# Drake Equation Calculator

Created by Davide Borchia and Rijk de Wet
Reviewed by Steven Wooding
Last updated: Jun 05, 2023

Calculating the Drake equation is a straightforward mathematical exercise but a wonderful, open question that makes us peek up to the night sky, thinking, "Where is everybody?" Join us on the journey in the Cosmos, where you will learn:

• What's the matter with the search for alien intelligence?

• What is the Drake equation?

• How do we calculate the Drake equation formula?

• What is the answer to the Drake equation?

And much more!

## The search for intelligent life in our Galaxy

From antiquity, our species entertained the thought of a Universe populated by all forms of life. From the first, almost mythological and religious ideas about the "extraterrestrial" life of ancient civilization, humanity always questioned whether we were alone.

With growing scientific knowledge, our methods to satisfy our curiosity also improved. From an increased awareness, reached after Copernicus removed Earth from the central place prescribed by the geocentric model to the development of telescopes and, centuries later, of radioastronomy, we reached a stage where we can scan our Galaxy, able to pick up the signs of any civilization that may exist out there.

Yet we found nothing! If only there were a way to quantify how likely we are to find someone out there...

## What is the Drake equation?

Is our search for civilization meaningful? Does it make sense to invest time and resources in the search for our galactic neighbors? The astronomer Frank Drake decided to find a mathematical argument for these questions in 1961, on the occasion of the first meeting related to the systematic search for extraterrestrial intelligence (SETI). The question is straightforward: Can we estimate the number of alien civilizations in our galaxy? The answer is given by a simple mathematical formula:

$\small N\! =\! R_* \!\times\!f_\mathrm{p} \!\times\!n_\mathrm{e} \!\times\!f_{\mathrm{l}} \!\times\! f_\mathrm{i} \!\times\!f_\mathrm{c} \!\times\!L$

where:

• $N$ — Number of expected alien civilizations;

• $R_*$Rate of stellar formation in our galaxy: a measure of how many stars are born in a given timeframe, usually years;

• $f_\mathrm{p}$Fraction of stars with a planetary system;

• $n_\mathrm{e}$Number of Earth-like planets, or planets able to sustain life in an average planetary system;

• $f_\mathrm{l}$Fraction of planets able to sustain life where life actually develops;

• $f_\mathrm{i}$Fraction of planets with life where intelligence eventually emerges;

• $f_\mathrm{c}$Fraction of intelligent civilizations that manage to develop a way to communicate beyond the planetary boundary; and

• $L$ — The average lifespan of such a civilization.

The result, $N$, is the number of civilizations existing in any given instant in our galaxy.

As you can see, calculating the Drake equation formula is a matter of a few multiplications. Where's the catch? The estimates of the Drake equation formula's factors are not easy to make: Our science is not advanced enough to determine the magnitude of almost all these variables!

## How to calculate the Drake equation: estimates for the Drake equation formula's factors

Each of the factors we use to calculate the Drake equation answers a particular question. We don't have any accurate answer for each of them, with results that vary from educated estimations to wild guesses.

Let's see the original values of the Drake equation, set by its creator in 1961.

• For the rate of stellar formation $N$, Drake chose $1$ new star per year.

• At the time, the number of planets per stellar system constituted a wild guess: the first confirmed exoplanet (a planet orbiting a star different from the Sun) was discovered only in 1995! Drake assessed it to be between $0.2$ and $0.5$ (from a fifth to a half of all starts harbor planets). If you want to learn more about exoplanets, visit our exoplanet discovery calculator!

• The number of Earth-like planets able to sustain life $n_\mathrm{e}$ is the first of the "controversial" values of the Drake equation. A straightforward definition of this quantity starts with eliminating non-rocky planets. We then constrain the position: we want these planets to be in the Goldilock zone (not too hot, not too cold, but just right) where water can consistently exist in a liquid state. We then have to factor in other variables such as the stellar neighborhood, the type of star the planet is orbiting, and other characteristics that would make a planet unsuitable for life. There isn't, and there won't be (at least in the foreseeable future) a consensus on these factors. Drake estimated $n_\mathrm{e}$ to be between $1$ and $5$.

💡 In 1961 we were just about to explore our Solar system. Sixty years later, we discovered that liquid water is much more common than we expected, present on many solar system moons. We also found that Titan, Saturn's largest moon, has a dynamic environment based on the cycle of methane rather than on water. Is Titan a candidate for simple extraterrestrial life? We may discover it soon, as NASA's Dragonfly mission to the moon is expected to launch in 2027 and arrive in 2034!

• For the following parameter, the fraction of planets able to sustain life, where life effectively develops ($f_\mathrm{l}$), Drake gave an optimistic answer: $1$. Life, at least in his view, is inevitable. Current estimates differ.

