density of earth = 5.515 g/cm3

density of ultramafic rock = 3.3 g/cm3

Radius of earth = 6370 km

Radius of core = 6370 - 2890 = 3480 km

Average density of entire core = (5.515-3.3)*(6370^3/3480^3)+3.3 = 16.9 g/cm3

https://www.wolframalpha.com/input/?...0%5E3%29%2B3.3
Radius of inner core = 1220 km

Density of outer core = density of liquid iron = 7.2 g/cm3

Density of inner core = (16.9-7.2)*(3480^3/1220^3)+7.2 = 232.3 g/cm3

https://www.wolframalpha.com/input/?...0%5E3%29%2B7.2
Density of iron = 7.87 g/cm3 (bcc) or 8.6 g/cm3 (fcc/close packing)

Density of inner core = 232.3 g/cm3 = 27 * 8.6 g/cm3

Exactly 27 times denser!

This means that the iron atoms in the inner core are exactly three times smaller

This is the expected result if electron shells are evenly spaced.

(Except it isn't clear why the iron atoms go from 3rd to 1st electron shell rather than 3rd to 2nd)

Iron has 4 electron shells but the 4th is already degenerate (which is why its a metal and a conductor)

Gravity at the top of the inner core = (232.3/5.515)*(1220/6370) = 8.067 g's

3/5 of earth gravity is due to density 3.3

Gravity = (0.6 g's * r/6370 km) + (0.4 g's * (6370 km)^2/r^2) for r=3480 to 6370 km

Gravity = 1.668 g's at r=3480 km

1.668*7.2/16.9 = 0.71 g's of that is due to density 7.2

Gravity = ( 0.71 g's * r/3480 km) + (0.958 g's * (3480 km)^2/r^2) for r=1220 to 3480 km

Gravity = 8.04 g's at r=1220 km

integral of [ (0.6 * r/6370) + (0.4 * (6370)^2/r^2)] for r=3480 to 6370 = 3456 g's * km

integral of [( 0.71 * r/3480) + (0.958 * (3480)^2/r^2)] for r=1220 to 3480 = 7260 g's * km

3456 g's * 1 km * 3.3 g/cm3 = 1.13 * 10^6 bar

7260 g's * 1 km * 7.2 g/cm3 = 5.13 * 10^6 bar

Pressure at top of inner core = 1.13 + 5.13 = 6.26 million bar