Question

The energy from the sun reaches just outside the earth's atmosphere at a rate of 1400 W m${}^{-2}$. The distance between the sun and the earth is $1.5\times {10}^{11}m$.
(a) Calculate the rate at which the sun is losing its mass.
(b) How long will the sun last assuming a constant decay at this rate? The present mass of the sun is $2\times {10}^{30}kg$

Answer 1

Given : $r=1.5\times {10}^{11}$ m $I=1400$ $W{m}^{-2}$ $m_o=2\times {10}^{30}$ kg

Power radiated by sun $P=I\left(4\pi r^2\right)=1400\times (4\pi \times (1.5\times {10}^{11}{)}^2)=3.96\times {10}^{26}$ $W$

Thus rate of losing mass $\frac{dm}{dt}=\frac{P}{c^2}=\frac{3.96\times {10}^{26}}{(3\times {10}^8{)}^2}=4.4\times {10}^9$ $kg/s$

Time for which the sun would last $t=\frac{m_o}{\frac{dm}{dt}}=\frac{2\times {10}^{30}}{4.4\times {10}^9}=4.55\times {10}^{20}$ s

$⟹$ $t=\frac{4.55\times {10}^{20}}{365\times 86400}=1.44\times {10}^{13}$ years