# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 3 Results and Discussions
### Performance Aspects
Since the error phase shifts due to wavefront aberrations result from the convolution between the distribution of atomic trajectories and the Raman beam wavefronts and, consequently, depend on parameters such as atomic temperature, atoms' initial position/velocity, and Raman laser beam shape, it is important to consider the related properties of the atomic source. If we simplify the problem and assume only a quadratic dependency of the wavefronts on the transverse position of the atoms, with an atomic cloud centered on the mirror and with no mean velocity, we can give here a rough estimation of acceleration stability. Considering a high-quality mirror of flatness (\(\lambda/300\)) over a diameter of 20 mm, corresponding to a radius of curvature of \(\approx\)19 km and a temperature variation of 100 nK, this leads to acceleration variations of \(5\times 10^{-10}\) m\(\cdot\)s\({}^{-2}\). Note that this model constitutes a huge simplification of the problem of wavefront aberration and that it does not reflect totally the achievable performances. It has to be taken as a best-case scenario. To evaluate more rigorously the stability of the acceleration measurement, we have to take into account the position and velocity stability of the atomic source along with the temperature stability. This leads to a complex analysis, which should be treated with dedicated simulations [13]. * Coupling with satellite rotation: All the acceleration terms coming from the satellite rotation, fundamentally linked to the centrifugal acceleration, to Coriolis acceleration, and to angular acceleration, could contribute to the measurement error of the atomic accelerometer. We give in Table 4 an estimation of the dominant errors terms. We consider here that the CAI, and more particularly the atomic cloud, is located by construction at the position of the satellite center of mass. Table 5 lists the assumptions concerning the atomic source allowing the numerical evaluation of acceleration errors gathered in Table 4. We can see that the Coriolis term, involving the transversal atomic velocity, has a largely dominant contribution to the instrument stability. Assuming a compensated satellite rotation around both the \(Y\) and \(Z\) axis at a level of 100 nrad/s, we can see that we reached a level of \(10^{-10}\) m\(\cdot\)s\({}^{-2}\) of acceleration stability. The error is dominated by the Coriolis acceleration and the angular acceleration terms. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Inertial Terms**} & **Acc. Stab.** & **Acc. Stab.** & **Acc. Stab** \\  & **without rot. Comp.** & **with rot. Comp.** & **with rot. Comp.** \\  & & **Around Y** & **Around Y** and \(Z\) \\  & \([\pm\)**m\(\cdot\)s\({}^{-2}]\)** & \([\pm\)m\(\cdot\)s\({}^{-2}]\) & \([\pm\)m\(\cdot\)s\({}^{-2}]\) \\ \hline \(2\cdot\Omega_{y}\cdot v_{z}\) & \(2.3\times 10^{-6}\) & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) \\ \(2\cdot\Omega_{z}\cdot v_{y}\) & \(2\times 10^{-7}\) & \(2\times 10^{-7}\) & \(2\times 10^{-10}\) \\ \(\Omega_{y}^{2}\cdot x\) & \(2.7\times 10^{-10}\) & \(2\times 10^{-18}\) & \(2\times 10^{-18}\) \\ \(\Omega_{y}\cdot z\) & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) \\ \(\Omega_{z}\cdot y\) & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) & \(2\times 10^{-10}\) \\ \(\Gamma_{xx}\cdot v_{x}\cdot T\) & \(3\times 10^{-9}\) & & \(3\times 10^{-9}\) \\ \(\Gamma_{xx}\cdot x\) & \(6\times 10^{-10}\) & & \(6\times 10^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Estimation of main errors coming from the coupling of the atomic source kinematics with the satellite rotation and the Earth gravity gradient. The estimations with rotation compensation are given considering a satellite rotation compensation with a residual at a level of 100 nrad/s.