• As for the development of life, in the view of Drake, the value in the equation that describes the emergence of intelligence is $f_\mathrm{i}=1$: intelligence is an unavoidable consequence of life. As the definition of intelligence is an open and unanswerable question, this estimation is subjected to many changes. Are dolphins intelligent? Undeniably so. Jays? To a surprising degree. Do you know that bees have a rudimental understanding of zero? Are bees intelligent?

• The development of detectable communication is, once again, a hard parameter to estimate. We did it: humanity is currently a gleaming beacon of rather ordinate radio signals that propagate in all directions. But what's the likelihood of other civilizations developing the same way we did? Drake used a conservative estimate of $f_\mathrm{c}$ between $0.1$ and $0.2$.

💡 Why would a civilization end up not developing detectable signs of its presence? This is a wonderful question. It's only a century since humanity started doing so: a blink of an eye in the scale of the entire existence of our species. The booming technological development of the last years may be a fluke. We are also witnessing a decisive self-destructive tendency in the collective behavior of humanity. Is communication associated with short-lived civilizations? Or maybe, silence is a choice?

• The last value of the Drake equation is a matter of debate: For how long can a communicating civilization exist? Drake estimated $L$ to lie between $1000$ and $100,\!000,\!000$ (one hundred million) years. Our current behavior puts us closer to the shorter end of this range unless significant changes happen in the next few years.

Now that we defined all the values of the Drake equation, it's time to compute the result. Let's start with the pessimistic extremes of the specified ranges:

$\small \begin{split} N &= 1\!\times\!0.2\!\times\!1\!\times\!1\!\times\!1\!\times\!0.1\!\times\!1000\\ &= 20 \end{split}$

Humanity is one of them! Let's now check the result of the optimistic estimates!

$\small \begin{split} N &= 1\!\times\!0.5\!\times\!5\!\times\!1\!\times\!1\!\times\!0.2\\ &\quad\times\!100,\!000,\!000\\ &= 50,\!000,\!000 \end{split}$

Fifty million civilizations! If our galaxy is teeming with life, where is everyone?

#### Modern estimates for the Drake equation

In the sixty years after the inception of the Drake equation, astronomers, astrobiologists, and planetary scientists improved the estimates for the factors appearing in the Drake equation formula.
A good set of values is:

• From $R^* = 1.5$ to $R^* = 3$.

• $f_\mathrm{p}= 1$ — The introduction of space telescopes and refined analytical techniques allowed scientists to discover an astonishing number of exoplanets. It appears that having planets is the norm rather than the exception!

• $n_\mathrm{e}$ between $0.5$ and $2$ — Though we discovered many exoplanets, most of the star systems to which they belong don't look as welcoming as our immediate neighborhood.

• $f_\mathrm{l}=1$ — The estimate of Drake remains valid: for most scientists, abiogenesis, the emergence of life is almost assured if the right conditions arise.

• The estimates for a fraction of life-bearing planets that see the emergence of intelligence are still a matter of intense controversy. For some, intelligence is unavoidable, which means $f_\mathrm{i}=1$. For others, the process that leads to complex brains is fragile and long, and the existence of intelligence on Earth is exceptional. Let's set up a range: from $f_\mathrm{i} = 0.0002$ (intelligence is the exception) to $f_\mathrm{p} = 1$ (intelligence is the norm).

• What is the fraction of intelligent civilization that develops detectable communications? The speculations here didn't change much, and a commonly accepted value is $f_\mathrm{c} = 0.2$.

• The lifespan of a communicating civilization, on the other hand, was the subject of some interesting studies: looking at the history of humanity, it seems like civilization lasts about $450$ years. But it's not that humanity disappears; it's instead a takeover. However, in the setting of a globalized civilization, is this estimate reasonable? On the other end of the spectrum, once a civilization reaches a certain level of development, is it possible that it achieved "immortality"?

Let's see another pair of pessimistic and optimistic results we can calculate for the Drake equation.

For the conservative set of values:

• $R^*=1$ new star per year;

• $f_\mathrm{p}\times n_\mathrm{e}\times f_\mathrm{l} = 10^{-5}$ (number of Earth-like planets that develop life);

• $f_\mathrm{i} = 10^{-9}$;

• $f_\mathrm{c} = 0.2$; and

• $L = 300$ years.

we find:

$\small \begin{split} N &= 1\!\times\!10^{-5} \!\times\!10^{-9}\!\times\!0.2\!\times\!300 \\ &=6.0\!\times\!10^{-13} \end{split}$

This result suggests that humanity is the only civilization that will ever wonder if it's alone in the Galaxy.

