Drunkard’s Walk

Before I launch into stuff here, check out Our Violent Sun – an article I wrote for the University Observer in November 2017.


Five billion years ago there was a cloud of gas and dust in space. It wasn’t up to much. The force of gravity then pulled this gas together and — poof — the solar system was born. This NASA infographic can fill in the details for those interested, but the important detail is this: about 99.8% of the gas and dust from the cloud ended up in the Sun, with the remaining 0.2% going on to form all of the planets.

All of this mass is what allows the Sun to shine. Einstein discovered that matter can be converted into energy, and vice versa. This is how the Sun is powered – it converts hydrogen into helium. Each second, it burns 564 million tonnes of hydrogen, resulting in 560 million tonnes of helium. The 4 million tons of matter that’s gone awry during this process was converted into light energy. To put this into context, the Great Pyramid of Giza weighs about 6.5 million tons, so the Sun loses a pyramid’s worth of hydrogen every 1.6 seconds.

Random walk

In the Sun’s core, light is generated. A piece of light is called a photon, and photons scatter off matter. When a photon is created in the centre of the Sun, it takes a long time for it to escape to the surface because it scatters off all of the particles inside the Sun. This video illustrates the concept nicely where the yellow dot is a photon at the centre of the Sun and the red dots are the atoms in the Sun.

A photon travels just a couple of centimetres before it encounters a particle. The photon then scatters off this particle in a completely random direction — perhaps even back from where it came. As the photon pinballs around inside the Sun, it can take thousands of years for it to reach the surface — and when it does escape, it’s by sheer chance. The path the photon takes is called a random walk.

Drunkard’s walk

This random walk has also been described as the drunkard’s walk which is a somewhat easier to visualise analogy. Imagine stumbling out of the pub late one night with the aim of walking home. Let’s say you live straight down the road, and you’re really intoxicated so you can’t even orientate yourself correctly.

A visualisation of the drunkard’s walk. The path he takes is anything but straight.

You take one step out of the pub. Then your next step will be in a completely random direction — it could be left, right, backwards, or forwards. You make a step, say, to the left. Again, you walk in another random direction, independent of that from which you came. Your first few steps could look like this: A step forwards, then left, left again, the forwards, two steps back, then forwards, a step to the left, and three steps to the right. In this case, you’ve taken eleven steps, but you end up just one step away from where you began.

Over a long enough timescale, the random nature of your travels will take you everywhere — including home. The question is, how long will this take? There’s a formula for this which states that in order to get N steps away from the pub (as the crow flies), after you take N² steps. For example, if one step is a metre, then getting 5 metres from the pub will take 5² = 25 random steps. To get a longer distance from the pub can take much longer, however: A distance of just 100 metres will take 100² = 100,000 random steps.

random-walk
This is an example of a drunkard’s walk. You start at the yellow star and finish at the blue star. The endpoints are separated by just five steps, as indicated by the red line, but it took a total of 25 random steps to get from the beginning to the end.

The Sun

Relating these principles back to the Sun, we find that it takes a photon thousands of years to escape the Sun:

  • Given the density of the Sun, we know that one “step” for a photon is about one centimetre. This is how far it gets before it bumps into a particle and changes direction randomly.
  • We also know that the Sun has a radius of 70 billion centimetres. This is how far the photon needs to get.
  • To reach this distance, it has to take (70 billion)² steps.
  • We also know the speed of light so we can work out how long it takes light to travel this distance.
  • The result is that it takes a photon over 5,000 years to leave the Sun. (A more rigorous approach shows that the actual escape time is even longer but this quick analysis isn’t far off).

This is actually a very long time. To reiterate, it takes a photon thousands of years to get 0.7 million kilometres from the centre of the Sun because it travels randomly. Once it’s in space, though, it travels unimpeded. So it traverses the next 150 million kilometres, from the Sun to Earth, in just eight minutes.

How and why things shine

The primary goal of astrophysics is to deduce how and why things shine. Such a seemingly straightforward question can get pretty complicated however, and part of the reason is because we’re not built to think in really big or really small numbers.

It isn’t intuitive how a lot of microscopic particles behave on colossal timescales. Applying statistics is our saviour, and it can do a lot of work for you in astrophysics. If astrophysics isn’t your cup of tea though, fear not. The real power of the statistical laws we’ve seen is that they’re universal. So even if you’re more interested in going to the pub than learning about stars, it’s reassuring to know that the same rules which dictate the path of photons in a giant ball of plasma billions of kilometres away can also be used to get you home from a bar.