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4.4.2 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

$\displaystyle \Sigma_d(r)=\Sigma_0\,\exp\left(-\frac{r}{r_s}\right)~,$ (4.9)

where $ \Sigma_0$ is the central surface brightness and $ r_s$ is the so called disk scale length. Equation 4.9 can be rewritten in a form similar to the one used for the $ r^{1/4}$ law as

$\displaystyle \Sigma_d(r)=\Sigma_e\,\exp\left[-1.6783\left(\frac{r}{r_e}-1\right)\right]$ (4.10)


$\displaystyle \Sigma_0=5.3567\,\Sigma_e~,~~~~~~r_s=\frac{r_e}{1.6783}~.$ (4.11)

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{displaymath}\begin{split}\Sigma_D(r)&=\Sigma_b(r)+\Sigma_d(r)=\\  &=\Sigm...
...,e}\,\exp \left(-\frac{1.6783\,r}{r_{d,e}}\right)~, \end{split}\end{displaymath} (4.12)

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 4.3. Therefore, the values of two other parameters must be given in order to completely determine the form of Equation 4.12. 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 Appendix D, 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{displaymath}\begin{split}\Sigma_D(r)&=\Sigma_b(r)+\Sigma_d(r)=\\  &=0.769...
...a_{D,e}\,\exp \left(-\frac{1.3945\,r}{r_e}\right)~. \end{split}\end{displaymath} (4.13)

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

$\displaystyle \mu_D(r)=\mu_{D,e}-2.5\,\log\left[ 0.76931\,\exp\left(-7.6692\,\l...
...1/4}-1\right]\right)+2.9343\,\exp \left(-\frac{1.3945\,r}{r_e} \right)\right]~.$ (4.14)

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

$\displaystyle F_{D,tot}=15.796\,\Sigma_{D,e}\,r_e^2~,$ (4.15)


$\displaystyle \mu_{D,e}=2.5\,\log(15.796)+5\,\log(r_{e,\mathrm{[as]}})+I_{\mathrm{[mag]}} ~~~\mathrm{[number~deg^{-2}~mag^{-1}]}~.$ (4.16)

Table 4.5 reports the values of $ <\!\mu\!>_e$, $ \mu_{E,e}$ and $ \mu_{D,e}$ for different total $ I$ magnitudes. The values therein listed in the second column will be used in Chapter 5 to estimate the number of galaxies that could be detected by GAIA ASM. On the other hand, the values listed in the third and fourth column will be used to estimate the standard error of GAIA broad-band photometry at the effective radius of a galaxy.

Table 4.5: Reprentative surface brightness levels of galaxies according to Equations 4.4, 4.8 and 4.16. Average inside the effective radius and at that radius for E and D galaxies. As discussed in Section 4.2, 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

next up previous contents
Next: 4.5 Model's Validity and Up: 4.4 Surface Brightness Distribution Previous: 4.4.1 Elliptical Galaxies   Contents
Mattia Vaccari 2000-12-05