AAstronomy & Astrophysics manuscript no. ms c (cid:13)
ESO 2018November 13, 2018
Evolution of galaxy habitability
R. Gobat and S.E. Hong School of Physics, Korea Institute for Advanced Study, Hoegiro 85, Dongdaemun-gu, Seoul 02455, Republic of Korea
ABSTRACT
We combine a semi-analytic model of galaxy evolution with constraints on circumstellar habitable zones and the distribution ofterrestrial planets in order to probe the suitability of galaxies of di ff erent mass and type to host habitable planets, and how it evolveswith time. We find that the fraction of stars with terrestrial planets in their habitable zone (known as habitability) depends only weaklyon galaxy mass, with a maximum around 4 × M (cid:12) . We estimate that 0.7% of all stars in Milky Way-type galaxies to host a terrestrialplanet within their habitable zone, consistent with the value derived from Kepler observations. On the other hand, the habitability ofpassive galaxies is slightly but systematically higher, unless we assume an unrealistically high sensitivity of planets to supernovae. Wefind that the overall habitability of galaxies has not changed significantly in the last ∼ ∼ ∼ × planets similar to present-dayEarth have existed so far in our galaxy. Key words.
Astrobiology – Planet and satellites:terrestrial planets – Galaxies:evolution – Galaxies:star formation – Galax-ies:abundances
1. Introduction
The idea of a plurality of worlds in which the universe is filledwith vast number of planets similar to our own has long beenpart of philosophical discourse and has acquired even more sub-stance with the dawn of modern astronomy and the final shatter-ing of the sphere of fixed stars. While Epicurus, Bruno or Her-schel might have had little doubt that all these worlds were in-habited, our explorations of the sky have made us finally realizehow unforgiving the cosmos could be for life, and we are begin-ning to wonder what other environments in the universe, if any,might actually support it.This question was partly answered in the last century withthe concept of the circumstellar habitable zone (HZ; e.g., Huang1959; Kasting et al. 1993), defined as the orbital belt around astar where the surface temperature of a planet would allow theexistence of a biosphere. More recently, the idea of the HZ hasby analogy been extended to that of the the galactic habitablezone (GHZ; Gonzalez et al. 2001), i.e., the region within a galaxywhere planets can form around stars and sustain a biosphere for asignificant amount of time. Initially mostly speculative, the GHZhas become increasingly quantifiable thanks to the data accumu-lated by exoplanet surveys during the last decade (e.g., Wright &Gaudi 2013, and references therein), as well as an increasinglybetter understanding of galaxy-scale physics. Modeling of theGHZ then tends to focus either on our own Milky Way (MW; e.g.Lineweaver et al. 2004; Prantzos 2008; Gowanlock et al. 2011)or specific nearby galaxies (e.g., Spitoni et al. 2014; Forgan etal. 2015). The interest in our own surroundings is naturally high,and these studies benefit from the large amounts of data we havecollected on them, allowing for fine-grained estimates. General-izing then the GHZ to cosmic scales and epochs, we can discussthe capacity of the universe (or even universe s ; e.g., Adams et al.2015) to sustain life up to the cosmic horizon, only limited by ourunderstanding of galaxy evolution. This comes at the cost of sim-plification, however, either treating star-forming galaxies with(semi-)analytic recipes of galaxy evolution (Dayal et al. 2015; Zackrisson et al. 2016) or only considering the global star forma-tion history (SFH) of the universe (Lineweaver 2001; Behroozi& Peeples 2015). This approach typically makes the simple as-sumption that the formation rate of habitable planets at any pointin space and time is determined only by the local star formationrate (SFR) and metal content of the interstellar medium (ISM).Nevertheless, other factors can be expected to influence the num-ber of habitable planets a galaxy can host, such as its stellar pop-ulation, supernova rate (SNR), and structure. Moreover, early-type galaxies (ETGs), which are very common in the local uni-verse, follow di ff erent scaling relations than star-forming disksand thus might have di ff erent habitability conditions.Here we take a semi-analytical approach that considers bothstar-forming and passive galaxies in a consistent way, as well asthe e ff ects of stellar evolution. This paper is structured as fol-lows: In Section 2 we describe the model and its underlyingassumptions, in Section 3 we show the results under di ff erentinitial parameters, while in Section 4 we summarize our conclu-sions.
