2.8. Deprojected quantities and dynamical terms
Ciotti (1991) provides an exact, numerical expression for the deprojected light-profile of the R1/n model, that is, the luminosity density profile. He additionally provides numerical expressions for the gravitational potential and also the spatial and line-of-sight velocity dispersions. These must however be solved by integration, which means they require considerably more computer time than analytical expressions. Ciotti does however provide analytical expressions for the behavior of the above expressions at both small and large radii. Luminosity-weighted aperture velocity dispersions have been used in Ciotti, Lanzoni, & Renzini (1996), and also in Graham & Colless (1997) where the radial profiles are shown for different values of the Sérsic index n. Ciotti (1991) additionally provides expressions for the distribution function and the normalised differential energy distribution.
An exact, analytical expression for the density profile was finally discovered a couple of years ago and is given in Mazure & Capelato (2002). It involves the use of somewhat complicated Meijer G functions. For those interested in a more simple, analytical approximation, an accurate expression is given in Prugniel & Simien (1997), which is developed slightly in Lima Nieto, Gerbal, & Márquez (1999) and Trujillo et al. (2002).
Mazure & Capelato (2002) also provide exact analytical expressions for the mass, gravitational potential, total energy, and the central velocity dispersion. For modellers interested in fast-to-compute, analytical approximations for not only the density profile but also the potential and force, such expressions, which additionally include optional power-law cores, can be found elsewhere (B. Terzic & A.W. Graham, in preparation).