Sérsic's (1963;
1968)
*R*^{1/n} model is most commonly expressed as an
intensity profile, such that

(1) |

where *I*_{e} is the intensity at the effective radius
*R*_{e} that encloses half of the total light from the model
(Caon et al. 1993;
see also Ciotti 1991,
his Equation 1).
The constant *b*_{n} is defined in terms of the third and final
parameter *n* which describes the `shape' of the light-profile, and
is given below. ^{2}

One can integrate Equation (1) over a projected area
*A* =
*R*^{2} to obtain the luminosity, *L*, interior to any
radius R. This is
simply a matter of solving the integral
^{3}

which yields, after substituting in
*x* = *b*_{n}(*R* /
*R*_{e})^{1/n},

(2) |

where (2n,x) is the incomplete gamma function defined by

(3) |

Replacing
(2*n*,
*x*) with
(2*n*) in
Equation (2) gives one the value of *L*_{tot}
(Ciotti 1991).

Thus, the value of *b*_{n} which we saw in Equation (1) is
such that

(4) |

where is the
(complete) gamma function
(Ciotti 1991).
Common values of *b*_{n} are
*b*_{4} = 7.669 and
*b*_{1} = 1.678. In passing we note a useful property of the
function, which
is, (2*n*) =
(2*n* - 1)!.

Analytical expressions which approximate the
value of *b*_{n} have been developed.
Capaccioli (1989)
provided one of the first such approximations such that
*b*_{n} = 1.9992*n* - 0.3271, for
0.5 < *n* < 10 (see also
Prugniel & Simien
1997,
their equation A3a).
Ciotti & Bertin (1999)
showed
*b*_{n}
2*n* - 1/3 for large values of *n*, and in practice this
provides a better fit for values of *n* greater than about 8
(see Graham 2001a,
his Figure 2).
Ciotti & Bertin (1999)
also provided an asymptotic expansion which, for values
of *n* > 0.36, is accurate to better than 10^{-4} and the
approximation of choice. For smaller values of *n*,
MacArthur, Courteau, &
Holtzman (2003)
provide a fourth order polynomial which
is accurate to better than two parts in 10^{3} (see their Figure
23). However, the exact value of *b*_{n} in Equatio (4) can
be solved numerically, and fast codes exist to do this,

For an exponential (*n* = 1) profile, 99.1% of the flux resides within
the inner 4 *R*_{e} (90.8% within the inner 4
scale-lengths) and
99.8% of the flux resides within the inner 5 *R*_{e} (96.0%
within the inner 5 scale-lengths).
For an *n* = 4 profile, 84.7% of the flux resides with the inner 4
*R*_{e} and 88.4% within the inner 5 *R*_{e}.

Multiplying the negative logarithm of the luminosity profile (Equation 2) by 2.5 gives the enclosed magnitude profile, known as the "curve of growth",

(5) |

which asymptotes to the total apparent magnitude *m*_{tot}
as *R* tends to infinity and, consequently,
(2*n*,
*x*)
(2*n*)
(Figure 1).

Multiplying the negative logarithm of Equation (1) by 2.5 yields the surface brightness profile (Figure 1), as used in Caon et al. (1993),

(6) |

**2.1.1. Asymptotic behavior for large n**

For large values of *n*, the Sérsic model tends to a power-law
with slope equal to 5. Substituting
*e*^{t} = *z* = *R* / *R*_{e} into
Equation (1), one has

Now, for large *n*, *e*^{t/n} is small, and so
one can use e^{t/n} ~ 1 + *t* / *n*. One can also use
*b*_{n} ~ 2*n* to give

Thus
*µ*(*z*) = - 2.5 log[*I*(*z*)] ~ 5 log(*z*).

For simplicity, the subscript `*n*' will be dropped from the term
*b*_{n} in what follows.

^{2} It is common for researchers studying dwarf
galaxies to replace the exponent 1/*n* with *n*. In this case, de
Vaucouleurs' model would have *n* = 0.25, rather than 4.
Back.

^{3} Obviously if one is using
a major- or minor-axis profile, rather than the geometric mean
(*R* = (*ab*)^{1/2}) profile, an ellipticity term will
be required.
This is trivial to add and for the sake of simplicity won't be
included here. The issue of ellipticity gradients is more difficult,
and interested readers should look at
Ferrari et al. (2004).
Back.