We even know the elastic modulus of the solid inner core: http://advances.sciencemag.org/content/4/6/eaar2538
At present day, Earth is losing heat at a rate of about 40 terawatts. This 40 terawatts drives mantle convection, plate tectonics, and every volcano out there.
As best we know, about 35% of that 40TW is radiogenic (decay of Th-232 with a half-life of 14 bilion years, U-238 with a half life of 4.5 billion years and K-40 with a half life of 1.25 billion years, among others). A fair amount of the decay energy is actually lost to neutrinos, but the rest is converted to heat and eventually lost by convection or conduction.
The other 65% or so is primordial -- essentially, gravitational potential energy from the accretion of the Earth, converted to heat and slowly lost over the past ~4.51 billion years since the moon-forming impact.
Earth's bulk thermal inertia (heat capacity times mass) is about 5.6E27 J/K. In other words, cooling the earth by one degree C releases 5.6E27 joules. That's a lot of joules!
Even so, there's a bit of a problem because geochemical evidence suggests that the mantle (which makes up most of the earth by volume) has probably not cooled by more than about 150C over the past 3.5 billion years, which means either one of our estimates is wrong, or the core is cooling pretty fast -- this would actually mean that the inner core might be pretty young, like < 1 billion years.
What are the error bars on that? It seems to be getting quite close to the emergence of complex life, enough that the radiation shielding may be a more important pre-requisite than previously believed.
One counterpoint that comes to mind is that the Ediacaran fauna is usually though to be marine, and water is a pretty good shield of radiation. There is one geologist who argues the ediacarans were terrestrial (Greg Retallack), but he interprets any sedimentary rock that's red as a paleosol, which is actually not completely silly, but is a bit more liberal than most others would be comfortable with.
In any case, I don't know, but it's not impossible!
edit: I should also say that the generation of Earth's magnetic field in the liquid outer core is a bunch of incredibly messy magnetohydrodynamics that is hard to simulate accurately, but we can probably have a pretty reasonable magnetic field with a purely liquid core (no solid inner core at all) -- it's the convection of the liquid outer core that provides the energy to sustain the magnetic field, so it's actually once the core becomes fully solid that we'll be in trouble
Thanks for the response, btw, been waiting for a geologist ama for a while!
While in neither case is diffusion the only process at work, it's a critical limiting factor in both cases. Diffusion of heat, diffusion of nutrients: it's all the same math!
There's a square cube relation between surface area and volume. Smaller planets have much more surface area compared to volume than larger planets. So it takes a lot longer for heat to escape the core. While smaller planets can radiate their heat away relatively quickly, in the Earth it just builds up.
Venus is of comparable size to Earth. But we don't know very much about Venus' geology. We know that there isn't any active vulcanism on Venus, its crust is all approximately the same age, and is (in geologic and astronomical terms) relatively young. The speculation is that the hot blanket of Venus' atmosphere actually keeps something of a lid on things. The magna isn't so much hotter than the atmosphere that vulcanism is an efficient means of letting off heat. So it just builds up for millennia (hundreds of millions of years? I do not recall the timescale.) until it reaches a critical temperature where most of the entire surface begins melting/venting/erupting. We can't prove it to any significant extent, but it's the model that best fits the evidence.
It might be resolved now, but for a while there was a bit of a mystery where tidal dissipation as estimated by the recession rate of the moon (as measured by bouncing lasers off a corner reflector left by the apollo astronauts) was significantly larger than we could account for by modelling oceanic and solid-earth tides.
There's also the issue that if you extrapolate the recession rate of the moon at the present rate, it ends up within the Roche limit less than 2 billion years ago. That's not necessarily catastrophic since solid bodies have cohesion (the ISS is well within the Roche limit, for instance). Nonetheless, it's likely that tidal dissipation is higher now than it was for much of Earth history. This paper may be paywalled but is a pretty recent take ont he subject: https://doi.org/10.1016/j.epsl.2016.12.038
While Earth is a bit unique within the solar system in having a (relatively) really huge moon, the reason Earth has plate tectonics and Venus doesn't is generally thought to have more to do with a proposed mutual dependence of liquid water and plate tectonics (e.g. https://doi.org/10.1029/GL010i011p01061), since Venus accreted more volatile-depleted than the earth. So while Venus has no shortage of what would be considered young volcanic activity by geologic standards (https://en.wikipedia.org/wiki/Volcanism_on_Venus), its crust is stuck in place (a "stagnant lid"). The fact that Earth is a little bit bigger probably doesn't hurt either.
But seriously, thanks for the additional links and confirming some suspicions.