The magazine of the Melbourne PC User Group
Saturn V - Apollo 11
 Part 3 - The Path to the Moon
Ken Holmes

As threatened, this month we will run last month's CASE 1 to 3 and this month's CASE 4 codes in a FOR...NEXT loop to plot four possible routes. As seen in Listing 3, we will take a working version of Listing 2 (from last month) and hack it to a new program. In each loop, we need to do the near-earth path calculations but we don't need to draw anything on screen, so we will adopt the big earth-to-moon WINDOW on which the four paths can be drawn. Colours 1 to 4, blue/green/cyan/red, are used to distinguish the four routes using different values for the notional apogee. Note that we have shortened the step between the 2nd and 3rd routes to get the particular ones desired; FOR...NEXT loops can be fiddled with like this without breaking anything.

I suggest, when you have apollo11.bas, Listing 2, working, you "copy as" apolloop.bas, then carefully modify this as detailed in Listing 3. Originally I drew the near-earth screen, on just the first run (colour = 1), but changed to this simpler approach. This explains the commented lines like "IF colour = 1..." in Listing 2; sorry for the confusion.


Figure 5

In Figure 5 we see that Route 1, blue, passes well in front of the moon and goes well beyond it before swinging back behind it to head for the earth. The moon, blue in this case, is only drawn when Apollo's x coordinate exceeds the moon's xm so we see where Apollo first did this from the (lowest) start of the thick blue moon trace. Unfortunately, this route heads for the left of the earth and is not a smart move since it would involve a re-entry against the earth's rotation, an 8% greater velocity relative to the atmosphere and a 16% increase of Kinetic Energy to be dissipated by the heat shield. This would probably spell disaster.

Route 2, green, passes closer beyond the moon and heads back to the right side of earth appropriate to a good re-entry with the earth's rotation. After the explosion 3 days into Apollo 13's outward journey they had no other option than to adjust to this route and survive another 3 days to go around the moon and line up for earth re-entry. All Apollos would of course need to return on similar courses. Note where the green moon trace starts (and is superimposed over the blue one).

Route 3, cyan, is the way to go since it passes the other side of the moon at about 110 km altitude, from where retrofiring can insert into moon orbit. It needed much trial and error to find a combination, of notional apogee and the timing of the TLI burn or, really, its exact position in earth orbit, to achieve this in approximately the same 77 hours as did Apollo 11. In the process, I changed the calculation time interval, ti, with an effect that we need to change the critical notional apogee from 549200 (in Listing 2) to 547220. You may note the cyan moon trace and beyond it the remnant of the first blue moon trace. We are not concerned about this return route, which would result in a highly eccentric earth orbit, since there is no intention to use it.

On Route 4, red, Apollo starts to curve downwards due to the approaching moon's gravity but the moon nips in front of it and pulls it sharply upwards, adding to its velocity and "whipping" it off into space. NOT recommended for Apollo, but very useful for free boosts, as in the Grand Tour of the planets and many other space projects. Next time you see such things on television, you will recognise the characteristic "elbow" in the paths passing the planets. The red moon trace continues until it goes off the screen.

Note the colour coded plots for Specific Energy; on approaching the moon there is a dip (as the moon's gravity retards Apollo) and, after passing, a recovery, to varying extents, near to the former values. However, Route 4 leaves with positive energy, or above escape velocity, so that it would never return to the earth/moon system but move to a solar orbit. Well, hardly ever; it is possible that after many years it might find itself in the earth's vicinity and have its orbit modified by the earth's gravity.

Note the table printed out for notional apogee, closest moon approach, time, required energy, Erq, and remnant fuel. The moon's radius is 1738 km, so the 1849 km from the centre represents an altitude of 111 km. The apogee variations appear large but, in fact, they entail quite small differences in required energy. The fuel flow rate during the Stage 3 TLI burn was 221 kg/sec and the remaining fuel only varied by 221 kg or 1 second over the extremes. We only burnt in whole second amounts and the code backed off velocity to exact Erq without "recovering" excess fuel burnt. This illustrates the great sensitivity of results to the performance and control of the burn; in practice it is not possible to achieve perfection and Apollo 11 had two correction burns scheduled in the middle of the journey (using the Service Module motor). The first was described as a modest correction and the second one required was so small that it was not conducted. Not worth any small risk of a malfunction or overcorrection. Bearing in mind that reality is 3D and it is necessary to approach the moon off the orbital plane, if the proposed landing site is not on the equator, to get into an inclined orbit which passes over the landing point. A polar orbit would be required if it is desired to inspect or land at the poles, which offer a less rigorous climate for a long term lunar base.

It is not difficult to extend the program to 3D and it would reliably calculate any paths required. However, a human mind is necessary to interpret the results and contrive variations to achieve a specific goal. 
To assist this, it needs versatile graphical presentation, preferably some form of stereo such as the mirror, red/green goggles or more sophisticated virtual reality devices. Also it needs facilities to vary parameters such as rocket thrusts, timings and directions over appropriate ranges on a trial and error basis, with printed accurate figures for critical desired results such as altitude above the lunar surface or orientation of an orbit. The possibilities in pre-planning are endless and, during an actual mission the path will diverge to some degree from that planned; enormous amounts of contingency planning would be necessary for guidance on what to do. In orbital dynamics, outcomes are often counter-intuitive and it needs lots of practice to inculcate the correct line of logic to apply in various circumstances. Extra information would be available from on-board instruments such as radar altimeters, attitude gyroscopes and astral navigation telescopes; these must be able to be fed into the program.

Notice the characteristic figure eight shape of trips to the moon; for fuel economy and re-entry energy, the only sensible way to leave earth, and return to it, is with the earth's rotation, and heading in front of the moon results in approaching it with least relative velocity, lessening the velocity change needed to get into orbit.
 
Next month we will revert to your working copy of the Listing 2 program as a basis to continue with the Apollo 11 journey to approach the moon and get into a near circular orbit of it.

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