## 2 Preliminaries
### Orbit Types  
Before we start studying the orbit determination problem, it is better to choose suitable orbits. There are two rules we should follow when we make our choice. First, the chief spacecraft should move on a practically stable orbit. Second, the orbits should be chosen such that the signal from the asteroid's non-spherical gravity is strong. In order to characterize the strength of the signal in the inter-satellite range data caused by the asteroid's non-spherical gravity, we introduce the index \(\mathbf{J}\) in Equation (5). For the fourth order and degree non-spherical gravity field listed in Table 1, we integrated two orbits and recorded the mutual distance \(\rho\) after an integration time. By adding a small error to the coefficients listed in Table 1, we integrated two orbits starting from the same initial conditions and recorded the mutual distance \(\rho^{\prime}\) after the same integration time. The index \(\mathbf{J}\) was, thus, defined as \(\rho^{\prime}-\rho\). It is an obvious fact that the influence is stronger for a larger \(\mathbf{J}\) value. Our studies find that the tesseral term has a more obvious influence on the mutual distance on short time scales. In addition, as we already mentioned in the introduction section, only the second order and degree gravity field will be recovered. Therefore, only the error to C\({}_{22}\) is considered. There are six orbital elements (\(a\) for the semi-major axis, \(e\) for orbit eccentricity, \(i\) for orbit inclination, \(\Omega\) for the longitude of the ascending node, \(\omega\) for the argument of perigee, and \(M\) for the mean anomaly) to be considered for one spacecraft. To limit the dimension of the problem, we should introduce some constraints. The most important factor should be the semi-major axis which directly determines the spacecraft's altitude. In order for the signal from the non-spherical gravity to be strong, the altitude should be sufficiently low. On the other hand, low-altitude orbits are generally unstable, which may lead to \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Mass (M) & \(6.6904\times 10^{13}\) kg \\ Unit Length (L) & 8420 m \\ Unit Time (T) & 1156.486 s \\ Rotation period/h & 5.27025547 \\ Main diameter/km & 16.840 \\ Average density/g: cm\({}^{-3}\) & 2.675 \\ \hline \end{tabular}
\end{table} 
Table 2: Physical parameters of Eros. the spacecrafts' collision. Based on this consideration and the highly elongated shape of Eros, the axes of both spacecrafts were chosen as \(3[L]\), which is a sufficiently low orbit and can avoid the spacecrafts' collision with the asteroid within a moderate time. Due to the low-altitude orbit, we also set the initial orbit eccentricity as \(e=0\). Moreover, considering the orbital stability of the chief spacecraft, the terminator orbit is a good choice [21]. In our work, since the SRP is not considered, we approximated the terminator orbit as a near-circular polar orbit. As a result, we set the orbital element of the chief spacecraft as \([a,e,i,\Omega,\omega,M]=[3,0,90^{\circ},60^{\circ},60^{\circ},0^{\circ}]\). For the deputy, we set \(a=3[L]\), \(e=0\), \(\omega=0^{\circ}\) and \(M=0^{\circ}\), and we changed the values of its longitude of ascending node \(\Omega\) and orbit inclination \(i\) to study the sensitivity of the inter-satellite range data to its orbital plane. Left frame of Figure 1 displays the contour maps of measurement sensitivity when different values of orbit inclinations and longitude of ascending node are chosen.The abscissa is the orbit inclination of the deputy, and the ordinate is the difference in the longitude of the ascending node between the chief and the deputy. The sensitivity index is calculated using the following equation  
\[\mathbf{J}=\sqrt{\frac{\sum(\rho_{i}^{\prime}-\rho_{i})^{2}}{n}} \tag{5}\]  
where \(n\) is the number of measurements.