Consider a spiral galaxy that is moving directly away from Earth with a speed V

Question

Consider a spiral galaxy that is moving directly away from Earth with a speed V = 4.300×105 m/s at its center, as shown in the figure(Figure 1) . The galaxy is also rotating about its center, so that points in its spiral arms are moving with a speed v = 6.500×105 m/s relative to the center.

part a: If light with a frequency of 8.245×1014 Hz is emitted in both arms of the galaxy, what frequency is detected by astronomers observing the arm that is moving toward the Earth? (Measurements of this type are used to map out the speed of various regions in distant, rotating galaxies.) express answer using four sig figs.

Ftoward =

part b: If light with a frequency of 8.245×1014 Hz is emitted in both arms of the galaxy, what frequency is detected by astronomers observing the arm that is moving away from Earth?

Express your answer using four significant figures.

Faway =

Explanation / Answer

The velocities given are much less than the speed of light, so ordinary doppler equations apply. For the arms moving toward the earth the (red shift) velocity of the emitters is 4.300*10^5 - 6.500*10^5 = -2.200*10^5 m/s. The doppler formula is

v/c =(fe - fo)/fo

fo*v/c = fe - fo

fo*(1 + v/c) = fe

fo = fe/(1 + v/c)

v/c = -2.2*10^5 / 2.998*10^8 = -0.7338*10^-3

fo = fe*1.0007338

fo = 8.251*10^14 Hz

For the other arm, v = 4.300*10^5 + 6.500*10^5 = 10.8*10^5 m/s

fo = fe/(1 + 0.003602) = 8.215*10^14 Hz

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