A galaxy is observed receding from Earth with a speed of 9800 km/s. A redshifted spectral line originating in the galaxy is compared to the same spectral line originating in the laboratory. If Hubble's law is valid for this galaxy, what is the ratio of the wavelength of the redshifted line to that of the unshifted line if the Hubble constant is 22 km/s per million light-years? (c = 3.00 × 10⁸ m/s, 1 ly = 9.461 × 10¹⁵ m, 1 y = 3.156 × 10⁷ s) A) 1.033 B) 1.052 C) 1.072 D) 1.091 E) 1.130

Question

Quizplus · Accepted Answer

The correct answer is A.Explanation:The redshift of the spectral line is given by:$$z = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}$$where $\lambda_{obs}$ is the observed wavelength, $\lambda_{emit}$ is the emitted wavelength, and z is the redshift.Hubble's law states that the redshift of a galaxy is proportional to its distance from Earth:$$z = H_0 \frac{d}{c}$$where $H_0$ is the Hubble constant, d is the distance to the galaxy, and c is the speed of light.In this problem, we are given that the speed of the galaxy is 9800 km/s. We can use this to find the distance to the galaxy:$$d = \frac{v}{H_0} = \frac{9800 \text{ km/s}}{22 \text{ km/s per million light-years}} = 445.5 \text{ million light-years}$$Now we can use Hubble's law to find the redshift of the galaxy:$$z = H_0 \frac{d}{c} = (22 \text{ km/s per million light-years}) \frac{(445.5 \text{ million light-years})}{(3.00 \times 10^8 \text{ m/s})} = 0.033$$Finally, we can use the redshift to find the ratio of the observed wavelength to the emitted wavelength:$$\frac{\lambda_{obs}}{\lambda_{emit}} = 1 + z = 1 + 0.033 = 1.033$$Therefore, the ratio of the wavelength of the redshifted line to that of the unshifted line is 1.033.

A Galaxy Is Observed Receding from Earth with a Speed