Trans-Neptunian objects (TNO) represent the leftovers of the formation of the Solar System. Their physical properties provide constraints to the models of formation and evolution of the various dynamical classes of objects in the outer Solar System. Based on a sample of 19 classical TNOs we determine radiometric sizes, geometric albedos and beaming parameters. Our sample is composed of both dynamically hot and cold classicals. We study the correlations of diameter and albedo of these two subsamples with each other and with orbital parameters, spectral slopes and colors. We have done three-band photometric observations with Herschel/PACS and we use a consistent method for data reduction and aperture photometry of this sample to obtain monochromatic flux densities at 70.0, 100.0 and 160.0 mu m. Additionally, we use Spitzer/MIPS flux densities at 23.68 and 71.42 mu m when available, and we present new Spitzer flux densities of eight targets. We derive diameters and albedos with the near-Earth asteroid thermal model (NEATM). As auxiliary data we use reexamined absolute visual magnitudes from the literature and data bases, part of which have been obtained by ground based programs in support of our Herschel key program. We have determined for the first time radiometric sizes and albedos of eight classical TNOs, and refined previous size and albedo estimates or limits of 11 other classicals. The new size estimates of 2002 MS4 and 120347 Salacia indicate that they are among the 10 largest TNOs known. Our new results confirm the recent findings that there are very diverse albedos among the classical TNOs and that cold classicals possess a high average albedo (0.17 +/- 0.04). Diameters of classical TNOs strongly correlate with orbital inclination in our sample. We also determine the bulk densities of six binary TNOs.
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