 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.
|
|
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
