The Lagrange point L1 between the Earth and the Moon

On a line connecting the Earth and the Moon there are three points where a small body may exist in an unstable equilibrium if it has a velocity giving it the same period as the Moon. In these points, the sum of the gravitational forces from the Earth and the Moon supplies just the centripetal force necessary for a circular orbit with the period of the Moon. These points are called Lagrange points. Also see the simulation ("The Moon and L4"). We assume that the Moon travels in a circular orbit at the distance a from the Earth. Let the distance from the Earth to L1 be  x and put  z = x/a. The ratio between the masses of the Moon and the Earth is k = 0.01229. One can show that z is a solution of the equation

(1) Derivation of this equation.

This 5. degree equation may be solved on a graphical calculator. We find  z = 0.849, giving x = 326054 km.

Formulas for calculating the relevant parameters of the Earth and the Moon are derived in the simulation "Orbit of the Moon". Using one of these formulas gives the Moon's period in our assumed circular orbit equal to T = 2352900 s. We want Body1 to travel in a circular orbit with this period about the center of mass. The distance from L1 to the center of mass is  326054 km - 4662 km = 321392 km. The velocity of Body1 should be 1. Go to Parameters. Decrease the number of bodies to 3 (without deleting Body4). The mass of Body1 is so small that it doesn't affect the Moon. Run the simulation. You will see that after about one half of a revolution the circular orbit of Body1 becomes unstable. A body in L1  can not follow a circular orbit  for any length of time without frequent course adjustments by a rocket motor. One might say that Body1 balances like a body on the edge of a knife in an unstable equlibrium. Any  small deviation from the correct velocity and position is rapidly amplified.

2. Go to Parameters and increase the number of bodies to 4 so that Body4 appears in the list. Body4 is outside L1, but also has the correct velocity to go in a circular orbit with the same period as the Moon. Run the simulation and compare the behavior of Body1 and Body4.

3. Calculate the velocity Body4 should have to go in a circular orbit around the Earth if there was no moon. You may test your velocity by setting the mass of the Moon to be 1000 kg, and run the simulation. Then reset the mass to the correct value and observe the influence of the Moon on body 4.

4. The NASA solar observational satellite SOHO is located near the L1 point between the Earth and the Sun. Its position is maintained by firing of rocket motors. Find the position of this point using eq. (1). Make a simulation to test your calculation.

5. Construct an equation to find L2 in the Earth-Moon system. Solve the equation and make a simulation to test your result.