## 1 Introduction
The highly non-spherical gravity obviously disturbs the spacecraft's orbit and efficiently removes the rank-deficiency problem, a fact which will be shown in our studies. Taking an audacious step further, we attempt to study the feasibility of simultaneously determining the asteroid's gravity field and the spacecraft's orbit using only the inter-satellite range data. This is definitely impossible for Earth satellites, because determining the spacecraft's orbit alone already requires the support from ground stations. But for asteroid missions it is possible due to the strong non-spherical gravity perturbation. To this end, we assume that the asteroid's rotation is already well established, i.e, rotational parameters such as the pole angles and angular speed are already known. In the current study we assume that the asteroid is uniformly rotating along its largest principal axis in space. As a very preliminary setting, we use a gravity field up to fourth order and degree to simulate the orbits and the inter-satellite range data. However, we only try to determine the gravity field up to the second order and degree. By doing so, we simulate the force model errors in the orbit determination and gravity field recovery. The gravity field of the asteroid 433 is taken as an example [1]. To simplify the notations, we call the problem of simultaneously determining the asteroid's gravity field and the spacecraft the complete-inverse problem. We divide the complete problem into two sub-problems. Sub-problem 1 is the orbit determination with a prescribed gravity field, and sub-problem 2 is the gravity-field recovery with prescribed orbit elements. In this paper, the feasibility of solving the complete problem using only the inter-satellite range data is analyzed. Our studies indicate that the complete-inverse problem is sensitive to errors in the initial estimation. It converges only when the initial estimation is close to the real value. When the initial estimation is poor, the iteration process usually fails. In our test, the accuracy of our best estimation is 3% for the \(C_{20}\) term and 8% for the \(C_{22}\) term. To help enhance the estimation accuracy, we propose two improved strategies. One is to extend the single link to multiple links by adding more deputies, and the other is to add the angle-measurement data between the chief and the deputy. Both strategies work fine in the sense that the convergence region is much wider. For the example case studied, the accuracy is 3% for the \(C_{20}\) term and 0.15% for the \(C_{22}\) term for the first improved strategy, and is 3% for the \(C_{20}\) term and 0.1% for the \(C_{22}\) term for the second improved strategy. Atthe end, we also discuss the influence of SRP and demonstrate the other perturbations only have little influence in our case. This paper is organized as follows: The coordinate system, dynamic model and measurement model are introduced in Section 2. Sub-problem 1 and sub-problem 2 are studied in Section 3 and Section 4, respectively. The complete inverse problem is studied in Section 5. Two improved strategies and the influence of SRP are discussed in Section 6. Section 7 concludes this study.