Home >> Science & Nature >> Astronomy

How to Calculate Speed in an Elliptical Orbit

Orbiting objects always trace out an ellipse as they revolve around a larger body. An ellipse is basically a flattened circle with different distances to its center along its two perpendicular axes. These distances are known as the semi-major and semi-minor axis lengths.

The speed of an orbiting object constantly varies as its altitude changes and some of the potential energy of its altitude is converted into motion. The maximum speed occurs at its perigee and the minimum at the apogee. Velocities are calculated with an equation that uses the object̵7;s altitude and some fixed parameters of the orbit.

Things You'll Need

Scientific calculator
Pencil and paper

Instructions

- 1
  Write down the two parameters that define the orbit: the maximum and minimum altitudes above the earth. Also make a note of the altitude of the object at the position where the velocity will be found. You can use kilometers or miles.
- 2
  Calculate the length of the orbiting object̵7;s semi-major axis by adding its maximum and minimum altitudes, dividing by two, and then adding the radius of the earth. The earth̵7;s radius is 6380 km or 3985 miles; use the figure that is consistent with the units you choose for the axis length.
- 3
  Find the inverse of the semi-major axis length.
- 4
  Determine the object̵7;s distance from the center of the earth at the position for the velocity you wish to find. This is its altitude above the earth̵7;s surface plus the radius of the earth.
- 5
  Find the inverse of the object̵7;s distance from the earth center and multiply the result by two.
- 6
  Subtract the result of Step 3 (the inverse of semi-major axis length) from the result of Step 5 (twice the inverse of the object̵7;s distance from the earth center).
- 7
  Multiply the result of Step 6 by the planetary gravitational constant. For the earth̵7;s gravity this number is approximately 400,000 cubic kilometers/square second. In U.S. units, this is about 1.27 trillion (1.27E+12) cubic miles/square hour.
  
  The gravitational constants may seem to have strange dimensions but they are designed to provide correct answers in the equations.
- 8
  Find the square root of the result of Step 7. This is the instantaneous velocity of the satellite at the point you chose.
  
  EXAMPLE: Find the maximum speed of a satellite with minimum and maximum altitudes of 180 and 2000 miles.
  
  The semi-major axis is (180 + 2000)/2 + 3985 = 5,075 miles. The inverse of this number is 0.000197 1/mile.
  
  The maximum speed occurs at the perigee (low point) of the orbit where the distance from the earth center is 180 + 3985 = 4165 miles. 2/4165 is 0.000502 1/mile
  
  0.000502 - 0.000197 = 0.000305 1/mile
  0.000305 x 1.27 trillion = 387,000,000 square miles/square hour
  The square root of this number is the maximum speed:19,680 miles/hour

Astronomy