Robert A. Braeunig
In 2005 I programmed a computer simulation of the ascent of Apollo 17's lunar module, Challenger, from the surface of the Moon to lunar orbit. Why Apollo 17? No particular reason other than it being the last of the six lunar landings. Thanks to some newly found data, I've been able to produce a new and improved simulation. Here I present my findings along with a description of my work.
In order to simulate a launch, we need to have a target to shoot for. Let's start by examining the following 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.
On the second line of the table we see the ascent burn. For now, the data we need to look at is the resulting apolune and perilune, which is listed as 48.5 and 9.1 nautical miles respectively. I have not been able to confirm Challenger's exact altitude at the time of ascent orbit cutoff, but the typical insertion point was downrange of the perilune at an altitude of about 10 n.mi, so let's use this altitude for lack of better data. Using the computational methods described in my Orbital Mechanics web page, we can calculate the following:
Altitude at insertion, h = 10 × 1,852 = 18,520 m
Radius at insertion, r = 18,520 + 1,738,000 = 1,756,520 m
Semi-major axis, a = (48.5 + 9.1) / 2 × 1,852 + 1,738,000 = 1,791,340 m
Eccentricity, e = (48.5 × 1,852 + 1,738,000) / 1,791,340 - 1 = 0.020366
True anomaly, n = arccos[ (1,791,340 × (1 - 0.0203662) / 1,756,520 - 1) / 0.020366 ] = 17.72o
Flight-path angle, f = arctan[ 0.020366 × sin(17.72) / (1 + 0.020366 × cos(17.72)) ] = 0.3484o
Velocity, v = SQRT[ 4.902794E+12 × 1,791,340 × (1 - 0.0203662) ] / (1,756,520 × cos(0.3484)) = 1,687 m/s
Therefore, our launch goal is to achieve engine cutoff at an altitude of 18,520 m, a flight-path angle of 0.3484o, and a velocity of 1,687 m/s. Doing so will place the LM in an orbit matching that attained by Challenger. Or course, for this simulation, we'll be happy with something that gets us within a small margin of this target goal.
Mass and Thrust
We must know the ascent stage mass, propellant mass, and engine thrust, which we obtain from the following sources:
From Selected Mission Weights, LM ascent stage at lunar liftoff = 10,997 lbm (4,988.2 kg), at orbit insertion = 6,042 lbm (2,740.6 kg).
From LM Ascent Stage Propellant Status, loaded propellant = 5,261.7 lbm (2,386.7 kg), consumed propellant = 4,977.2 lbm (2,257.6 kg).
From Apollo Spacecraft News Reference (Lunar Module), Grumman Aerospace Corp., nominal engine thrust = 3,500 lbf (15,569 N)
Although a total of 4,977.2 lbm propellant was consumed, not all of this was consumed during ascent. The ascent propulsion system (APS) was also used to perform a 3.2-second, 53.8 ft/s TPI burn (see Table 7 above). From Selected Mission Weights, the mass of the LM ascent stage at terminal phase initiation (TPI) was 5,970 lbm, and, from Astronautix.com, the ascent engine specific impulse was 311 seconds. Using the computational methods described in my Rocket Propulsion web page, we can calculate the amount of propellant used to produce a Dv of 53.8 ft/s:
Effective exhaust gas velocity, C = 311 × 32.174 = 10,006 ft/s (3,050 m/s)
Propellant consumed = 5,970 - 5,970 × e-(53.8 / 10,006) = 32.0 lbm (14.5 kg)
Therefore, the propellant consumed during ascent was 4,977.2 - 32.0 = 4,945.2 lbm (2,243.1 kg).
From Table 7 above, we see the ascent burn time was 440.9 seconds; therefore, the average propellant mass flow rate was:
Mass flow rate, q = 4,945.2 / 440.9 = 11.216 lbm/s (5.088 kg/s)
Let's now check the engine thrust based on the propellant mass flow rate and specific impulse:
Average thrust, F = 11.216 × 311 = 3,488.2 lbf (15,516 N)
Since this number is a little lower than the nominal thrust of 3,500 lbf, we'll use the lower figure to be conservative.
As we saw earlier, NASA reports the mass of the LM as 10,997 lbm at launch and 6,042 lbm at insertion, indicating a mass reduction of 4,955 lbm. This reduction is 9.8 lbm more than the mass of APS propellant consumed (as interpreted above). As a consequence, the LM's dry mass (total mass less APS propellant) is 9.8 lbm lighter at insertion than at launch. The simulation will progressively decrease the dry mass from its launch value to its insertion value:
LM dry mass @ launch = 10,997 - 5,261.7 = 5,735.3 lbm (2,601.5 kg)
LM dry mass @ insertion = 6,042 - (5,261.7 - 4,945.2) = 5,725.5 lbm (2,597.0 kg)
Steering the LM to the desired insertion point is achieved by controlling the thrust direction. The LM starts with the thrust directed vertically downward, i.e. a pitch angle of zero degrees. As the LM rises, it pitches over through a series of preprogrammed maneuvers until the thrust is near horizontal, i.e. a pitch angle of 90 degrees.
