(See page 461 in your text)
When Johannes Kepler developed his Third Law (chapter 1), all he had to work with was the planets going around the Sun. Thus is formulation of the law applies only to planets in our Solar System. But when Isaac Newton revised Kepler's Laws about 150 years later, he realized that they were of a more universal nature, and could be slightly modified to apply to other circumstances. In particular, he revised Kepler's Third Law to apply to satellites orbitting planets, stars orbitting other stars, and even the Sun orbitting the center of the Milky Way (although neither Kepler nor Newton knew that they Milky Way was a rotating galaxy). Put very simply, this is how Newton reformulated Kepler's Third Law:
M1 + m2 = a3/P2
where M1 is the mass of one of the two objects, and m2 the other. The small a represents the average distance between them in Astronomical Units or AU, and P is the orbital period in years. In this simplification, we consider that m2 is negligible compared to the other mass and can be ignored. Thus it will disappear from the equation. (If we were looking at two objects of similar mass, we could not ignore either of the masses, and although the solution would be more difficult, it could still be found.) Strictly speaking, the a is the semi-major axis of the orbit, but if you consider M1 vastly more massive than m2, and the orbit as being circular, then we can simply consider it the distance between the centers of mass of each object.
According to your book, the distance of the Sun to the center of the Milky Way (a) is about 8.5 kiloparsecs or 28,000 light years (page 444). This rather large number must be converted to Astronomical Units. You could multiply 28,000 by the number of kilometers in a light year, and then divide by 149.6 million (the number of kilometers in an AU). Or you could just multiply 28,000 by 63,000 (the approximate number of AU in a light year). Doing the latter we get about:
a = 1.8 *109 (the figure has been rounded to the most significant figures.)
Using the concept of Doppler Shift (see pages 112 & 463), astronomers have measured the motions of other stars relative to the Sun to find the Sun's speed around the Galactic center. They have found that the Sun hurtles around the Milky Way core at 220 kilometers per second. (That's like flying from Los Angeles to San Francisco in 3 seconds!)
Remember the formula for the Circumference of a circle, C = 2&piR, where R is the radius of the circle (a in our case). Knowing the circumference of the orbit and how fast the Sun is traveling, you can figure out how long it takes to go around once, P. In this case it is about 2.4 * 108 years.
So, finally, to find the Galactic mass in terms of the Sun's mass, we must cube a and divide it by P squared:
M1 = (1.8 * 109)3 / (2.4 * 108)2
which is:
M1 = 1.83 * 10(9+9+9) / 2.42 * 10(8+8)
M1 = 5.8 * 1027 / 5.8 * 1016
M1 = 1 * 1011
Thus the mass of the Galaxy is at least 100 billion times that of the Sun!
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