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I'm trying to understand more about tidal deformation in planets, and I frequently encounter Love numbers. However, sometimes the fluid Love number is mentioned, and sometimes the tidal Love number is mentioned. As I currently understand, the fluid Love number is the long-term behavior of the body and the tidal Love number is related to tidal frequencies. Does it mean that the fluid Love number assumes the whole body is fluid and the tidal assumes a viscoelastic behavior? If someone can expand on the physical differences between the two I would really appreciate it.

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In short, you're understanding is correct in the essentials.

I think the reason it is challenging to get a grasp on this is because the literature is not always careful it its usage of "fluid Love number" and "tidal Love number". I've seen many cases where these two terms are used somewhat interchangeably or without context and I've even seen the term "fluid tidal Love number".

Nevertheless, the Love numbers are dimensionless quantities which measure the susceptibility of a body to deform due to tidal forces. The nature of any deformation is a function of the body's viscoelastic behavior and rigidity.

The fluid Love numbers "describe the tidal response of a fluid nonrotating planet" (Padovan et al. (2018)). In that way, the fluid Love numbers are from an idealized case where the planet under the influence of tidal forces does not have any rigidity or viscoelasticity. Because of that, any response to tidal forces is immediate (on the proper timescale) and, as you say, would be the long-term state of a rigid body, given enough time to settle and no other influences.

The tidal Love numbers describe the tidal response of a rigid planet. Once viscosity (or, more generally, anelasticity) is in the mix, the body's response to tidal forces is complicated by the fact that it takes time for the body to respond to tidal forces. If you consider, for example the Earth/Moon system, the Moon is constantly orbiting Earth, causing the tidal force to constantly change. The solid Earth's response to that change is dictated by (a) much it can deform, and (b) how fast it can deform. The magnitude of its deformation is defined by an interplay of its rheology (mainly, rigidity) and self-gravitation, and is parameterised with the tidal Love number. The lag of its deformation (i.e., how quickly the solid Earth can deform in respond to loading) is defined mainly by the interplay of its mantle's rigidity and viscosity.

For further reading, see these Earth Science course lecture notes.

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