Lunar Module Descent Simulation

Robert A. Braeunig
© December-2009


In 2006 I programmed a computer simulation of the descent of Apollo 17's lunar module, Challenger, from lunar orbit to the surface of the Moon. This was done as a follow-on project to my simulation of Challenger's ascent. I've revisited the simulation and updated it with some newly found information. Here I present my findings along with a description of my work.

Descent, Approach and Landing

Below is Apollo 17's lunar orbit summary from NASA's mission reports:


From Post Launch Mission Operations Report, No. M-933-72-17, 19-Dec-1972.
Republished in APOLLO 17, The NASA Mission Reports, Volume One, Apogee Books.

The Command/Service Module (CSM) performed most of the listed maneuvers. The maneuvers performed by the Lunar Module (LM) were DOI-2 (Descent Orbit Insertion #2), PDI (Powered Descent Initiation) and Landing. DOI was a maneuver to insert the spacecraft in the correct orbit from which to initiate descent. DOI-1 was performed by the CSM with the LM still docked. The LM, using its RCS thrusters, performed DOI-2. PDI was the maneuver that brakes the LM out of lunar orbit and lands it softly on the surface of the Moon. This was the only maneuver to use the main engine of the Descent Propulsion System (DPS).

We are concerned only with the PDI maneuver. The simulation begins with powered descent initiation and ends with the landing on the Moon. As we can see in Table 6, PDI was pre-planned as a 720-second, 6,702 ft/s burn (2,042.8 m/s); with the actual burn being 717 seconds and 6,698 ft/s (2,041.6 m/s). Since information on the pre-planned descent is more readily available than the actual descent, the simulation will be based on the pre-planned profile.

Below is diagram and summary of Apollo 17's powered descent:


From Apollo 17 Press Kit, Release No. 72-220K, 26-Nov-1972.
Republished in APOLLO 17, The NASA Mission Reports, Volume One, Apogee Books.

The DPS engine was ignited at 10% throttle and held there for 26 seconds to allow the DPS engine gimbal to be aligned through the spacecraft center of gravity before throttling up to maximum thrust. The braking phase was designed for efficient reduction of orbit velocity and, therefore, used maximum thrust for most of the phase; however, the DPS was throttled during the final two minutes of this phase. The DPS was throttleable only between 10% and 60%.

The approach phase provided visual monitoring of the approach to the lunar surface. At "high gate" (from old aircraft-pilot parlance meaning the beginning of the approach to an airport in a landing path) the LM pitched forward to give the command pilot a view of the moon. "Low gate" was the start of the landing phase, and was the point for making a visual assessment of the landing site to select either automatic or manual control.

Below we have additional information regarding the approach and landing phases. (The LPD angle, which was read from the guidance computer by the LM pilot, tells the Commander where to look along the scale scribed on his window to see where the computer thinks the LM will land.)


From Apollo 17 Press Kit, Release No. 72-220K, 26-Nov-1972.
Republished in APOLLO 17, The NASA Mission Reports, Volume One, Apogee Books.

Control of the thrust vector is key to both the landing and the simulation. A program of throttle settings and pitch angles must be derived to guide the LM to a successful landing. The following is an example showing how the LM pitched forward and the engine was throttle back during the approach (visibility) and landing phases of a mission. This is just a generic diagram and is not specific to any particular mission. The pitch angles, thrust settings, velocities, altitudes and distances shown may be different for Apollo 17.


Apollo Spacecraft News Reference (Lunar Module), Public Affairs, Grumman Aerospace Corporation, 1971.
Republished by Apogee Books, 2005.

Mass and Thrust

We must know the lunar module mass, propellant mass, and engine thrust, which we obtain from the following sources:

From Selected Mission Weights, LM at powered descent initiation = 36,686 lbm (16,640.5 kg), at lunar landing = 18,305 lbm (8,303.0 kg).
From LM Descent Stage Propellant Status, loaded propellant = 19,564.2 lbm (8,874.2 kg), consumed propellant = 18,248.9 lbm (8,277.6 kg).
From Launch Vehicle/Spacecraft Key Facts, LM descent stage maximum rated thrust = 9,870 lbf (43,904 N).
From Apollo Spacecraft News Reference (Lunar Module), Grumman Aerospace Corp., minimum engine thrust = 1,280 lbf (5,594 N)

The difference between the LM starting and ending mass, 36,686 - 18,305 = 18,381 lbm, is 132.1 lbm more than mass of DPS propellant consumed, which is likely the mass of RCS propellant used. To account for this difference, the simulation will progressively reduce the LM dry mass (total mass less DPS propellant) from its PDI value to its landing value:

LM dry mass @ PDI = 36,686 - 19,564.2 = 17,121.8 lbm (7,766.3 kg)

LM dry mass @ landing = 18,305 - (19,564.2 - 18,248.9) = 16,989.7 lbm (7,706.4 kg)

Propellant consumption rate will be determined from the thrust. From Astronautix.com, the descent engine specific impulse was 311 seconds. Using equation (1.23) from my Rocket Propulsion web page, we can calculate the propellant mass flow rate at any throttle setting:

Propellant mass flow rate, q = (43,904 × Throttle setting) / (311 × 9.80665), in kg/s.

