Matrix-propagator approach to compute fluid Love numbers and applicability to extrasolar planets

Abstract

Context. The mass and radius of a planet directly provide its bulk density, which can be interpreted in terms of its overall composition. Any measure of the radial mass distribution provides a first step in constraining the interior structure. The fluid Love number k 2 provides such a measure, and estimates of k 2 for extrasolar planets are expected to be available in the coming years thanks to improved observational facilities and the ever-extending temporal baseline of extrasolar planet observations. Aims. We derive a method for calculating the Love numbers k n of any object given its density profile, which is routinely calculated from interior structure codes. Methods. We used the matrix-propagator technique, a method frequently used in the geophysical community. Results. We detail the calculation and apply it to the case of GJ 436b, a classical example of the degeneracy of mass-radius relationships, to illustrate how measurements of k 2 can improve our understanding of the interior structure of extrasolar planets. We implemented the method in a code that is fast, freely available, and easy to combine with preexisting interior structure codes. While the linear approach presented here for the calculation of the Love numbers cannot treat the presence of nonlinear effects that may arise under certain dynamical conditions, it is applicable to close-in gaseous extrasolar planets like hot Jupiters, likely the first targets for which k 2 will be measured.

Liquid Iron Equation of State to the Terapascal Regime From Ab Initio Simulations

“…Beyond the mass‐radius relations in use today, moment of inertia ( I ) and fluid Love number ( k 2 ) will not be affected by differences of the magnitude suggested here (Padovan et al, ). The robust behavior of the iron EOS offers the opportunity to reliably invert observational data of I and k 2 , which are within reach of astronomical observations (Batygin et al, ; Biersteker & Schlichting, ; Csizmadia et al, ), for first‐order planetary structure such as R core / R planet (Padovan et al, ).…”

Retrieval of the Fluid Love Number k2 in Exoplanetary Transit Curves

“…We considered a planetary Love number k 2 = 0.5 based on recent models (Wahl et al 2016), and forced k 3 = k 4 = 0. Indeed, Padovan et al (2018) showed that k 2 has the strongest dependence on central mass concentration among the three. We highlight that only synthetic light curves were created to prove the feasibility of our method, not real data, thus our results should not be physically interpreted in terms of, e.g., internal structure.…”

“…In particular, the second-degree Love number, k 2 , is an indication of mass concentration towards the body's center, providing additional information about the interior (see, e.g., Kellermann et al 2018). A value of k 2 = 0 indicates a mass-point approximation (a.k.a., Roche model), k 2 = 1.5 corresponds to a fully homogeneous body, and its full derivation depends on the internal radial den-arXiv:1905.03171v1 [astro-ph.EP] 8 May 2019 sity profile, (see, e.g., Padovan et al 2018). As a result of tidal and rotational deformations, the stellar eclipsed area during transit will differ from a transiting sphere, modifying the corresponding transit light curve ).…”

Machine-learning Inference of the Interior Structure of Low-mass Exoplanets

“…They depend on the timescale of the perturbation and on the rheological properties of the interior (i.e., radial density profile, elastic properties, viscosity). For example, the rotational figure of the Earth is relaxed, i.e., it attained hydrostatic equilibrium and accordingly, its shape can be parameterized using fluid Love numbers (Padovan et al 2018). At the same time, on the shorter tidal lunar and solar timescales, the response is described by tidal Love numbers (e.g., Petit & Luzum 2010).…”