SPROUTS: Spectral Modelling of Planetary Interiors for BepiColombo and JUICE

Probing the deep interiors of planetary bodies relies on forward modelling that can be confronted with and constrained by space mission observations. In this regard, modelling global planetary deformation under tides raised by surrounding celestial objects is paramount. Planetary deformation and the associated changes in the gravity field are encoded in the Love numbers, which are directly comparable to observational data. Their computation rests on solving the same set of gravito-elastic equations relevant to normal-mode seismology. Accurate and efficient forward modelling of these quantities is one of the keystones of ongoing missions such as BepiColombo and JUICE, which will provide high-precision measurements of the tidal response of Mercury and the Galilean moons.

Classically, this has been achieved by integrating these equations as a system of ordinary differential equations (ODEs) from the centre outward — the so-called shooting method. A seed solution is prescribed at the centre of coordinates, numerical integration proceeds by iterating small radial steps, appropriate junction conditions are enforced at physical discontinuities such as the Core-Mantle Boundary, and the solution is finally checked for compatibility with surface boundary conditions. In case of mismatch, the process is repeated with adjusted initial conditions. While this approach has proven its validity over decades, the mathematical problem is more naturally cast as a boundary value problem rather than an initial value problem, and this reformulation opens the door to significantly more efficient numerical strategies.

Here we present SPROUTS (Symbolic Parser for ROUnd objecTS), a publicly available solver that exploits this perspective. SPROUTS implements an optimal spectral discretisation of the boundary value problem. This yields a system of sparse matrices that are computationally inexpensive to assemble and invert even on low-end computers. Beyond efficiency, the spectral formulation offers superior numerical accuracy and flexibility for handling physical discontinuities and boundary conditions.

We demonstrate the capabilities of SPROUTS by computing the free oscillation modes of interior models of the Earth and Mercury. The associated Love numbers are computed and benchmarked against state-of-the-art methods. We further discuss how SPROUTS is being developed with direct applications to the interpretation of tidal observations from BepiColombo and JUICE in mind, with the goal of better constraining the interior structure of Mercury and the icy Galilean moons.