Hubble's law: 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

Using Hubble's Law, we have:

v = H*d

where v is the observed recessional velocity, H is the Hubble constant, and d is the distance to the galaxy from Earth.

We can rearrange this equation to solve for d:

d = v/H

In this problem, we are given v = 9800 km/s and H = 22 km/s per million light-years. We need to convert v to units of light-years, so we can use the conversion factor:

1 ly = 9.461 × 10^15 m

1 y = 3.156 × 10^7 s

v = 9800 km/s * (1000 m/km) * (1 ly/9.461 × 10^15 m) * (3.156 × 10^7 s/1 y)

= 10.44 million ly/year

Now we can plug in v and H to solve for d:

d = v/H = (10.44 million ly/year) / (22 km/s per million ly)

= 475.1 million ly

The redshift factor z can then be found using the equation:

z = Δλ/λ = v/c

where Δλ is the shift in wavelength, λ is the unshifted wavelength, v is the recessional velocity (positive for redshift), and c is the speed of light.

Solving for Δλ/λ, we get:

Δλ/λ = z = v/c = (10.44 million ly/year) / (3.00 × 10^8 m/s)

= 0.0348

Therefore, the ratio of the redshifted wavelength to the unshifted wavelength is:

λ_observed/λ_unshifted = 1 + Δλ/λ = 1 + z = 1.0348

Rounding to three significant figures, we get:

λ_observed/λ_unshifted ≈ 1.03

So the answer is A) 1.033.

Hubble's Law: a Galaxy Is Observed Receding from Earth with a Speed