Question

Stars and galaxies in the distance universe are receding from us with a speed proportional to their distance away, v = Hr, where H ≈ 73 m/s/Mpc is the Hubble constant. (a) What is the value of H in SI units? SR units? (Mpc = Mega parsec.) (b) Because they are receding from us, light received from distant stars will be red-shifted. It is common to define the dimensionless red-shift parameter z as λ/λ0 ≡ 1 + z. There is a specific spectral line (due to a nuclear transition) in hydrogen called the 21 cm line, named after it’s wavelength.1 Suppose that light from a distant star is observed to have a red-shift of z = 0.08. What would be the observed wavelength of the 21 cm line? (c) What would be this star’s recessional velocity (use the relativistic Doppler formula), and its distance from us? Express the distance in seconds and light-years.2

Answer 1

(b) We know that, ( / ₀) = (1 + z)

Suppose that light from a distant star is observed to have a red-shift of z = 0.08.

Therefore, the observed wavelength of a 21 cm line which will be given as -

₀ = / (1 + z)

₀ = (21 cm) / [1 + (0.08)]

₀ = [(21 cm) / (1.08)]

₀ = 19.4 cm

(c) This star’s recessional velocity which will be given by -

z = v / c v = z c

v = [(0.08) (3 x 10⁸ m/s)]

v = 2.40 x 10⁷ m/s

It's distance from us which will be given by -

v = H₀ D D = v / H₀

D = [(2.40 x 10⁷ m/s) / (73 m/s/Mpc)]

D = 328767.1 Mpc

converting Mpc into ly :

D = 1.07 x 10¹² ly