PX158 - K1 - hubble's law

in $1929$ , Hubble found that the radial velocity of galaxies increased with distance
these velocities are almost always red shifts, ie: galaxies are moving away

v = H_{0} d

where, $H_{0} =$ hubble's constant

$v$ was measured by observing shifts in atomic absorprtion/emission lines in galaxy spectrum
the best fit line gives:

H_{0} = 72 k m s^{- 1} M p c^{- 1}

this implied that the universe is expanding
the expansion is space-time itself, not an explosion into space
the expansion can be measured by using a term called redshift:

1 + z = \frac{λ}{λ_{0}}

where, $λ =$ apparent wavelength, $λ_{0} =$ rest wavelength
- $z = 0 :$ local
- $z \sim 0.1 :$ local universe
- $z > 1 :$ high-redshift universe

v = c z

where, $v =$ recessional velocity

eg: if galaxy shows features at $498 n m$ , $λ_{0} = 486 n m$ , what is $z, v, d$ and apparent magnitude of a cepheid variable with $M = - 7$ ? $\begin{align*} 1-z = \frac{\lambda}{\lambda_{0}} &\implies z \approx 0.0247 \\ v = cz &\approx 7\,400\,km\,s^{-1} \\ v=H_{0}d &\implies d=103\,Mpc \\ m-M=5\log_{10}\left(\frac{d}{10\,pc}\right) &\implies m=28.1$
- the apparent magnitude is very faint (to even be detected by a telescope)