# How complex was the math and physics necessary to place Apollo 11 on the moon?

Specifically, I'm interested in how closely the models used to calculate the various burns and course corrections represented reality. Was standard Newtonian mechanics sufficient or were relativistic effects included? Were the Earth, Moon, and spacecraft modelled as point masses or more complicated bodies? Were forces such as solar wind included in the calculations?

Approaching the question a different way: to what extent was the control of the spacecraft pre-calculated, vs. calculated in real time by spacecraft computers, vs. done manually by the astronauts?

• Dec 3, 2017 at 16:41

Was standard Newtonian mechanics sufficient or were relativistic effects included?

Relativistic effects didn't have to be modeled; other sources of error would have swamped the effects of relativity, and midcourse corrections were made.

Were the Earth, Moon, and spacecraft modelled as point masses or more complicated bodies?

The moon's gravity was modeled with the L-1 potential model consisting of "5 coefficients out to a maximum degree of 3". I don't know a lot about geopotential modeling; it's discussed a little bit in this Q/A and the "Tindallgrams" linked therefrom. Apparently the model was updated between Apollo 11 and 12 with better data obtained from Lunar Orbiter data.

A gravity field is modeled (and can be visualized as) deviations from the gravity of a perfect sphere. The visualization of the L-1 Lunar potential looks like this (with red indicating increased gravity and blue decreased):

And the Apollo 12 gravity model looks like this:

And, for reference our current lunar gravity model, based on GRAIL data, looks like this:

The spacecraft's attitude control system required good estimates of its mass, center of mass location, and moment of inertia in order to maneuver efficiently. Without that level of detail in the modeling, the attitude control system would likely have been less responsive or more wasteful of propellant or both.

I'm not sure about their Earth gravitational model; they may not have needed much detail since they would only be in parking orbit for a few hours on the way out and re-entering aerodynamically on the way back.

I don't believe solar wind was factored into navigation, again because any effect it would have was easily corrected for.

Approaching the question a different way: to what extent was the control of the spacecraft pre-calculated, vs. calculated in real time by spacecraft computers, vs. done manually by the astronauts?

Mostly navigation was done by computers on Earth. The position of the spacecraft was accurately tracked throughout the mission, and correction maneuvers would be calculated on ground-side computers and called up to the crew to execute as needed; the guidance computer would execute the maneuver with a crewman ready to hit the shutdown button if necessary.

A few parts of the flight were flown manually. The terminal phase of the moon landing was one of them; the capability existed to land semi-automatically using a radar altimeter, with the commander able to adjust the targeted landing spot, but in every landing the computer was switched into a more manual mode at around 500 feet altitude and landed with the commander's hands on the controls while the LM pilot managed the computer and called out altitude and speed figures to the commander. Lovell intended to try to use the automatic mode on Apollo 13, but didn't get the chance to attempt the landing.

The transposition-docking-extraction maneuver to pull the LM away from the booster was flown manually by the command module pilot (CMP).

After the LM separated in lunar orbit, the commander would manually turn the LM to allow the CMP to visually inspect it; on the LM's return from the moon's surface, the commander would manually fly the very last part of the approach and docking, with the CMP ready to take over the active role if a problem developed on the LM.

• I remember a story from Apollo, the time of engine shutdown was calculated using analog computing. Available digital computing of engine shutdown was not able to calculate it in real time. But digital computing had better precision, so both methods were used and the remaining error from analog computing was corrected later by a short additional course correction burn.
– Uwe
Dec 3, 2017 at 10:02
• Not sure what you’re referring to. The duration of the burn would be computed digitally well in advance along with all the other parameters of the burn, and the countdown to engine cutoff was trivial, done digitally by the onboard guidance computer. Are you thinking of the Apollo 13 post-acccident midcourse corrections, which were timed by stopwatch since the computer was shut down? Dec 3, 2017 at 13:01
• Calculation of the burn time in advance may use only the specified thrust. But there might be a difference between specified and actual thrust, planned acceleration profile and actual acc. profile. But how big are those differences and how large is their influence on burn time? Fractions of a second or some seconds?
– Uwe
Dec 3, 2017 at 13:23
• Small fraction of a second. They occasionally did a correction with the small RCS thrusters after a main engine burn, but more often than not it wasn’t necessary. Dec 3, 2017 at 15:29
• Sorry, I assumed we were talking about the Apollo spacecraft rather than the Saturn booster. @prl is correct that the first and second stage cutoffs were not time-based. I believe the third stage orbital insertion and trans-lunar injections would have been done on a velocity-to-go basis rather than time-based; we've discussed those in some other Q/A here that I can't be bothered to hunt for. Dec 3, 2017 at 20:05

As a complement to the good answers provided above, I remember learning in a control system's graduate course that the Saturn V launch was modelled via a 23rd order equation. This gives an indication of the complexity of the math involved in the control of the full up Saturn V.