Mistakes in the Drake Equation

Juggling all the factors impacting the emergence of extraterrestrial civilizations is no easy task, which is why the Drake equation has become such a handy tool. But are there assumptions locked inside it that need examination? Robert Zubrin thinks so, and in the essay that follows, he explains why, with a particular nod to the possibility that life can move among the stars. Although he is well known for his work at The Mars Society and authorship of The Case for Mars, Zubrin became a factor in my work when I discovered his book Entering Space: Creating a Spacefaring Civilization back in 2000, which led me to his scientific papers, including key work on the Bussard ramjet concept and magsail braking. Today’s look at Frank Drake’s equation reaches wide-ranging conclusions, particularly when we begin to tweak the parameters affecting both the lifetime of civilizations and the length of time it takes them to emerge and spread into the cosmos.

by Robert Zubrin

There are 400 billion other solar systems in our galaxy, and it’s been around for 10 billion years. Clearly it stands to reason that there must be extraterrestrial civilizations. We know this, because the laws of nature that led to the development of life and intelligence on Earth must be the same as those prevailing elsewhere in the universe.

Hence, they are out there. The question is: how many?

In 1961, radio astronomer Frank Drake developed a pedagogy for analyzing the question of the frequency of extraterrestrial civilizations. According to Drake, in steady state the rate at which new civilizations form should equal the rate at which they pass away, and therefore we can write:

Equation (1) is therefore known as the “Drake Equation.” Herein, N is the number of technological civilizations is our galaxy, and L is the average lifetime of a technological civilization. The left-hand side term, N/L, is the rate at which such civilizations are disappearing from the galaxy. On the right-hand side, we have R∗, the rate of star formation in our galaxy; f p , the fraction of these stars that have planetary systems; n e , is the mean number of planets in each system that have environments favorable to life; f l the fraction of these that actually developed life; f i the fraction of these that evolved intelligent species; and f c the fraction of intelligent species that developed sufficient technology for interstellar communication. (In other words, the Drake equation defines a “civilization” as a species possessing radiotelescopes. By this definition, civilization did not appear on Earth until the 1930s.)

By plugging in numbers, we can use the Drake equation to compute N. For example, if we estimate L=50,000 years (ten times recorded history), R∗ = 10 stars per year, f p = 0.5, and each of the other four factors n e , f l , f i , and f c equal to 0.2, we calculate the total number of technological civilizations in our galaxy, N, equals 400.

Four-hundred civilizations in our galaxy may seem like a lot, but scattered among the Milky Way’s 400 billion stars, they would represent a very tiny fraction: just one in a billion to be precise. In our own region of the galaxy, (known) stars occur with a density of about one in every 320 cubic light years. If the calculation in the previous paragraph were correct, it would therefore indicate that the nearest extraterrestrial civilization is likely to be about 4,300 light years away.

But, classic as it may be, the Drake equation is patently incorrect. For example, the equation assumes that life, intelligence, and civilization can only evolve in a given solar system once. This is manifestly untrue. Stars evolve on time scales of billions of years, species over millions of years, and civilizations take mere thousands of years.

Current human civilization could knock itself out with a thermonuclear war, but unless humanity drove itself into complete extinction, there is little doubt that 1,000 years later global civilization would be fully reestablished. An asteroidal impact on the scale of the K-T event that eliminated the dinosaurs might well wipe out humanity completely. But 5 million years after the K-T impact the biosphere had fully recovered and was sporting the early Cenozoic’s promising array of novel mammals, birds, and reptiles. Similarly, 5 million years after a K-T class event drove humanity and most of the other land species to extinction, the world would be repopulated with new species, including probably many types of advanced mammals descended from current nocturnal or aquatic varieties.

Human ancestors 30 million years ago were no more intelligent than otters. It is unlikely that the biosphere would require significantly longer than that to recreate our capabilities in a new species. This is much faster than the 4 billion years required by nature to produce a brand-new biosphere in a new solar system. Furthermore, the Drake equation also ignores the possibility that both life and civilization can propagate across interstellar space.

So, let’s reconsider the question.

Estimating the Galactic Population

There are 400 billion stars in our galaxy, and about 10 percent of them are good G and K type stars which are not part of multiple stellar systems. Almost all of these probably have planets, and it’s a fair guess that 10 percent of these planetary systems feature a world with an active biosphere, probably half of which have been living and evolving for as long as the Earth. That leaves us with two billion active, well-developed biospheres filled with complex plants and animals, capable of generating technological species on time scales of somewhere between 10 and 40 million years. As a middle value, let’s choose 20 million years as the “regeneration time” t r . Then we have:

where N and L are defined as in the Drake equation, and n s is the number of stars in the galaxy (400 billion), f g is the fraction of them that are “good” (single G and K) type stars (about 0.1), f b is the fraction of those with planets with active biospheres (we estimate 0.1), f m is the fraction of those biospheres that are “mature” (estimate 0.5), and n b , the product of these last four factors, is the number of active mature biospheres in the galaxy.

If we stick with our previous estimate that the lifetime, L, of an average technological civilization is 50,000 years, and plug in the rest of the above numbers, equation (2) says that there are probably 5 million technological civilizations active in the galaxy right now. That’s a lot more than suggested by the Drake equation. Indeed, it indicates that one out of every 80,000 stars warms the home world of a technological society. Given the local density of stars in our own region of the galaxy, this implies that the nearest center of extraterrestrial civilization could be expected at a distance of about 185 light years.

