Fermi estimate: S of the Sun vs the Earth

Problem

Knowing that the Sun radiates a power $P\approx 4\cdot 10^{26}\;\text{W}$ at a surface temperature $T_\text{Sun}\approx 5800\;\text{K}$, estimate the rate of entropy production associated with the flow of energy from the Sun into cold space (assume $T_\text{space}\approx 3\;\text{K}$).

Solution

Setup: the Sun emits the power $P$ as radiation at its temperature $T_\text{Sun}$, losing entropy at a rate $P/T_\text{Sun}$. That same energy ends up absorbed by cold space at $T_\text{space}$, which gains entropy at a rate $P/T_\text{space}$. The net production rate is the difference: $\frac{dS}{dt}\approx\frac{P}{T_\text{space}}-\frac{P}{T_\text{Sun}}=\frac{4\cdot 10^{26}}{3}-\frac{4\cdot 10^{26}}{5800}\approx 1{,}3\cdot 10^{26}\;\text{J/(K·s)}$

The cold term clearly dominates, because $1/3\gg 1/5800$: it is the fact that solar energy is “degraded” from $5800\;\text{K}$ down to $3\;\text{K}$ that makes the entropy production so large. $\ev{dS/dt\approx +1{,}3\cdot 10^{26}\;\text{J/(K·s)}}$

Linked atoms

Topics: Entropia e secondo principio Concepts: Entropia Skills: Stime di Fermi · Bilancio entropico Objects: Corpo nero