Thermal Radiation And Stefan's Law

A black body radiates energy at a rate governed by the Stefan–Boltzmann law, $P=\sigma A{T}^4$, where the emitting area $A=4\pi R^2$ for a sphere and $T$ is the absolute surface temperature. Combining these, the total power is $P=\sigma (4\pi R^2)T^4$, so $P\propto R^2{T}^4$. Forming the ratio for the two stars, $\frac{P_2}{P_1}={\left(\frac{R_2}{R_1}\right)}^2{\left(\frac{T_2}{T_1}\right)}^4=(2{)}^2\times (2{)}^4=4\times 16=64$. The value 8 ignores the temperature exponent and uses $R^{2}T$. The value 16 keeps only the $T^4$ factor and forgets the area. The value 32 mixes an incorrect $R^2{T}^3$ combination. This strong dependence, especially the fourth power of absolute temperature, is the essence of the NCERT treatment of black-body radiation, and it must use the Kelvin scale because $T$ measures the true thermal energy content of the radiator. Wien's displacement law accompanies Stefan's law and tells us the hotter star also radiates its peak at a shorter wavelength, appearing bluer than its cooler companion. As a plausibility check, doubling the temperature alone already multiplies power sixteenfold, and the larger surface adds another factor of four, so a total ratio of 64 is the physically expected, sharply amplified result for a hotter, bigger star.