V1135 Herculis: a double-lined eclipsing binary with an Anomalous Cepheid
BVR light curves and radial velocities for the double-lined eclipsing binary V1135\,Her were obtained. The brighter component of V1135\,Her is a Cepheid variable with a pulsation period of 4.22433$\pm$0.00026 days. The orbital period of the system is about 39.99782$\pm$0.00233 days, which is the shortest value among the known Type\,II Cepheid binaries. The observed B, V, and R magnitudes were cleaned for the intrinsic variations of the primary star. The remaining light curves, consisting of eclipses and proximity effects, are obtained. Our analyses of the multi-colour light curves and radial velocities led to the determination of fundamental stellar properties of both components of the interesting system V1135\,Her. The system consists of two evolved stars, G1+K3 between giants and supergiants, with masses of M$_1$=1.461$\pm$0.054 \Msun ~and M$_2$=0.504$\pm$0.040 {\Msun} and radii of R$_1$=27.1$\pm$0.4 {\Rsun} and R$_2$=10.4$\pm$0.2 {\Rsun}. The pulsating star is almost filling its corresponding Roche lobe which indicates the possibility of mass loss or transfer having taken place. We find an average distance of d=7500$\pm$450 pc using the BVR magnitudes and also the V-band extinction. Location in the Galaxy and the distance to the galactic plane with an amount of 1300 pc indicate that it probably belongs to the thick-disk population. Most of the observed and calculated parameters of the V1135\,Her and its location on the color-magnitude and period-luminosity diagrams lead to a classification of an Anomalous Cepheid.