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6. table

Appendix A: Circularized half-luminosity 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 bulge-to-total 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 half-luminosity radius R_e of the resulting galaxy model as follows. We determine the flux inside a circular aperture of radius R (the so-called 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 bulge-plus-disk models, and (A.4)for the Sersic fits numerically. In general, the resulting R_e agree within 1% with the half-luminosity 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 image-based 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 half-luminosity radius derived from circular curves of growth at high ellipticities. On the other hand, the effective surface brightness within the ellipse of semi-major 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 edge-one (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 half-luminosity radius Re(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 Rave = 0.5(ae + be) (top) and harmonic mean (bottom) as a function of the ellipticity 1 − be/ae. The simple mean approximates more accurately Re(Ser).
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