# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 3 Results and Discussions
### Hybrid Instrument Simulator
#### 3.3.3 Satellite Rotation Compensation  
Due to satellite rotation around the Earth, the atoms are submitted to centrifugal and Coriolis accelerations. Because of the initial position and velocity dispersion of the atomic cloud, each atom should provide different outputs in terms of acceleration. The resulting interferometric phase difference for each atom induces a contrast loss in the interferometer fringes. This effect is taken into account in the simulator (see Figure 23) through the random inertial position and velocity affecting the atoms of the cloud. We considered for the simulation a standard deviation for the position and velocity distribution of, respectively, 2 mm and 2.5 mm/s. This velocity dispersion corresponds to an effective atomic temperature of 70 nK coming from the velocity selectivity of the foreseen detection scheme (see Section of Global Architecture). With an orbital angular velocity of \(\approx\)1.1 mrad/s and an interrogation time of \(T=1\) s, the atomic signal disappears almost completely. It has been proposed to overcome this limitation in CAIs by rotating the CAI reference mirror, which acts as the acceleration measurement reference [55; 69], and compensating the detrimental effect of rotation. In our specific hybrid design, the EA's proof-mass is used as the reference mirror for the CAI. Because the atomic interferometer measurement is not affected by the motion of the mirror between the laser pulses, whose durations are considered negligible, we chose to move the proof-mass angular position following a sine  
Figure 22: ASD of the acceleration noise, for the electrostatic accelerometer in black, the atom interferometer in blue, and the resulting combination in red. These data are provided by the developed hybrid instrument simulator. motion rather than with a constant angular velocity in order to have a smoother control (see Figure 24). If a saw-tooth signal had been used, the control of the PM would have been challenging in order to be fast enough to allow the repetition of the command for every interferometric cycle while allowing such a large motion range. A PID-type controller was used to control the electrostatic proof-mass. With a large bandwidth of 27 Hz and a high control gain, the PM was able to accurately follow the compensation command, on the condition of correcting the command's amplitude and time delay due to the control loop. A simulation was run with the computation of the acceleration with 100 atoms, with the electrostatic proof-mass acting as the CAI reference mirror while compensating the satellite rotation. As shown in Figure 23 (purple curve), the contrast of the interferometric fringes, which was completely lost when using a fixed Raman mirror, was fully recovered. For an efficient correction, the real orbital rotation shall be known with an accuracy better than 1%. As seen previously (cf. Figure 21), the ability to rotate the EA proof-mass induces an increase of EA noise on the radial axis due to the common electrodes for the control of translation along \(Z\) and the rotation around \(Y\). To decrease this impact, a new design of the electrodes with a separation between linear and angular control could be studied in the future. For the \(X\) axis, along-track, the main impact of the rotation is due to the fact  
Figure 23: Interferometric signal when using a fixed CAI reference mirror (yellow) and when using the electrostatic proof-mass, acting as the reference mirror, to compensate for the satellite rotation (purple). The simulation is performed with a simulated atomic cloud of 100 atoms, a standard deviation of the atomic velocities of 2.5 mm/s, of the initial atomic positions of 2 mm, and an interrogation time of \(T=1\) s.  
Figure 24: Control of the electrostatic proof-mass angular position during an interferometric cycle. The angular velocity to be compensated between \(t_{1}\) (first CAI laser pulse) and \(t_{3}\) (last CAI laser pulse) is represented in red.