Question

Consider a galaxy cluster of physical (visible) size 1.0 Mpc. What angle would this galaxy cluster subtend if situated at redshift 1.0? Redshift 5.0? Do the calculation in a matter-dominated flat universe.

Express your answer in arcminutes, where 1 arcminute = (pi/180)*(1/60) radians.

Answer 1

For solving this we will use hubble's law, according to which

V=H_{​​​​​​0}​​​​​/D

where

• V is the recessional velocity, typically expressed in km/s.
• H₀ is Hubble's constant
• D is the distance

also   V= cz

where c is speed of light

and z is redshift number.

so  D = cz/H₀

for redshift 1 :(taking H₀​​​​​=70 km/s/Mpc)

D= (3×10⁵ km/s × 1)/70 km/s/Mpc

= 4285.71 Mpc ........(1)

For redshift z=5:

D= (3×10⁵×5)/70 Mpc

=21428.5 Mpc............(2)

Where θ is the angle subtended by galaxy on the earth and 'd' is given to us i.e. 1Mpc

So for redshift 1:

θ=d/D rad

= (1/4285.71)× (180/π)×60

= 0.802 arcminutes

For redshift z=5

θ = (1/21428.5)×(180×60)/π

= 0.1604 arcminutes