2. Method
In this section, we describe the di ff erent assumptions that un-derlie our estimates of galaxy habitability and its evolution withredshift. In particular, we rely on a treatment of galaxy evolutionthat includes the quenching of star formation. This is motivatedby the predominance of passive galaxies in the local Universe(e.g., Fukugita et al. 1998); while their structure and composi-tion can not be directly equated with those of low star formationactive systems, an analysis that considers both galaxy types re-quires that their evolution be approached in a consistent manner.Our objective here is not to recreate an accurate model of galaxy For reference, we assume a Λ CDM cosmology with H =
70 km s − Mpc − , Ω M = .
27, and Ω Λ = .
73. However, the choiceof cosmological parameters has little impact on our analysis or conclu-sions. Article number, page 1 of 10 a r X i v : . [ a s t r o - ph . GA ] J un & A proofs: manuscript no. ms evolution and planetary formation, but to generate simple esti-mates that nevertheless self-consistently account for complexi-ties (such as di ff erent stellar and galaxy types, that follow di ff er-ent evolutionary paths). We base our model of galaxy evolution on three main ingredi-ents: a mass-dependent “universal” SFH, a single type of stel-lar initial mass function (IMF), and observed galaxy mass func-tions. We first assume that all galaxies start on the SFR - stellarmass relation (SFR-M (cid:63) , or “main sequence” of star formation;e.g., Brinchmann et al. 2004; Daddi et al. 2007; Rodighiero etal. 2010), and describe the SFH of main sequence (MS) galaxiesusing the 2-SFM formalism (Béthermin et al. 2012; Sargent etal. 2014). The slope and evolution of the SFR-M (cid:63) relation implya SFH peaking at redshift z = −
2, with the SFR (here as afunction of redshift) given by Ψ ( z ) = sSFR MS , M (cid:63) ( z ) (cid:32) M (cid:63) ( z )10 M (cid:12) (cid:33) α MS − (1 + min( z , z MS )) γ MS , (1)where γ MS = z MS = .
5, sSFR MS , = − . yr − is the spe-cific SFR (sSFR) of a 10 M (cid:12) galaxy at z =
0, and α MS = (cid:63) relation (Abramson et al. 2014;Schreiber et al. 2015). The stellar mass M (cid:63) of the galaxy isthen given by the time-integrated SFR minus losses due to stellardeath. It is thus dependent on the choice of IMF (see Sect. 2.1.2).This SFH also requires a “seed” mass as initial condition. Herewe assume a single burst of star formation at z in =
10, withM (cid:63), in = − × M (cid:12) . While the upper limit would yieldan unrealistically high final galaxy mass at z = ∼ M (cid:12) ,such a large seed mass is necessary in this context to generatethe highly star-forming progenitors of massive z ∼ r SB = . × (1 + min( z , . (Sargent etal. 2012; Béthermin et al. 2012). However, we here take the ap-proximation that SBs do not contribute significantly to the M (cid:63) ofthe galaxy and only use the SB boost as a correction term to theSFR when estimating SNRs (Sect. 2.3.1). In this paper, all galax-ies referred to as star-forming (SF) are either on the MS or arepart of the SB fraction.As long as a galaxy stays on the MS, we assume that itsgas-phase metallicity Z g is given by the fundamental mass-metallicity relation (FMR; Mannucci et al. 2010). A stellar pop-ulation forming at time t will then have a metallicity equal to Z g ( t ), with the average stellar metallicity of the galaxy Z (cid:63) beinggiven by Z (cid:63) ( t ) = (cid:82) t Z g ( t (cid:48) ) Ψ ( t (cid:48) ) f (cid:63) ( t − t (cid:48) )d t (cid:48) (cid:82) t Ψ ( t (cid:48) ) f (cid:63) ( t − t (cid:48) )d t (cid:48) f (cid:63) ( t ) = (cid:90) m max m min m φ ( m , t ) H ( t MS ( m ) − t )d m , (2)where H is the Heaviside function, φ ( m , t ) the IMF, and t MS ( m )the stellar main sequence (sMS) life-time of a star of mass m (Sect. 2.1.2); m min and m max are, respectively, the lower and up-per limit on star masses. Here we choose m min = . (cid:12) (see Fig. 1. (A):
SFHs of a passive galaxy quenching at z = . (cid:63) = . × M (cid:12) .The gray dot shows the estimated SFR of the MW (Licquia & Newman2015). (B): calculated SNII (green), SNIa (yellow), and total SN rate(black) for the M (cid:63) = . × M (cid:12) SF galaxy. The estimated MWvalue is shown as a gray dot (Cappellaro et al. 1993).
