Distant Retrograde Orbits Research Articles

Location and stability of distant retrograde orbits around the Moon

ABSTRACT Recently has grown the interest of placing natural or artificial objects in the neighbourhood of the Moon. We numerically investigate a region of retrograde orbits around the Moon associated with the C Family of periodic orbits and the quasi-periodic orbits that oscillate around them (Broucke 1968; Winter 2000). We have given continuity to Winter (2000) investigations by introducing a more realistic dynamical scenario, one based on the four-body Sun–Earth–Moon–particle problem. Our results showed that the region of stability diminished to approximately 4 ${{\ \rm per\ cent}}$, the original size encountered for the circular-restricted three-body problem (CRTBP), mainly due to the Sun’s gravitational perturbations. None the less, the size of the region continues to be significant and we were able to found distant retrograde orbits (DROs) around the Moon with eccentricity following e = 2.259 63 × 10−6a + 0.238 45 (standard error of 1 ${{\ \rm per\ cent}}$) and semimajor axis values of the initial osculating orbits, varying between 110 000 and 185 000 km, remaining stable for a time span of 104 lunar periods. This set of distant orbits from the Moon are characterized by a narrow range of acceptable initial positions (0.8–0.83) and velocities of ∼0.5, in the rotating Earth–Moon frame. The out of plane amplitude oscillations of $\pm 15\, 000$ km presented by these DROs are a natural outcome of the significant Moon’s inclination of 5.15°. Some results presented on this work can be useful for lunar missions, such as the ones that would require prolonged stays around the satellite and use stable distant orbits as ‘parking’ orbits, such as the advanced concepts of NASA’s Asteroid Redirect Mission, proposed a few years ago.

Coverage Analysis of Lunar Communication/Navigation Constellations Based on Halo Orbits and Distant Retrograde Orbits

AbstractWith more and more missions around the Moon, a communication/navigation constellation around the Moon is necessary. Halo orbits, due to their unique geometry, are extensively studied by researchers for this purpose. A dedicated survey is carried out in this work to analyse the coverage ability of halo orbits. It is found that a two-satellite constellation is enough for continuous one-fold coverage of the north or the south polar regions but never both. A three-satellite constellation is enough for continuous one-fold coverage of both north and south polar regions. A four-satellite constellation can cover nearly 100% of the whole lunar surface. In addition, the coverage ability of another special orbit – distant retrograde orbit (DRO) – is analysed for the first time in this study. It is found that three satellites on DROs can cover 99·8% of the lunar surface, with coverage gaps at polar caps. A four-satellite constellation moving on spatial DROs can cover nearly the whole lunar surface. By combining halo orbits and DROs, we design a five-satellite constellation composed of three halo orbit satellites and two DRO satellites. This constellation can provide 100% continuous one-fold coverage of the whole lunar surface.

Joint navigation performance of distant retrograde orbits and cislunar orbits via LiAISON considering dynamic and clock model errors

Joint navigation performance of distant retrograde orbits and cislunar orbits via LiAISON considering dynamic and clock model errors

The use of vertical instability of $$L_1$$ and $$L_2$$ planar Lyapunov orbits for transfers from near rectilinear halo orbits to planar distant retrograde orbits in the Earth–Moon system

This paper highlights natural transport pathways between in-plane and out-of-plane states associated with the vertical instability of planar Lyapunov orbits around the Lagrange points \(L_1\) and \(L_2\) in the Earth–Moon circular restricted three-body problem. Computations of invariant manifolds associated with the vertical instability of planar periodic orbits, “vertically” stable and unstable manifolds, enable quantitative analyses of inclination changes. This study finds that multiple lunar flybys gradually change the orbital elements of vertically stable and unstable manifolds, and that the distributions of the affected orbital elements depend on the Jacobi constant and on the associated Lagrange point \(L_1\) or \(L_2\) of the planar Lyapunov orbits. As an application, this study uses the vertically stable manifolds of the planar Lyapunov orbits as initial guesses for optimizing transfers from near rectilinear halo orbits to planar distant retrograde orbits. Significant \({\varDelta }v\) savings as compared with the known solutions demonstrate the usefulness of the vertical instability in spacecraft trajectory designs.

Hybrid Differential Dynamic Programming in the Circular Restricted Three-Body Problem

This paper presents the application of differential dynamic programming to low-thrust trajectory optimization in the three-body problem along with supporting techniques. A penalty method is suggested for path inequality constraints and used to enforce a minimum radius constraint. Time variable formulations are presented for the flight time and a moving target. The latter is used to access all points along a desired periodic orbit as valid terminal states. Multiphase differential dynamic programming is applied to discontinuous initial guesses that combine segments of different periodic orbits. Numerical examples include multirevolution transfers between distant retrograde orbits, a heteroclinic transfer between planar Lyapunov orbits, and transfers between L1 and L2 halo orbits in the Earth–moon circular restricted three-body problem.

