Calculating Love numbers (tidal deformation) for non-homogeneous planets

Question

I have a question about planetology. I'm not sure if it belongs here, but the astronomy stack exchange seemed a bit odd, too.

I would like to enhance my understanding of Love numbers. Love numbers quantify the deviation from equilibrium tides on a planet. For a homogeneous body, you can write, for example $$ k_2 = \frac{3}{2}\Big(1 + \frac{19}{2}\frac{\mu}{\rho gR}\Big)^{-1}$$ where $\mu$ is the rigidity, $\rho$ is the density, $R$ is the radius of the planet and $g$ is the gravity constant. I have been taught that for non-homogeneous bodies, e.g. planets with a core etc, there is no simple analytical expression and you need numerical techniques. I have been reading a paper about exactly that, but I don't see what equations are used to obtain $k_2$. In this article, the interior of the planet is described by Maxwell models and a pseudo-period Andrade model. These rheological models are used in conjunction with so called structural models for the interior. It's not clear to me what these are, either. I think they are equations for pressure and temperature. The article then says: "Based on these, the tidal Love number $k_2$ is calculated and compared against measurement."

Does anyone know what equation needs to be solved to obtain the $k_2$ Love number? Or what steps need to be taken to do such a calculation? Or if you know a relevant reference, anything is greatly appreciated, because I don't know where to start.

EDIT: I found a calculation for planets in hydrostatic equilibrium (see my own answer below). If someone knows the equations for a terrestrial planet with viscoelastic behaviour, I would still be interested in that.

Answer 1

I actually just found an article that explains such a calculation in detail, using a matrix propagator method: Padovan et al. 2018 This is a calculation for gas giants or bodies in hydrostatic equilibrium. The calculation of the Love numbers for terrestrial planets that have viscoelastic behaviour requires the rheology (which includes models for the rigidity). I did not find the equations that need to be solved in that case, but a similar matrix propagator method can be used apparently.

I will not repeat the matrix propagation method here, as it is quite well elaborated in the paper I just mentioned, except for the equation you have to start from, which is explained in another article, but looks a bit too complicated for me at this point.