# Next-Generation Gravity Missions: Sino-European Numerical Simulation Comparison Exercise
## 2 Closed-Loop Simulations and Input Data
aspect, these drifting orbits have the additional advantage that they produce a dense ground-track distribution over longer periods of time, which might improve the spatial resolution for the gravity retrieval of long-term static gravity solutions. Table 2 provides an overview of the force and noise models of the true and reference world. Static, tidal, and non-tidal gravity models are used up to SH degree and order (d/o) 120. The ISL is assumed to be realized by a Laser Ranging Interferometer (LRI). The assumed spectral noise behavior of the LRI in terms of range rate is expressed by the following Amplitude Spectral Density (ASD) [22]:  
\[d_{range\text{-}rates}\ =\ 2\ \times\ 10^{-8}\times 2\pi f\times\sqrt{\left( \frac{10^{-2}\text{Hz}}{f}\right)^{2}+1}\frac{\text{m}}{\text{s}\sqrt{\text{Hz}}} \tag{1}\]  
where \(f\) is the linear frequency. The resulting error spectrum is shown in Figure 3(a) (dark blue curve), and the corresponding noise realization in light blue color. The non-gravitational forces are typically sensed by the on-board accelerometers located in the center-of-mass of the satellite. In our simulations, an electrostatic accelerometer is assumed with two high-sensitive axes oriented in the flight direction (largest signal) and radial direction, and one \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **True World** & **Reference World** \\ \hline Static gravity field & GOC005s [18] & GOC005s \\ Non-tidal time-variable gravity field & ESA AOHIS [19] & — \\ Ocean tides & EOT11a [20] & GOT4.7 [21] \\ Inter-Satellite Link (ISL) noise & Equation (1) & — \\ Accelerometer (ACC) noise & Equations (2) and (3) & — \\ Star tracker (STR) noise & — & — \\ Orbit noise & 1 cm white noise & — \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Force and noise models of the “true” and “reference” world used in the Next Generation Gravity Mission (NGGM) simulations. Figure 3: Amplitude spectral density (ASD) of (**a**) relative distant measurement errors in terms of range-rates, and (**b**) accelerometer errors. The dark blue and red curves show the analytical noise models in accordance with Equations (1) to (3), and the light blue and green curves the corresponding noise realizations, which have been generated scaling the spectrum of normally distributed random time-series with their individual spectral model. low-sensitive axis in the cross-track direction (see Figure 3b). The accuracy level in terms of accelerations is expressed by [22]:  
\[d_{acc.~{}along}~{}=~{}d_{acc.~{}radial}~{}=~{}10^{-11}\sqrt{\left(\frac{10^{-3} \mathrm{Hz}}{f}\right)^{4}}/\left(\left(\frac{10^{-5}\mathrm{Hz}}{f}\right)^{4} +1\right)+1+\left(\frac{f}{10^{-1}\mathrm{Hz}}\right)^{4}\frac{\mathrm{m}}{s^{ 2}\sqrt{\mathrm{Hz}}} \tag{2}\]  
\[d_{acc.~{}across}~{}=~{}10\,\times\,d_{acc.~{}along}, \tag{3}\]  
It has a noise level of \(10^{-11}\) m/s\({}^{2}\) in the measurement bandwidth, and a \(1/f^{2}\) slope at frequencies from \(10^{-3}\) Hz and lower. Equation (3) expresses the fact that electrostatic accelerometers, as they have been used in all previous and current satellite gravity missions, usually have two high-sensitive axes and one less-sensitive axis. In our set-up, we assume a degraded performance of the less-sensitive axis by one order of magnitude, and that it is oriented in an across-track direction. In this study, we further assume that the attitude, which is usually a minor error contributor compared to ISL and ACC, is known perfectly. Orbit errors are simulated by white noise with a standard deviation of 1 cm. The generation of the ISL and ACC noise time-series was done by scaling the spectrum of normally distributed random time-series with their individual spectral analytical noise model, as given in Equations (1) to (3).