A distant galaxy is simultaneously rotating and receding fro

A distant galaxy is simultaneously rotating and receding from the earth. As the drawing shows, the galactic center is receding from the earth at a relative speed of u_G = 2.30 × 10⁶ m/s. Relative to the center, the tangential speed is v_T = 0.390 × 10⁶ m/s for locations A and B, which are equidistant from the center. When the frequencies of the light coming from regions A and B are measured on earth, they are not the same and each is different from the emitted frequency of 6.549 × 10¹⁴ Hz. Find the measured frequency for the light from (a) region A and (b) region B. (Give your answer to 4 significant digits. Use 2.998 × 10⁸ m/s as the speed of light.)

Solution

A. Tangential speed at A is toward earth:
v = 1.91E6 m/s
ß = v/c = 6.3709139426E-3
fA/f0 = sqrt((1+ß)/(1-ß)) = 1.006
fA = 6.588294E14 Hz
B. Tangential speed at B is away from earth:
v = -2.690E6 m/s
ß = v/c = -9.002677E-3
fB/f0 = sqrt((1+ß)/(1-ß)) = 0.99103
fB = 6.490E14 Hz