The Drake equation: first estimates

The Drake equation aims to estimate the number of communicating civilizations and is given, in its original form, as

(2)$$N_{{\rm CC}} = R_\ast \,f_{\rm p} n_{\rm e}\, f_{\rm l}\,f_{\rm i}\,f_{\rm c}\,L,\; $$

where $R_\ast$ is the formation rate of stars in our Galaxy averaged over all times, $f_{\rm p}$ the fraction of stars that host planetary systems, $n_{\rm e}$ the average number of planets per system suitable for life, $f_{\rm i}$ is the fraction of such planets where intelligent life has developed, $f_{\rm c}$ is the fraction of such planets where life has reached the communicative phase, and $L$ is the average duration of the communicative phase.

Some of these parameters are very well-known; we have a star formation rate of $\sim 1.9$ M$_\odot$ year$^{-1}$ (Twarog Reference Twarog1980; Meusinger Reference Meusinger1991; Chomiuk and Povich Reference Chomiuk and Povich2011). As shown by the work of Kroupa and Weidner (Reference Kroupa and Weidner2003), the initial mass function of stars is peaked in the low-mass regime, such that the average number of stars per one solar mass is about 2. This implies the formation of roughly 4 stars per year. In the Milky Way galaxy, overall a rather quiescent evolution seems to have taken place over its history (Kennicutt and Evans Reference Kennicutt and Evans2012). To reproduce the $\sim 10^{11}$ stars in the Milky Way today, one needs to assume on average a rate $R_\ast \sim 10$ year$^{-1}$ for the last $10$ billion years (Prantzos Reference Prantzos2013).

We focus here predominantly on planets in the super-Earth regime with two up to ten Earth masses, as defined by Stevens and Gaudi (Reference Stevens and Gaudi2013). An upper limit on their abundance can be obtained from the analysis of Mayor et al. (Reference Mayor, Marmier, Lovis, Udry, Ségransan, Pepe, Benz, Bertaux, Bouchy, Dumusque, Lo Curto, Mordasini, Queloz and Santos2011), who estimate the fraction of planets with less than 30 Earth masses as about $10\%$. Another estimate has been made by Selsis (Reference Selsis2007), requiring stars to be sufficiently long-lived for the development of complex life forms (with main-sequence lifetimes larger than 4.5 Gyr, and thus masses less than $1.1$ M$_\odot$) while having sufficiently high masses to have habitable zones outside the tidally locked regions, where always the same side of the planet would be exposed to radiation from the star. Based on these considerations, they arrived at a fraction of about $10\%$. We also assume here that within the habitable zone, we will only have about one planet per system. As these are independent requirements that need to be simultaneously fulfilled, we may then estimate $f_{\rm p}n_{\rm e}\sim 0.1\cdot 0.13 = 0.013$ (see also Prantzos Reference Prantzos2013, for a similar discussion).

From astrophysical considerations, we thus arrive at a rate $R_{\rm astro} = R_\ast f_{\rm p}n_{\rm e}\sim$0.052 habitable planets per year. We can further group the biological terms, $f_{\rm b} = f_{\rm l}f_{\rm i}$, describing the fraction of those planets where intelligent life developed. The only reference system we can consider here is Earth, which has an approximate total age of $4.5$ Gyrs, while the first hominids appeared about $6$ million years ago. Based on these considerations, one could estimate $f_{\rm b}\sim {0.006\over 4.5}\sim 10^{-3}$, which might be considered as an upper limit. To estimate the fraction of planets where life has reached the communicative phase, again our only reference point is Earth, where radio signals exist since roughly the 1950s. Based on this consideration, one might infer a possible minimum value of the corresponding fraction, of course under the assumption that typical life will behave similarly as on Earth, which would be $f_{\rm c, min}\sim {70\over 6\times 10^6}\sim 10^{-6}$. On the other hand, this fraction may also be considerably larger, and also the human society may keep communicating for a considerable amount of time. With the average duration of the communication phase $L$, we could make an improved estimate $f_{\rm c}\sim {L\over 6\times 10^6{\rm \ years}}$, implying $f_{\rm c}L\sim {L^2\over 6\times 10^6{\rm \ years}}$.

Putting the numbers together, we thus obtain

(3)$$N_{{\rm CC}}\sim8.7\times10^{-12}\left({L\over {\rm years}} \right)^2.$$

The number of communicating civilizations thus depends very strongly on their assumed lifetime, and of course it is still important to note that it would require a very advanced civilization to detect the radio signals from Earth over a large distance. Nonetheless, for lifetimes of $L\sim 10^6$ years, we could then expect $N_{{\rm CC}}\sim$9 communicating civilizations in our Galaxy.

