Robert A. Braeunig
© July-2010, Revised Dec-2013
My first attempt to program a Saturn V simulation was back in 1995. The results were encouraging and the knowledge gained lead me to launch my web site Rocket & Space Technology, though I never released the simulation publicly. Since that time my knowledge, experience and database has increased substantially. Combined with the fact the original simulation was written in a now obsolete programming language, I figured now was a good time to give it another try. In this article I present a new simulation along with a description of my work. I have selected Apollo 11 as my subject mission.
The first order of business is data gathering. In order to simulate a launch, we must know the sequence and timing of key launch events. This we obtain from Apollo by the Numbers. The following is the Apollo 11 timeline starting with S-IC engine start command through Earth orbit insertion, given in hours, minutes and seconds Ground Elapsed Time (GET):
Apollo by the Numbers - Apollo 11 Timeline
S-IC engine start command: -000:00:08.9
S-IC engine ignition (#5): -000:00:06.4
All S-IC engines thrust OK: -000:00:01.6
Range zero: 000:00:00.00
All holddown arms released (1st motion): 000:00:00.3
Liftoff (umbilical disconnect) (1.07g): 000:00:00.63
Tower clearance yaw maneuver started: 000:00:01.7
Yaw maneuver ended: 000:00:09.7
Pitch and roll maneuver started: 000:00:13.2
Roll maneuver ended: 000:00:31.1
Mach 1 achieved: 000:01:06.3
Maximum dynamic pressure (735.17 lb/ft2): 000:01:23.0
Maximum bending moment (33,200,000 lbf-in): 000:01:31.5
S-IC center engine cutoff command: 000:02:15.2
Pitch maneuver ended: 000:02:40.0
S-IC outboard engine cutoff: 000:02:41.63
S-IC maximum total inertial acceleration (3.94 g): 000:02:41.71
S-IC maximum Earth-fixed velocity; S-IC/S-II separation command: 000:02:42.30
S-II engine start command: 000:02:43.04
S-II ignition: 000:02:44.0
S-II aft interstage jettisoned: 000:03:12.3
Launch escape tower jettisoned: 000:03:17.9
Interactive guidance mode initiated: 000:03:24.1
S-IC apex: 000:04:29.1
S-II center engine cutoff: 000:07:40.62
S-II maximum total inertial acceleration (1.82 g): 000:07:40.70
S-IC impact (theoretical): 000:09:03.7
S-II outboard engine cutoff: 000:09:08.22
S-II maximum Earth-fixed velocity; S-II/S-IVB separation command: 000:09:09.00
S-IVB 1st burn start command: 000:09:09.20
S-IVB 1st burn ignition: 000:09:12.20
S-IVB ullage case jettisoned: 000:09:21.0
S-II apex: 000:09:47.0
S-IVB 1st burn cutoff: 000:11:39.33
S-IVB 1st burn maximum total inertial acceleration (0.69 g): 000:11:39.41
Earth orbit insertion; S-IVB 1st burn maximum Earth-fixed velocity: 000:11:49.33
Furthermore, the S-IVB stage of the Saturn V was restarted for Translunar Injection (TLI). Below are the events associated with this maneuver:
S-IVB 2nd burn restart preparation: 002:34:38.2
S-IVB 2nd burn restart command: 002:44:08.2
S-IVB 2nd burn ignition (STDV open): 002:44:16.2
S-IVB 2nd burn cutoff: 002:50:03.03
S-IVB 2nd burn maximum total inertial acceleration (1.45 g): 002:50:03.11
S-IVB 2nd burn maximum Earth-fixed velocity: 002:50:03.5
S-IVB safing procedures started: 002:50:03.8
Translunar injection: 002:50:13.03
The Saturn V
Bulk Parameters & Thrust
From Apollo by the Numbers, we obtain the following launch vehicle mass data:
Apollo by the Numbers - Ground Ignition Weights
Ground Ignition Time: -6.4 sec
S-IC stage, dry: 287,531 lbm
S-IC stage, oxidizer: 3,305,786 lbm
S-IC stage, fuel: 1,424,889 lbm
S-IC stage, other: 5,442 lbm
S-IC stage, total: 5,023,648 lbm
S-IC/S-II interstage, dry: 11,477 lbm
S-II stage, dry: 79,714 lbm
S-II stage, oxidizer: 819,050, lbm
S-II stage, fuel: 158,116 lbm
S-II stage, other: 1,260 lbm
S-II stage, total: 1,058,140 lbm
S-II/S-IVB interstage, dry: 8,076 lbm
S-IVB stage, dry: 24,852 lbm
S-IVB stage, oxidizer: 192,497 lbm
S-IVB stage, fuel: 43,608 lbm
S-IVB stage, other: 1,656 lbm
S-IVB stage, total: 262,613 lbm
Total Instrument Unit: 4,275 lbm
Spacecraft/Lunar Module Adapter: 3,951 lbm
Lunar Module: 33,278 lbm
Command and Service Module: 63,507 lbm
