This is the third post in my weeklong series about space travel.  Yesterday’s post can be found here and in it I explained the basics of getting a spaceship from low Earth orbit to the surface of the moon using the simple concepts of a prograde and retrograde burns.  Remember that burning prograde means firing your rockets in such a way that you increase your velocity in the direction of your motion, relative to the body you are orbiting.  Burning retrograde decreases your velocity in that direction.  If you are orbiting around the Earth’s equation, burning prograde means pointing your rocket in the direction or your current motion and executing a burn to gain more velocity in that direction.  

Now that we’ve been to the surface of the moon we can play a few holes of moon golf, and then once finished we can leave the surface of the moon and return to Earth.

The trip from the moon’s surface to low lunar orbit is much like the trip we took in Part 1 from the surface of the Earth to low Earth orbit, only this time there’s no atmosphere to drag us down.  So we only need to gain enough altitude to clear any lunar mountains, then burn horizontally from the lunar surface until we have enough horizontal velocity that gravity bends our trajectory around the planet and into an orbit.  If we have too little horizontal velocity, our trajectory will be bent back down to the planet’s surface, and if we have too much horizontal velocity we will escape the moon’s orbit.  Escaping the moon’s orbit is actually our next step though, so once in orbit we can burn prograde to gain velocity relative to the moon and escape its orbit.  

Once we escape the moon’s orbit, where will we be?  Back in orbit around the earth.  Remember that the moon itself orbits the Earth, and so anything orbiting the moon is also itself orbiting the Earth.  Escaping the moon’s orbit will likely bring us to an elliptical orbit with Earth as its focus.  We gained a lot of velocity relative to both the moon and the earth in order to escape the moon, but we still haven’t escaped the Earth’s orbit.  That’s actually good, we don’t want to escape the Earth (yet), personally I need to get back home.  So now that we’re out of the moon’s orbit and back into an Earth orbit, how do we get back to Earth?  Simply burn retrograde to reduce our velocity relative to Earth.  Doing this will shrink our orbit, just as burning prograde expanded our orbit in part 2.  And once we’ve shrunk our orbit to the point that our orbital trajectory crosses into Earth’s atmosphere, we’re basically guaranteed to get home.  The Earth’s atmospheric drag will slow our craft down, sapping it of horizontal momentum, until our trajectory no longer maintains an orbit but instead is bent towards the planet’s surface by gravity.  

This was something we couldn’t do on the moon because the moon doesn’t have an atmosphere, but it does bring back the Apollo 13 dilemma that I discussed all the way back in Post 2.  To recap: the Apollo 13 dilemma was about how Apollo 13 would navigate the Earth’s atmosphere to ensure it got home safely.  The astronauts needed to burn retrograde to lose enough velocity such that Earth’s atmosphere would slow them down and they would land on Earth with their parachutes, but how much should they slow down?  If they slowed down too much, they would take a steep plunge through the atmosphere, the intense heat from re-entry might destroy their capsule, and even if it didn’t the steep trajectory might not give their craft enough time to slow down enough for a safe landing.  However if they slowed down too little, then they would take a very shallow trajectory.  This shallow trajectory would mean they would not pass through enough of earth’s dense atmosphere, meaning they would not be slowed down significantly by the atmosphere, meaning their trajectory would not be bent into a surface-crossing one.  As they passed through the atmosphere, they would be slowed down, but it would not be enough and they would continue on their elliptical earth orbit.  Their orbit would still cross Earth’s atmosphere, and so each time their obit passed through the atmosphere the craft would be slowed more and more until their trajectory was bent into a surface-crossing one and it hit the ground.  The problem for the Apollo 13 astronauts was by then it would be too late.  Their elliptical orbit took days to complete and they didn’t have enough food, water or oxygen to survive for that long.  They needed to come down to earth in a single pass.

This dilemma is similar to the one we would face coming back from the moon, we need to burn retrograde such that we will pass through the earth’s atmosphere and let it take enough of our momentum so we can safely land with our parachutes.  Again the calculus for figuring this out is diabolical, and it’s the reason NASA employed so many people just to do calculations during the Apollo program.  But once we are slowed down enough by the atmosphere, our trajectory will be bent into one which crosses the surface of the earth, and from there it’s simply a matter of deploying a parachute at the right time and our craft can gently float down to land on the surface.  Mission accomplished.