Hubble's law relates a galaxy's recession speed to its distance, and gives us the age of the Universe

Astrophysics | AQA A-Level Physics

Key Definition

Hubble's law: The recessional velocity of a galaxy is proportional to its distance from Earth. Expressed mathematically as $v = Hd$.

The equation

$$v = Hd$$

$v$ = recessional velocity of the galaxy (km s$^{-1}$)
$H$ = the Hubble constant (km s$^{-1}$ Mpc$^{-1}$)
$d$ = distance from the Earth to the galaxy (Mpc)

Hubble's law tells us that the further away a galaxy is, the faster it is moving away from us. Closer galaxies recede more slowly.
A graph of recession velocity ($v$) against distance ($d$) gives a straight line through the origin. The gradient of this line is the Hubble constant $H$.

The Hubble constant

The Hubble constant $H$ is the constant of proportionality in Hubble's law:

$$H = \frac{v}{d}$$

Its value has been estimated using data from thousands of galaxies and standard candles.
The current best estimate, based on CMB observations by the Planck satellite, is approximately:

$$H \approx 67.4 \pm 0.5 \text{ km s}^{-1} \text{ Mpc}^{-1}$$

This value is constantly under review as more data is collected.

Estimating the age of the Universe

Hubble's law can give us an estimate of the age of the Universe. The key part is the reasoning:
If all matter was at the same point at the start of the Big Bang ($t = 0$), then a galaxy now at distance $d$, moving at speed $v$, has been travelling for time $t = d / v$.
Since $v = Hd$, we can substitute to get:

$$t = \frac{d}{v} = \frac{d}{Hd} = \frac{1}{H}$$

This assumes that the recessional speed has been constant throughout the history of the Universe.
Using the current estimate of $H \approx 67.4$ km s$^{-1}$ Mpc$^{-1}$, converting to SI units and taking the reciprocal gives an age of approximately 14.6 billion years.

Common Mistake

Unit conversions in Hubble constant calculations catch many students out. To find $1/H$ in seconds, you must first convert $H$ from km s$^{-1}$ Mpc$^{-1}$ to s$^{-1}$. That means converting km to m (multiply by 1000) and Mpc to m (1 pc = $3.1 \times 10^{16}$ m, so 1 Mpc = $3.1 \times 10^{22}$ m). Only then can you take the reciprocal to get a time in seconds.