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Appendix A: Optical light curves

	Fig. A.1 Photometric periodogram, and photometric light curves for SDSS0853+0720, SDSS 2243+3122, SDSS 1611+4640 and SDSS 2112+1014 phase folded with the orbital period corresponding to the highest peak in the periodogram (bottom panels), Porb = 0.1503 d, Porb = 0.11954 d, Porb = 0.0823 d and Porb = 0.0923 d respectively. Double-sine fits are shown in red.
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Photometric follow-up observations were carried out for a total of seven WDMS binaries where the available spectroscopy suggested a short orbital period. Telescope, filter, and duration of the observations are listed in Table 1. Four out of the seven observed systems showed ellipsoidal modulation: SDSS 0853+0720, SDSS 1611+4640, SDSS 2112+1014, and SDSS 2243+3122. We determined their orbital periods by computing periodograms and fitting sine curves to the phase folded light curves (see Fig. A.1). We estimated P_orb = 3.6,1.9,2.2, and 2.8 h respectively. Spectroscopic follow-up observations were carried out for SDSS 0853+0720, SDSS 1611+4640, and SDSS 2243+3122 covering a full cycle, and confirming the measured periods. We could connect the spectroscopic and photometric runs for SDSS 1611+4640 and SDSS 2243+3122, without cycle count aliases, which allowed us to refine their orbital periods.

The light curves of SDSS 0853+0720 and SDSS 2243+3122 both show two uneven maxima at phases 0.25 and 0.75. For SDSS 0853+0720 the first maximum is brighter than the second one, while for SDSS 2243+3122 is the other way round. In both systems equal minima at phases 0 and 0.5 are observed, and the maximum variation is almost 0.2 mag in the r and the i bands, respectively. Ellipsoidal modulation and spots in one of the hemispheres of the secondary star could explain the uneven maxima (O’Connell effect, see Liu & Yang 2003). Photometry carried out on 15 August 2009 for SDSS 2243+3122 revealed a flare with  ~25 min length in the decay, and rising by 0.35 mag in the i band (see Fig. A.2). This would strengthen the argument of magnetic activity being the cause of the observed disparity in the maxima (see Fig. A.1).

	Fig. A.2 Differential photometry in the i band obtained for SDSS 2243+2122 on 15 August 2009. A flare of  ~25 min length with a relative enhancement of  ~0.4 mag was observed.
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Although no radial velocity curve is available for SDSS 2112+1014, the five available spectra show radial velocity variations of more than 300 km   s^-1 in less than 40 min, indicating a rather short orbital period, consistent with the measured photometric orbital period of 2.2 h.

Appendix B: Notes on individual systems

SDSS 0225+0054, the primary star was classified as a DC by Kleinman et al. (2004). Although it is true that no absorption lines are visible in the primary star we consider that the signal-to-noise of the residual white dwarf after subtracting of the best-fitted secondary template is insufficient for any other classification than white dwarf (see Rebassa-Mansergas et al. 2007, for details on the spectral analysis). SDSS 0225+0054 and SDSS 0853+0720 are WDMS binaries consisting of cold DC white dwarfs as primary stars and an M4.5 and M3 companion star, repectively. These are clear examples of the old PCEBs predicted by Schreiber & Gänsicke (2003) that would not have been found in any search for blue objects.

SDSS 0307+3848 (SDSS J030716.44+384822.8) has a red star nearby – within 5′′ – (SDSS J030716.16+384821.4) and is slightly brighter in the r band (16.70 versus 15.17). To be sure that there was no contamination in our spectra by this source we rotated the slit so as to take spectra simultaneously and compared the radial velocities with those of this system. No variations in the radial velocity of this visual companion were observed. The optical spectrum of SDSS 0307+3848 is dominated by the secondary star, and it presents strong emission lines. We measured the radial velocities from the Hα  line in the SDSS-subspectra. The Na   I  and the Hα  lines are in phase, and the amplitudes were consistent with each other within the errors.

SDSS 1006+0044, SDSS 1523+4604, and SDSS 1844+4120 have very cold WDs, T_eff = 7800,8400,7600 K, late secondary star spectral types M9, M7, M6, and are among the closest PCEBs in the sample, d_wd = 60,76,73 pc respectively. SDSS 1844+4120 is the system with the highest systemic velocity, and it has Ca II H&K in absorption; i.e., the white dwarf is a DAZ. This photospheric absorption could be produced by some wind from the secondary star that is being accreted by the WD (see e.g. LTT560 Tappert et al. 2007). Looking at the SDSS images of SDSS 1523+4604, aka CSO749 (Sanduleak & Pesch 1989), the components of this binary seem to be slightly separated, also noted by Heller et al. (2009), and there seems to be a third red object towards the north-east. We measured an orbital period of 9.93 h, so it could be that SDSS 1523+4604 is a triple system formed by a close binary and a wide companion.

