Question

Suppose that you have obtained spectra of several galaxies and have measured the observed wavelength of the H-alpha line (rest wavelength = 656.3nm) to be:

Galaxy 1: 659 nm. Galaxy 2: 666.5 nm. Galaxy 3: 676.6 nm.

10. [4pt] Calculate the redshift, z, for each of these galaxies.

11. [4pt] Calculate the radial velocity of each of these galaxies.

12. [4pt] Assuming a Hubble Constant of 69.6 km/s/Mpc, cal- culate the distance to each of these galaxies (answer in Mpc).

Answer 1

10. redshift z = /

where = rest wavelength

= observed wavelength - rest wavelength

for Galaxy 1:   = 659-656.3 = 2.7nm

z1 = 2.7nm / 656.3 = 0.0041

for Galaxy 2   = 666.5-656.3 = 10.2 nm

z2 = 10.2nm / 656.3 = 0.0155

for Galaxy 3:   = 676.6-656.3 = 20.3nm

z3 = 20.3 / 656.3 = 0.031

11. Velocity and redshift relation = v/c = ((z+1)² -1 )/ ((z+1)² +1)

For Galaxy 1 :

(z1+1)² = 1.0041² = 1.008

v/c = (1.008 - 1) / (1.008+1) = 0.008/2.008 = 0.0039

v1=0.0039c = 0.0039*3*10⁸ = 1.2x 10⁶ m/s

For Galaxy 2 :

(z2+1)² = 1.015² = 1.03

v/c = (1.03- 1) / (1.03+1) = 0.03/2.03 = 0.0147

v2=0.0147c = 0.0147*3*10⁸ = 4.4x 10⁶ m/s

For Galaxy 3 :

(z3+1)² = 1.03² = 1.061

v/c = (1.061- 1) / (1.061+1) = 0.061/2.061 = 0.0296

v3=0.0296c = 0.0296*3*10⁸ = 8.8x 10⁶ m/s

12. according to Hubble's law, v = HD

H= Hubble constant =69.6 km/s/Mpc = 0.0696m/s/Mpc

D= distance of galaxy from observer

v= velocity of galaxy

Hence, D= v/H

Distance of galaxy1 = D1 = v1/H = 1.2 x10⁶ / 0.069 Mpc = 17.39 x 10⁶ Mpc

Distance of galaxy2 = D2 = v2/H = 4.4x12⁶ / 0.069 Mpc = 63.7 x 10⁶ Mpc

Distance of galaxy3 = D3 = v3/H = 8.8x10⁶ / 0.069 Mpc = 127.5 x 10⁶ Mpc