5. Model fitting
According 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 fitting
An 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 z axis. We do so in Fig. 1, where we have collapsed the galaxy between the radial distances 15 kpc and 25 kpc and the average
value of the surface brightness in the I band as a function of z is shown. It is evident that for large the surface brightness falls exponentially with , with a scaleheight = 1.2 kpc (for the assumed distance, see end of Sect. 6.) derived from and = 1.4 kpc derived from . None of the two values is accurate (as we will see below), but they are good initial guesses for a global fit. The inaccuracy of is due to both the bulge light and the exponential distribution of the dust. Between the two, the bulge light, which is present at faint levels even at large distances from the center, has a larger effect in the determination of . For an estimate of the scalelength of the stars in the disk, we collapse the galaxy perpendicular to the z axis. By avoiding the bulge and the dust lane, the rest of the galaxy should be well described by Eq. (4). Integrating this equation over z, we then ask for that value of that will make the ratio
approximately constant at all R.
For each trial value of we evaluate the ratio (9) for every R and compute the mean value and the standard deviation. Fig. 2 shows the ratio of the standard deviation to the mean value as a function of trial . At the minimum we have the best estimate of the true . At the assumed distance, this is 9.4 kpc in the I band. Again, this value is not accurate enough due to the contamination by the bulge and the dust. Having derived estimates for and , we use Eq. (4) to fit the surface brightness of the galaxy at a few positions away from the dust and the bulge. In this way, the central luminosity density and thus the central surface brightness of the disk is estimated and has been found to be mags/arcsec2 in the I band.
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/arcsec2 and = 0.55 kpc for the Hubble profile, while mags/arcsec2 and = 1.9 kpc for the profile in the I band.
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 fitting
For 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/arcsec2. The optical depth is the central optical depth of the galaxy if it were to be seen face-on.
Table 1. Global model fit parameters for UGC 2048. The Hubble profile is used for the bulge.
Table 2. Global model fit parameters for UGC 2048. The law is used for the bulge.
5.3. Dust content in the galaxy
Having 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-3, the total mass of grain material is given by :
where S is the surface area of the galaxy projected in the sky, is the optical depth in the V-band and is the extinction cross-section of a single grain and it is (see, e.g. Whittet 1992).
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).
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