# Gravity Field Recovery Using High-Precision, High-Low Inter-Satellite Links
## 2 Mission Concept
### Instrumentation
2015 [32], and is expressed by:  
\[\mathrm{d_{acc.\ x}=d_{acc.\ z}=10^{-11}\sqrt{\left(\frac{10^{-3}Hz}{t}\right)^ {4}/\left(\left(\frac{10^{-5}Hz}{t}\right)^{4}+1\right)+1+\left(\frac{f}{10^{- 1}Hz}\right)^{4}}\frac{m}{s^{2}\sqrt{Hz}}} \tag{2}\]  
\[\mathrm{d_{acc.\ y}=10\cdot d_{acc.\ z}} \tag{3}\]  
with x denoting along-track, y across-track, and z (close to) the radial direction. Based on the heritage of previous gravity missions for MOBILE, we seek a resolution on the level of \(10^{-11}\) m/s\({}^{2}\), which is the same as assumed for the along-track and radial axes of the accelerometer on-board the low-low satellite pair, but ideally with the same performance in all three directions. Furthermore, the slope at frequencies from \(10^{-3}\) Hz and lower is pressed down from \(1/f^{2}\) for the low-low pair to \(1/f\) for MOBILE. The performance of the relative acceleration measurement error is displayed in Figure 4 (green curve). While an accelerometer is mandatory for the MOBILE LEO satellite, for the MEOs, less stringent requirements might apply because of the substantially smaller amplitude of the signal and the fact that non-conservative forces can be modelled much more accurately in high altitudes. Also, the design of the MEO could be optimized for the high predictability of non-gravitational forces, e.g., by implementing very simple geometrical surfaces wherever radiative pressure is relevant. In spite of these facts, in the MOBILE concept, the implementation of accelerometers is proposed. Figure 3: Amplitude spectral density (ASD) of the relative distant measurement errors in terms of range-rates for the low–low pair formation (dark blue) and for MOBILE (dark green). The generation of the noise time-series was done by scaling the spectrum of normally distributed random time-series with their individual spectral model (light blue and light green curves). Geo-location of satellite observations, as well as gravity retrieval, require highly accurate continuous orbit determination, making GNSS space receivers on all satellites obligatory. In our simulations, we assume an absolute kinematic positioning on a cm level. Using a laser ranging instrument as the main measurement system requires exact pointing of the tracking antenna in the order of 10 \(\upmu\)rad or less, and therefore the implementation of systems for attitude determination and control. We assume star camera sensor errors for all satellites represented as rotation angles around the along-track (roll), cross-track (pitch), and radial (yaw) axes, expressed by the ASD of the following analytical noise models [32]:  
\[\mathrm{d}_{\mathrm{roll}}=10^{-5}\sqrt{\left(\frac{10^{-3}\mathrm{Hz}}{I} \right)^{4}/\left(\left(\frac{10^{-5}\mathrm{Hz}}{I}\right)^{4}+1\right)+1 \frac{\mathrm{rad}}{\sqrt{\mathrm{Hz}}}}, \tag{4}\]  
\[\mathrm{d}_{\mathrm{pitch}}=\mathrm{d}_{\mathrm{yaw}}=2\times 10^{-6}\sqrt{ \left(\frac{10^{-3}\mathrm{Hz}}{I}\right)^{2}/\left(\left(\frac{10^{-1} \mathrm{Hz}}{I}\right)^{2}+1\right)+1\frac{\mathrm{rad}}{\sqrt{\mathrm{Hz}}}}. \tag{5}\]  
In addition, for the MOBILE LEO satellite, a drag-reduction system needs to be implemented in order to maintain the orbit, and not to saturate the accelerometers due to non-gravitational accelerations. The MEO satellites will very likely require an electrical propulsion system to move to their target orbit from the lower separation altitude achievable with a low-cost launcher.