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 increase 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.

Launch Events

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
Fuel: RP-1
Oxidizer: LO₂
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
Fuel: LH₂
Oxidizer: LO₂
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
Fuel: LH₂
Oxidizer: LO₂
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.

Propellant Utilization

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 lbm/sec
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
Total, lbm	4,730,675	70,535	4,660,140	27,734.0
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
Total, lbm	977,166	14,354	962,812	2,505.9
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
Total, lbm	236,105	166,880	69,225	470.5
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
Total, lbm	164,141	7,462	156,679	451.7
Oxidizer-Fuel Ratio:
S-IC Stage	2.320	---	2.343	---
S-II Stage	5.180	---	5.536	---
S-IVB Stage, 1st burn	4.414	---	4.831	---
S-IVB Stage, 2nd burn	4.597	---	4.758	---

And from Lunar Landing Mission Press Kit, July 6, 1969, Release No. 69-83K:

S-IC thrust buildup propellant used: 85,745 lbm (from ignition to first motion).

The following is a description of the second and third stage propellant utilization subsystem, from Technical Information Summary, Apollo-11 (AS-508), Apollo Saturn V Space Vehicle, S&E-ASTR-S-101-69, June 25, 1969:

S-II Stage
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.

S-IVB Stage
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.

Apollo 11 transcripts show that Neil Armstrong reported the S-II PU mixture ratio shift at 000:08:22 GET.

Ullage Motors

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-508), Apollo Saturn V Space Vehicle, 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

S-IVB Stage
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.

Data Analysis

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

S-IC Stage

The propellant use table shows a S-IC propellant burn rate of 27,734.0 lbm/s, however this is an average value. As we can see from the list of launch events, first motion occurs at 000:00:00.3 GET with all five engines operating at full thrust until the center engine cuts off at 000:02:15.2 GET. The remaining four engines continue to burn until cutoff at 000:02:41.63 GET. The total seconds of full-thrust engine operation is:

134.9 × 5 + 26.43 × 4 = 780.22 s

We know that 4,660,140 lbm of propellant was burned in total, and we're told that 85,745 lbm was burned during thrust buildup prior to first motion. Therefore, the propellant consumption rate per second of full-thrust engine operation is:

(4,660,140 – 85,745) / 780.22 = 5,862.96 lbm/s

The actual sea level thrust of the first stage is given as 7,552,000 lbf, or 1,510,400 lbf per engine. Checking the specific impulse, we have:

1,510,400 / 5,862.96 = 257.6 s (sea level)

Please note that this specific impulse is lower than all published values for the F-1 engine. According to The Saturn V News Reference, August 1967, the F-1 had a sea level specific impulse of 260 seconds minimum. Other sources report a value as high as 265 s (304 s vacuum). We'll examine this situation in more detail later in this article, as an upward adjustment will likely be necessary to attain the reported performance. For now we'll use the numbers calculated here.

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:

GET, hhh:mm:ss		Time sec.	Thrust, lbf		Propellant, lbm			Rate lbm/s
Start	End		S.L.	Vac.	Start	End	Used
-000:00:06.4	000:00:00.30	6.70	Varies		4,730,675	4,644,930	85,745	Varies
000:00:00.30	000:02:15.20	134.90	7,552,000	8,667,031	4,644,930	690,367	3,954,563	29,314.78
000:02:15.20	000:02:41.63	26.43	6,041,600	6,933,625	690,367	70,535	619,832	23,451.82

S-II Stage

Analyzing the second stage is a bit more complex because, in addition to early cutoff of the center engine, there was also the PU mixture ratio shift. S-II ignition occurred at 000:02:44.0 GET, center engine cutoff occurred at 000:07:40.62 GET, and outboard engine cutoff occurred at 000:09:08.22 GET. The exact time of the PU MR shift is not recorded in the Apollo 11 timeline, however Neil Armstrong verbally reported the shift at 000:08:22.0 GET. Since it probably took Armstrong a second or two to react, we'll assume the PU MR shift occurred at 000:08:20.0 GET.

