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

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

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

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

Gauge invariant perturbations of self-similar Lema\^itre-Tolman-Bondi spacetime: even parity modes with l≥2l\geq 2

In this paper we consider gauge invariant linear perturbations of the metric and matter tensors describing the self-similar Lema\^itre-Tolman-Bondi (timelike dust) spacetime containing a naked singularity. We decompose the angular part of the perturbation in terms of spherical harmonics and perform a Mellin transform to reduce the perturbation equations to a set of ordinary differential equations with singular points. We fix initial data so the perturbation is finite on the axis and the past null cone of the singularity, and follow the perturbation modes up to the Cauchy horizon. There we argue that certain scalars formed from the modes of the perturbation remain finite, indicating linear stability of the Cauchy horizon.Comment: 16 pages, 4 figure

P5_6 A Race in Space

By comparing solar and laser radiation sources, each driving a nano-satellite of mass 1 gram, we find that the solar sail is more advantageous up to 9.5 AU, and the laser sail is better beyond this distance. We also find that the laser sail has a constant acceleration throughout, but the solar sail acceleration decreases at a velocity of 150,000 ms-1

Solar sail dynamics in the three-body problem: homoclinic paths of points and orbits

In this paper we consider the orbital previous termdynamicsnext term of a previous termsolar sailnext term in the Earth-Sun circular restricted three-body problem. The equations of motion of the previous termsailnext term are given by a set of non-linear autonomous ordinary differential equations, which are non-conservative due to the non-central nature of the force on the previous termsail.next term We consider first the equilibria and linearisation of the system, then examine the non-linear system paying particular attention to its periodic solutions and invariant manifolds. Interestingly, we find there are equilibria admitting homoclinic paths where the stable and unstable invariant manifolds are identical. What is more, we find that periodic orbits about these equilibria also admit homoclinic paths; in fact the entire unstable invariant manifold winds off the periodic orbit, only to wind back onto it in the future. This unexpected result shows that periodic orbits may inherit the homoclinic nature of the point about which they are described

Spatially resolved spectroscopy of the globular cluster RZ 2109 and the nature of its black hole

We present optical HST/STIS spectroscopy of RZ 2109, a globular cluster in the elliptical galaxy NGC 4472. This globular cluster is notable for hosting an ultraluminous X-ray source as well as associated strong and broad [OIII] 4959, 5007 emission. We show that the HST/STIS spectroscopy spatially resolves the [OIII] emission in RZ 2109. While we are unable to make a precise determination of the morphology of the emission line nebula, the best fitting models all require that the [OIII] 5007 emission has a half light radius in the range 3-7 pc. The extended nature of the [OIII] 5007 emission is inconsistent with published models that invoke an intermediate mass black hole origin. It is also inconsistent with the ionization of ejecta from a nova in the cluster. The spatial scale of the nebula could be produced via the photoionization of a strong wind driven from a stellar mass black hole accreting at roughly its Eddington rate.Comment: 7 pages, 4 figures - accepted for publication in Ap