…or, how well is the Earth magnetic field mapped?
Three hallotrons (simple, cheap, robust, tiny, low-power devices) aligned in three axis can be used to determine the direction of local vector of Earth magnetic field.
GPS can be used to determine the location of the satellite. (This is trickier due to mechanisms that disable all commercial GPS chips at these speeds and altitudes, to prevent military use, but still possible.)
Knowing the location, direction of Earth magnetic field at the location, and the measurement from the hall sensors, one should be able to determine attitude of the satellite (excluding spin around precisely the axis of the magnetic field vector) quite accurately.
How precisely is this shape mapped, and how constant/variable is it over time?
…or did I miss something else; is this a viable method of attitude determination at all?