The magazine of the Melbourne PC User Group The Lagrange Points - Part 3 Ken Holmes

When colonies are established on the moon and at L.4, there may well be traffic between them for freight and passengers. Perhaps your frequent flyer points will get you the three leg round trip, or metals may be refined on the moon for manufacture of components to extend the L4 structure. There will be the same trade-off between journey time and fuel consumption as with freight by sea and passengers by air. Passengers may not wish to spend more than a few days; this would entail greater fuel burns to get greater transit speed and more fuel would be needed on approach to L4 to wash off the excess energy. With freight it will be desirable to conserve fuel by working with the gravity field, but this could entail times of a month or two.

Planning to conserve fuel is the more challenging mode so, naturally, that's the way we'll go. We will learn more, just as you do sailing a boat rather than using a motor. Figure 4 reminds us of the contours of the Potential Energy field of the moon's (flared cone) gravity well, sitting on the side of the earth's ditto. Near L4 is a (dynamically) flattish area where we wish to arrive with exactly the right energy to avoid an overshoot. We would like to burn all our fuel at launch and coast to L4, but this is trickier than your average mini-golf shot; in fact it is impossible and we will need to do correction burns along the way. There is no friction or resistance in space to dispose of energy and ease up to our destination; it needs fuel to slow down.

As we need to get 60 degrees ahead of the moon, our experience with L1 bodies in Part 1 suggests we should try for a smaller orbit of the earth with a shorter period, in effect taking a short cut to get ahead of the moon. Perhaps also it may be more economical to slip over the saddle-shaped lower lip of the moon's well towards the earth so as to need to acquire the least Total Energy.

Let's Launch

We will assume the launch site to be at the centre of the moon's face towards earth and the direction will be towards L4 to give a clockwise moon orbit - as viewed from the North Star or on Fig 4. All moon orbits are in this direction; when the Apollos arrived, they went in front of the moon which slowed them so that they needed less fuel in a deceleration burn on the far side to get into orbit. On leaving, an acceleration burn on the far side from a similar orbit sent them towards earth. We will assume our freight ship has been launched, whether vertically or along a launch ramp, and has a horizontal speed relative to the moon of 3330 km/br, 2km above the surface. Since the moon is travelling at 3670 km/hr along its orbit, the body is actually moving at 7000 km/hr relative to earth. It needs this latter speed to be 6050 km/hr for a close moon orbit of 1.8 hrs duration but we wish to get the apolune well out on the far side. Apollos had to fire near the centre of the far side, as do we, but a little later to send us on a suitable orbit rather than directly towards earth; the amount of fuel and the direction of thrust are the result of a great deal of trial and much error to get something to work. Arrays of programs, varying all parameters, could doubtless optimise this whole program.

The Trip

We see stages 1 (a half orbit), 2 (burn) and 3 (coasting) in the moon inset and stage 4 (still coasting) is on the big picture where both absolute position and position relative to the moon are plotted in black. The absolute orbit is highly eccentric and, if allowed to coast, the body would move out across the moon's path and slow down, possibly being recaptured by it. In stage 5, plotted in red, we institute a control scheme of burns to apply critical velocity damping in the radial direction to prevent it going outside the L4 orbit. If it does, when we are trying not to exceed L4 Total Energy, then larger radius means higher Potential Energy which means lower Kinetic Energy which means lower speed than L4 speed, so that we won't catch it unless we burn more fuel. We have deserted the continuous correction as impractical and use here firings at 12 hour intervals. You may note that the first thrust lines are large to get radial velocity under control and then fall off rapidly. In practice, firings would be individually calculated and would occur only on a few occasions with carefully chosen timings.

The moon is still tugging backwards so we now need to start feeding in forward thrust components to ensure escape. As in Part 2, we make the correction proportional to our deficit from L4 Total Energy so that by the time we reach L4 we have that TE. By staying at smaller radii, or lower PE, we maintain higher KE, or positive overtaking speed. Obviously, you can play tunes with all these variables to vary trip time and fuel consumption. Note that the thrust lines are predominantly radially inwards and most of the fuel is burnt to change the eccentric orbit to a circular orbit, ie. L4's orbit. Forward thrust, along orbit, is relatively small and is thus cheaply available from a small change of thrust direction.

Arrival

The Fig 4 printout shows that at the end of stage 5, the body was 1404 km inside L4 with an overtaking speed (Delta V) of 13 km/hr. If allowed to coast now, it would perform a neat elliptical circuit of L4 as in the hypothetical scenario used in Part 2 to explain the monthly sine wave components; there happen to be no 3-monthly components from this particular starting point and TE. In the L4 inset, we apply a control scheme, similar to Part 2, causing a slow spiral inwards, each circuit being one month. Freight consignments could be parked in circuit and brought in as required with a couple of deft squirts of thrust.

Fuel Consumption

The figures here are only a rough indication since they ignore the appreciable changes of mass. They are in fact an integration of the velocity changes for all firings, which are the same as the accelerations multiplied by burn durations. Accelerations should really be multiplied by ship mass (at that moment) to get thrust and hence fuel usage. At launch, we needed a horizontal velocity change of 3330 krn/hr plus a considerable fuel burn to take off and provide vertical support until orbital speed was achieved. Stage 2 burn gave a velocity change of 2213 km/hr and stage 5 added another 2035 km/hr for a post-launch total of 4248. The 3 month spiral in to L4 only added 23 but a more rapid retrieval would use more. Note that the trip time to end of stage 5 was approaching two months (1154 hours).

The Program

Real programmers would have preferred lots of SUB programs, longer variable names, longer comments (your comments are these articles), one statement per line, the odd vacant line or string of dashes, etc. I hope that the less experienced will find it easier to see the whole picture with the one piece approach and easier to type into QBasic. The common code calculating accelerations, velocities and positions forms the backbone and the various stages are well handled by the SELECT CASE structure. I have had a lot of fun with this and writing the articles has led me to a more thorough development of a program written several years ago. The more you investigate a subject like this, the more things suggest themselves for investigation and better ways to do it keep thrusting themselves forward. You don't ever expect to be completely satisfied with a program you have finished with (for the moment). Nevertheless, I hope that copying these programs might entice more of our members towards this rewarding pastime.

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

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