Quasi-satellite orbits (QSOs) are considered by JAXA’s MMX mission, in which CNES is involved, for the scientific observation of the Martian moon Phobos prior to landing and sample return operations. These periodic orbits, originally defined in the Mars-Phobos circular restricted three-body problem, generically lose periodicity once the eccentricity of Phobos’ orbit is taken into account. In this case, QSOs are replaced by quasi-periodic tori. Recent work on MMX project includes, among many others, station-keeping strategies around QSOs exploiting invariant tori in the elliptic Hill problem. In the present paper, a resonant QSO will be first computed in the Mars-Phobos circular restricted three-body problem. Then, by continuation on the eccentricity of the secondary, this QSO will be converted into a periodic resonant QSO in the Mars-Phobos elliptic restricted three-body problem. The latter problem is more precise than the Hill problem at far distance from the secondary and thus more adapted to handle Mars-Phobos transfers. Notice that considering resonant orbits enables to preserve the periodicity of the QSOs when the eccentricity is nonzero. Then, a family of invariant tori surrounding the resonant QSO in the Elliptic Restricted Three-Body Problem will be obtained by continuation on the frequency. In the next step, stable invariant manifolds emanating from the tori will be computed. Finally, two-impulse transfers between a parking orbit around Mars and the tori surrounding the QSO will be presented with the objective to minimize the total velocity increment. Interesting transfer trajectories will be shown allowing for a ballistic approach to Phobos. These trajectories lead to a reduction in the total velocity increment in comparison with classical Mars to QSOs transfers. Here, instead of targeting directly the resonant QSO, the spacecraft will reach first an invariant manifold before coasting along this manifold until encountering a torus surrounding the QSO. Thus, the spacecraft will reach a quasi-periodic QSO inside the torus that is close to the periodic resonant QSO.

Numerical exploration of the restricted problem. VI. Hill's case: Non-periodic orbits.

Low-fuel transfers from Mars to quasi-satellite orbits around Phobos exploiting manifolds of tori

The return of human space missions to the Moon puts the Earth–Moon system (EMS) at the center of attention. Hence, studying the periodic solutions to the circular restricted three-body problem (CR3BP) is crucial to ease transfer computations, find new solutions, or to better understand these orbits. This work proposes a novel continuation method of periodic families using differential algebra (DA) mapping. We exploit DA with automatic control of the truncation error to represent each family of periodic orbits as a set 2D Taylor polynomial maps. These maps guarantee the access to any point of the family without any numerical propagation, providing a continuous abacus. When applied to the halo family at $$L_1$$ and $$L_2$$ , the planar Lyapunov at $$L_1$$ and $$L_2$$ , the distant retrograde orbit (DRO) family, and the butterfly family, we show that the DA-based 2D mapping is asymptotically more efficient than point-wise methods by at least two orders of magnitude, with controlled accuracy. To assist the computation of family of periodic orbits, we propose a novel DA-based automatic bifurcation detection algorithm that enables the continuous mapping of the family’s bifurcation criteria. A bifurcation study on the halo $$L_2$$ shows identical results as point-wise methods while highlighting two undocumented bifurcations. 

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Differential algebra methods applied to continuous abacus generation and bifurcation detection: application to periodic families of the Earth–Moon system

Progression of f-type family of periodic orbits, their nature, stability and location nearer the smaller primary for different mass ratios in the framework of circular restricted three-body problem is studied using Poincare surfaces of section. The orbits around the smaller primary are found to decrease in size with increase in Jacobian Constant C, and move very close towards the smaller primary. The orbit bifurcates into two orbits with the increase in C to 4.2. The two orbits that appear for this value of C belong to two adjacent separate families: one as direct orbit belonging to family g of periodic orbits and other one as retrograde orbit belonging to family f of periodic orbits. This bifurcation is interesting. These orbits increase in size with increase in mass ratio. The elliptic orbits found within the mass ratio 0 0.1 are elliptical orbits with eccentricity above 0.2. Deviations in the parameters: eccentricity, semi-major axis and time period of these orbits with solar radiation pressure q are computed in the frame work of photogravitational restricted Three-body problem in addition to the restricted three-body problem. These parameters are found to decrease with increase in the solar radiation pressure.

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Numerical exploration of the restricted problem. VI. Hill's case: Non-periodic orbits.

Citations

Low-fuel transfers from Mars to quasi-satellite orbits around Phobos exploiting manifolds of tori

Differential algebra methods applied to continuous abacus generation and bifurcation detection: application to periodic families of the Earth–Moon system

Progression of bifurcated family f type periodic orbits in the circular restricted three-body problem

Momentum Maps and Transport Mechanisms in the Planar Circular Restricted Three-Body Problem

Searching for centrally symmetric generating solutions in Hill problem

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