Total Launch Escape System: 8,910 lbm
Total Spacecraft (CSM): 109,646 lbm
Total Vehicle: 6,477,875 lbm
From the following pages we obtain key dimensional and thrust data:
Apollo by the Numbers - Launch Vehicle/Spacecraft Key Facts, Page 1 and Page 2
First Stage (S-IC)
Diameter, base: 33.000 ft
Diameter, top: 33.000 ft
Height: 138.030 ft
Engines, type/number: F-1/5
Rated thrust each engine: 1,522,000 lbf
Rated thrust total: 7,610,000 lbf
Thrust at 35 to 38 sec: 7,552,000 lbf
Second Stage (S-II)
Diameter: 33.000 ft
Height: 81.500 ft
Engines, type/number: J-2/5
Rated thrust each engine: 230,000 lbf
Rated thrust total: 1,150,000 lbf
Thrust, engine start command +61 sec: 1,155,859 lbf
Thrust, outboard engine cutoff: 625,751 lbf
Third Stage (S-IVB)
Diameter, base: 33.000 ft
Diameter, top: 21.667 ft
Height: 58.630 ft
Engines, type/number: J-2/1
Rated thrust total: 230,000 lbf
Thrust, 1st burn: 202,603 lbf
Thrust, 2nd burn: 201,061 lbf
The above is compiled from Saturn launch vehicle flight evaluation reports. Thrust for S-IC stage is at sea level and for the S-II and S-IVB stages is at altitude. Thrust listed at "35 to 38 sec", "Engine Start Command + 61 seconds", and at "Outboard Engine Cutoff" is actual thrust as flown.
Not the entire initial propellant load is useable – there is always a small portion remaining in the tanks, feed lines and engines. The following table gives the actual quantity of propellant used as recorded in the Saturn V vehicle flight evaluation reports:
Apollo by the Numbers - Launch Vehicle Propellant Use
|Event||Burn start||Burn end||Burn time||Burn rate
|S-IC Burn, sec||-6.4||161.63||168.03||---|
|Oxidizer (LOX), lbm||3,305,786||39,772||3,266,014||19,437.1|
|Fuel (RP-1), lbm||1,424,889||30,763||1,394,126||8,296.9|
|S-II Burn, sec||164.00||548.22||384.22||---|
|Oxidizer (LOX), lbm||819,050||3,536||815,514||2,122.5|
|Fuel (LH2), lbm||158,116||10,818||147,298||383.4|
|S-IVB 1st Burn, sec||552.20||699.33||147.13||---|
|Oxidizer (LOX), lbm||192,497||135,144||57,353||389.8|
|Fuel (LH2), lbm||43,608||31,736||11,872||80.7|
|S-IVB 2nd Burn, sec||9,856.20||10,203.03||346.83||---|
|Oxidizer (LOX), lbm||134,817||5,350||129,467||373.3|
|Fuel (LH2), lbm||29,324||2,112||27,212||78.5|
|S-IVB Stage, 1st burn||4.414||---||4.831||---|
|S-IVB Stage, 2nd burn||4.597||---||4.758||---|
And from Saturn V Launch Vehicle Flight Evaluation Report - AS-506 Apollo 11 Mission, Report No. MPR-SAT-FE-69-9, September 20, 1969, page 5-3: "The best estimate of propellants consumed between ignition and HDA release was 39,374 kilograms (86,803 lbm)."
The following is a description of the second and third stage propellant utilization subsystem, from Technical Information Summary, Apollo-11 (AS-506), Apollo Saturn V Space Vehicle, Document No. S&E-ASTR-S-101-69, June 25, 1969:
The propellant utilization (PU) subsystem controls the mixture ratio (MR) of the LOX/LH2. The PU subsystem consists of a rotary valve, to control the amount of LOX flowing to the engine, and electrical controls for the valve. At engine start, the PU value is in the neutral position and supplies a MR of 5.0:1. Approximately 5 seconds after start, electrical signals from the LVDC in the IU commands the PU valve to supply a nominal MR of 5.5:1. Five minutes and 20 seconds after engine start, the LVDC commands the PU valve to a MR of 4.5:1 for the remainder of the S-II stage burn. Capacitance probes in the propellant tanks provide telemetered data to ground stations so the propellant consumption can be monitored.
Prior to engine start the PU valve is commanded to the neutral position to obtain a MR of 5.0:1. The PU valve remains at the 5.0:1 position during the first burn. Prior to engine restart (first opportunity) the PU valve is commanded by the switch selector to a MR of 4.5:1 and remains at this position for approximately one minute and 55 seconds of the S-IVB burn. The PU valve is then commanded to the neutral position (5.0:1). If the S-IVB restart is delayed to the second opportunity, the MR is shifted from 4.5:1 to 5.0:1 when the engine reaches 90 percent thrust.
