Effective Stability of Quasi-Satellite Orbits in the Spatial Problem for Phobos Exploration

Abstract

The generation of bounded trajectories complying with operational constraints in the complex dynamic environment surrounding Phobos is not an easy task. The vicinity of Phobos is dominated by the gravity field of Mars; consequently, orbiting on a Keplerian orbit about this moon is not feasible. The quasi-satellite orbit (QSO) is a means to orbit Phobos in the sense of relative motion. In particular, the three-dimensional QSO (3D QSO) has been recently suggested as an approach for better meeting mission objectives, such as global mapping. However, the linear stability of QSOs concluded in the simplified three-body model cannot sufficiently ensure a stability domain for operations. In this context, this paper investigates the strategy for designing bounded orbits with desired stability properties and characteristics for observation. Families of periodic 3D QSOs are first computed in the circularrestricted three-body problem. The sensitivity of the QSOs to the initial epoch and operational errors is analyzed, revealing effective stability levels and region that can guide trajectory and operation design. The stability levels are then validated by a dispersion analysis in the full dynamics. Furthermore, being guided by effective stability, a preliminary attempt to maintain low-altitude and high-inclination QSOs in the full dynamics has proven successful.

Cold Atom Interferometry for Enhancing the Radio Science Gravity Experiment: A Phobos Case Study

“…In light of the MMX mission, Refs. [44][45][46] have advanced QSO and maintenance strategies. To gain a physical understanding of the dynamics for planning efficient manoeuvres, these typically adopt the elliptic Hill formulation for a timeinvariant system, or omit higher-order terms of Phobos' gravity field.…”

“…A grid search with a differential corrector step (details in [44]) set out to find a set of initial conditions leading to orbits which stay bound to Phobos for one week. This timespan can guarantee an acceptable geodetic solution accuracy, without incurring excessive costs in terms of orbit maintenance and computational runtime.…”

Mothership-Cubesat Radioscience for Phobos Geodesy and Autonomous Navigation

“…The retrograde quasi-satellite orbit (QSO) found in the Mars-Phobos three-body problem is a means for spacecraft to orbit around Phobos in the sense of relative motion [31,32]. Previous work has computed a database of periodic three-dimensional QSO (3D QSO) around Phobos in the circular-restricted three-body problem (CR3BP) [33,34]. Figure 2 presents families of periodic QSO described by the x-amplitude, A x , and z-amplitude, A z .…”

“…Figure 2 presents families of periodic QSO described by the x-amplitude, A x , and z-amplitude, A z . The sensitivity of the orbits to the injection epoch, presumed ground-based orbit determination error (i.e., 1-σ 50 m and 3 cm/s on each component), and ∆v execution error (i.e., 1-σ 1.4 cm/s on each component) has been investigated by propagating the orbits for one week in the presence of the perturbations [34,35]. The color scale in the Figure 2 represents the effective stability of the QSO according to results of sensitivity analyses, which reveals the stability region of bounded orbits around Phobos.…”

“…Therefore, a low-altitude and high-inclination 29 × 50 × 21 km (in A x × A y × A z format, where A y is the y-amplitude) 3D QSO, which is marked with a black circle and indicated hyper-stable in Figure 2 and shown to be easily maintainable in Refs. [34,36], is chosen as the candidate science orbit for the CubeSat. For the mothership, a high 100 × 200 km (A x × A y ) planar QSO, which is safe from escape and impact, is adopted.…”