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

Options

A. The planets surface temperature vaires 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 proporional to the fourth power of distance from the sun.

Explanation

To solve the problem, we need to analyze the relationship between the heat radiated by the planet and the heat received from the sun. Let's break it down step by step.

### Step 1: Understand the Heat Radiation from the Planet
The planet radiates heat at a rate proportional to the fourth power of its surface temperature \( T \). This relationship is given by the Stefan-Boltzmann law:
\[ P = \sigma T^4 \]
where \( P \) is the power (rate of heat loss), \( \sigma \) is the Stefan-Boltzmann constant, and \( T \) is the absolute temperature of the planet.

### Step 2: Heat Received from the Sun
Let \( Q \) be the total energy emitted by the Sun per second. This energy spreads out uniformly over the surface area of a sphere with radius \( d \) (the distance from the Sun to the planet). The rate of heat received by the planet per unit area is:
\[ \text{Rate of gain of heat per unit area} = \frac{Q}{4 \pi d^2} \]

### Step 3: Steady State Condition
At steady state, the heat lost by the planet is equal to the heat gained from the Sun:
\[ \sigma T^4 = \frac{Q}{4 \pi d^2} \]

### Step 4: Rearranging the Equation
From the equation above, we can express the temperature \( T \) in terms of \( Q \) and \( d \):
\[ T^4 = \frac{Q}{4 \pi \sigma d^2} \]
Taking the fourth root of both sides gives:
\[ T = \left( \frac{Q}{4 \pi \sigma d^2} \right)^{1/4} \]

### Step 5: Analyzing the Relationship
From the equation \( T = \left( \frac{Q}{4 \pi \sigma d^2} \right)^{1/4} \), we can see that:
\[ T \propto \frac{1}{d^{1/2}} \]
This indicates that the surface temperature \( T \) of the planet is inversely proportional to the square root of the distance \( d \) from the Sun.

### Conclusion
Thus, the correct statement is that the planet's surface temperature varies inversely as the square root of its distance from the Sun.