1. Use Stefan–Boltzmann law for total power radiated by the Sun

   The total radiant power (luminosity) of a star is $L = 4\pi R^2 \sigma T^4$ where:

   • $R$ = radius of the Sun
   • $T$ = surface temperature
   • $\sigma$ = Stefan–Boltzmann constant

2. Initial luminosity

   Initially, $L_1 = 4\pi R^2 \sigma T^4$

3. New luminosity when radius and temperature change

   New radius $= 2R$ and new temperature $= 2T$.

   So, $L_2 = 4\pi (2R)^2 \sigma (2T)^4$

   Simplify: $L_2 = 4\pi \cdot 4R^2 \sigma \cdot 16T^4$ $L_2 = 64\pi R^2 \sigma T^4$

4. Ratio of new luminosity to old luminosity

   $\frac{L_2}{L_1} = \frac{64\pi R^2 \sigma T^4}{4\pi R^2 \sigma T^4} = 16$

   More directly, $\frac{L_2}{L_1} = \left(\frac{2R}{R}\right)^2 \left(\frac{2T}{T}\right)^4 = 2^2 \cdot 2^4 = 4 \cdot 16 = 64$

   The simplification in step 3 should be compared correctly with the original expression: $L_1 = 4\pi R^2 \sigma T^4$ $L_2 = 4\pi (2R)^2 \sigma (2T)^4 = 4\pi \cdot 4R^2 \cdot \sigma \cdot 16T^4 = 64\,(4\pi R^2 \sigma T^4)?$

   Let us compute carefully: $L_2 = 4\pi \times 4 \times 16 \times R^2 \sigma T^4 = 256\pi R^2 \sigma T^4$

   Since $L_1 = 4\pi R^2 \sigma T^4,$ therefore $\frac{L_2}{L_1} = \frac{256\pi R^2 \sigma T^4}{4\pi R^2 \sigma T^4} = 64$

5. Energy received on Earth

   The radiant energy received per unit time on Earth is proportional to the Sun’s luminosity divided by $4\pi d^2$, where $d$ is the Earth–Sun distance.

   Since $d$ remains unchanged, $\text{received energy} \propto L$

   Hence the ratio of energy received on Earth is also $64$

6. Check options

   • A: $32$ ❌
   • B: $16$ ❌
   • C: $4$ ❌
   • D: $64$ ✅

Therefore, the correct answer is D.