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lecture-06 - These lecture notes were prepared for Rutgers...

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These lecture notes were prepared for Rutgers Physics 341/342: Principles of Astrophysics by Prof. Chuck Keeton, and modified by Profs. Saurabh Jha and Eric Gawiser. All rights reserved. c 2011 Lecture 6: Supermassive Black Holes in Other Galaxies I. Technical Background: Doppler Effect (See § 4.3 of Carroll & Ostlie for more details.) You are probably familiar with the Doppler effect: when a train/airplane/racecar passes by, the sound shifts from a high pitch to a low pitch. In 1842, Christian Doppler explained this effect by noting that as a source of sound moves toward you, the sound waves get compressed so the frequency (pitch) increases; while when the source moves away, the waves get stretched out so the frequency goes down. He showed that the change in wavelength can be written as λ obs - λ rest λ rest = v c s where c s is the speed of sound, and v is the speed with which the source is moving away from you (so v < 0 if the source is moving toward you). Figure 1: (From http://odin.physastro.mnsu.edu/˜eskridge/astr101/kauf5 23.JPG.) The same thing happens for light, as shown above, but the details are slightly different because of relativity. Specifically, when we take relativity into account (as we will see later 1
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in the semester), we arrive at the exact relativistic Doppler formula : λ obs λ rest = s 1 + v/c 1 - v/c When the speed of the source is slowed compared with the speed of light, we can use our favorite Taylor series approximation 1 to write: λ obs λ rest = 1 + v c 1 / 2 1 - v c - 1 / 2 = 1 + 1 2 v c + O v c 2 1 + 1 2 v c + O v c 2 = 1 + 1 2 v c + 1 2 v c + O v c 2 = 1 + v c + O v c 2 So when v c , we derive the non-relativistic Doppler formula λ obs λ rest 1 + v c We sometimes write this in terms of the redshift z , defined by z λ obs - λ rest λ rest Given this definition, note that z = λ obs rest - 1, and in the non-relativistic case z v/c . Now the idea is that if we measure the observed wavelength λ obs of some light, and we know the corresponding rest wavelength λ rest (e.g., from lab measurements), then we can use these formulas to determine the speed with which an object is moving toward or away from us. II. NGC 4258 = M106 After the Milky Way, the galaxy where we have the best constraints on a supermassive black hole is NGC 4258. Radio observations reveal that there are water masers moving in circular orbits around the center of the galaxy. Maser stands for microwave (or molecular) amplification by stimulated emission of radiation; it is basically the same thing as laser (which after all is an acronym for light amplification by stimulated emission of radiation). A maser emits light at very specific wavelengths, so we can use the Doppler effect to measure the velocity. 1 As always, (1 ± x ) α = 1 ± αx + O ( x ) 2 . So if x 1, then (1 ± x ) α 1 ± αx .
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