The Apollo missions, like most all missions since, used a heat shield to keep from disintegrating in the atmosphere. This approach had its flaws, however.

For one, if your approach was too shallow, you could do this:

enter image description here

                                            This is know as a bad day. Img credit me.

I'm not sure what would happen if it was too steep, but as I remember that scenario isn't fun either.

My question is this: For Apollo 13, what was the ideal sliver of atmosphere they needed to hit?

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    $\begingroup$ The cartoon depicts something that is not necessarily a bad day. Apollo had a skip entry capability, which though never used was available for use to increase access to alternate landing sites. A very carefully controlled skip, using the lifting guidance of the Apollo capsule, could result in a safe second entry much further downrange. $\endgroup$ – Mark Adler Dec 6 '13 at 17:47
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    $\begingroup$ Neat! They did think of almost everything, didn't they? $\endgroup$ – Undo Dec 6 '13 at 17:48
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    $\begingroup$ Yes, the artwork is cute. However that's not what a skipout trajectory looks like. It curves down, not up. It just doesn't curve down as much as the atmosphere is curved, resulting in an exit. $\endgroup$ – Mark Adler Dec 9 '13 at 5:35
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    $\begingroup$ @Mark Yes, I was trying to exaggerate a wee bit to show it clearly ;) $\endgroup$ – Undo Dec 9 '13 at 6:02
  • $\begingroup$ Entry angle is a big thing, if you want to splash down where you intend to and not somewhere else. Once you jettison the CM engine or in 13's case the LEM engine if your not on the right path you can probably obtain the correct entry angle with the thrusters on the CM capsule, but you will miss your targeted splashdown location, perhaps even hitting land which would not have been good! $\endgroup$ – user3459 Jun 3 '14 at 4:12

From Apollo 13 by Jim Lovell, Jeffrey Kluger, the following is stated:

In order to reenter Earth's atmosphere safely, Apollo 13 had to approach at an inclination no shallower than 5.3 degrees, and no steeper than 7.7 degrees. Come in at 5.2 degrees or below, and the blunt-ended command module would skip off the top of the atmosphere and boing straight back into space, entering a permanent orbit around the sun. Come in at a 7.8 degree or above, and the spacecraft would be able to reenter all right, but at so steep an angle and with such a high g force that the crew would probably be crushed well before they ever hit the water.

The quote is not quite correct about going into a "permanent orbit about the sun". They were in orbit about the Earth before entry, and would remain in orbit about the Earth if they skipped out or missed completely. Apollo 13's entry velocity is documented in Apollo by the Numbers. With that and the entry interface of 400,000 ft altitude, it is straightforward to compute that the specific energy of the return trajectory was negative, about $-0.4\,\mathrm{{km}^2\over s^2}$. It was therefore in orbit and not on an escape trajectory.

One could argue that Earth is in orbit about the Sun, so then anything in orbit around the Earth is as well. However in this case, the orbit is definitely not permanent. If they skipped out the first time, they would reencounter Earth's atmosphere on the next pass, since orbits are closed, and enter a final time. That would eliminate the possibility of a lunar gravity assist (which might result in an escape), since the moon wouldn't be there on the next orbit.

This video provides an interesting account of what would have happened had they not been able to target an Earth entry at all, missing by 2500 miles. In that case, perturbations from the Moon cause the vehicle to very steeply reenter Earth's atmosphere five weeks later.

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    $\begingroup$ How could it go into orbit of the sun when they were only coming from the moon? That wouldn't happen for a transfer orbit from the moon's distance. Plus, even if the angle is too shallow, they still lose some speed from drag. $\endgroup$ – AlanSE Dec 6 '13 at 15:21
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    $\begingroup$ From the numbers I can pull, Apollo 13 was actually cruising almost right at Earth escape velocity, remember they did a few burns to pick up speed on the way back. I think it's still enough that they wouldn't have orbited the sun, but it was no doubt a close call. $\endgroup$ – PearsonArtPhoto Dec 6 '13 at 15:26
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    $\begingroup$ @MarkAdler That video addresses this detail extremely well. You should post that in another answer, it's too valuable to just leave as a comment. I also think the question title should be edited, since it doesn't summarize the question very well. If you don't post it, I'll try to come back and put it as community wiki, to get this cleaned up. $\endgroup$ – AlanSE Dec 8 '13 at 16:54
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    $\begingroup$ By the way, all the drama about too shallow or too steep is just that, drama. There was really no difficulty in targeting the desired entry flight path angle, assuming that your thrusters were working. All of the Apollo lunar returns were within about a tenth of a degree of -6.5° (halfway between -5.3° and -7.7°), except for Apollo 13, which was within a quarter of a degree. $\endgroup$ – Mark Adler Dec 9 '13 at 5:43
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    $\begingroup$ @Mark Adding to really old comments ... at the time of the lunar missions I was a young Uni student doing digital orbit re-entry simulations. If the retros fire on time, at the required thrust, for the required duration, and the vehicle mass is accurate, then there's little concern. But with the known error rates in the ignition command time, the ignition start delay, the thrust magnitude, the cutoff timer, the thrust termination delay, and possible error in the re-entry mass, there was actually reasonable concern. But yes, it was a trifle overblown. $\endgroup$ – user8406 Feb 11 '15 at 6:06