Sec. 2.2) and m max =
100 M (cid:12) . The parameter f (cid:63) is then the re-maining stellar mass fraction at time t , after accounting for lossesdue to stellar death. In this model, passive galaxies follow the MS SFH until the startof quenching at t q , after which we follow a simple closed boxformalism, with initial gas masses and SFR e ffi ciencies based onthe 2-SFM parameterization (Sargent et al. 2014). The evolutionof the SFR Ψ , gas content M g , and metallicity Z after t q are thengiven by the following equations, which we solve numerically insteps of ∆ t =
10 Myr: M g ( t > t q ) = M g ( t q ) − (cid:90) tt q Ψ ( t (cid:48) )( d ) t (cid:48) Ψ ( t > t q ) = (cid:15) ( t − ∆ t ) M g ( t − ∆ t ) Z g ( t > t q ) = Z g ( t q ) − y ln (cid:32) M g ( t ) M g ( t q ) (cid:33) log (cid:15) ( t ) = (1 − β ) log Ψ ( t ) − α log M g ( t q ) = α + β log Ψ ( t q ) (3)where (cid:15) ( t ) and M g ( t q ) are, respectively, the SFR e ffi ciency(SFE) at time t and molecular gas mass at the start of quenching. Article number, page 2 of 10obat & Hong: The habitability of galaxies
As a result, the SFR will then decrease more or less rapidlydepending on galaxy mass. We then consider that a galaxyhas become passive when its SFR is 1.5 dex below that of asame-mass galaxy on the MS. An example of a quenched SFHis shown in Fig. 1. In our case, the values for the slope β andintercept α of the M g -SFR and SFE-SFR relations given inSargent et al. (2014) would preclude the existence of passivegalaxies with M (cid:63) below ∼ × M (cid:12) at z ≥
2, in contradictionwith observations (e.g., Ilbert et al. 2013). This is not entirelysurprising, since environmental e ff ects play a more significantrole in the quenching of low-mass galaxies than high-massones even at high redshift (Strazzullo et al. 2013; Gobat etal. 2013; Newman et al. 2014; Gobat et al. 2015) and can beexpected to have di ff erent timescales than the simple case weare considering here. Letting both parameters vary, we find thatvalues of α = . β = ∼
3, consistent with the valuereported at low redshift (Martig et al. 2013) and suggested byobservation at z ∼ . z < y ∼ .
9, regardless of the IMF (Sect. 2.1.2).While we assume that SF galaxies constitute a homogeneouspopulation that always follows MS evolution, passive galax-ies clearly form a composite population of objects that havequenched at di ff erent times and thus have di ff erent SFHs. To in-clude this fundamental aspect of the ETG population, we useobserved galaxy mass functions (GMFs) from Ilbert et al. (2013,at z = . − .