On the other hand, for an optimistic view of the problem, we'll use the following values:

• $R^*=3$ new stars per year;

• $f_\mathrm{p}=1$;

• $n_\mathrm{e}=0.2$;

• $f_\mathrm{l} = 0.13$;

• $f_\mathrm{i} = 1$;

• $f_\mathrm{c} = 0.2$; and

• $L = 10^9$ years (a billion years: a quarter of Earth's existence).

Substituting these values in the formula for the Drake equation gives us the following:

$\small \begin{split} N&=3\!\times\!1\!\times\!0.2\!\times\!0.13\!\times\!1\!\times\!0.2\!\times\!10^9\\ &=6.0\!\times\!10^{-13}\\ &=15,\!600,\!000 \end{split}$

Who's right?

## Are we alone? The answer to Drake equation

There is no answer to the Drake equation: for now, we can only listen patiently and make estimates. Humanity is getting good at this: at the current stage, our skills at looking for signs of life in the Solar System and beyond are almost optimal. There is a timid consensus that we will find signs of life outside Earth in about a generation. Not finding them would be a harrowing shadow, suggesting that the Universe is less lively than we'd like.

#### The rare Earth hypothesis

The promoters of this hypothesis postulate that regardless of the likelihood of the emergence of intelligence, the bottleneck in the development process of civilization lies in the appearance of the correct conditions. It appears that too many things went "right" in the case of Earth: a stable star, an orbit in the correct position with very low eccentricity (if you want to learn more about it, visit our conic sections calculator and our Earth orbit calculator), even the presence of Jupiter that, with its mass, shielded us from many dangerous asteroid impacts. Maybe we are alone because the Universe is not as hospitable as we think.

#### Great filters

It's possible that during the history of civilization, some extreme events represent almost insurmountable steps. The presence of such great filters would be a final argument confirming that our Planet is even more precious than we think. This theory leaves us with a bittersweet taste: the great filters may lie behind us (after all, life on Earth survived multiple mass extinctions and maybe even more life-ending scenarios), leaving us unaware of our peculiarity, or ahead of us (are nuclear weapons and climate change one-way tickets for humanity's extinction?): in this case, the Universe may be filled with life, waiting to destroy itself.

#### The dark forest theory

Alien civilizations exist and are widespread, but they refuse to communicate due to the fear of an unknown galactic threat. As much as this theory sounds fictional, it's as good as many other attempts to explain the great silence that envelops us.

## FAQ

### What is the Drake equation?

The Drake equation is a probabilistic equation that sets a mathematical way to estimate the number of alien civilizations able to communicate inhabiting our galaxy at any given moment.

Using a small set of parameters, the Drake equation allows us to restrict the number of suitable planets to the ones hosting an intelligent and technologically capable civilization and then to restrict this number according to the lifespan of a society.

### How do I calculate the Drake equation?

To calculate the Drake equation, follow these easy steps:

1. Find the average rate of stellar formation.

2. Multiply by the fractions of stars hosting planets.

3. Multiply by the number of Earth-like planets in a stellar system.

4. Multiply by the fraction of such planets of developing life.

5. Multiply by the fraction of life-bearing planets where intelligence emerges.

6. Multiply by the number of intelligent civilizations that develops interstellar communication.

7. Multiply by the average lifespan of such civilizations.

The result is the number of civilizations in our galaxy at any given time.

### What does the Drake equation attempt to predict?

The Drake equation attempts to predict the number of civilizations that we can expect to discover in our galaxy. To do so, we need to estimate a set of parameters that describe all the necessary steps for the emergence of an intelligent civilization able to communicate beyond the boundaries of its home planet.

### How many habitable planets orbit an average star?

Although the estimates vary, the consensus is that a Sun-like star can host in its stellar system between 0.4 and 0.9 planets with Earth-like size in the "Goldilocks zone", the distance where we can expect to find liquid water on average conditions at the surface.

This estimate will change with the improvement in our ability to detect such planets, with some scientists saying that some star systems may host seven or more worlds suitable to host life as we know it.

Davide Borchia and Rijk de Wet
R* — Rate at which stars suitable for the development of intelligent life are formed:
stars per year
fp — Fraction of those stars with planetary systems:
%
ne — Number of planets, per solar system, with an environment suitable for life:
planets per system
fl — Fraction of suitable planets on which life actually appears:
%
fi — Fraction of life bearing planets on which intelligent life emerges:
%
fc — Fraction of civilizations that develop a technology that produces detectable signs of their existence:
%
L — Average length of time such civilizations produce such signs:
yrs
N — Number of civilizations in the Milky Way galaxy whose electromagnetic emissions are detectable:
N = R* × fp × ne × fl × fi × fc × L
civilizations
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