# Drag and Attitude Control for the Next Generation Gravity Mission
## 4 Drag and Attitude Control Impacts on Mission Performance
### Error Budget for the Non-Gravitational Acceleration Measurement  
The mission's control requirements regarding drag and attitude/pointing control flow down from the coupling (C) and transformation (T) contributors to the measurement error. On the other hand, the intrinsic accelerometer performance (I) is ensured through the use of the MicroSTAR accelerometer described in Section 3.2. The stability between the CoM and accelerometer feeds into the satellite (S) contributor. When assuming that said control requirements are exactly met through the use of fine pointing attitude control and full (3-axis) drag-free propulsive control at the mission's altitude range, the baseline performance in the measurement of the non-gravitational differential acceleration between the CoMs of the satellites (\(\Delta a_{\text{D}}\)) is achieved as shown in Figure 17.  
While this is the only configuration explored that is capable of fulfilling the performance requirements, several performance degradation scenarios of interest have been explored in the context of the NGGM's Phase A, as part of the ongoing mission tradeoff analysis, in the interest of cost, complexity, and other practical parameters of interest. By selecting the uncontrolled drag accelerations provided in Section 3.4 (Figure 16) to feed into the performance model, while assuming the fulfillment of the attitude control requirements, the following scenarios have been studied obtaining the performance shown in Figure 18:  
Figure 17: Nominal non-gravitational acceleration measurement performance (green line) in the case of perfect matching of control requirements (see Section 3.3) and realistic conservative allocations for the values of the parameters subject to calibration (see Section 3.1), in comparison to mission performance requirements (light blue) and goals (magenta). Shown in dark blue, red, yellow, and purple are the (I), (C), (S), and (T) uncertainty contributors to the overall budgeted performance, respectively. * no drag-free control along any of the axes;
* drag-free control along the flight direction (satellite X axis) only, no control along Y and Z axes. As expected, the impact on the measurement of the differential acceleration caused by the total or partial lack of drag-free control is more important on the lower orbit where the aerodynamic forces experienced by the satellite are larger. Note that, since the Earth's gravitational potential decreases with the orbit radius \(r\) (from Earth's center) very rapidly for large values of the spherical harmonic degree (proportionally to \(1/r^{(\text{SHD}+1)}\), as deduced from the spherical harmonic expansion of the gravitational potential \(U(r,\theta,\varphi)=R(r)P(\theta)Q(\varphi)\) of a non-symmetrical mass distribution satisfying Laplace's equation \(\
abla^{2}U=0\), where the solution for the radial component takes the form \(R(r)=r^{-(l+1)};l\in[0,\infty)\) for points located outside the mass distribution), a worse measurement on a low orbit degrades the gravity field retrieval at a high spatial resolution more than a better measurement on a high orbit. Alternatively, while assuming the fulfillment of the drag-free requirements along any axis, the effect of a coarse attitude control has been studied. For this purpose, in-flight data from the GFO's attitude have been employed (from ref. [32]) and fed into the performance model obtaining the performance shown in Figure 19. Note that, in principle, this scenario is independent of orbital altitude since the triaxial drag-free control capability is retained. This kind of coarse attitude control produces, in particular, a variation of the satellite pitch/yaw angles in the measurement bandwidth which translates into a transformation error degrading the \(\Delta a_{\text{D}}\) measurement performance by about two orders of magnitude around 1 mHz independently on the orbit altitude. A similar degradation is produced by the lack of drag-free control on the lowest orbit considered for the NGGM scenario.