The magazine of the Melbourne PC User Group The Lagrange Points - Part 2 A Space Colony Ken Holmes

We are about to establish a habitat at the Lagrange L4 point. Monsieur L. has told us roughly where this region of stability is, leading the moon by about 60 degrees in its orbit, but we would like to calculate its exact position, where the nett gravitational acceleration towards the earth and moon provides the correct centripetal acceleration towards the centre of rotation. The obvious way is to write a (separate) trial-and-error program which prints out the difference between gravitational and centripetal accelerations for ranges of radius and lead angle until we find a value of zero. It involves some trigonometry with which I won't bother you, unless you are interested enough to contact me.

For our simplified system, the radius is 381922.1 km, 2356 km outside the moon's orbit, and the lead angle is 60.61 degrees (1.057785 rad). A surprising fact emerging from the calculation is that the earth, moon and L4 do form a perfect equilateral triangle, just as Lagrange stated it. This is not at all obvious from basic principles and is a good illustration of Occam's razor, the assertion that behind most seemingly complex phenomena lie simple truths or solutions. Knowing this fact, it is a simpler geometry problem to locate L4 and, with earth/moon mass ratio and, hence, the centre of rotation, known, to determine the lead angle and the radii of the orbits of the three bodies. The lead angle is not 60 deg. since it is the angle subtended at the centre of rotation, not at earth's centre.

To generalise for different mass ratios, if we make the moon smaller, the lead angle trends towards 60. At the other extreme, if the "moon" equals the earth's mass, L4 lies on the bisector, as does the centre of rotation (as you would expect from the symmetry), and thus the lead angle is 90 deg. Another fact is that the L4 orbital radius is greater than that of the moon and less than the earth/moon separation.

We know how to launch stuff into space and we would be able to put components into the stability region, but we would wish to get the station as close to L4 as possible to provide a "fixed" target to send cargo and people. Listing 2 is a QBasic program to plot the sort of orbit we would find if not exactly positioned and it will investigate a control scheme to dampen the oscillations and place space ships exactly at L4. A "relative" plot with earth and moon fixed on the x axis, see Part 1, is the way to go and you will recognise the main calculation to be the same as in Part 1, Listing 1. Extra code is to plot the sine waves and to institute a control scheme and see how it works. Figure 3 shows the results.

Uncontrolled Orbits

A body stable at L4 will have a certain potential energy due to its distances from the earth and the moon and a certain kinetic energy due to its orbital velocity. Any body not meeting both criteria will move around the L4 region. In Fig 3 we start a body 2000 km outside L4 with the same orbital angular velocity as the moon; it will have excess potential energy and excess kinetic energy and we see (green) the uncontrolled orbit during the first ten lunar orbits (months). Plotting the radius and lead angle variations at the top left gives an insight to these basic elements; in both, we see a sine wave with a period of slightly over one month since, relative to the earth, we have a slightly eccentric or elliptic orbit.

To expand on this, consider a situation without the moon present, with a body in circular orbit at the L4 distance from the earth; this would have a period slightly over one month. Now add another body with the same period but a small eccentricity, and plot its position relative to the first. This would give a small ellipse around L4; at its apogee, outside L4, it would be slower and it would fall behind L4 as it moved inwards, picking up speed so that it would be going faster at its perigee, inside L4. Thus we see that the differential position has sine wave components in both the radial and the orbital directions. We can argue a similar effect for the moon and, indeed, we see oscillations with just over three months period in both directions on our plot. We can think about this as the body being attached to the earth by a gravitational spring and to the moon by a much softer spring which has a natural frequency about one third of the earth's spring or about three times the period. We also note that our "relative" plot is extended along the orbit; this is akin to the bigger movements of the wheels you get in a car with softer springing.

You note that the shape of both plots changes slightly over the ten orbits due to the fact that the ratio is not exactly 3 and there is a small change in the phase difference between the two oscillations. The ratio has a direct relationship to the earth/moon mass ratio and would become larger with a smaller moon, and vice versa. With equal bodies it would be 1.

