Estimating Detectable Alien Life: Drake Equation

To begin thinking about what equation we would need to construct to show the number of potentially detectable alien civilizations we must first begin by considering the factors that will affect this number.


Before I begin explaining how we go about constructing the equation I want to first say that this equation is a slightly altered version of the Drake equation, it will yield the exact same results (comment for an explanation as to why) but I personally feel this version is more intuitive and far easier to grasp as a concept. I also need to say that although this is an equation of sorts, it does not have an implicit use that will give a correct answer, the reason for this is just we do not know enough about a lot of the variables to make them constants, so they will change depending on each interpretation.


Well let's start by thinking about what we need as the variables in the equation. The obvious first thing to consider is the number of star systems in our galaxy, we will denote this S. Now our best estimate for this is anything from 200 billion to 600 billion stars, with the increase in the power of our telescopes we get more and more accurate estimates constantly.


A lot of these star systems are simply devoid of any planets at all, they are just a star with no planets at all. So the next variable that we need to consider is the fraction of stars with planets, we will denote this as P. Now this really is a pretty massive estimate, we could never know the exact fraction of stars that have planets, but again we do have methods of checking if a star has a planet and currently it is thought that around 50% of stars have planets too.


Now we currently will have an approximation of the number of stars with planets orbiting them, the next thing we will want to do to narrow down this number to the number of detectable alien life is for what fraction of these star systems lies a planet that is capable of supporting life, moreover we want a planet that could be Earth-like, this reason is for what we know now is that life can only develop on planets that are Earth-like. However for all we now their may be incredibly intelligent gaseous beings on a distant planet that has developed a highly technical civilization, we just do not know. However, we will denote this variable as E, for Earth-like planets. To put a number onto this I will just pluck a number completely out of the air and say only 10% of planets could be capable of sustaining life.


Just because the tools are there it does not mean that it will result in their being life. This variable is the fraction of Earth-like planets that do develop life, we will denote this L. However, I think a lot of the planets that are Earth-like will evolve life in some form, there is life at the very deepest depths of the ocean, there is life where a human would be completely obliterated within seconds. For this reason I think that if the planet has the means to sustain life, it often will, I estimate it will at least half of the time, so to be conservative I will put this variable at 0.5.


Again, just because there is life it doesn't mean that it will ever become 'intelligent' enough. Life may not have evolved as well on Earth had the dinosaurs not been wiped out, they may have never developed to an intelligent enough state to communicate using radio waves, they may have just stayed as a less intelligent being. This variable will then be the fraction of life that will develop into intelligent life, we will denote this as I. Now this will be far more rare than their just being life, and in fact there is not even a rough number we can apply to this, but just for the sakes of this let's say that 1% of life will at one stage become intelligent.


The next variable we need to consider is the fraction of these civilizations that communicate via a means that we will be able to detect, we will denote this variable as C. For example humans have been around and intelligent for around thousands of years, but only for the last 80 or so would we be able to be detected, this is because of the discovery of radio waves. So before that we were, as a civilization, undetectable. Also we may discover that in a few hundred years there are far more efficient and productive methods to communicate, other alien civilizations may already be using this. So let's say again, pulling numbers completely at random, that 10% of intelligent civilizations develop a means of communication that we can detect.


The last variable that we need to consider is the fraction the average time the civilization is able to communicate takes up of the average age of star system, we will call this T. How long a civilization is able to communicate is something that although we do not know (as we still exist, just!) we can estimate. We have only been able to communicate via radio waves for 80 years, and every single one of those has been riddled with war. However, I remain optimistic that we, and all intelligent life, should be able to last about 10,000 years in the state of communication. The average age of a start system is around 10 billion years. So the calculation to find T is 10,000/10,000,000,000, which as a decimal is 0.000001.


The equation that we now have, after considering all the things we need to look for is:
Number of Alien Civilizations = Number of Stars * Fraction of stars with Planets * Fraction of Earthlike Plants * * Fraction of Planets with Life * Fraction of Intelligent Planets * Ability to Communicate * (Lifetime of Planet)/(Lifetime of Star)


This is a lot of text for maths, so to put it algebraically:
N = S*P*E*L*I*C*T


Now if we input the estimates that I designated earlier we get:
N = 200,000,000,000 * 0.5 * 0.01 * 0.5 * 0.1 * 0.1 * 0.000001


Now if we do this calculation we get that in this case N = 5. So there should be about 5 alien civilizations that are detectable from my very estimated estimates.


