2.3. Concentration
For many years astronomers have had an interest in how centrally concentrated a galaxy's stellar distribution is (e.g., Morgan 1958, 1959, 1962; Fraser 1972; de Vaucouleurs 1977). Sérsic (1968) used de Vaucouleurs (1956) somewhat subjective size ratio between the main region of the galaxy and the apparent maximum dimension of the galaxy as a measure of concentration.
Trujillo, Graham & Caon (2001c) defined a useful, objective expression for concentration, such that, in pixelated form
![]() | (20) |
Here, E(Re) means the isophote which encloses
half of the total light, and
E(
Re) is the
isophote at a radius
(0 <
< 1) times
Re. This is a flux ratio.
For a Sérsic profile which extends to infinity,
![]() | (21) |
This expression is a monotonically increasing function of n, and
for = 1/3 it's values are
shown in Figure (3).
An often unrealized point is that if every elliptical galaxy had an
R1/4 profile then they would all have exactly the same
degree of concentration. Observational errors aside, it is only because
elliptical galaxy light-profiles deviate from the R1/4
model that a range of concentrations exist. This is true for all
objective concentration indices in use today.
![]() |
Figure 3. The central concentration CRe(1/3), as defined by Trujillo et al. (2001c), is a monotonically increasing function of the Sérsic index n. |
It should be noted that astronomers don't actually know where the edges of elliptical galaxies occur; their light-profiles appear to peeter-out into the background noise of the sky. Due to the presence of faint, high-redshift galaxies and scattered light, it is not possible to determine the sky-background to an infinite degree of accuracy. From an analysis of such sky-background noise sources, Dalcanton & Bernstein (2000, see also Capaccioli & de Vaucouleurs 1983) determined the limiting surface brightness threshold to be ~ 29.5 B-mag arcsec-2 and µ ~ 29 R-mag arcsec-2. Such depths are practically never achieved and shallow exposures often fail to include a significant fraction of an elliptical galaxy's light. One would therefore like to know how the concentration index may vary when different galaxy radial extents are sampled but no effort is made to account for the missed galaxy flux. The resultant impact on CRe and other popular concentration indices is addressed in Graham, Trujillo, & Caon (2001).
It was actually, in part, because of the unstable nature of the
popular concentration indices that
Trujillo et al. (2001c)
introduced the notably more stable index given in Equations (20) and (21).
The other reason was because the concentration index
C() =
i,j
E(
)
Iij /
i,j
E(1)
Iij,
where E(
)
denotes some inner radius which is
(0 <
< 1)
times the outermost radius which has been normalized to 1
(Okamura, Kodaira, &
Watanabe 1984;
Doi, Fukugita, &
Okamura 1993;
Abraham et al. 1994),
should tend to 1 for practically every profile that is sampled to
a large enough radius. It is only because of measurement errors,
undersampling, or the presence of truncated profiles such as the
exponential disks in spiral galaxies, that this index deviates from a
value of 1.