Answer to Question #166938 in Astronomy | Astrophysics for Beth

Question

a) A particular emission line is originally emitted with a wavelength of 556.3 nm from a gas cloud. At our telescope, we observe the wavelength of the emission line to be 556.6 nm. How fast is this gas cloud moving toward or away from Earth?

b) Two stars have the same size and are the same distance from us. Star A has a surface temperature of 5500 K, and star B has a surface temperature twice as high, 11,000 K. How much more luminous is star B compared to star A?

c) What is the value of power received in orbit around Mars by a 1 m² solar panel? Assume Mars is 1.5 A.U. from the Sun at the time of measurement. 

Answer 1

a) Due to the Doppler effect, the increase of a wavelength corresponds to the movement away from the observer.

$v_r = \dfrac{\Delta\lambda}{\lambda} c = \dfrac{0.3}{556.3}\cdot 3\cdot10^5\,\mathrm{km/s} = 162\,\mathrm{km/s}.$

b) According to the Stephen-Boltzmann law, luminosity is proportional to temperature powered to 4 and to the area of surface

$L_1 = 4\pi R_1^2 \sigma T_1^4, \;\; L_2 = 4\pi R_2^2 \sigma T_2^4, \;\; \dfrac{L_2}{L_1} = \dfrac{R_2^2}{R_1^2} \cdot \dfrac{T_2^4}{T_1^4} = 16.$

c) The illuminance is

$E = \dfrac{L_{\odot}}{4\pi a^2} = \dfrac{4\cdot10^{26}\,\mathrm{W}}{4\cdot \pi \cdot (1.5\cdot1.5\cdot10^{11}\,\mathrm{m})^2} = 629 \,\mathrm{W/m^2}.$