Chapter 16, Problem 16.74P

DATA Supernova! (a) Equation (16.30) can be written as

$f_{\text{R}}=f_{\text{S}}{\left(1-\frac{\upsilon }{c}\right)}^{1/2}{\left(1+\frac{\upsilon }{c}\right)}^{-1/2}$

where c is the speed of light in vacuum, 3.00 × 10⁸ m/s. Most objects move much slower than this (υ/c is very small), so calculations made with Eq. (16.30) must be done carefully to avoid rounding errors. Use the binomial theorem to show that if υ ≪ c, Eq. (16.30) approximately reduces to f_R = f_S [1 − (υ/c)]. (b) The gas cloud known as the Crab Nebula can be seen with even a small telescope. It is the remnant of a supernova, a cataclysmic explosion of a star. (The explosion was seen on the earth on July 4, 1054 C.E.) Its streamers glow with the characteristic red color of heated hydrogen gas. In a laboratory on the earth, heated hydrogen produces red light with frequency 4.568 × 10¹⁴ Hz; the red light received from streamers in the Crab Nebula that are pointed toward the earth has frequency 4.586 × 10¹⁴ Hz. Estimate the speed with which the outer edges of the Crab Nebula are expanding. Assume that the speed of the center of the nebula relative to the earth is negligible. (c) Assuming that the expansion speed of the Crab Nebula has been constant since the supernova that produced it, estimate the diameter of the Crab Nebula. Give your answer in meters and in light-years. (d) The angular diameter of the Crab Nebula as seen from the earth is about 5 arc-minutes ( 1 arc-minute = $\frac{1}{60}$ degree ). Estimate the distance (in light-years) to the Crab Nebula, and estimate the year in which the supernova actually took place.