A Breath of Fresh Vacuum

This past week was LISACon 8, a meeting of the LISA Consortium, similar to the LIGO meetings I've attended in the past. One of the presenters showed this daVinci-esque drawing of the LISA constellation:

ESA

A couple weeks ago, I talked about how I was working on a model of the rotation of the LISA satellites. Nominally, the satellites form an equilateral triangle, with 60° angles, but over the course of a year's orbit, those angles "breathe", getting wider and narrower as the spacecraft move along their orbits. That means that we need to change the angles the lasers point, so they can hit the distant sensors.

The laser beams are sent and received by Movable Optical Sub-Assemblies (MOSAs), the tubes in the picture above. We need to rotate those MOSAs to track the other satellites, but there's a problem: Angular momentum is conserved. Usually, we can count on the Earth to absorb extra angular momentum, but that's not possible in orbit. When we turn one of the MOSA, the spacecraft will turn under it. We can figure out how much using Newton's laws:

This says that the sum of the torques on each MOSA and the body of the spacecraft have to cancel out – "Every action has an equal and opposite reaction." We can use another of Newton's laws to express those torques in terms of the angular acceleration:

Here, the Is are the rotational inertia of the MOSAs and the spacecraft. The accelerations are measured in the inertial frame where all three bodies are rotating, but the MOSAs move within a range on the spacecraft, so we really want relative accelerations. We can get this by regrouping things:

We can make a bare-bones model by imagining two rods rotating in a solid disk to get I_M and I_S, then integrate to get the angles:

where m_S and m_M are the masses of the spacecraft and each MOSA, and φ1 and 2 are the angles of the MOSA relative to the spacecraft.

I thought this would be another fun problem to make into an HTML5 demo. Below, you'll find a diagram of this simple LISA satellite where you can control each MOSA by holding down the buttons and see how the rest of the satellite turns. You can also adjust the mass ratio – For the actual spacecraft the total mass is planned to be 480 kg, and the MOSAs are 75 kg each, giving a ratio around 2.2. You can try that below, but the differences are a little clearer if you keep the ratio small.

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MOSA 1		MOSA 2
Mass Ratio