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ABSTRACT 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. 

Abstract 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 Schrödinger 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 Schrödinger 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 Schrödinger equation. This methodology provides a connection between celestial and quantum mechanics. 

For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of “diffusing orbits” along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system obtained is not symplectic but conformally symplectic. We provide explicit conditions on the dissipation parameter, so that the resulting system still exhibits energy growth. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. In this work, the coupling is carefully chosen, but the mechanism we present can be adapted to general couplings, and we will deal with the general case in future work. 

We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on A^3. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are –dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable manifolds. These enable us to define chains of cylinders, i.e., finite, ordered families of cylinders where each cylinder admits homoclinic connections, and any two consecutive cylinders in the chain admit heteroclinic connections. Our main result is on the existence of diffusing orbits which drift along such chains of cylinders, under precise conditions on the dynamics on the cylinders – i.e., the existence of Poincaré sections with the return maps satisfying a tilt condition – and on the geometric properties of the intersections of the center-stable and center-unstable manifolds of the cylinders – i.e., certain compatibility conditions between the tilt map and the homoclinic maps associated to its essential invariant circles. We give two proofs of our result, a very short and abstract one, and a more constructive one, aimed at possible applications to concrete systems. 

We consider a restricted four-body problem, with a precise hierarchy between the bodies: two larger bodies and a smaller one, all three of oblate shape, and a fourth, infinitesimal body, in the neighborhood of the smaller of the three bodies. The three heavy bodies are assumed tomove in a plane under theirmutual gravity, and the fourth body to move in the three-dimensional space under the gravitational influence of the three heavy bodies, but without affecting them.We first find that the triangular central configuration of the three heavy oblate bodies is a scalene triangle (rather than an equilateral triangle as in the point mass case). Then, assuming that these three bodies are in such a central configuration, we perform a Hill approximation of the equations of motion describing the dynamics of the infinitesimal body in a neighborhood of the smaller body. Through the use of Hill’s variables and a limiting procedure, this approximation amounts to sending the two larger bodies to infinity. Finally, for the Hill approximation, we find the equilibrium points for the motion of the infinitesimal body and determine their stability. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter, and the Jupiter’s Trojan asteroid Hektor, which are assumed to move in a triangular central configuration. Then, we consider the dynamics of Hektor’s moonlet Skamandrios.