The spectrum of a star is used to measure its radial velocity with respect to the Sun, the component of the star's velocity vector that is parallel to a vector connecting the star's centre to the centre of the Sun. The measurement relies on the Doppler effect in which radiation is shifted in frequency when the source is moving towards or away from the observer. When a star emitting electromagnetic radiation of frequency v moves with a speed s relative to an observer, the observer detects radiation of frequency v receding= v f or v approaching = v/f, where f= {(1 – s/c)/(l + s/c)}1/2 and c is the speed of light. It is easy to see that v receding < v and a receding star is characterized by a red shift of its spectrum with respect to the spectrum of an identical, but stationary source. Furthermore, v approaching > v and an approaching star is characterized by a blue shift of its spectrum with respect to the spectrum of an identical, but stationary source. In a typical experiment, v is the frequency of a spectral line of an element measured in a stationary Earth-bound laboratory from a calibration source, such as an arc lamp. Measurement of the same spectral line in a star gives v star and the speed of recession or approach may be calculated from the value of v and the equations above.

(a) Three Fe I lines of the star HDE 271 182, which belongs to the Large Magellanic Cloud, occur at 438.882 nm, 441.000 nm, and 442.020 nm. The same lines occur at 438.392 nm, 440.510 nm, and 441.510 nm in the spectrum of an Earth-bound iron arc. Determine whether HDE 271 182 is receding from or approaching the Earth and estimate the star's radial speed with respect to the Earth.

(b) What additional information would you need to calculate the radial velocity of HDE 271 182 with respect to the Sun?