# Drag and Attitude Control for the Next Generation Gravity Mission
## 3 Drag and Attitude Control Role in the Measurement of the Non-Gravitational Accelerations  
### The Measurement of the Non-Gravitational Accelerations and the Main Error Terms  
The NGGM will measure the non-gravitational differential acceleration between the CoMs of the satellites in each pair (\(\Delta a_{\rm D}\)) through the use of highly sensitive electrostatic accelerometers. This kind of accelerometer operates by nulling the relative motion (linear and angular) between a proof mass (PM) and its \"electrode cage\", through the use of electrostatic forces and torques applied on it. The acceleration corresponding to the force applied to the PM for nulling its motion (\(\mathbf{a}\)) is the quantity nominally measured by the accelerometer. It is equivalent to the relative acceleration between the PM and the cage (\(\mathbf{a}=\mathbf{a}_{\rm cage}-\mathbf{a}_{\rm PM}\)), and can be expressed as:  
\[\mathbf{a}=-\Big{(}[\rm U]-[\Omega^{2}]-[\dot{\Omega}]\Big{)}\mathbf{R}+2\;[\Omega] \dot{\mathbf{R}}\;+\bar{\mathbf{R}}+\;\mathbf{a}_{D}-\;\mathbf{a}_{\rm para}=\;\mathbf{a}_{D}+\delta \mathbf{a}_{D}\]  
Here, \(\mathbf{a}_{\rm D}\) encompasses all environmental non-gravitational accelerations (atmospheric drag, radiation pressure, and propulsion system thrusting) experienced by the satellite's CoM, which are also the linear accelerations of the accelerometer cage being rigidly connected to the satellite, and \(\mathbf{a}_{\rm para}\) refers to the PM acceleration relative to the cage produced by forces internal to the satellite and to the accelerometer (self-gravity forces, thermo-molecular forces, magnetic forces, etc.). The term (\([\rm U]-[\Omega^{2}]-[\dot{\Omega}]\))\(R\) is the acceleration of the PM relative to the cage due to the coupling of Earth's gravity gradient [U], the spacecraft centrifugal acceleration [\(\Omega^{2}\)] and angular acceleration [\(\dot{\Omega}\)] around its CoM with the position vector \(\mathbf{R}\) of the center of the PM relative to the satellite's CoM. The term \(2\;[\Omega]\dot{\mathbf{R}}\) is the Coriolis acceleration of the PM relative to the cage induced by the coupling of the rate of change of the position vector \(\mathbf{R}\) with the satellite angular velocity [\(\Omega\)] around its CoM. Since the PM is kept motionless relative to the cage, the first and second time derivative of the position vector \(\mathbf{R}\) (\(\bar{\mathbf{R}}\), \(\bar{\mathbf{R}}\)) result from variations of the accelerometer cage offset relative to the CoM produced by causes internal to the satellite itself (thermo-elastic deformations of the structure to which the cage is connected, vibrations originated inside the satellite and propagating through the structure up to the cage, etc.). A schematic illustration of the contributors to the acceleration nominally measured by the accelerometer on board of the satellite is shown in Figure 6.  
Figure 6: Contributors to the acceleration \(\mathbf{a}\) nominally measured by the accelerometer. The above equation would reduce to \(\mathbf{a}=\mathbf{a}_{\rm D}\) were it to be assumed that \(\delta\mathbf{a}_{\rm D}=0\), namely that the PM control is perfect (and the PM was perfectly centered within the electrodes cage at all times), vibrations between the electrodes and the rest of the spacecraft are neglected, and the center of the PM was perfectly coincident with the spacecraft's CoM or the angular rates and accelerations were tightly controlled. On the contrary, some terms accounting for the imperfections need to be added to the equation in order to allow for systematic uncertainties in the accelerometers, beyond misalignments in the reference frames: a scale factor, misalignment and coupling matrix [M] = 1 + [dM], a quadratic factor matrix [K2], a matrix for the coupling between linear and angular accelerations [C], the accelerometer's bias \(\mathbf{b}\), and the measurement's random noise \(\mathbf{n}\).