Observable properties of the power-law galaxies

Abstract

The power-law models are the largest known set of axisymmetric galaxies with simple even distribution functions (DFs) depending on the two integrals of binding energy E and angular momentum Lz. This paper develops a natural choice for the odd part of the DF, which yields models with streaming laws akin to the oblate isotropic rotators. All the projected kinematic properties - the means and dispersions in the radial and proper motions - are found to be elementary. The velocity profiles and proper motion distributions can be evaluated easily. This speeds the comparison of model predictions with observables. It allows for convenient analysis of the effect of changes in the model parameters. Three applications are presented. First, the scale-free model with ρ ∼ r-1.8 provides a good representation of the density profile of the inner 500 pc of the Galactic Bulge, fitting the line-of-sight dispersions and proper motions in Baade's Window. Our simple DF yields predictions for the distributions of radial and proper motions that compare well with the available velocity histograms. The inclusion of a 2 x 106 MΘ central mass allows the variation of the dispersions along major and minor axes in the inner 10 pc to be recovered as well. Secondly, the velocity profiles (VPs) of the models are not Gaussian and are skewed towards velocities less than the mean. This is the same behaviour as observed in rotating galaxies. We present the range of Gauss-Hermite moments for power-law models compatible with the same observables - that is, the same apparent ellipticity ϵ and the same ratio of peak streaming to central velocity dispersion (νmax)/σ0. This shows real variety in the fourth Gauss-Hermite moment h4, much less so in the third moment h3. It suggests h4 is a useful discriminant between the two-integral power-law models, whereas h3 can tell us when more general three-integral DFs are needed. Thirdly, Illingworth and Binney have popularized the diagram of (νmax)/σ0 against ellipticity ϵ. The line occupied by the oblate isotropic rotators is of practical importance, as it divides galaxies flattened by rotation from those distorted by anisotropic velocity dispersion. Its position is conventionally estimated with the classical spheroids. We show that the isotropic rotating power-law galaxies fall below the classical spheroids - and significantly so at large ellipticities. Many of the conclusions derived from the diagram are none the less probably correct, as kinematic quantities derived by fitting Gaussians to VPs themselves yield overestimates for (νmax)/σ0