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If something would have gone wrong after a successful Moon landing requiring to ascent as soon as possible, what about the launch window for the lunar rendezvous maneuver? Could they ascent at any time or only in a short interval of the orbit period of the Command Module?

One lunar orbit of Apollo 8 took about two hours with 45 minutes without radio contact to Earth. See the answer to this question for the average orbit periods of all Apollo lunar orbits very close to two hours.

The launch window for LM might have been about 15 to 30 minutes every two hours, is that right?

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    $\begingroup$ To avoid giving readers the wrong impression, this question is about an emergency launch from the Moon. The timing of actual launches from the Moon were planned in advance by computers back in Houston, the parameters were radioed to the astronauts who typed them into the guidance computer, and the guidance computer performed the launch at the optimal time. $\endgroup$
    – DrSheldon
    Commented Jul 13, 2021 at 14:02

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The launch window for LM might have been about 15 to 30 minutes every two hours, is that right?

Not necessarily. Assuming the lunar module could sustain the astronauts for the needed time, the lunar and command service modules could have performed orbit phasing to rendezvous with each other with either the LM or CSM performing the orbital adjustments to slow down or speed up to intercept each other. This would take more time than desired, but very possible and the ability of the CSM to perform rescue maneuvers was designed into the mission. This would not be as optimal as taking off in the ideal window of time, but would be possible.

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    $\begingroup$ But orbit phasing requires orbits within the same plane. But what about LM and CSM in different orbital planes? $\endgroup$
    – Uwe
    Commented Jun 25, 2020 at 14:08
  • $\begingroup$ @PcMan Good point, the Moon rotates slowly. The Sidereal rotation period is 27.321661 days, that is 13.176 degrees per day or 0.549 degrees per hour. Your 6 degree number seems to be wrong? $\endgroup$
    – Uwe
    Commented Jul 14, 2021 at 10:46
  • $\begingroup$ @PcMan 12 degrees per day is equivalent 30 days per revolution. But the Moon period is 27.321 days, not 30. $\endgroup$
    – Uwe
    Commented Jul 15, 2021 at 10:36
  • $\begingroup$ @uwe you like nitpicking unimportant factoids, hmm? Well, your figure of 13.176 degrees per day is wrong in the 7th digit. $\endgroup$
    – CuteKItty_pleaseStopBArking
    Commented Jul 15, 2021 at 10:39
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    $\begingroup$ @PcMan An error of 9.9 % and nitpicking are two very different things to me $\endgroup$
    – Uwe
    Commented Jul 15, 2021 at 10:59

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