The illustration to the right shows the first 16 seconds of the LM's ascent. Pitch over begins at 10 seconds, with a pitch of 40o being achieved at 14 seconds – a pitch rate of 10o/s – and a pitch of 52o at 16 seconds. Little information has been found regarding the pitch angle after 16 seconds. It is necessary, through educated guesses and trial and error, to derive a series of pitch adjustments that gets us to our targeted insertion point.
Although many solutions to the pitch program are possible, large deviations are not possible without producing results radically different from those required. Any solution found, therefore, will no doubt be a reasonably close facsimile of the actual trajectory flown.
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 pitch angle per preprogrammed commands.
(3) Calculate gravity as a function of altitude, g=GM/r2
(4) Calculate acceleration as a function of mass, velocity, thrust and gravity (see below).
(5) Update velocity as a function of acceleration and time.
(6) Update flight path angle, f=arctan(Vv/Vh).
(7) 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 described earlier, the pitch angles are determined by trail and error. To reach the targeted insertion point, I derived the following pitch program:
|0 to 10||0|
|10 to 15||0 to 50 at 10o/s|
|16 to 59||52|
|60 to 119||57|
|120 to 179||63|
|180 to 239||69.3|
|240 to 319||76|
|320 to 419||87.242|
|420 to cutoff||90|
The simulation yields the following at orbit insertion:
For a second-by-second print out of the complete simulation, see here: Ascent of Apollo 17 Lunar Module
Although the insertion orbit matches favorably the 9.1 × 48.5 n.mi. orbit reported by NASA, the simulation required an additional 5.2 seconds burn time and 58.3 lbm (26.5 kg) of propellant. Encouragingly, the velocity change, Dv, is almost exactly that reported – 6,077.8 ft/s for the simulation versus 6,075.7 ft/s as shown in Table 7. This near perfect Dv match confirms the validity of the simulation methodology and formulae.
The reason the simulation required more propellant and a longer burn time is likely due to the APS specific impulse. Attaining the indicated Dv using the amount of propellant shown in the NASA documents requires a specific impulse of about 316 seconds. It's possible the actual engine performance was greater than that commonly reported. Reworking the simulation using the higher specific impulse results in a near perfect match to the Apollo 17 ascent and orbit insertion.
The moon landing conspiracy theorists often complain that the LM was not capable of performing the maneuvers attributed to it. Their arguments rarely consist of anything more than incredulity and appeals to common sense. For example, conspiracy filmmaker Bart Sibrel posts the following on his website:
"The top portion of the lunar module which landed on the moon supposedly popped up off the moon with two astronauts aboard, entered lunar orbit 60 miles up, and docked with the command module in lunar orbit. To look at its design and think such could have actually occurred is absolutely ludicrous. The fuel tanks were nowhere near one-sixth the size of those on the space shuttle as one would expect to achieve lunar orbit." — Bart Sibrel
One would expect the propellant tanks to be "one-sixth the size of those on the space shuttle" only if one has no idea what he's talking about. A rocket scientist knows better. Sibrel's claim is apparently inferred from the fact that lunar surface gravity is one-sixth that of Earth. Unfortunately for Mr. Sibrel, this has little to do with it. The mass of propellant required is an exponential function of velocity change and exhaust gas velocity. As we've seen above, the Dv required to attain lunar orbit is about 1,852 m/s. Knowing the APS exhaust gas velocity is 3,050 m/s, the mass ratio (initial mass divided by final mass) is calculated as follows:
Mass ratio = e(Dv / Ve) = e(1,852 / 3,050) = 1.835
This means that 0.835 part propellant is required for every one part dry mass.
Attaining orbit around Earth is a far more difficult task, typically requiring a Dv of about 9,000 m/s. The required mass ratio is, therefore:
Mass ratio = e(9,000 / 3,050) = 19.12
Meaning 18.12 parts propellant are needed per one part dry mass. Or course, Earth launchers use multiple stages that lower the required mass ratio. For instance, the two-stage Titan II missile, which is probably the closest Earth-launch analogue to the LM, has a mass ratio of 15. Launching from Earth, therefore, requires about 15 / 0.835 = 18 times more propellant than launching from the Moon. This means Sibrel's amateurish claim is off by a factor of three.
Furthermore, what is most important is the mass of the propellant, not its volume. The LM used high-density hypergolic propellants, while the Space Shuttle's large external tank contains very low-density liquid hydrogen and LOX. A given mass of liquid hydrogen requires more than twelve times the storage volume of an equal mass of hypergolic fuel.
