D.4 Bulge+Disk Profile

The surface brightness radial profiles of disk galaxies are usually modelled as the sum of a bulge and a disk component. The resulting bulge+disk profile can in general be written as

$$\begin{split}\Sigma_{b+d}(r)&=\Sigma_b(r) + \Sigma_d(r)=\\ &... ...,e}\,\exp \left(-1.6783\,\frac{r}{r_{d,e}}\right)~, \end{split}$$ (D.16)

where quantities subscripted with $b$ and $d$ refer to the bulge and disk component, respectively. The total brightness predicted by this profile is

$$F_{b+d,tot}=F_{b,tot}+F_{d,tot}=22.665\,\Sigma_{b,e}\,r_{b,e}^2+ 11.948\,\Sigma_{d,e}\,r_{d,e}^2~.$$ (D.17)

The relative importance of the bulge and disk component in terms of the brightness they contribute to the overall profile can be quantified by the bulge/disk ratio $B/D$, which, from Equations D.6 and D.12, can be expressed in terms of the bulge and disk parameters as

$$\frac{B}{D}=\frac{k_4\,\Sigma_{b,e}\,r_{b,e}^2}{k_1\,\Sigma_{d,e}... ...}{\Sigma_{d,e}} \,\frac{r_{b,e}^2}{r_{d,e}^2}=1.8969\,\rho_{\Sigma}\,\rho_r^2~,$$ (D.18)

where

$$\rho_{\Sigma}=\frac{\Sigma_{b,e}}{\Sigma_{d_e}}~,~~~~~~~~ \rho_r=\frac{r_{b,e}}{r_{d,e}}~.$$ (D.19)

$B/D$ is related to the more frequently used bulge/bulge+disk ratio $B/T$ by the

$$\frac{B}{D} = \frac{B/T}{1-B/T}~.$$ (D.20)

The relation between the radii $r_{b,e}$ and $r_{d,e}$ and the effective radius of the bulge+disk profile $r_{b+d,e}$, can be determined using Equations D.2 and D.3, whose combination yields

$$\frac{k_4}{\Gamma(8)}\,\Sigma_{b,e}\,r_{b,e}^2\,\gamma\left(8\, ,... ...frac{k_4}{2}\,\Sigma_{b,e}\,r_{b,e}^2+ \frac{k_1}{2}\,\Sigma_{d,e}\,r_{d,e}^2~,$$ (D.21)

which, after trivial modifications, becomes

$$\frac{B}{D}\,\left[\frac{1}{\Gamma(8)}\, \gamma\left(8\, , \,b_4\... ...t(2\, , \,b_1\,\rho_r\,\frac{r_{b+d,e}}{r_{b,e}}\right) -\frac{1}{2}\right]=0~.$$ (D.22)

When values for the ratios $B/D$ and $\rho _r$ are assumed, Equation D.22 can be solved numerically to obtain the corresponding values of the ratios $r_{b,e}/r_{b+d,e}$ and $r_{d,e}/r_{b+d,e}$. Numerical values of the latter two ratios are given in Table D.2 for some values of the former two.

Table D.2: Values of $r_{b,e}/r_{b+d,e}$ and $r_{d,e}/r_{b+d,e}$ for some values of $B/D$ and $\rho _r$. Values calculated via Newton integration (see Section 9.7 in [Press et al. 1996]) of Equation D.22.

$\frac{r_{b,e}}{r_{b+d,e}}$		$B/D$
		0.333	0.444	0.555	0.666	0.777
$\rho _r$	0.3	0.37776903	0.39659331	0.41501285	0.43296121	0.45038387
	0.4	0.47738606	0.49489417	0.51165418	0.52767183	0.54296003
	0.5	0.57169562	0.58717625	0.60178337	0.61556793	0.62857967
	0.6	0.66216968	0.67514232	0.68725805	0.69858888	0.70920016
	0.7	0.74977666	0.75987986	0.76924321	0.77794075	0.78603748

$\frac{r_{d,e}}{r_{b+d,e}}$		$B/D$
		0.333	0.444	0.555	0.666	0.777
$\rho _r$	0.3	1.2592301	1.3219777	1.3833761	1.4432040	1.5012795
	0.4	1.1934651	1.2372354	1.2791355	1.3191796	1.3574001
	0.5	1.1433912	1.1743525	1.2035667	1.2311359	1.2571593
	0.6	1.1036161	1.1252372	1.1454301	1.1643148	1.1820002
	0.7	1.0711095	1.0855427	1.0989189	1.1113440	1.1229107

