Life on other planets and a maths joke

Life on other planets and a maths joke

Is there life on other planets?

The wool of a black sheep is just as warm from Flickr via Wylio
© 2012 Mikael Moiner, Flickr | CC-BY | via Wylio

Here’s another maths joke: An astronomer, a physicist and a mathematician were travelling by train from England into Wales. As they looked out of the window, they saw a black sheep in a field. “Ah”, the astronomer said dreamily, “Wales is full of black sheep.”

“You can’t say that astronomer my friend,” said the physicist. “All you can say is that there are some black sheep in Wales”.

“Physicist, my friend,” said the mathematician “You can’t even say that. All you can say is that in Wales there is at least one field, with at least one sheep, and at least half of that sheep is black.” (I once saw this joke actually published in a maths textbook. The point of the joke is not that the mathematician has friends by the way!)

What is needed for life?

Over the last few years there has been a lot of media excitement with the discovery of planets orbiting other stars and the speculation that one of them may contain life. So what is needed for there to be life on other planets?

Hubble Witnesses Massive Comet-Like Object Pollute Atmosphere of a White Dwarf from Flickr via Wylio
© 2017 NASA Goddard Space Flight Center, Flickr | CC-BY | via Wylio

The first is water in liquid form. Water is liquid in a relatively narrow range of temperature. That means a planet needs to be a certain distance away from its star to allow water to be liquid. Too close and all the water will be steam. Too far and it will be ice. Scientists refer to the Goldilocks or habitable zone around each star. Not too hot, not too cold, but just right.

The habitable zone

Let’s consider our solar system as an example. Now that Pluto has been relegated to a dwarf planet, the furthest planet to orbit the sun is Neptune. Neptune is 30.1 AU from the sun. (An astronomical unit is the distance from the earth to the sun, so Neptune is 30 times further from the sun than earth.) The habitable zone for our solar system is from about 0.95 AU to 1.32 AU from the sun. This distance is only a little over 1% of the distance to Neptune.

Elliptical orbits

Another factor to consider is that the orbits of planets are not circular, but elliptical. Earth can sustain life because its orbit is almost circular. Its orbital eccentricity is only 0.0167. (This measure is 0 for a circle and parabolic at 1. In other words for an ellipse it must be between 0 and 1). If this number is too high then even if a planet is inside the habitable zone for part of its orbit, it leaves the habitable zone for a proportion of its orbit. Of the eight planets in our solar system, five have an orbital eccentricity that would be too big to remain in the habitable zone.

Axial tilts and tidal locking

The next factor is the time it takes for a planet to spin on its axis

Planet from Flickr via Wylio
© 2012 garysan97, Flickr | CC-BY-SA | via Wylio

and its axial tilt. The earth is perfect for life, spinning every 24 hours at an axial tilt of 23 degrees. Some planets spin very slowly. For example a ‘day’ on Venus is 243 earth days, a ‘day’ on Mercury is 58 days. If these planets were in the habitable zone then any water would be ice during the very long ‘night’ and then steam during the long ‘day’. The largest planets in our solar system spin very quickly. This causes very violent storms which would make them inhospitable.

The axial tilts of Mercury, Uranus and Neptune are almost perpendicular to the sun, which again means that the change in temperature from one part of the orbit to another would result in water not being liquid very often. Scientists describe some planets as being tidally locked. This means that the same side of the planet always faces its star. For example, the moon is tidally locked in its orbit of earth. In other words half of the moon is always impossible to see from earth. Tidal locking would render habitable planets inhospitable.

Other factors

So far we have only considered five physical factors of a planet’s potential to contain life: the habitable zone, elliptical orbit, length of a ‘day’, axial tilt and whether it is tidally locked. Already the improbabilities are stacking up. If we

Earth render 7 from Flickr via Wylio
© 2017 Alex Antropov, Flickr | PD-CC0 | via Wylio

go on to consider other factors, like the large amount of water on earth, one of the very few liquids that float when ice. Or the chemical makeup of earth, with substantial amount of nitrogen needed for amino acids. Or the relative stability of our sun in terms of damaging radiation. Or the earth’s magnetic core that gives protection from the sun’s most harmful radiation. Or the size of earth that enables it to retain an atmosphere, the probabilities for life continue to stack up.

Conditions needed for DNA to evolve

All of the above are factors to sustain life, but what about the conditions needed for life to evolve? Life, as we understand it, requires DNA. DNA is an incredibly complex molecule. If evolution is an entirely random process, DNA would have had to ‘assemble’ by chance. It is too big a molecule to be formed with its constituent parts freely suspended in a liquid. Half a DNA molecule is not any good, and couldn’t reproduce. My understanding is that it would need some kind of scaffold, possible a clay-like substance or small ice crystals. Scientists do not yet really understand, but suffice to say it is very difficult to replicate. The conditions required have a huge number of variables.

The probability of it happening is minuscule.

But the universe is vast

The argument goes that the universe is a very big place. We are only now appreciating the vast number of planets in the universe. We

Galaxy from Flickr via Wylio
© 2017 kristian fagerström, Flickr | CC-BY-SA | via Wylio

know the universe is not infinite, it is expanding. But let’s say it is so big that we can consider the number of planets as being virtually infinite. With that number of planets the conditions for life must have been replicated on at least one of them, somewhere in the universe. As a result some astronomers give the impression that life on other planets is inevitable. Wales is full of black sheep. A mathematician replies that you can’t say that because it is based on a fundamental misunderstanding of how number and probability work.

Clearly not inevitable

In a previous blog, ‘Infinity, a maths joke, and another infinity’, I explained how the infinity of numbers between 0 and 1 is much much bigger than the infinity of starting to count and never stopping. These are called countable and uncountable infinities. The number of planets, however big that number is, clearly belongs on the scale of countable infinity. Probability, as any GCSE student will tell you, has a scale between 0 and 1 and belongs on a scale of a much bigger infinity.

The lottery provides a good example. In the lottery there are only 6 variables (balls) that can only take values between 1 to 50. With this relatively simple situation, the chances of winning are in the order of tens of millions. The variables needed for life are a great deal more than 6, and the probability of many of these variable is much smaller than the 2% chance of a lottery ball being selected. Even if the universe is infinite, the countable infinity of planets is not big enough to contain the uncountable infinity of probability.

So is there life on other planets? The astronomer says discovering life on other planets is inevitable. The Physicist says it is probable. But as with the black sheep, the mathematician says that you can’t say that. All we can say is that there’s one planet in the universe that does have life. The chances of even that happening are so remote, it is amazingly awe inspiring that it should have happened once.  The fact that the earth is so perfectly suited for life and that life is so incredibly varied, and that there are intelligent beings like us increases that wonder.

Is there life on other planets? Almost definitely not – but don’t let that diminish the wonder of life on earth.

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