Direct observations provide limited information about the deep internal structures and basic properties of gaseous planets, even in our own Solar system. However, more can be learned from how planets interact with their surroundings. I will introduce research focused on interpreting the satellite Cassini's observations of Saturn's gravitational interactions with its rings and satellite moons. Observing these interactions yields information about Saturn's internal oscillation modes and tidally excited waves, the successful inversion of which may provide our best hope for constraining the planet's deep internal structure and rotation state.

In this talk, I will review the different kernel-based inversion techniques that have been used in asteroseismology. In particular, I will describe regularised least-squares (RLS) as well as optimally localised averages (OLA) type inversions. These have been applied to rotation and structural profiles as well as to integrated quantities such as the total kinetic rotation energy, the mean density, the acoustic radius, and various evolutionary phase and convective region indicators. I will also briefly show how inverse techniques can lead to more subtle constraints such as inequalities on rotational splittings if one makes certain assumptions on the rotation profile.

The Juno and Cassini spacecrafts have measured the gravity fields of Jupiter and Saturn with exquisite precision. The gravity field can then be projected onto the Legendre polynomials to obtain the gravitational moments, signatures of the density distribution in the planet as a function of radius and angle. In the past few years, a lot of effort has thus been dedicated to create precise methods to calculate gravitational moments from synthetic models and optimise the retrieval of internal structure by comparing with Juno and Cassini data.

In this talk, I will detail how we tackled this inverse problem in the case of Jupiter and Saturn. I will quickly introduce the concentric Maclaurin spheroid method used to calculate gravitational moments, before detailing the recovered internal structures. I will expose the dominant influence of winds on the gravity field and how planetary oscillations can constrain further the recovered density profiles. These results have strong implication for the formation and evolution of the giant planets and solar system in general.

The understanding of the internal stellar rotation and its evolution are important ingredients for the construction of accurate stellar models. We use asteroseismology, the study of global stellar oscillations, to probe the interior rotation of stars, particularly that of red giants. Large systematic errors previously hindered the accurate determination of near-surface rotation rates in evolved red giants e.g. [1]. We have developed a method of effectively eliminating these systematic errors by introducing an extension to a currently used rotational inversion method for red-giant stars [2].

We demonstrate the ability of the new inversion technique to compute accurate envelope rotation rates of stars along the red giant branch (RGB). Furthermore, we show the resulting improvement of our new method compared to other seismic inversion methods. Subsequently, we aim at quantifying systematic uncertainties in asteroseismic rotational inversions occurring due to inaccurate stellar modelling (Ahlborn et al. in prep). More accurate surface rotation rates for evolved red giants will be an important probe to understand the loss of angular momentum in red-giant cores, and an important milestone to improve the theory of rotation in stellar models.

[1] F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and G. C. Angelou. Asteroseismic sensitivity to internal rotation along the red-giant branch, Astronomy and Astrophysics, 639:A98, 2020. https://doi.org/10.1051/0004-6361/201936947

[2] F. Ahlborn, E. P. Bellinger, S. Hekker, S. Basu and D. Mokrytska. Improved asteroseismic inversions for red-giant surface rotation rates, Astronomy and Astrophysics, 668:A98, 2022. https://doi.org/10.1051/0004-6361/202142510