The shift in mixture ratio was achieved by opening bypass valves that reduced the amount of LOX flowing to the engines. The total propellant flow rate decreases by an amount greater than we would expect simply by decreasing the proportion of LOX from 5.5 to 4.5. This is because the PU valve is located upstream of the turbopump gas generator. Opening the PU valve reduces propellant flow to the gas generator, thereby decreasing pump speed and capacity. This results in an overall throttling down of the engine as it reestablishes a new steady state equilibrium.

Reducing the mixture ratio actually increases the specific impulse of the engines. According to The Saturn V New Reference, August 1967, the J-2 engine specific impulse was 424 s at MR 5.5:1. My calculations indicate that we should see an increase of about 10 seconds in specific impulse going from MR 5.5:1 to MR 4.5:1, thus giving a specific impulse of about 434 s at the lower MR. The vacuum thrust of the second stage is given as 1,155,859 lbf with all five engines burning at full thrust, which is reduced 20% at center engine cutoff. At outboard engine cutoff, the stage thrust was reported as being 625,751 lbf with four engines burning at the reduced propellant flow rate.

We can calculate the theoretical propellant use based on the burn time, thrust, and specific impulse, as follows:

( 296.62 × 1,155,859 + 39.38 × 924,687 ) / 424 + ( 48.22 × 625,751 ) / 434 = 964,018 lbm

The actual propellant use was 962,812 lbm, therefore we factor the theoretical consumption rate by 962,812 / 964,018 = 0.998749 to obtain the actual rate. The propellant flow rate per engine operating at full thrust is:

0.998749 × ( 1,155,859 / 424 ) / 5 = 544.53 lbm/s

After the PU MR shift, the per engine flow rate is:

0.998749 × ( 625,751 / 434 ) / 4 = 360.00 lbm/s

The revised specific impulses are 424.5 s at MR 5.5:1 and 434.5 s at MR 4.5:1.

The simulation will ignore thrust buildup and thrust decay, assuming the engines turn instantly full on or full off.

Please note that the thrust at engine cutoff and the resulting propellant flow rate appears to be unusually low. After seeing how the stage performs during simulation test trials, we may need to revisit these calculations and consider modifications. In the meantime, the following is a summarization of the S-II operation as determined above:

GET, hhh:mm:ss		Time sec.	Thrust lbf (vac)	Propellant, lbm			Rate lbm/s
Start	End			Start	End	Used
000:02:44.00	000:07:40.62	296.62	1,155,859	977,166	169,567	807,599	2,722.67
000:07:40.62	000:08:20.00	39.38	924,687	169,567	83,792	85,775	2,178.14
000:08:20.00	000:09:08.22	48.22	625,751	83,792	14,354	69,438	1,440.02

S-IVB Stage

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:

GET, hhh:mm:ss		Time sec.	Thrust lbf (vac)	Propellant, lbm			Rate lbm/s
Start	End			Start	End	Used
000:09:12.20	000:11:39.33	147.13	202,603	236,105	166,880	69,225	470.50
002:44:16.20	002:50:03.03	346.83	201,061	164,141	7,462	156,679	451.75

Ullage

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, 000:02:42.11 GET, and continue to 3.31 s after separation, 000:02:45.61 GET. 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, 000:09:08.95 GET, and continue to 0.75 s after separation, 000:09:12.95 GET. 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:

GET, hhh:mm:ss		Time sec.	Thrust, lbf (vac)		Propellant
Start	End		Total	Forward	Used, lbm	Rate, lb/s
000:02:42.11	000:02:45.61	3.5	92,000	90,602	1,361	388.86
000:09:08.95	000:09:12.95	4.0	6,800	6,697	118	29.50

Dry Mass

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:

Event	GET hhh:mm:ss	Dry Mass, lbm
		Before	After	Change
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

External Factors

Earth's atmosphere plays a large role in the launch of a space vehicle. First, it affects the thrust of the engines, and second, it exerts a drag force on the launch vehicle. The properties of the atmosphere can be obtained from the following web source. The simulation will use the standard atmosphere for altitudes up to 85 kilometers, and the MSISE-90 data at mean solar activity for altitudes above 85 kilometers:

Rocket & Space Technology - Atmosphere Properties

Pressure Thrust

The basic thrust equation is F = qV_e+(P_e–P_a)A_e, where F is the thrust, q is the propellant mass flow rate, V_e is the exhaust gas velocity, P_e is the pressure at the nozzle exit, P_a is the ambient air pressure, and A_e 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 P_aA_e. Alternatively, an engine's thrust at altitude is equal to its sea level thrust plus (P_SL–P_a)A_e.

