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

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3. The modelling

3.1. Mass density profile

The method used by B91 to obtain a local [FORMULA] profile for NGC 5077 consisted of two distinct steps: the determination of viewing angles and intrinsic axial ratios from the observed mean values for the ellipticity and position angle of the stellar component and the observed range for the inclination angle [FORMULA] allowed by the apparent axis ratio of the gas component. For each of the two possible configurations (gas disk perpendicular to the short axis or perpendicular to the long axis), B91 fit the gas velocity field and obtained a set of parameters related to the total mass, to the core radius and to the asymptotic behaviour of the velocity curves respectively. The derived values then fix the intrinsic density profile. In this paper our aim is essentially the same, determining the intrinsic shape of the ellipsoid and then fitting the observed velocity field.

To model the observed velocity field it is necessary to know the intrinsic axial ratios of the density distribution and the viewing angles. There is not a unique way to deproject the image of a triaxial galaxy projected on the sky since there are more unknown quantities than observables. The [FORMULA] -models allow us to correlate [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA] to the observed ellipticity and position angles of the stellar major axis at small and large radii ([FORMULA], [FORMULA], [FORMULA], [FORMULA]). Hence four quantities out of eight are free to cover the entire range of values in which they are defined. We will determine the value of these four parameters from the velocity field fit. We now discuss the procedure step by step.

The first step consists of finding the proper center and systemic velocity for all the velocity curves. This is done by fitting a simple (circular) velocity field to the data. The next step is to find all the possible combinations of the parameters, [FORMULA] and [FORMULA] that reproduce the observed shape of the galaxy.

We thus calculate [FORMULA] on a 4-dimensional grid obtained varying the three viewing angles and [FORMULA] with [FORMULA], [FORMULA], [FORMULA]. [FORMULA] is chosen to be within a reasonably range around the apparent minor axis of the gaseous disk. Among all the possible combinations of the above parameters, some lead to non-physical models because [FORMULA], the orbital ellipticity in the center, becomes greater than 0.4 and/or the mass density along the short axis becomes negative for some values of the radius r. We then selected the sets of parameters that matches the two above criteria. At this point, for each set of the parameters so far obtained, the best fit model has been derived by means of a least squares fit to the observed velocity field. That is, for each set of parameters [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA] that reproduce the observed [FORMULA] and that are physically allowable we found the value for [FORMULA] and [FORMULA] that best reproduced the observed velocity field minimizing the root-mean-square. The set of parameters that, among all, give the minimum r.m.s. represents the best fit of the velocity field.

Because the selected sets of parameters are usually very large, this procedure is iterated using smaller and smaller steps for [FORMULA] and [FORMULA] around the best fitting parameter-set, as mentioned above. The model velocities are convolved for the seeing before the comparison with the observed values. A gaussian PSF has been used.

An extra constraint on [FORMULA] can by found from the morphology of the ionized gas. Since the gas orbits become nearly circular at large radii, the apparent axis ratio of the gas disk is directly related to the inclination angle [FORMULA]. Assuming the disk intrinsically circular, the observed axis ratio is equal to [FORMULA]. We do not use this constraint because we don't know how the emissivity of the gas varies with position and we do not know whether the disk is flat and circular or thick, warped and elliptical. We thus consider this value of [FORMULA] as only a rough check for our determination.

3.2. Light density profile

To derive the luminosity density of a galaxy we have to fit the surface brightness profile [FORMULA] taking into account the triaxial shape of the galaxy. Using the value of the three viewing angles previously determined by means of the kinematical fit, we fit the surface brightness profile using the constraints imposed by the twisting of the isophotes and the variation in the ellipticity profile of the stellar component as we did in the previous section. The [FORMULA] model gives an explicit analytical form for the projected surface density:


where [FORMULA] is the position angle of the stellar major axis, [FORMULA] is again the generic point of the galaxy projected on the plane of the sky and [FORMULA] is the azimuthally averaged surface density profile. Expression can be found in ZC. The fit to the observed surface brightness profile involves, as usual, ten parameters, [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. The three viewing angles [FORMULA] and [FORMULA] are now fixed to the values given by the mass density model. Because [FORMULA] depend not only on [FORMULA] but also on [FORMULA], it is possible for their values to be different from the values found in the kinematical fit. This fitting procedure allows one to fit the surface brightness profile and at the same time to roughly reproduce the observed ellipticity and position angle radial profile.

3.3. Statistical errors in the velocity field modeling

We have not derived in a rigorous way the statistical uncertainties of the parameters obtained for the dynamical model. To this aim we would need the standard deviation of each velocity measurement while only an estimate of it is available. (Paper II). For this reason we cannot obtain the [FORMULA] of our models. Nevertheless it is important to have an estimate of how much each parameter is constrained by the fit. Using the value [FORMULA] km/s as the standard deviation of all the velocity measurements, for each best fitting set of parameters ([FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]) we computed the [FORMULA] confidence region. We did not include the axial ratios [FORMULA], [FORMULA], [FORMULA], [FORMULA] in the above parameter list because the error on this four quantities depends mostly on the error in the measured [FORMULA], [FORMULA]. The error estimate of the model parameters is given only in the cases of NGC 1453 and NGC 2974.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998