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Astron. Astrophys. 335, 449-462 (1998)

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Appendix A: model construction

We assume a galaxy to consist of several physically homogeneous components with masses M, mass-to-light ratios f and fixed colour indices. In order to fit model parameters simultaneously by using both photometrical data and kinematical data more practical is to start with sufficiently flexible spatial density distribution [FORMULA]. Otherwise we need to calculate the observable dynamical functions (rotational velocities and velocity dispersions) of the model via the surface density distribution, which is more complicated. The density distribution of each component we approximated as an inhomogeneous ellipsoid of rotation with a constant axial ratio [FORMULA]. All components, except the corona, form an optically visible part of the galaxy, and their volume densities are described by a modified exponential law

[EQUATION]

allowing a description of both the light profiles of disks and of spheroidal components by simply varying the structure parameter N. Similar density distribution law for surface densities was introduced by Sersic (1968) and independently for spatial densities by Einasto (1969b). This is a simple law allowing sufficiently precise numerical integration and has a minimum number of free parameters (two scale parameters and a density gradient parameter). The remaining component - the invisible massive corona - is represented by a modified isothermal law

[EQUATION]

In these formulae [FORMULA] is the central density, a is the distance along the major axis, [FORMULA] - the core radius, [FORMULA] - the harmonic mean radius, [FORMULA] - the outer cutoff radius for the corona, and h and k are normalizing parameters, depending on the parameter N (Paper IV, Appendix B).

For the disk and the flat components we use the density distribution in the following form

[EQUATION]

where subindices "-" and "+" denote density distributions (1) with negative and positive masses, respectively. In this way we have density distributions with central density depression. If we demand that the density is zero at [FORMULA] and positive elsewhere, the following relations must hold between the parameters of components [FORMULA] and [FORMULA] [FORMULA] [FORMULA] [FORMULA] where [FORMULA] is a parameter which determines the relative size of the hole in the centre of the disk. The structural parameters [FORMULA] and [FORMULA] were assumed to be equal.

The density distributions for visible components were projected along the line of sight and their superposition gives us the surface brightness distribution of the model

[EQUATION]

where A is the major semiaxis of the equidensity ellipse of the projected light distribution, n is the number of components and [FORMULA] are their apparent axial ratios.

The masses of the components can be determined from the rotation law

[EQUATION]

[EQUATION]

where G is the gravitational constant and R is the distance in the equatorial plane of the galaxy.

For "hot" components the masses were determined from the virial theorem for multicomponent systems connecting the mean line-of-sight velocity dispersion of the k -th component with the masses of all components [FORMULA] (q.v. Paper IV, Appendix A)

[EQUATION]

In this formula m is the total number of components and [FORMULA] are dimensionless coefficients depending on the mass distribution laws (parameter N) of the components k and l. It is evident that all subsystems contribute to the mean velocity dispersion of a particular component, but often the influence of the component [FORMULA] is more or less dominating.

For a point mass within a spherical stellar nucleus the virial theorem for multicomponent systems has a form

[EQUATION]

where parameter [FORMULA] depends on the structure parameter N of the nucleus and M is mass.

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© European Southern Observatory (ESO) 1998

Online publication: June 18, 1998
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