Controlling the Beast

We won't claim to have the optimum control procedure here as this is food for computers and teams of engineers/programmers. Short bursts of thrust at strategic times would probably provide the most fuel efficient techniques and the choice would be influenced by engineering aspects such as thrust relative to mass, fuel efficiency at different thrust levels, attitude control (pointing the thrust) and so on. We adopt a continuous scheme, switched on at the 10th orbit since it gives a simpler program. Firstly, we adjust the total energy to the L4 value by thrusting along orbit to change the orbital velocity, and secondly, we calculate the radial velocity and apply radial thrust (ie. acceleration) to provide velocity damping, similar to the shock absorbers of your car. If we get radial velocity to zero, we have a circular orbit and, if we get total energy to the L4 value, we must be at L4. It sounds like blind faith but it does happen. Instead of using two separate thrusts, fuel is saved by applying one thrust at an angle so that the resolved components are correct. A separate program was used to explore appropriate levels for the two components, le. the fga and fgr coefficients for our program. We have plotted (solid) the levels of the two thrust components; note how the radial thrust curve is 90 degrees out of phase with the radial position curve, ie. maximum thrust outwards when it is moving fastest inwards due to the velocity damping we are applying. The along-orbit, energy adjustment relationship is more obscure. The continuous thrusts called for are probably too low for the chemical rockets which would be used in supply ships. Short bursts of higher thrust would be used, but our simple continuous approach does suggest a starting point in choosing the timing and size of these. However, the large space station might well use continuous thrust from solar or nuclear powered small thrusters to maintain position.

Let's Get Energetic

Kinetic Energy per unit mass is given by half of velocity squared. For Potential Energy on earth we might take sea level as the reference zero and the PE at a height would be +g times height, or the work required to lift unit mass to it against gravity. In space this is not so convenient, certainly not where we have PE in the gravity fields of two bodies. So infinity is the reference zero and points have a negative PE, or a PE deficit, being the work necessary to move a body to infinity against gravity. To calculate this involves an integration of small distances against the (inverse square law) gravity; calculus tells us this is minus a gravity constant over the distance; we have to escape from both bodies so we add the deficits. Thus, in Listing 2, we are able to derive the total energy per unit mass (enL4) required at L4. By the way, if the total energy is zero, we normally say that a body has "escape velocity", le. it has enough kinetic energy to make up the PE deficit and can escape to infinity, or, at least, out of the gravity fields we happen to be considering.

After control is invoked, in each program do-loop the current energy is calculated and the energy error is multiplied by the coefficient (fga) to determine the orbital thrust (acceleration) to be applied. To determine the radial velocity we note the old radius (rdo) each time so that we can get the change during the time interval and then use another arbitrary coefficient (fgr) to determine the radial thrust necessary for radial velocity damping.

Result

It can be done, as shown by the black plot near the orbit and in the black traces in the top right graphs. The fuel consumption table, and the red thrust plots at top right, show that most fuel is used in the first orbits and it rapidly tapers off as L4 is approached. You can experiment with the arbitrary coefficients to fine time the optimum fuel consumption, or to see how much you can vary them before you lose the ship out of the stability zone. It is worth repeating, however, that our approach is probably not the most suitable. Since the moon is about 1% as significant as the earth, it could be ignored and fairly simple maths can deal with two body orbits, le. body around earth. From any point, a transfer orbit to L4 could be defined and the velocity adjustment determined to change to that orbit. The necessary thrust impulse could be delivered in a short time, measured in minutes, and on nearing L4 another velocity adjustment could match the velocity of L4 to arrive with dignity. But it wouldn't be as much fun.

The code is not too long; you have no excuse for not tapping it into QBasic or QuickBasic, and can be downloaded from Listing 2. 

Reprinted from the August 2000 issue of PC Update, the magazine of Melbourne PC User Group, Australia

[About Melbourne PC User Group]