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Astron. Astrophys. 325, 135-143 (1997)

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4. Model

In our model we assume that the stellar distribution consists of two components, the disk and the bulge. For the light distribution in spiral disks, two laws seem to be generally accepted so far. The sech2 z law (van der Kruit & Searle 1981) and the exponential law (Freeman 1970; Wainscoat et al. 1989). Both of these laws provide a good representation of the vertical distribution of light in the galactic disks. For the radial distribution, an exponential law is used. In the present work, we use the exponential law in both the radial and vertical direction.

For the bulge several profiles have been introduced. The well known [FORMULA] law (de Vaucouleurs 1953; Young 1976), the Hubble-Reynolds law (Reynolds 1913; Hubble 1930), the Hernquist law (Hernquist 1990) and the exponential law (Andredakis & Sanders 1994) are good representations of bulges. For our model both the [FORMULA] law and the Hubble profile are used and a comparison is made. A good description of the luminosity densities is given by Christensen (1990) for the [FORMULA] law and by Binney & Tremaine (1987) for the Hubble law.

The stellar emissivity is then described by

[EQUATION]

[EQUATION]

where R and z are the cylindrical coordinates, [FORMULA] is the stellar emissivity at the center of the disk and [FORMULA] and [FORMULA] are the scalelength and scaleheight respectively of the stars in the disk. The second term in this equation gives the two different types of bulge luminosity density profiles, the first being the Hubble law and the second the [FORMULA] law, with [FORMULA] a normalization constant and

[EQUATION]

where [FORMULA] is the effective radius of the bulge and a and b are the semi-major and semi-minor axis respectively of the bulge. Because the [FORMULA] profile has an infinite luminosity density at the center, and in order to avoid computational problems, the luminosity density in a small sphere of radius 0.2 kpc around the center was given a constant value and equal to that at 0.2 kpc. To be consistent, this region was excluded when the fitting procedure was done.

For the extinction coefficient we also use a double exponential law, namely

[EQUATION]

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.

If the above model galaxy is seen edge-on, and for the moment we ignore completely the effects of dust, the surface photometry due to the disk alone, after integration of the first term on the right hand side of Eq. (1), is

[EQUATION]

where [FORMULA] is the modified Bessel function of the second kind, first order (Abramowitz & Stegun 1965), with a central value of

[EQUATION]

The central value of the bulge surface brightness is

[EQUATION]

for the modified Hubble profile and

[EQUATION]

for the de Vaucouleurs [FORMULA] profile (Christensen 1990).

The optical depth through the disk, in directions parallel to the plane of the disk is

[EQUATION]

Thus, the central optical depth of an edge-on galaxy is [FORMULA] and the central optical depth of the same galaxy seen face-on is [FORMULA].

The radiative transfer is done in the way described by Kylafis & Bahcall (1987). The intensity I reaching a pixel is thought of as the sum of [FORMULA], where [FORMULA] consists of the photons that suffered no scattering between their point of emission in the galaxy and the pixel, [FORMULA] consists of the photons that suffered one scattering between their point of emission and the pixel, [FORMULA] consists of the photons that suffered two scatterings between their point of emission and the pixel, and so on. The term [FORMULA] is proportional to the albedo [FORMULA] of the dust, the term [FORMULA] is proportional to [FORMULA] and so on. Since [FORMULA] (say [FORMULA], see below), the contribution to the intensity I of the terms [FORMULA], with [FORMULA], is generally small compared to [FORMULA]. Thus, to save computer time, we compute [FORMULA] and [FORMULA] very accurately and approximate the sum [FORMULA] with [FORMULA], where [FORMULA] (see Eq. 19 of Kylafis & Bahcall 1987). The error that this approximation introduces to the intensity I is typically less than 1%.

For specific values of the parameters in Eqs. (1) - (3), a 2D image of a model galaxy is produced. The goal is to find those values of the parameters which create an image of the model galaxy as close as possible to the image of the observed galaxy. A >Henyey-Greenstein phase function has been used for the scattering of the dust (Henyey & Greenstein 1941). A mean value 0.4 was used for the anisotropy parameter g, while the value 0.6 was used as an average albedo [FORMULA] for all B, V, I bands. The effects of changing these parameters within the limits given by (Bruzual et. al. 1988) have been explored. It has been found that varying these parameters within the above limits has no important changes in the intensity because of the low optical thickness that has been found for this galaxy.

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© European Southern Observatory (ESO) 1997

Online publication: May 5, 1998

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