If the radius of a star is \(R\) and it acts as a black body, what would be the temperature of the star at which the rate of energy production is \(Q\)?
1. \(\frac{Q}{4\pi R^2\sigma}\)
2. \(\left(\frac{Q}{4\pi R^2\sigma}\right )^{\frac{-1}{2}}\)
3. \(\left(\frac{4\pi R^2 Q}{\sigma}\right )^{\frac{1}{4}}\)
4. \(\left(\frac{Q}{4\pi R^2 \sigma}\right)^{\frac{1}{4}}\)

The total radiant energy per unit area, normal to the direction of incidence, received at a distance \(R\) from the centre of a star of radius \(r,\) whose outer surface radiates as a black body at a temperature \(T\) K is given by: (Where \(\sigma\) is Stefan’s constant):
1. \(\frac{\sigma r^{2}T^{4}}{R^{2}}\)

2. \(\frac{\sigma r^{2}T^{4}}{4 \pi R^{2}}\)

3. \(\frac{\sigma r^{2}T^{4}}{R^{4}}\)

4. \(\frac{4\pi\sigma r^{2}T^{4}}{R^{2}}\)

A black body at \(227^{\circ}~\mathrm{C}\) radiates heat at the rate of \(7~ \mathrm{cal-cm^{-2}s^{-1}}\).  At a temperature of \(727^{\circ}~\mathrm{C}\), the rate of heat radiated in the same units will be:
1. \(60\)
2. \(50\)
3. \(112\)
4. \(80\)