Regular and irregular geodesics on spherical harmonic surfaces

The behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex. In this paper we use the Hamiltonian formulation of the geodesic equations to analyze their integrability properties. In particular, we examine the behavior of geodesics on surfaces defined by the spherical harmonics. Using the Morales-Ramis theorem and Kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable, and we use Poincar\'{e} sections to demonstrate the breakdown of regular motion.Comment: Accepted Physica D : Nonlinear Phenomena; 25 pages, 3 figure

Evolution of the L1 halo family in the radial solar sail CRTBP

We present a detailed investigation of the dramatic changes that occur in the $\mathcal{L}_1$ halo family when radiation pressure is introduced into the Sun-Earth circular restricted three-body problem (CRTBP). This photo-gravitational CRTBP can be used to model the motion of a solar sail orientated perpendicular to the Sun-line. The problem is then parameterized by the sail lightness number, the ratio of solar radiation pressure acceleration to solar gravitational acceleration. Using boundary-value problem numerical continuation methods and the AUTO software package (Doedel et al. 1991) the families can be fully mapped out as the parameter $\beta$ is increased. Interestingly, the emergence of a branch point in the retrograde satellite family around the Earth at $\beta\approx0.0387$ acts to split the halo family into two new families. As radiation pressure is further increased one of these new families subsequently merges with another non-planar family at $\beta\approx0.289$, resulting in a third new family. The linear stability of the families changes rapidly at low values of $\beta$, with several small regions of neutral stability appearing and disappearing. By using existing methods within AUTO to continue branch points and period-doubling bifurcations, and deriving a new boundary-value problem formulation to continue the folds and Krein collisions, we track bifurcations and changes in the linear stability of the families in the parameter $\beta$ and provide a comprehensive overview of the halo family in the presence of radiation pressure. The results demonstrate that even at small values of $\beta$ there is significant difference to the classical CRTBP, providing opportunity for novel solar sail trajectories. Further, we also find that the branch points between families in the solar sail CRTBP provide a simple means of generating certain families in the classical case.Comment: 31 pages, 17 figures, accepted by Celestial Mechanics and Dynamical Astronom

New periodic orbits in the solar sail restricted three body problem

In this paper we consider periodic orbits of a solar sail in the Earth-Sun restricted three-body problem. In particular, we consider orbits which are high above the ecliptic plane, in contrast to the classical Halo orbits about the collinear equilibria. We begin with the Circular Restricted Three-Body Problem (CRTBP) where periodic orbits about equilibria are naturally present at linear order. Using the method of Lindstedt-Poincaré, we construct nth order approximations to periodic solutions of the nonlinear equations of motion. In the second part of the paper we generalize to the Elliptic Restricted Three Body Problem (ERTBP). A numerical continuation, with the eccentricity, e, as the varying parameter, is used to find periodic orbits above the ecliptic, starting from a known orbit at e = 0 and continuing to the required eccentricity of e = 0:0167. The stability of these periodic orbits is investigated

Cauchy horizon stability in self-similar collapse: scalar radiation

The stability of the Cauchy horizon in spherically symmetric self-similar collapse is studied by determining the flux of scalar radiation impinging on the horizon. This flux is found to be finite.Comment: 10 pages. To appear in Phys Rev

Integrable geodesic flows on tubular sub-manifolds

<p>In this paper we construct a new class of surfaces whose geodesic flow is integrable (in the sense of Liouville). We do so by generalizing the notion of tubes about curves to 3-dimensional manifolds, and using Jacobi fields we derive conditions under which the metric of the generalized tubular sub-manifold admits an ignorable coordinate. Some examples are given, demonstrating that these special surfaces can be quite elaborate and varied.</p

Control of solar sail periodic orbits in the elliptic three-body problem

A solar sail essentially consists of a large mirror that uses the momentum change due to photons reflecting off the sail for its impulse. Solar sails are therefore unique spacecraft, as they do not require fuel for propulsion [1]. In this Note we consider using the solar sail to continuously maintain a periodic orbit above the ecliptic plane using variations in the sail's orientation. Positioning a spacecraft continuously above the ecliptic would allow continuous observation and communication with the poles

Invariant manifolds and orbit control in the solar sail three-body problem

In this paper we consider issues regarding the control and orbit transfer of solar sails in the circular restricted Earth-Sun system. Fixed points for solar sails in this system have the linear dynamical properties of saddles crossed with centers; thus the fixed points are dynamically unstable and control is required. A natural mechanism of control presents itself: variations in the sail's orientation. We describe an optimal controller to control the sail onto fixed points and periodic orbits about fixed points. We find this controller to be very robust, and define sets of initial data using spherical coordinates to get a sense of the domain of controllability; we also perform a series of tests for control onto periodic orbits. We then present some mission strategies involving transfer form the Earth to fixed points and onto periodic orbits, and controlled heteroclinic transfers between fixed points on opposite sides of the Earth. Finally we present some novel methods to finding periodic orbits in circumstances where traditional methods break down, based on considerations of the Center Manifold theorem

Stability of naked singularities in self-similar collapse

Certain classes of solutions to Einstein’s field equations admit singularities from which light can escape, known as ‘naked5 singularities Such solutions contradict the Cosmic Censorship hypothesis, however they tend to occur in spacetimes with a high degree of symmetry. It is thought that naked singularities are artifacts of these symmetries, and would not survive when the symmetry is broken. In particular, a rich source of naked singularities is the class of self-similar spherically symmetric spacetimes. It is the purpose of this thesis to test the stability of these solutions and examine if the naked singularity persists. We first consider the propagation of a scalar field on these background spacetimes and then study gauge-invariant perturbations of the metric and matter tensors. We exploit the spherical symmetry of the background to decompose the angular part of the perturbation in terms of spherical harmonics. Then we perform a Mellin transform of the field to reduce the problem to a set of coupled ordinary differential equations, and seek solutions for the individual modes. The asymptotic behaviour of these modes near singular points of the ODE’s is used to calculate a set of gauge invariant scalars, and we examine the finiteness of these scalars on the naked singularity’s horizons. The background spacetimes we examine are the self-similar null dust (Vaidya) solution, the self-similar timelike dust (Lemaitre-Tolman-Bondi) solution, and finally a general self-similar spacetime whose matter content is unspecified save for satisfying the dominant energy condition. In each case examined we find the Cauchy horizon, signalling the presence of a naked singularity, is stable to linear order, a surprising result that suggests naked singularities may arise in physical models of gravitational collapse. The second future similarity horizon which follows the Cauchy horizon is unstable, which suggests that the naked singularity is only visible for a finite time