Armstrong in Orbit

If you've been reading ScienceBlogs for a while, you might remember this little physics blog I used to write. It and I sort of vanished off the internet for a long while. More than a year, I'm sad to say. Long story short, being a grad student takes up a huge amount of time. While I sort of thought being done with classes and going to pure research would give me some breathing room, it did the opposite. Also, I got married. Being married is nice. But did it add to my quantity of free time? It did not. And after a while of not doing something which is not mandatory - like writing a blog - you start to feel guilty, which paradoxically makes you not want to think about it. And thus is the story of why my little corner of the web has lain fallow for so long.

On the other hand I do at least have something to show for my displaced effort. I've got a few papers out, won a fellowship that will fund the rest of my grad school(!), have an anticipated graduation date(!!) and even have a job waiting for me(!!!). So now I really do have some breathing room, so I'm writing here again. Since I want to make sure it is sustainable, I'm sticking to a schedule. For the moment I'm going to be writing one regular post per week, to appear each Monday. I'll end up writing more than that, I think, but the Monday post is the bread-and-butter for now.

To get back in the groove, we'll start with something easy. This particular post is in honor of Neil Armstrong and his two still-living crew-mates, Buzz Aldrin and Michael Collins. While Neil and Buzz are at least commonly-known names, their accomplishments other than those few hours on the moon are less well-known. But all three had very extensive aerospace accomplishments before Apollo 11, and all three flew in space on Project Gemini, where all three worked on the very difficult task of orbital rendezvous. That's one of the things you REALLY REALLY have to get right. After all, when Neil and Buzz took off from the lunar surface they had to find and dock with Michael, who was holding down the fort in the command module that would take them back home. But that command module was orbiting thousands of miles away from Tranquility Base, and moving at thousands of miles per hour.¹ They didn't have a lot of room for error, and it would have been impossible without the earlier hard-won experience of Project Gemini.

Consider Michael Collins a few years before, orbiting in his tiny Gemini 10 spacecraft with John Young. Their task was to dock with an unmanned Agena rocket which would propel them into a higher orbit. We can pretend both spacecraft were in circular orbits 200km above the surface of the earth. Young and Collins see the Agena in the distance ahead of them. Gemini 10 fires its rockets to catch up, and starts heading toward the Agena.

View from Gemini 10 prior to docking

Well, they're in circular orbits, so they're experiencing the acceleration of uniform circular motion. That acceleration is being provided by gravity. Put the acceleration of uniform circular motion on the right, and the acceleration due to gravity on the left:

G is the gravitational constant, M the mass of the earth, v the orbital velocity. We can solve for v and find their speed as a function of r. Here r is the distance from the center of mass of the earth, so their above-ground altitude is r minus the radius of the earth. At their altitude, they're scooting along at a velocity of perhaps 7500 m/s. We notice that v and r are the only variables here, so as one changes so must the other. Solve for r:

They key point is that they are related one-to-one. Change v, and r changes. If you are properly lined up in altitude with your docking target, and you increase your v to catch up, you will screw up your altitude. Mathematically we can differentiate, and we can see how r changes as v increases due to Collins hitting the throttle:

Plug in a change in velocity of (say) 1 meter per second. The corresponding change in r is something like -1.8 km. The thrust was applied to move the spacecraft forward in its orbital direction, and this will raise the orbit. But the equation told us that higher v means a lower orbit by almost two kilometers. The math is telling us that this "let's accelerate and catch up" maneuver won't work. It produces a higher altitude and thus ends up reducing v. So Gemini 10 moves forward, and simultaneously it starts rising to a higher orbit. Higher orbits are slower. Gemini 10, despite starting off heading toward Agena, finds itself rising above and drifting backwards from its target. Further attempts to get close to Agena make the problem worse, and the mission fails.

Of course this didn't happen, because NASA knew orbital mechanics very well. If you're behind your target in an orbit, you fire your thrusters to seemingly move away from your target. This moves you to a lower orbit, you catch up, and then you fire your thrusters the other way once you're where you want to be. (Real orbital mechanics is more complicated still: after firing thrusters you're no longer in a circular orbit. But we'll not worry about that.) Indeed one of the reasons the US eventually passed the Soviets in the space race was Russian difficulty with orbital rendezvous. Not because the Russians didn't understand the math of course, but because what you know in theory isn't always easy to do in engineering practice. Collins and Young accomplished their docking mission pretty much without a hitch. The Gemini 8 mission flown by Armstrong and David Scott did not go so well, though not for reasons or orbital mechanics. While docked they had a thruster stick in the "on" position, and the resulting spin would probably have ended up killing both crew members without some very quick and precise reactions by Armstrong. He was quite a pilot, and the world will miss him.

[1]On this blog I usually use metric, but once in a while I'll lapse into US Customary when not doing calculations. Hey, Apollo-era NASA used feet per second.