According to The Saturn V New Reference, August 1967, the maximum nozzle exit diameter of the F-1 engine is 11 feet 7 inches, and that of the J-2 engine is 6 feet 5 inches, thus yielding exit areas of 15,175 square inches and 4,657 square inches respectively. The thrust of the F-1 engines increase about 15% between liftoff and their cutoff at an altitude of about 66 kilometers.

Atmospheric Drag

Drag is the resistance offered by a gas or liquid to a body moving through it. The drag force, F_D, on a body acts in the opposite direction of the velocity vector and is given by the equation F_D = C_Drv²A/2, where C_D 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 Apollo 11 Ascent Phase, we see that maximum dynamic pressure (also called maximum q) occurred at an altitude of 7.326 nautical miles (44,512 feet) and an Earth fixed velocity of 1,653.4 ft/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.) From the standard atmosphere, we estimate the density and temperature of the atmosphere at 44,512 feet are 0.0004733 slug/ft³ and 390^o R. Since the acoustic velocity at 390^o R is 968 ft/s, the Mach number at maximum q is 1,653.4 / 968 = 1.71. We now have enough information to calculate the drag coefficient:

C_D = 460,000 × 2 / (0.0004733 × 1,653.4² × 1,216) = 0.585

This value is close to what we see above for a round nose projectile, therefore we'll use C_D = 0.585 at N_M = 1.71 as a data point.

Something not considered to this point is angle of attack. Since our drag coefficient calculation is based on the total drag force, the product C_DA is correct even if the individual values of C_D 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 is rolled into and considered 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 our one known data point and assuming we have a curve similar to those seen above for the sharp nose and round nose projectiles, we can guesstimate a C_D versus N_M curve to use in our simulation. We'll try the following and modify it later if we have to:

The increase in C_D below a Mach number of 0.5 is due to the occurrence of non-laminar (turbulent) airflow at low velocity. This curve is defined by a set of four polynomials, each describing a separate part of the curve.

Earth Rotation

Thanks to Earth's rotation, our launch vehicle starts out with a due east velocity before it ever leaves the launch pad. We can take advantage of this free ride to reduce the amount of velocity the rocket must provide to reach orbit, however this complicates the simulation.

The simulation is greatly simplified if we can treat the launch as a two-dimensional problem, where we have only vertical (radial) and horizontal (tangential) accelerations, velocities, and distances within the orbital plane. Unfortunately, the initial velocity we get from Earth's rotation turns it into a three-dimensional problem. From Ascent Data we see that Apollo 11's flight azimuth was 72.058 degrees. Because the flight azimuth is in a direction other than the initial velocity, we are in a condition where the thrust vector is continually modifying the orbital plane.

Fortunately there is a simply trigonometric solution we can use to reduce the simulation to a two-dimensional problem. From Apollo 11 Ascent Phase we see that the 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. 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. The loss is the DV required to produce the plane change.

We will, therefore, treat the simulation as a two-dimensional analysis with the initial space fixed horizontal velocity set equal to 1,278.8 ft/s. This is an approximate solution but it's good enough for our purposes.

Test Trials

Initial simulation test trials discovered a few problems with our previous calculations and assumptions, which was not unanticipated. Let's now revisit these items and make the modifications necessary to improve the simulation.

S-IC Stage

We saw earlier that our calculated value for the F-1 engine specific impulse, 257.6 seconds, is lower than all published values. Not unexpectedly, trial simulations based on this low number were unsuccessful as the S-IC stage is unable to attain the reported burnout velocity.

The calculated specific impulse is based on the S-IC thrust of 7,552,000 lbf as documented here. This is, of course, just a snapshot for the brief period between 35 and 38 seconds after launch. There is sufficient evidence to suggest this thrust measurement, as well as others, may not be representative of the thrust averaged over the entire burn duration. It is not wise to base the simulation on a single questionable data point. Since we should expect the F-1 engine to perform up to specifications, we are justified in making the corrections necessary to match the anticipated performance.