A very relevant uncertainty can be removed, however, if we employ a modified version of the Drake equation where we aim to estimate the total number of civilizations, irrespective of whether they are communicating or not. We denote the number of total civilizations as $N_{\rm c}$. It can be estimated by removing the factor $f_{\rm c}$ from the Drake equation and by replacing the lifetime $L$ of the communicative phase with the total lifetime $L_{\rm tot}$ of an average civilization. We obtain the following estimate:

(4)$$N_{{\rm c}}\sim5.2\times10^{-5}\left({L_{\rm tot}\over {\rm years}} \right).$$

If we take $L_{\rm tot}\sim 10^7$ years, which can be easily justified with the timescale since hominids are living on Earth, we arrive at $N_{\rm C}\sim$520 civilizations.

These estimates in particular do not yet take the possible process of colonization into account. While our civilization has not started such a process yet, it is worth noting that Earth formed only $4.5$ Gyrs ago, and other civilizations may have formed even Gyrs before ours. A more complete model thus needs to take this process into account. We further point out another important limitation of the Drake equation: being tied to the star formation rate, it assumes civilizations to exist only on planets around recently formed stars. It is not obvious that this assumption is correct and even in the complete absence of current star formation, it would be conceivable that extraterrestrial civilizations exist on planets that formed billions of years ago. Another improvement of a dynamical framework is thus that it allows to take both the recently formed planets plus the already existing ones into account.

A non-equilibrium model

We now go beyond the Drake equation and present a dynamical non-equilibrium model which takes into account the process of colonization as well as astrophysical destruction mechanisms of civilizations. Within this framework, we will consider the number of habitable planets $N_{\rm h}$ which provide conditions for life to develop though it has not developed yet. $N_{\rm l}$ denotes the number of systems where simple forms of life have developed, $N_{\rm a}$ the number of systems including animal life forms on land and finally $N_{\rm c}$ the number of systems with civilizations. The total number of habitable planets at a given time is thus given as $N_{\rm h} + N_{\rm l} + N_{\rm a} + N_{\rm c}$, which is increasing over time due to ongoing star formation.

The evolution equations in our model are given as (see also Fig. 1 for an illustration)

(5)$$\dot{N}_{\rm h} = R_{\rm astro}( t) -\epsilon_{\rm l} N_{\rm h}-\epsilon_{\rm col}N_{\rm c}N_{\rm h}/N,\; $$

(6)$$\dot{N}_{\rm l} = \epsilon_{\rm l} N_{\rm h}-\epsilon_{\rm a} N_{\rm l} + \epsilon_{\rm ast, a}( t) N_{\rm a}-\epsilon_{\rm col}N_{\rm c}N_{\rm l}/N,\;$$

(7)$$\eqalign{\cr \dot{N}_{\rm a}& = \epsilon_{\rm a} N_{\rm l}-\epsilon_{\rm c} N_{\rm a} + \epsilon_{\rm d} N_{\rm c} + \epsilon_{\rm ast, c}( t) N_{\rm c}\cr & \quad-\epsilon_{\rm ast, a}( t) N_{\rm a}-\epsilon_{\rm col}N_{\rm c}N_{\rm a}/N,\;}$$

(8)$$\eqalign{\dot{N}_{\rm c} & = \epsilon_{\rm c} N_{\rm a}-\epsilon_{\rm d}N_{\rm c} + \epsilon_{\rm col}N_{\rm c}( N_{\rm h} + N_{\rm l} + N_{\rm a}) /N \cr & \quad-\epsilon_{\rm ast, c}( t) N_{\rm c},\;} $$

where $N = N_{\rm h} + N_{\rm l} + N_{\rm a} + N_{\rm c}$ is the total number of systems, $R_{\rm astro}( t)$ is the astrophysical formation rate of habitable systems and $\epsilon _{\rm l}$, $\epsilon _{\rm a}$, $\epsilon _{\rm c}$ $\epsilon _{\rm col}$, $\epsilon _{\rm d}$, $\epsilon _{\rm ast, c}$ and $\epsilon _{\rm ast, a}$ denote various efficiency parameters that are introduced in further detail below.

Figure 1. Illustration of the dynamical model described via equations (5)–(8). A symbols legend is also reported for clarity.

