How much of Earth's mass is created by the energy of the core?

Question

I've read that higher energy means higher mass, and in atomic systems, the kinetic energy and potential energy actually contributes more mass than the actual particles themselves (or so I've read). So, how much of Earth's mass is created by the energy in the molten core? What would be the difference in mass between an almost identical Earth with no molten core and the Earth that we actually have?

Answer 1

According to Table 2.17 from page 109 of Chemistry of the Climate System by Detlev Möller, the heat content of the inner core of the Earth is $\sim 3.6\times 10^{30}$ J, and the outer core is $\sim 1.5\times 10^{31}$ J. The total heat content of the Earth is $\sim 2\times 10^{31}$ J. The author stresses that these are only crude estimates based on theories that give mean temperature and composition for the various layers.

Using $E=mc^2$, the mass equivalence of the inner core heat is $\sim 4\times 10^{13}$ kg, the outer core is $\sim 1.67\times 10^{14}$ kg, so the total for the core is around $2.1\times 10^{14}$ kg.

For comparison, the Earth's mass is $\sim 5.9722\times 10^{24}$ kg. So the core heat contributes around 1 part per 29 billion of the total mass.

---

Here's the contents of Möller's table.

region	distance	mean T	density	matter	heat
	(km)	°C	$g/cm^3$		(J)
crust	0-30*	350	3.5	rocks	$2×10^{22}$
outer mantle	30-300	2000	4	rocks	$5.6×10^{28}$
inner mantle	300-2890	3000	5	rocks	$2.2×10^{30}$
outer core	2890-5150	5000	8	Fe-Ni	$1.5×10^{31}$
inner core	5150-6371	6000	8.5	Fe	$3.6×10^{30}$

• Continental crust, oceanic crust is 5-10 km depth.

It's surprisingly difficult to find this geothermal energy data. Wikipedia gives a figure of $10^{31}$ J for the internal heat content of the Earth, linking to a report which quotes a figure of $12.6×10^{24}$ MJ from What is Geothermal Energy by Dickson & Fanelli (2004), but that article gives no details for the calculation.