Where does Love's number come from in $\delta m = \frac{1}{2}k_2M_2\lbrace\frac{R_1}{r_\star}\rbrace^3$, the formula for the mass of a tidal bulge?

Question

My question refers to page 165 of this paper. Specifically, how does one derive the equation $\delta m = \frac{1}{2}k_2M_2\lbrace\frac{R_1}{r_\star}\rbrace^3$ for the mass of a tidal bulge? A full derivation would be much appreciated, and also an explanation as to where Love's number $k_2$ comes from.

Answer 1

Let us begin with the basics. If an exterior perturber generates the potential W, then an originally spherical planet gets distorted. Owing to this distortion, its potential acquires an increment U which is usually called the additional tidal potential or, simply, the tidal potential.

Now let us fix a point on the surface of the planet. Let $\gamma$ be the angular separation between the planetocentric vectors aimed at this point and the perturber.

The perturbing potential W in that point can be expanded into a series over the Legendre polynomials $P_l(\cos\gamma)$. This is an almost trivial fact, see equations (9 - 11) in https://link.springer.com/article/10.1007%2Fs10569-009-9204-7

https://arxiv.org/abs/0803.3299

In those equations, $r^*$ is the position of the perturber, while R is the surface point.

As it turns out, a degree-$l$ term in that equation contains a factor of $(R/r^*)^{l}$.

The additional tidal potential U in that point can be expanded over the same $P_l(\cos\gamma)$.

Moreover, you can write down the potential U in a point r located right above R (i.e., having the same angular separation from $r^*$ as R has). This expansion is rendered by equation (19). In that equation, you need only the first line. Ignore the rest for now.

As we see from this formula, the degree-$l$ term of U is equal to the degree-$l$ term of W multiplied by $(R/r)^{l + 1}$ and by an additional factor $k_{l}$ named the degree-$l$ Love number.

Derivation of this formula is boring. It is a solution to a boundary-value problem for an elastic medium, and the factors $k_{l}$ emerge as some functions of the shear elastic modulus of the material of which the planet is composed.

This is how it is for static tides. In real life we, of course, have tides evolving in time, and all this machinery has to be generalised. We have to expand both W and U into Fourier harmonics and to write down all this formalism for each harmonic separately. Then you get different lags for different harmonics involved. Also, the dynamical analogues of the Love numbers will be functions of other rheological parameters (like the viscosity). However, the central idea will be more or less the same. I can give you more references, if you are interested.

Jérémy Leconte in his work tried to substitute the degree-2 part of U with an equivalent potential generated by two equal masses located oppositely. You can think of this as two tidal bulges approximated with point masses. This was done for illustrative purposes solely and, in my opinion, bears no practical value.