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Orbital periods and Accretion disc structure of four AM CVn systems

Phase-resolved spectroscopy of four AM CVn systems obtained with the William Herschel Telescope and the Gran Telescopio de Canarias (GTC) is presented. SDSS\,J120841.96+355025.2 was found to have an orbital period of 52.96$\pm$0.40\,min and shows the presence of a second bright spot in the accretion disc. The average spectrum contains strong Mg\,{\sc i} and Si\,{\sc i/ii} absorption lines most likely originating in the atmosphere of the accreting white dwarf. SDSS\,J012940.05+384210.4 has an orbital period of 37.555$\pm$0.003 min. The average spectrum shows the Stark broadened absorption lines of the DB white dwarf accretor. The orbital period is close to the previously reported superhump period of 37.9\,min. Combined, this results in a period excess $ε$=0.0092$\pm$0.0054 and a mass ratio $q=0.031\pm$0.018. SDSS\,J164228.06+193410.0 displays an orbital period of 54.20$\pm$1.60\,min with an alias at 56.35\,min. The average spectrum also shows strong Mg\,{\sc i} absorption lines, similar to SDSS\,J120841.96+355025.2. SDSS\,J152509.57+360054.50 displays an period of 44.32$\pm$0.18\,min. The overall shape of the average spectrum is more indicative of shorter period systems in the 20-35 minute range. The accretor is still clearly visible in the pressure broadened absorption lines most likely indicating a hot donor star and/or a high mass accretor. Flux ratios for several helium lines were extracted from the Doppler tomograms for the disc and bright spot region, and compared with single-slab LTE models with variable electron densities and path lengths to estimate the disc and bright spot temperature. A good agreement between data and the model in three out of four systems was found for the disc region. All three systems show similar disc temperatures of $\sim$10\,500 K. In contrast, only weak agreement between observation and models was found for the bright spot region.