## 5. Model fittingAccording to Eqs. (1) - (3), a fit to the surface photometry of a spiral galaxy should produce values of the parameters a) (or equivalently ), and for the stars in the disk, b) (or equivalently ), and for the stars in the bulge, c) (or equivalently ), and for the dust and d) the inclination angle of the galactic disk with respect to our line of sight. The search for a minimum (in a least-squares sense) in a space of ten (the number of parameters) dimensions is not only time consuming but also contains the danger of ending up in a local minimum rather than the global one. For these reasons, it is helpful to get good estimates of as many parameters as possible before attempting a global fit of the galaxy. ## 5.1. Partial fittingAn inspection of the image of UGC 2048 reveals that this galaxy is
seen approximately edge-on (the exact value of the inclination angle
will be determined below). For an edge-on disk galaxy, the surface
brightness away from the dust lane is proportional to
at all radial distances R (see Eq. 4). Thus,
excluding the central part of UGC 2048, which is affected by the
bulge, the rest of the galaxy can be collapsed into one dimension
parallel to the
value of the surface brightness in the I band as a function of
approximately constant at all For each trial value of we evaluate the
ratio (9) for every
Subtracting the derived image of the disk from the image of the
galaxy, we are left with the image of the bulge away from the dust.
From it we have found ,
mags/arcsec Then, assuming that , we fitted the analytic solution of the radiative transfer equation in the edge-on case (neglecting scattering) at a few points of the galaxy away from the bulge. From this we were able to determine and , which were found to be 0.36 kpc and 0.25 respectively in the I band. Finally, using the numerical model, which is able to deal with any inclination angle, we fit the surface brightness at a few cuts of the galaxy parallel to the z axis by changing only the inclination angle. By trial and error, this parameter has been found to be approximately degrees. Again, these values are not accurate, but they are good initial guesses for the I-band data. ## 5.2. Global fittingFor the global fit we used the Levenberg-Marquardt algorithm (see Press et al. 1986) embedded in the IMSL MATH/LIBRARY. During this procedure, the radiative transfer is performed taking into account both absorption and scattering and a model galaxy is formed. Then, the observed surface brightness is compared with the computed surface brightness from the model and a new set of parameters is found. This is repeated until a minimum in the value is reached. A confidence interval on the regression parameters is also calculated, using the inverse of the Student's t distribution function. Having at hand good initial guesses for the parameters, it was fairly easy to find the minimum. Tests were then made, with the initial values set more than off the original values, in order to make sure that the minimum is indeed global. In all runs it was found that the final values derived from the fit were in the confidence interval calculated. The values of the parameters derived for UGC 2048 are shown in
Table 1 for the case where the Hubble profile is used and in Table 2
for the law. All lengths are in kpc (see,
however, our comment on the distance to the galaxy at the end of Sect.
6), while the central luminosity densities for
the stars in the disk and for the stars in the
bulge are given in terms of the central surface brightnesses
and respectively (see
Eqs. 5 - 7) in units of mags/arcsec
## 5.3. Dust content in the galaxyHaving derived the distribution of dust, it is straightforward to calculate the total amount of dust in the galaxy. Assuming that the grains can be approximated by spheres of radius
m and material density
kgr m where Using Eq. (8), which gives the optical depth through the disk in directions parallel to the plane of the disk and integrating over the whole projected surface of the galaxy, the dust mass is then given by : with given in kpc. Substituting the values of and calculated from the model, the dust mass of the galaxy at the assumed distance of 63 Mpc is (see, however, our comment at the end of Sect. 6). Using Eq. (2) of Devereux & Young (1990) and the published
value for the flux (Huchtmeier & Richter
1989) Jy km s at the assumed distance. Unfortunately, we have not been able to locate a measurement for the 2.6 mm CO line for this galaxy in order to calculate the mass of molecular hydrogen . However, a crude approximation is to assume the same mass for molecular hydrogen as the mass we derived for atomic hydrogen (see, e.g. Table 1 in Devereux & Young 1990). If we do so, the total gas mass is approximately Finally, from the above calculations, the gas to dust ratio for this galaxy is found to be which is close to the value of that is widely adopted for our Galaxy (Spitzer 1978, p.162). © European Southern Observatory (ESO) 1997 Online publication: May 5, 1998 |