5) and Baldry et al. (2012, at z ∼ z is then given by the combination of the ETGs presentat z + ∆ z , and the systems that quenched between z + ∆ z and z ,with their relative weights given by the value of the GMF, x ( M (cid:63) , z j ) = (cid:80) z i ≤ z max i = j w i , j x ( M (cid:63) , z i ) (cid:80) i w i , j w i , j = Φ ( M (cid:63) , z i ) − Φ ( M (cid:63) , z i − ) Φ ( M (cid:63) , z j ) , (4)where Φ is the GMF and x a derived observable (such asluminosity, total M (cid:63) , or number of stars or planets as in Eq. 9)for a passive galaxy of mass M (cid:63) quenched at z i . Here wechoose z max =
2, making the approximation that all z ∼ ∆ z = . ff er much fromthat of a galaxy of equivalent combined mass, and our assumedSFH can produce massive ETGs at high redshift without the needfor mergers, with peak SFRs in their progenitors that are large(e.g., ∼
600 M (cid:12) yr − for (cid:38) × M (cid:12) ETG at z = −
20% in the case ofa merger, than that of a non-remnant galaxy. Furthermore, sincehigher mass galaxies are typically hosted in larger halos, withmore surrounding substructure (i.e., satellites), galaxy mergers can be expected to play a more significant role in the case oflarge central galaxies (e.g., Feldmann et al. 2010; Carollo et al.2013), and we can thus expect our estimates to match observ-ables less well at large ( > M (cid:12) ) masses. For ease of comparison with the rest of the literature, moststudies assume a single universal stellar IMF, typically eitherthe canonical Salpeter (1955) IMF or a “bottom-light” function(Kroupa 2001; Chabrier 2003, these two IMFs have slightlydi ff erent slopes and parameterization of the low-mass regime,but otherwise yield very similar M (cid:63) and SFRs). The choice ofIMF typically only determines the scaling of derived quantitiessuch as M (cid:63) or SFR, although it has a greater impact on theamount of metals produced and returned to the ISM. On theother hand, recent studies have shown evidence for a varyingIMF in massive elliptical galaxies (e.g., van Dokkum & Conroy2011, 2012; Cappellari et al. 2012, 2013; Spiniello et al. 2014),implying that the IMF of their SF progenitors are not universaleither. Furthermore, the IMFs mentioned above have all beenderived by measuring the distribution of stars in localized re-gions of the MW and might therefore not be directly applicableto extragalactic environments.Here we adopt the integrated galactic IMF (IGIMF; Kroupa& Weidner 2003), i.e., a convolution of the IMF of individual starclusters with a distribution of star cluster masses. The IMF of in-dividual clusters is assumed to correspond to the Kroupa (2001)function below m = . (cid:12) , with a slope at m > . (cid:12) andmaximum stellar mass m max proportional to the embedded clus-ter mass (Pflamm-Altenburg et al. 2007; Weidner et al. 2011).The cluster mass function is, in turn, described by a power lawwith index β = (cid:12) . While clearly more complex, theIGIMF can reproduce some observables better than a fixed IMF(see Kroupa et al. 2013, and references therein). In particular,since more massive galaxies experience higher peak SFRs thanlower mass ones, the variation of the IGIMF slope with SFR nat-urally translates into a variation with final M (cid:63) when integratedover the galaxy’s SFH. As an example, we show in Fig. 2 theratio of M (cid:63) to bolometric luminosity (M / L) of passive galax-ies compared to those derived from observations of ETGs in thenearby universe. While not a tight fit, the IGIMF reproduces rea-sonably well the observed o ff set in M / L compared to the Salpetercase for low-mass galaxies. It also yields an increase in M / L athigher M (cid:63) (although the latter is steeper than observed, possi-bly due to our neglect of galaxy mergers). This occurs becausethe IGIMF is by design very similar to the Kroupa IMF at lowSFRs, if slightly top-light due to the cuto ff M (cid:63) , but becomes top-heavy at SFR >
