The period-luminosity and period-radius relations of Type II and anomalous Cepheids

Method: In an accompanying paper (arXiv: 1705.00886) we determined luminosity and effective temperature for the 335 T2Cs and ACs in the LMC and SMC discovered in the OGLE-III survey, by constructing the spectral energy distribution (SED) and fitting this with model atmospheres and a dust radiative transfer model (in the case of dust excess). Building on these results we study the PL and PR relations. Using existing pulsation models for RR Lyrae and classical Cepheids we derive the period-luminosity-mass-temperature-metallicity relations, and then estimate the pulsation mass. Results: The PL relation for the T2Cs does not appear to depend on metallicity, and, excluding the dusty RV Tau stars, is $M_{\rm bol}= +0.12 -1.78 \log P$ (for $P < 50$ days). Relations for fundamental and first overtone LMC ACs are also presented. The PR relation for T2C also shows little or no dependence on metallicity or period. Our preferred relation combines SMC and LMC stars and all T2C subclasses, and is $\log R = 0.846 + 0.521 \log P$. Relations for fundamental and first overtone LMC ACs are also presented. The pulsation masses from the RR Lyrae and classical Cepheid pulsation models agree well for the short period T2Cs, the BL Her subtype, and ACs, and are consistent with estimates in the literature, i.e. $M_{\rm BLH} \sim 0.49$ \msol\ and $M_{\rm AC} ~\sim 1.3$ \msol, respectively. The masses of the W Vir appear similar to the BL Her. The situation for the pWVir and RV Tau stars is less clear. For many RV Tau the masses are in conflict with the standard picture of (single-star) post-AGB evolution, the masses being either too large ($\gtrsim$ 1 \msol) or too small ($\lesssim$ 0.4 \msol).