point.
  
  For the example problem, the transfer orbit velocity is 8569 meters per second and the initial velocity is 7755 meters per second; so the difference is 8569 - 7755 = 814 meters per second.
- 5
  Calculate the final velocity of the satellite in the transfer orbit. That is, how fast the satellite will be going when it's travelling in its transfer orbit out to the final orbit radius. The equation is the same as that in Step 3, except that the "r_initial"s and "r_final"s change places.
  
  For the example problem, this becomes:
  
  sqrt (3.968 x 10^14 x (2/10.378 x 10^6 - 2/(10.378 x 10^6 + 6.628 x 10^6)) = 5472 meters per second.
- 6
  When the satellite is at its desired final radius, apply another delta-V, this time equal to the difference between the desired final velocity calculated in Step 2 and the transfer orbit velocity at that same point, calculated in Step 5.
  
  For the example problem, this becomes:
  
  6197 - 5472 meters per second = 725 meters per second.

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