New Cepheid variables in the young open clusters Berkeley 51 and Berkeley 55

As part of a wider investigation of evolved massive stars in Galactic open clusters, we have spectroscopically identified three candidate classical Cepheids in the little-studied clusters Berkeley 51, Berkeley 55 and NGC 6603. Using new multi-epoch photometry, we confirm that Be 51 #162 and Be 55 #107 are bona fide Cepheids, with pulsation periods of 9.83+/-0.01 d and 5.850+/-0.005 d respectively, while NGC 6603 star W2249 does not show significant photometric variability. Using the period-luminosity relationship for Cepheid variables, we determine a distance to Be 51 of 5.3(+1.0,-0.8) kpc and an age of 44(+9,-8) Myr, placing it in a sparsely-attested region of the Perseus arm. For Be 55, we find a distance of 2.2+/-0.3 kpc and age of 63(+12,-11) Myr, locating the cluster in the Local arm. Taken together with our recent discovery of a long-period Cepheid in the starburst cluster VdBH222, these represent an important increase in the number of young, massive Cepheids known in Galactic open clusters. We also consider new Gaia (data release 2) parallaxes and proper motions for members of Be 51 and Be 55; the uncertainties on the parallaxes do not allow us to refine our distance estimates to these clusters, but the well-constrained proper motion measurements furnish further confirmation of cluster membership. However, future final Gaia parallaxes for such objects should provide valuable independent distance measurements, improving the calibration of the period-luminosity relationship, with implications for the distance ladder out to cosmological scales.