New insights into the structure of early-type galaxies: the Photometric Plane at z~0.3

21 pages, 10 figures, MNRAS in pressWe study the Photometric Plane (PHP), namely the relation between the effective radius re, the mean surface brightness within that radius e, and the Sersic index n, in optical (R and I) and near-infrared (K) bands for a large sample of early-type galaxies (ETGs) in the rich cluster MS1008-1224 at z=0.306. The PHP relation has an intrinsic dispersion of ~32% in re, and turns out to be independent of waveband. This result is consistent with the fact that internal colour gradients of ETGs can have only a mild dependence on galaxy luminosity (mass). There is no evidence for a significant curvature in the PHP. We show that this can be explained if this relation origins from a systematic variation of the specific entropy of ETGs along the galaxy sequence, as was suggested from previous works. The intrinsic scatter of the PHP is significantly smaller than for other purely photometric relations, such as the Kormendy relation and the photometric Fundamental Plane, which is constructed by using colours in place of velocity dispersions. The scatter does not depend on the waveband and the residuals about the plane do not correlate with residuals of the colour-magnitude relation. Finally, we compare the coefficients of the PHP at z~0.3 with those of ETGs at z~0, showing that the PHP is a valuable tool to constrain the luminosity evolution of ETGs with redshift. The slopes of the PHP do not change significantly with redshift, while the zero-point is consistent with cosmological dimming of the surface brightness in an expanding universe plus the passive fading of galaxy stellar populations with a high formation redshift (z_f >1-2)