100 M (cid:12) yr − . Since Eq. 1, Eq. 3,and the FMRand GMFs, assume a Chabrier IMF, we evaluate these expres-sions using the appropriate IMF and convert all quantities to theIGIMF afterwards. For comparison, in Sec. 3, we also give re-sults with a Salpeter IMF. As noted in Sec. 2.1, here we evaluatethe IMF between 0.1 and 100 M (cid:12) . We assign a probability of hosting life-sustaining planets to thestars in each galaxy based on their age, metallicity and mass.What constitutes a habitable world, in the sense of a planetary
Article number, page 3 of 10 & A proofs: manuscript no. ms
Fig. 2. (A): stellar metallicity of ETGs, as a function of M (cid:63) , predicted by our model (orange to red) and observed at z ∼ z ∼ . z = (B): M (cid:63) to bolometric light ratio (M / L) for z = (cid:63) into velocity dispersions using Eq. 5 in Cappellari et al. (2013). body that can sustain life for some fraction of its lifetime, issomewhat conjectural as we are still limited by our incompleteunderstanding of the emergence and possible types of life. Inprinciple, any environment allowing for liquid water and anenergy source, whether internal or external, and stable overgeological times could sustain life. These could include notonly the usually considered icy moons of gas giants, but alsomore exotic environments such as free-floating planets (e.g.,Stevenson 1999; Laughlin & Adams 2000; Abbot & Switzer2011). Furthermore, biochemistries di ff erent from our owncould exist, in which case the range of conditions favorable tothe emergence of life might be much greater than what we cancurrently envision, and all the more unconstrained.For the purpose of this work, we limit ourselves to environ-ments similar to our own, i.e., terrestrial planets around stars onthe sMS, with a combination of atmospheric pressure and tem-perature allowing them to sustain liquid water on their surface.We adopt the definition of circumstellar HZ given by Kopparapuet al. (2013), in this case the region bracketed by the “moistgreenhouse” and “maximum greenhouse” limits(respectively,inner and outer). Kopparapu et al. (2013) define the HZ fore ff ective temperatures between 2600 and 7200 K, correspondingto a mass interval of ∼ (cid:12) . We accordingly adopt a lowermass limit of 0.1 M (cid:12) and make the ad hoc assumption that onlystars with ≤ . (cid:12) can host habitable planets. Furthermore,we only consider stars older than t min = ff ectivetemperature changes. We take the e ff ect of stellar aging intoaccount by calculating the inner and outer HZ radii at eachtime step, using the fitting formulae of Hurley et al. (2000) toestimate accurate luminosities and e ff ective temperatures.The probability of a star hosting a terrestrial planet withinits HZ is then given by the width of the HZ, the probability ofplanet formation, and the distribution of orbital periods. Despiterecent advances, our knowledge of the distribution of low-mass or long-period exoplanets is still limited. In this paper, we mainlyrely on the observed properties of short-period planets and useextrapolation when necessary. We assume that the distributionof the orbital periods P of terrestrial planets follows a simplepower law of the form d N ∝ P ( r , m ) β P d P ( r and m are, respec-tively, the radius of the orbit and mass of the host star) and set β P = − . f T ,depends principally on the host star’s metallicity rather than itsmass. We consider two cases for this metallicity dependence: f T ( Z ) = H ( Z − Z min ) (cid:40) f T , ( Z / Z (cid:12) ) α P (Case 1) f T , − f HJ ( Z / Z (cid:12) ) α P (Case 2) . (5)Case 1 is a straightforward power law, based on the historical as-sumption that planet formation requires an abundance of heavyelements, and represents the simplest form of metallicity depen-dence for f T . While Case 1 does not appear to fit very well thedistribution of currently known terrestrial planets, it still de-scribes accurately the observed correlation between the occur-rence of giant planets and the host star metallicity (e.g., Fischer& Valenti 2005; Gonzalez 2014; Gaidos & Mann 2014). Becausethe value of the exponent is still somewhat uncertain, we choosea mid-range α P = f HJ ( Z / Z (cid:12) ) α P , and therefore assume that f T is weakly anticorre-lated with metallicity (see, e.g., Prantzos 2008; Zackrisson et al.2016). We take f HJ = .
