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The Drake Equation
Posted: 24 May 2010

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How many advanced extraterrestrial civilisations are there in our Galaxy? Some critics argue that you might as well ask how many angels can dance on the point of a needle, but Frank Drake thinks otherwise. The SETI pioneer’s famous equation seeks an answer to that question and although it has proven controversial over the years, Drake still stands by it and it continues, rightly or wrongly, to form the heart of our estimates about ET.

Dr Frank Drake and his famous equation.

The Drake Equation has held sway over SETI for the last fifty years, so only a brief reminder of its contents and history should be required. Created by Drake to provide talking points for the first ever SETI conference at the Green Bank radio observatory in West Virginia in November 1960, the equation collects a roster of variables that Drake deemed relevant to the evolution of intelligent life, and multiplies them to produce an estimate of the number of advanced civilisations in the Galaxy. The equation is:

N = R* x f(p) x n(e) x f(l) x f(i) x f(c) x L

Where N is the total number of civilisations, R* is the star-formation rate (SFR) in the Milky Way, f(p) the fraction of stars with planets, n(e) is the number of habitable planets per star, f(l) is the fraction of habitable planets with life on them, f(i) is the fraction of those planets where intelligent life arises, f(c) is the fraction of those planets that develop technological civilisations, and L is the lifetime of those civilisations.

The reason for its controversial status is that the majority of these factors are complete unknowns. This was especially so back in the day when Drake formulated the equation that has become his greatest legacy. Ten years into the twenty-first century we’re better placed to assess at least some of these factors, and understand what we need to learn to better appreciate the remaining unknowns. The question is, ultimately is there any usefulness to the Drake Equation, and could it be improved further?

Part 1: Star birth

Let’s begin with the factors that we know best – the astrophysical stuff. If life needs planets, then planets need stars. The rate of star birth, R*, is supposedly an easy one; even in 1960 it was possible to make a pretty good estimation. Drake estimated that ten stars per year were born in our Milky Way; fifty years later and estimates of the SFR have remained in the same ballpark. There are two ways to measure the star-formation rate; indirectly, through some by-product of star-birth and star-death, or directly, by counting up the number of baby stars in the Galaxy.

The former, indirect, method has been largely employed over recent decades and its most accurate findings came in 2006 thanks to the European Space Agency’s INTEGRAL (International Gamma-Ray Astrophysics Laboratory) spacecraft. Boldly claimed by ESA to be the most sensitive gamma-ray observatory ever launched, it has measured the amount of aluminium-26 in the Galaxy, a rare isotope produced by core-collapse supernova explosions (the destruction of massive stars). Aluminium-26 is radioactive, with a half-life of 750,000 years, and emits gamma-rays with an energy of 1.809 MeV, producing a faint glow at this energy for INTEGRAL to detect. And when we say rare, we mean it – judging by INTEGRAL’s observations, there is only 2.8 solar masses worth of this stuff floating around in the Milky Way. Given that each supernova throws out about 0.0001 solar masses of aluminium-26, the Milky Way must have witnessed 20,000 supernovae over the past million years or so to arrive at the current quantity (coincidentally, that works out on average as two supernovae per century).

The Orion Nebula, where stars are being born. Image: NASA/C R O’Dell and S K Wong (Rice University).

The supernova rate gives us some indication of how many stars are being born, because for every massive star that eventually blows up, thousands of lower mass are created in stellar nurseries like the famous Orion Nebula. “If you form 100 stars then it is quite unlikely that one of them will be a massive star,” says Professor Tim Naylor of the University of Exeter. “But if you form a very large association of thousands of stars then some of them are going to be more massive stars.” This population distribution is called the ‘initial mass function’ and it basically counts the number of stars of a given mass; you can imagine totting them up on a histogram with mass running from left to right. If we did, we’d find that the graph slopes down to the right, with the lowest mass stars far outweighing the more massive stars. And that’s good for life. More massive stars don’t last very long – those that explode as supernovae remain for just a few million years or so before they deplete their fuel stores and explode. Lower mass stars like our Sun can survive for ten billion years or so, and as we have seen on Earth that is plenty of time for intelligent life to arise. Red dwarfs, the smallest stars, can survive even longer, perhaps 50 billion years, and red dwarfs are the most common type of star in the Galaxy.