Now, the general bulge+disk profile can be written as

$$\begin{split}\Sigma_{b+d}(r)&=\frac{\Sigma_{b,e}}{\Sigma_{b+d... ...c{r_{b+d,e}}{r_{d,e}}\,\frac{r}{r_{b+d,e}}\right)~. \end{split}$$ (D.23)

In much the same way as Equation D.22, Equation D.23 can be numerically solved with respect to the ratios $\Sigma _{b,e}/\Sigma _{b+d,e}$ and $\Sigma _{d,e}/Sigma_{b+d,e}$ if values of the ratios $B/D$ and $\rho _r$ are assumed. Numerical values of the two former ratios are given in Table D.3 for the same values of the latter two as in Table D.2.

Table D.3: Values of $\Sigma _{b,e}/\Sigma _{b+d,e}$ and $\Sigma _{d,e}/Sigma_{b+d,e}$ for the same values of $B/D$ and $\rho _r$ as in Table D.2. Values calculated from Equations D.16 and D.18, and from Table D.2.

$\frac{\Sigma_{b,e}}{\Sigma_{b+d,e}}$		$B/D$
		0.333	0.444	0.555	0.666	0.777
$\rho _r$	0.3	1.5059000	1.6695642	1.7862309	1.8669555	1.9202392
	0.4	0.90250402	1.0169289	1.1059437	1.1750876	1.2286100
	0.5	0.61171687	0.69840479	0.76930751	0.82759966	0.87573973
	0.6	0.44733235	0.51643686	0.57490397	0.62470279	0.66737834
	0.7	0.34431760	0.40137789	0.45089086	0.49413744	0.53213802

$\frac{\Sigma_{d,e}}{\Sigma_{b+d,e}}$		$B/D$
		0.333	0.444	0.555	0.666	0.777
$\rho _r$	0.3	0.57903215	0.51357003	0.45788132	0.41020644	0.36917465
	0.4	0.61692617	0.55611501	0.50399454	0.45900376	0.41992150
	0.5	0.65336297	0.59676211	0.54778829	0.50511037	0.46768035
	0.6	0.68801330	0.63543886	0.58948209	0.54903740	0.51322600
	0.7	0.72080756	0.67220776	0.62927470	0.59111205	0.55699899

Tables D.2 and D.3, combined with Equation D.23, allows to write the appropriate bulge+disk profile for different values of $B/D$ and $\rho _r$. The total brightness predicted by the law can then be written in terms of the bulge+disk quantities $r_{b+d,e}$ and $\Sigma_{b+d,e}$ as

$$F_{D,tot}=\left[k_4\,\frac{\Sigma_{b,e}}{\Sigma_{b+d,e}} \,\left(... ...ght]\,\Sigma_{b+d,e}\,r_{b+d,e}^2 \equiv k_{b+d}\,\Sigma_{b+d,e}\,r_{b+d,e}^2~,$$ (D.24)

and, proceeding like for the pure bulge and disk profiles, $\mu_{b+d,e}$ can be written as

$$\mu_{b+d,e}=\,2.5\,\log(k_{b+d})+5\,\log(r_{e,\mathrm{[as]}})+ I_{\mathrm{[mag]}}~~~\mathrm{[mag~arcsec^{-2}]}~.$$ (D.25)

Values of $k_{b,d}$ are given in Table D.4.

Table D.4: Values of $k_{b+d}$ for the same values of $B/D$ and $\rho _r$ as in Tables D.2 and D.3. Values calculated from Equation D.24 and from Tables D.1, D.2 and D.3.

$k_{b+d}$		$B/D$
		0.333	0.444	0.555	0.666	0.777
$\rho _r$	0.3	15.841388	16.675985	17.443019	18.140889	18.770256
	0.4	15.161182	15.816591	16.415234	16.959955	17.454163
	0.5	14.737528	15.291208	15.795789	16.255446	16.674198
	0.6	14.458169	14.948771	15.395577	15.803140	16.175547
	0.7	14.268144	14.717733	15.127242	15.501275	15.843825