These assumptions are made in the rest of this paper:
1. At one time, the Earth's metallic core was entirely liquid.
2. Since the formation of Earth, the solid part of the Earth's core formed by solidification of the liquid core.
3. That when a portion of the liquid core solidifies, it increases in density and decreases in volume.
4. That there is negligible difference in composition between the outer surface of the inner (solid) core and the inner surface of the outer (liquid) core.
5. Inner core is growing by circa 0.1 mm per year.
[A brief comment on scientific notation: Due to the limitations of the medium, scientific notation is used in Excel standard format. *10 to the 4th power is represented by E+4, etc. Superscripts are not permitted in the current format.]
As the core of the Earth cools, the liquid outer core gradually solidifies, adding to the surface layer of the solid inner core. At ambient pressure on the boundary between liquid and solid cores, the solidification of iron-nickel alloy results in an increase in density of about 4.7%. An increase in density means a decrease in volume. Hence, the layer of liquid iron directly outside the solid inner core must contract as it solidifies and is assimilated into the inner core. The contraction is not large, since the liquid iron-nickel is under immense pressure, and is far denser than liquid iron-nickel at atmospheric pressure, having an estimated density of 12.2. Yet, the contraction is NON-ZERO as the atoms of liquid metal join together in highly ordered crystals having a density of roughly 12.8 at the surface of the solid core. The resulting contraction is (12.8-12.2)/12.8=0.047 or 4.7%. This small contraction has huge effects, because the entire outer core plus the mantle and crust are being supported by the outer core. Therefore, the entire Earth above the outer core moves slightly closer to the center of the Earth, releasing gravitational potential energy. Furthermore, this energy does NOT propagate as heat from the core, but produces heat from the entire liquid core plus the mantle as these layers are compressed by contraction of the planet Earth as a whole. This effect propagates at the speed of sound.
The pressure at the surface of the inner core is circa 3,000,000 atmospheres times 100000 pascals/per atmosphere = 3E+11 pascals = 3E+11 newtons/meter squared.
The area of the inner core = 4·π·R2 (where the radius = 1221 km) = 1.221E+12 meters squared (assuming approximate sphericity). This gives a total area of 1.87345E+13 meters squared. Thus, the total force on the surface of the inner core is
3E+11 newtons/meters squared · 1.221E+12 meters squared = 3.663E+23 newtons. If the core grows by .1 mm / year, then the total energy being released in a year is 0.047 · 0.1 · 3.663E23 newton meters. Since a newton meter=1 joule, and there are 3.156 E7 seconds in a year, the average power = 1.72161E21 watts.
By way of comparison, the total geothermal heat delivered to the surface of the Earth is generally estimated at 42 million megawatts. Depending on whose values you accept for the density of the solid core versus adjacent liquid core (I used 12.8 & 12.2 respectively), the thermal power of the contraction is circa 17 million megawatts, or 17 terawatts about 40% of 42 terawatts, making it a major source of geothermal heat, comparable to all heat released by radioactive decay.
This is NOT to be confused with the crackpottery wherein the Earth is pictured as having shrunk by 50% or more. The tiny amount of loss of radius is on the order here of only a few hundreds of meters over 4.5 billion years, much less than 0.01%. This slow contraction may be measurable by modern Geodetic Science.
The above is only a first approximation, and ignores subtleties and other layers of the Earth's structure.
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