Ars Technica reviews "STS-107 In-Flight Options Assessment" in Appendix D.13 of the Columbia accident report which was NASA's own detailed assessment of what a rescue mission would have looked like.
The scenarios were to assume that a decision to repair or rescue the Columbia crew would be made quickly, with no regard to risk.
Columbia's LiOH canister supply to cover a total of 30 days of mission time without breaking that CO2 threshold. However, doing so would require the crew to spend 12 hours of each day doing as little as possible—sleeping, resting, and doing everything they could to keep their metabolic rates low.
The amount of liquid oxygen on board could be stretched past the CO2 scrubbers' 30-day mark by drastically cutting down Columbia's power draw.
The remaining three consumable categories consisted of food, water, and propellant. Assuming that the crew would be moving minimally, food and water could stretch well beyond the 30-day limit imposed by the LiOH canisters. To preserve propellant, the orbiter would be placed into an attitude needing minimal fuel to maintain.
Planning the inspection EVA would have taken most of Flight Day 4 (January 19), but the hard deadline of the lithium hydroxide canisters remained set at Flight Day 30 (February 15) regardless of what happened on the ground. Work would simultaneously have had to begin at the Kennedy Space Center to accelerate the processing of Atlantis.
"Accelerate" is a prosaic word for the herculean effort that would have been needed. Activities that normally take place across weeks or months would have to happen in hours or days. Civil servants and contractors at KSC would have to begin 24/7 shift work, keeping the lights on and the process running every hour of every day, for a minimum of 21 days, to power Atlantis through checkout and make it ready to launch.
Three unceasing, brutal weeks of 24/7 shift work—and that's with absolutely no margin factored in for errors or failures.
An oft-asked question is whether or not Columbia could have docked with the ISS, which would have had consumables to spare. There are numerous reasons why this would not have been possible, but the overriding one comes down to simple physics: Columbia would have had to execute what is known in orbital mechanics terminology as a "plane change" maneuver—applying thrust perpendicular to its orbital track in order to shift to match the ISS' inclination.
The plane change would have taken 25 times more fuel than Columbia had.
Weather is one of the major unknowns when planning a shuttle launch—not just at the launch site but also at the multiple places around the world that must be kept ready for an emergency landing if the orbiter needs to abort its attempt to reach orbit. The CAIB report shows that luck would have been on NASA's side here; a review of observed weather conditions on the proposed launch days showed that there was nothing happening in the atmosphere that would have hindered the launch.
More worrying, though, was that the three windows all opened at night. A night launch would substantially reduce NASA's ability to observe foam damage during Atlantis' flight to orbit, which was particularly ominous in light of the reason behind the rescue mission. Because of this, an additional EVA was added for Atlantis' crew after reaching Columbia—they would carefully examine Atlantis' wings and tiles for any damage.
About 8.5 hours of EVAs would be needed to transfer the crew from Columbia.
Atlantis would back slowly away from its sister ship, and its crew of 11 would busy themselves preparing for their own crowded reentry ordeal—never before has an orbiter landed with 11 crewmembers, and even simple things like seating would be complicated. Some crew would literally have to sit strapped to the floor during reentry.
The mission to rescue Columbia represents the kind of task that NASA, since its beginnings, has demonstrated an unswerving ability to execute. There would have been a clear goal, there would have been hard timing requirements, and the agency's massive pool of engineering talent would be empowered to accomplish the goal at any cost and without restriction.
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