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In the flight path of Apollo 13 (data taken from Wikipedia), the first mid course correction was done at 30:40:50 and cryogenic oxygen tank incident occurred at 55:54:53. I have three small questions here

  1. The Apollo 13 path had a crossover around the moon i.e it went around the moon and it reached a point in space that it had already passed before (just after the trans-earth injection). Is it really necessary to choose such a path in space travel? What is the problem in going around the moon and not having a crossover in onward and return flight path?

  2. If the cryogenic oxygen incident had occurred at 30:40:50, then what would have been the flight path?

  3. There were four mid course corrections made during the flight, two each in onward and return path. Why was the span between command module/service module separation and fourth mid course correction so small?

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The Apollo 13 path had a crossover around the moon i.e it went around the moon and it reached a point in space that it had already passed before (just after the trans-earth injection). Is it really necessary to choose such a path in space travel? What is the problem in going around the moon and not having a crossover in onward and return flight path?

If the earth-to-moon trajectory passes in front of the moon's path, the moon's gravity will tend to slow the trajectory relative to Earth, and thereby make the return flight shorter in duration; this gives the "figure 8" trajectory that you're referring to. The alternative, passing behind the moon's path, tends to accelerate the spacecraft, making the return path much longer (this is the same as the "gravity assist" used by some interplanetary missions).

The figure-8 trajectory used by the Apollo missions is referred to as a "free return" trajectory: if the spacecraft doesn't slow down to enter lunar orbit, it will return to Earth quickly "for free". (Actually, on most of the missions, the trajectory wasn't free-return for most of the flight, but a small correction could make it free-return.)

If the cryogenic oxygen incident had occurred at 30:40:50, then what would have been the flight path?

The mid-course correction was fairly minor, so the trajectory wouldn't have been greatly different.

In this case, though, the power crisis faced by Apollo 13 would have started an entire day earlier, so it would have been necessary to use all the LM's descent stage fuel to accelerate the return trajectory even further after rounding the moon -- this was considered at the time, but rejected, because they would have landed in the Atlantic instead of the Pacific, and no recovery ships were immediately available. The fastest possible return trajectory that didn't use the (untrusted) service module engine would have only cut 9 hours off the trajectory they did use, so they would have had to stretch their resources another 16-17 hours to survive the crisis; this might have been survivable, but it was already a pretty close thing.

In particular, at 30 hours, I think a direct abort (thrusting directly back towards the Earth without continuing on to the moon) wouldn't have been any more feasible. With the LM attached, there wasn't enough propellant available to stop and reverse the trajectory; without the LM attached, there wasn't enough battery power to keep the crew alive.

There were four mid course corrections made during the flight, two each in onward and return path. Why was the span between command module/service module separation and fourth mid course correction so small?

During the return flight, mission control was puzzled by an ongoing unexplained "shallowing" of the trajectory. The 3rd MCC (i.e. the first one made on the return trip) put the spacecraft right on target, so that it would neither re-enter too steeply and burn up, nor graze the atmosphere without slowing down enough for a prompt reentry. It was thought (and hoped) at that point that no additional correction would be needed. Making a correction burn was a big challenge at that point: the spacecraft computer was shut down, the orientation of the spacecraft was uncertain, and the crew was exhausted. When the trajectory started shallowing again, they put off the 4th MCC until the last moment, so there wouldn't be enough time left in the flight for the spacecraft to go significantly off course again.

After the flight, they worked out that the problem was thrust produced by water sublimating off the LM's heat exchanger, pushing the spacecraft off course. The LM was shut down on the outbound flights, so not producing any heat, and not present for the other Apollo return flights, so the phenomenon hadn't been noticed or expected.

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    $\begingroup$ Yes -- let me rewrite that (though technically it's not a reversal of trajectory, just a very enthusiastic lowering of apogee....) $\endgroup$ – Russell Borogove Apr 29 at 20:25

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