This measurement is performed with the control scheme described above, namely:

• The satellite is drag-free controlled to maintain the TM1 in a nominally centered position with respect to the satellite, using interferometer measurement of the TM1 displacement relative to the optical bench (o1)
• TM2 is then controlled, with a weak electrostatic force, to fix the distance between the two TMs, using the differential interferometer (o12)

From the measured signals, particularly that measuring the differential displacement of the two TMs (o12), we can obtain the difference in the stray forces per unit mass acting on the two TMs.  This quantity, which we call Δg = g2 – g1, is the differential acceleration that would be present between the two TMs in the absence of any applied forces or any elastic coupling to the satellite, observationally equivalent to a difference in the local gravitational field.

To calculate Δg from the measured signals o1 and o12, we simply apply Newton’s equations to the system of the satellite and the two TMs, considering the applied electrostatic forces on TM2, F_ES, and the effective resonant angular frequencies associated with the elastic coupling – associated with any steady force gradients – between each TM and the spacecraft, ω1p2 and ω2p2. This yields the value according to the force equation shown in the graphic.

The final term in square brackets involves the noise of the differential interferometer, n12, and of the TM1-SC interferometer n1. This enters directly into the “nominal” differential acceleration signal (via angular acceleration) but also into the evaluation of the elastic coupling terms.

The official LISA Pathfinder mission goal requires that the noise in the differential acceleration, Δg, be less than 30 fm/s²/√Hz at 1 mHz. Our best estimate – based on extensive ground testing and analysis of the flight hardware systems and known environmental noise – estimates that we will achieve roughly 3 times lower – and even better in the envisioned free-fall mode.