Reference

This is the theory behind Obliqua: where the forcing comes from, how a material responds to it, and how that response is actually solved for across a planet's interior. Read it start to finish for the full story, or jump straight to the model you're configuring — each page stands on its own and links back to the ones it builds on.

1. Setting the stage: what's forcing the tides?

A planet in an eccentric orbit feels tidal forcing at more than just one frequency. Before any material response can be computed, we first need to know which frequencies matter and how strongly each one drives the response.

• Forcing Frequency — which harmonics and Fourier modes actually contribute, and why we can usually stop at $n=2$.
• Tidal potentials — normalizing the forcing itself, and turning a Love-number spectrum into a heating rate.

2. How a material responds

• Rheology — Maxwell, Andrade, and the purely elastic limit: the three ways Obliqua lets shear and bulk moduli become frequency-dependent (and dissipative).

3. The solid interior

The bulk of Obliqua's machinery lives here: propagating the tidal response equations through a radially-structured solid mantle, from a homogeneous zeroth-order estimate up to a fully poro-viscoelastic, core-inertia-aware solver.

• Solid-Phase — the shared theory: $y$-functions, the motion matrix $\pmb{A}_n(r)$, the core boundary matrix $\pmb{I}_C$ and its four core options, and the shooting vs. relaxation split.
• Solid-Phase - solid0d — the zero-dimensional, single-layer approximation. Fast, and a good sanity check for everything else.
• Solid-Phase - solid1d — radially resolved, shooting method.
• Solid-Phase - solid1d-relax — the same physics, but solved with a Henyey-style relaxation scheme instead — generally far more stable.
• Solid-Phase - solid1d-mush and Solid-Phase - solid1d-mush-relax — their poro-viscoelastic extensions, for a mantle with a partially molten layer.
• Solid-Phase (Equilibrium) - solid1d-equil-relax — the reduced, two-component equilibrium-tide limit used automatically at very low forcing frequencies.

4. The mushy transition

• Mush layer - interp — a lightweight, purely interpolated treatment of dissipation across a thin mushy or transitional region, for when the full poro-viscoelastic machinery above is overkill.

5. The fluid response

• Liquid-Phase — the Laplace tidal equations for a magma ocean or other fully fluid layer, and the family of radial dissipation profiles (including the energy-conserving dynamic_interp default) used to distribute that heating with depth.

6. Reading out the answer

• Surface Loading — turning surface boundary conditions into tidal and load Love numbers, and how the solid and fluid contributions are currently combined into one global $k_n$.