Does the thickness of earth's crust accurately correspond with the current proposed age of the earth?

Question

The current model is that earth started as a molten ball of liquid around 4.5 billion years ago. Today, the earth's crust ranges from about 5-50KM in thickness. This is just a layman's opinion, but to me it seems like 4.5 billion years is quite a short time for the earth to cool enough to produce 50KM of crust, especially given that there are still internal sources adding energy to the earth (such as radioactive decay). Do models of the formation of the crust match the proposed age of 4.5 billion years, or are there discrepancies?

Answer 1

Just a short back of the envelope: Geothermal heat flow is approx. $0.087W/m^2$. Thermal conductivity of basalt is approx. $3.5W/km$. Basalt melts around 1000-1200 degrees Celsius, i.e. the total temperature gradient in the crust is about 600-900K. This predicts (assuming linear heat conduction WITHOUT convection) a thickness of between

${{600K\times 3.5W/Km}\over {0.087W/m^2}}\approx 24km$

and

${{900K\times 3.5W/Km}\over {0.087W/m^2}}\approx 36km$.

Throw in convection (which we know happens) to increase the effective thermal conductivity and the average thickness of the crust makes sense.

Answer 2

Consider a very simplified model of the earth as a uniform sphere at a temperature $T_{0}$ at time $t=0$.

Lets say the earth's specific heat capacity $s\approx 1000 J Kg^{-1} K^{-1}$, which is in the right range for common rocks. The mass of the earth $M\approx 6\times 10^{24} Kg$. So the total heat energy in the earth is $E=sMT_{0}$

To investigate the case of the earth cooling with no heat sources (ignoring tidal heating, radioactive heating and energy from the sun), the only thing affecting the earth's temperature is radiative cooling. The rate of energy loss is then:

$\frac{dE}{dt}=\epsilon\sigma A T^4$ with A being the surface area of the earth, sigma being the Stefan-Boltzmann constant ($\approx5.67\times 10^{-8} W m^{-2} K^{-4}$) and $\epsilon$ is the emissivity (in the range 0..1, 1 being for a perfect emitter).

It turns out that the time taken to cool to a temperature $T \ll T_0$ is independent of $T_0$ and is given by

$t=\frac{MS}{12\pi R_{e}^{2}\epsilon\sigma T^3}$

Plugging the numbers in (and $R_{e}\approx 6.4\times 10^{6} m$ for the radius of the earth) and guessing that the earth is an inefficient emitter (lets make $\epsilon=0.1$) gives about 800,000 years to cool to $300K$ ($27^{\circ} C$) more or less independent of the starting temperature (the time takes is mostly the time to cool the last little bit, the very rapid initial cooling adds very little to the overall time).

So, the earth cooling with no other factors affecting it (and ignoring the whole issue of temperature gradients - this is a very roughly ball-park figure) says that whatever the starting temperature, the earth can reach its current surface temperature in very roughly 1 million years of cooling, which is way, way less than the age of the earth (and this would actually be an earth solid all the way through with no molten core).

Obviously the various heating mechanisms are what stops the temperature being far below that now.

But the point is that your gut feeling that 4.5 billion years isn't long enough for the earth to cool is wrong by a factor of several thousand. It is plenty long enough, and it's not even close.

Incidentally a more accurate estimate of the time taken for the earth to cool from molten rock to is present state was made by Lord Kelvin back in the late 19th century, paying attention to the issues of heat conduction from the center to the edges etc. and being more detailed than what I've done here, and came up with 20-400 million years (he later refined it to 20-40 million years). See for example Lord kelvin on the Age of the Earth

Answer 3

First of all, let me correct a misconception. According to your question, your understanding of the structure of the Earth is that there's a solid crust of 5 to 50 km and below it there is liquid. This is not correct. The Earth is mostly solid. It's solid almost all the way down: you can drill down 3000 km without seeing any liquid at all. The liquid you do see eventually is liquid metal, not rock. Molten rock exists only in very localised anomalies throughout the crust and mantle, and sometimes we see their effect on the surface as volcanoes.

As already answered by others, there was more than enough time to cool down any liquids that existed. However, heat is constantly being produced by radioactive decay. It is also effectively removed from the Earth by processes like volcanic and hydrothermal activities. You also have to remember that the material is heterogeneous, and the melting points of rocks vary from 600 to 1500 degrees Celsius. Assuming a simple model like you described does not work.