*Astron. Astrophys. 335, 449-462 (1998)*
## 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 . 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 . All
components, except the corona, form an optically visible part of the
galaxy, and their volume densities are described by a modified
exponential law
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
In these formulae is the central density,
*a* is the distance along the major axis,
- the core radius, - the harmonic mean radius,
- 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
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 and positive
elsewhere, the following relations must hold between the parameters of
components and
where is a parameter
which determines the relative size of the hole in the centre of the
disk. The structural parameters and
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
where *A* is the major semiaxis of the equidensity ellipse of
the projected light distribution, n is the number of components and
are their apparent axial ratios.
The masses of the components can be determined from the rotation
law
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 (q.v. Paper IV, Appendix A)
In this formula *m* is the total number of components and
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 is more or less
dominating.
For a point mass within a spherical stellar nucleus the virial
theorem for multicomponent systems has a form
where parameter depends on the structure
parameter *N* of the nucleus and *M* is mass.
© European Southern Observatory (ESO) 1998
Online publication: June 18, 1998
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