Monday, 4 June 2012

What is the Drake Equation? (Guest post from B.C.)

This a guest post from the blog B.C.. The blog describes itself as "of Prehistory, Palaeontology and the Past". If you enjoy this post, go check them out here!

1969 is best remembered for the lunar landing. For the first time, we had truly left the surface of the Earth and stood upon a body which had gone unchanged and lifeless for over 4 billion years; truly one small step for man, but giant leap for mankind. What is rather interesting is that this event completely eclipsed a second NASA mission operating on a far larger scale.

For years cosmologists had been developing probes to land on the Red Planet. While their missions did not involve human cargo, the engineering challenges they faced were even greater. The craft would have to cross hundreds of millions of miles across a hostile, freezing void blasted with solar radiation, descend through a thin atmosphere without burning up and land without smashing on the red, iron rich rocks.
The first successful landing took place in 1972 by what was actually a Soviet probe, but the Americans soon followed up with many missions of their own. It had been a long held view that there were living organisms on Mars. Astronomers had seen, through telescopes, landforms which could only have been created by liquid water, an essential component for life from an Earth perspective.

Some went as far as to claim that channels on the surface of the planet were canals made by intelligent aliens. Dark areas recorded by the primitive black and white cameras of early fly-by probes were thought to have been lush, verdant forests. When the first probes reached the surface and sent back colour images, scientists found not a thriving world, but a barren wilderness of red rocks and sand under red skies.

Subsequent mission have failed to find any trace of Martian life, either in a fossil record or simple microbes under rocks or within the ice at the poles. These first images of the Red Planet forced cosmologists to rethink all their ideas about life on other planets. Today the subject is still contentious and without any solid physical evidence, but what little we have was sparked off by an equation created by a NASA scientist, Frank Drake, just a few years before the lunar mission.

Bring out the maths!

While this might look rather complex, the variables involved are actually quite simple.

  • R* is the average rate of star formation per year in a galaxy,
  • fp is the fraction of those stars which have planets,
  • ne the average number of planets that can potentially support life per star that has planets,
  • fl is the fraction of those planets which will go on to develop life at some point,
  • fi is the fractions of those again whose life will become intelligent,
  • fc is the fractions of those again which will develop a technology which can release detectable signs of their existence into space 
  • L which is the length of time for which such civilisations release detectable signals. 

When the numbers are multiplied together, they produce N, the number of civilisations in a galaxy with which communication might be possible with each other. Since its creation by one Frank Drake, this equation has been altered to take into account variables such as the time needed for life to develop to an intelligent, technologically advanced stage (in Earth’s case 3.8 billion years) and how long these civilisations may last (taken as 10,000 years).
The idea of time scales can be put into the form of:

In this Tg is the age of the galaxy in question. Assuming that R* is a constant, then this can be simplified to:

This can now be placed back into the original Drake equation to provide a more rounded version which takes into account time scales:

So what we are going to do is crunch the numbers as Drake did back in 1961 for our own galaxy, the Milky Way. 

  • R* = 10 stars formed per year, 
  • fp = 0.5 (half the stars will have planets)
  • ne = 2 (Stars with planets will have two capable of sustaining life)
  • fl = 1 (all these planets will develop life),
  • fi= 0.01 (A hundredth of all this life will be intelligent),
  • fc = 0.01 (one hundredth of those will be able to communicate with us 
  • L = 10,000 (whose civilisation will last for 10,000 years) 
  • Tg = 13.0 billion years old, (though this value appears twice in the equation, the other time within N* as a multiplication and a division, they cancel each other out and do not affect the rest of the mathematics, remaining only as a reminder of the time scale involved.)

Overall this gives us a value of just 10 detectable civilisations within our galaxy. Only 10! Considering that the Milky Way is 13 billion years old with a diameter of 120,000 light years and over 100 billion stars in the spiral arms, this is very small. While some discredit the Drake equation as a simple and inaccurate estimation tool which glosses over many other important factors, its significance extends far beyond mere statistics. 

What is interesting is just how different our estimates for the different variables can be. For example, the value of R* can be anything from the 10 stars that Drake used, down to 1 star, however, even then we are assume these are stars of solar mass. Modern figures suggest the real number could be anything from 0.14 to 7! 

In fact, the values for almost all the variables are in debate. Some research suggests that fp could be 0.4, but other research seems to find that the number tends to 1! And even if we discover the real number, we still have the fl, fi and fc. These are the variables to do to with the likelihood of life developing. Seeing as the only life we have ever observed was on earth, it is very difficult to estimate these. This means that the resulting value of the Drake Equation can be nearly anything, ranging all the way from not much above 0 up to 182 million contactable life forms!  

However, what we should remember is that the maths is not wrong. The equation uses multiplying probabilities, a reliable and common mathematical method. However, due to the number of difficult-to-predict variables, it is very difficult to get a reliable result. The maths can only help when the numbers are right. 

Because the Drake Equation has a large number of variables, even small errors in the values of the variables can lead to very large variances in N. For example, if we say that each variable in an equation is accurate ±10%, then with two variables, to find the resultant range of N, we have to find the highest and lowest possible value of N, assuming that the variables are as wrong as they can be within the error bars. So, if we let both variables be 1, then to find the upper bound we do
(1+(10%×1)) × (1+(10%×1))=1.12=1.21
And to find the lower bound, we do
(1-(10%×1)) × (1-(10%×1))=0.92=0.81
So this means that although multiplying the original figures would give 1, the errors mean that the actual result could be as low as 0.81 or as high as 1.21. This gives a variance of 0.4, which is relatively high.

However, at more variables to this equation, at it gets worse. If, like the Drake Equation, it has 7 variables, then the variance will be much higher. Again, we will take all variables to be 1 ±10%.

The upper bound is 1.17=1.949 (to 3 decimal places) and the lower bound is 0.97=0.478 (to 3 d.p.). This results in the huge variance of over 1.47, larger than the value itself! This means the actual value could be almost double or less than half the stated ones. Many variables in the Drake Equation have much less certainty than the ±10% used here, so this could be even worse. However, it does at least give us an idea: with ±10% accuracy on all variables, you will still get a result in the right order of magnitude.

Another important point, is that it only accounts for advanced civilisations. What it cannot give us an idea of is of any kind of life. To do this requires looking at the conditions and elemental composition of other solar systems. This is something which mathematics simply cannot do as the variables involved would simply be too complex to reduce to single, simple terms. For life to form on a planet, three different things are needed. 

The first are the right elements. From an Earth view, these are Carbon, Hydrogen, Nitrogen and Oxygen. The Carbon is the linchpin of the quartet as its chemical diversity facilitates the intricate and precise reactions which fuel living organisms. The second factor is an abundant energy source. This could be anything from heat or ultraviolet radiation to lightning strikes. All have their merits and their abundance will be decided by the third factor: the conditions present on the planet. The ability to analyse reflected light from far-off worlds can give us some idea of its elemental composition, but until we develop the technology to travel across interstellar space or an asteroid brings hard evidence of alien life to Earth, we will have to console ourselves with this equation and Bowie’s famous words ringing in our ears ‘is there really life on Mars’? 

If you are still confused, here is Carl Sagan going through the equation:

We hope you liked this guest post. B.C. posts about historical biology and the world when it was very young. If you like the post, check out the blog!

Check out our last two posts:
The Physics of Gymnastics - The forces exerted on a gymnasts body during some routines are extreme, so how does it work?
The Physics of Field Athletics: Why do the shot-put, the javelin, the discus and the hammer fly differently? - All these events have the same aspects, but they all get different distances. How does this work? 

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