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.