Appendix A

Generalized exponential brightness distributions and conversion factors

Assuming circular symmetry, we define a generalized exponential bulge as a surface brightness distribution:

$$I(r) = I_0 ~\exp \left[-\left(\frac{r}{r_0}\right)^{1/n}\right]$$ (A 1)

where r is the projected radius, I₀ the central brightness, r₀ the exponential folding length, and n an integer index ($n\geq1$). With the change of variable xⁿ=r/r₀, it is easily seen that the total luminosity L is:

$$L = \int_{0}^{\infty}I(r)~2 \pi r~{\rm d}r = \pi (2n)!~I_0 r_0^2.$$ (A 2)

It is customary to introduce instead of r₀ an effective radius $r_{\rm e}$,which encircles half of the luminosity, and instead of I₀ an effective brightness $I_{\rm e} = I(r_{\rm e})$. We derive in the following the relations between these quantities in the case of a generalized exponential.

The relation defining $r_{\rm e}$ is:

$$\int_{0}^{r_{\rm e}}I(r)~2 \pi r~{\rm d}r = \frac{L}{2}~~ ;$$ (A 3)

with the above mentioned change of variable, this integral yields:

$${\rm e}^{-\alpha_n} \sum_{i=0}^{2n-1} \frac {\alpha_n^{2n-1... ...~~ \alpha_n=\left(\frac{r_{\rm e}}{r_0}\right)^{\frac{1}{n}},$$ (A 4)

which we write as:

 

$${\rm e}^{-\alpha_n} \sum_{i=0}^{m} \frac {\alpha_n^{i}}{i!} = \frac{1}{2}\hspace{1.5cm}{\rm with}~~~~m=2n-1.$$ (A 5)

The left hand side of this last equation is a cumulative Poisson distribution with parameter $\alpha_n$. This can be written in terms of $\chi^{2}$ probability functions (Abramowitz & Stegun 1971):

$${\rm e}^{-\alpha_n} \sum_{i=0}^{m} \frac {\alpha_n^{i}}{i!} = Q(\chi^2\vert~\nu)\, ,$$ (A 6)

with $\chi^2=2 \alpha_n$ and $\nu=2(m+1)=4n$.

$\alpha_n$ is therefore easily evaluated from $\chi^{2}$ tables. In any case, for large $\nu$ values, $(\chi^2/\nu)^{1/3}$ is

approximately normally distributed:

$$Q\left(\chi^2\vert~\nu\right) \simeq Q(x_2)\, ,$$ (A 7)

$${\rm with} \hspace{0.5cm}x_2=\frac{\left(\frac{\chi^2}{\nu}\... ...}{3}} -\left(1-\frac{2}{9\nu}\right)}{\sqrt{\frac{2}{9\nu}}}\,.$$ (A 8)

Since in our case $Q(x_2)=\frac{1}{2}$, it follows x₂=0 and $\chi^2\simeq\nu(1-\frac{2}{3\nu})$. Then, with the approximations adopted:

 

$$\alpha_n\simeq 2n-\frac{1}{3} , ~~~~~~~{\rm or}~~~~~~~r_{\rm e}\simeq r_0 \left( 2n-\frac{1}{3}\right)^n .$$ (A 9)

It turns out that the approximation is already quite good for n = 1, with a relative error of 0.7%. Table A1 reports the solution of Eq. (A5) and its approximation with Eq. (A9) for various values of n.

For the effective surface brightness we have:

$$I_{\rm e} = I(r_{\rm e})=I_0~\exp\left[-\left(\frac{r_{\rm e}}{r_0}\right)^{\frac{1}{n}}\right]$$ (A 10)

$$=I_0~{\rm e}^{-\alpha_n}\simeq I_0~\exp~\left(-2n+\frac{1}{3}\right)\, .$$

These results are easily generalized to elliptical distributions and they are found to hold in the same form if $r_{\rm e}$ is the major semiaxis of the ellipse encircling half luminosity.

  

Table 5: $r_{\rm e}$ approximation