# Maths Bite: Hubble’s Constant

In today’s post I want to take a small diversion into the realm of Physics, in particular Astrophysics, to look at an extremely significant constant: Hubble’s constant.

Edwin Hubble measured the speed of galaxies and their distance from Earth and obtained the following graph:

As the graph is a straight line through the origin it shows that velocity is directly proportional to the distance from Earth. This is known as ‘Hubble’s law’.

### Measuring Distances

Hubble’s Constant can be used to measure astronomical distances, which are too big to be measured by parallax or by using standard candles.

Hubble’s Constanthas a value of 2.3 x 10-18 s^-1 or 72 km s^–1 Mpc^–1

[Mpc = megaparsec = 3.26 million light years]

As:

and v = zc (where z is the redshift of the galaxy):

we can find the distance for any distant galaxy, provided we can measure its redshift (z).

### Assumptions

We need to assume the straight line remains linear as the redshift becomes bigger and bigger – Hubble’s law holds universally.

This is not true, for example, with Hooke’s law, as it has a limit of proportionality (elastic limit).

### Age of the Universe

All distant objects are moving away from us, suggesting that the Universe as a whole is expanding. If we turn back time, then the Universe would contract to a single point. This moment is called the Big Bang.

If we can find the Hubble constant, it will tell us how quickly the Universe is expanding, and from this we can work out how old our universe is.

If the universe was created at a time T ago, for a galaxy that has been moving away from us at a steady rate v for a time T, its distance d from us will now be vT.

Hubble’s Law tells us v = Hd, so v = HvT, which gives us HT = 1. Hence, the age of the universe can be given by:

T = 1/H

This gives an estimate of 14 billion years to 2 significant figures.

However, there are great uncertainties involved with this estimate:

• Gravitational forces will mean that the present rate of expansion is less than in the past, so T < 1/H
• Although the value for Hubble’s constant has become more accurate since the launch of the Hubble Space Telescope, the current value is only considered accurate to within 5%, so there is an uncertainty to the value for T.

### Fate of the Universe?

• CLOSED UNIVERSE: since gravity works against expansion, if the density were large enough then the expansion would stop and the universe would collapse in a ‘big crunch’. (Ω > 1)
• OPEN UNIVERSE: If the density is small enough, then the expansion would continue forever – steady increase in Hubble’s constant. (Ω < 1)

Hope you enjoyed the post. Let me know what you think of more physics-based posts! M x