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 uG = 2.30 × 106 m/s. Relative to the center, the tangential speed is vT = 0.350 × 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 5.118 × 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

According to the guven problem,

Using formula,

f/f₀ = (1+ß)/(1-ß), where ß = v/c.

a.) Tangential speed at A is toward earth:

v = v_G - v_T = 1.68*10⁶ m/s

ß = v/c = 5.6037 *10^-3

f_A/f₀ = (1+ß)/(1-ß) = 1.00562

f_A = 5.146* 10¹⁴ Hz

b.) Tangential speed at B is away from earth:

v = -v_G - v_T = -2.65 *10⁶ m/s

ß = v/c = -8.839 *10^-3

f_B/f₀ = (1+ß)/(1-ß) = 0.9911995

f_B = 5.073 *10¹⁴Hz