Question
(a) State what is meant by luminosity of a star.
(b) The luminosity of the Sun is \(3.83 \times 10^{26}\) W. The distance between the Earth and the Sun is \(1.51 \times 10^{11} m\).
Calculate the radiant flux intensity F of the Sun at the Earth. Give a unit with your answer.
(c) Use data from (b) to calculate the mass that is converted into energy every second in the Sun.
(d) The radius of the Sun is \(6.96 \times 10^{8}\) m.
Show that the temperature T of the surface of the Sun is \(5770\) K.
(e) The wavelength \(λ_{max}\) of light for which the maximum rate of emission occurs from the Sun is \(5.00 \times 10^{–7}\)m.
The temperature of the surface of the star Sirius is \(9940\) K.
Use information from (d) to determine the wavelength of light for which the maximum rate of emission occurs from Sirius.
Answer/Explanation
Ans:
(a) total power of radiation emitted (by the star)
(b) \(F\) = \(\frac{L}{4\pi d^{2}}\)
= \(\frac{3.83\times 10^{26}}{4\times \pi 1.51\times 10^{112}}\)
= \(1340 W\, m^{-2}\)
(c) \(m = \frac{E}{c^{2}}\)
= \(\frac{3.83\times 10^{26}}{3.00\times 10^{82}}\)
= \(4.26 \times 10^{9}\) kg
(d) \(L = 4\pi \sigma r^{2}T^{4}\)
\(3.83 \times 10^{26}\) = \(4\times \pi 5.67\times 10^{-8}\times 6.96 \times 10^{82}\times T^{4} leading to T\) = \(5770\) K
(e) \(\lambda _{(max)} \frac{1}{T}\)
\(\frac{5.00 \times 10^{-7}}{\lambda} = \frac{9940}{5770}\)