First, we aim to derive a time-dependent model for the formation rate of habitable planets. As the evolution of star formation in the Milky Way overall was rather quiescent (e.g. Kennicutt and Evans Reference Kennicutt and Evans2012), we assume only a moderate time-dependence, following an exponential decay over a long timescale of $\sim 5$ Gyrs. We choose our normalization such that over a timescale of $10$ Gyrs, the same number of systems will be produced with the time-averaged rate we adopted in the Drake equation above. We then have

(9)$$R_{\rm astro}( t) = 0.20\times10^9{\rm \ Gyr}{\rm \ \exp}\left(-{t\over 5{\rm \ Gyr}} \right).$$

Here $t$ is the time coordinate in our model, which is chosen such that at the present day we have $t = 0$.

As initial conditions, we assume here that $N_{\rm h} = N_{\rm l} = N_{\rm a} = N_{\rm c} = 0$, so all the systems form via the production term $R_{\rm astro}( t)$. $\epsilon _{\rm l}$ is the efficiency parameter describing the transition from a habitable planet to a planet with simple forms of life (cells or bacteria), which we parametrize as $\epsilon _{\rm l} = T_{\rm l}^{-1}$. $T_{\rm l}$ is the characteristic timescale for a simple form of life to develop. We adopt here $T_{\rm l} = 10^9$ years, comparable to the time it took for the first life forms to develop on Earth. The term $\epsilon _{\rm a}$ describes the transition from a planet with simple forms of life to planets with animal-type forms of life that live on land. We consider this as a particularly important intermediate stage that was a precondition for the subsequent development of civilizations. We adopt here $\epsilon _{\rm a} = T_{\rm a}^{-1}$, with $T_{\rm a}\sim 3$ Gyr considering the evolutionary time on Earth.

For the development of intelligent life and civilizations, similar as Annis (Reference Annis1999), we assume that roughly $0.1$ Gyr are required to develop sufficiently complex base classes of land-dwelling creatures, and subsequently another 0.1 Gyr to develop intelligence once sufficiently complex base classes were being established. For the parameter $\epsilon _{\rm c}$ describing the transition to intelligent life and civilizations, we thus adopt $\epsilon _{\rm c} = T_{\rm c}^{-1}$, with $T_{\rm c}\sim 0.2$ Gyr.

Astrophysical destruction mechanisms

We now consider astrophysical effects that can potentially destroy civilizations and even affect animal life. We parametrize the effect on civilizations via $\epsilon _{\rm ast, c}( t) = \epsilon _{\rm AGN}( t) + \epsilon _{{\rm gb}}( t) + \epsilon _{\rm SNIa}( t) + \epsilon _{\rm SNII}( t)$, including feedback from the galactic supermassive black hole, gamma-ray bursts as well as type Ia and type II supernovae. The effect of such feedback on animal life is incorporated as well, which we parametrize via $\epsilon _{\rm ast, a} = a\epsilon _{\rm ast, c}$ with $a\sim 0.25$. The parameter $a$ is of course very uncertain, though we consider here that more simple types of small animals should have higher chances to potentially survive catastrophic events.

For up to the first two billion years in our Galaxy, the supermassive black hole has likely been active, and we parametrize the effect of its feedback following Balbi and Tombesi (Reference Balbi and Tombesi2017), who calculated that it would very severely affect planetary atmospheres and ecosystems up to $3$ kpc from the centre of the Galaxy. Considering a typical size of $30$ kpc of the Galactic disk, the fraction of systems affected via the feedback thus amounts to $p_{\rm AGN}\sim 0.01$. We adopt a characteristic timescale of $T_{\rm AGN}\sim 10^{-3}$ Gyrs for the feedback events from the black hole, and thus arrive at an efficiency $\epsilon _{\rm AGN} = p_{\rm AGN}/T_{\rm AGN}$ during the first $2$ Gyrs, and subsequently $\epsilon _{\rm AGN} = 0.$