The published value of the F-1 engine specific impulse varies from source to source, but according to Astronautix.com, the specific impulse was 265 s at sea level and 304 s vacuum. It has been found through simulation trials that setting the vacuum specific impulse to 304 s attains to performance necessary to match the ascent data of Apollo 11. This makes the sea level specific impulse 265.8 s, which is slightly higher than that given in the referenced source due to a greater than nominal propellant flow rate. The thrust will be that as determined by propellant flow rate and specific impulse.

After further test trails it was found that, in comparison to Apollo 11, the simulated first stage was overpowered during the first half of the burn and underpowered during the last half. This could be explained by a propellant flow rate that increases as the burn progresses (we assumed a constant flow rate). A closer match to the actual Apollo 11 data can be achieved by a small 1/2 percent decrease in the propellant flow rate at the beginning of the burn. The flow rate will then be increased at a constant rate until center engine cutoff, after which we'll assume a constant rate until outboard engine cutoff. (As rocket acceleration increases, the propellant pressure at the pump suction increases, which likely explains the rise in propellant flow rate.)

Making the above change we update the operation of the S-IC as follows:

GET, hhh:mm:ss		Time sec.	Thrust, lbf		Propellant, lbm			Rate lbm/s
Start	End		S.L.	Vac.	Start	End	Used
-000:00:06.4	000:00:00.30	6.70	Varies		4,730,675	4,644,930	85,745	Varies
000:00:00.30	000:02:15.20	134.90	7,752,102 to 7,830,585	8,867,134 to 8,945,616	4,644,930	690,367	3,954,563	29,168.20 to 29,426.37
000:02:15.20	000:02:41.63	26.43	6,264,468	7,156,493	690,367	70,535	619,832	23,541.10

S-II Stage

Although the simulated S-II stage produced the expected overall performance, it was overpowered during the first part of the burn with a noticeable performance drop-off after center engine cutoff. This was not entirely unexpected as it was noted earlier that the reported thrust at S-II outboard engine cutoff, and the propellant flow rate calculated from it, seemed unusually low.

As we saw with the S-IC stage, the documented thrust measurements do not seem to be representative of performance across the entire period of operation, for if they were, the simulation would have more closely mirrored the Apollo 11 ascent data. Our first calculations derived propellant mass flow rates of 544.53 lbm/s at a MR of 5.5:1 and 360.00 lbm/s at MR 4.5:1, but this was determined from the thrust measurements and not actual propellant usage. The most reliable flow rate calculation we have for the J-2 engine comes from the first burn of the S-IVB, in which the rate was 470.50 lbm/s at MR 5.0:1.

We should attempt to determine propellant flow rates that replicate the actual ascent data. For our first attempt, let's try the following and see what results we get: Suppose we assume that going from MR 5.5:1 to MR 5:1 results in the same change in flow rate as going from MR 5:1 to MR 4.5:1. We can then say the propellant flow rate at MR 5.5:1 is 470.5 + X and the flow rate at MR 4.5:1 is 470.5 – X, where 470.5 is the neutral flow rate at MR 5.0:1 and X is the change resulting from the MR shift. From the known propellant mass and burn times, we can solve for X as follows:

(470.5 + X) × 5 × 296.62 + (470.5 + X) × 4 × 39.38 + (470.5 – X) × 4 × 48.22 = 962,812

X = 69.18 lbm/s

Therefore,

Flow rate at MR 5.5:1 = 470.5 + 69.18 = 539.68 lbm/s
Flow rate at MR 4.5:1 = 470.5 – 69.18 = 401.32 lbm/s

Using the specific impulse numbers previously determined, the engine thrusts at the new propellant flow rates are:

Thrust at MR 5.5:1 = 539.68 × 424.5 = 229,093 lbf
Thrust at MR 4.5:1 = 401.32 × 434.5 = 174,375 lbf

Test trails using the above revision results in a much closer reproduction of the Apollo 11 ascent data. The new results are good enough that further refinements are likely to produce negligible improvement. Tabulating the updated numbers we have the following:

