The Drake Equation as a Function of Spectral Type and Time Jacob Haqq-Misra1,2,* and Ravi Kumar Kopparapu1,2,3,4 1 Blue Marble Space Institute of Science, 1001 4th Ave, Suite 3201, Seattle, WA 2 NASA Astrobiology Institute’s Virtual Planetary Laboratory, P.O. Box 351580, Seattle, WA 3 NASA Goddard Space Flight Center, 8800 Greenbelt Road, Mail Stop 699.0 Building 34, Greenbelt, MD 4 Department of Astronomy, University of Maryland, College Park, MD *Email: [email protected] Published in Habitability of the Universe Before Earth, R. Gordon and A. Sharov (Eds.), Elsevier Abstract This chapter draws upon astronomical observations and modeling to constrain the prevalence of communicative civilizations in the galaxy. We discuss the dependence of the Drake equation pa- rameters on the spectral type of the host star and the time since the galaxy formed, which allow us to examine trajectories for the emergence of communicative civilizations over the history of the galaxy. We suggest that the maximum lifetime of communicative civilizations depends on the spectral type of the host star, which implies that F- and G-dwarf stars are the best places to search for signs of technological intelligence today. 1. Introduction The Drake equation is a probabilistic expression for the number of communicative civilizations in the galaxy (Drake, 1965). This equation typically takes the form = , (1) ∗ where R∙* is ∙the rate∙ ∙ of ∙star ∙ formation, fp is the fraction of stars with planets, ne is the number of habitable planets per system, fl is the fraction of habitable planets that develop life, fi is the fraction of inhabited planets that develop intelligence, fc is the fraction of planets with intelligent life that develop technology capable of interstellar communication, and L is the average communicative lifetime of technological civilizations (c.f. Ćirković and Vukotić, 2008). The original formulation of this equation was developed by Frank Drake during a 1961 conference at Green Bank, West Virginia, with the goal of identifying the key factors needed for a planet to develop life and tech- nology that could communicate with Earth. Since this discussion at Green Bank, the Drake equa- tion has seen wide use among scientists and educators, with some studies proposing methods for constraining the Drake equation with observable quantities (Frank and Sullivan, 2016) or statistical methods (Maccone, 2010; Glade et al., 2012) in order to guide future search for extraterrestrial intelligence (SETI) surveys. Others have extended the traditional representation of the Drake equa- tion to include time-dependent quantities, in an effort to understand the distribution of civilizations across the history of the galaxy (Ćirković, 2004; Glade et al., 2012; Prantzos, 2013). In this chapter we discuss the functional dependence of the Drake equation parameters on the spectral type of the host star and the time since the galaxy formed. We discuss observations, plan- etary formation models, and climate models that provide constraints on the values of R*, fp, and ne, as well as looser bounds on fl and fi. We also examine common assumptions for estimating the parameter L and develop two approaches for calculating the maximum value, Lmax, as a function of spectral type. We use these constraints to develop an expression for N(s,t), the number of com- municative civilizations in the galaxy as a function of spectral type s and time since galaxy for- mation t. We also discuss and critique the possibility that N has distinct phases in time, such that certain events are impossible or unlikely until a particular galactic era. Ćirković and Vukotić (2008), fol- lowing Annis (1999), describe this behavior as a ‘phase transition,’ suggesting that phenomenon such as supernovae, gamma ray bursts, or other physical phenomena could have been more prev- alent during the early phase of the galaxy compared to today. The idea of a phase transition indi- cates that astrophysical factors may have inhibited the emergence of communicative civilizations until relatively recently in galactic history. We argue that phase transitions for N are most likely to emerge from the dependence of Lmax on stellar spectral type. 2. Constraints from Observations In this section we discuss observational constraints on the parameters R*, fp, and ne along with their dependence on spectral type and time. We consider F-, G-, K-, and M-dwarf stars in the majority of our analysis, so that { , , , }, as these present the most likely candidates for supporting intelligent life. However, casting the Drake equation as a function of spectral type also allows for consideration of O-, B-, and∈ A-dwarf stars (which will explicitly show why such stars are deemed unlikely for advanced life) and even planets around evolved stars. 