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

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.

• This would be in fact also appropriate on the astro.se, as a part of (exo)planetary science is concerned with love numbers as well. Here is one paper that goes through the entire apparatus to calculate $k_2$ for gas giants: academic.oup.com/mnras/article/450/4/3952/1749966 and if you're looking for a ton of references to start, the protostars&planets 6 chapter here should be useful too: www2.mpia-hd.mpg.de/homes/ppvi/chapter/baraffe.pdf Commented Apr 24, 2021 at 11:51