GET, hhh:mm:ss		Time sec.	Thrust lbf (vac)	Propellant, lbm			Rate lbm/s
Start	End			Start	End	Used
000:02:44.00	000:07:40.62	296.62	1,145,464	977,166	176,771	800,395	2,698.39
000:07:40.62	000:08:20.00	39.38	916,371	176,771	91,761	85,010	2,158.71
000:08:20.00	000:09:08.22	48.22	697,499	91,761	14,354	77,407	1,605.29

Drag Coefficient

Trial simulations using our previously estimated drag coefficient resulted in a maximum drag force of about 500,000 lbf, which is too high compared to the 460,000 lbf number quoted in The Saturn V News Reference, August 1967. To bring the drag force down to the neighborhood of the published number, the drag coefficient was recalculated to lower the maximum from 0.60 to 0.56. Below is the revised drag coefficient curve:

Computer Simulation

For a second-by-second print out of the complete simulation, see here: Saturn V Launch Simulation.

Overview

The simulation progresses through a series of small time steps, with the Saturn V's state vector updated at each step. The physics behind the calculations is not especially difficult, but the many interdependencies between the variables produce some complexity. A detailed description of the simulation is more than I care to present in this article; only a general overview is provided.

Before examining the results of the simulation, let's explain the meaning of each column, from left to right:

Time (s) - Ground elapsed time; simulation starts at first motion (0.3 seconds)
Vehicle Mass
  Dry (kg) - Total mass of all parts not yet jettisoned, excluding propellants
  Fuel (kg) - Mass of stage 1, 2 & 3 liquid propellants not yet used or jettisoned
  Total (kg) - Sum of dry + propellant mass
Frontal Area (m²) - Maximum cross-sectional area of parts not yet jettisoned
Total Thrust (N) - Total instantaneous thrust, main engines + ullage (when applicable)
Stage 1
  Dry (kg) - Mass of stage 1 less propellants (goes to 0 when jettisoned)
  Fuel (kg) - Mass of stage 1 liquid propellants less amount used
  Thrust (N) - Thrust of stage 1 engines burning at current time
ISA Mass (kg) - Mass of stage-1/2 interstage adapter (goes to 0 when jettisoned)
Stage 2
  Dry (kg) - Mass of stage 2 less propellants (goes to 0 when jettisoned)
  Fuel (kg) - Mass of stage 2 liquid propellants less amount used
  Thrust (N) - Thrust of stage 2 engines burning at current time
ISA Mass (kg) - Mass of stage-2/3 interstage adapter (goes to 0 when jettisoned)
Stage 3
  Dry (kg) - Mass of stage 3 less propellants
  Fuel (kg) - Mass of stage 3 liquid propellants less amount used
  Thrust (N) - Thrust of stage 3 engines burning at current time
IU Mass (kg) - Mass of instrument unit
Payload Mass (kg) - Mass of payload, including CSM, LM and SLA
LES Mass (kg) - Mass of launch escape system (goes to 0 when jettisoned)
Vehicle Angles
  Pitch (degrees) - Pitch of rocket's thrust vector measured from vertical
  f (degrees) - Space fixed flight path angle
  AoA (degrees) - Angle of attack
Radius Vector (m) - Rocket's distance from center of Earth
Altitude (m) - Rocket's altitude above sea level
Range (m) - Horizontal distance of rocket from launch point
Acceleration
  g (m/s²) - Acceleration of Earth's gravity at rocket's radius vector
  Vert. (m/s²) - Rocket's vertical (radial) acceleration
  Horz. (m/s²) - Rocket's horizontal (tangential) acceleration
  Load (g) - Acceleration load (excludes gravity)
Space Fixed Velocity
  Vert. (m/s) - Rocket's vertical (radial) space fixed velocity
  Horz. (m/s) - Rocket's horizontal (tangential) space fixed velocity
  Total (m/s) - Rocket's total space fixed velocity (relative to stars)
Earth Fixed V (m/s) - Rocket's total Earth fixed velocity (relative to rotating Earth)
Atmosphere
  Temp. (K) - Air temperature at current altitude
  Density (kg/s³) - Air density at current altitude
  Press. (Pa) - Air pressure at current altitude
  Mol. Wt. (g/mol) - Average molecular weight of air at current altitude
  g - Specific heat ratio of air at current altitude
Nm - Mach number at current velocity and ambient atmosphere
Cd - Drag coefficient at current Mach number
Drag (N) - Total aerodynamic drag force
Heating (kW/m²) - Aerodynamic heating

