Question

Radiant energy from the Sun, approximately$\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{\times }{\mathbf{10}}^{11}\mathbf{m}$away, arrives at Earth with an intensity of$\mathbf{1}\mathbf{.}\mathbf{5}\mathbf{kW}/{\mathbf{m}}^2$. At what rate is mass being converted in the Sun to produce this radiant energy?

Short answer

4.7 million tons of mass being converted into radiation energy per second.

Step-by-step solution

Define Radiation energy and Radiation flux

Radiant energy or also usually called luminosity and is defined as the total energy radiated by the sun per unit time. The units of luminosity are W or J/s.

Intensity or also called radiation flux is defined as the energy flowing through per unit time per unit area.The unit is${W/m}^2$. The radiant fluxF of a star, is related to its luminosity L as follows:

$F=\frac{L}{4\pi r^2}$

Determine Mass converted per second

F is the intensity which is$1.5kW/m^2$and the distance r to the earth from sun is$1.5\times {10}^{11}m$, putting the values, we get luminosity L as$4.24\times {10}^{26}W$. It means that every second sun is radiating$4.24\times {10}^{26}J$of energy. By using Einstein’s mass-energyrelation one can determine the amount of mass in the core of the sun converted into radiative energy per second as follows:

$\begin{array}{c}E=m{c}^2\\ m=\raisebox{1ex}{$E$}\!\left/ \!\raisebox{-1ex}{$c^2$}\right.\\ m=\frac{4.24\times {10}^{26}\mathrm{J}}{(3\times {10}^8\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.)}\\ =4.7\times {10}^{9}kg\end{array}$

Therefore, 4.7 million tons of mass is being converted into this radiant energy every second in the sun’s core.