According to the Saturn V Launch Vehicle Flight Evaluation Report - AS-506 Apollo 11 Mission, the S-II PU mixture ratio shift occured at a range time of 498.0 seconds.
The S-IC/S-II interstage adapter contained four solid propellant ullage motors to provide artificial gravity by momentarily accelerating the second stage forward after first stage burnout. This moment of forward thrust made certain the liquid propellant was properly positioned to be drawn into the pumps prior to starting the second stage engines. The ullage motor nozzles were canted 10 degrees to reduce exhaust impingement against the interstage structure.
Similarly, the S-IVB stage contained two solid propellant ullage motors to provide positive stage acceleration during separation, to position the liquid propellant toward the aft end of the tanks, and to force propellant boil-off vapors to the forward end for venting. The S-IVB ullage motor assemblies were jettisoned after use. According to Thiokol, the mass of each TX-280 ullage motor was 29.7 lbm.
From Technical Information Summary, Apollo-11 (AS-506), Apollo Saturn V Space Vehicle, Document No. S&E-ASTR-S-101-69, June 25, 1969:
S-II Aft Interstage
Ullage motors: 4 each
Thrust: ≈23,000 lbs per each
Burn time: ≈3.5 sec
Ullage motors: 2 each
Thrust: ≈3,400 lbs per each
Burn time: ≈4 sec
The Lunar Landing Mission Press Kit, July 6, 1969, Release No. 69-83K gives the following ullage propellant data:
S-II ullage propellant used: 73 lbm before S-IC separation, 1,288 lbm after S-IC separation.
S-IVB ullage propellant used: 96 lbm before S-IVB ignition, 22 lbm after S-IVB ignition.
The above gives us all the launch vehicle data needed to program a simulation; however, some further analysis is needed to make sense of all this information. Let's start by studying the thrust and propellant flow rates for each of the Saturn V stages.
Thrust & Flow Rate
The propellant use table above shows a S-IC propellant burn rate of 27,734.0 lbm/s, however this is an average value. To determine the actual burn rate versus time, we refer to the following, from Saturn V Launch Vehicle Flight Evaluation Report - AS-506 Apollo 11 Mission, Report No. MPR-SAT-FE-69-9, September 20, 1969, page 5-4:
The top chart shows the total stage propellant flow rate versus range time (synonymous to GET). We see that the flow rate was approximately constant from liftoff to the 70-second mark, after which the rate steadily increased to a maximum value at the time of center engine cutoff, followed by an approximate 20% drop. Getting an accurate measurement from the graph is difficult, but I estimate a flow rate of about 29,125 lbm/s for the first 70 seconds, increasing to a maximum of 29,600 lbm/s, and then falling of to about 23,600 lbm/s for the remainder of the burn. Referring to our list of launch events, we see that first motion occured at 0.30 s, center engine cutoff at 135.2 s, and outboard engine cutoff at 161.63 s. We can, therefore, calculate the total propellant consumed based on these burn times and our estimated flow rate values:
(70.00 – 0.30) × 29,125 + (135.2 – 70.00) × (29,125 + 29,600) / 2 + (161.63 – 135.20) × 23,600 = 4,568,195 lbm
This total is about 0.1% less than the reported propellant consumption. We know that a total of 4,660,140 lbm of propellant was burned, with 86,803 lbm consumed during thrust buildup prior to first motion. Therefore, the total propellant burned during ascent was, 4,660,140 – 86,803 = 4,573,337 lbm. We conclude from this that our flow rates are too low and must be increased. We'll bump up each flow rate proportionately until our calculated propellant total equals the known total consumption.
(70.00 – 0.30) × 29,157.78 + (135.2 – 70.00) × (29,157.78 + 29,633.32) / 2 + (161.63 – 135.20) × 23,626.56 = 4,573,337 lbm
To determine the thrust we again refer to Figure 1. The bottom chart shows stage specific impulse versus range time. At liftoff the specific impulse was 264.6 s (sea level), and at cutoff the specific impulse was 304 s (vacuum). The thrust is simply the propellant flow rate times the specific impulse. For example, the sea level thrust at liftoff is,
29,157.78 lbm/s × 264.6 s = 7,715,150 lbf
Also note that there is a brief period of thrust decay at engine cutoff, which has been ignored. For the simulation it is assumed that engine cutoff occurs instantly.
We can summarize the S-IC operation as follows:
|Thrust, lbf||Propellant, lbm||Rate|
To analyze the second stage we again refer to Saturn V Launch Vehicle Flight Evaluation Report - AS-506 Apollo 11 Mission, Report No. MPR-SAT-FE-69-9, September 20, 1969, page 6-6, for the following:
We can distinctly see abrupt changes in propellant flow rate and specific impulse resulting from center engine cutoff (CECO) and engine mixture ratio shift (EMR). Between these changes we see a constant flow rate. I estimate a flow rate of about 2,725 lbm/s prior to CECO, about 2,180 lbm/s between CECO and EMR, and about 1,620 lbm/s after EMR. S-II ignition occurred at 164.0 s, center engine cutoff occurred at 460.62 s, the EMR shift occured at 498.0 s, and outboard engine cutoff occurred at 548.22 s. As before, we calculate the total propellant consumed based on estimated flow rates and times:
(460.62 – 164.00) × 2,725 + (498.0 – 460.62) × 2,180 + (548.22 – 498.0) × 1,620 = 971,134 lbm
This is about 1% higher than the reported propellant consumption of 962,812 lbm. We'll reduce the flow rates as neccessary to attain the actual total consumption.
