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I had never noticed it before, but when I was watching Apollo 13, the path of the rocket looked like this. Obviously since it's going around the moon, it won't actually be elliptical, but that doesn't even look close around the other parts. Is this an accurate model of the orbit? And if so, how can that path be modelled mathematically? Sorry if its a basic question, I'm new to this corner of stack exchange. enter image description here

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    $\begingroup$ This isn't the first time that a synodic frame point of view has confused people. However, an Earth-centered inertial point of view might well be equally confusing. $\endgroup$ – David Hammen Jun 6 '17 at 1:03
  • $\begingroup$ @DavidHammen isn't that already an Earth-centered inertial representation? Just like this one: braeunig.us/apollo/free-return.htm ? I'm looking mostly at the thin, faint line. The thick marker/crayon-like line, that's just hand drawn, right? I mean, it has corners! $\endgroup$ – uhoh Jun 7 '17 at 13:34
  • $\begingroup$ @Vedvart1 can you give us any clue where this is from? A snapshot of a slide in a talk? Simulator software? A bad 1970 SciFi movie? Enquiring minds want to know! $\endgroup$ – uhoh Jun 9 '17 at 3:00
  • $\begingroup$ @DavidHammen please check this answer - your hint was really helpful! $\endgroup$ – uhoh Jun 9 '17 at 3:35
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    $\begingroup$ @uhoh For the record, I was just looking for pictures of illustrated orbit paths for a project I was doing on rocketry, and saw this in google images and got curious. It appears to be from a movie, I want to say Apollo 13 but I have a feeling that's not quite it. $\endgroup$ – Vedvart1 Jun 9 '17 at 3:51
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Orbital paths are elliptical in the case where only one massive body is affecting a spacecraft's path gravitationally. In this case, the Earth and the moon each have significant gravity, so you could approximate the path as a partial ellipse from Earth to the point where the moon's gravity dominates, then a different partial ellipse going around the moon, then another partial ellipse returning to Earth. As Hohmannfan notes, this is the "patched conic" approximation, because conic trajectories are patched together.

Here's some more info and animations for the circumlunar free return trajectory and the normal Apollo-style trajectory.

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  • $\begingroup$ I like that Earth-Centered Inertial view in the linked article. It shows very much why the patched conic approximation is valid. On the outbound leg, the Moon is, for the most part very far from the spacecraft. It's gravity is but a minor perturbation; the orbit is very close to an elliptical orbit about the Earth with a very high eccentricity. The same applies in the return leg of the flight. But in those interim four orbits about the Moon, it's the Moon's gravity that dominates. $\endgroup$ – David Hammen Jun 6 '17 at 9:25
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The trajectory in your image is actually elliptical. It is just projected in a way that makes it appear differently.

How? This is a rotating frame of reference; that is, our point of view is rotating along with the Moon's orbit so both the Earth and the Moon appears stationary.

Here is another example of how this works: Stand in the middle of the room and start spinning. Then place your hand on your face and move it outwards. From your point of view it is moving in a straight line away from you, but for another person in the room that is not spinning it looks like your hand is moving in a spiral.

The simplest way of modelling this orbit is the patched conic approximation. It is sufficient in most cases, and assumes we are only affected by the gravity of one object at a time. In this case, the first part of the trajectory is an ellipse with respect to Earth, then it is a hyperbola with respect to the Moon, and then an ellipse again. A more advanced model would be to always consider the gravitational influence of both the Earth and the Moon. This, however is known as the three-body problem and can only be solved by numerical simulations. (Though restricted versions can have usable analytical approaches.)

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  • $\begingroup$ "The trajectory in your image is actually elliptical." No it's not. It is a figure eight in an inertial frame. There is a cross-over in an inertial frame. It does not even look like two ellipses and two hyperbolas patched. It's purely a 3-body orbit. You might call it a patched CR3BP orbit because the Moon's orbit is elliptical, not circular, but I'm pretty sure two ellipses and two hyperbolae would not make for a survivable reentry. Look at the inertial-frame animation in the answer by @RussellBorogove, it's great! $\endgroup$ – uhoh Jun 7 '17 at 12:23
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TL;DR Patched Conics solutions look WEIRD when viewed in synodic frames. Since they do not reflect gravitationally correct trajectories, representing them in rotating frames will accentuate their "rough edges."

The kernel for this answer is here.

From page 437 of An Introduction to the Mathematics and Methods of Astrodynamics by Richard H. Battin, as seen in google books:

An adequate approximate trajectory may be had by matching in both position and velocity at the junction points (1) an ellipse from Earth to the sphere of influence of the moon whose focus is at the center of the earth, (2) a hyperbola around the moon, and (3) an ellipse from the sphere of influence back to Earth. The simplified problem, though itself fairly complex, is tractable when then relevant parameters and independent variables are identified.

The faint line in the drawing in the question and the drawing from the linked reference (the line drawings below, not the photos), is shown with the moon fixed. This is a frame rotating with the moon's orbit around the Earth.

The hand-drawn yellow-orange arc to the right of the moon in the drawing highlights what bothered me, that there does not seem to be an easy way to smoothly patch a hyperbolic trajectory around the far side of the moon to either of the elliptical conics with the Earth as the focus.

The drawing shows corners where the elliptical segments are patched to the hyperbola.



In my opinion the drawing highlights a problem understanding how this orbit can be approximated with two elliptical segments and one hyperbolic segment that are smoothly patched - preserving both continuity and velocity at the patch points simultaneously.

The resolution to this apparent problem is that at each patching, there is another reference frame shift. Another way to say that is that "the moon is moving!" So when you patch back to the 2nd elliptical segment, you have to remember that you've shifted reference frames twice!

The drawing in the question is provocative, it certainly gave me something to think about, but without the accompanying talk, or text, or subtext, it is does not teach us much more than to dig in to the issue more deeply because drawings can be deceiving and those Apollo engineers were really clever!

Take a look at Exploration of Lunar Free-return Trajectories by Brian Michels, ASEN 5050, University Colorado, Aerospace Sciences where there is a nice review of the patched conic analysis of the free-return orbit, and includes a lot of raw, historical data from that analysis:

http://ccar.colorado.edu/asen5050/projects/projects_2012/michels/

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