 The magazine of the Melbourne PC User Group Saturn V - Apollo 11 Part 4 - Lunar Orbit Injection (LOI) Ken Holmes |
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After our diversion to the multiple paths to the moon, we will revert to a
single path (fairly) similar to that of Apollo 11. Putting aside your successful Apolloop.bas based on Listing 3, make a copy of your working Apollo11.bas based on Listing 2, calling it, say, Apoleorb.bas, and also put it aside. It is wise
to keep copies at various stages as they can prove useful if some mysterious problem arises with your further
efforts. We will now modify Apollo11.bas in accordance with Listing
4. Incidentally, I have learnt that you can get QBASIC.EXE and QBASIC.HLP from your Windows 98 CD in
the \Tools\oldmsdos directory.
Firstly, we need to adjust the "notional apogee" again to 547460 to achieve the desired orbit around the
moon. Figure 6 is a dog's breakfast but there is a lot of info. to convey and I will accept any criticism.
Whilst these programs run on a black screen, I need to use a modified copy to produce the Figs for printing
on white paper and also cram in more info. Your new program will not show the earth orbit at top left as the
second screen will start when Apollo is clear of the earth. Also, if you wish, you need not print all of the
stuff on screen; instead you might print it on the top line only, with pauses (DO: LOOP WHILE INKEY$ = "") to
read it; this will also slow down the program which runs too quickly anyhow. During the long trip, just
printing the distance to the moon will also slow it nicely.
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The Mechanics
The outward journey is shown with hour and 10-hour graduations. Apollo slows to about 0.77 km/s on the way
but the moon then accelerates it to about 1.5 km/s downwards as it passes the far side. The moon is
travelling at 1.02 km/s upwards in its orbit and we need to retrofire to reduce the relative velocity if we
are to get into orbit. It would be too risky to attempt an immediate circular orbit at 112 km altitude so the
first orbit was 112 X 550; ie. moon radius 1738, perilune 1850, apolune 2288. At earth launch, we used the
apogee/perigee equation to determine the velocity and energy needed at TLI, ignoring moon gravity, and it
served our purposes for a notional calculation on a trial and error basis. At the moon, with the greater
relative significance of the earth's gravity, this calculation is hopeless for a real apolune, so we might as
well use burn duration on a trial and error basis. The Service Module's fuel is hydrazine, a hypergolic,
single component, fuel, with a Specific Impulse of 310 and a flow rate of 30 kg/sec, and we find that a burn
time of 340 secs will do it.
We wish to pass through the lowest altitude, or perilune, in about the middle of the burn and so need to
anticipate the exact time to start it. I don't know the exact technique used but will assume we have a radar
altimeter giving distance from the surface and rate of approach. Looped programs could produce look-up tables
of the relationship between these two variables for different lowest altitudes. These could warn whether the
lowest altitude would be not as required and also determine when to start the LOI 1 burn. I tried various
original TLI notional apogees, LOI 1 burn times and trigger approach rates to start the burn until it more or
less fitted the needs, with thrust aligned backwards parallel to the moon's surface. In reality, an
unacceptable approach might be corrected by firing early and/or at an angle to the moon horizontal. Also,
astral observations from Apollo and radar from earth kept a close watch on the path and Houston decided the
exact timing and direction of thrust, including possibly lateral components to control the orbit achieved.
The burn button was still on board; remember that Apollo is on its own at this time, behind the moon and out
of contact with Houston for about one hour.
The firing is marked by red radials and we next start trying to detect the apolune, when radial distance
will start decreasing again. Since we are only using single precision (7 significant figures), there is some
scatter in the successive radial distances, so we look for a definite decrease, greater than 0.06 km in each
one second time interval, rather than zero. Although we detect it about 3 minutes late, the apolune value is
only 1 km less than the true 545 - not worth using double precision to avoid. Also, we don't achieve the
exact orbit attempted since we limited ourselves to whole seconds in burn time.
Now we start looking for post-perilune, by an altitude increase of 0.05 km/sec, to start the LOI 2 burn. As
we are again a bit late, we need to direct the 31 secs of thrust backwards and inwards to force Apollo
towards the 112 km circular orbit. By T&E, we used "adj = -0.15" radians (8.5 deg.) to depress the thrust
line below horizontal. In practice it would be more efficient to fire exactly at perilune and horizontally,
and I'm sure this would have been done by a more accurate anticipation of perilune by the combined
navigational resources of Houston and Apollo. Note that the Total Energy varies sinusoidally in both orbits
since the velocity in fixed space increases on the near side and decreases on the far side due to the moon's
speed. The second orbit has a shorter period and greater energy variation since it is closer to the moon.
At 30 minute intervals after the LOI 2, altitude was printed on Figure 6, giving 110, 107, 110 and 112 - not
perfect but good enough. Note that the 2 hour point overlaps LOI 2 by about a minute, indicating the orbit
period to be 119 minutes; a theoretical period for a 110 km circular orbit, using our values and ignoring
earth gravity, gives 118.8 minutes. So, our measurement has not indicated any marked effect of earth gravity
on the period. Strictly speaking, you cannot get an exactly circular orbit since the additive earth gravity
on the far side gives a smaller radius curve and the subtractive earth gravity on the near side gives a
flatter curve.
It is slightly egg-shaped with the little end on the far side; perhaps the moon is populated by Swift's
Lilliputian Big-endians and Little-endians?
Apollo 11 did 13 orbits, 26 hours, after LOI 1, observing the moon surface and establishing an orbit passing
over the landing site in the Sea of Tranquility, just north of the equator and 30 degrees right of centre.
From our southern hemisphere, when the moon is due north, it is half way out to the left of centre and a
little below. Also there were domestic chores, eating and sleeping before Armstrong and Aldrin transferred to
the Lunar Expedition Module and separated to fly alongside the Command Module until the appropriate moment
for the Descent Orbit Injection burn. The LEM also uses hydrazine fuel and only needed to burn about 100 kg
to get a descent to 16 km over the landing area. We fix the time at 10 secs and throttle the engine to 72%
power to get an accurate perilune before landing; I don't know the actual procedure.
Note that the landing site is about 10 degrees from the edge of darkness between the sunlit lower half and
the dark upper half. On landing, the sun was 10 degrees above the horizon so temperatures were moderate and
more comfortable than a day later, on departure, when it would have risen by an extra 13 degrees (the sun,
that is). Next month we will start a brand new program for the landing and take-off with simplification by
ignoring earth gravity and using a new set of local reference axes; in effect, similar to the earth launch
program in Listing 1, Part 1.
Reprinted from the February 2001 issue of PC Update, the
magazine of Melbourne PC User Group, Australia
