[Astrogazer (OP)]

Can anyone do some fag-packet calculations for me please?  We know that the solid inner core of the Earth began to form about 565 million years ago.  We also know that the heat energy produced by the radioactive decay of K, U and Th is not sufficient to account for the maintenance of the temperature at the Earth’s inner core.  As the solid inner core grows, it must be releasing iron/nickel’s latent heat of fusion.  My question is:- can this release of heat to form a body-centred cubic crystals, plus the heat from radioactive decay account for the maintenance of the core’s temperature?  How much heat is currently being released from this solidification process?
We know the size of the inner core, we know it’s mass therefore, we know the latent heat of fusion of iron, we know how much heat is being lost from the core, if we assume that the rate of heat released is constant over the last 565million years, do the figures make sense?   The calculations involved of a growing sphere would fry my mind so I’d appreciate some help with this please.

[evan_au]

It seems that there are still major disagreements about the properties of iron at the temperatures and pressures in the Earth's core.
- It is very hard to measure these properties in the lab, since the only thing that can generate the required pressures (diamond) tends to dissolve in molten iron
- There are also greatly varying estimates of the types and quantities of radioactive minerals in the Earth's core; radioactive decay provides a heat input in addition to heat released by crystallization
- Some descriptions have suggested that there is a steady "snow" of solidified crystals gently falling on the outside of the solid core.

Note that the crystal structure of Earth's core is not necessarily body-centered cubic; as the OP notes, the evidence suggests the presence of a significant amount of nickel, which tends to form a face-centered cubic structure.
- Evidence of the crystal structures that might exist comes from metallic meteorites.
- But it looks like the Widmanstätten pattern is produced at much lower temperatures than is expected at the core of the Earth.
See: https://en.wikipedia.org/wiki/Widmanst%C3%A4tten_pattern#Lamellae_formation_mechanism

A space mission is planned to the metallic asteroid Psyche, which may reveal more (launch is planned for 2022, with a 4-year trip time).
See: https://en.wikipedia.org/wiki/16_Psyche

[gem]

Hi all

Can anyone do some fag-packet calculations for me please?  We know that the solid inner core of the Earth began to form about 565 million years ago.  We also know that the heat energy produced by the radioactive decay of K, U and Th is not sufficient to account for the maintenance of the temperature at the Earth’s inner core.  As the solid inner core grows, it must be releasing iron/nickel’s latent heat of fusion.  My question is:- can this release of heat to form a body-centred cubic crystals, plus the heat from radioactive decay account for the maintenance of the core’s temperature?  How much heat is currently being released from this solidification process?
We know the size of the inner core, we know it’s mass therefore, we know the latent heat of fusion of iron, we know how much heat is being lost from the core, if we assume that the rate of heat released is constant over the last 565million years, do the figures make sense?   The calculations involved of a growing sphere would fry my mind so I’d appreciate some help with this please.

I believe W Thomson AKA Lord Kelvin had a good stab at this https://en.wikipedia.org/wiki/William_Thomson,_1st_Baron_Kelvin
which would be a good starting point to the calculation, given he calculated an age for the earth due to temperature/cooling.

but as Evan-au as hinted at there is still a lot that is not known about the dynamics within the earth.

I remember reading quite a few years ago a Russian geologist concluding the majority of heat within the earth was generated by friction, which raises yet more questions.   

[Kryptid]

I'm interested in looking into this. I'll do a bit of research and get back to you.

EDIT: the latent heat of fusion of iron is about 0.21 kilojoules per gram, whereas the latent heat of fusion of an iron-nickel alloy called FeNi36 is 0.28 kilojoules per gram. I need to point out that the heat of fusion of iron and iron alloys is very probably different under extreme pressure than under atmospheric pressure. How much different, I don't know. Just to see where the calculations go, I'll just use these numbers.

The mass of the inner core is about 10²⁶ grams. If the inner core was right at its melting point (it isn't, but we're making simplifying assumptions), then going from a molten state to a solid state would involve the release of 2.1 x 10²⁵ kilojoules of heat (if it's pure iron) or 2.8 x 10²⁵ kilojoules of heat (if it's the iron-nickel alloy I'm using).

The Earth's inner core is estimated to be growing at a rate of about 1 millimeter per year (I assumed that was a radius increase of 1 millimeter for the calculations. Maybe I should have made that half a millimeter?). After crunching the numbers using that metric, I calculated that the inner core is currently losing heat at a rate of between 4.9 x 10¹⁶ and 6.5 x 10¹⁶ kilojoules per year (depending on whether one uses iron or the iron-nickel alloy for the math).

You said to assume that the rate of heat loss is constant (it isn't, but again, let's see where the math goes). If we do assume a constant rate of heat loss, you end up with the inner core right at its melting point going from pure liquid to its current size in about 430 million years. I'm honestly surprised that number turned out to be as close to your stated figure of 565 million years as it was.

But, as I pointed out before, my math made a lot of simplifying assumptions. It doesn't take into account the fact that the only place in the core that has a temperature around the core's melting point is roughly the boundary between the inner and outer core (obviously, since that's where it's solidifying). It doesn't take into account changes in heat of fusion due to pressure. It doesn't take frictional heating or radioactive heating into consideration either. Nor does it take the changing rate of heat loss over time into consideration.

[Astrogazer (OP)]

Thank you Kryptid, that’s very kind of you to do that rough calc for me, that’s really cheered me up.   I’m pleased the amount of heat released and the growth of the core is within about an order of magnitude of what seems to be actually happening.   Yes, of course, there are loads of details that we don’t know, but nevertheless to get the right order of magnitude is pleasing as the papers I was reading couldn’t account for the supplementary heat need to maintain the Earths magnetic field and I haven’t come across a single reference that said that the crystallisation of the core would release a substantial amount of heat.    There was one reference that said that to crystallise the iron would require heat - I couldn’t get my head around that statement at all, another statement I’ve read elsewhere said that to start the crystallisation would require impurities in the liquid iron to start off the process and he couldn’t see how there could be any impurities at the centre of the Earth, I find this statement weird too.