Cores and Cusps in the Dwarf Spheroidals

We consider the problem of determining the structure of the dark halo of nearby dwarf spheroidal galaxies (dSphs) from the spherical Jeans equations. Whether the dark halos are cusped or cored at the centre is an important strategic problem in modern astronomy. The observational data comprise the line-of-sight velocity dispersion of a luminous tracer population. We show that when such data are analysed to find the dark matter density with the spherical Poisson and Jeans equations, then the generic solution is a dark halo density that is cusped like an isothermal. Although milder cusps (like the Navarro-Frenk-White 1/r cusp and even cores are possible, they are not generic. Such solutions exist only if the anisotropy parameter beta and the logarithmic slope of the stellar density gamma satisfy the constraint gamma = 2 x beta at the centre or if the radial velocity dispersion falls to zero at the centre. This surprisingly strong statement is really a consequence of the assumption of spherical symmetry, and the consequent coordinate singularity at the origin. So, for example, a dSph with an exponential light profile can exist in Navarro-Frenk- White halo and have a flat velocity dispersion, but anisotropy in general drives the dark halo solution to an isothermal cusp. The identified cusp or core is therefore a consequence of the assumptions (particularly of spherical symmetry and isotropy), and not the data.Comment: MNRAS, in pres

Modified Virial Formulae and the Theory of Mass Estimators

We show how to estimate the enclosed mass from the observed motions of an ensemble of test particles. Traditionally, this problem has been attacked through virial or projected mass estimators. Here, we examine and extend these systematically, and show how to construct an optimal estimator for any given assumption as to the potential. The estimators do not explicitly depend on any properties of the density of the test objects, which is desirable as in practice such information is dominated by selection effects. As particular examples, we also develop estimators tailored for the problem of estimating the mass of the Hernquist or NFW dark matter haloes from the projected positions and velocities of stars.Comment: 9 pages, MNRAS, in pres

A theorem on central velocity dispersions

It is shown that, if the tracer population is supported by a spherical dark halo with a core or a cusp diverging more slowly than that of a singular isothermal sphere, the logarithmic cusp slope 'g' of the tracers must be given exactly by g=2b where b is their velocity anisotropy parameter at the center unless the same tracers are dynamically cold at the center. If the halo cusp diverges faster than that of the singular isothermal sphere, the velocity dispersion of the tracers must diverge at the center too. In particular, if the logarithmic halo cusp slope is larger than two, the diverging velocity dispersion also traces the behavior of the potential. The implication of our theorem on projected quantities is also discussed. We argue that our theorem should be understood as a warning against interpreting results based on simplifying assumptions such as isotropy and spherical symmetry.Comment: submitted to Ap

The Tilt of the Halo Velocity Ellipsoid and the Shape of the Milky Way Halo

A sample of roughly 1,800 halo subdwarf stars with radial velocities and proper motions is assembled, using the repeated multi-band Sloan Digital Sky Survey photometric measurements in Stripe 82. Our sample of halo subdwarfs is extracted via a reduced proper motion diagram and distances are obtained using photometric parallaxes, thus giving full phase space information. The tilt of the velocity ellipsoid with respect to the spherical polar coordinate system is computed and found to be consistent with zero for two of the three tilt angles, and very small for the third. We prove that if the inner halo is in a steady-state and the triaxial velocity ellipsoid is everywhere aligned in spherical polar coordinates, then the potential must be spherically symmetric. The detectable, but very mild, misalignment with spherical polars is consistent with the perturbative effects of the Galactic disk on a spherical dark halo. Banana orbits are generated at the 1:1 resonance (in horizontal and vertical frequency) by the disk. They populate Galactic potentials at the typical radii of our subdwarf sample, along with the much more dominant short-axis tubes. However, on geometric grounds alone, the tilt cannot vanish for the banana orbits and this leads to a slight, but detectable, misalignment. We argue that the tilt of the stellar halo velocity ellipsoid therefore provides a hitherto largely neglected but important line of argument that the Milky Way's dark halo, which dominates the potential, must be nearly spherical.Comment: Submitted to Ap

The Chang-Refsdal Lens Revisited

This paper provides a complete theoretical treatment of the point-mass lens perturbed by constant external shear, often called the Chang-Refsdal lens. We show that simple invariants exist for the products of the (complex) positions of the four images, as well as moment sums of their signed magnifications. The image topographies and equations of the caustics and critical curves are also studied. We derive the fully analytic expressions for precaustics, which are the loci of non-critical points that map to the caustics under the lens mapping. They constitute boundaries of the region in the image domain that maps onto the interior of the caustics. The areas under the critical curves, caustics and precaustics are all evaluated, which enables us to calculate the mean magnification of the source within the caustics. Additionally, the exact analytic expression for the magnification distribution for the source in the triangular caustics is derived, as well as a useful approximate expression. Finally, we find that the Chang-Refsdal lens with the convergence greater than unity can exhibit third-order critical behaviour, if the reduced shear is exactly equal to \sqrt{3}/2, and that the number of images for N-point masses with non-zero constant shear cannot be greater than 5N-1.Comment: to appear in MNRAS (including 6 figures, 3 appendices; v2 - minor update with corrected typos etc.