4.2 Disk Galaxies

As first suggested by [de Vaucouleurs(1959)], the surface brightness radial profile of disk galaxies can be interpreted as the sum of two components, the so called bulge component following the $r^{1/4}$ law and the so called disk component following the exponential law

$$\Sigma_d(r)=\Sigma_0\,\exp\left(-\frac{r}{r_s}\right)~,$$ (9)

where $\Sigma_0$ is the central surface brightness and $r_s$ is the so called disk scale length. Equation 9 can be rewritten in a form similar to the one used for the $r^{1/4}$ law as

$$\Sigma_d(r)=\Sigma_e\,\exp\left[-1.6783\left(\frac{r}{r_e}-1\right)\right]~,$$ (10)

where

$$\Sigma_0=5.3567\,\Sigma_e~,~~~~~~r_s=\frac{r_e}{1.6783}~.$$ (11)

Since the first systematic study of [Freeman(1970)], this law has been known to fit the profiles of the outer regions of a large class of disk galaxies and has, in fact, come to define the typical surface brightness profile of the intrinsically flat component of disk galaxies, to the extent that deviations from these profile are generally ascribed to the existence of other components or to the effects of dust. Assuming this analytical form for the profile of the disk component, one can then try to disentangle the contributions of the bulge and disk components by means of fitting techniques. The methods for doing this have in time undergone a great development, from the simple one-dimensional fitting procedure along the galaxy major axis first adopted by [Freeman(1970)] to the bidimensional decomposition techniques currently being developed, which are applied to whole galaxy images (see e.g. [Byun and Freeman(1995)]). In this study, a bulge+disk profile will be considered, combining an $r^{1/4}$ law with an exponential law, thus not considering contributions from components such as spiral arms, bars, rings, lenses, or the photometric effcts of dust. The most general form of such a profile

$$\begin{split}\Sigma_D(r)&=\Sigma_b(r)+\Sigma_d(r)=\\ &=\Sigma... ...,e}\,\exp \left(-\frac{1.6783\,r}{r_{d,e}}\right)~, \end{split}$$ (12)

depends on two pairs of parameters characterizing the bulge and disk components, respectively. The only independent variable we have so far introduced in our model is the total magnitude, which however determines also the effective radius through Equation 3. Therefore, the values of two other parameters must be given in order to completely determine the form of Equation 12. A convenient choice are two quantities derived from bulge/disk decompositions and frequently reported in the literature, namely the bulge/bulge+disk ratio $B/T$, or the ratio between the brightness contributed by the bulge component and the total brightness, and the ratio $r_{b,e}/r_{d,e}$ between the effective radii of the bulge and disk components. As shown in the Appendix, these two parameters completely determine the bulge+disk profile. As for $B/T$, [Kent(1985)] found that in intrinsically luminous D galaxies this is tightly correlated with the morphological type, falling from a mean value of 0.65 for S0 to a mean value of 0.15 for Sc and later types. Recently, [Ratnatunga et al.(1999)] found a mean $B/T$ of 0.4 from the bulge/disk decomposition of the MDS galaxies. The same two references then agree in fixing to about 0.5 the mean value of $r_{b,e}/r_{d,e}$. Such an excellent agreement between parameters obtained from galaxies of largely different magnitudes suggested to consider the two parameters as fixed, without introducing in the model the complications of other free parameters. Values of $B/T=0.4$, corresponding to a bulge/disk ratio $B/D=0.666$, and $r_{b,e}/r_{d,e}=0.5$ were therefore assumed. With our choices for $B/T$ and $r_{b,e}/r_{d,e}$ the bulge+disk profile becomes

$$\begin{split}\Sigma_D(r)&=\Sigma_b(r)+\Sigma_d(r)=\\ &=0.7693... ...a_{D,e}\,\exp \left(-\frac{1.3945\,r}{r_e}\right)~. \end{split}$$ (13)

In general, the bulge and disk components dominate the profile at small and large radii, respectively. Note, however, that due to the analytical form of the two profiles, at very large radii the bulge contribution eventually exceeds that of the disk. In fact, practically all (99%) of the brightness predicted by the disk profile falls within 4 effective radii, but for the bulge profile only 85% of the light is within 4 effective radii, and the model needs to extend out to about 19 effective radii to contain 99% of it. Shifting to a magnitude scale, the analytical expression for $\mu_D$ is not as simple as that derived for ellipticals and is

$$\mu_D(r)=\mu_{D,e}-2.5\,\log\left[ 0.76931\,\exp\left(-7.6692\,\l... ...1/4}-1\right]\right)+2.9343\,\exp \left(-\frac{1.3945\,r}{r_e} \right)\right]~.$$ (14)

The total brightness emitted by the galaxy and the effective surface brightness can be written as

$$F_{D,tot}=15.796\,\Sigma_{D,e}\,r_e^2~,$$ (15)

and

$$\mu_{D,e}=2.5\,\log(15.796)+5\,\log(r_{e,\mathrm{[as]}})+I_{\mathrm{[mag]}} ~~~\mathrm{[number~deg^{-2}~mag^{-1}]}~.$$ (16)

Table 5 reports the values of $<\!\mu\!>_e$, $\mu_{E,e}$ and $\mu_{D,e}$ for different total $I$ magnitudes.

Table 5: Reprentative surface brightness levels of galaxies according to Equations 4, 8 and 16. Average inside the effective radius and at that radius for E and D galaxies. As discussed in Section 2, D galaxies are four times more frequent than E galaxies.

$I$	$<\!\mu\!>_e$	$\mu_{E,e}$	$\mu_{D,e}$
mag	mag/arcsec$^2$	mag/arcsec$^2$	mag/arcsec$^2$
10	20.2481	21.6410	21.2490
11	20.2800	21.6730	21.2809
12	20.3067	21.6996	21.3076
13	20.3321	21.7250	21.3330
14	20.3616	21.7545	21.3625
15	20.4023	21.7952	21.4032
16	20.4628	21.8557	21.4637
17	20.5531	21.9461	21.5540
18	20.6851	22.0780	21.6860
19	20.8718	22.2648	21.8727
20	21.1281	22.5210	22.1290
21	21.4702	22.8632	22.4711
22	21.9161	23.3090	22.9170
23	22.4851	23.8780	23.4860
24	23.1981	24.5911	24.1991