[ExI] anders' sea of pu

[Anders Sandberg]

On 2013-12-02 22:31, Anders Sandberg wrote:
> The temperature distribution of Plutonia is also pretty interesting. 
> If you solve the heat diffusion equation for a sphere with internal 
> heat generation you get T(r) = T_S + (q/6k)(R^2-r^2) where T_0 is the 
> surface temperature, q is the volume heat generation and k thermal 
> conductivity. It is hottest in the centre with a temperature of 
> T(0)=T_S + R^2 q/6k.
>
> T_S can be calculated due to energy conservation: the heat production 
> is balanced by emission to space. 4 pi R^2 sigma T_S^4 = 4 pi q R^3 / 
> 3 gives T_S =  (q R / 3 sigma)^.25. So the surface temperature grows 
> slowly with radius, but the core temperature increases with the square 
> of the radius.

Cool insight: if the surface has T_S=912 K and q=10.61 W/m^3 and is in 
radiative equilibrium, then R=11 km. (Unless I miscalculated q)

If we use R=6371 km, then T_S=4464 K. This is actually above the boiling 
point, so eventually Plutonia will turn somewhat gaseous. However, I 
think leakage of the plutonium atmosphere to space will be slow. 
Especially given the 3 times higher surface gravity and the fact that 
this gas is *really* heavy - Earth can retain even nitrogen n the 
exosphere, and plutonium has a 17 higher atomic mass.

Meanwhile, at the core... k=6.74 W/mK, so T(0)= 10^13 K. Whoa. I think 
we can clearly expect that even if there is never any criticality 
anywhere, the core will tend to bubble up as very hot gas. In fact, I 
suspect that we are going to be in a way convective mode as the planet 
gets older. This will cool the core more efficiently. Given the 
radiative equilibrium above, I guess actual the final state of Plutonia 
is a fuzzy cloud of plutonium/helium mist, shining nicely orange.

Occasionally comets hit, shine brightly of Cherenkov radiation and 
finally act as neutron reflectors as they go out in spectacular flashes.

It makes quite a sight from its moon Urania. Which is slowly spiralling 
in due to tidal dissipation...

-- 
Dr Anders Sandberg
Future of Humanity Institute
Oxford Martin School
Oxford University