Design of distant retrograde orbits based on a higher order analytical solutiont

Because the distant retrograde orbits dynamics inherently depends on special functions, approximate analytical solutions in the literature are commonly constrained to providing rough approximations of the qualitative behavior. We rely on perturbation methods and succeed in computing a higher order analytical solution that extends the range of applicability of previous solutions to the problem of relative, quasi-satellite orbits with large librations. Besides, the analytical solution provides two design parameters that are effectively used in the computation of periodic, distant retrograde orbits without constraining to the typical 1:1 resonance.

Analytical expression for distant retrograde orbits around a small natural satellite

An analytical expression for distant retrograde orbits (DROs) is obtained in this study. Owing to the fact that a planar DRO is a closed orbit and can be expressed as an approximately elliptical orbit, respective geometries and periods of DROs are analytically calculated. A switching point, where various properties of planar DROs change abruptly with an increase in the orbital radius, is determined. The Mars–Deimos system is taken as a case study in this work. The proposed method can be applied to cases where the Hill’s approximation of the restricted three-body problem is valid. Numerical calculations are performed to validate the proposed method.

A transfer network linking Earth, Moon, and the triangular libration point regions in the Earth-Moon system

In the near future, several space applications in the Earth-Moon system may require a spacecraft to hold a stable motion, but the transfer trajectory infrastructure to access such stable motions has not been fully investigated yet. The triangular libration points, L4 and L5, in the Earth-Moon system have long been thought of as potential locations for a communications satellite. Recently, Distant Retrograde Orbits (DROs) and Near-Rectilinear Halo Orbits (NRHOs) in the vicinity of the Moon have been identified as orbits of interest for manned and unmanned missions with a focus on operations in cislunar space. The triangular libration points, as well as lunar DROs and NRHOs describe special types of possible motion for a spacecraft/satellite that is influenced solely by the gravitational fields of the Earth and the Moon. What is common to the three types of solutions is that they are practically stable, that is, a spacecraft/satellite can naturally follow the solution for extended periods of time without requiring significant course adjustment maneuvers. This investigation proposes the lunar region as the central link to a transfer network that enables travel throughout the Earth-Moon system, connecting the lunar region to the vicinity of the Earth and the neighborhood of the triangular libration points. The work presented here also contributes to the infrastructure supporting such a network by expanding the transfer options available between these regions. Several new transfer options between regions of stability are presented and discussed, including transfer options between Low Earth Orbit (LEO) and lunar DRO, lunar DRO and periodic orbits near L4 and L5, as well as lunar DRO and L2 NRHOs. Underlying dynamical mechanisms enabling transfers between selected orbits are analyzed, and sample itineraries are provided.

Nonlinear librations of distant retrograde orbits: a perturbative approach—the Hill problem case

The non-integrability of the Hill problem makes that its global dynamics must be necessarily approached numerically. However, the analytical approach is feasible in the computation of relevant solutions. In particular, the nonlinear dynamics of the Hill problem close to the origin, and the libration point dynamics have been thoroughly investigated by perturbation methods. Out of the Hill sphere, the analytical approach is also feasible, at least in the case of distant retrograde orbits. Previous analytical investigations of this last case succeeded in the qualitative description of the dynamics, but they commonly failed in providing accurate results. This is a consequence of the essential dependance of the dynamics on elliptic functions, a fact that makes to progress in the perturbation approach beyond the lower orders of the solution really difficult. We propose an alternative perturbation approach that allows us to provide a very simple low order analytical solution in trigonometric functions, on the one hand, and, while still depending on special functions, to compute higher orders of the solution, on the other.

Ballistic capture into distant retrograde orbits around Phobos: an approach to entering orbit around Phobos without a critical maneuver near the moon

This paper discusses an approach for designing missions to Phobos that do not require a critical maneuver in proximity of the moon. A low-energy transfer is designed that utilizes the aspherical mass distribution of Phobos to capture a spacecraft into a distant retrograde orbit (DRO) for the mission duration. The process for generating stable DROs in the Mars–Phobos system is discussed along with the method for generating and surveying a set of ballistic capture trajectories (BCTs) for DROs with altitudes between 0.5 and 14 km above Phobos. Statistical analysis of the BCT set reveals options for designing a mission to the desired DRO. This approach can be used in any three-body system when a significant perturbation is present, such as Phobos’ aspherical co-rotating gravity field.

