Given that the Earth is in radiative equilibrium and the equivalent black body temperature of the Earth without having any atmosphere is T1​. If the Sun’s emission changes abruptly such that the solar constant increases to four times its previous value, the new equivalent black body temperature of the Earth will be:

Correct option is D

Introduction:

• For a planet to maintain a stable temperature, it must be in Radiative Equilibrium, meaning the energy it absorbs from the Sun must equal the energy it radiates back into space.
• Earth absorbs solar radiation (shortwave) over its cross-sectional area ($$\pi R^2$$​ and emits terrestrial radiation (longwave) over its entire surface area ($$4\pi R^2$$​).
• According to the Stefan-Boltzmann Law, the energy flux ($E$) emitted by a black body is directly proportional to the fourth power of its absolute temperature (T):
• ​$$E = \sigma T^4$$​​
• where $$\sigma$$​ is the Stefan-Boltzmann constant ($$5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4$$​). By equating the incoming and outgoing energy, we establish the relationship between the Solar Constant (S) and the Earth's equilibrium temperature (T):
• ​$$S(1 - A) \pi R^2 = \sigma T^4 (4\pi R^2)$$​​
• ​$$\frac{S(1-A)}{4} = \sigma T^4$$​​
• This implies that $$T \propto S^{1/4}$$​​

Information Booster:

• The "equivalent black body temperature" $$T_1$$​ is derived from the initial solar constant ($$S_1$$​). We can determine the new temperature ($$T_2$$​) when the solar constant increases to four times its original value ($$S_2 = 4S_1$$).​
• Step-by-step Derivation:

1. Initial State:

   ​$$T_1 = \left( \frac{S_1(1-A)}{4\sigma} \right)^{1/4}$$​​

2. New State $$S_2 = 4S_1$$​​

   ​$$T_2 = \left( \frac{4S_1(1-A)}{4\sigma} \right)^{1/4}$$​​

3. Ratio Calculation:

   ​$$\frac{T_2}{T_1} = \frac{(4S_1)^{1/4}}{(S_1)^{1/4}} = 4^{1/4}$$​​

4. Simplifying the Power:

   Since $$4 = 2^2, we can write 4^{1/4} as (2^2)^{1/4} = 2^{2/4} = 2^{1/2}$$​​

   ​$$T_2 = 2^{1/2} T_1 = \sqrt{2} T_1$$​​

   The new temperature is $$\sqrt{2} T_1$$​ (approximately 1.41 $$T_1$$​).