(460.62 – 164.00) × 2,701.65 + (498.0 – 460.62) × 2,161.32 + (548.22 – 498.0) × 1,606.11 = 962,812 lbm
Table 6-1 of Saturn V Launch Vehicle Flight Evaluation Report - AS-506 Apollo 11 Mission gives the specific impluses of the five engines at 61 s after startup as 425.2, 425.3, 424.8, 423.9, and 424.5 s, for an average of 424.74 s. From the bottom chart in figure 2 above, we see a small 1 to 2 s decrease in specific impulse after CECO, to an estimated value of 423 s. After the EMR shift, the specific impulse increased to about 427 s due to the lower mixture ratio. Using these specific impulses, we calculate the thrust.
The simulation will ignore thrust buildup and thrust decay, assuming the engines turn instantly full on or full off.
The following is a summarization of the S-II operation as determined above:
For the first burn of the S-IVB stage (launch), the PU valve was placed in and remained in the neutral position, i.e. mixture ratio of 5.0:1. Therefore, the engine thrust and propellant flow rate was constant throughout the burn.
For the second burn (TLI), the engine started with the PU valve in the open position, MR 4.5:1, and then switched to the neutral position, MR 5.0:1, while the burn was in progress. This shift resulted in a change of thrust and propellant flow rate; however, a close enough approximation can be achieved by assuming the average thrust and average propellant flow rate are constants over the duration of the burn.
We therefore have:
According to the pre-launch plan, the S-II ullage propellant burned before/after S-IC separation was 73 lbm/1,288 lbm. Given a burn time of 3.5 seconds, and assuming a constant burn rate, ullage motor ignition would begin 0.19 s prior to separation and continue to 3.31 s after separation, that is, between the range times of 162.11 s and 165.61 s. The 1,361 lbm of solid propellant is included in the mass of the S-IC/S-II interstage adapter.
The S-IVB ullage propellant burned before/after S-IVB ignition is given as 96 lbm/22 lbm. Given a burn time of 4 seconds, and assuming a constant burn rate, ullage motor ignition would begin 3.25 s prior to ignition and continue to 0.75 s after separation, i.e. from 548.95 s to 552.95 s. The 118 lbm of solid propellant is included in the mass of the S-IVB stage.
The nozzles of the ullage motors are canted 10 degrees to angle the exhaust stream away from the interstage/stage wall, thereby reducing the effective forward thrust. The forward thrust is equal to the total thrust times cosine(10).
The ullage events are as follows:
|Thrust, lbf (vac)||Propellant|
|Start||End||Total||Forward||Used, lbm||Rate, lb/s|
In the preceding section we determined the changes in propellant mass that took place during launch and TLI. We will now examine all other changes in mass. The total pre-launch mass of Apollo 11 was 6,477,875 lbm, of which 5,943,946 lbm was liquid propellant. The remaining 533,929 lbm included the payload, the dry mass of the Saturn V, and "other" mass. Of this amount, 131,341 lbm attained orbit, while the rest was jettisoned during ascent (or burned in the case of ullage propellant).
Below is a table of all mass-changing events:
|Dry Mass, lbm|
|All holddown arms released (1st motion)||000:00:00.30||533,929||---|
|S-II ullage (propellant used)||000:02:42.11 to 000:02:42.30||533,929||533,856||73|
|S-IC/S-II separation command||000:02:42.30||533,856||240,883||292,973|
|S-II ullage||000:02:42.30 to 000:02:45.61||240,833||239,595||1,288|
|S-II aft interstage jettisoned||000:03:12.30||239,595||229,479||10,116|
|Launch escape tower jettisoned||000:03:17.90||229,479||220,569||8,910|
|S-IVB ullage||000:09:08.95 to 000:09:09.00||220,569||220,567.5||1.5|
|S-II/S-IVB separation command||000:09:09.00||220,567.5||131,517.5||89,050|
|S-IVB ullage||000:09:09.00 to 000:09:12.95||131,517.5||131,401||116.5|
|S-IVB ullage case jettisoned||000:09:21.00||131,401||131,341||60|
There are several factors outside of the rocket itself that affects its flight and performance. Earth's atmosphere affects a rocket in two ways: (1) the ambient air pressure reduces the thrust of the engines, and (2) the resistance offered by the air produces a restraining drag force. Earth's rotation also affects the rocket as it provides an initial velocity, the effective magnitude of which is dependent on the location of the launch site and the direction of flight.
