The phase space structure of r¼ galaxies: Are these galaxies ‘isothermal’ after all?

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Abstract
The distribution function f(E) that self-consistently generates a galaxy whose surface density profile obeys de Vaucouleurs' r¼ law is calculated. This distribution function is such that the number N(E)dE of stars having binding energies near E is well approximated by the Boltzmann formula N(E) = exp(βE) over the range of values of E for which there is secure observational support for de Vaucouleurs' law. King models do not have N(E) curves of this form. The surface density and velocity dispersion properties are presented of a galaxy whose N(E) curve is exactly given by the Boltzmann formula at all possible binding energies. Like the r¼ model, this new model has only two fitting parameters – a radius and a surface density – yet it fits the photometry of the standard elliptical NGC 3379 even better than the r¼ model. In particular the model possesses a nucleus-like feature at its centre in which the velocity dispersion is essentially the same as in the body of the galaxy. It is argued that a fundamental understanding of the structure of spheroidal components is more likely to be found in terms of some special form of N(E) than in terms of the distribution function f(E). An interpretation is offered of the Boltzmann form of N(E) in terms of the form of the density fluctuation in the early Universe which may have given rise to the modern galaxy.

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