# Hybrid Electrostatic-Atomic Accelerometer for Future Space Gravity Missions
## 3 Results and Discussions
### Performance Aspects
Satellite rotation is also responsible for contrast loss through centrifugal acceleration, Coriolis acceleration, or angular acceleration. To estimate the impact of satellite rotation on the CAI contrast (see Equation (6)), we assumed an angular acceleration \(\Omega_{sat}=2\times 10^{-6}\) rad\(\cdot\)s\({}^{-2}\), corresponding to the maximum angular acceleration measured in GOCE, an angular velocity \(\Omega_{sat}=1.16\) mrad\(\cdot\)s\({}^{-1}\), an atomic velocity dispersion of \(\approx\)1 cm/s (\(\approx\)1 \(\upmu\)K), and an initial atomic position dispersion of 2 mm. Figure 27 illustrates that the dominant contribution is by far the Coriolis acceleration, which limits drastically the possibility to increase the interrogation time \(T\). The two other terms are nevertheless non-negligible for \(T\) approaching 1 s.  
To allow measurements with the CAI, it appears fundamental to reduce the apparent satellite rotation. The contrast drop is at the level of \(\approx\)40% by reducing the effective angular velocity down to a level of 5 \(\upmu\)rad/s and, at the same time, by reducing the effective atomic temperature to 0.1 \(\upmu\)K, which seems possible with velocity selection during the detection phase and/or with a preliminary Raman velocity selective pulse before the interferometer phase. Without atomic velocity selection, the residual angular velocity should be reduced down to a level of 1 \(\upmu\)rad/s. * Scale factor: Considering the CAI parameters given in Table 3, the relative scale factor uncertainty was estimated at the level of \(4\times 10^{-7}\). Considering linear drag accelerations with amplitudes lower than \(10^{-6}\) m\(\cdot\)s\({}^{-2}\) and a noise in the measurement bandwidth (1-100 mHz) lower than \(5\times 10^{-9}\) m\(\cdot\)s\({}^{-2}\)[1], the scale factor uncertainty should impact the acceleration measurement at the level of \(4\times 10^{-13}\) m\(\cdot\)s\({}^{-2}\) with a noise impact of \(2\times 10^{-15}\) m\(\cdot\)s\({}^{-2}\). Note that in some specific mode of operation, where the CAI is forced to operated at mid-fringe, the scale factor uncertainty could be even further reduced to \(<\)10\({}^{-9}\). * Magnetic fields: Even if the atoms are in one of the \(m_{F}=0\) first-order insensitive magnetic states, they are still affected by the second-order Zeeman effect [21]. Considering the uniformity of the magnetic field on the order of 0.4% over \(\approx\)1 cm and assuming a bias magnetic field during the interferometer phase of 10 mG, we can anticipate a parasitic gradient magnetic field of typically \(\approx\)40 \(\upmu\)G/cm. For an interrogation time of \(T=1\) s, this gradient would lead to an acceleration error of \(\approx\)10\({}^{-10}\) m\(\cdot\)s\({}^{-2}\). The rotating Earth magnetic field on the order of \(\approx\)0.4 G, which creates magnetic field rate variations up to 0.5 mG/s, should not have a significant impact on the measurement error thanks to the double-diffraction scheme, but should be kept significantly lower than the bias  
Figure 27: Contrast loss due to the different rotation terms for a satellite in orbit. Figure 26: (**a**) Dependence of the acceleration sensitivity according to the interrogation time \(T\). The dashed red line shows the sensitivity decrease, scaling as \(1/T^{2}\). (**b**) Dependence of the number of detected atoms according to the interrogation time \(T\). magnetic field to keep the atom spins aligned along the quantization axis during the interferometric phase. The foreseen double layer of magnetic shield should allow this objective. * Light shifts: The effect of one- and two-photon light shifts [21; 78] occurs during each Raman light pulse and could be responsible for an energetic displacement of the two involved hyperfine ground states, which has an impact on the phase at the output of a single-diffraction Mach-Zehnder interferometer. Here, a double-diffraction scheme should be implemented, and in principle, no impact of light shifts is expected, at least for differential AC Stark shifts.