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Please watch 45 seconds of this demonstration performed by no one less than Wernher von Braun himself! https://www.youtube.com/watch?v=1ZImSTxbglI#t=1800

The numbers along the illustrated ellipse between Earth and the Moon are obviously in units of hours travel time from/to LEO. So that makes no physical sense. According to the illustration in the movie, the slowest speed is in lunar periapsis. The fastest speed at the nodes (minor axis end-points) of the ellipse. And this about 350 years after Johannes Kepler had proven that the opposite is true.

Yes, it is a Disney movie. But an educational documentary one, in this part at least. And purportedly, Wernher und Walt were personally good friends. Why make this mistake in public? Or is it I who make a fool out of myself now?

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Old Wernher got it right. Of course. It all makes perfect sense, at least the transit to the Moon part and the relative velocities.

I calculate 121 hours for half of an Earth orbit with a periapsis 400 km above the equator and an apoapsis 100 km (60 mi as he stated) above the far side of the Moon.

At apoapsis, the vehicle will be going about 0.2 km/s in the Earth frame. The Moon is going about 1.0 km/s around the Earth. So for the vehicle to encounter the Moon, it will need to be ahead of the Moon, and the Moon will catch up to it from behind at 0.8 km/s.

Therefore as the Moon approaches the vehicle (or as the vehicle approaches the Moon — same thing), the Moon will slow down the vehicle as it pulls on it from behind (and even reverse its motion), in the reference frame of the Earth. As the Moon passes, it will then be in front and pull the vehicle forward, speeding it up again.

Only in the frame of reference of the Moon will it appear that the vehicle speeds up as it approaches the Moon. You cannot think of this problem as the Moon and Earth being immobile with the vehicle flying around them. The Moon is moving, in fact much faster than the vehicle as it approaches.

The velocity at Earth closest approach is close to 11 km/s, so yes, as Wernher describes, the speed is slower at the Moon and faster at Earth. Much faster.

As already noted, the orbit's shape and speed is determined almost entirely by the Earth, with only a very small portion of the trajectory dominated by the Moon's gravity. So you can use a two-body (Earth and vehicle) Kepler orbit to approximate the trajectory to the Moon very well, ignoring the Moon's gravity.

If you like, you can then correct for the Moon as a perturbation using patched conics, considering the Lunar flyby as a hyperbolic pass with a $V_\infty$ of 0.8 km/s, the difference between the vehicle velocity and the Lunar velocity. In the Lunar reference frame, the initial and final velocities at $\infty$ have the same magnitude, but different directions. When that is converted to the Earth frame, it will result in a $\Delta V$ to the vehicle which will cause it to depart from the initial Kepler orbit about the Earth.

Here's what the trajectory actually ends up looking like in the Earth frame, where the trajectory is orange and the path of the Moon is blue (Earth is at 0, 0):

loop around Moon, then to much larger orbit

After that close lunar flyby on the far side, the vehicle is in a much larger orbit of Earth with an apoapsis almost ten times the distance of the Moon. So Wernher didn't have the post-flyby state correct. The orbit he showed is what would happen if the Moon weren't there when you got to apoapsis.

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    $\begingroup$ In the lunar reference frame, the vehicle is moving retrograde as it passes the moon? $\endgroup$
    – Jerard Puckett
    Dec 2 '14 at 5:30
  • $\begingroup$ And at the moon's transit, the vehicle describes a loop in its orbit relative to the Earth? $\endgroup$
    – Jerard Puckett
    Dec 2 '14 at 5:35
  • $\begingroup$ Yes, and yes. See added plot. $\endgroup$
    – Mark Adler
    Dec 2 '14 at 6:23
  • $\begingroup$ Orbital mechanics isn't mild to intuition or improvisation. "Words without thoughts never to heaven go." $\endgroup$
    – LocalFluff
    Dec 2 '14 at 15:45
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Over most of the flight, Earth's gravity dominates, so what you're referring to as lunar periapsis is better thought of, for Keplerian purposes, as Earth apoapsis, and so is rightfully the slowest speed. The highest speeds shown are at the minor axis endpoints, but in order to reach the first of those after ~15 hours, and the far point of the ellipse at 121 hours, you must be going quite a bit faster in the first leg of the journey.

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  • $\begingroup$ As if the Moon had no gravity? That the slowest speed of the spacecraft would be achieved when it were at the nearest of the Moon, on its far side. Which in reality is when it has its highest speed and one therefor want's to take advantage of the Oberth effect. He does say: "As the Earth's gravity begins to slow the rocket down". But okay, as an Earth orbit, ignoring the existence of the Moon, it does make sense. $\endgroup$
    – LocalFluff
    Dec 1 '14 at 22:04
  • $\begingroup$ Try applying the same logic you use in your question to the return leg of the flight, where the end of the trip is Earth periapsis. Which end of the ellipse should be the fast one in that case? $\endgroup$
    – Russell Borogove
    Dec 1 '14 at 22:57
  • $\begingroup$ Sorry, I don't understand. The cartoon looks like one single ellipse with Earth and the Moon in each one focus, which is impossible. Perigeo and periluna must be the two fastest points in each part of each an ellipse. The mid-transits must be the slowest. Which is not what is illustrated. $\endgroup$
    – LocalFluff
    Dec 1 '14 at 23:34
  • $\begingroup$ I didn't realize you were arguing for both endpoints being fast, so disregard my comment. To my knowledge, Kepler's work considered orbits around a single massive body, not around two. Earth's gravity is vastly dominant in this situation, Earth having 80 times the mass of the moon, so to a first approximation you can ignore the moon when reasoning about the elliptical orbit. $\endgroup$
    – Russell Borogove
    Dec 1 '14 at 23:54
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They didn't have computers readily available at the time. I think they used an earth orbit for this simplified illustration, because you can calculate it by hand. They could have used patched conics, but it really doesn't matter for this kind of presentation.

This may be a going out on a limb, but it may point to von Braun's ability to use "good enough" solutions for a set of requirements instead of over-engineering.

To answer the question, Wernher von Braun was very aware of the mathematics of the two body problem. You can read about this in "First Men to the Moon", where he suggest that the moon mission may have to carry a large set of course correction maneuvers stored on magnetic tape, because he expects the computers to be too big to fit on a spacecraft.

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