Relation between the moment of inertia and the k₂ Love number of fluid extra-solar planets

¹ Dipartimento di Fisica e Astronomia “Augusto Righi” (DIFA), Alma Mater Studiorum Università di Bologna, Viale Berti Pichat 8, 40127 Bologna, Italy
e-mail: anastasia.consorzi2@unibo.it
² Istituto Nazionale di Geofísica e Vulcanologia, Via di Vigna Murata 605, 00143 Roma, Italy

Received: 8 March 2023
Accepted: 6 June 2023

Abstract

Context. Tidal and rotational deformation of fluid giant extra-solar planets may impact their transit light curves, making the k₂ Love number observable in the near future. Studying the sensitivity of k₂ to mass concentration at depth is thus expected to provide new constraints on the internal structure of gaseous extra-solar planets.

Aims. We investigate the link between the mean polar moment of inertia N of a fluid, stably layered extra-solar planet and its k₂ Love number. Our aim is to obtain analytical relations valid, at least, for some particular ranges of the model parameters. We also seek a general approximate relation useful for constraining N once observations of k₂ become available.

Methods. For two-layer fluid extra-solar planets we explore the relation between N and k₂ via analytical methods, for particular values of the model parameters. We also explore approximate relations valid over the entire range of two-layer models. More complex planetary structures are investigated by the semi-analytical propagator technique.

Results. A unique relation between N and k₂ cannot be established. However, our numerical experiments show that a rule of thumb can be inferred that is valid for complex, randomly layered stable planetary structures. The rule robustly defines the upper limit to the values of N for a given k₂, and agrees with analytical results for a polytrope of index one and with a realistic non-rotating model of the tidal equilibrium of Jupiter.