For example, the propellant mass flow rate at a throttle setting of 60% is (43,904 × 0.60) / (311 × 9.80665) = 8.637 kg/s. Although throttling an engine may cause a small reduction in the specific impulse, we'll ignored this for the simulation.

Although most sources list the minimum thrust setting as 10%, we'll use the thrust value of 1,280 lbf (approximately 13%) given in the Grumman documentation.

Computer Simulation

The simulation progresses through a series of small time steps, with the LM's state vector updated at each step. The physics is pretty basic, so only a brief description is included here. Each step of the simulation includes the following operations:

(1) Calculate mass of propellant remaining.
(2) Set throttle per preprogrammed commands and calculate thrust.
(3) Set pitch angle per preprogrammed commands.
(4) Calculate gravity as a function of altitude, g=GM/r2
(5) Calculate acceleration as a function of mass, velocity, thrust and gravity (see below).
(6) Update velocity as a function of acceleration and time.
(7) Update flight path angle, f=arctan(Vv/Vh).
(8) Update altitude as a function of vertical velocity and time.

Although most operations are self-explanatory, a brief description of acceleration is warranted. Acceleration is broken down into vertical and horizontal components, where the vertical component is normal to the Moon's surface. The instantaneous vertical and horizontal accelerations of a moving body in reference to this surface are Av=Vh2/r and Ah=-VhVv/r. To this we add the accelerations resulting from gravity and thrust. Gravity, of course, acts vertically downward. The vertical and horizontal components of thrust are a function of the pitch angle.

To calculate the velocity change over a period of time, the accelerations at the start of the period and the end of the period are averaged. Likewise, altitude is calculated using the average vertical velocity. Due to the interdependency of all the variables, this averaging technique requires the problem be solved by iteration.

As mentioned earlier, throttle settings and pitch angles were derived, via trial and error, to produce a working simulation:

Throttle Settings & Pitch Angles
Time
(seconds)
Throttle
(%)
Pitch
(degrees)
0 to 25 12.97 90
26 to 59 100 90
60 to 119 100 84
120 to 217 100 78
218 to 337 100 73
338 to 439 100 68
440 to 448 100 61.1
449 to 559 60 61.1
560 to 579 60 31
580 to 599 53 20
600 to 619 48 20
620 to 639 43 20
640 to 659 40 18
660 to 679 34.3 15.5
680 to 709 32.33 3.6
710 to 720 32.33 0.1

The simulation yields the following at landing:

For a second-by-second print out of the complete simulation, see here: Descent of Apollo 17 Lunar Module

Encouragingly, the velocity change, Dv, is very close to that reported – 6,672.7 ft/s for the simulation versus 6,698 ft/s as shown in Table 6. This difference is insignificant, therefore the simulation nicely confirms the numbers reported by NASA and Grumman.

The simulation's propellant consumption, however, is quite a bit less than that reportedly consumed – 17,822.0 lbm versus 18,248.9 lbm. The difference appears to be due to the DPS specific impulse. Attaining the indicated Dv using the amount of propellant shown in the NASA documents results in a specific impulse of about 302 seconds. It's possible a less than nominal engine performance was attained. Another possibility is that the engine suffers a significant loss in performance when operating in a throttled down condition, though my own calculations indicate that such a large loss should not be expected. Reworking the simulation using the lower specific impulse results in a close match to the reported propellant use.

Let's now spot check the simulation against points in the Apollo 17 pre-launch plan, which we saw earlier:

LM Powered Descent Summary
Event Time
(min:sec)
Horiz. Velocity
(ft/s)
Vert. Velocity
(ft/s)
Height
(ft)
  Powered Descent Initiation 0:00 5,568
5,568
-67
-67
56,544
56,544
  Throttle to Maximum Thrust 0:26 5,542
5,537
-65
-63
54,823
54,851
  DPS Throttle Recovery 7:20 1,202
1,272
-90
-89
25,746
23,746
  High Gate 9:20 311
328
-177
-185
8,159
8,217
  Low Gate 10:40 81
80
-25
-26
709
722
  Landing 12:00 0
0
-5
-3
6
0

BLACK = Pre-Launch Plan — RED = Simulation

Below is the simulated approach overlaid on the Apollo 17 planned approach. The simulation doesn't follow the planned approach exactly, but the correlation is strong. A closer match could be obtained with further tweaking of the throttle settings and pitch angles, but a greater degree of correlation is unnecessary for this demonstration. The simulation has satisfactorily confirmed the feasibility of the descent/landing plan and the correctness of the numbers reported by NASA.

Conspiracy Claims

The moon landing conspiracy theorists often argue the lunar module was incapable of landing on the moon, sometimes claiming an insufficient propellant load, but usually simply expressing disbelief and denial. Since incredulity is not an argument, no response is needed. As for the amount of propellant, this simulation clearly demonstrates that the propellant load was adequate for landing on the moon.


Also see: Lunar Module Ascent Simulation.