How much of Earth's internal heat comes from primordial heating and how has it changed since Earth's formation?

Question

I'm trying to get a general idea of how Earth and other planets behave over time for a project. While I can find plenty of good data about how much radiative heating there was, I can't find anything on primordial heat.

What kind of relationship does internal heat and time have?

cheers.

Answer 1

William Thomson's calculation is a poor first order approximation: take a molten blob of earth and calculate how long it takes too cool to near zero K. What Thomson couldn't know, for one thing, is how high initial temperatures got during the initial accretionary phase of earth's formation. Another thing that Thomson couldn't know in the mid 19th century is that the mantle is an excellent insulator, preventing much of that primordial heat from escaping.

Despite uncertainties in composition and masses of specific chemistries of the deep earth, modern diamond anvil work has refined thermal conductivity values of earth materials at high temperatures and pressures. According to Quentin Williams of UC Santa Cruz, who, among many others, has done a lot of work constraining heat flow values in earth materials at depth:

It takes a rather long time for heat to move out of the earth. This occurs through both "convective" transport of heat within the earth's liquid outer core and solid mantle and slower "conductive" transport of heat through nonconvecting boundary layers, such as the earth's plates at the surface. As a result, much of the planet's primordial heat, from when the earth first accreted and developed its core, has been retained.

You can find and read more of his work here.

Answer 2

William Thompson, the first baron Kelvin worked out that the Earth was around 20 to 400 million years old, based on the rate of cooling from an assumed molten state. This would imply that the Earth's primordial heat would already be essentially lost, so you could argue the answer is zero (assuming Kelvin's calculation remains reasonable). However once energy is "thermalised" it loses any memory of where it is from, which makes that answer questionable.

[I assume that by "radiative heating" you mean heating from radioctive decay in the mantle, rather than radiative heating of the surface by the sun. There is also the Kelvin-Helmholtz heating from the iron sinking to form the core, but presumably that is finished now?]