## 2 Orbit Selection Approach
Generally, the satellites for measuring the gravity field need to orbit as low as possible because the magnitude of the gravity field is stronger at lower altitudes. To spend as much time as possible at low altitudes, circular orbits are preferred. To enable the retrieval of gravity field models with a high resolution in space and time, we need to achieve a dense spatial sampling with a minimum number of orbital revolutions. To analyze the denseness of the sampling, we exploit the fact that the ground tracks of two arbitrary revolutions of a circular orbit differ mainly in longitude at the equator. We simplify by considering only the ascending tracks, i.e., the half of the ground track where the satellite is orbiting northward. Obviously, the spatial sampling is as dense as possible when the intersections  
Figure 1: Amplitude spectral density of the spherical harmonics of the time-variable gravitational potential due to nontidal mass change in the atmosphere, oceans, land hydrology, land cryosphere and solid Earth inferred from the ESM. of the ascending tracks at the Earth equator are evenly distributed along the equator. This is illustrated in Figure 2, which shows 31 ascending tracks of a circular orbit with a semimajor axis of 6,718,085 m and an inclination of 70\({}^{\circ}\). The intersections of the ascending tracks and the equator, to which we refer as ascending equator crossings in the following, are marked by red dots. Since they are almost evenly distributed, the spatial sampling is near homogeneous; i.e., it is almost as dense as possible for that number of orbital revolutions. To quantify the denseness of the spatial sampling at a predefined number of orbital revolutions, we define the ground track homogeneity \(h_{l}\). \[h_{l}=\frac{\Delta\lambda_{max,l}}{\Delta\lambda_{min,l}} \tag{1}\]  
\(h_{l}\) is the ratio of the largest and the smallest difference between adjacent ascending equator crossings, denoted by \(\Delta\lambda_{max,l}\) and, respectively, where \(l\) is the number of orbital revolutions. The value \(h_{l}=1\) indicates that the ground track repeats every \(l\) orbital revolutions, which is the definition of a repeat orbit. Values larger than unity, but still close to unity, indicate near-repeat orbits. In the following, we describe how to calculate the homogeneity \(h_{l}\). We start with the difference in longitude between one ascending equator crossing and another after \(n\) orbital revolutions,  
\[\Delta\lambda_{n}=\Big{(}\hat{\Omega}-\omega\Big{)}nT, \tag{2}\]  
where \(\hat{\Omega}\) is the secular motion of the ascending node, \(\omega\) is Earth's mean angular velocity and \(T\) is the orbital period. The secular motion of the ascending node is defined as  
\[\hat{\Omega}=-\frac{3}{2}\frac{J_{2}R^{2}\sqrt{GM}}{a^{7/2}{(1-e^{2})}^{2}}\cos i, \tag{3}\]  
where \(GM\) is Earth's gravitational constant, \(J_{2}\) is Earth's dynamical form factor, \(R\) is the equatorial radius of the Earth (6.3781366 \(\times\) 10\({}^{6}\) m), is the semimajor axis of the orbit, \(e\) is the eccentricity of the orbit and \(i\) is the inclination of the orbit. The orbital period is  
\[T=2\pi\sqrt{\frac{a^{3}}{GM}}, \tag{4}\]  
Figure 2: Ascending tracks (blue lines) of a circular orbit with a semimajor axis of 6,718,085 m and an inclination of 70\({}^{\circ}\). Ascending equator crossings are marked by red dots. which remains practically constant as an orbit maintenance and/or drag compensation system can ensure the spacecraft's target altitude. Let \(\lambda_{k}\) denote the longitude of the \(k\)th ascending equator crossing, which we may freely select. Then, the longitudes of all other ascending equator crossings can be calculated by  
\[\lambda_{k+n}=\lambda_{k}+\Delta\lambda_{n}. \tag{5}\]  
Alternatively, we could determine the longitudes of the ascending equator crossings through orbit integration. Successively, we sort the first \(k\) longitudes in ascending order, so that they form a monotonously increasing sequence.