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Astron. Astrophys. 353, 117-123 (2000)

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2. Method

The method that we follow in this work consists of two basic steps. In the first step, model galaxies with realistic spiral structure are constructed. After a visual inspection of the face-on appearance of these models to see the spiral pattern, we create their edge-on images which are now treated as real observations. In the second step we fit these "observations" with a galaxy model where now the galactic disk is described by the widely used plain exponential model. In this way, a comparison between the parameters derived from the fitting procedure and those used to produce the artificial "observations" can be made and thus a quantitative answer about the validity of the plain exponential model as an approximation to galactic disks can be given.

2.1. Artificial spiral galaxies

We adopt a simple, yet realistic, distribution of stars and dust in the artificial galaxy. A simple expression is needed in order to keep the number of free parameters as small as possible and thus have a better control on the problem. A realistic spiral structure is that of logarithmic spiral arms (Binney & Merrifield 1998). Thus, a simple but realistic artificial spiral galaxy is constructed by imposing the logarithmic spiral arms as a perturbation on an exponential disk. In this way, the azimuthally averaged face-on profile of the artificial galaxy has an exponential radial distribution.

For the stellar emissivity we use the formula


In this expression the first part describes an exponential disk, the second part gives the spiral perturbation and the third part describes the bulge, which in projection is the well-known [FORMULA]-law (Christensen 1990). Here R, z and [FORMULA] are the cylindrical coordinates, [FORMULA] is the stellar emissivity per unit volume at the center of the disk and [FORMULA] and [FORMULA] are the scalelength and scaleheight respectively of the stars in the disk.

The amplitude of the spiral perturbation is described by the parameter [FORMULA]. When [FORMULA] the plain exponential disk is obtained, while the spiral perturbation becomes higher with larger values of [FORMULA]. Another parameter that defines the shape of the spiral arms is the pitch angle p. Small values of p mean that the spiral arms are tightly wound, while larger values produce a looser spiral structure. The integer m gives the number of the spiral arms.

For the bulge, [FORMULA] is the stellar emissivity per unit volume at the center, while B is defined by


with [FORMULA] being the effective radius of the bulge and a and b being the semi-major and semi-minor axis respectively of the bulge.

For the dust distribution we use a similar formula as that adopted for the stellar distribution in the disk, namely


where [FORMULA] is the extinction coefficient at wavelength [FORMULA] at the center of the disk and [FORMULA] and [FORMULA] are the scalelength and scaleheight respectively of the dust. Here [FORMULA] gives the amplitude of the spiral perturbation of the dust. Note that the angle [FORMULA] here need not be the same as that in Eq. (1). The stellar arm and the dust arm may have a phase difference between them.

For the parameters describing the exponential disk of the stars and the dust as well as the bulge characteristics we use the mean values derived from the B-band modeling of seven spiral galaxies presented in Xilouris et al. (1999). Note that, instead of the face-on optical depth [FORMULA] used by Xilouris et al., we parameterize the dust content of the galaxy using the edge-on optical depth [FORMULA], which is much larger than the face-on one. In the exponential disk model the two optical depths are related by the equation [FORMULA].

Since the most dominant spiral structure in galaxies is that of the two spiral arms (Kennicutt 1981; Considere & Athanassoula 1988; Puerari & Dottori 1992) we only consider models where [FORMULA]. Galaxies with strong one-arm structure do exist, but they constitute a minority (Rudnick & Rix 1998).

For the parameter [FORMULA] we take the value of 0.4. With this value the optical depth calculated in the arm region is roughly twice as much as in the inter-arm region. This is in good agreement with studies of overlapping galaxies (e.g. White & Keel 1992).

For [FORMULA] we use the value of 0.3 resulting (with the extinction effects included) in a spiral arm amplitude of [FORMULA] mag, which is a typical amplitude seen in radial profiles of face-on spiral galaxies and reproduces the desired strength for the spiral arms (Rix & Zaritsky 1995).

For the pitch angle p we consider the cases of [FORMULA] and [FORMULA], which give a wide variety of spiral patterns from tightly wound to loosely wound. All the parameters mentioned above are summarized in Table 1.


Table 1. Parameters used to describe a typical spiral galaxy.

The radiative transfer is done in the way described by Kylafis & Bahcall (1987; see also Xilouris et al. 1997). As described in detail in these references, the radiative transfer code is capable of dealing with both absorption and scattering of light by the interstellar dust and also of allowing for various distributions for the stars and the dust.