The basic thrust equation is F = qVe+(Pe–Pa)Ae, where F is the thrust, q is the propellant mass flow rate, Ve is the exhaust gas velocity, Pe is the pressure at the nozzle exit, Pa is the ambient air pressure, and Ae is the area of the nozzle exit. From this we see that the thrust produced by an engine in an atmosphere is equal to the thrust produced in a vacuum less the product PaAe. Alternatively, an engine's thrust at altitude is equal to its sea level thrust plus (PSL–Pa)Ae.
According to The Saturn V New Reference, August 1967, the maximum nozzle exit diameter of the F-1 engine was 11 feet 7 inches, and that of the J-2 engine was 6 feet 5 inches. For the J-2 engine this yields an exit area of 4,657 square inches per engine. For the F-1 engine we can't trust the accuracy of the 11'-7" dimension since the engine was upgraded after the date of this publication. Instead we can solve for the total nozzle exit area of the S-IC stage. Dividing the difference between the vacuum thrust and the sea level thrust by the atmospheric pressure at sea level, we get the total exit area.
(8,863,960 – 7,715,150) lbf ÷ 14.696 lb/in2 = 78,172 in2
From this we can see that the effective nozzle diameter was 141.1 inches.
Drag is the resistance offered by a gas or liquid to a body moving through it. The drag force, FD, on a body acts in the opposite direction of the velocity vector and is given by the equation FD = CDrv2A/2, where CD is the drag coefficient, r is the air density, v is the body's velocity, and A is the area of the body normal to the flow.
The cross-sectional area of the second and third stages is simply the area of a circle with a diameter equal to that of the stage. The first stage, however, is more complex because, in addition to the stage core, the S-IC included fairings covering its four outboard engines and fins for stabilization. An end view of the S-IC stage is shown in the illustration below. This diagram shows the I-IC's general arrangement and is not drawn to scale.
The primary stage diameter was 33 feet. The radius of the fairings was 100 inches and the aft lip of the fairings spanned on arc of 180 degrees. The fins were 14.4 inches thick at the root, 4 inches thick at the end, and 103 inches wide at the aft end. Based on these dimensions, the total stage cross-section is calculated to be approximately 1,216 square feet.
The drag coefficient is dependent on the geometric form of the body and is generally determined by experiment. The drag coefficient is not constant but varies as a function of the Mach number. Below are sample drag coefficients for some common shapes:
No information has been found regarding the drag coefficient of the Saturn V, however at least one data point can be calculated – that at the time of maximum dynamic pressure. The Saturn V News Reference, August 1967, states "At approximately 69 seconds into the flight the vehicle experiences a condition of maximum dynamic pressure. At this time, the restraining drag force is approximately equal to 460,000 pounds." From Ascent Data we see that, for Apollo 11, maximum dynamic pressure (also called maximum q) occured at a time of 83.0 s, an altitude of 44,512 feet, and had a value of 735.17 lb/ft2. There are two methods we can use to estimate the drag coefficient at maximum q.
First, given that q = rv2/2, we substitue q for rv2/2 in the drag force equation, obtaining FD = qCDA. We now rearrange the equation, plug in our known values (assuming 460,000 lbf is applicable to Apollo 11), and solve for CD.
CD = 460,000 / (735.17 × 1,216) = 0.515
For the second method we refer ahead to the Reference Atmosphere for July/30oN, from which we estimate the density and temperature of the atmosphere at 44,512 feet (13,567 m) to be 0.0005303 slug/ft3 (0.2733 kg/m3) and 384.3o R (213.5 K). We also obtain from Apollo/Saturn V Postflight Trajectory - AS-506, Document No. D5-15560-6, October 6, 1969 that Apollo 11's earth fixed velocity at maximum q was 1,654 ft/s (504.0 m/s). (We use Earth fixed velocity because, since the atmosphere rotates with Earth's surface, this represents the speed of the rocket in relation to the air.) Using the drag force equation, we solve for CD.
CD = 460,000 × 2 / (0.0005303 × 1,6542 × 1,216) = 0.521
Since the acoustic velocity at 384.3o R is 961 ft/s, the Mach number at maximum q was about 1,654 / 961 = 1.72.
We can glean further information from NASA document Detailed Test Objectives for NASA Mission MA-6, Report No. TOR-930(21011)-3, November 10, 1961, which contains a detailed pre-flight simulation of the MA-6 mission. The report includes enough simulated data to allow for the calculation of the drag coefficient for the Mercury-Atlas launch vehicle. Although not a Saturn V, it at least gives us some reasonable numbers to use as a guideline. (Note that the Mercury-Atlas has a shape somewhat similar to the Saturn V, with a conical interstage and aft engine fairings.)
