Hubble's law: The H_α spectral line of hydrogen has a wavelength of 656.3 nm as measured in the laboratory. The H_α spectral line is observed in light from a distant star at a redshifted wavelength of 789.0 nm. If Hubble's law is valid for this star, estimate how far is it from Earth 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)

Question

Quizplus · Accepted Answer

Hubble's law states that the redshift of a galaxy is proportional to its distance from Earth. The redshift is the difference between the observed wavelength of light from a galaxy and the wavelength of the same light as measured in the laboratory. The Hubble constant is the constant of proportionality between the redshift and the distance.In this problem, the observed wavelength of the Hα spectral line is 789.0 nm and the laboratory wavelength is 656.3 nm. The redshift is therefore:$$z = \frac{\lambda_{observed} - \lambda_{laboratory}}{\lambda_{laboratory}} = \frac{789.0 nm - 656.3 nm}{656.3 nm} = 0.201$$The distance to the star is then:$$d = \frac{cz}{H} = \frac{(3.00 	imes 10^8 m/s)(0.201)}{22 km/s per million light-years} = 2.7 	imes 10^8 ly$$Therefore, the correct answer is C.

Hubble's Law: the Hα Spectral Line of Hydrogen Has a Wavelength

Hubble's Law: the H_α Spectral Line of Hydrogen Has a Wavelength