Rotation curve and Doppler shift

The differential galactic rotation produces Doppler shifts in spectral lines that we observe from gas at different distances from the galactic center than the Sun. This is illustrated in Fig. 16.4. In Fig. 16.4(a), we look at five test particles at different distances along the line of sight. In each case the Doppler shift depends on the relative radial velocity of the test particle and the Sun. That is, we take the line of sight component of the particle's motion and subtract the line of sight component of the Sun's motion.

In Fig. 16.4(b) we look at the Doppler shifts for each test particle. Point 1 is slightly closer to the center than the Sun. It is moving slightly faster than we are, so there will be a small Doppler shift. It is moving away from us so that the shift will be to longer wavelength (redshift). Point 2 is where our line of sight crosses the same circle. The speed is the same as at point 1, and the angle with the line of sight is the negative of that at point 1. Since the line of sight component depends on the cosine of that angle, that component is the same. The Doppler shift for point 2 is therefore the same as for point 1. Point 3 is where the line of sight passes closest to the center. Material is moving fastest around that circle. It is also moving directly away from us, so that point has the largest Doppler shift to the red. Point 4 is in the same orbit as the Sun. Our distance from that object is constant, so our relative radial velocity must be zero, meaning that the Doppler shift is zero. Point 5 is farther from the center than the Sun, and is moving u

Line of Sight

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