Question

The temperature of earth is maintained by a dynamic equilibrium between Sun and Earth. Sun & Earth can be assumed to be black bodies :

• A. If the power output of sun would double with changing the temperature, equilibrium temperature of earth also doubles.
• B. If the radius of sun doubles without changing its' power, its surface temperature would decrease by factor of $\sqrt{2}$.
• C. If the radius of earth doubles without any change in sun, it's equilibrium temperature would increase by factor of $\sqrt{2}$.
• D. If the distance between earth and sun would decrease by a factor of 2, the equilibrium temperature of earth would increase by factor of $\sqrt{2}$.

Answer 1

Answer: (B), (D)

(B), (D)

Answer 2

The Earth's equilibrium temperature is determined by the balance between the power absorbed from the Sun and the power radiated by the Earth. Assuming both Sun and Earth are black bodies:

The power radiated by the Sun (a black body of radius $R_S$ and surface temperature $T_S$) is $P_S = 4\pi R_S^2 \sigma T_S^4$, where $\sigma$ is the Stefan-Boltzmann constant.

At a distance $d$ from the Sun, the intensity of solar radiation is $I_S = \frac{P_S}{4\pi d^2}$.

The Earth intercepts this radiation over its cross-sectional area $\pi R_E^2$. Since the Earth is assumed to be a black body, it absorbs all the incident radiation. The power absorbed by the Earth is $P_{abs} = I_S (\pi R_E^2) = \frac{P_S}{4\pi d^2} (\pi R_E^2) = \frac{P_S R_E^2}{4d^2}$.

The Earth radiates energy as a black body with surface temperature $T_E$ over its entire surface area $4\pi R_E^2$. The power radiated by the Earth is $P_{rad} = 4\pi R_E^2 \sigma T_E^4$.

In equilibrium, $P_{abs} = P_{rad}$: $\frac{P_S R_E^2}{4d^2} = 4\pi R_E^2 \sigma T_E^4$

We can cancel $R_E^2$ from both sides: $\frac{P_S}{4d^2} = 4\pi \sigma T_E^4$

Solving for $T_E$: $T_E^4 = \frac{P_S}{16\pi d^2 \sigma}$ $T_E = \left(\frac{P_S}{16\pi d^2 \sigma}\right)^{1/4}$

Alternatively, substituting $P_S = 4\pi R_S^2 \sigma T_S^4$: $\frac{4\pi R_S^2 \sigma T_S^4}{4d^2} = 4\pi \sigma T_E^4$ $\frac{R_S^2 T_S^4}{d^2} = 4 T_E^4$ $T_E^4 = \frac{R_S^2 T_S^4}{4d^2}$ $T_E = \left(\frac{R_S^2 T_S^4}{4d^2}\right)^{1/4} = \frac{(R_S^2)^{1/4} (T_S^4)^{1/4}}{(4)^{1/4} (d^2)^{1/4}} = \frac{R_S^{1/2} T_S}{\sqrt{2} d^{1/2}}$.

Let's re-derive from the $P_S$ formula. $T_E^4 = \frac{P_S}{16\pi d^2 \sigma}$. $T_E \propto P_S^{1/4} d^{-1/2}$.

Let's check the statements based on $T_E \propto P_S^{1/4} d^{-1/2}$ and $P_S = 4\pi R_S^2 \sigma T_S^4$.

(A) If the power output of sun would double ($P_S' = 2P_S$), equilibrium temperature of earth also doubles. $T_E' \propto (P_S')^{1/4} \propto (2P_S)^{1/4} = 2^{1/4} P_S^{1/4} \propto 2^{1/4} T_E$. $T_E' = 2^{1/4} T_E$. Since $2^{1/4} \approx 1.189 \neq 2$, the temperature does not double. Statement (A) is false.

(B) If the radius of sun doubles ($R_S' = 2R_S$) without changing its' power ($P_S' = P_S$), its surface temperature would decrease by factor of $\sqrt{2}$. $P_S = 4\pi R_S^2 \sigma T_S^4$. $P_S' = 4\pi (R_S')^2 \sigma (T_S')^4$. Given $P_S' = P_S$ and $R_S' = 2R_S$: $4\pi R_S^2 \sigma T_S^4 = 4\pi (2R_S)^2 \sigma (T_S')^4$ $R_S^2 T_S^4 = 4R_S^2 (T_S')^4$ $T_S^4 = 4 (T_S')^4$ $(T_S')^4 = \frac{1}{4} T_S^4$ $T_S' = \left(\frac{1}{4}\right)^{1/4} T_S = \frac{1}{4^{1/4}} T_S = \frac{1}{\sqrt{2}} T_S$. The surface temperature decreases by a factor of $\sqrt{2}$. Statement (B) is true.

(C) If the radius of earth doubles ($R_E' = 2R_E$) without any change in sun, it's equilibrium temperature would increase by factor of $\sqrt{2}$. The formula $T_E = \left(\frac{P_S}{16\pi d^2 \sigma}\right)^{1/4}$ shows that $T_E$ is independent of $R_E$. Alternatively, the equilibrium equation $\frac{P_S R_E^2}{4d^2} = 4\pi R_E^2 \sigma T_E^4$ shows that $R_E^2$ cancels out, so $T_E$ does not depend on $R_E$. If $R_E$ doubles, $T_E$ remains unchanged. Statement (C) is false.

(D) If the distance between earth and sun would decrease by a factor of 2 ($d' = d/2$), the equilibrium temperature of earth would increase by factor of $\sqrt{2}$. $T_E \propto d^{-1/2}$. $T_E' \propto (d')^{-1/2} \propto (d/2)^{-1/2} = (d^{-1} 2)^{1/2} = \sqrt{2} d^{-1/2} \propto \sqrt{2} T_E$. $T_E' = \sqrt{2} T_E$. The equilibrium temperature increases by a factor of $\sqrt{2}$. Statement (D) is true.