Technological civilizations, if they last any time at all, will become starfaring. In our own case (and our own case is the only basis we have for most of these estimations), the gap between development of radiotelescopes and the achievement of interstellar flight is unlikely to span more than a couple of centuries, which is insignificant when measured against L=50,000 years. This suggests that once a civilization gets started, it’s likely to spread. Propulsion systems capable of generating spacecraft velocities on the order of 5 percent the speed of light appear possible. However, interstellar colonists will probably target nearby stars, with further colonization efforts originating in the frontier stellar systems once civilization becomes sufficiently well-established there to launch such expeditions.

In our own region of the galaxy, the typical distance between stars is five or six light years. So, if we guess that it might take 1,000 years to consolidate and develop a new solar system to the point where it is ready to launch missions of its own, this would suggest the speed at which a settlement wave spreads through the galaxy might be on the order of 0.5 percent the speed of light. However, the period of expansion of a civilization is not necessarily the same as the lifetime of the civilization; it can’t be more, and it could be considerably less. If we assume that the expansion period might be half the lifetime, then the average rate of expansion, V, would be half the speed of the settlement wave, or 0.25 percent the speed of light.

As a civilization expands, its zone of settlement encompasses more and more stars. The density, d, of stars in our region of the galaxy is about 0.003 stars per cubic light year, of which a fraction, f g , of about 10 percent are likely to be viable potential homes for life and technological civilizations. Combining these considerations with equation 2, we can create a new equation to estimate C, the number of civilized solar systems in our galaxy, by multiplying the number of civilizations N, by, n u , the average number of useful stars available to each.

For example, we have assumed that the average lifespan, L, of a technological species is 50,000 years, and if that is true, then the average age of one is half of this, or 25,000 years. If a typical civilization has been spreading out at the above estimated rate for this amount of time, the radius, R , of its settlement zone would be 62.5 light years (R = VL/2 = 62.5 ly), and its domain would include about 3,000 stars. If we multiply this domain size by the number of expected civilizations calculated above, we find that about 15 billion stars, or 3.75 percent of the galactic population, would be expected to lie within somebody’s sphere of influence. If 10 percent of these stars are actually settled, this implies there are about 1.5 billion civilized stellar systems within our galaxy. Furthermore, we find that the nearest outpost of extraterrestrial civilization could be expected to be found at a distance of 185-62.5 = 122.5 light years.

The above calculation represents my best guess as to the shape of things, but there’s obviously a lot of uncertainty in the calculation. The biggest uncertainty revolves around the value of L; we have very little data to estimate this number and the value we pick for it strongly influences the results of the calculation. The value of V is also rather uncertain, although less so than L, as engineering knowledge can provide some guide. In Table 1 we show how the answers might change if we take alternative values for L and V, while keeping the other assumptions we have adopted constant.

Table 1 The Number and Distribution of Galactic Civilizations

V=0.005 c V=0.0025 c V=0.001 c L=10,000 years N (# civilizations) 1 million 1 million 1 million C (# civilized stars) 19.5 million 2.4 million 1 million R (radius of domain) 25 ly 12.5 ly 5 ly S (Separation between civilizations) 316 ly 316 ly 316 ly D (distance to nearest outpost) 291 ly 304 ly 311 ly F (fraction of stars within domains) 0.048% 0.006% 0.0025% L=50,000 years N (# civilizations) 5 million 5 million 5 million C (# civilized stars) 12 billion 1.5 billion 98 million R (radius of domain) 125 ly 62.5 ly 25 ly S (Separation between civilizations) 185 ly 185 ly 185 ly D (distance to nearest outpost) 60 ly 122.5 ly 160 ly F (fraction of stars within domains) 30% 3.75% 0.245% L=200,000 years N (# civilizations) 20 million 20 million 20 million C (# civilized stars) 40 billion 40 billion 18 billion R (radius of domain) 500 ly 250 ly 100 ly S (Separation between civilizations) 131 ly 131 ly 131 ly D (distance to nearest outpost) 0 ly 0 ly 31 ly F (fraction of stars within domains) 100% 100% 44%

In Table 1, N is the number of technological civilizations in the galaxy (5 million in the previous calculation) , C is the number of stellar systems that some civilization has settled (1.5 billion, above), R is the radius of a typical domain (62.5 ly above), S is the separation distance between the centers of civilization (185 ly above), D is the probable distance to the nearest extraterrestrial outpost (122.5 ly, above), and F is the fraction of the stars in the galaxy that are within someone’s sphere of influence (3.75% above).

Examining the numbers in Table 1, we can see how the value of L completely dominates our picture of the galaxy. If L is “short” (10,000 years or less), then interstellar civilizations are few and far between, and direct contact would almost never occur. If L is “medium” (~50,000 years), then the radius of domains is likely to be smaller than the distance between civilizations, but not much smaller, and so contact could be expected to happen occasionally (remember, L, V, and S are averages; particular civilizations in various localities could vary in their values for these quantities). If L is a long time (> 200,000 years), then civilizations are closely packed, and contact should occur frequently. (These relationships between L and the density of civilizations apply in our region of the galaxy. In the core, stars are packed tighter, so smaller values of L are needed to produce the same “packing fraction,” but the same general trends apply.)

Any way you slice it, one thing seems rather certain: There’s plenty of them out there.

What are these civilizations like? What have they achieved?

It would be good to know.