A Modification of the Drake Equation

My version of the Drake Equation, which uses the total number of stars in our galaxy instead of the rate of star formation. Many of these variables have been narrowed down using advanced space telescopes, but the final term, the lifetime of a civilization, winds up being the critical factor.

by David Black

It has been two years since Lily wrote the article in my previous post and I am only now putting together the 4th edition of our Ad Astra Per Educare student newsletter which will include her article. At the time that Frank Drake created the equation in 1961, hardly anything could be answered about any of the variables in the equation except perhaps the first variable about the rate of star formation per year in the Milky Way galaxy, which at the time was estimated to be about three stars per year. With additional data and the advent of space telescopes, we are beginning to get ever better estimates of some of the variables.

One thing that has always puzzled me about Drake’s equation is the inclusion of this first term. Since it is a yearly estimate, the final answer must be the number of communicating civilizations that come into existence per year, which seems an odd way of looking at it. We want to know the total civilizations out there that we might converse with, not just the newbies like us. Carl Sagan spoke of how, given how many of these factors were considered to be close to 1 (or 100%) if given enough time, the truly limiting factor is the final one, the life-span of a civilization where they are able and willing to communicate over interstellar distances. This is why he was so adamant about preserving the Earth and getting rid of nuclear weapons. He wanted us to last long enough to become part of some great Encyclopedia Galactica, a galactic storehouse of the wisdom of all civilizations.

If we do want to estimate the total communicating civilizations, I suggest a modification of the Drake Equation. Here is my own version of it:

N = Stot • fm • fFGK • fp • f HZ • fl • fi • fc • L

N = the total number of communicating civilizations at any one time.

Stot = the total number of stars in our galaxy, which is around 200 billion based on mass estimates.

fm = the fraction of those stars that have high metallicity, such as our sun. These are primarily Population I stars compared with the metal poor, older Population II stars. For life to exist, the proper elements including metals must be present, and the older metal-poor stars are poor candidates. That gets rid of about half the stars, as fm is about 0.5.

fFGK = the fraction of those stars that are like our sun, with long enough life spans for intelligence life to evolve and stable enough to not have UV flares or small habitable zones like red dwarf stars. Counting the number of such stars in the space around us out to 15 light years (this is where we pulled out our 3D model), we see there is one F type star, three G stars, and five K stars out of about 45 nearby stars, or 9/45 or 0.2.

fp = the fraction of those stars that actually have planets, which we know is near 100%, probably about 0.9 to be conservative, based on all the planets we are currently finding.

My astrophysics students created these exoplanet paintings using spray paint and various circular masks like bowls and platters. This shows an orange dwarf star with a purple exoplanet orbiting.

fHZ = the fraction of those planets found inside the habitable zone (HZ) of that star. Based on various planetary systems we have detected, if appears this number is about one in four or 0.25.

fl = the faction of those planets that actually evolved life on them. This becomes hard to estimate since we only have one example of life evolving so far. However, we do know that as soon as conditions settled down after the Late Heavy Bombardment ended, life evolved rather quickly within a hundred million years or so. So this number also approaches 1, given enough time. To be conservative, let’s say it is about 0.8.

fi = the fraction of planets with life that evolve intelligent life. This is where I disagree with Drake’s initial estimate that if life hangs on long enough, it will eventually evolve into intelligence. There is no proof of that and it took a rather lengthy time to happen on Earth, despite several near misses. Certainly there was impetus for intelligence during the Mesozoic, what with large predators running amok, but the rodent-like creatures were too small (a necessity to avoid the large predators) and intelligence never happened. So there have been intelligent creatures capable of using tools for the last three million out of 3.8 billion years since life first evolved. This gives us a limiting factor of about 3/3800 or 0.00079.

fc = is that fraction of intelligent creatures that develop technology capable of sending messages over interstellar distances, which for us occurred in 1932 with the first television broadcast capable of reaching beyond the ionosphere. It was sent from the opening ceremonies of the summer olympics, were held in Berlin that year, and hosted as master of ceremonies by none other than Adolf Hitler himself, with a parade of goosestepping Nazis. Yes, Hitler is our ambassador to the stars and there is nothing we can do about it. That gives us 90 years that we’ve had the technology to send signals, or 90 out of 3 million years, a factor of 0.00003.

L = the number of years a civilization will be capable of sending out or detecting signals. If it is only a hundred years or so for us, if we destroy ourselves sometimes soon, then the numbers look grim. But if we can overcome our adolescent tendencies for self-destruction then we might keep radio technology around for a long time, perhaps 10,000 years. This is the real kicker – it all depends on surviving long enough to become a part of the intragalactic conversation.

As a project in my STEAM class this summer my students learned pyrography, or wood burning. I created this little saying (Bonus points if you know its origin) as a demonstration.

Putting all these numbers together, we get:

200,000,000,000 • 0.5 • 0.2 • 0.25 • 0.9 • 0.8 • 0.00079 • 0.00003 • 100 and we get: 8532 communicating civilizations in our galaxy. That’s still a decent sized number. Tweak the numbers such as adding to the lifetime of a civilization and there could be more. Other estimates put the number at less than 1.0, which would mean we are an oddity and possibly alone in the galaxy.

Enrico Fermi was famous for asking a question that is now known as the Fermi Paradox: If there are so many civilizations out there, why haven’t we heard from them by now? Why the great silence?

If you take 8532 worlds and spread them out randomly throughout the spiral arms of the Milky Way (where the metal-rich stars are found), which is 100,000 light years in diameter and is basically disk-shaped, you can use the formula for the volume of a cylinder with a radius of 50,000 light years and a thickness (height) of about 1000 light years. This gives an overall volume of about 7.85 trillion cubic light years. In all that space, 8532 civilizations will be greatly spread out, probably thousands of light years apart. The answer to Fermi’s Paradox becomes obvious: we haven’t heard from anyone because our little Nazi parade has only been traveling for 90 years. No one has heard us yet, if our signal is even strong enough to be picked up. Maybe that’s a good thing.

There have been proposals to send out other, stronger signals and to send them in a tight beam to probable star systems instead of broadcasting in all directions. We’ve sent plaques out on the first space probes to go beyond the solar system, the Pioneer 10 and 11 and Voyager 1 and 2 probes. But it will take tens of thousands of years for them to reach even the closest star systems. We may have to wait awhile before we join the conversation.

I have now completed the fourth edition of Ad Astra Per Educare and it can be downloaded here:

A green exoplanet painted by one of my astrophysics students.

About davidvblack

I teach courses in multimedia, 3D animation, Earth science, physics, biology, 8th grade science, chemistry, astronomy, engineering design, STEAM, and computer science in Utah. I've won numerous awards as an educator and am a frequent presenter at state and national educator conferences. I am part of the Teachers for Global Classrooms program through the U.S. Department of State and traveled to Indonesia in the summer of 2017 as an education ambassador. I learned of the Indonesian education system and taught classes in astronomy and chemistry at a high school near Banjarmasin in southern Borneo. I am passionate about STEAM education (Science, Technology, Engineering, Arts, and Mathematics); science history; photography; graphic design; 3D animation; and video production. This Spaced-Out Classroom blog is for sharing lessons and activities my students have done in astronomy. The Elements Unearthed project (http://elementsunearthed.com) will combine my interests to document the discovery, history, sources, uses, mining, refining, and hazards of the chemical elements.
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