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Appendix A Generalized exponential brightness distributions and conversion factors

Assuming circular symmetry, we define a generalized exponential bulge as a surface brightness distribution:
 (A 1)

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

 (A 2)
It is customary to introduce instead of r0 an effective radius ,which encircles half of the luminosity, and instead of I0 an effective brightness . We derive in the following the relations between these quantities in the case of a generalized exponential.

The relation defining is:

 (A 3)

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

 (A 4)

which we write as:

 (A 5)

The left hand side of this last equation is a cumulative Poisson distribution with parameter . This can be written in terms of probability functions (Abramowitz & Stegun 1971):

 (A 6)
with and .

is therefore easily evaluated from tables. In any case, for large values, is

approximately normally distributed:

 (A 7)

 (A 8)

Since in our case , it follows x2=0 and . Then, with the approximations adopted:

 (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:
 (A 10)

These results are easily generalized to elliptical distributions and they are found to hold in the same form if is the major semiaxis of the ellipse encircling half luminosity.

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