012 for solar-type stars (Wright et al.2012) and, in both cases, set f T , = . Z = Z (cid:12) followingPrantzos (2008) and in accordance with the occurrence rate of Article number, page 4 of 10obat & Hong: The habitability of galaxies planets with a radius of < R ⊕ and a period of P = −
500 daysderived by Petigura et al. (2013) from the
Kepler sample. In bothcases, we adopt a low-metallicity threshold of 0 . Z (cid:12) under whichthe probability of forming terrestrial planets is zero (e.g., Fischer& Valenti 2005; Buchhave et al. 2012). The fraction of stars of agiven mass, age, and metallicity with habitable planets can thenbe expressed as w h ( m , Z , t ) = f T ( Z ) H (1 . − m ) C (cid:90) r h,o ( m , Z , t ) r h,i ( m , Z , t ) P ( r , m ) β P d P d r d r , (6)where C is a normalization constant so that C (cid:82) N d P d r d r = The habitable stellar fraction given in Eq. 6 can be consideredan upper limit, as energetic phenomena such as supernovae (SN)and active galactic nuclei (AGN) can sterilize the surface of plan-ets if they happen close enough.
The majority of supernova events falling into either the type Iaor type II category (e.g., Hakobyan et al. 2014; Cappellaro et al.2015), we only consider these two classes (hereafter, SNIa andSNII). Both the SNII and SNIa rate depend on the IMF throughthe fraction of stars with, respectively, m ≥ (cid:12) and m < (cid:12) .The SNII rate is directly proportional to the instant SFR, the life-time of a m ≥ (cid:12) star being shorter than the typical timescalenormally considered for star formation in galaxies ( ∼
100 Myr).On the other hand, SNIa are significantly delayed by the longertime spent on the sMS by their progenitors and the additionaltime elapsed between the formation of the white dwarf (WD)and its detonation. The rates of the two SN types can thus beexpressed as:SNR II ( t ) = Ψ ( t ) (cid:90) m max (cid:12) m φ ( m , t )d m SNR Ia ( t ) = η WD (cid:90) (cid:12) m min m (cid:90) t τ Ia + t MS (8) Ψ ( τ ) φ ( m , τ )d t (cid:48) d m , (7)where η WD = .
01 is the WD conversion rate (Pritchet, Howell& Sullivan 2008) and τ = t (cid:48) − t MS ( m ) − τ Ia , with t MS ( m ) and τ Ia =
500 Myr being, respectively, the main sequence lifetimeof a progenitor of mass m and the average delay time betweenstellar death and the detonation of the WD (Raskin et al. 2009).The impact of SN on galaxy habitability depends principallyon two parameters: the distance r SN at which a SN can sterilizea planet and / or render it uninhabitable (hereafter the lethalradius) and the time required by a planet to recover from thee ff ects of the SN, t rec . There is circumstantial evidence that SNevents might be implicated in some of the mass extinctions ofspecies that have happened on Earth throughout its history (e.g.,Ellis & Schramm 1993; Detre et al. 1998; Melott & Thomas2009; Thomas et al. 2015), but the full range of SN e ff ects onlife-bearing worlds has, to our knowledge, not been extensivelyprobed. The emphasis is usually put on hard radiation (e.g.,gamma rays) owing to its disruptive e ff ect on atmospheric ozone, which leads to an increase in biologically hazardous UVirradiance. For example, Gowanlock et al. (2011) and Forganet al. (2015) use the ozone-depletion estimates of Gehrels et al.(2003) to set the lethal radius at 8 pc. However, the expectedrate of SN within 8 pc (see Gehrels et al. 2003) suggests that theEarth might have experienced several during the last few Gyrwhile still ultimately retaining its capability to support life. Suchan event would also only a ff ect the unprotected areas of a planet:photosynthesis-independent benthic and lithoautotrophic organ-isms would be scarcely a ff ected by hard radiation on the surfaceand the approximately tenfold decrease (in the case of Earth)in ocean water absorbance between UVB wavelengths and thefirst peak of chlorophyll absorbance ( ∼
300 nm and ∼