Approximate Drag Coefficient
Mercury-Atlas Launch Vehicle
First motion: 0.3
Mach 1: 0.4
Maximum (Mach 1.2): 0.55
Minimum (Mach 5): 0.2
High-hypersonic (>Mach 10): 0.26
Something not considered to this point is angle of attack. Since our drag coefficient calculation is based on the total drag force, the product CDA is correct even if the individual values of CD and A are not. If we define A to be the cross-section of the rocket at zero angle of attack, then any effects of non-zero angle of attack are included in the valve of the drag coefficient. This simple method yields sufficient accuracy for our purposes. Greater accuracy requires more information then we currently have.
Taking into consideration all the data we now have and applying some judgement, we can guesstimate a CD versus NM curve to use in our simulation. We'll use the following, which, we will see later, yields seemingly good simulated results:
This curve is defined by a set of five polynomials, each describing a separate part of the curve.
Thanks to Earth's rotation, our launch vehicle starts out with a due east velocity before it ever leaves the launch pad, which we can take advantage of to reduce the amount of velocity the rocket must provide to reach orbit. Full use of this free ride is attained with a due east launch, but this results in an orbit with an inclination equal to the latitude of the launch site, 28.447o. Since Apollo 11 required insertion into a parking orbit with an inclination of 32.5o, it was necessary to launch in an east-northeast direction.
We can estimate how much Earth's rotation contributed to the final orbital velocity by applying some simple vector mathematics. From Apollo 11 Ascent Phase we see that Apollo 11's initial space fixed velocity was 1,340.7 ft/s with a heading of 90 degrees and the velocity at S-IVB first burn cutoff was 25,561.6 ft/s. Furthermore, from Ascent Data we see that Apollo 11 was launched on a flight azimuth of 72.058o. Therefore, we have a vector diagram that looks like that to the right (not drawn to scale).
Using law of cosines, we can determine that the DV, i.e. the velocity the rocket must provide to attain orbital velocity, is 24,282.8 ft/s. Therefore, the effective "free ride" received from Earth's rotation is the difference between final orbital velocity and the DV provided by the launch vehicle, 25,561.6 – 24,282.8 = 1,278.8 ft/s (390 m/s). The loss is the DV required to produce the orbital plane change.
In order to simulate the flight of a rocket, we must have a model of how atmospheric temperature, pressure and density changes over the wide range of altitudes that the rocket will operate. In most cases we can use the International Standard Atmosphere (ISA) or the U.S. Standard Atmosphere, which is a modified version of the ISA. These standards cover all altitudes from sea level to 86 km. For higher altitudes we can use MSISE-90 data. Tabulated data from each model can be obtained from the following web source:
Rocket & Space Technology - Atmosphere Properties
The U.S. Standard Atmosphere is an idealized, steady-state representation of mean annual conditions of Earth's atmosphere at latitude 45o N. Actual atmospheric conditions will very depending upon factors such as latitude, time of year, and, for very high altitudes, solar activity. From Launch Weather we see that, for Apollo 11, the pressure and temperature were measured to be 14.798 psi and 84.9o F (302 K). We can also derive, from the fact that Apollo 11 was traveling 321.3 m/s at the time it achieved Mach 1, that the temperature was 257 K at an altitude of 7.84 km. From this we can see that temperatures in the troposphere were about 14-20 K warmer than standard. Other than this, not much can be gleaned from the Apollo data – certainly not enough to develop an atmospheric model.
Fortunately, sets of mean monthly Reference Atmospheres for altitudes up to 90 km have been developed for 15o intervals of latitude to provide information on the seasonal and latitudinal variations in the thermodynamic properties of the atmosphere. Apollo 11 launched July 16 and crossed latitude 30o N during its ascent; therefore, we use the following:
|Reference Atmosphere - July - 30oN|
Using the tabulated data above, we use curve fitting to derive equations to estimate temperature and pressure as a function of altitude. Density is calculated from temperature and pressure using the ideal gas law, PV = nRT, from which we derive ρ = P/(RT), where ρ is density, P is pressure, T is temperature, and R is the specific gas constant equal to 287 J/kg-K for air.
For altitudes higher than 90 km we use the MSISE-90 model of Earth's upper atmosphere. As before, we can fit a curve to the known data points to estimate points in between. The exospheric temperature in a 1969 was about 1,100 K; therefore we use "mean solar activity" as our basis. We need only estimate atmospheric properties to 200 km, as Apollo 11's parking orbit remained below this altitude. In reality, the atmosphere above 90 km is so thin that it has negligible affect on the simulation (assuming a perfect vacuum above 90 km would produce barely noticeable changes in the results).
