 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2015/8/e/solution

v_{∥}

within the group, cosmological and Doppler factors multiply:

1 + z_{obs} = (1 + z_{g}) (1 + z_{pec}), z_{pec} ≃ v_{∥} / c .

At fixed group redshift this gives

δz = (1 + z_{g}) v_{∥} / c

. Therefore

σ_{v} = \frac{c σ _{z}}{1 + z _{g}} = 99.9 km s^{- 1} .

Using

c σ_{z}

directly would give a detector-frame redshift interval without removing the cosmological stretch. This is the one-dimensional line-of-sight velocity dispersion requested, not the three-dimensional RMS speed. Redshift measurement errors or a large unbound line-of-sight extent would require additional corrections; neither is supplied here.

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