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,
*I _{0}* the central brightness,

(A 2) |

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) |

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 *x _{2}*=0 and .
Then, with the approximations adopted:

(A 9) |

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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