If f_x is the difference in the specific attraction of the earth on the two proof masses and x₀ is the difference in the semimajor axes of the two orbits, the combination (f_x + 3 n² x₀) always occurs together in the equations describing the separation versus time of the two proof masses where n is the mean orbital angular velocity in radians/second. f_x and x₀ cannot be distinguished even in principle no matter how small the measurement noise is. x₀ must be known to measure f_x.

To get an idea of the tolerance in in measuring x₀, consider a satellite EP experiment designed to measure an EP violation as small as 10^-18 g_e which is 10^-17 meters/sec². For typical satellite orbits, n is about 10^-3 radians/second; so that f_x / (3 n²) is 10^-17 meters/sec² / (3 x 10^-6 /sec²) = 3 x 10^-12 meters. This distance is one hundredth the size of a Hydrogen atom and is about 10,000 times smaller than the tolerance with which the sphere and the shell can be manufactured.

Second Problem: A Violation of the EP Only Causes the Proof Masses to Separate Proportional to t Not t ²

The equation for the separation between the proof masses versus time is (f_x + 3 n² x₀) t / n. When two bodies fall on the earth (such as in the famous Leaning Tower of Pisa Experiment) they separate proportional to t² . This is important for two reasons:

1. For experiment times longer than one orbit period, an effect which grows as t² becomes much larger that one which only grows as t.

2. One of the biggest disturbances, collisions between the proof mass and gas molecules left over from the residual vacuum, grows as t^3/2. If the effect of an EP violation only grew as t, the measurement would get worse (not better) as t^1/2 . Taking a measurement for a long time would not help the experiment accuracy but would worsen it.

   Solving Both of These Problems At Once with Either DC or AC Cancellation

   It is possible to use cancellation masses to change the orbit dynamics of a Drag-Free satellite such that the degeneracy between f_x and x₀ vanishes or is at least suppressed and such that an EP violation causes the proof masses to separate proportional to 3 / 7 g_e t² in the radial direction (toward the earth). This method was first presented at the 8th Marcel Grossmann Meeting in Jerusalem and is known as DC Cancellation (PDF).

   A second method of solving this problem known as AC Excitation (PDF) was also presented at MG8. AC Excitation does not work as well as DC Cancellation, but it is possible to combine the two into a method known as AC Cancellation which has the advantages of both. In AC Cancellation, the gravity-gradient cancellation mass is shaped as a toroid; and the rotation of the satellite provides an AC Excitation signal as well as changing the dynamics with DC Cancellation. AC Cancellation is also discussed in the MG8 paper on AC Excitation.

   STEP Solves the First Problem with a Spinning Single-Axis Differential Accelerometer

   The STEP design uses a spinning differential accelerometer to solve the problem of the two proof masses having different semimajor axes. If the gravitation centers of the two proof masses are not identical, the differential accelerometer gives a force rebalance error which is at two times the satellite spin frequency. On the other hand, a violation of the EP gives a signal at the spin frequency. Thus the two proof masses may be servoed to reduce the centering error to almost zero, and the EP violation may be read off from the spin-frequency signal.

   The problem with STEP, however, is that it must be operated at low temperatures (primarily because the cylindrical form is much more sensitive to the Radiometer Effect) and forces and torques must be deliberately applied to the proof masses.