Constraining the interior of extrasolar giant planets with the tidal Love number k2 using the example of HAT-P-13b

Context. Transit and radial velocity observations continuously discover an increasing number of exoplanets. However, when it comes to the composition of the observed planets the data are compatible with several interior structure models. Thus, a planetary parameter sensitive to the planet's density distribution could help constrain this large number of possible models even further. Aims. We aim to investigate to what extent an exoplanet's interior can be constrained in terms of core mass and envelope metallicity by taking the tidal Love number k(2) into account as an additional, possibly observable parameter. Methods. Because it is the only planet with an observationally determined k(2), we constructed interior models for the Hot Jupiter exoplanet HAT-P-13b by solving the equations of hydrostatic equilibrium and mass conservation for different boundary conditions. In particular, we varied the surface temperature and the outer temperature profile, as well as the envelope metallicity within the widest possible parameter range. We also considered atmospheric conditions that are consistent with nongray atmosphere models. For all these models we calculated the Love number k(2) and compared it to the allowed range of k(2) values that could be obtained from eccentricity measurements of HAT-P-13b. Results. We use the example of HAT-P-13b to show the general relationships between the quantities temperature, envelope metallicity, core mass, and Love number of a planet. For any given k(2) value a maximum possible core mass can be determined. For HAT-P-13b we find M-core < 27 M-circle plus, based on the latest eccentricity measurement. We favor models that are consistent with our model atmosphere, which gives us the temperature of the isothermal region as similar to 2100 K. With this external boundary condition and our new k(2)-interval we are able to constrain both the envelope and bulk metallicity of HAT-P-13b to 1-11 times stellar metallicity and the extension of the isothermal layer in the planet's atmosphere to 3-44 bar. Assuming equilibrium tidal theory, we find lower limits on the tidal Q consistent with 10(3)-10(5). Conclusions. Our analysis shows that the tidal Love number k(2) is a very useful parameter for studying the interior of exoplanets. It allows one to place limits on the core mass and estimate the metallicity of a planet's envelope.