Question

Assuming that the temperature at the surface of the sun is 6000 K, find out the size of a virtual star (in terms of the size of the sun) whose surface termperature is 3000 K and the power radiated by the virtual star is 25 times the power radiated by the sun. Treat both the sun and virtual star as a blackbody.

Answer 1

Given: T_sum = 6000 k, T_star = 3000 K,

P_star = 25 × P_sun

Power radiated by the sun,

${\displaystyle P_{sun}={\left(\frac{dQ}{dt}\right)}_{sun}}$

${\displaystyle =\sigma A_{sun}{T}_{sun}^4=\sigma 4\pi r_{sun}^2{T}_{sun}^4}$

Power radiated by the virtual star,

${\displaystyle P_{\text{star}}={\left(\frac{dQ}{dt}\right)}_{\text{star}}}$

${\displaystyle =\sigma A_{\text{star}}{T}_{\text{star}}^4=\sigma 4\pi r_{\text{star}}^2{T}_{\text{star}}^4}$

∴ ${\displaystyle \frac{P_{\text{star}}}{P_{sun}}=\frac{\sigma 4\pi r_{\text{star}}^2{T}_{\text{star}}^4}{\sigma 4\pi r_{sun}^2{T}_{sun}^4}}$

∴ ${\displaystyle 25=\frac{r_{\text{star}}^2{\left(3000\right)}^4}{r_{sun}^2{\left(6000\right)}^4}}$

∴ ${\displaystyle \frac{r_{\text{star}}^2}{r_{sun}^2}=25\times \frac{{\left(6000\right)}^4}{{\left(3000\right)}^4}=400}$

or ${\displaystyle r_{\text{star}}=20\times r_{sun}}$