 The magazine of the Melbourne PC User Group Saturn V - Apollo 11 Part 6: Ascent Module Rendezvous Ken Holmes |
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Click here for Program Listing
Last month, the Ascent Module had reached 18 km altitude with the energy
necessary for a 16 X 83 km orbit as the engine was shut down. The task is now to marry it up with the Command
Module in its 112 km circular orbit.
Orientation.
The CM orbit had been adjusted to pass over the LEM and take-off occurred 70
seconds after it passed overhead. The AM must remain in the CM's orbital plane; no small task, be we will
assume it taken care of to reduce our program to two dimensions. Also, we ignore earth gravity to allow use
of simple orbital dynamics to calculate correctional engine burns; these are very short events and we will
apply them "instantaneously" so that the common code applying in the predominantly coasting conditions is
very simple, only involving moon gravity accelerations.
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If you refer to Figure 6
in Part 4, you may imagine that the day on the moon would have extended the "pigtail" of the CM's path
and, of course, the moon to the top of the screen, with the earth directly to the left. The axes of the
present Figure 9 are similarly oriented and you may note from the sun-illuminated moon hemisphere that the
sun has "moved" about 5 degrees anticlockwise in the 5 days since earth launch. The orbit diagram around the
moon gives the general picture, with the AM's eccentric transition orbits in blue and its circular portion in
green, but we also plot magnified separation distances in the lower left graph to understand what is
happening, and, indeed, to achieve a rendezvous when concocting the code. At lower right is shown the
relative position of the AM during final approach, an intriguing loop.
The Code
In Listing 6, we need to determine the velocity vector angle at 18 km in a
16 X 83 orbit. This was most easily done by trial and error, plotting altitude while varying the angle from
horizontal; this showed that lifting the path by .005 radians at point (a), in line 20, caused the altitude
to crest near 83 km at point (b). Notice that the code in each sector in SELECT CASE is mainly watching for
the end criterion while the common code calculates the current orbit; it then sets conditions for the next
sector.
Point (b) has the rather grandiose title of Co-elliptic Sequence Initiation, CSI, a generalised description
for establishing concentric near-circular orbits. The aim is to have the AM 29 km lower in altitude than the
CM while its higher angular velocity on the inner orbit lets it catch up. A short burst of thrust gives it
the right velocity, which is vectored to be horizontal, for a circular orbit. The black plot of Height
Difference shows a sinusoidal variation of a km or so, indicating that we didn't get an exact circle - a
little more interesting than a straight line! At (c), was an opportunity for Correction - Differential
Height, CDH, and we haven't done anything there. Apollo 11 burnt 5 kg of fuel here to correct the orbit since
there are difficulties in real life in assessing true orbit and metering fuel flow in the previous burns.
There may have been a lateral component of thrust if they had strayed from the CM's orbital plane; this could
be true of any of the firings.
The concept of "lookup angle" was central to the technique of reuniting the vehicles. From the AM, the angle
of the CM above the moon horizon was monitored and, as it passed through 26.6 degrees, Terminal Phase
Initiation, TPI, was invoked with the motor fired to adjust to a 83 X 112 km transition orbit. We use a
simpler calculation of the ratio of height difference / slant range and started TPI when this equalled
sin(22.127) - determined by T & E. At the altitude, the moon horizon is depressed a few degrees and the
paths' curvatures make true lookup angle calculation unnecessarily complicated for our purposes.
In the lower graph, as the AM nears the CM altitude, the blue plot of the horizontal difference illustrates
an interesting feature. We see clearly that the AM actually overtakes 10 km beneath the CM due to its higher
angular velocity, the lookup angle cresting at 90 deg, and it then gets 4 km ahead; as it rises to 112 km,
its forward velocity becomes less than the CM velocity and it drops back. As it passes backwards underneath,
there is a blip in the lookup and this is the moment for the AM to burn 5 kg of fuel to match the CM
velocity.
In Figure 9 we allow the AM to continue in its eccentric orbit since, although it makes a mess of the graph,
it helps to understand what happens. Our critical lookup for the TPI timing has been adjusted to give a near
perfect intercept, with (e) 180 deg. from (d), and the apolune is set at 111 km to ensure the final approach
from underneath, as in Apollo 11. The retrograde motion relative to the CM reverses after it falls back 4 km
and it then again overtakes underneath at a 10 km lower altitude to continue its orbit back to 83 km, as we
see from the (black) sinusoidal height difference. The (green) slant range, being an absolute value,
completes a symmetrical shape as distinct from the anti-symmetrical S-shape. The (red) lookup angle also
produces a nearly symmetrical plot; with a working programme, it is fun to fiddle with the lookup decision
point and the apolune to better understand the results. It may be necessary to relax the approach distance
criterion to ensure point (e) is recognised. The values here get an approach within 0.2 km since the apolune
creeps out from 111 to 111.8 km due to accumulated errors in the calculations every second. The approach path
is shown at bottom right and is a typical epicycloidal loop; the code for this is not in Listing 6 but you
could easily plot it if you are typing in the Listing.
The docking procedure calls for the services of the pilot, ably assisted by the computer, to use the small
thruster rockets to line up the hatches and, gently, close the gap. This occurred 3.5 hours after take-off;
the two astronauts, with rock samples and film, transferred to the CM and the AM was later discarded to crash
back onto the moon. The CM continued for several orbits whilst the astronauts rested, ate, carried out many
checks on equipment and procedures and, with Houston, established their exact orbit. Ahead lay the critical
moment when, alone behind the moon and 10 hours after take-of, they would fire the Service Module engine to
accelerate out of moon orbit into a return path to earth. This will be our subject next month.
Reprinted from the April 2001 issue of PC Update, the magazine of Melbourne PC User Group,
Australia
