Imagine that you have obtained spectra for several galaxies and have measured the observed wavelength of a hydrogen emission line that has a rest wavelength of 656.3 nanometers. Here are your results: Galaxy 1: Observed wavelength of hydrogen line is 659.7 nanometers. Galaxy 2: Observed wavelength of hydrogen line is 667.7 nanometers. Galaxy 3: Observed wavelength of hydrogen line is 683.7 nanometers. A) Calculate the redshift, z, for each of these galaxies.
B) From their redshifts, calculate the speed at which each of the galaxies is moving away from us using the Doppler formula. Give your answers as a fraction of the speed of light.
C) Estimate the distance to each galaxy from Hubbleʹs law.

A. z (galaxy 1) = -0.00515, z (galaxy 2) = -0.01707, z (galaxy 3) = -0.04008

B. v (galaxy 1) = -0.00517c, v (galaxy 2) = -0.01707c, v (galaxy 3) = -0.0401c

C. d (galaxy 1) = -21 Mpc,d (galaxy 2) = -69.4 Mpc, d (galaxy 3) = -163 Mpc

Explanation:

We begin by listing out the parameters we were given:

λ (obsv) = 656.3 nm, λ (emit 1) = 659.7 nm, λ (emit 2) = 667.7 nm,

λ (emit 3) = 683.7 nm

A) Using the Redshift formula, we have:

z = [λ (obsv) - λ (emit)]  ÷ λ (emit)

For galaxy 1:

z = [λ (obsv) - λ (emit 1)]  ÷ λ (emit 1)

z = (656.3 - 659.7) ÷ 659.7 = -0.00515

z = -0.00515

For galaxy 2:

z = [λ (obsv) - λ (emit 2)]  ÷ λ (emit 2)

z = (656.3 - 667.7) ÷ 667.7 = -0.01707

z = -0.01707

For galaxy 3:

z = [λ (obsv) - λ (emit 3)]  ÷ λ (emit 3)

z = (656.3 - 683.7) ÷ 683.7 = -0.04008

z = -0.04008

B) Using the Doppler formula, we have:

(Δλ ÷ λ) = v ÷ c

v = c * (Δλ ÷ λ)

but, z = (Δλ ÷ λ)

⇒ v = c * z

speed of light (c) = 3 x m/s

For galaxy 1:

v = c * z

Substitute z into the equation calculated from A) above

v =  3 x * (-0.00515)

v = -1.55 x m/s

v = -0.00517c

For galaxy 2:

v = c * z

Substitute z into the equation calculated from A) above

v =  3 x * (-0.01707)

v = -5.12 x m/s

v = -0.01707c

For galaxy 3:

v = c * z

Substitute z into the equation calculated from A) above

v =  3 x * (-0.04008)

v = -12.03 x m/s

v = -0.0401c

N.B: the negative value of velocity connotes that the galaxies are moving away from us (not towards us)

C) Using Hubbleʹs law, we have:

v = H · d

where:

v = velocity of a galaxy (km/s), d = distance (Mpc),

H = Hubble's constant (km/s/Mpc) = 73.8 km/sec

d = v ÷ H

We use the velocities calculated in B) above

For galaxy 1:

v = -1.55 x km/s

d = -1.55 x ÷ 73.8

d = -21 Mpc

For galaxy 2:

v = -5.12 x km/s

d = -5.12 x ÷ 73.8

d = -69.4 Mpc

For galaxy 3:

v = -12.03 x km/s

d = -12.03 x ÷ 73.8

d = -163 Mpc

N.B: distance cannot be a negative value