Axisymmetric Self-Similar Equilibria of Self-Gravitating Isothermal Systems

All axisymmetric self-similar equilibria of self-gravitating, rotating, isothermal systems are identified by solving the nonlinear Poisson equation analytically. There are two families of equilibria: (1) Cylindrically symmetric solutions in which the density varies with cylindrical radius as R^(-alpha), with 0 <= alpha <= 2. (2) Axially symmetric solutions in which the density varies as f(theta)/r^2, where `r' is the spherical radius and `theta' is the co-latitude. The singular isothermal sphere is a special case of the latter class with f(theta)=constant. The axially symmetric equilibrium configurations form a two-parameter family of solutions and include equilibria which are surprisingly asymmetric with respect to the equatorial plane. The asymmetric equilibria are, however, not force-free at the singular points r=0, infinity, and their relevance to real systems is unclear. For each hydrodynamic equilibrium, we determine the phase-space distribution of the collisionless analog.Comment: 13 pages, 7 figures, uses emulateapj.sty. Submitted to Ap

The Milky Way Galaxy as a Strong Gravitational Lens

We study the gravitational lensing effects of spiral galaxies by taking a model of the Milky Way and computing its lensing properties. The model is composed of a spherical Hernquist bulge, a Miyamoto-Nagai disc and an isothermal halo. As a strong lens, a spiral galaxy like the Milky Way can give rise to four different imaging geometries. They are (i) three images on one side of the galaxy centre (`disc triplets'), (ii) three images with one close to the centre (`core triplets'), (iii) five images and (iv) seven images. Neglecting magnification bias, we show that the core triplets, disc triplets and fivefold imaging are roughly equally likely. Even though our models contain edge-on discs, their image multiplicities are not dominated by disc triplets. The halo has a small effect on the caustic structure, the time delays and brightnesses of the images. The Milky Way model has a maximum disc (i.e., the halo is not dynamically important in the inner parts). Strong lensing by nearly edge-on disc galaxies breaks the degeneracy between the relative contribution of the disc and halo to the overall rotation curve. If a spiral galaxy has a sub-maximum disc, then the astroid caustic shrinks dramatically in size, whilst the radial caustic shrinks more modestly. This causes changes in the relative likelihood of the image geometries, specifically (i) core triplets are now 9/2 times more likely than disc triplets, (ii) the cross section for threefold imaging is reduced by a factor of 2/3, whilst (iii) the cross section for fivefold imaging is reduced by 1/2. Although multiple imaging is less likely (the cross sections are smaller), the average total magnification is greater.Comment: MNRAS, in pres

Planetary Stability Zones in Hierarchical Triple Star Systems

A symplectic integrator algorithm suitable for hierarchical triple systems is formulated and tested. The positions of the stars are followed in hierarchical Jacobi coordinates, whilst the planets are referenced purely to their primary. The algorithm is fast, accurate and easily generalised to incorporate collisions. There are five distinct cases -- circumtriple orbits, circumbinary orbits and circumstellar orbits around each of the stars in the hierarchical triple -- which require a different formulation of the symplectic integration algorithm. As an application, a survey of the stability zones for planets in hierarchical triples is presented, with the case of a single planet orbiting the inner binary considered in detail. Fits to the inner and outer edges of the stability zone are computed. Considering the hierarchical triple as two decoupled binary systems, the earlier work of Holman & Wiegert on binaries is shown to be applicable to triples, except in the cases of high eccentricities and close or massive stars. Application to triple stars with good data in the multiple star catalogue suggests that more than 50 per cent are unable to support circumbinary planets, as the stable zone is almost non-existent.Comment: 16 pages, MNRAS, in pres

Microlensing Halo Models with Abundant Brown Dwarfs

All previous attempts to understand the microlensing results towards the Large Magellanic Cloud (LMC) have assumed homogeneous present day mass functions (PDMFs) for the lensing populations. Here, we present an investigation into the microlensing characteristics of haloes with spatially varying PDMFs and anisotropic velocity dispersion tensors. One attractive possibility -- suggested by baryonic dark cluster formation in pregalactic and protogalactic cooling flows -- is that the inner halo is dominated by stellar mass objects, whereas low mass brown dwarfs become more prevalent on moving outwards. The contribution to the microlensing rate must be dominated by dark remnants (of about 0.5 solar masses) to recover the observed timescales of the microlensing experiments. But, even though stellar remnants control the rate, they do not dominate the mass of the baryonic halo, and so the well-known enrichment and mass budget problems are much less severe. Using a simple ansatz for the spatial variation of the PDMF, models are constructed in which the contribution of brown dwarfs to the mass of the baryonic halo is 55 % and to the total halo is 30 %. An unusual property of the models is that they predict that the average timescale of events towards M31 is shorter than the average timescale towards the LMC. This is because the longer line of sight towards M31 probes more of the far halo where brown dwarfs are the most common constituent.Comment: 17 pages, 1 figure, in press at The Astrophysical Journal (Letters

Condensation Transition in Polydisperse Hard Rods

We study a mass transport model, where spherical particles diffusing on a ring can stochastically exchange volume $v$, with the constraint of a fixed total volume $V=\sum_{i=1}^N v_i$, $N$ being the total number of particles. The particles, referred to as $p$-spheres, have a linear size that behaves as $v_i^{1/p}$ and our model thus represents a gas of polydisperse hard rods with variable diameters $v_i^{1/p}$. We show that our model admits a factorized steady state distribution which provides the size distribution that minimizes the free energy of a polydisperse hard rod system, under the constraints of fixed $N$ and $V$. Complementary approaches (explicit construction of the steady state distribution on the one hand ; density functional theory on the other hand) completely and consistently specify the behaviour of the system. A real space condensation transition is shown to take place for $p>1$: beyond a critical density a macroscopic aggregate is formed and coexists with a critical fluid phase. Our work establishes the bridge between stochastic mass transport approaches and the optimal polydispersity of hard sphere fluids studied in previous articles

Simulations of the Population of Centaurs II: Individual Objects

Detailed orbit integrations of clones of five Centaurs -- namely, 1996 AR20, 2060 Chiron, 1995 SN55, 2000 FZ53 and 2002 FY36 -- for durations of 3 Myr are presented. One of our Centaur sample starts with perihelion initially under the control of Jupiter (1996 AR20), two start under the control of Saturn (Chiron and 1995 SN55) and one each starts under the control of Uranus (2000 FZ53) and Neptune (2002 FY36) respectively. A variety of interesting pathways are illustrated with detailed examples including: capture into the Jovian Trojans, repeated bursts of short-period comet behaviour, capture into mean-motion resonances with the giant planets and into Kozai resonances, as well as traversals of the entire Solar system. For each of the Centaurs, we provide statistics on the numbers (i) ejected, (ii) showing short-period comet behaviour and (iii) becoming Earth and Mars crossing. For example, Chiron has over 60 % of its clones becoming short-period objects, whilst 1995 SN55 has over 35 %. Clones of these two Centaurs typically make numerous close approaches to Jupiter. At the other extreme, 2000 FZ53 has roughly 2 % of its clones becoming short-period objects. In our simulations, typically 20 % of the clones which become short-period comets subsequently evolve into Earth-crossers.Comment: 10 pages, in press at MNRA