SDSS 1143+0009, aka WD1140+004, is an X-ray emitting source (Agüeros et al. 2009) from the Rosat All Sky Survey (Voges et al. 1999). The X-ray to optical flux ratio (log f_X/f_g) is 0.22, a value too high to be explained by the coronal emission of the dMe star (see Maccacaro et al. 1988; Motch et al. 1998, for typical values), and the SDSS spectrum presents some marginal Hα  emission (although this can be partially masked by the flux of the white dwarf). The high flux ratio could be explained by some wind from the secondary star being accreted by the white dwarf. If so, some Ca II H&K absorption would then be expected, which is not visible in the spectrum. A small fraction of isolated white dwarfs show hard (higher than 0.5 keV) X-ray emission (O’Dwyer et al. 2003; Chu et al. 2004; Bilíková et al. 2010); nevertheless, the effective temperature of the white dwarf in SDSS 1143+0009 (~17 000 K) is too cold to explain its hard X-ray emission. Briggs et al. (2007) list six other PCEBs X-ray emitters: BPM 71214, RR Cae, UZ Sex, EG UMa, Feige 24, and WD 1541-381 . We investigated the X-ray fluxes of these sources within the XMM-Newton, making use of xcat-DB⁸ (Michel et al. 2004). None of them show an X-ray spectrum as hard as that of SDSS 1143+0009. It has a counterpart the Galaxy Evolution Explorer (GALEX) database (Martin et al. 2005), with detection only in the FUV flux, 17.97 ± 0.08 mag, consistent with the effective temperature of the white dwarf. We conclude that there is no straightforward explanation of the hard X-ray emission of SDSS 1143+0009.

SDSS 1352+0910 contains a hot WD, T_eff = 36   200 K and no companion visible in the SDSS spectrum. Nevertheless, emission in the Hα  line was observed in the SDSS spectrum, pointing to a binary nature for the system, and triggering further spectroscopy, which revealed it to be a close binary. We investigated infrared surveys to learn more about the secondary star. We found a counterpart in the UKIDSS catalogue, ULAS J135228.14+091039.0. For details on the UKIDSS project, camera, photometric system, calibration, pipeline processing, and science archive see Lawrence et al. (2007); Casali et al. (2007); Hewett et al. (2006); Hodgkin et al. (2009) and Hambly et al. (2008). Magnitudes found in the seventh data release are Y = 18.24 ± 0.03, J = 18.04 ± 0.03, H =  + 17.68 ± 0.04, K = 17.49 ± 0.06 mag, yielding infrared colours J − H = 0.36 and H − K = 0.19 mag. We subtracted the infrared contribution of the white dwarf and, using the empirical luminosity-mass relation

from Delfosse et al. (2000), we estimated a mass of 0.175 M_⊙ for the secondary star. With the spectral type mass radius relation from Rebassa-Mansergas et al. (2007) this is consistent with a spectral type of M6.5.

SDSS 1434+5335 (SDSS J143443.24+533521.2) has a background or a nearby red object (SDSS J143443.19+533525.0), classified as a galaxy by the SDSS. The background galaxy is too faint, and our object is bright enough that the integration time was relatively short and a possible contamination of our spectra is negligible.

SDSS 1436+5741 and SDSS 2156–0002 present a slight blue excess in the SDSS spectra with respect to single dM stars. Spectroscopic observations carried out for these two systems (see Table 1) showed significant radial velocity variation, confirming their binary nature. We explored GALEX with the aim of learning about the primary stars and found that SDSS 2156–0002 has GALEX J215614.5-000237 as a counterpart. Only flux in the NUV was detected, with a magnitude of 21.99    ±    0.34, consistent with a cold white dwarf.

SDSS 1559+0356 has a hot white dwarf, T_eff = 48   212 K, and no companion is visible in the SDSS spectrum. It has a strong Hα  emission line from which we were able to measure the radial velocities. SDSS 1559+0356 was listed as a CV candidate by Szkody et al. (2007), but there is no obvious sign of mass transfer in the spectrum. With its orbital period of 2.2 h, this system is located in the CV orbital period gap. If it were a detached CV, one would expect a cooler white dwarf (Townsley & Gänsicke 2009). We investigated infrared surveys to have more information for the secondary star, but no counterpart was found for the object.

SDSS 1611+4640 has the shortest period among the PCEBs, with P_orb = 1.97 h, and it has a cold white dwarf, T_eff ~ 10   300 K and a late secondary star spectral type, M5.

SDSS 1731+6233 has a red nearby star, and care was taken not to contaminate our spectra with any contribution from it. Unbeknownst to us at the start of our spectroscopic campaign, this system had been previously classified a probable binary by Pourbaix et al. (2005) on the basis of radial velocity variations of  ~ 40 km   s^-1.

SDSS 2149–0717 has SDSS photometry, but it was not an SDSS spectroscopic target. It was identified as a WDMS by Southworth et al. (2007) during some pilot follow-up studies of WDMS candidates in the SDSS footprint at the UK Schmidt telescope with the 6dF spectrograph. The spectral type of the secondary star is less certain than for SDSS spectra since the 6dF only covers up to 7500 Å.

SDSS 2208+1221 contains the hottest white dwarf in our sample, T_eff ~ 100   000 K and the earliest spectral type secondary star, a K7 star (Silvestri et al. 2006).

SDSS 2243+3122 has an M5 ± 1 so that it could be a detached CV (dCV) or hibernating CV, which has relaxed to its equilibrium radius. The SDSS spectrum presents the Balmer lines in emission. Radial velocities from the Hα  emission line showed no difference in phasing from the Na   I  absorption doublet; i.e. both sets of lines must originate in the secondary, which argues against a CV nature for the system.

	Fig. 3 SDSS spectra of the 58 systems with orbital period measured in this paper. Systems are sorted in right ascension from left to right and from top to bottom.
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	Fig. 4 Phase-folded radial velocities curves and sine fits to the data (dashed lines). As for Fig. 3, systems are sorted in right ascension from left to right and from top to bottom.
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	Fig. 5 SDSS spectra of the 58 systems with orbital period measured in this paper. Systems are sorted in right ascension from left to right and from top to bottom.
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	Fig. 6 Phase-folded radial velocities curves and sine fits to the data (dashed lines). As for Fig. 3, systems are sorted in right ascension from left to right and from top to bottom.
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© ESO, 2011