Using the model described above and the parameters given in Table 1 we produce the images shown in Fig. 1. The top three panels of this figure show the face-on surface brightness distribution of such a galaxy for the three values of the pitch angle ([FORMULA] and [FORMULA]) from left to right. The spiral structure is evident in these images with the spiral arms being more tightly wound for [FORMULA] and looser when [FORMULA].

[FIGURE] Fig. 1. Face-on surface brightness of model galaxies (top panel) with different values of the pitch angle ([FORMULA] and [FORMULA] from left to right). The distribution of the face-on optical depth for these galaxies is shown in the middle, while their edge-on appearance (looking from [FORMULA]) is shown in the bottom panel.

In the middle three panels of Fig. 1 we show the distribution of the optical depth when the galaxy is seen face-on for the three different values of the pitch angle mentioned above. In these pictures one can follow the spiral pattern all the way to the center of the galaxy since the bulge is assumed to contain no dust.

Finally, in the last three panels of Fig. 1 one can see the corresponding edge-on appearance of the galaxies shown face-on in the top three panels.

One thing that is very obvious from Fig. 1 (top and middle panels) is that the galaxy is no more axisymmetric as it is the case in the plain exponential disk model. The spiral structure that is now embedded in the model as a perturbation in the disk has broken this symmetry. Thus, in order to do a full analysis of the problem we have to examine the galaxy from different azimuthal views (position angles). To do so we have created nine edge-on model galaxies (for each of the three different pitch angles considered here), covering the range from [FORMULA] to [FORMULA] with a step of [FORMULA] for the position angle. For the definition of the position angle, see Fig. 2. Since the galaxy has exactly the same appearance in the interval from [FORMULA] to [FORMULA], we only consider the range of position angles mentioned above.

[FIGURE] Fig. 2. Schematic representation of the position angle. The value of [FORMULA] is as shown in the figure, while the position angle increases when the line of sight moves counterclockwise.

To demonstrate this asymmetry more quantitatively we have computed the central edge-on optical depth ([FORMULA]) for all these nine model galaxies. Unlike the plain exponential disk model where [FORMULA] can be calculated analytically ([FORMULA]), here we have to perform numerical integration of Eq. (3) along the line of sight that passes through the center. The value of [FORMULA] is shown in Fig. 3 as a function of the position angle. In order to have the full coverage in position angle (from [FORMULA] to [FORMULA]), the values calculated in the interval ([FORMULA] - [FORMULA]) were repeated in the interval ([FORMULA] - [FORMULA]). In this figure, the three models constructed with pitch angles [FORMULA] and [FORMULA] are denoted with circles, squares and diamonds respectively. In all three cases, a variation of the optical depth with position angle is evident. The largest variation is found for the case where the pitch angle is [FORMULA] and it is [FORMULA]. It is obvious that all the values are around the true value of 27, used to construct the galaxy (see Table 1).

[FIGURE] Fig. 3. The central edge-on optical depth of the model spiral galaxy as a function of position angle. Circles, squares and diamonds refer to three different models with pitch angle [FORMULA], [FORMULA] and [FORMULA] respectively.

2.2. The fitting procedure

The edge-on images created as described earlier are now treated as "observations" and with a fitting procedure we seek for the values of the parameters of the plain exponential disk that gives the best possible representation of the "observations". The fitting algorithm is a modification of the Levenberg-Marquardt routine taken from the Minipack library. The whole procedure is described in detail in Xilouris et al. (1997). For the sake of completeness we note here that our fitting criterion was the minimization of the sum of the squares of the differences between the model and the "observation". Tests that we have done using other criteria (such as the sum of the squares of the relative differences) indicate that the best fit parameters are independent of the minimized quantity. However, using the sum of the squares of the differences proved the fastest way for the Levenberg-Marquardt algorithm to converge to the best fit.

Preliminary tests have shown that the derived values of the parameters describing the bulge are essentially identical to the real values used to construct the model images. In order to simplify the fitting process and since we are only interested in the disk, the bulge parameters were kept constant during the fit. Six parameters are now free to vary. These are the scalelength and scaleheight of the stellar disk with its central surface brightness ([FORMULA], [FORMULA] and [FORMULA] respectively) as well as the scalelength and scaleheight of the dust and the central edge-on optical depth ([FORMULA], [FORMULA] and [FORMULA] respectively).

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© European Southern Observatory (ESO) 2000

Online publication: December 8, 1999