(21) Kepler's Third Law

Besides the Moon, Earth now has many artificial satellites, put up by us earthlings for a variety of purposes. The calculation applied by Newton to the Moon can also be used for them.

This is the velocity required by the satellite to stay in its orbit ("1" in the drawing). Any slower and it loses altitude and hits the Earth ("2"), any faster and it rises to greater distance ("3"). For comparison, a jetliner flies at about 250 m/sec, a rifle bullet at about 600 m/sec.

We again need a notation for square root. Since the HTML language does not provide one, we use the notation SQRT found in some computer languages. The square root of 2, for instance, can be written

SQRT(2) = 1. 41412. . .

If the speed V of our satellite is only moderately greater than V_o curve "3" will be part of a Keplerian ellipse and will ultimately turn back towards Earth. If however V is greater than 1. 4142. . .times V_o the satellite has attained escape velocity and will never come back: this comes to about 11.2 km/sec.

The value of 11.2 km/s was already derived in the section on Kepler's 2nd law, where the expression for the energy of Keplerian motion was given (without proof) as

where for a satellite orbiting Earth at distance of one Earth radius R_E, the constant k equals k=gR_E². The energy is negative for any spacecraft captured by Earth's gravity, positive for any not held captive, and zero for one just escaping. Let V₁ be the velocity of such a spacecraft, located at distance R_E but with zero energy, i.e. E=0. Then

Kepler's Third Law for Earth Satellites

The velocity for a circular Earth orbit at any other distance r is similarly calculated, but one must take into account that the force of gravity is weaker at greater distances, by a factor (R_E/r)². We then get

V²/r = g (R_E r)² = g R_E²/r²

Let T be the orbital period, in seconds. Then (as noted earlier), the distance 2 πr covered in one orbit equals VT

and by the earlier equality

4 π²r/T² = g R_E²/r²

Get rid of fractions by multiplying both sides by r²T²

4 π²r³ = g R_E² T²

To better see what we have, divide both sides by g R_E², isolating T²:

T² = (4 π²/g R_E²) r³

What's inside the brackets is just a number. The rest tells a simple message--T² is proportional to r³, the orbital period squared is proportional to the distance cubes. This is Kepler's 3rd law, for the special case of circular orbits around Earth.

If you are not yet tired of the calculation, you may click here for turning the above into a practical formula.