A spiritual grandfather of the Math Club blog was Robert Osserman. He was a well regarded mathematician at Stanford University. In recent years, though, he was equally well known for his efforts to connect math and mathematicians with everyday people. In his role with the Mathematical Sciences Research Institute in Berkeley, CA he organized a large number of public events about math. Everything from Fermat Fest in 1993 celebrating the 2 month old proof of Fermat’s Last Theorem, to public discussions with Steve Martin (which began Bob Osserman: “I’ve got a book here from 1612. . . .” Steve Martin, in a deadpan drawl: “And I thought this was going to be dull.” and ended with a cameo by Robin Williams) and Alan Alda.
Unfortunately Dr. Osserman passed away recently. In his honor we thought we’d share an essay he cowrote about using math to fix the crippled Mir space station. He wrote the article with Michael Foale (an astronaut who was part of the ill fated Mir mission) and Marianne Freiberger.
In 1997 a failed docking of a supply ship with Mir caused it to lose 1/3 of its solar panels and starting tumbling in space. The problem was to use the limited amount of power left to stabilize Mir in such a way as to point the remaining solar panels towards the sun.
Where’s the math? Well, after several hours of trial and error, the cosmonauts were able to get Mir temporarily pointed in the right direction. The problem, though, is that its position wasn’t a stable solution — a slight nudge and it would start tumbling again. The solution? As even Landry Jones would tell you, you’ve got to put some spin on it!
Euler‘s equations of rotational motion show that any rigid body in three dimensions has three principal axes. If you rotate around two of these (the stable axes), then the position of Mir becomes stable. If you rotate around some random other axis, then it becomes unstable. If you think of a football, this makes sense. There is a reason why a spiral travels so nicely, but a badly thrown ball tumbles erratically through the air.
They explain this nicely in the article:
Euler’s equations of rotational motion describe exactly how the movement of a rotating body evolves over time, and capture how this movement changes if you apply certain external forces. His equations show that every body possesses three very special axes, and that the body’s movement is crucially determined by three numbers called the principal moments of inertia.
When you set an object spinning about an axis, you need to apply a certain amount of force to get it going in the first place. This amount depends on the way in which the object’s mass is distributed with respect to the axis: a wheel, for example, is harder to spin when most of its mass is located around its rim, than when it is concentrated close to the centre. Mathematicians use a number to measure the resistance of the object to being spun, and this number is called a moment of inertia. Each axis has its own moment of inertia.
Euler noticed that, no matter which object you are spinning, whether it’s a space station, a potato or an elephant, the same basic pattern always occurs. The axis that gives you the smallest moment of inertia (the least resistance to being spun) is perpendicular to the axis giving you the greatest moment of inertia. These two axes, together with the line that is perpendicular to both of them, are known as the object’s principal axes of inertia.
When an object has some symmetry, the principal axes often coincide with the symmetry axes, and this is the case for Mir: its principal axes of inertia are exactly the x, y and z-axes shown in the image [above]. What Euler’s equations also show — and what Michael Foale knew — is that the two axes associated to the smallest and the largest moment of inertia give rise to stable rotations: spin Mir around one of these axes and it will keep on spinning nicely in the same direction. A spin about the third axis, however, is not stable.
Not only did they have the problem of rotation, but Mir had to spend enough time with its solar panels pointed towards the sun or it would all be for naught.
The position of the thrusters meant that rotating around the x-axis was out. And rotating around the z-axis would not have allowed the panels to face the sun. So the y-axis in the picture was their best option:
“The best thing we could do was to rotate about the y-axis and hope that it was stable.
Eventually, we managed to get a rough spin about the y-axis and to orient it in what we thought was the direction of the Sun. We could not know for sure because at the time we were passing the dark side of the Earth. But then we saw the rim of the Earth go blue, then red, and then the Sun popped out — and low and behold it was roughly in the direction of the y-axis.
At that point we managed to charge the station’s batteries continually while the Mir passed on the Earth’s sunny side, which takes about an hour. That was enough to bring the station back to life and to bring up communications with the ground. With Moscow’s help we could get accurate control of our orientation. After 30 hours we could finally relax, get some sleep and start dealing with other consequence of the collision.”
The y-axis, as it turned out, was not stable. “I noticed that our rotation was stable for only about one and a half hours. Then the Mir would do a flip and the solar arrays were pointing exactly in the opposite direction. What we had discovered was that the y-axis was the one with the middle moment of inertia.” Luckily though, with batteries charged and communications to the ground re-established, this did not turn out to be a major problem.
— Michael Foale quoted in the article.
All’s well that ends well!