Showing an Apollo mission free return trajectory in jsOrrery
The show-stopper of the whole jsOrrery project, to me anyway, is Apollo. In a previous post, I told you that I never found the orbital elements for any manmade object, but that's not entirely true.
While building the project, I was reading books about the Apollo missions, notably "How Apollo flew to the Moon" by David Woods and "A man on the Moon" by Andrew Chaikin. Both are a tremendous read by the way. I became very interested in the physics involved in that voyage, and I thought it would be fantastic if I could observe it in my simulation.
When Apollo missions were sent to the Moon, the spacecraft was not firing its engine all the way to the destination. As there is no friction in space, you don't have to keep spending energy to maintain your speed. The only thing that alter your velocity, you know by now, is the gravity of the celestial bodies that surround you. The engineers that devised the trajectories were clever enough to use that gravity to their advantage, and limit the amount of fuel that was necessary to get to the Moon and back. The way they used it is amazing. As you can imagine, the Apollo missions were very high risk, and the resources that could be sent into space were limited. They had to find a way to minimize risk and fuel usage. In particular, they wanted a way to get the spacecraft back to Earth safely, even if everything broke down en route to the Moon.
Adding velocity to the direction an object is travelling in orbit results in changing the shape of its orbit. It makes it become a longer ellipse. The Apollo spacecraft were launched to a normal, mostly circular in low Earth orbit, which they kept for 1 and a half revolution, with the last stage of the rocket still attached but its engine turned off. At a certain point, the engine would fire for another 160 seconds or so (called the Trans-lunar injection), which would result in putting Apollo on a very long elliptical orbit that would extend way past the Moon's distance. This orbit was called the free-return trajectory.
The path of the orbit roughly intersected that of the Moon's orbit. At the time of trans-lunar injection, the Moon was not at the point of intersection, since it also is constantly moving. It would get there roughly 3 days later, by which time Apollo would be there also, so that the it would catch the spacecraft in its gravitational field. But it would not catch it so that it would stay in orbit there. Rather, the trajectory was calculated so that Apollo would pass behind the Moon, and the effect of gravity from the Moon would swing it back to the Earth without any burn from the engine. That trajectory is called free return because it does not require any fuel to get back.
In a normal mission, engine burns would be required near the Moon for Apollo to change course and stay in orbit there, and a few days later another burn would put it on a trajectory back to its home planet. The genius of the free return trajectory payed its dividend on a single mission, you know, that one with Tom Hanks. There is a lot of precision involved in the calculations, and it is difficult to imagine for a computer-age person like me that they managed to do it back then.
To be able to simulate a free-return trajectory in jsOrrery, I needed two things : a very precise position of the Moon and complete numbers for Apollo missions orbits. For the Moon, I already had the best I could achieve, but I still did not know if it was precise enough. As for Apollo's numbers, once again Internet came to my rescue.
Now came the moment of truth. The first mission numbers I tried are Apollo 8's. When I hit the play button, I was in complete awe before what was happening in my browser. Apollo was leaving the Earth, going towards a point where it would get caught in the passing Moon's gravity and swinging back towards the Earth just as it was supposed to. To be honest, it misses the Earth by a margin when it gets back, but considering that all the numbers, as well as the methods for using them, come from different sources, it is a major achievement that it works at all. I did'nt cheat the numbers to make things better, what you see comes directly from the raw data.
I tried the numbers for every Moon-bound Apollo misison, and most work but not all are precise. Apollo 17 does not even remotely work, but I suspect that there is an error in the numbers (I can tell because the Earth orbit does not match the free-return orbit at all). I don't know how the trajectory that you can see in jsOrrery is representative of the real-life Apollo trajectories, but I am quite satisfied nonetheless to get something that is at least plausible. It is beyond all the expectations I had when launching the project.
Click to see Apollo 8's free return trajectory in jsOrrery.