**2.2. Surface brightness, radial scale, and absolute magnitude**

From the value of
*µ*_{e}, the `effective surface brightness' at
*R*_{e}, and knowing the value of *n*, one can compute
both the central surface brightness
*µ*_{0} and the average/mean surface
brightness <*µ*>_{e} within the effective radius.

At the centre of the profile one has, from Equation (6),

(7) |

The difference between *µ*_{e} and
*µ*_{0} is shown
in Figure (2) as a function of the Sérsic
index *n*.

The `mean effective surface brightness', often simply referred to as
the `mean surface brightness', is computed
as follows. The average intensity, <*I*>_{e},
within the effective radius is obtained by integrating the intensity
over the area *A* =
*R*_{e}^{2} such that

Letting
*x* = *b*(*R* / *R*_{e})^{1/n}, one has

where

Now as *b* was chosen such that *R*_{e} is the radius
containing half of the total light, one has

(8) |

Thus,

(9) |

The difference between *µ*_{e} and
<*µ*>_{e}
is shown in Figure (2) as a function of the
Sérsic index *n*
(Caon et al. 1994;
Graham & Colless
1997).

Substituting equation 9 into Equation 5, one has, at
*R* = *R*_{e},

(10) |

and thus ^{4}

(11) |

This expression can be rewritten in terms of the absolute magnitude,
*M*_{tot}, the effective radius
in kpc, *R*_{e, kpc}, and the *absolute* effective
surface brightness,
<*µ*>_{e, abs}, (i.e., the mean
effective surface brightness if the galaxy was at a distance of 10pc):

(12) |

The apparent and absolute mean effective surface brightnesses are related by the cosmological corrections:

(13) |

where *z*, *E*(*z*), and *K*(*z*) are the
redshift, evolutionary correction, and K-correction respectively (e.g.,
Driver et al. 1994
and references therein).

Another transformation arises from the use of scale-lengths *h*
rather than effective radii *R*_{e}. When the
*R*^{1/n} model is written as

(14) |

(e.g., Ellis & Perry
1979,
their page 362;
Davies et al. 1988),
where *I*_{0} = *I*(*R* = 0), one has

(15) (16) |

It's straightforward to show that

(17) |

and

(18) |

Given the small scale-lengths associated with the *n* = 4 model, and
the practical uncertainties in deriving a galaxy's central brightness,
one can appreciate why Equation (1) is preferred over Equation (14).

If one is modelling a two-component spiral galaxy, consisting of an
exponential disk and an *R*^{1/n} bulge, then the bulge-to-disk
luminosity ratio is given by the expression

(19) |

where *h* and *I*_{0} are respectively the
scale-length and central intensity of the disk, and
*R*_{e}, *I*_{e}, and *n*
describe the Sérsic bulge profile. Noting that 2*n*
(2*n*) =
(2*n* + 1) =
(2*n*)!, the above equation can be simplified for
integer values of 2*n*. For those who are curious,
the first term on the right hand side of the
equality can be seen plotted as a function of *n* in
Graham (2001a).

^{4} Using empirical measurements within
some suitably large aperture, one has from simple geometry that
<*µ*>_{1/2} = *m*_{tot, ap} +
2.5 log(2
*R*_{1/2}^{2}).
Expressions to correct these approximate values -- due to the missed
flux outside of one's chosen aperture -- are given in
Graham et al. (2005).
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