2.4. Profile slopes
Given HST's ability to resolve the inner light-profiles of nearby galaxies, the slope of a galaxy's nuclear (the inner few hundred parsec) stellar distribution has become a quantity of interest. Defining ^{5}
(22) |
Rest et al. (2001, their Equation 8) used this to measure the nuclear slopes of `core' and `power-law' galaxies. From Equation (1) one can obtain
(23) |
This is approximately 2(R / R_{e})^{1/n} (see section 2.1). Thus, at constant (R / R_{e}), is a monotonically increasing function of the Sérsic index n (Graham et al. 2003b).
It turns out Equation (23) is appropriate for the so-called `power-law' galaxies which are now known to possess Sérsic profiles down to their resolution limit (Trujillo et al. 2004) and would be better referred to as `Sérsic' galaxies as they do not have power-law profiles. A modification is however required for the luminous `core galaxies', and is described in Section 2.7.
Another logarithmic slope of interest is that used by Gunn & Oke (1975) and Hoessel (1980), and is defined as
(24) |
From Equation (2) one has
(25) |
where, as before, x = b(R / R_{e})^{1/n} (Graham et al. 1996, their equation 8).
Figures (4) and (5) show how (R) and (R) vary with normalised radius R / R_{e} for a range of different profile shapes n.
Figure 4. The slope of the Sérsic profile (Equation 23) is shown as a function of profile shape n for R / R_{e} = 0.01, 0.05, 0.1, and 0.2. |
Figure 5. The slope (Equation 25) is shown as a function of normalised radius R / R_{e} for n = 0.5, 1, 2, 4, 7, and 10. |
^{5} This should not be confused with the incomplete gamma function seen in Equation (3). Back.