Question

The sun (and the earth with it) are traveling at 2.00×104 km/s toward a star that is 45.0 light years away. (A light year is the distance light travels in one year.) If an astronomer on earth measures light of wavelength 522.4 nm from this star, what frequency would an observer at rest with respect to the star measure for the light it emits? If an astronomer on earth measures light of wavelength 522.4 nm from this star, what wavelength would an observer at rest with respect to the star measure for the light it emits?

Answer 1

velocity of observer = 2*10⁷ m/s

wavelength observed by moving observer = 522.4 nm

beta = - v/c = -(2*10⁷) / (3*10⁸) = -2/30

wavelength emitted = wavelength observed * sqrt [ (1 - beta) / (1 + beta) ]

=> wavelength emitted = 522.4 *  sqrt [ (1 + (2/30)) / (1 - (2/30)) ] = 558.47 nm

frequency observed by an observer at rest with respect to the star = c/lambda = 3*10⁸ / (558.47*10^-9) = 5.37*10¹⁴ Hz

wavelength observed by an observer at rest with respect to the star = wavelength emitted = 558.47 nm