During the simulation the rocket executes a pitch program using user-selected control points. These control points, each consisting of a time and a pitch angle, determine the rocket's thrust vector and, therefore, are the means by which we control the rocket's trajectory. Between points the rocket varies its pitch at a constant rate. From Apollo 11 Ascent Phase we have several altitude, velocity and flight path angle target points to shoot for. By carefully controlling pitch we can steer the rocket through, or at least near to, these target points. By trial and error, the following pitch angles were found to approximate the ascent of Apollo 11:
|0.3 to 30||0||0|
|30 to 80||0 to 37.40||0.7280000|
|80 to 135||37.40 to 62.23||0.4696364|
|135 to 165||62.23 to 71.14||0.2970000|
|165 to 185||71.14 to 60.57||–0.5285000|
|185 to 320||60.57 to 64.75||0.0309630|
|320 to 460||64.75 to 77.35||0.0900000|
|460 to 480||77.35 to 74.59||–0.1380000|
|480 to 550||74.59 to 81.39||0.0971429|
|550 to 570||81.39 to 77.25||–0.2070000|
|570 to 640||77.25 to 85.07||0.1117143|
|640 to 705||85.07 to 88.23||0.0486154|
In addition to providing pitch commands, we must also command the vehicle to fly a preprogrammed azimuth. During the period immediately after "pitch over", Apollo 11 flew in a nearly straight line along its initial flight azimuth of 72.058o. However, after a short time, the earth-fixed velocity vector began to curve toward the east as the rocket naturally followed the surface path of a great circle. At the same time the rocket's flight azimuth also turned to the east.
At this point in time the trajectory had not yet attained the required 32.5o inclination. The rocket must maintain thrusting in a direction north of the velocity vector to continue pushing the orbit into a higher and higher inclination. It is possible to set and maintain the flight azimuth, i.e. the thrust vector, in a fixed relationship to the earth-fixed velocity vector for the remainder of the ascent. The small angle separating the thrust and velocity vectors is that necessary to attain the final orbital inclination, and is determined by trail and error. Denoting the azimuth of the earth-fixed velocity vector aEF, the following has been found to approximate the flight of Apollo 11:
|0.3 to 90||72.058|
|over 90||aEF – 0.1368o|
The purpose of this page is not to explain how the simulation works, but rather, to report the results. The actual internal workings of the simulation are in accordance with the techniques described in the following article:
Rocket & Space Technology - Space Flight Simulations
For a second-by-second tabulation of the simulated results, see here: Saturn V Launch Simulation.
Below is an explanation of each data column in the above referenced report:
Time (s) - Ground elapsed time; simulation starts at first motion (0.3 seconds)
Dry (kg) - Total mass of all parts not yet jettisoned, excluding propellants
Propellant (kg) - Mass of stage 1, 2 & 3 liquid propellants not yet used or jettisoned
Total (kg) - Sum of dry + propellant mass
Flow rate (kg/s) - Liquid propellant mass flow rate
Total Thrust (N) - Total instantaneous thrust, main engines + ullage (when applicable)
Thrust Accel (g) - Thrust acceleration during powered flight
Radius Vector (m) - Radius vector magnitude, distance to center of Earth
GCLat (degrees) - Geocentric latitude
Long (degrees) - Geographic longitude (negative indicates W longitude)
AZ (degrees) - Azimuth of thrust vector, measured from north
Pitch (degrees) - Pitch of thrust vector, measured from radius vector
AZ (degrees) - Azimuth of earth-fixed velocity vector, measured from north
F-Path (degrees) - Flight path angle of earth-fixed velocity vector
Vel (m/s) - Total earth-fixed velocity (relative to rotating Earth)
Head (degrees) - Heading of space-fixed velocity vector, measured from north
F-Path (degrees) - Flight path angle of space-fixed velocity vector
Vel (m/s) - Total space-fixed velocity (relative to stars)
Range (m) - Surface distance from launch point to sub-vehicle point
Altitude (m) - Altitude above reference ellipsoid
Temperature (K) - Air temperature at current altitude
Pressure (Pa) - Air pressure at current altitude
Mach # - Mach number, earth-fixed velocity ÷ local speed of sound
Dynamic Press (Pa) - Dynamic pressure
Drag (N) - Total aerodynamic drag force
Up until now everything we've done has been in U.S. Customary Units (FPS system). This is not my preferred system but it is used in nearly all historical NASA documentation from the Apollo era – it has just been easier to use the units presented in the reference documents. However, for the simulation I've converted everything to metric units (MKS system). To convert units in the simulation back to FPS units, use the following conversion factors:
Simulation vs. Apollo 11
The Saturn V Launch Simulation produced results very close to those of the actual launch of Apollo 11, though not an exact match. The following table compares the simulation to real Apollo 11 ascent phase data. Apollo 11 data is in black and the simulated data is in red.
|Mach 1 achieved||01:06.3
|Maximum dynamic pressure||01:23.0
|S-IC center engine cutoff||02:15.20
|S-IC outboard engine cutoff||02:41.63
|S-II center engine cutoff||07:40.62
|S-II outboard engine cutoff||09:08.22
|S-IVB 1st burn cutoff||11:39.33
|Earth orbit insertion||11:49.33
Let's now take a look at some interesting graphs. Below we have plots of acceleration, altitude, range, earth fixed velocity, space fixed velocity, space fixed flight path angle, space fixed heading, and dynamic pressure & Mach number versus time. The red "+" signs indicate known Apollo 11 data points, from which you can see just how closely the simulation reproduced the Apollo 11 results.
