The tidal stripping of satellites

We present an improved analytic calculation for the tidal radius of satellites and test our results against N-body simulations. The tidal radius in general depends upon four factors: the potential of the host galaxy, the potential of the satellite, the orbit of the satellite and {\it the orbit of the star within the satellite}. We demonstrate that this last point is critical and suggest using {\it three tidal radii} to cover the range of orbits of stars within the satellite. In this way we show explicitly that prograde star orbits will be more easily stripped than radial orbits; while radial orbits are more easily stripped than retrograde ones. This result has previously been established by several authors numerically, but can now be understood analytically. For point mass, power-law (which includes the isothermal sphere), and a restricted class of split power law potentials our solution is fully analytic. For more general potentials, we provide an equation which may be rapidly solved numerically. Over short times (\simlt 1-2 Gyrs $\sim 1$ satellite orbit), we find excellent agreement between our analytic and numerical models. Over longer times, star orbits within the satellite are transformed by the tidal field of the host galaxy. In a Hubble time, this causes a convergence of the three limiting tidal radii towards the prograde stripping radius. Beyond the prograde stripping radius, the velocity dispersion will be tangentially anisotropic.Comment: 10 pages, 5 figures. Final version accepted for publication in MNRAS. Some new fully analytic tidal radii have been added for power law density profiles (including the isothermal sphere) and some split power law

The mass of dwarf spheroidal galaxies and the missing satellite problem

We present the results from a suite of N-body simulations of the tidal stripping of two-component dwarf galaxies comprising some stars and dark matter. We show that recent kinematic data from the local group dwarf spheroidal (dSph) galaxies suggests that dSph galaxies must be sufficiently massive ($10^9 - 10^{10}$M$_\odot$) that tidal stripping is of little importance for the stars. We discuss the implications of these massive dSph galaxies for cosmology and galaxy formation.Comment: 4 pages, 1 figure, to appear in the proceedings of the IAUC198 "Near-Field Cosmology with Dwarf Elliptical Galaxies", H. Jerjen & B. Binggeli (eds.). Comments welcom

Alcohol: taking a population perspective

Alcohol consumption is a global phenomenon, as is the resultant health, social and economic harm. The nature of these harms varies with different drinking patterns and with the societal and political responses to the burden of harm; nevertheless, alcohol-related chronic diseases have a major effect on health. Strong evidence exists for the effectiveness of different strategies to minimize this damage and those policies that target price, availability and marketing of alcohol come out best, whereas those using education and information are much less effective. However, these policies can be portrayed as anti-libertarian and so viewing them in the context of alcohol-related harm to those other than the drinker, such as the most vulnerable in society, is important. When this strategy is successful, as in Scotland, it has been possible to pass strong and effective legislation, such as for a minimum unit price for alcohol

Further Evidence for a Merger Origin for the Thick Disk: Galactic Stars Along Lines-of-sight to Dwarf Spheroidal Galaxies

The history of the Milky Way Galaxy is written in the properties of its stellar populations. Here we analyse stars observed as part of surveys of local dwarf spheroidal galaxies, but which from their kinematics are highly probable to be non-members. The selection function -- designed to target metal-poor giants in the dwarf galaxies, at distances of ~100kpc -- includes F-M dwarfs in the Milky Way, at distances of up to several kpc. Thestars whose motions are analysed here lie in the cardinal directions of Galactic longitude l ~ 270 and l ~ 90, where the radial velocity is sensitive to the orbital rotational velocity. We demonstrate that the faint F/G stars contain a significant population with V_phi ~ 100km/s, similar to that found by a targeted, but limited in areal coverage, survey of thick-disk/halo stars by Gilmore, Wyse & Norris (2002). This value of mean orbital rotation does not match either the canonical thick disk or the stellar halo. We argue that this population, detected at both l ~ 270 and l ~ 90, has the expected properties of `satellite debris' in the thick-disk/halo interface, which we interpret as remnants of the merger that heated a pre-existing thin disk to form the thick disk.Comment: Accepted, Astrophysical Journal Letter

