Question

A planet radiates heat at a rate proportional to the fourth power of its surface temperature $T$ . If the steady temperature of the planet is due to an exactly equal amount of heat received from the sun then which of the following statements is true?

• A. The planet's surface temperature varies inversely as the distance of the sun
• B. The planet's surface temperature varies directly as the square of its distance from the sun
• C. The planet's surface temperature varies inversely as the square root of its distance from the sun
• D. The planet's surface temperature is proportional to the fourth power of distance from the sun

Answer 1

Answer: (C)

Rate of loss of energy by unit area of the planet $= \sigma \text{T}^{4}$ , where $\sigma $ is the Stefan's constant. Let $Q$ be the total energy emitted by the sun every second. If $d$ is the distance of the planet from sun, then $Q$ falls uniformly over the inner surface of the sphere of radius $d$ . Rate of gain of heat by unit area of planet,
$=\frac{\text{Q}}{4 \pi \text{d}^{2}}$
For steady temperature of planet,
$\sigma \text{T}^{4} = \frac{\text{Q}}{4 \pi \text{d}^{2}}$
$\text{T}^{4} = \frac{\text{Q}}{4 \pi \sigma \text{d}^{2}}$ or $\text{T} = \left(\frac{\text{Q}}{4 \pi \sigma \left(\text{d}\right)^{2}}\right)^{1 / 4}$
$\text{T} \propto \frac{1}{\sqrt{\text{d}}}$