The simulation was able to attain Apollo 11's altitude and velocity but required a longer than nominal burn. Apollo 11 exceeded the simulation's velocity by 21.0 m/s at S-IC cutoff, 43.9 m/s at S-II cutoff, and 48.5 m/s at the 699.33 s mark. An additional burn time of 7.52 seconds was necessary to reach the required orbital velocity, resulting in 1,605 kg (3,538 lbm) of additional propellant being used.
The exact reason for the simulation's under performance is unknown; however, a deviation from actual results is not unexpected. Given that this was an amateur project developed from often sketchy or incomplete information, the closeness of the results is encouraging. The simulation's launch duration was only 1.0% longer than Apollo 11's and the amount of additional propellant burned was only 0.06% of the total consumption. And, as we will later see, the simulation had enough propellant remaining to successfully complete the S-IVB second burn.
Aside from the duration of the burn, the only other noteworthy difference between actual and simulated data is the maximum dynamic pressure. Earlier we saw that Apollo 11's maximum q was 735.17 lb/ft2 (35,200 Pa). We now see that the simulation produced a maximum q of 706.72 lb/ft2 (33,838 Pa). Although this is lower than Apollo 11, it is typical of a Saturn V launch. The average maximum q for the ten manned Apollo-Saturn launches was 702.5 lb/ft2.
It was also previously noted that The Saturn V News Reference, August 1967 states that the Saturn V's maximum restrained drag force was 460,000 lbf. The simulation's maximum drag compares favorably to this with a value of about 462,600 lbf.
Below is a summary of the velocities and losses as determined by the simulation:
|Launch vehicle DV||9,194 m/s|
|Earth rotation||390 m/s|
|Total velocity||9,584 m/s|
|Loss due to gravity||–1,743 m/s|
|Loss due to drag||–48 m/s|
|Burnout velocity||7,793 m/s|
Gravity and drag losses for launch vehicles are typically 1,500-2,000 m/s, so we are right about mid-range. For the Saturn V, the loss is nearly all due to gravity, which is because of the vehicle's low thrust-to-weight ratio. Low thrust-to-weight causes gravity losses to be high because the vehicle spends more time in ascent, while high thrust-to-weight causes drag losses to be high because of the higher velocities achieved in the atmosphere.
S-IVB 2nd Burn
The second burn of the S-IVB stage was to send Apollo 11 on a trajectory to the Moon, a maneuver know as translunar injection, or TLI. This has nothing to do with our launch simulation, but since we're studying the capabilities of the Saturn V rocket, let's examine it.
Since TLI takes place with the vehicle already in Earth orbit, we don't need to worry about gravity and drag losses – we simply use Tsiolkovsky's rocket equation to calculate the velocity change and/or propellant usage.
According to the numbers we derived earlier, the total dry mass placed in Earth orbit was 131,341 lbm. This number might be slightly reduced by the time of TLI due to use of consumables, but not nearly enough to concern us here. The amount of S-IVB propellant prior to TLI was 164,141 lbm and the amount remaining after TLI was 7,462 lbm. (The amount of propellant remaining immediately after S-IVB 1st burn was 166,880 lbm, but during the approximate 2 1/2 hours between orbit insertion and TLI, 2,739 lbm was lost due to boil-off of the cryogenic propellants.) According to Post Launch Mission Report No. M-932-69-11, the TLI DV was 10,441 ft/s (3,182 m/s). Given the initial mass, final mass, and velocity change, we solve for the specific impulse:
DV = Ve × LN( Mo / Mf ), where Ve = Isp × go
10,441 = Isp × 32.174 × LN[ (131,341 + 164,141) / (131,341 + 7,462) ]
Isp = 429.51 s
According to the Saturn V launch simulation, we had to burn the S-IVB a little longer than the actual Apollo 11 in order to attain the same orbit. The longer burn left us with 3,538 lbm less propellant. How does this affect the TLI burn? Let's reduce the amount of propellant available for TLI by the extra amount used during launch, 164,141 – 3,538 = 160,603 lbm. We can now calculate how much propellant is left over after completing the TLI burn:
10,441 = 429.51 × 32.174 × LN[ (131,341 + 160,603) / (131,341 + Mp) ]
Mp = 5,800 lbm
Apollo 11 had 7,462 lbm remaining after TLI, but is there enough useable propellant in the S-IVB stage that we can draw it all the way down to 5,800 lbm remaining and still be able to complete the burn? The least amount of S-IVB propellant remaining among all the Apollo missions was 5,795 lbm on Apollo 13; therefore, even though we're cutting it close, we can confirm that we fall within the capability of the S-IVB propulsion system.
The primary purpose for the creation of this simulation is twofold:
In December 2013 the simulation was updated and this web page revised accordingly. Three major changes were made: