Apollo's Slingshot

This story is probably one of mankind's greatests feats, a story where math, theoretical physics, space travel and actual experience meet in an exciting way.

From left to right: Lovell, Swigert, Haise

April 11th 1970, 13:13 CST, Jim Lovell (Commander), Fred Haise (Lunar Module Pilot) and Jack Swigert (Command Module Pilot) lift off the Kennedy Launch Pad 39A, on the 7th manned mission to land on the Moon. Mission Apollo XIII starts off on the wrong foot, as only 5 minutes 32 seconds in, the crew feels a little vibration and inboard (center) engine cuts two minutes earlier than planned. The crew continue as everything looks ok, but the inmediate consequence of this is that they are going to have to burn extra fuel of the remaining engines to be able to exit the Earth's orbit. This minor setback was not significant in the outcome of the mission. Two hours into the flight, both the crew and ground control are preparing for the Translunar Injection (TLI). This is a propulsive maneuver to set the spacecraft on a trajectory towards the Moon. Calculations have been made and everything runs smoothly. The mathematical expression that gives you the escape velocity to leave the Earth's atmosphere and free from its gravittional pull is:

Where G is the Universal Gravitational Constant (G = 6.67×10⁻¹¹ m³ kg⁻¹ s⁻²), M is the mass of the celestial body, R its radius and g the gravitational constant in Earth. The first part is the general expression and the second the particular case for Earth. If you wonder how much this is worth for our planet, it is 11,2 km/s or 40320 km/h.

Now back to the mission. For two days everything is normal, a couple of midcourse corrections have been made and the crew is on their way to the Moon. On the second day they are requested by Mission Control to stir up the cryo tanks. The astronauts had to turn on the stirring fans in their Hydrogen and Oxigen tanks, in order to destratisfy their cryogenic contents and increase the accuracy of their quantity readings. After a couple of minutes of silence, the next thing to be said will remain in history books and in all of our memories: "Houston, we've had a problem".

 For a transcript of the whole mission, visit Spacelog. To read this particular part, here.

As you can hear in the recording, when the crew performed the operation, they heard a loud bang. They don't know what it is yet, but they are worried about the readings from their oxygen supply. Tank 2 is reading Quantity Zero, and 1 and 3 seem to be dropping pressure. To make things worse, Jim Lovell is quoted in the transcript: "...and it looks to me, looking out the hatch, that we are venting something. We are venting something out into the...into space." 
The crew starts troubleshooting, working neck to neck with Mission Control to find out what's going on. Definitely, Tank 2 is gone, and the venting is coming from Tank 1. It will later be found that damaged teflon insulators on the wires to the stirring fans from Tank 2 had caused a short circuit, igniting this insulation. This had caused the tank to explode, damaging also Tank 1, which vented its contents in space.
With this situation,the success of the mission no longer resides in landing in the Moon, but in bringing the crew back home safe and sound. And their odds don't seem to be very high. Out of the three supply tanks, 2 are gone. Command Module and Service Module will soon no longer be useful, and the idea of moving to the Lunar Module starts to seem like their only chance of survival.

The Moon as Apollo passed behind it.

First, the crew and Mission Control need to get the LM working in the lowest possible energy consumption set up. Their water and oxigen supply is short and they will need to save as much as possible. 

Then they need to make the course corrections to slingshot the Moon. Out of the return possibilities, this is the one chosen due to the fact that they can use the Moon's gravity for a "free return" to Earth. In this maneuver, the spacecraft will go around the Moon and head back towards our planet.

But still one issue will present itself. The Lunar Module was intended to support 2 men for a day and a half, and now it will have to do it with 3 men during almost 3 days. But the problem is that there are not enough carbon dioxide filters. Both the Lunar Module and the Command Module use lithium hydroxide canisters to filter the carbon dioxide, but the stock in the LM is limited and they can't use the ones in the CM because they can't fit the cube-shaped CM canisters in the cylindrical LM socket.
Engineers on the ground, with a list of what's available onboard, figure out a way of making this possible, with a rig that will draw air from the canisters with a suit return hose. Astronauts called the invention "the mailbox".

Swigert with the "mailbox"

With this last obstacle saved, the crew can now focus on their return. One more thing they will have to do is jettison (separate) the Service Module before the re-entry. They do this in order to take pictures for the analysis. In fact it was shocking for the crew to see the hole that the explosion had caused on the hatch, they would have never imagined such a damage...

Apollo SM after separation

At this point, there is some concern on the ground that the heat shield has suffered damage from the tank explosion. The spacecraft must be able to cope with temperatures in excess of 3,000 ºC. During the re-entry, there is usually an interval of around three minutes where Ground Control and the crew don't have any communication, as the electrostatic forces around the ship make it impossible for any signal to be transmitted or received.
In the case of Apollo 13, after the fourth and the fifth minute went by, people on the ground were starting to think the worst, but on the sixth minute they can hear Jack Swigert's voice and the excitement is general.
The recovery team is waiting for the astronauts in the Pacific Ocean and the splashdown goes smoothly.
The mission will be regarded as a "successful failure", as even though the main goal of the mission, which was to land on the Moon, was not achieved, the epic return the crew in a very delicate situation made it a huge success.

Apollo's trajectory