From LM Ascent Stage Propellant Status, we see that the Apollo 17 lunar module was loaded with 5,261.7 lbm of propellant, consisting of 2,026.9 lbm fuel and 3,234.8 lbm oxidizer. According to Apollo Spacecraft News Reference (Lunar Module), Grumman Aerospace Corporation, each ascent stage propellant tank had a volume of 36 cubic feet. The fuel was Aerozine-50 (a mixture of hydrazine and UDMH) and the oxidizer was nitrogen tetroxide (N2O4). From Astronautix.com, Aerozine-50 and nitrogen tetroxide have densities of 0.903 g/cc and 1.45 g/cc respectively. Therefore, each tank had the holding capacity of:
|From Virtual LM, Apogee Books, Scott P. Sullivan.|
Fuel capacity = 36 × 62.428 × 0.903 = 2,029 lbm
Oxidizer capacity = 36 × 62.428 × 1.45 = 3,259 lbm
Since 2,029 > 2,026.9 and 3,259 > 3,234.8, the tanks had the capacity to hold the reported mass of propellant. (The oxidizer-to-fuel mixture ratio of 1.6 is common for this propellant.)
Given a volume of 36 cubic feet, the spherical tanks would have an inside diameter of 4.1 feet (1.25 meters). We can see this size is proportionally consistent with the cut-away illustration to the right, where the height of the LM is 12 feet 4 inches (3.76 meters). The fuel tank is shown in orange and the oxidizer tank in blue.
To finish off this analysis, let's check the ascent stage's overall Dv capability versus that required to complete the mission requirements. Since the LM's mass at lunar liftoff was 10,997 lbm, and 5,261.7 lbm of this was propellant, the lunar module's mass ratio was:
Mass ratio = 10,997 / (10,997 - 5,261.7) = 1.917
Thus, the attainable Dv of the ascent propulsion system was:
APS Dv = 3,050 × LN(1.917) = 1,985 m/s
The Apollo 17 APS was used twice to produce two velocity changes. From Apollo 17 Lunar Orbit Summary, Table 7, the velocity change of the ascent burn was 6,075.7 ft/s and that of the TPI burn was 53.8 ft/s, for a total Dv of 6,129.5 ft/s (1,868 m/s). My simulation has conclusively verified that the ascent burn Dv is correctly reported, and the TPI burn is an easily verified orbit altitude change. There is no question the velocity changes reported by NASA are correct and true.
Since 1,985 m/s > 1,868 m/s, the ascent propulsion system had more than enough capacity to produce the reported velocity changes.
Launch Pad Acceleration (Added 25-March-2016)
Since publishing this article in 2009, some conspiracy theorists have used it to claim that the video record of Apollo 17's launch from the surface of the moon is not authentic. The argument made is that the lunar module is seen to rise off the launch pad with greater acceleration than suggested by the results of this simulation. That is entirely true, but instead of asking why it is true, the conspiracists jump immediately to the explanation they want to be true, i.e. the video is fake. A competent investigator would research other possible explanations for the observation that don't require extraordinary claims, but that's not what we get from conspiracy theorists. A conspiracist is not interested in seeking the truth, his objective is to find any evidence that seemingly supports his belief while ignoring or rejecting anything that contradicts it.
It should be stated that the objective of this simulation was not to precisely recreate the behavior of the lunar module at the moment of liftoff. The goal of the exercise was to see if the LM could attain lunar orbit given its reported mass and propellant load. Any momentary extra "push" given to the LM at liftoff would have inconsequential effects on the final outcome of the simulation; therefore, no attempt was made to simulate the exact liftoff conditions. The opening sequence of the simulation does not compare to the real-life launch in terms of velocity and altitude versus time, thus it is expressly unsuitable for the type of analysis performed by the conspiracy theorists. (It is telling that no conspiracy theorist has ever asked me about it.)
So why does the lunar module rise off its launch platform at a seemingly greater than expected acceleration? There are two main contributing factors that come into play.
First, when a rocket engine is fired, there is a brief period, called the ignition transient, during which extreme conditions can occur, such as high pressure and temperature peaks. For the LM's ascent engine, the ignition transient lasted for about 350 milliseconds, during which stronger than normal thrust was produced.
The second factor can be seen in the illustration to the right (back view). Notice that the exit of the ascent engine nozzle sat tight against the upper deck of the descent stage. On start-up, the gas pressure at the nozzle exit rose to higher than normal values due to the constricted flow of exhaust gas. This produced a high degree of transient pressure thrust just at the moment of liftoff. Once the LM climbed high enough that the exhaust could flow from the nozzle unrestricted, the pressure and thrust fell to nominal levels.
These factors combined to give the ascent stage a brief but significant spike in thrust immediately after engine ignition. This extra "kick" caused the LM to jump off the launch platform, attaining greater altitude and speed within the first second of flight than otherwise possible, and producing the faster than expected initial climbout observed in the Apollo 17 video. As explained, this was only a transitory condition, after which the LM's acceleration was consistent with the steady-state operation of its ascent engine.
Also be advised that time measurements made from Internet-posted videos should be considered suspect. Most of these videos have gone through format and framerate conversions of unknown type and origin. These manipulations can change the playback speed, rendering the videos unreliable for making time and velocity measurements.
Also see: Lunar Module Descent Simulation.