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In this note, an interesting region about the Sun in phase space is described where the permanent capture of an object, $P$, of small mass from interstellar space can occur, under the gravitational perturbation of the resultant mass of the Galaxy. $P$ is never ejected back into interstellar space and won't collide with the Sun. It cycles about the Sun for all time, asymptotically approaching the capture set. In addition to being permanently captured, $P$ is also weakly captured. A recent result in \cite{Belbruno:2024} describes this region in general. It has a fractal structure. This is applied to the permanent capture of rogue planets and other objects from interstellar space, Comment: 13 pages, 0 figures

The geometry of the weak stability boundary region for the planar restricted three-body problem about the secondary mass point has been an open problem. Previous studies have conjectured that it may have a fractal structure. In this paper, this region is studied for infinitely many cycles about the secondary mass point, instead of a finite number studied previously. It is shown that in this case the boundary consists of a family of infinitely many Cantor sets and is thus fractal in nature. It is also shown that on two-dimensional surfaces of section, it is the boundary of a region only having bounded cycling motion for infinitely many cycles, while the complement of this region generally has unbounded motion. It is shown that that this shares many properties of a Mandelbrot set. Its relationship to the non-existence of KAM tori is described, among many other properties. Applications are discussed., Comment: 30 pages, 5 figures

We consider the Hill four-body problem where three oblate, massive bodies form a relative equilibrium triangular configuration, and the fourth, infinitesimal body orbits in a neighborhood of the smallest of the three massive bodies. We regularize collisions between the infinitesimal body and the smallest massive body, via McGehee coordinate transformation. We describe the corresponding collision manifold and show that it undergoes a bifurcation when the oblateness coefficient of the small massive body passes through the zero value.

A resultant gravitational force due to the current estimates of the virial mass of the Milky Way galaxy, dominated by dark matter, is estimated near the Sun and is described in two different analytical models yielding consistent results. One is a two-step Hernquist model, the other is a Navarro-Frenk-White model. The effect of this force is estimated on trajectories for spacecraft sufficiently far from the Sun. The difficulty of detecting this force is studied. It is concluded that its effect should be considered for certain spacecraft missions. Its effect on the Pioneer and New Horizons spacecrafts is discussed. A future mission is discussed that may be able to detect this force. Implications of this force are discussed with its impact for problems in planetary astronomy and astrophysics., Comment: 11 pages, 2 figures

It is shown that a class of approximate resonance solutions in the three-body problem under the Newtonian gravitational force are equivalent to quantized solutions of a modified Schr\"odinger equation for a wide range of masses that transition between energy states. In the macroscopic scale, the resonance solutions are shown to transition from one resonance type to another through weak capture at one of the bodies, while in the Schr\"odinger equation, one obtains quantized wave solutions transitioning between different energies. The resonance transition dynamics provides a classical model of a particle moving between different energy states in the Schr\"odinger equation. This methodology provides a connection between celestial and quantum mechanics., Comment: 40 pages, 7 figures

In this paper we study the limit behavior of a family of chords on compact energy hypersurfaces of a family of Hamiltonians. Under the assumption that the energy hypersurfaces are all of contact type, we give results on the Omega limit set of this family of chords. Roughly speaking, such a family must either end in a degeneracy, in which case it joins another family, or can be continued. This gives a Floer theoretic explanation of the behavior of certain families of symmetric periodic orbits in many well-known problems, including the restricted three-body problem., Comment: 21 pages, 1 figure

A family of periodic orbits is proven to exist in the spatial lunar problem that are continuations of a family of consecutive collision orbits, perpendicular to the primary orbit plane. This family emanates from all but two energy values. The orbits are numerically explored. The global properties and geometry of the family is studied., Comment: 25 pages, 16 figures

We show how to regularize the big bang singularity in the presence of random perturbations modeled by Brownian motion using stochastic methods. We prove that the physical variables in a contracting universe dominated by a scalar field can be continuously and uniquely extended through the big bang as a function of time to an expanding universe only for a discrete set of values of the equation of state satisfying special co-prime number conditions. This result significantly generalizes a previous result \cite{Xue:2014} that did not model random perturbations. This result implies that the extension from a contracting to an expanding universe for the discrete set of co-prime equation of state is robust, which is a surprising result. Implications for a purely expanding universe are discussed, such as a non-smooth, randomly varying scale factor near the big bang., Comment: 21 pages, 4 figures

We consider a contracting universe and its transition to expansion through the big bang singularity with a time varying equation of state $w$, where $w$ approaches $1$ as the universe contracts to the big bang. We show that this singularity is non-regularizable. That is, there is no unique extension of the physical quantities after the transition, but rather infinitely many. This is entirely different from the case of $w > 1$ studied in \cite{Xue:2014}, where $w$ approaches a constant value $w_c > 1$ as the universe contracts. In that case a continuous transition through the big bang to yield a unique extension was possible only for a discrete set of $w_c$ satisfying coprime conditions. We also show that there exists another time variable, $N$, at the big bang singularity itself, at $t=0$, where $w$, varies as a function of $N$. This defines an {\em extended big bang state}. Within it, $H$ is infinity. In the extended state, $w$ varies from a universe dominated by the cosmological constant to $1$. After $w$ reaches $1$ then the big bang occurs and time $t >0$ resumes. This gives a dynamical mechanism for the big bang that is mathematically complete as a function of $N$. Dynamical systems methods are used with classical modeling., Comment: arXiv admin note: text overlap with arXiv:1403.2122

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.

