Geometry of Weak Stability Boundaries

The notion of a weak stability boundary has been successfully used to design low energy trajectories from the Earth to the Moon. The structure of this boundary has been investigated in a number of studies, where partial results have been obtained. We propose a generalization of the weak stability boundary. We prove analytically that, in the context of the planar circular restricted three-body problem, under certain conditions on the mass ratio of the primaries and on the energy, the weak stability boundary about the heavier primary coincides with a branch of the global stable manifold of the Lyapunov orbit about one of the Lagrange points

A Dynamical Systems Approach to Schwarzschild Null Geodesics

The null geodesics of a Schwarzschild black hole are studied from a dynamical systems perspective. Written in terms of Kerr-Schild coordinates, the null geodesic equation takes on the simple form of a particle moving under the influence of a Newtonian central force with an inverse-cubic potential. We apply a McGehee transformation to these equations, which clearly elucidates the full phase space of solutions. All the null geodesics belong to one of four families of invariant manifolds and their limiting cases, further characterized by the angular momentum L of the orbit: for |L|>|L_c|, (1) the set that flow outward from the white hole, turn around, then fall into the black hole, (2) the set that fall inward from past null infinity, turn around outside the black hole to continue to future null infinity, and for |L|<|L_c|, (3) the set that flow outward from the white hole and continue to future null infinity, (4) the set that flow inward from past null infinity and into the black hole. The critical angular momentum Lc corresponds to the unstable circular orbit at r=3M, and the homoclinic orbits associated with it. There are two additional critical points of the flow at the singularity at r=0. Though the solutions of geodesic motion and Hamiltonian flow we describe here are well known, what we believe is a novel aspect of this work is the mapping between the two equivalent descriptions, and the different insights each approach can give to the problem. For example, the McGehee picture points to a particularly interesting limiting case of the class (1) that move from the white to black hole: in the limit as L goes to infinity, as described in Schwarzschild coordinates, these geodesics begin at r=0, flow along t=constant lines, turn around at r=2M, then continue to r=0. During this motion they circle in azimuth exactly once, and complete the journey in zero affine time.Comment: 14 pages, 3 Figure

Earth--Mars Transfers with Ballistic Capture

We construct a new type of transfer from the Earth to Mars, which ends in ballistic capture. This results in a substantial savings in capture $\Delta v$ from that of a classical Hohmann transfer under certain conditions. This is accomplished by first becoming captured at Mars, very distant from the planet, and then from there, following a ballistic capture transfer to a desired altitude within a ballistic capture set. This is achieved by manipulating the stable sets, or sets of initial conditions whose orbits satisfy a simple definition of stability. This transfer type may be of interest for Mars missions because of lower capture $\Delta v$, moderate flight time, and flexibility of launch period from the Earth

To the Moon from a B-52: Robotic Lunar Exploration using the Pegasus Winged Rocket and Ballistic Lunar Capture

A subset of the presently-defined NASA robotic lunar exploration objectives may be achievable with a new mission architecture involving the Pegasus winged rocket, small satellites, and a new class of Earth-Moon trajectories incorporating ballistic lunar capture. Enabling this potentially low-cost method of lunar exploration - perhaps for a few tens of millions of dollars per mission - is the application of the Weak Stability Boundary Theory developed by Belbruno during 1987-89, which leads to ballistic ( maneuverless ) Earth-Moon trajectories. On such a path, a spacecraft could be orbited at the Moon for little additional ∆ V (\u3c 50 m/s for minor trajectory correction maneuvers) beyond that supplied by the Pegasus for the initial Earth departure burn, resulting in a significant propellant savings. (Additional maneuvers would then be required to establish a more useful lunar orbit.) The price for this savings is an extended trip time to the Moon of 3-5 months. This type of trajectory is presently being demonstrated for the first time by the Japanese Hiten spacecraft, using an application developed in 1990 by Belbruno and James K. Miller at JPL; it may also be employed for the Japanese Lunar-A penetrator mission in 1996. If conventional Hohmann-like Earth-Moon transfers are employed, present versions of the Pegasus - even if outfitted with a small fourth stage can deliver only modest-sized spacecraft to the Moon (\u3c 50 kg), most likely not big enough to address presently-defined NASA robotic lunar exploration objectives. In contrast, if the ballistic capture technique is employed in conjunction with four-stage. versions of Pegasus, an additional 15 to 30 kg or more of spacecraft mass is gained, resulting in 65-80 kg small satellites which may be able to accomplish some meaningful objectives at the Moon, including gravity field determination, magnetospheric studies, and other related fields, particles and waves objectives. Advertised growth versions of the Pegasus combined with recent developments in small-satellite technology may allow for more capable satellites to reach the Moon, perhaps enabling the achievement of more demanding objectives. In the current tight budgetary climate, this new mission architecture may allow for incremental achievement of some NASA lunar science objectives by enabling significant enhancements in delivered small lunar satellite mass and capability while at the same time reducing the total mission costs for simple lunar missions. This lower-cost way of reaching the Moon may also provide an avenue for pursuing attractive commercial lunar activities and interesting lunar-based small-satellite constellation concepts

Chaotic exchange of solid material between planetary systems: implications for lithopanspermia

We examine a low energy mechanism for the transfer of meteoroids between two planetary systems embedded in a star cluster using quasi-parabolic orbits of minimal energy. Using Monte Carlo simulations, we find that the exchange of meteoroids could have been significantly more efficient than previously estimated. Our study is relevant to astrobiology as it addresses whether life on Earth could have been transferred to other planetary systems in the solar system's birth cluster and whether life on Earth could have been transferred here from beyond the solar system. In the solar system, the timescale over which solid material was delivered to the region from where it could be transferred via this mechanism likely extended to several hundred million years (as indicated by the 3.8-4.0 Ga epoch of the Late Heavy Bombardment). This timescale could have overlapped with the lifetime of the Solar birth cluster (~100-500 Myr). Therefore, we conclude that lithopanspermia is an open possibility if life had an early start. Adopting parameters from the minimum mass solar nebula, considering a range of planetesimal size distributions derived from observations of asteroids and Kuiper Belt Objects and theoretical coagulation models, and taking into account Oort Cloud formation models, the expected number of bodies with mass > 10 kg that could have been transferred between the Sun and its nearest cluster neighbor could be of the order of 1E14-3E16, with transfer timescales of 10s Myr. We estimate that of the order of 3E8 x l(km) could potentially be life-bearing, where l(km) is the depth of the Earth crust in km that was ejected as the result of the early bombardment.Comment: Accepted by Astrobiology. Submitted: Sep. 21, 2011. Accepted: May 2, 2012. 39 pages. 21 figures. arXiv admin note: substantial text overlap with arXiv:0808.326