2.1 Rate of Star Formation Common analyses of the Drake equation cite a value of 7 to 10 star/yr as an estimate of R*. Because R* is difficult to measure directly, one way of calculating this rate is to write = , where n* is the current number of stars in the galaxy and t0 ≈ 13 Gyr is the current age of the galaxy. ∗ ∗ 0 However, this uniform approach assumes that the rate of star formation has remained constant⁄ over the history of the galaxy (Ćirković, 2004), proportional to the stellar population today. As an im- provement to previous estimates that offer a single value of R*, we calculate values of R* that depend on spectral type. Analysis of Spitzer data (Robitaille and Whitney, 2010) finds that the stellar mass formation rate is 0.68 to 1.45 Msun/yr (where Msun is the mass of the sun). To convert this number into a value for R*(s), we separate the spectral types into mass ranges based on the initial mass function categories of Kroupa (2001). We assume an initial mass of 1 Msun for F- and G-dwarfs, 0.5 Msun for K-dwarfs, and 0.1 Msun for M-dwarfs. We then divide the observed stellar mass formation rate by this initial mass function to give R*(s), with units of number of stars formed per year. F- and G-dwarfs have the lowest rate of R*(F) ~ R*(G) ~ 1 star/yr. K-dwarfs have a rate of R*(K) ~ 1.3 to 3 star/yr (we use R*(K) = 2.2 star/yr in our analysis), while M-dwarfs have the highest rate of R*(M) ~ 7 to 14 star/yr (we use R*(M) = 10.5 star/yr in our analysis). We summarize these values for R* in Table 1. Star R* fp ne fl·fi·fc L* F-dwarf 1 star/yr 1 0.2 1 4 Gyr 0 Gyr 2 Gyr G-dwarf 1 star/yr 1 0.2 1 10 Gyr 6 Gyr 5 Gyr K-dwarf 2.2 star/yr 1 0.2 1 30 Gyr 26 Gyr 15 Gyr M-dwarf 10.5 star/yr 1 0.2 1 100 Gyr 96 Gyr 50 Gyr Table 1: Summary of Drake equation parameters discussed in this chapter. 2.2 Fraction of Stars with Planets Ground- and space-based observations all suggest that terrestrial planets are commonplace around all stars. Microlensing observations indicate that typical stellar systems contain one or more bounded planets (Cassan et al., 2012). Furthermore, analysis of Kepler data indicates that there are ~2 planets per cool star (Teff < 4000 K) with periods <150 days (Morton and Swift, 2014). This orbital period encompasses the habitable zone (HZ) of cool stars. Therefore, we can safely assume that fp = 1 for all spectral types F, G, K, and M (c.f. Frank and Sullivan, 2016). Lineweaver et al. (2004) developed a model for the evolution of the galaxy and found that rocky planets could not have formed until about t = 4 Gyr, due to the lower metallicity of stars at the time. If planet formation were lower during the early history of the galaxy, then this would suggest a phase transition for the emergence of planets in the galaxy. However, these results of Lineweaver et al. (2004) are inconsistent with observations of planets orbiting stars nearly as old as the galaxy itself. The Kepler-444 system is one of the oldest known to host exoplanets, with five planets smaller than Earth orbiting this 11.2 Gyr old K-dwarf. At least for the long-lived K- and M-dwarf stars, fp = 1 for t > 1 Gyr, which is close to the age of the galaxy. Even if F- and G-dwarf stars develop planets slightly later than this due to metallicity limitations, we remain skeptical that dif- ferences in metallicity over the history of the galaxy are responsible for a phase transition in the Drake equation. We therefore maintain the expression fp(s, t) = 1. 2.3 Number of Habitable Planets Per System A common approach to estimating ne is to focus on the related quantity “eta-Earth”, ηearth, which is defined as the fraction of stars with at least one terrestrial mass/size planet within the HZ. The value of ηearth can be estimated from Kepler data (Dressing and Charbonneau, 2013, 2015; Kop- parapu, 2013; Gaidos, 2013; Petigura et al., 2013; Morton and Swift, 2014; Foreman-Mackey et al., 2014), with current estimates varying between 22% (Petigura et al., 2013) to 2% (Foreman- Mackey et al., 2014) for G- and K-dwarfs, and 20% for M-dwarf stars (Dressing and Charbonneau, 2015). In reality, and ηearth are describing the same quantity: the average number of habitable planets (which we will assume means terrestrial planets located within the HZ) per system, is the same as the fraction of stars with at least one terrestrial planet in the HZ. This means that is in the range of 0.02 to 0.22 for G- and K-dwarfs, and 0.20 for M-dwarfs.