The same series of operations are performed each step of the simulation, which are summarized below:

(1) Update dry mass and frontal area as parts are jettisoned.
(2) Calculate mass of propellant remaining as a function of previously calculated flow rates.
(3) Update thrust as a function of stage, engines, and air pressure (add ullage when applicable).
(4) Set pitch angle per preprogrammed commands (see below).
(5) Calculate gravity as a function of radius vector.
(6) Calculate air temperature, density, pressure, molar weight and g as a function of altitude.
(7) Calculate Mach number as a function velocity, temperature, molecular weight and g.
(8) Update drag coefficient as a function of Mach number.
(9) Calculate drag force as a function of velocity, air density, frontal area and Cd.
(10) Calculate heating as a function of velocity and air density.
(11) Calculate accelerations as a function of total mass, velocity, pitch, thrust, drag and gravity (see below).
(12) Update velocities as a function of acceleration and time.
(13) Update flight path angle and angle of attack.
(14) Update radius vector and altitude as a function of vertical velocity and time.
(15) Update range as a function of horizontal 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 Earth's surface. The instantaneous vertical and horizontal accelerations of a moving body in reference to this surface are A_v=V_h²/r and A_h=–V_hV_v/r. To this we add the accelerations resulting from gravity, drag and thrust. Gravity, of course, acts vertically downward. Drag acts in the opposite direction of the velocity vector. 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. The simulation uses a time step of 1/2 second, though the provided print out displays data in one-second intervals.

During a 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:

Unit Conversions

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 Results

The Saturn V Launch Simulation produced results very close to those of the actual launch of Apollo 11, though not an exact match. There were enough approximations and simplifications made that an exact reproduction was never to be expected. 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.

The simulation was able to attain Apollo 11's altitude and velocity, but doing so required a 2.57-second extension of the burn and 1,209 lbm of additional propellant used. Below is a summary of the velocities and losses as determined by the simulation:

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.

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, and aerodynamic drag & heating 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. On the drag chart, the green "+" signifies the position of maximum q according to The Saturn V News Reference, August 1967.

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.) Given the thrust and propellant flow rate of the second burn, the specific impulse was 201,061 / 451.75 = 445.07 s. Therefore, the calculated velocity change is:

DV = V_e × LN( M_o / M_f ), where V_e = I_sp × g_o

DV = 445.07 × 32.174 × LN((131,341 + 164,141) / (131,341 + 7,462)) = 10,819 ft/s

This is actually too much DV as it would place Apollo 11 on a hyperbolic escape trajectory. The problem seems to be another case of the reported thrust not being indicative of the performance over the entire burn duration. According to Post Launch Mission Report No. M-932-69-11, the actual TLI DV was 10,441 ft/s (3,182 m/s). Assuming the propellant amounts are correct, let's calculate the engine specific impulse required to produce the reported DV:

10,441 = I_sp × 32.174 × LN( 295,482 / 138,803 )
I_sp = 429.51 s

This value agrees very well with earlier calculations in which the specific impulse of the J-2 engine was found to be 424.5 s at MR 5.5:1 and 434.5 s at MR 4.5:1. The mixture ratio of the S-IVB 2nd burn is recorded as 4.758.

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 1,209 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 – 1,209 = 162,932 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 + 162,932) / (131,341 + M_p))
M_p = 6,894 lbm

Apollo 11 had 7,462 lbm remaining after TLI, therefore the longer burn at launch actually cost us only 7,462 – 6,894 = 568 lbm more propellant. The least amount of S-IVB propellant remaining among all the Apollo missions was 5,795 lbm on Apollo 13, therefore we fall well within the capability of the S-IVB propulsion system.

Closing Comments

The simulation has closely reproduced the launch of Apollo 11 and has confirmed the capability of the Saturn V launch vehicle.