Earth-Mars transfers through Moon Distant Retrograde Orbits

This paper focuses on the trajectory design which is relevant for missions that would exploit the use of asteroid mining in stable cis-lunar orbits to facilitate deep space missions, specifically human Mars exploration. Assuming that a refueling “gas station” is present at a given lunar Distant Retrograde Orbit (DRO), ways of departing from the Earth to Mars via that DRO are analyzed. Thus, the analysis and results presented in this paper add a new cis-lunar departure orbit for Earth-Mars missions. Porkchop plots depicting the required C3 at launch, v∞ at arrival, Time of Flight (TOF), and total ΔV for various DRO departure and Mars arrival dates are created and compared with results obtained for low ΔV Low Earth Orbit (LEO) to Mars trajectories. The results show that propellant-optimal trajectories from LEO to Mars through a DRO have higher overall mission ΔV due to the additional stop at the DRO. However, they have lower Initial Mass in LEO (IMLEO) and thus lower gear ratio as well as lower TOF than direct LEO to Mars transfers. This results in a lower overall spacecraft dry mass that needs to be launched into space from Earth's surface.

Long term evolution of distant retrograde orbits in the Earth-Moon system

This work studies the evolution of several Distant Retrograde Orbits (DROs) of varying size in the Earth-Moon system over durations up to tens of millennia. This analysis is relevant for missions requiring a completely hands off, long duration quarantine orbit, such as a Mars Sample Return mission or the Asteroid Redirect Mission. Four DROs are selected from four stable size regions and are propagated for up to 30,000 years with an integrator that uses extended precision arithmetic techniques and a high fidelity dynamical model. The evolution of the orbit’s size, shape, orientation, period, out-of-plane amplitude, and Jacobi constant are tracked. It has been found that small DROs, with minor axis amplitudes of approximately 45,000 km or less decay in size and period largely due to the Moon’s solid tides. Larger DROs (62,000 km and up) are more influenced by the gravity of bodies external to the Earth-Moon system, and remain bound to the Moon for significantly less time.

Distant retrograde orbits and the asteroid hazard

Distant Retrograde Orbits (DROs) gained a novel wave of fame in space mission design because of their numerous advantages within the framework of the US plans for bringing a large asteroid sample in the vicinity of the Earth as the next target for human exploration. DROs are stable solutions of the three-body problem that can be used whenever an object, whether of natural or artificial nature, is required to remain in the neighborhood of a celestial body without being gravitationally captured by it. As such, they represent an alternative option to Halo orbits around the collinear Lagrangian points L1 and L2. Also known under other names (e.g., quasi-satellite orbits, cis-lunar orbits, family-f orbits) these orbital configurations found interesting applications in several mission profiles, like that of a spacecraft orbiting around the small irregularly shaped satellite of Mars Phobos or the large Jovian moon Europa. In this paper a basic explanation of the DRO dynamics is presented in order to clarify some geometrical properties that characterize them. Their accessibility is then discussed from the point of view of mission analysis under different assumptions. Finally, their relevance within the framework of the present asteroid hazard protection programs is shown, stressing the significant increase in warning time they would provide in the prediction of impactors coming from the direction of the Sun.

Results of Long-Duration Simulation of Distant Retrograde Orbits

Distant Retrograde Orbits in the Earth–Moon system are gaining in popularity as stable “parking” orbits for various conceptual missions. To investigate the stability of potential Distant Retrograde Orbits, simulations were executed, with propagation running over a thirty-year period. Initial conditions for the vehicle state were limited such that the position and velocity vectors were in the Earth–Moon orbital plane, with the velocity oriented such that it would produce retrograde motion about Moon. The resulting trajectories were investigated for stability in an environment that included the eccentric motion of Moon, non-spherical gravity of Earth and Moon, gravitational perturbations from Sun, Jupiter, and Venus, and the effects of radiation pressure. The results indicate that stability may be enhanced at certain resonant states within the Earth–Moon system.

Extension of Earth-Moon libration point orbits with solar sail propulsion

This paper presents families of libration point orbits in the Earth-Moon system that originate from complementing the classical circular restricted three-body problem with a solar sail. Through the use of a differential correction scheme in combination with a continuation on the solar sail induced acceleration, families of Lyapunov, halo, vertical Lyapunov, Earth-centred, and distant retrograde orbits are created. As the solar sail circular restricted three-body problem is non-autonomous, a constraint defined within the differential correction scheme ensures that all orbits are periodic with the Sun’s motion around the Earth-Moon system. The continuation method then starts from a classical libration point orbit with a suitable period and increases the solar sail acceleration magnitude to obtain families of orbits that are parametrised by this acceleration. Furthermore, different solar sail steering laws are considered (both in-plane and out-of-plane, and either fixed in the synodic frame or fixed with respect to the direction of Sunlight), adding to the wealth of families of solar sail enabled libration point orbits presented. Finally, the linear stability properties of the generated orbits are investigated to assess the need for active orbital control. It is shown that the solar sail induced acceleration can have a positive effect on the stability of some orbit families, especially those at the $L_{2}$ point, but that it most often (further) destabilises the orbit. Active control will therefore be needed to ensure long-term survivability of these orbits.