For gamma-ray bursts, it is harder to make precise estimates, as their explosions are not spherical but highly beamed. In principle there are uncertainties given their absolute event rates, as they can only be detected from Earth if Earth falls into the beam of the gamma-ray burst. In case of smaller beams, the event rate is thus higher than typically estimated, and smaller for larger beams. Regarding the overall efficiency of the feedback, this uncertainty nevertheless is balanced out as a higher event rate would then correspond to an accordingly smaller part of the Galaxy affected by the event. For simplicity, what we adopt here is the estimate by Melott et al. (Reference Melott, Lieberman, Laird, Martin, Medvedev, Thomas, Cannizzo, Gehrels and Jackman2004), who assume that planetary systems would be strongly affected within $3$ kpc of the gamma-ray burst, thus corresponding to a fraction of $p_{{\rm gb}} = 0.01$ of the Galactic disk. Again following Melott et al. (Reference Melott, Lieberman, Laird, Martin, Medvedev, Thomas, Cannizzo, Gehrels and Jackman2004), we assume that currently in the Universe, such events occur roughly at a period of $P_\gamma = 170$ Myrs. This rate should be expected to evolve over time, and we assume it to decay over a characteristic timescale of $t_\gamma \sim 5$ Gyr, consistent with cosmological observations (Bromm and Loeb Reference Bromm and Loeb2002). We thus arrive at the following event rate

(10)$$\nu_\gamma( t) = \nu_0{\rm exp}\left(-{t\over t_\gamma} \right)$$

with $\nu _0 = 1/P_\gamma$ and the corresponding efficiency parameter is thus given as

(11)$$\epsilon_{{\rm gb}}( t) = p_{{\rm gb}}\nu_\gamma( t) .$$

For the rate of type Ia supernovae, we adopt the estimated rate of $\nu _{\rm SNIa} = 4\times 10^{-3}$ yr$^{-1}$ from Gowanlock et al. (Reference Gowanlock, Patton and McConnell2011) for the last billion years, consistent also with Meng and Yang (Reference Meng and Yang2010). The type Ia supernova rates are expected to evolve as the Galactic star formation rate, though with a delay time of $1$ Gyr due to stellar evolution effects. We assume here a similar decay law as for the gamma-ray bursts, though with a different frequency. For the type Ia supernovae, Gowanlock et al. (Reference Gowanlock, Patton and McConnell2011) derived a sterilization radius of 19 pc, corresponding to a fraction $p_{\rm SNIa}\sim 4\times 10^{-7}$ of the Galactic disk. Accounting for the delay time of $1$ Gyr, we write

(12)$$\epsilon_{{\rm SNIa}}( t) = p_{{\rm SNIa}}\nu_{{\rm SNIa}}\ {\rm exp}\left(-{t-1{\rm \ Gyr}\over t_\gamma} \right).$$

For the type II supernovae, we recall the sterilization radius derived by Gowanlock et al. (Reference Gowanlock, Patton and McConnell2011), corresponding to $8$ pc or a fraction of $p_{\rm SNII} = 0.008^2/30^2\sim 7\times 10^{-8}$ of the Galactic disk. For their event rate, we adopt a value of $\nu _{\rm SNII} = 2\times 10^{-2}$ yr$^{-1}$, which is in-between the Gowanlock et al. (Reference Gowanlock, Patton and McConnell2011) estimates. We thus have

(13)$$\epsilon_{\rm SNII}( t) = p_{\rm SNII}\nu_{\rm SNII}\ {\rm exp}\left(-{t\over t_\gamma} \right),\; $$

assuming the time dependence to be roughly the same as for the gamma-ray bursts.

Colonization

We finally arrive at the uncertainties that are set by the potential behaviour or evolution of intelligent civilizations themselves, which we will regard here as essentially unknown, and we will aim to avoid here any a priori assumptions to the degree possible. Assuming that advanced civilizations will aim to survive catastrophic destruction events of their host stellar system due to the finite lifetime of the star, one could assume that at least a few colonization events should occur over the characteristic stellar evolution time of $\sim 10$ Gyrs in case of a solar mass star, and we parametrize here the colonization efficiency as

(14)$$\epsilon_{\rm col} = {N_{\rm new}\over 10{\rm \ Gyr}},\; $$

where $N_{\rm new}$ the number of new colonies a typical civilization would produce over a time of $10$ Gyrs (assuming they exist long enough). Depending on the typical behaviour of such civilizations, of course $N_{\rm new}$ could be considerably larger than a factor of a few, and we will later explore the dependence on this parameter. The other relevant and basically unknown factor is the typical lifetime of a civilization. On Earth, the first hominids appeared roughly $6$ million years ago, which sets a possible lower limit for the existence of an intelligent species (though most of that time had not been spent at a technologically advanced stage). It may be the case that the lifetime of a civilization becomes shorter once a significant level of technological advance has been reached, but is also not completely obvious. We therefore will regard it as an unknown parameter as well, and parametrize it as $\epsilon _{\rm d} = T_{\rm d}^{-1}$.