# Impact of Groundtrack Pattern of a Single Pair Mission on the Gravity Recovery Quality
## 2 Simulation Environment  
### RST Gravity Recovery Simulation Software  
For the gravity recovery simulation, a nominal circular orbit simulation package of so-called Reduced-Scale Tool, (RST) (previously called Quick-look Tool, QLT [8; 17]) is employed. The software assumes that the satellite orbits are nominal, i.e., the only perturbing force is \(J_{2}\) effect by the Earth flattening, hence the semi-major axis, inclination angle and eccentricity of the satellites are not time-variable parameters [8]. The observation equation is built up by  
\[\ddot{\rho}-\frac{1}{\rho(t)}(\left(\Delta\dot{r}_{12}\right)^{2}-\dot{\rho}^ {2}(t))=e_{12}.(V(r_{2}(t))-V(r_{1}(t))) \tag{3}\]  
where \(\rho\) is range, \(\dot{\rho}\) is range rate and \(\ddot{\rho}\) is range acceleration. \(\Delta\dot{r}_{12}\) is the relative velocity vector and \(e_{12}\) is the unit vector of the inter-satellite distance of the satellite pair. The potential gradient at the first and second satellite positions for the epoch \(t\) are \(V(r_{1}(t))\) and \(V(r_{2}(t))\), respectively.  
By the RST software, the right side of the equation above is calculated at the nominal positions of the satellites for the time interval of interest. The time-variable potential of the Earth at the positions of the satellites is calculated by the provided time-variable gravity field models at those epochs. Afterwards, the calculated values for the right side of the equation are set to the left side as the observables at those epochs, and the gravitational potential, in terms of spherical harmonics coefficients, is estimated through the system of equations for that time interval. It is important to mention that although the assumption of keeping the satellites in a perfect nominal orbit is not realistic, the simulation software provides a quick comparison of gravity recovery solutions in different mission scenarios, which can later be studied in more details by full-scale recovery tools. An evaluation of the RST has been made with the ll-SST acceleration approach applied to orbits from real orbit integration by [8], which shows a good agreement between the results of the two approaches for gravity recoveries, despite the differences between the two methods.  
The equatorial bulge of the Earth (the Earth flattening effect) is responsible for change of the Keplerian elements \(\dot{\omega}\) (argument of perigee rate), \(\dot{\Omega}\) (rate of right ascension of the ascending node) and \(\dot{M}\) (mean anomaly rate). For circular orbits, the argument of perigee is not defined, but the other two elements change with the following rates:  
\[\begin{split}\dot{\Omega}=\frac{1}{na^{2}\sqrt{1-e^{2}sin\dot{ \Omega}}}\frac{\partial T}{\partial\dot{\Omega}}\stackrel{{ for circular orbit}}{{\Rightarrow}}\dot{\Omega}=\frac{1}{na^{2}}\frac{\partial T}{ \partial\dot{\Omega}}\\ \dot{M}=n-\frac{1-e^{2}}{na^{2}e}\frac{\partial T}{\partial e}- \frac{2}{na}\frac{\partial T}{\partial a}\stackrel{{ for circular orbit}}{{\Rightarrow}}\dot{M}=n-\frac{2}{na}\frac{\partial T}{ \partial a}\end{split} \tag{4}\]  
where \(a\), \(e\) and \(I\) are respectively the semi-major axis, eccentricity (here sets to zero for the circular orbits) and inclination angle. The parameter \(n=\sqrt{\frac{GM}{a^{3}}}\) stands for the mean motion of the satellite and \(T\) is the disturbing potential field.