But the point of this equation is not to come up with a concrete number of civilizations that we must be able to communicate with right now. But this equation gives us the types of data that we should be looking for if we want to know the potential amount of aliens in the galaxy. If we know what to be looking for to know how many potentially intelligent aliens could be in our galaxy we can hone our efforts in on what data we need to look for. Also, it is pretty cool to be able to estimate how many aliens are out there and able to be detected in our own galaxy, and what happens if we change certain variables, etc.


If you fancy estimating how many detectable lives our out there using the original Drake equation, try it out for yourself at WolframAlpha.

Maths and the Real World: Linear Programming

Linear Programming may be bread and butter to you or it may be an entirely new concept. But it is one of the most applicable pieces of maths that is used in every day life by business and companies alike, this of course is minimising costs and maximising profits.


I will propose a problem to you, you are a company that sells two types of fruit drinks that consists of fruit juice and sugar syrup. Juice A consists of 0.3 litres of fruit juice and 0.5 litres of syrup and Juice B consists of 0.6 litres of juice and 0.4 litres of syrup. You have 30,000 litres of juice and 40,000 litres of syrup already in your stock. The profit for Juice A is 20p and the profit for Juice B is 30p. Given this scenario, you wish to maximise your profit.


How would you go about doing this? Well let's begin by putting the information we have in a table and go from there.

	Fruit Juice (in litres)	Syrup (in litres)	Profit (in pence)
A	0.3	0.5	20
B	0.6	0.4	30
Total	20,000	30,000

Now, from this information we need to construct the constraints of the problem into mathematical terms. What inequality will represent the amount of fruit juice that is allowed to be used? Well it must be less than or equal to 30,000 that is clear, it also depends on how much of it is used by Juice A and Juice B, so if 0.3 of A is used each time Juice A is created, and 0.6 of B is used when Juice B is created. This then means that 0.3A+0.6B ≤ 30,000. Using the same rules we must concur that 0.5A+0.4B ≤ 40,000. Also we want to maximise the amount of profit that we make, this means that P = 20A+30B. But there are other less obvious constraints that we must consider. We can not use a negative amount of juice or syrup so A ≥ 0 and B ≥ 0.

To get an idea of what sort of values we can have we plot these inequalities onto a graph which will give us an idea of what values of A and B are actually possible to obtain.

The blue shaded region is the answers that are within
 the constraints of our inequalities. This is called the
region of feasibility.

So we have the region that the answers must be within, now we want to go about maximising the profit which has an equation of, P = 20A+30B. This will be the last point that the line P = 20A+30B touches on the region of feasibility, this means that what P actually equals is arbitrary as we only need to gradient of that line and it will then be moved until it touches the last point it possible can on the region of feasibility. So we will choose a number that is convenient to plot for us, I'll be using P = 600.

The line begins to gain opacity as it moves closer to the
further point on the region of feasibility. Point 'A' is the
 maximum point within the reason of feasibility, therefore
this is the maximum value.

We could try to read this point of the graph but it would far more accurate to solve this using where the point is met by the two equations and solve simultaneously. So we are solving the simultaneous equations of 0.3A+0.6B = 30,000 and 0.5A+0.4B = 40,000.

This then in the context means that to optimise the profit within our constraints we should make 200,000/3 litres of Juice A and 50,000/3 of Juice B. This then equates to about £6333, which is the maximum profit we can achieve from the circumstances we have been given.

This is just an example of how to use linear programming to optimise finances, but it is easily transferable to almost any situation. The only thing you may need to watch out for is if A and B are number of items that need to be sold, they must be whole numbers (obviously) so you may need to round and then check that this will still lie in the region of feasibility.

Again, I hope you find this interesting and in fact very applicable to real life. If you have any questions on anything I have done, how it works or even how I create my images, please comment and I will reply.