In this article we have compiled and developed equations pertaining to the Sérsic profile in a purely analytical manner. To mention just one of many important uses of the Sérsic profile is its potential for deriving accurate total magnitudes. This need is motivated by a growing community-wide awareness of the complex nature of galaxy photometry, and in particular the large amounts of flux which can be missed by isophotal, aperture, Petrosian, or Kron magnitudes (e.g., Figures 7 & 10). Cross et al. (2004) recently compared APM, SuperCosmos, SDSS and MGC photometry for several thousand galaxies and concluded that the photometric errors are mainly dominated by the systematics of the methodology rather than the random errors. The Sérsic magnitude provides a logical standard and is derived by evaluating Equation 5 at R = given µe, n, and Re derived from a fit to the available light-profile.
In practice various "facts-of-life" issues remain; these are not specific problems to the Sérsic model, but generic to studies of galaxy photometry. The most obvious ones are: the smoothing effect of the point-spread function (PSF); profile truncation; multiple component systems; dust and asymmetric profiles. All of these can act to exacerbate or ameliorate the amount of missing flux. While a detailed discussion of these issues is beyond the scope of this paper we provide some suitable references below.
The smearing effect of the PSF will cause the observed profile to tend to n 0.5 (i.e., Gaussian), this is dealt with straightforwardly by incorporating PSF convolution into the model fitting process, e.g., Andredakis et al. (1995; their Equation 10) and Trujillo et al. (2001a, b). Disk truncation is trickier (see the reviews by van der Kruit (2001) and Pohlen et al. 2004) and is assumed to be related to the minimum gas density required for star-formation (Kennicutt 1989; Kregel & van der Kruit 2004). Initially truncation was reported to occur at around 4 scale-lengths for exponential disks (van der Kruit 1979; van der Kruit & Searle 1981). More recent studies have argued that the truncation is better described by a broken exponential fit (e.g., de Grijs et al. 2001; Pohlen et al. 2002). Others argue that truncation may actually be a manifestation of poor background modelling or simply due to intrinsic variations in the disk (Narayan & Jog 2003). Certainly some recent studies find no discernible truncation to extremely faint isophotal limits; for example, NGC 300 is a pure exponential out to 10 scale-lengths (Bland-Hawthorn et al. 2005). The net result is that in practice it is not clear exactly how far out one should integrate the Sérsic profile for disk galaxies. As discussed in Section 2.3, there is no evidence for truncation in the elliptical galaxy population. From Figure 1 we see that this issue is far more significant for high Sérsic index systems. A new generation of publicly available 2D fitting codes, in particular GIM2D (Marleau & Simard 1998), GALFIT (Peng et al. 2002), and BUDDA (de Souza, Gadotti, & dos Anjos 2004), can routinely deal with multiple-component profiles. Dust opacity (Disney, Davies, & Phillipps 1989, Davies et al. 1993) can also lead to changes in the light-profile because of the more centrally concentrated dust distribution. Modelling opacity is non-trivial as there are strong inclination dependencies (Choloniewski 1991; Jones, Davies & Trewhella 1996; Graham 2001b) however models are being developed to provide detailed corrections (e.g., Pierini et al. 2004; Tuffs et al. 2004). From the dust attenuation studies of Calzetti (2001, and references therein) and many others, dust issues can be minimised via structural analysis at near-IR wavelengths. Non-biaxial asymmetry of galaxy images can be readily identified via the use of the `asymmetry' measures (e.g., Schade et al. 1995; Rix & Zaritsky 1995; Conselice 1997; Kornreich, Haynes, & Lovelace 1998).
We kindly thank Valeria Coenda for faxing us a copy of Sérsic's 1963 article, referenced in his 1968 Atlas. We are additionally grateful to Steve Phillipps for refereeing this work, and Massimo Capaccioli for his helpful and rapid response to our request for information and references.