What does it mean for a planet to have an "icy" core?

My understanding of the physics of planets leans heavily on my understanding of the physics of stars - of course there are important differences.

One important difference that I've found is the nature of the core of a planet. I've read that exoplanet cores may be hot or cold/icy. I've also even seen this said about Saturn and Jupiter.

My questions: in principle, how could the core of a planet be icy and cold (or are "icy" and "cold" not mutually inclusive here?). What is meant by "icy core." And how could it possibly be physically reasonable? Could only planets with low enough effective temperature possibly have an icy/cold core? Am I correct in thinking that a hot planet (i.e. one with high effective temperature) could not have a icy core? So, for a terrestrial example, is Jupiter (or Saturn also) cold enough that its core could be icy? At what temperature is the boundary between cold and hot cores? I'm having difficulty understanding heat transport in such an object. Is there a thermal gradient? If so, is its descent just very shallow?

Perhaps I've oversimplified something crucial, or I've totally misunderstood the use of these words in this context? Any clarification is greatly appreciated.

• Definitively not sure about the established semantics here, but my impression would be that "icy" refers to composition (eg "icy" materials such as water, ammonia, methane vs iron) while hot/cold refers to temperature ("hot" meaning way hotter than the surface and cold somewhat similar in temperature wrt the surface; might also be simply that hot cores are molten and cold are solid?). I am almost certain there is no clear cut numerical limit what would be called cold vs hot Jun 4 '21 at 21:20

While for low core masses ($$m_c<\sim 17_{\oplus}$$) the mass-radius relationship is near-identical for all pure compositions, and can be approximated by a modified polytrope (see e.g. Seager et al. (2007)), for larger core masses the different polytropic indices of Ice vs. Silicates vs. Iron give rise to modelled interior compositions of gas giant cores.