On the tidal heating of a moon. What is the second Love number?

Question

Years ago it was asked here how to calculate the tidal heating of a moon orbiting another body with a simple equation. The answer is very detailed. They explain the equation, its shortcomings, the aproximations made, the constants and variables used, etc.

The entire equation is proportional to a variable: $-Im(k_2)$. The answerer explains that it is the second Love number, part of a family of numbers which "are dimensionless parameters that measure the rigidity of a planetary body and the susceptibility of its shape to change in response to a tidal potential." [Wikipedia]. This variable is hard to calculate, so they advice using known values, like for example the Moon or Io, which correspond to $-Im(k_2) = 0.02$

The problem is that I couldn't found values for more Solar System bodies, or any reference on how $-Im(k_2)$ changes in function of density, mass, composition, or structure. I only found values for $k_2$, but nowhere I looked I found $-Im(k_2)$. Is there a useful source somewhere where I can find a table of $-Im(k_2)$ values for the planets, or studies that estimate the number in function of the type of planet?

Answer 1

So, I've found a source for the offending tidal heating equation in the form of Tidally Heated Terrestrial Exoplanets: Viscoelastic Response Models. This has two versions of the tidal energy formula, one which is the same and one under "Fixed Q Tidal Model" which is similar though notably instead of $-Im(k_2)$, it has $\frac{k_2}{Q}$, where $k_2$ is not a complex-valued number.

The paper goes on, however:

Using equation 1 to calculate global tidal heat is useful for estimates, however it ignores the frequency dependence of a material’s response to loading. $Q$ and $k_2$ are neither constant nor entirely independent parameters, (Kaula 1964; Zschau 1978; Segatz et al. 1988). In its most general form for a viscoelastic body, the ratio $k_2/Q$ is replaced by the imaginary part of the complex Love number $−I⁢m⁢(k_2)$, which characterizes the material’s viscous phase lag.

So that's why you can find the non-complex values, because they're simpler estimations and the associated equation is a lot simpler to work with. HDE 226868 did clearly warn you that this stuff was complicated, and they weren't kidding... honestly, their answer is by far the most readable bit of work on the subject that I've been able to find (though its a shame that the Haussman et al 2010 paper they referenced no longer has a free copy online). I'm not totally sure where they felt the imaginary portions of $k_2$ values could be easily found though, because by far the most common values I've come across are the real-valued coefficients.

Perhaps, then, you should just use the version of the equation that takes those, and be content with that approximation.

$$\dot{E}_{t⁢i⁢d⁢a⁢l}=\frac{21}{2}\frac{⁢k_2}{Q}\frac{⁢G⁢M_{p⁢r⁢i}^2⁢R_{s⁢e⁢c}^5⁢n⁢e^2}{a^6}$$

where:

• $\dot{E}_{t⁢i⁢d⁢a⁢l}$: Tidal heat production rate, w⁢a⁢t⁢t⁢s
• $a$: Semi-major axis
• $e$: Eccentricity
• $G$: The gravitational constant
• $M_{p⁢r⁢i}$: Mass of the primary
• $R_{s⁢e⁢c}$: Radius of the secondary
• $k_2$: Second-order Love number of the secondary
• $Q$: Quality factor of the secondary

There's some discussion of the derivation of $Q$ in the paper, though it is a little unwieldy. It does link to some papers which estimate values for various solar-system bodies though, which might be a good starting point for your needs. Coming up with your own values for $k_2$ though remains a bit of a non-starter, though it does at least give a few examples from the solar system for you to be getting on with.