2.6. Kron magnitudes

Kron (1980) presented the following luminosity-weighted radius, R₁, which defines the `first moment' of an image

(31)

He argued that an aperture of radius twice R₁, when R₁ is obtained by integrating to a radius R that is 1% of the sky flux, contains more than ~ 90% of an object's total light, making it a useful tool for estimating an object's flux.

It is worth noting that considerable confusion exists in the literature in regard to the definition of R₁. To help avoid ambiguity, we point out that g(x) in Kron's (1980) original equation refers to xI(x), where x is the radius and I(x) the intensity profile. Infante (1987) followed this notation, but confusingly a typo appears immediately after his Equation 3 where he has written g(x) ~ I(x) instead of g(x) ~ xI(x). Furthermore, Equation 3 of Bertin & Arnouts (1996) is given as R₁ = RI(R) / I(R) where the summation is over a two-dimensional aperture rather than a one-dimensional light-profile. In the latter case, one would have R₁ = R² I(R) / RI(R).

Using a Sérsic intensity profile, and substituting in x = b(R / R_e)^1/n, the numerator can be written as

Using Equation (2) for the denominator, which is simply the enclosed luminosity, Equation (31) simplifies to

(32)

The use of `Kron radii' to determine `Kron magnitudes' has proved very popular, and SExtractor (Bertin & Arnouts 1996) obtains its magnitudes using apertures that are 2.5 times R₁. Recently, however, it has been reported that such an approach may, in some instances, be missing up to half of a galaxy's light (Bernstein, Freedman, & Madore 2002; Benitez et al. 2004). If the total light is understood to be that coming from the integration to infinity of a light-profile, then what is important is not the sky-level or isophotal-level one reaches, but the number of effective radii that have been sampled.

For a range of light-profiles shapes n, Figure (8) shows the value of R₁ (in units of R_e) as a function of the number of effective radii to which Equation (31) has been integrated. Given that one usually only measures a light-profile out to 3-4 R_e at best, one can see that only for light-profiles with n less than about 1 will one come close to the asymptotic value of R₁ (i.e, the value obtained if the profile was integrated to infinity). Table (1) shows these asymptotic values of R₁ as a function of n, and the magnitude enclosed within 2R₁ and 2.5R₁. This is, however, largely academic because observationally derived values of R₁ will be smaller than those given in Table (1), at least for light-profiles with values of n greater than about 1.

Figure 8. Kron radii R1, as obtained from Equation (32), are shown as a function of the radius R to which the integration was performed. Values of n range from 0.5, 1, 2, 3,... 10.

From Figure (8) one can see, for example, that an R^1/4 profile integrated to 4R_e will have R₁ = 1.09R_e rather than the asymptotic value of 2.29R_e. Now 2.5 × 1.09R_e encloses 76.6% of the object's light rather than 90.4% (see Table 1). This is illustrated in Figure (9) where one can see when and how Kron magnitudes fail to represent the total light of an object. This short-coming is worse when dealing with shallow images and with highly concentrated systems having large values of n (brightest cluster galaxies are known to have Sérsic indices around 10 or greater, Graham et al. 1996).

Figure 9. Kron luminosity within 2.5R1, normalised against the total luminosity, as a function of how many effective radii R1 corresponds to. Values of n range from 0.5, 1, 2, 3,... 10.

To provide a better idea of the flux fraction represented by Kron magnitudes, and one which is more comparable with Figure (7), Figure (10) shows this fraction as a function of light-profile shape n. The different curves result from integrating Equation (31) to different numbers of effective radii in order to obtain R1. If n=4, for example, but one only integrates out to 1R_e (where R_e is again understood to be the true, intrinsic value rather than the observed value), then the value of R₁ is 0.41R_e and the enclosed flux within 2.5R₁ is only 50.7%. If an n = 10 profile is integrated to only 1R_e, then R₁ = 0.30R_e and the enclosed flux is only 45.0% within 2.5R₁. It is therefore easy to understand why people have reported Kron magnitudes as having missed half of an object's light.

Figure 10. Kron luminosity within 2.5R1, normalised against the total luminosity, as a function of the underlying light-profile shape n. The different curves arise from the different values of R1 obtained by integrating Equation (31) to (i) 1Re, (ii) 2Re, (iii) 4Re, and (iv) infinity.