## 4 Instrument Sensitivity, Accelerometer and Drag Compensation Assessment
The driving design choices for the space segment are related to the _range observations_ for the relative distance measurements between a pair of satellites and the _acceleration observations_ for the measurement of nongravitational accelerations acting on the individual satellites, and they are combined at formation orbiting satellite pair level. Depending on the design and control of the satellites, additional information such as attitude knowledge may be required to apply corrections to the aforementioned observations. From both observables, the sensitivity level of the instrument/sensor is expressed in terms of amplitude spectral density (ASD) of their correlated noise. For sake of the preliminary sensitivity analysis, the noise ASD of the observables can be simplified as follows: For the satellite-to-satellite tracking instrument devoted to the range observations, the following simplified equation can be written, as a function of the frequency \(f\):  
\[ASD(\delta d(f)_{range})=\left\{ \begin{array}{cc}k_{r}&for\ f\ \geq 10\ \mathrm{mHz}\\ \frac{k_{r}}{f}&for\ f\leq 10\ \mathrm{mHz}\end{array} \right.\qquad\left[ \mathrm{m}/\sqrt{\mathrm{Hz}}\right] \tag{15}\]  
where \(k_{r}\) expresses the parametric performance of the tracking instrument. Similarly, for the accelerometers, we can write  
\[ASD(\delta\ddot{d}(f)_{acc})=\left\{ \begin{array}{cc}\frac{k_{a}}{f^{2}}& for\ f\ \leq 1\ \mathrm{mHz}\\ k_{a}&for\ 1\leq f\leq\ 100\ \mathrm{mHz}\\ k_{a}f^{2}&for\ f\ \geq 100\ \mathrm{mHz}\end{array} \right.\qquad\qquad\left[ \mathrm{m}/\sqrt{\mathrm{Hz}}\right] \tag{16}\]  
where \(k_{a}\) expresses the white noise performance in the measurement bandwidth. More complicated and realistic models can be used for refined analyses later on (see [29] and Section 4.1). Focusing on the definition of the requirements for the sensors and instruments, several matrices were produced where the observable noises are represented as a function of the altitude and of the satellite separation, as in Table 5. The selected set of user requirements--assumed during the ESA Phase 0 studies--that are linked directly with the example in Table 5 are stated in terms of geoid accuracy, namely 1 mm accuracy at 3-day intervals with 500 km spatial resolution and 10-day intervals with 150 km spatial resolution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Altitude [km]**} & \multicolumn{4}{c}{**Inter-Satellite Distance [km]**} \\ \cline{2-5}  & **50** & **100** & **200** & **300** \\ \cline{2-5}  & \(>\)**10** & \(>\)**10** & \(<\)**10** & \(<\)**10** \\ \cline{2-5}  & \(<\)**10** & \(<\)**10** & \(<\)**1** & \(<\)**1** \\ \cline{2-5}  & \(<\)**1** & \(<\)**1** & \(<\)**1** & \(<\)**1** \\ \cline{2-5}  & \(<\)**1** & \(<\)**1** & \(<\)**1** & \(<\)**1** \\ \cline{2-5}  & \(<\)**1** & \(<\)**1** & \(<\)**1** & \(<\)**1** \\ \hline \multirow{2}{*}{Altitude [km]} & \multicolumn{4}{c}{Inter-Satellite distance [km]} \\ \cline{2-5}  & 50 & 100 & 200 & 300 \\ \hline
300 & o(E-12) & o(E-11) & o(E-11) & o(E-11) \\
350 & o(E-12) & o(E-12) & o(E-11) & o(E-11) \\
400 & o(E-13) & o(E-12) & o(E-12) & o(E-12) \\
450 & \(<\)o(E-13) & \(<\)o(E-13) & o(E-12) & o(E-13) \\
500 & \(<\)o(E-13) & \(<\)o(E-13) & \(<\)o(E-13) & \(<\)o(E-13) \\
600 & \(<\)o(E-13) & \(<\)o(E-13) & \(<\)o(E-13) & \(<\)o(E-13) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Preliminary satellite-to-satellite tracking (top) and differential accelerometer (bottom) sensitivity requirements for a selected set of user requirements [34]. Instrument noise is expressed as nm/\(\sqrt{\mathrm{Hz}}\), accelerometer noise as orders of m/s\({}^{2}\)/\(\sqrt{\mathrm{Hz}}\). “Heat” colors mean “higher” instrument/sensor complexity. The symbol o(X) indicates the order of magnitude (smaller than or equal to the X value, if not specifically specified) of the differential accelerometer measurement requirements.