By deducing the Milky Way’s massive star population based on how often they explode, and working back to deduce how many lower mass stars were born alongside them, INTEGRAL’s observations imply that on average four solar masses worth of gas is converted into new stars each year. In practice this could add up to as many as seven stars, if we bear in mind that many are less massive than the Sun.

More recently, as revealed in a paper in the 10 February 2010 edition of Astrophysical Journal Letters, a more direct method may have refined this value even further. Many young stars are still hidden inside their nebulae, swathed by a cloak of gas and dust. Ordinarily we can’t see through this cloak, but infrared light can penetrate it. As part of the GLIMPSE survey (the Galactic Legacy Infrared Mid-Plane Survey Extraordinaire), NASA’s Spitzer infrared space telescope, under the guidance of astronomers Thomas Robitaille of the Harvard–Smithsonian Center for Astrophysics and Barbara Whitney of the Space Science Institute in Colorado, counted the number of young stellar objects (YSOs) in the 2 x 130 degree survey field. YSOs are protostars, still encased within their nebulous wombs and easily visible to infrared telescopes like Spitzer. Inputting the observed population of YSOs into a computer model that describes galactic star birth, Robitaille and Whitney came up with a figure for R* between 0.68 and 1.45 solar masses per year. They argue that because they are measuring the number of young stars directly, it is more accurate than estimating the star formation rate by indirect methods.

However, the Galaxy couldn’t have grown to its current size by creating just one new star per year, or even seven or ten stars per year. There is somewhere in the region of 250–500 billion stars in the Milky Way, many of which are far older than the Sun (which is 4.6 billion years old). Indeed, far fewer stars are being born today than in the past. The most massive stars had their heyday around eleven billion years ago; Sun-like stars reached a peak six billion years ago; today we are in the age of low mass red dwarf stars, and even their formation rate peaked two billion years ago. So R* is variable with time, a fact not considered by the Drake Equation. It’s no good trying to figure out what the star formation rate is today when looking for extraterrestrial intelligences that have spent billions of years evolving. The star-formation rate five or six billion years ago has been estimated to be six times what it is today, which is problematic because the Spitzer GLIMPSE survey says that one star per year is born in the current era, which would mean that six stars a year were being born five billion years ago. However, if INTEGRAL’s aluminium-26 observations turn out to be more accurate, then five billion years ago 42 stars per year were being born. That’s almost an order of magnitude difference.

We could go even further back. A trio of telescopes – Spitzer, the Hubble Space Telescope and the Japanese-built Subaru Telescope – have observed a galaxy that existed at a lookback time (the time it has taken for its light to reach us) of 12.3 billion light years that has a star-formation rate estimated to be 4,000 solar masses per annum. In ‘only’ 50 million years (on the cosmological timescale, very little time indeed) this galaxy will be able to grow to the size of modern day massive galaxies, assuming it keeps up this ferocious bout of activity. If, as we expect, our Galaxy formed in pretty much the same fashion, then the Milky Way would have one day had a similar star-formation rate. The Drake Equation discounts this, however, because its next term is the fraction of stars that have Earth-like planets, and to build a terrestrial world, large amounts of ‘metals’ – astronomer-speak for elements more massive than hydrogen and helium, formed inside stars – are required. These metals, such as carbon, oxygen, nitrogen, silicon and iron, were only available en masse around six or seven billion years ago after there had been sufficient generations of stars to produce them, although that doesn’t discount the possibility that some terrestrial worlds could have formed earlier, but just with less frequency. So what figure should we give R*? The one that makes most sense when looking for intelligent life today is the rate from six billion years ago, which could be anywhere between 6 and 42. Take your pick.

The star formation rate is often deemed one of the more straight-forward factors in the Drake Equation, and already we have found some problems with it. Can our search for exoplanets help make the next factor, the fraction of stars with terrestrial planets, f(p), any more accurate?

More SETI articles available here