Natural periodic orbit-attitude behaviors for rigid bodies in three-body periodic orbits

Trajectory design increasingly leverages multi-body dynamical structures that are based on an understanding of various types of orbits in the Circular Restricted Three-Body Problem (CR3BP). Given the more complex dynamical environment, mission applications may also benefit from deeper insight into the attitude motion. In this investigation, the attitude dynamics are coupled with the trajectories in the CR3BP. In a highly sensitive dynamical model, such as the orbit-attitude CR3BP, periodic solutions allow delineation of the fundamental dynamical structures. Periodic solutions are also a subset of motions that are bounded over an infinite time-span (assuming no perturbing factors), without the necessity to integrate over an infinite time interval. Euler equations of motion and quaternion kinematics describe the rotational behavior of the spacecraft, whereas the translation of the center of mass is modeled in the CR3BP equations. A multiple shooting and continuation procedure is employed to target orbit-attitude periodic solutions in this model. Application of Floquet theory and Poincaré mappings to identify initial guesses for the targeting algorithm are described. In the Earth–Moon system, representative scenarios are explored for axisymmetric vehicles with various inertia characteristics, assuming that the vehicles move along L1/L2 Lyapunov orbits as well as distant retrograde orbits. A rich structure of possible periodic behaviors appears to pervade the solution space in the coupled problem. The stability analysis of the attitude dynamics for the available families is included. Among the computed solutions, marginally stable and slowly diverging rotational behaviors exist and may offer interesting mission applications.

Distant Retrograde Orbits for space-based Near Earth Objects detection

We analyse a concept for the detection of Potentially Hazardous Asteroids (PHAs) from a space-based network of telescopes on retrograde Distant Periodic Orbits. Planar periodic orbits are designed in the Sun–Earth circular restricted three-body problem, starting from initial conditions in the Hill’s problem available from the literature. A family of retrograde orbits centred at the Earth is selected as baseline, based on their maximum distance from Earth, larger than the Earth–L2 distance. Indeed, spacecraft on such orbits can detect PHAs incoming from the Sun direction, which could not otherwise be monitored from current Earth-based systems. A trade-off on the orbit amplitude, asteroid diameter to be detected, and the constellation size is performed considering current visible sensor telescope technology. The Chelyabinsk meteor scenario is studied and the potential warning time that could be gained with a space-based survey system with respect to an Earth based-survey system is shown.

Mission Considerations for Transfers to a Distant Retrograde Orbit

This paper discusses the applications of Distant Retrograde Orbits (DROs) about the Moon in support of advanced concepts such as NASA’s Asteroid Redirect Mission. It studies how to build a direct transfer from a low Earth orbit to a DRO, paying attention to the navigation challenges of each transfer option. The characteristics of planar DROs in the Earth-Moon system are examined. The paper focuses on a DRO that is in a 2:1 resonance with the lunar synodic period. Trade studies illustrate the relationships between the transfer trajectory duration, required launch energy, and DRO orbit insertion Δv cost. In addition, powered lunar flyby transfers are explored to determine potential cost savings in Δv. These investigations are conducted in both the Circular Restricted Three-Body system and in a high-fidelity model of the solar system.

Application of two special orbits in the orbit determination of lunar satellites

Using inter-satellite range data, the combined autonomous orbit determination problem of a lunar satellite and a probe on some special orbits is studied in this paper. The problem is firstly studied in the circular restricted three-body problem, and then generalized to the real force model of the Earth-Moon system. Two kinds of special orbits are discussed: collinear libration point orbits and distant retrograde orbits. Studies show that the orbit determination accuracy in both cases can reach that of the observations. Some important properties of the system are carefully studied. These findings should be useful in the future engineering implementation of this conceptual study.

Transfer to long term distant retrograde orbits around the Moon

Distant retrograde orbit (DRO), serving for future astrophysical observatories and communications, has received much attention, including those around the Moon. Instead of high-order analytical approximate solutions, a numerical algorithm is deduced to find DRO, with a given Jacobi value or given amplitude. The goal is to determine the DRO around the Moon and the Lyapunov orbits which are tangential to the DRO. Considering the force of Sun׳s gravity, the long term DRO remains bounded at some moments which are expropriated as the launch window. After determining the launch window of a long term DRO, two different types of transfers to the DRO are obtained, namely interior transfers and exterior transfers. Numerical results indicate that exterior transfers need less fuel consumption while interior transfers always achieve quicker transfers. By the comparison of these transfer strategies, the interior transfers with non-tangential injection can be accomplished with as less fuel consumption as the exterior transfers approximately, about 549m/s less than the transfers in the previous works.