 The magazine of the Melbourne PC User Group Saturn V - Apollo 11 Part 5 - Lunar Landing and Departure Ken Holmes |
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Click here for Program Listing
We left the LEM after it had separated from the Command Module, orbitting at 112 km, and decelerated into an orbit with a perilune at 15 km. Our graphics screen, Figure 7, is 400 km wide to cover the retrofire and landing; at the start we assume the LEM is -.15 radians (8.6 deg. or 270 km) to the left of centre screen around the moon's surface, moving parallel to the surface as full reverse thrust (4455 kg force) is applied to its 14735 kg mass, giving a deceleration of 0.3 x earth g or nearly 2 x moon g. The CM is lagging at -.276 radians since the LEM's inner orbit has been quicker. We are only considering moon gravity with a constant, mgc = 4897, which is divided by radius squared to give gravitational acceleration; the CM/SM's circular orbit has an angular velocity, cw (0.00088 rad/sec) and we plot its position each 5 secs or cw5 (0.0044 rad. or 0.25 deg.).
Landing
Let's visualise this from the astronauts point of view. Seating is minimal and they are standing above the engine, providing its modest apparent g, looking at the moon surface as a vertical wall, 15 km away, nicely illuminated by the sun above and about 20 degrees behind their heads. The CM is about 250 km above them around the moon's curvature and the earth is behind them about 20 degrees below the normal to the moon's surface. The landing spot is 400 km further below and the wall is moving upwards at 6200 kmph. Over 7 minutes, 6000 kg of hydrazine burns, reducing mass to 8800 kg so that the deceleration due to the constant thrust rises to about 0.5 earth g.
The autopilot is controlling thrust and, via small thrusters, the attitude and the path; it will achieve something like a critically velocity-damped simple harmonic motion, in both the horizontal and the vertical directions, to arrive with dignity. For this program, we concoct coefficients for the acceleration towards target and against velocity to achieve much the same effect. The coefficients vary according to rather strange formulae which arose from pure trial and error to achieve a result. I have deliberately allowed a little overshoot horizontally to make the path shape a little more interesting - but this is not recommended vertically. Note that, at the start of retrofire, our thrust is tilted about 30 degrees; in actuality, thrust was probably kept parallel to the moon surface as this would likely be the most fuel efficient approach.
The "wall" has now folded about 25 deg. away from their apparent vertical and is about 3 km away as their path arcs towards the designated target, visible below the pilot's feet. It now rapidly falls away further towards the horizontal and the thrust decreases rapidly aiming to just reach the surface as horizontal velocity is nil, vertical velocity is very small and thrust is just supporting the LEM's moon weight, ie. hovering. They are descending nearly vertically at 1000 ft when Armstrong notices many large boulders at the target. He selects a better area about a mile away and inputs to the autopilot to move to it; fuel is running low. Hovering on a rocket is inherently highly unstable, much more so than a helicopter, and primary control needs an autopilot to predict behaviour and apply corrections, somewhat like the calculations performed in this program. No instrument is absolutely accurate and fuel gauges and flow meters are not; they were about 20 seconds from a decision to abort when the sensor on a leg touched the surface and cut the motor; there was some excitement among participants and onlookers.
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On Figure 7, the red fan gives a history of the thrust acceleration's magnitude and direction, matching the red lines on the periodic acceleration polygons along the bottom. These latter show the relationship between accelerations - thrust (red), centrifugal (magenta) and gravitational (grey). Also, velocity is shown in green. This diagram is constantly redrawn at right centre as the program runs and samples have been drawn along the bottom purely for Figure 7. You may, instead, use the bottom for various information which is commented out in Listing 5; also, you may wish to vary the timer delay for it to run slower. The "angles to horizontal" are a product of the arbitrary coefficients used and, I'm sure, bear inexact relationship to actuals.
Abort or Sojourn?
The astronauts' cabin is in the ascent stage atop the LEM. Had it been necessary to abort, explosive bolts would have separated the vehicles and the ascent engine would have fired up. After reorienting itself, the ascent stage would have climbed back into orbit to rejoin the CM. Contingency plans were in place to achieve this at any time of the retrofire.
Happily, in the event, the Eagle landed at 2.17 pm Florida time, 100 hrs 15 mins after blast-off. The astronauts slept, ate and carried out the EVA before preparing for take-off, which happened at 1.54 pm nearly 24 hours later.
Take-off
Whilst not initially as difficult as the landing, this was still "quite a ride"; however it had to be carefully timed to ensure joining up with the CM later. Departure was 70 seconds after the CM had passed directly overhead; the CM would get well ahead as the ascent stage climbed into orbit but the latter in its inner orbits would steadily catch up.
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With a mass of 4500 kg and engine thrust of 1364 kg.wt, the acceleration was again 0.3 earth g or nearly 2 moon g, so it lifted off gently and rolled over to a shallow climb angle. On reaching initial orbit, 2000 kg of fuel had burnt, bringing the mass to 2500 kg, so acceleration would have built again to 0.5 g. Looking down at the surface just after the roll-over must have been the ultimate in low flying. As with the earth launch, judicious use of acceleration and damping coefficients was necessary to smoothly enter the required orbit. Figure 8 is similar in presentation to Figure 7 and needs little elaboration.
The manoeuvering to overtake and join up with the CM is quite
impressively tricky and will be the subject for next month.
Reprinted from the March 2001 issue of PC Update, the
magazine of Melbourne PC User Group, Australia
