Issue 
A&A
Volume 524, December 2010



Article Number  A6  
Number of page(s)  33  
Section  Extragalactic astronomy  
DOI  https://doi.org/10.1051/00046361/201014703  
Published online  19 November 2010 
Online material
6. table
The FP parameters of cluster galaxies.
The FP parameters of field galaxies.
The structural parameters of galaxies with measured velocity dispersions and HST photometry derived from Sersic fits to HST images and bulge+disk fits to VLT images.
Appendix A: Circularized halfluminosity radii
GIM2D delivers bulge a_{e} and disk a_{h} scale lengths along the major axis, bulge apparent flattening (b/a)_{B} and disk inclination angles i (corresponding to an apparent flattening (b/a)_{D} = 1 − cosi), and bulgetototal ratios B/T. When fitting Sersic profiles, GIM2D delivers the n Sersic index, the major axis , and the flattening (b/a)^{Ser}. We compute the circularized halfluminosity radius R_{e} of the resulting galaxy model as follows. We determine the flux inside a circular aperture of radius R (the socalled curve of growth) of a model of apparent flattening b/a and surface density distribution constant on ellipses as (A.1)Using , we derive (A.2)We perform the angular integration numerically, using , where z = 7.67(r/R_{eB})^{1/4}, and , and x = R/h for the normalized de Vaucouleurs and exponential density laws, respectively. For a Sersic profile of given n, we use , where P is the incomplete Γ function and X = k(r/R_{eSer})^{1/n} and k = 1.9992n − 0.3271 (Simard et al. 2002). We determine R_{e} by solving the equation (A.3)for the bulgeplusdisk models, and (A.4)for the Sersic fits numerically. In general, the resulting R_{e} agree within 1% with the halfluminosity radii derived by measuring the curves of growth directly from (ACS HST like) images generated by GIM2D with the fit parameters and no PSF convolution, but the imagebased method overestimates R_{e} by up to 10% when it is smaller than 4 pixels (0.2 arcsec).
Figure A.1 illustrates that a more accurate approximation of the circularized radius R_{e}(Ser) of Sersic profiles, more accurate than 2%, is obtained by taking the simple mean R_{ave} = (a_{e} + b_{e})/2 of the major and minor axis scale lengths a_{e} and b_{e} instead of the harmonic mean . This is surprising only at a first sight, since R_{har} goes to zero as the flattening increases, while R_{ave} does not. Therefore R_{ave} should provide a closer approximation of the halfluminosity radius derived from circular curves of growth at high ellipticities. On the other hand, the effective surface brightness within the ellipse of semimajor and minor axis a_{e} and b_{e} is constant regardless of the flattening, while this is not true for the surface brightness within the circle of radius R_{e}(Ser). Since in this exercise the total luminosity L is kept constant, we have , with and . This is almost orthogonal to the FP (see Eq. (1)), making the choice of method unimportant, as long as not too many disks seen edgeone (i.e. of very high flattening) are present in the sample (see Fig. 5 and discussion in Sect. 2.2).
Fig. A.1
The circularized halfluminosity radius R_{e}(Ser) of the sample of EDisCS galaxies with HST photometry and velocity dispersions computed according to Eqs. (A.2) and (A.4) compared to the simple mean R_{ave} = 0.5(a_{e} + b_{e}) (top) and harmonic mean (bottom) as a function of the ellipticity 1 − b_{e}/a_{e}. The simple mean approximates more accurately R_{e}(Ser). 

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