We study the classical dynamics of the universe undergoing a transition from contraction to expansion through a big bang singularity. The dynamics is described by a system of differential equations for a set of physical quantities, such as the scale factor $a$, the Hubble parameter $H$, the equation of state parameter $w$, and the density parameter $\Omega$. The solutions of the dynamical system have a singularity at the big bang. We study if the solutions can be regularized at the singularity in the sense of whether they have unique branch extensions through the singularity. In particular, we consider the model in which the contracting universe is dominated by a scalar field with a time varying equation of state $w$, which approaches a constant value $w_c$ near the singularity. We prove that, for $w_c > 1$, the solutions are regularizable only for a discrete set of $w_c$ values that satisfy a coprime number condition. Our result implies that the evolution of a bouncing universe through the big bang singularity does not have a continuous classical limit unless the equation of state is extremely fine-tuned., Comment: minor changes to published version

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:0808.3268

The singularity for the big bang state can be represented using the generalized anisotropic Friedmann equation, resulting in a system of differential equations in a central force field. We study the regularizability of this singularity as a function of a parameter, the equation of state, $w$. We prove that for $w >1$ it is regularizable only for $w$ satisfying relative prime number conditions, and for $w \leq 1$ it can always be regularized. This is done by using a McGehee transformation, usually applied in the three and four-body problems. This transformation blows up the singularity into an invariant manifold. The relationship of this result to other cosmological models is briefly discussed., Comment: 22 pages, 0 figures

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.

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 Figures

The exchange of meteorites among the terrestrial planets of our Solar System is a well established phenomenon that has triggered discussion of lithopanspermia within the Solar System. Similarly, could solid material be transferred across planetary systems? To address this question, we explore the dynamics of the transfer of small bodies between planetary systems. In particular, we examine a dynamical process that yields very low escape velocities using nearly parabolic trajectories, and the reverse process that allows for low velocity capture. These processes are chaotic and provide a mechanism for minimal energy transfer that yield an increased transfer probability compared to that of previously studied mechanisms that have invoked hyperbolic trajectories. We estimate the transfer probability in a stellar cluster as a function of stellar mass and cluster size. We find that significant amounts of solid material could potentially have been transferred from the early Solar System to our nearest neighbor stars. While this low velocity mechanism improves the odds for interstellar lithopanspermia, the exchange of biologically active materials across stellar systems depends greatly upon the highly uncertain viability of organisms over the timescales for transfer, typically millions of years., Comment: 30 pages, 5 figures, submitted to AJ. Changes: Two new references in section 1 and two new figures (Figs. 5 and 6)

The current standard theory of the origin of the Moon is that the Earth was hit by a giant impactor the size of Mars causing ejection of iron poor impactor mantle debris that coalesced to form the Moon. But where did this Mars-sized impactor come from? Isotopic evidence suggests that it came from 1AU radius in the solar nebula and computer simulations are consistent with it approaching Earth on a zero-energy parabolic trajectory. But how could such a large object form in the disk of planetesimals at 1AU without colliding with the Earth early-on before having a chance to grow large or before its or the Earth's iron core had formed? We propose that the giant impactor could have formed in a stable orbit among debris at the Earth's Lagrange point $L_4$ (or $L_5$). We show such a configuration is stable, even for a Mars-sized impactor. It could grow gradually by accretion at $L_4$ (or $L_5$), but eventually gravitational interactions with other growing planetesimals could kick it out into a chaotic creeping orbit which we show would likely cause it to hit the Earth on a zero-energy parabolic trajectory. This paper argues that this scenario is possible and should be further studied., Comment: 64 pages, 27 figures, accepted for publication in AJ

A method is described for constructing a new type of low energy transfer trajectory from the Earth to the moon leading to ballistic capture. This is accomplished by utilizing the nonlinear Earth-moon-sun perturbations on a point mass in three dimensions. The interaction of the gravitational fields of the bodies defines transition regions in the position-velocity space where the dynamic effects on the point mass tend to balance. These are termed weak stability boundaries. The transfer is obtained by the use of trajectories connecting the weak stability boundaries. It uses approximately 18 percent less Delta-V than the Hohmann transfer to insert a spacecraft into a circular orbit about the moon. The use of this transfer has recently been demonstrated by Japan's Hiten spacecraft, which arrived at the moon on October 2, 1991. Application of the transfer method is also made to the Lunar Observer Mission.

The present concept for robotic lunar missions combines Pegasus-launched small vehicles with the weak-stability boundary (WSB) or 'ballistic capture' trajectory suggested by Belbruno (1991). The existence of the earth's WSB, as a result of a distance from the sun of about 1.5 million km, yields a transfer that is analogous to the parabolic case but in which the earth-to-moon transit time is only 3-5 months. The WSB technique has been demonstrated by the Japanase Hiten spacecraft, which reached the lunar vicinity after 5 months in transit.