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) A) 2.5 × 10⁹ ly B) 9.1 × 10⁸ ly C) 2.7 × 10⁸ ly D) 9.1 × 10⁷ ly E) 2.7 × 10⁷ ly

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The correct answer is A because the redshift of the Hα spectral line is due to the Doppler effect, which is caused by the star moving away from Earth. The redshift is given by the formula z = (λ_observed - λ_emitted) / λ_emitted, where λ_observed is the observed wavelength, λ_emitted is the emitted wavelength, and z is the redshift. The Hubble constant is given by the formula H = v / d, where v is the velocity of the star and d is the distance to the star. The velocity of the star is given by the formula v = c * z, where c is the speed of light. Substituting these formulas into the Hubble constant formula, we get H = c * z / d. Solving for d, we get d = c * z / H. Substituting the given values into this formula, we get d = (3.00 × 10^8 m/s) * (789.0 nm - 656.3 nm) / (656.3 nm) / (22 km/s per million light-years) * (1000 m/km) * (1 ly / 9.461 × 10^15 m) * (1 million ly / 10^6 ly) = 2.5 × 10^9 ly.

The Hα Spectral Line of Hydrogen Has a Wavelength of 656.3

The H_α Spectral Line of Hydrogen Has a Wavelength of 656.3