LOW EARTH CIRCULAR ORBITS
A circular orbit is mathematically the simplest orbit, although it can be difficult to achieve exactly in practice. Only one parameter is needed to specify the size of the orbit, and that is the radius r, the constant distance from the centre of the Earth to the satellite.
If the radius of the Earth is denoted by Re, then the altitude or height of the orbit (and the satellite) above the Earth's surface is h = r - Re.
Unfortunately for our simplicity, the radius of the Earth is not a constant, varying from 6357 km at the poles to 6378 km at the equator. If our satellite has a low inclination (ie it spends most of its time near the equator), then we might use the equatorial radius. Otherwise we could use the mean radius of 6371 km. Of course, for high accuracy applications we need to consider the varying shape of the Earth at each point beneath the satellite's orbit.
If we know the period of the satellite (the time it takes to make one complete orbit around the Earth) we can compute its height, and vice versa. We can also compute its orbital velocity (v) and a quantity known as its mean motion, the number of orbits it completes in one day.
Isaac Newton provided the formulae we need for these calculations. Starting with the famous second law (N2) of motion F = ma where F is the force on the satellite, m is its mass and a is its acceleration.
Newton's law of gravitation F = G M m / r2 is the required force that the Earth exerts on the satellite.
A body moving in a circle is actually being accelerated toward the centre of the circle (by the Earth's gravitational force). The acceleration is referred to as the centripetal (directed towared the centre) acceleration, and the force that produces the acceleration is known as the centripetal force.
The magnitude of the acceleration is given by a = v2 / r .
If we substitute for the force on the left hand side of N2 and the
acceleration on the right hand side we end up with:
In this system of units, the values for the gravitational constant and the mass of the Earth is given as:
A table of values computed from the above formulae is shown below. Note that the units have been converted into the more typically used quantities before display.
LEO CIRCULAR ORBIT PARAMETERS Height Velocity Period Mean Motion (km) (km/s) (mins) (revs/day) 200 7.79 88.4 16.30 300 7.73 90.4 15.93 400 7.67 92.4 15.58 500 7.62 94.5 15.24 600 7.56 96.5 14.92 700 7.51 98.6 14.60 800 7.46 100.7 14.30 900 7.40 102.8 14.00 1000 7.35 105.0 13.72 1100 7.30 107.1 13.44 1200 7.26 109.3 13.18 1300 7.21 111.4 12.92 1400 7.16 113.6 12.67 1500 7.12 115.8 12.43And the figure below displays them in graphical form.
The red curve shows the orbital period in minutes, and its corresponding scale is on the left hand side of the graph. The right hand side scale is used for both the orbital velocity (yellow) which ranges in value from 7 to 8 km/sec, and the mean motion (blue) which varies from 12 to 17 orbits per day.
Australian Space Academy