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Astron. Astrophys. 323, 349-356 (1997)

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2. Mass models

B91 used the family of separable triaxial models with constant density cores described by ZF to reproduce the ionized gas velocity field of NGC 5077 with a constant [FORMULA] model. Hubble Space Telescope observations indicate that elliptical galaxies generally have cusped central surface brightness profiles (Crane et al. 1993; Jaffe et al. 1994). Accordingly, we adopt the same approach as in B91, but we use a family of non-rotating triaxial mass models with a central density cusp. They are triaxial generalization of the so-called [FORMULA] -models (e.g., Carollo 1993; Dehnen 1993; Tremaine et al. 1994) introduced and described in detail by de Zeeuw & Carollo (1996, hereafter ZC). These models allow for variations of ellipticity and position angle of the major axis of the isophotes with radius.

The mass models introduced by ZC are based on the spherical [FORMULA] -models with density profile:


This spherical density, and its associated gravitational potential, are functions of three parameters: [FORMULA]. The first of these, [FORMULA] ([FORMULA]), measures the "cuspiness" of the central density profile which diverges as [FORMULA]. For high values of [FORMULA] we have a steep central profile while the case [FORMULA] corresponds to the model with finite central density and a central density gradient intermediate between the King and Hubble profiles, similar to the separable models described by ZF, and applied by B91. [FORMULA] is the total model-mass of the galaxy, and [FORMULA] is a scale-length, which is related to [FORMULA] (Dehnen 1993).

The models are made triaxial by adding two low order spherical harmonics terms to the potential of the spherical [FORMULA] models, with appropriately chosen radial functions, given in Eq. (2.5) of ZC. This introduces four free parameters in addition to [FORMULA], [FORMULA] and [FORMULA], which can be chosen as the intrinsic axis ratios [FORMULA], [FORMULA], [FORMULA], [FORMULA] of the triaxial surfaces of constant density at small and large radii, respectively.

A stable configuration of cold gas in a nonrotating triaxial galaxy can only be a ring or a disk in one of two principal planes, either perpendicular to the long axis of the figure or perpendicular to the short axis (Tohline, Simonson & Caldwell, 1982; Merritt & de Zeeuw, 1983). We use the convention that the Z axis is always perpendicular to the gaseous disk, and that the X axis is longer than the Y axis. From these definitions it follows that we have two distinct cases: when [FORMULA] the gas is perpendicular to the short axis, and when [FORMULA] the gas is perpendicular to the long axis. The seven parameters [FORMULA] allow us to determine the 3-dimensional density distribution and the orbital velocities on either of the two relevant principal planes. We use the properties of the velocity field as derived by ZC by means of the first order epicyclic approximation. The projected surface density and projected velocity field, which are the observable quantities, are derived once we know the three viewing angles [FORMULA] and [FORMULA], defined as the standard spherical coordinates of the line-of-sight in the galaxy coordinate system, and [FORMULA], the position angle of the projected Z axis, measured eastwards of North. The gas moves on closed orbits whose properties are fully determined by the model. The orbits become circular at large radii ([FORMULA]) whereas the ellipticity [FORMULA] reaches a maximum value for small radii ([FORMULA]) which is the smaller the steeper the central density gradient (higher [FORMULA]). For fixed central flattening ([FORMULA]) [FORMULA] is a function of the intrinsic axis ratio. [FORMULA] is the upper limit beyond which the epicyclic approximation is no longer valid and the description of the gas on closed orbits breaks down (see fig. 1 of ZC).

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998