Calculation of the unitary part of the Bures measure for N-level quantum systems

We use the canonical coset parameterization and provide a formula with the unitary part of the Bures measure for non-degenerate systems in terms of the product of even Euclidean balls. This formula is shown to be consistent with the sampling of random states through the generation of random unitary matrices

Kinematically Cold Populations at Large Radii in the Draco and Ursa Minor Dwarf Spheroidals

We present projected velocity dispersion profiles for the Draco and Ursa Minor (UMi) dwarf spheroidal galaxies based on 207 and 162 discrete stellar velocities, respectively. Both profiles show a sharp decline in the velocity dispersion outside ~30 arcmin (Draco) and ~40 arcmin (UMi). New, deep photometry of Draco reveals a break in the light profile at ~25 arcmin. These data imply the existence of a kinematically cold population in the outer parts of both galaxies. Possible explanations of both the photometric and kinematic data in terms of both equilibrium and non-equilibrium models are discussed in detail. We conclude that these data challenge the picture of dSphs as simple, isolated stellar systems.Comment: 5 pages, accepted for publication in ApJ Letter

The Tamm-Dancoff Approximation as the boson limit of the Richardson-Gaudin equations for pairing

A connection is made between the exact eigen states of the BCS Hamiltonian and the predictions made by the Tamm-Dancoff Approximation. This connection is made by means of a parametrised algebra, which gives the exact quasi-spin algebra in one limit of the parameter and the Heisenberg-Weyl algebra in the other. Using this algebra to construct the Bethe Ansatz solution of the BCS Hamiltonian, we obtain parametrised Richardson-Gaudin equations, leading to the secular equation of the Tamm-Dancoff Approximation in the bosonic limit. An example is discussed in depth.Comment: Submitted to the proceedings of the Group28 conference (Newcastle-upon-Tyne, UK). Journal of Physics: Conference Serie

A Dynamical Fossil in the Ursa Minor Dwarf Spheroidal Galaxy

The nearby Ursa Minor dwarf spheroidal (UMi dSph) is one of the most dark matter dominated galaxies known, with a central mass to light ratio roughly equal to 70. Somewhat anomalously, it appears to contain morphological substructure in the form of a second peak in the stellar number density. It is often argued that this substructure must be transient because it could not survive for the > 10 Gyr age of the system, given the crossing time implied by UMi's 8.8 km/s internal velocity dispersion. In this paper, however, we present evidence that the substructure has a cold kinematical signature, and argue that UMi's clumpiness could indeed be a primordial artefact. Using numerical simulations, we demonstrate that substructure is incompatible with the cusped dark matter haloes predicted by the prevailing Cold Dark Matter (CDM) paradigm, but is consistent with an unbound stellar cluster sloshing back and forth within the nearly harmonic potential of a cored dark matter halo. Thus CDM appears to disagree with observation at the least massive, most dark matter dominated end of the galaxy mass spectrum.Comment: Astrophysical Journal (Letters), in pres

First Clear Signature of an Extended Dark Matter Halo in the Draco Dwarf Spheroidal

We present the first clear evidence for an extended dark matter halo in the Draco dwarf spheroidal galaxy based on a sample of new radial velocities for 159 giant stars out to large projected radii. Using a two parameter family of halo models spanning a range of density profiles and velocity anisotropies, we are able to rule out (at about the 2.5 sigma confidence level) haloes in which mass follows light. The data strongly favor models in which the dark matter is significantly more extended than the visible dwarf. However, haloes with harmonic cores larger than the light distribution are also excluded. When combined with existing measurements of the proper motion of Draco, our data strongly suggest that Draco has not been tidally truncated within ~1 kpc. We also show that the rising velocity dispersion at large radii represents a serious problem for modified gravity (MOND).Comment: to be published in ApJL; 5 pages, 4 figure