Cooling of the Earth

How long does it take for the Earth to cool down to $3K$ if the sun was gone?

Assumptions: The Earth only loses heat to space and does not gain any heat. Heat transfer occurs only through radiation. The mass of the Earth is $5.972\cdot10^{24}\text{ kg}$ . The specific heat capacity of the Earth is $1260 \text{ J/kg} K$ . The current temperature of the Earth is $300K$ . The Earth is a perfect blackbody. The surface area of the Earth is $5.1\cdot10^{14}m^2$ . Do not take the latent heat of fusion or vaporisation into account. Assume heat is always evenly distributed around the Earth and the Earth is a uniform sphere made of the same material with the specific heat capacity as stated above.

#Mechanics

Easy Math Editor

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Comments

From the Stephan-Boltzmann Law, $\frac{dQ}{dt} = \sigma A T^4 = \sigma A T^4$ $Q$ is heat radiated by the earth, $T$ is the temperature of the earth, $A$ is the surface area.
By the definition of specific heat,
$\frac{dQ}{dT} = -mc$ $m$ is the mass of the earth, $c$ is the specific heat.
The negative sign is because the temperature is reducing.
Dividing,
$\frac{dT}{dt} = \frac{-\sigma A}{mc} \cdot T^4$ Rearranging,
$\int_{300 K}^{3 K} \frac{-dT}{T^4} = \int_0^t \frac{\sigma A}{mc} dt$

$\frac{1}{3} \left(\frac{1}{T^3}\right)\biggl|_{300}^{3} = \frac{\sigma A t}{mc}$ so,
$t = \frac{mc}{3\sigma A} \left(\frac{1}{27} - \frac{1}{27 \cdot 10^6}\right)$

$t \approx 3.23 \cdot 10^{18} \text{ seconds } \approx 10^{11} \text{ years }$ That is a long time!

Ameya Daigavane - 4 years, 12 months ago

Amazing solution sir , +1!

Rishabh Tiwari - 4 years, 12 months ago

Hey Jerry , its interesting to know that we can even calculate this , Can you explain how ? Anyone ? @Pranshu Gaba , @Swapnil Das , @Arjen Vreugdenhil please help .

Rishabh Tiwari - 4 years, 12 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$