Question

Assume earth is revolving around sun in circular orbit. If radius of earth's orbit around sun is increased 6%. Find percentage decrement in temperature of earth.

• A. 1%
• B. 2%
• C. 3%
• D. 4%

Answer 1

Answer: (C)

3%

Answer 2

1. The equilibrium temperature of Earth is determined by the balance between the absorbed solar radiation and the emitted thermal radiation. Since the solar flux $S$ varies as

   $$S \propto \frac{1}{r^2},$$

   and the temperature $T$ from the Stefan-Boltzmann law follows

   $$T \propto S^{1/4},$$

   we get

   $$T \propto \frac{1}{r^{1/2}}.$$

2. When the radius of the Earth’s orbit is increased by 6%, the new distance is

   $$r' = 1.06\,r,$$

   so the new temperature $T'$ becomes

   $$T' = \frac{T}{\sqrt{1.06}}.$$

3. The fractional change in temperature is

   $$\frac{T' - T}{T} = \frac{1}{\sqrt{1.06}} - 1.$$

   Estimating,

   $$\sqrt{1.06} \approx 1.02956 \quad \Longrightarrow \quad \frac{1}{\sqrt{1.06}} \approx 0.9713.$$

   Thus,

   $$\frac{T' - T}{T} \approx 0.9713 - 1 = -0.0287,$$

   which is a decrease of about $2.87\%$ (approximately $3\%$).

In summary: Increased orbital radius decreases the solar flux, leading to a decrease in equilibrium temperature by approximately 3%.