Another go at the Drake equation…

The Drake equation is a very famous bit of fun. Drake tried to use the rate of star formation and the likelihood of planetary formation to get an estimate of how many radio communicable civilisations were in existance. However, at this time not a single exoplanet had been observed, all the planets we knew of were in orbit of the Sun. This meant quite a few of the variables in the equation were guesses, or at best, wild estimates.

As the years passed, the field of exoplanetology has grown into one that encompasses the discovery of hundreds of planets, and with the results of the Kepler satellite, will no doubt grow even further. Whilst none of these have been so far good enough to appear alongside a large neon arrow saying “we are here”, they do at least allow some refining of the variables of Drake’s equation. We can make better guesses of how many planets could be host to civilisations.

When David Kipping (who tweets here) tried this last year, he came up with a figure of 50 million habitable exoplanets and 25 million habitable exomoons. In a new study, Gou et al estimate there are 45.5 billion habitable zone exoplanets in the Milky way galaxy in orbit of stars of masses ranging from 0.08 to 4 times the mass of the Sun.

The paper begins by defining the upper and low limits of what a terrestrial planet is – below 0.3 Earth masses and gravitational effects, both tidal and atmospheric, are insufficient, above ten Earth masses and the planet starts to become something else – most likely a gas giant. They also discuss the habitable zone, which is the distance from the star that receives enough energy through starlight to be warm enough for light, but not so much to burn it away again. They add in the dimension of stellar effective temperature as an important factor – no point having a bright star deliver the wrong type of photons. A star heavy in infrared emissions can warm a planet with a greenhouse effect far more efficiently than a brighter star emitting less infrared radiation – and there is a fine balance with the amount of ultraviolet radiation that can be tolerated.

The authors introduce the planetary evolution code they will be using, that has in the past been used to calculate the masses, orbits and numbers of planets formed at whatever distances from stars of different masses. They also introduce ‘Eggleton’s code’, which is a stellar evolution code. Whereas previous studies looked at habitable zones in terms of stellar effective temperature alone, this code looks at Teff and luminosity and calculates it at the beginning and the end of the main sequence of the star.

The authors state that they’re looking for stars that maintain habitable planets for at least 200 million years (noting that others have estimated planets may need five times this amount, plus a bit extra to cover any heavy bombardment period, where rubble in the early solar system struck the early planets). They start their search by using the Initial Mass Function of the Milky Way to determine the proportionality of stars of different masses in the range 0.08-4 times solar mass. They then use their stellar evolution code to determine the link between stellar luminosities and effective temperatures over the same mass range at the start of the main sequence. The habitable zone is defined as ranging from outside the distance at which water is destroyed by starlight until the distance at which carbon dioxide falls as snow. They calculate the habitable zones at the beginning and end of the stellar main sequence over the mass range considered according to this definition. The habitable zone narrows with decreasing stellar mass, and it is interesting to note that beneath 0.6 solar masses, the zone migrates outward so far that it doesn’t overlap with the initial zone at all (the other stellar masses all have some kind of overlap). However, lower mass stars have longer lives and planets do migrate outwards over the course of their star’s life as well as those that migrate in.

Having done all that, they then use their own planetary evolution code to define an initial mass function for planets – that is decide, over the range of stellar masses considered, the probabilities of planets of a given mass appearing in the habitable zones calculated. The planetary masses considered were 0.3-10 Earth masses, as mentioned earlier. They find stars beneath about 0.4 solar masses have a fairly uniformly low probability of producing planets in the right place for the right length of time. There is a leap about the 0.6 solar mass level and above 0.8 solar masses, again the probabilities are rather uniform, but at a much higher level.

This graph of habitable planet probability versus stellar mass is then multiplied by the initial mass function, which defines probabilities of stellar mass produced in the Milky Way. The result is a graph that shows a high probability of planets around abundant low mass stars, falling to a minimum around 0.4 solar masses of around one third that of stars like our Sun’s own probability of having planets in the habitable zone for long enough. Then the graph exhibits a sharp rise to reach a maximum at 0.6 solar masses of around twice the one solar mass probability. Thereafter, there is an exponential drop in the probability with increasing mass.

With there being around three hundred billion stars in the Milky Way, of which half are in binary or larger groups (which the authors point out some speculate could hold habitable planets), the probability function can be converted into hard numbers, which is where the 45.5 billion habitable planet number comes, including 11.548 billion around M type stars (less than 0.45 solar masses), 12.930 billion around K type stars (0.45-0.8 solar masses), 7.622 around G type stars (0.8-1.04 solar masses) and 5.566 around F type stars (1.04-1.4 solar masses).

The authors then cite research done on terrestrial evolution that identifies five steps required to get to complex, intelligent life and their probabilities (on this planet etc). Using that as the basis for a rough calculation, they estimate that there are 4.3 billion planets in the Milky Way with some kind of life. 3.7 million with complex life forms and 36o thousand with intelligent life forms.

In their discussion of results, they note that there are many planets seen with orbits within 0.03 AU and that this would put them in the habitable zones of 0.1 solar mass or less stars (under the mass considered in this study), openning up other possibilities as these stars are the most abundant out there in the Milky Way. They mention their planetary evolution code doesn’t include the effect of tidal interactions between type 1 protoplanetary embryos, which include migration and clearing of the inner solar system. They also haven’t fully considered the effect of deadly but necessary ultraviolet radiation in the habitable zone, which they reserve for a future study. In my view, they also take a rather optimistic view of every variable they come across (lower limits for planets, time for evolution to take hold, stability of planetary orbits in binary or above systems, even the use of terrestrial evolution as an analogue for the probabilities of intelligent life taking hold on a planet), and they don’t take account of stellar winds and surface activity, which recent stellar seismology surveys show is more of a problem even for Sun type stars than our own, suggesting magnetic fields are more important in other systems, but they do a thorough job in other respects, which will benefit from the UV study and further information on the variables covered.


2 responses to “Another go at the Drake equation…

  1. that was a fun read… must admit, that i started reading it and was quite engaged… then i trailed off. i can’t imagine what it must have been like to read the full study and then boil it down the way u did.

    kudos! (i’ll finish the article later)

    • Yes, these paper summaries are a little different from the main news items. I should put a health warning on them. But they’re fun if you’re already into the kinds of things the article’s talking about and so recognise all the weird terms from different areas of astrophysics. Bit of a curate’s egg though. I hope you enjoy the rest of the article!

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