# The Drake equation

Is there life on Mars? Or anywhere else?

Do aliens exist? It’s a rather controversial question.

Is it possible that life exists beyond Earth? Maybe. Possibly. Probably. It’s highly unlikely that we will ever get a universally acknowledged yes or no answer.

How many alien civilisations could there be? As far as I know no-one particularly wants to give an estimate.

A long time ago in a galaxy far far away….

However, the question ‘will we ever contact an extra-terrestrial civilisation?’ has been broached. The Drake equation sums up a probabilistic argument that estimates the number extra-terrestrial civilisations which broadcast signals into space which we may feasibly pick up. To fully understand this, let’s delve into some special relativity.

Firstly let’s condense three-dimensional space down to two dimensions (say $x$ and $y$) and add in a vertical time axis. Suppose that you are sending out a signal with a torch shining out in all directions. At any given time the points this signal would have reached will trace out a circular disk in our two-dimensional spatial coordinates. As time increases, so does the radius of our disk and thus in our three-dimensional space-time, the signal will trace out an upside down cone with its vertex at the torch. This is known as the future light cone of your position. If instead of shining a light you are sending out a signal by shouting, the upside down cone in 3D space-time would have steeper sides as sound travels slower than light.

Diagram of future and past light cone in 3D space-time of someone standing at the origin.

But what if you are receiving a light signal? For a given present time we can work calculate both the time at which and the associated set of spatial coordinates from which the signal must have been released to reach our position. In three-dimensional space-time these points also trace out a cone again with its tip at our position. This is known as the past light cone of our position. If you are picking up a signal travelling slower than the speed of light (like sound) then the set of three-dimensional points, from which this signal could have been produced, traces out a cone with steeper sides, sitting inside our past light cone.

So, back to the Drake equation. When we talk about signals that we may pick up, we are looking for civilisations sitting inside the Earth’s past light cone with the technology to produce, for example, radio signals, which are the signals the people searching for extra-terrestrial life typically look for. However, a consequence of this is that we are limited to our galaxy only since radio signals from other galaxies would be too weak to be received.

The Drake equation is

$$N=R \cdot f_p \cdot n_e \cdot f_l \cdot f_i \cdot f_c \cdot L,$$

where:

$N$ is the number of civilisations in our galaxy with which radio communication may be possible;

$R$ is the average rate of star formation in the Milky Way (in planets per year);

$f_p$ is the fraction of stars that have planets;

$n_e$ is the average number of planets that could potentially support life, per star that has planets;

$f_l$ is the fraction of the planets that could support life that do so;

$f_i$ is the fraction of planets on which there is life that develop intelligent life-forms;

$f_c$ is the fraction of planets with intelligent life that go on to develop civilisations that can broadcast radio signals;

$L$ is the length of time in years during which these civilisations release radio signals into space.

This version is the simplest and original version of the Drake equation. There are countless modifications and scenarios that can be considered: for example, what if life develops multiple times on one planet? What if a civilisation colonises other star systems and broadcasts from each planet it settles on? What if there are civilisations capable of radio transmissions who do not want to meet other beings?

But let’s consider some possible values in the Drake equation. NASA and others suggest that the average rate of star formation is about 7 stars per year. It has also been suggested that most stars in the Milky Way have planets, so let’s say say $f_p=0.95$.

It is at this point that we move deeper into the realm of guess work. The Kepler space mission data suggests that $n_e \cdot f_l=0.4$. Since the only evidence of a planet developing life is Earth, we can suggest that $f_i=1$, and if we estimate that about 10% of the civilisations on Earth survived for long enough to have access to radio technology, we can say that $f_c=0.1$. If we say Earth will broadcast radio signals for 200 years then $N\approx53$. But this is completely dependent on your opinion of the validity of the parameter values. Other estimates of the parameter values result in values of $N$ that range from less than one (incorrect since we know of one planet which broadcasts radio signals, don’t we?) to several million (which leads to the question why haven’t we heard from anyone?). This range of possible values is the main criticism of the Drake equation. With an error margin as large as that, is it possible to draw any conclusions at all? Not really, so what is the point of the Drake equation?

In reality, the equation was never meant to be used for quantifying the number of alien civilisations out there. Instead it was written by Dr Frank Drake in 1961 as a way to stimulate the scientific debate on the search for extra-terrestrial intelligence (SETI).

It has done exactly this, so we could say it’s been successful. Nevertheless, we’re not going to be finding Ewoks anytime soon.

I spy something beginning with E.

Eleanor is a maths PhD student at UCL. She is interested in the applications of mathematical modelling to biological and medical problems.
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