Did any Apollo missions have enough fuel left to bring back the LM ascent stage?

Question

A recent question asks why the Apollo lunar module wasn't routinely brought back when returning from the Moon, so it could be used as an Apollo 13 style lifeboat. The answers correctly state that after ascent from the Moon, the LM is essentially depleted of supplies and therefore would not be helpful. However, there is also an implication (particularly in the comments) that there was not enough fuel to bring back the added mass of the LM ascent stage.

I suspect that some missions actually did have enough fuel (or at least had capacity to add enough fuel) to bring back their ascent stage. Remember, the service module was designed for a "direct mode" mission, without a lunar module. The CSM was to land on the Moon atop a descent stage; the CSM would then take off from the Moon and return to Earth. When the mode was changed to lunar orbit rendezvous, the SM therefore had excess delta-v capability, particularly for Apollo 10-14. This excess capability was taken advantage of during Apollo 15-17 to add the service module experiment bay, a lunar rover, supplies for more days and more EVAs, and more moon rocks brought back.

Did some of the Apollo missions have enough fuel left to bring back the ascent stage?

• Consider Apollo 10-12 and 14-17. We know it worked for 13. The other missions either had no crew, did not go to the Moon, or did not have a LM. Apollo 10 could have performed the lifeboat scenario, even if it didn't get all the way to the lunar surface.
• Assume that the mission has already landed the LM on the Moon, performed its EVAs, launched from the Moon, and rendezvoused in lunar orbit.
• The mass to return should include the CM, SM, LM ascent stage, whatever residual propellants exist in these three modules, the 3 astronauts, and the lunar samples.
• There needs to be enough propellant to perform the trans-Earth injection. Your choice if you also want to consider any correction burns. Your choice if you want to burn the residual propellant in the ascent stage.
• Please use the actual statistics of the individual missions (available in Apollo by the Numbers), rather than the nominal or theoretical statistics of Apollo hardware.
• Don't worry about why you would do this (that's the purview of the previous question), just whether it is simply possible.

Answer 1

Let's look at A11 for simplicity. Some key points from Apollo by the Numbers, p 307, rounded a bit:

• "Empty" LM at undocking - 5460 lbs (including everything left on board)
• CSM at start of TEI - 36970 lbs (including everything on board)
• CSM at cut-off - 26790 lbs (10180 lbs used)

The mission report, p 99, gives the overall propellant values, again rounding:

• 40800 lbs loaded
• 35740 lbs consumed throughout mission
• 5060 lbs remaining

This indicates a very generous safety margin by the time you get to the last burn - Apollo 11 brought back almost as much unused propellant (5060 lbs) as would have been taken up by the lunar module (5460 lbs). In fact, if you had been able to vent or burn up some of the LM's unused consumables (400 lbs ascent propellant, 300 lbs RCS, & 50 lbs water) then the LM mass would have been less than the mass of unused propellants that A11 brought back to Earth.

I have not run the exact calculations (I am not confident in not making a stupid mistake somewhere...), but on the basis of these figures I think you would be safe in saying that had you actually used the surplus service module propellant rather than having it be a couple of tons of ballast, you would have had enough delta-V to bring everything back to Earth.

Of course, this still raises the question of why you would actually do this: you would be sacrificing a lot of your safety margins to get a bit of extra elbow-room for two days, and the LM would still have to be discarded on arrival.

You would also be introducing a new and tricky failure mode in that last step: on the normal mission plan, if you tried to undock the LM in lunar orbit and found it was stuck, you'd be able to idle in orbit for a while, have a nap and a sandwich, and let the engineers on the ground figure out a solution.

But on the Bring-It-Home plan, you'd be on a much narrower time window, as by the time you tried to jettison the LM you would be locked into a re-entry trajectory, meaning there would be much less room for working out any problems. It would probably work fine - but why introduce the extra risk?