The power-law galaxies

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Abstract
Axisymmetric galaxy models with simple distribution functions (DFs) that are sums of powers of the binding energy E and angular momentum component |$L_z$| are presented. They are useful for representing elliptical galaxies, as well as the bulges and haloes of spiral galaxies. The equipotentials are stratified on similar concentric spheroids. The rotation law can be asymptotically rising, flat or declining. The intrinsic and projected velocity dispersions are elementary. In the limit of vanishing core radius, the galaxies are scale-free, and the surface brightness is cusped. Composite models, in which a population of stars with a spheroidal density distribution is embedded within the halo, are easy to construct. For the oblate models, the generic orbits are short-axis tubes, while for the prolate models, they are inner and outer long-axis tubes. The orbital structure of the cored models is simple, with almost all stars moving on tori. When the core radius vanishes, the tube orbits passing close to the centre are replaced by higher order resonant families. This effect is particularly pronounced for prolate models, where the infinitesimally thin inner long-axis tubes undergo a sequence of period-multiplying bifurcations. This causes their complete replacement by resonant families at low |$L_z$|⁠. The simplicity of the DFs enables the line profiles (LPs) and the higher Gauss-Hermite moments to be found. Composite models with constant fractions of counter-streaming stars are useful for analysing the LPs of stars in the outer parts of galaxies. When the streaming is greatest, the LPs are skewed towards velocities greater than the mean, in contradiction with the observations. Models with 1/4 of the stars counter-streaming with respect to the rest of the galaxy have LPs with more realistic asymmetries. The LPs of the self-consistent coreless haloes are scale-free and invariant along the rays of constant position angle on the plane of the sky. The effects of flattening of the mass model on the skewness and bimodality of the LP are analysed. The error caused by approximating the LP of an axisymmetric galaxy by a sperical model can be serious even for ellipticities as mild as E2. In the flattened limit, the coreless models generate a useful family of scale-free discs with rising, flat or declining circular velocity curves. There are uninterrupted sequences of power-law DFs that connect the cold, centrifugally supported discs with the hot, pressure-supported discs. All the intrinsic properties of the discs are simple – in particular, the Toomre and Ostriker–Peebles criteria for local and global stability can be exactly calculated. The spherical scale-free limit also provides a set of star cluster models with sequences of simple power-law DFs.

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