450 nm,respectively; see Shifrin 1988) suggests that even shallow ma-rine ecosystems would be somewhat more resistant than surfaceones. We can then surmise that, at 8 pc, t rec is of the order of therecovery time of biodiversity after mass extinctions as inferredfrom the fossil record, i.e., small with respect to the timescalesrelevant to galaxies. In the { r SN = , t rec ∼ } case, the e ff ectof SN on habitability averaged over a whole galaxy is thenlikely negligible for all but the most active star-forming galaxies.Instead, we focus here on thermal e ff ects, i.e., how far aSN can occur from a planet and still push it out of the HZ ofits host star. Since this distance depends on the position of aplanet within the HZ and on the width of the HZ, we define thelethal radius as the distance at which 50% of planets normallywithin the HZ of their host star find themselves out of it forthe duration of the event. We consider two di ff erent sourcesof heating: the initial bolometric radiation emitted by the SNover ∼ erg forSNII and 3 × erg for SNIa (Scalzo et al. 2014). In thesecond, we consider ∼
10 M (cid:12) of ejecta for SNII (e.g., Woosleyet al. 2002) and 1.4 M (cid:12) for SNIa, adiabatically expandingwith an initial kinetic energy of 10 erg. We assume that theblastwave propagates in a surrounding ISM of pure hydrogenwith an average density of 1 atom / cm (see, e.g., Draine 2011).We further posit that the ejected shell can only interact witha planet as long as its pressure is larger than that of the windof the host star, for which we assume a constant density of 10atoms / cm and a speed of 500 km / s, and that the entire kineticenergy of the gas will be converted into heating the planet’satmosphere. Because the bolometric radiation is mostly releasedin the days following the detonation, but the ejected materialexpands much more slowly, these two e ff ects are consideredindependently. For our choice of d N / d P , we find a lethal radiusof r Ia = . r II = . t rec toa value greater than the Hubble time t H . On the other hand,feedback processes in the planet’s atmosphere might, dependingon its composition, compensate for the increased energy input.However, ascertaining the actual impact of the SN would thenrequire complex modeling that is beyond the scope of this paper.We then estimate a fractional irradiated volume to use as acorrection term to the time-varying habitable fraction. We ap-proximate passive galaxies as spheres and SF galaxies as disks,both with constant stellar density and redshift-dependent radiusset by the observed mass-size relation (van der Wel et al. 2014).We assume that the disks have a thickness of h d = . × r e athigh redshift, where r e is the e ff ective radius of the galaxy, and Article number, page 5 of 10 & A proofs: manuscript no. ms transition to a thin disk with h d = . × r e (based on the struc-ture of the MW disk; Bovy et al. 2012) after the peak of theirSFH (Lehnert et al. 2014). The fractional irradiated volumes forpassive (P) and SF galaxies are then V irr ( t ) = H ( t rec − t ) SNR Ia ( t ) r r ( t ) (P) h d ( t ) r ( t ) (SNR II ( t ) r + SNR Ia ( t ) r ) (SF) . (8)Since we do not include the growth of passive bulges in diskgalaxies, we simply assume that the volume of ETGs is givenby the disk case during their SF phase and the passive one afterquenching, with a simple linear transition between both duringquenching. We assume internal homogeneity for simplicity.This is however a clear limitation, especially in the case ofSF galaxies, as we can expect SNII and SNIa to preferentiallyhappen in comparatively dense star clusters (e.g. Shara & Hurley2002; Maoz et al. 2010), which would increase SN lethality.In more extreme environments, such as massive star clustersor very compact galaxies, nearby main sequence stars mightcontribute enough to the ambient radiation field to adverselya ff ect the HZ (Thompson 2013; Adams et al. 2015). We estimate a lethal radius r AGN resulting from the activity ofa galaxy’s central supermassive black hole (SMBH) in a sim-ilar manner. Although this activity is expected to be sporadic,we assume it to be frequent enough to define an exclusion re-gion where planets cannot stay habitable over the long term.In this case, only thermal radiative e ff ects are considered. Weassume that the SMBH radiates at Eddington luminosity anduse the local relation of Reines & Volonteri (2015) for ellipti-cals and bulges to derive the SMBH mass from a galaxy’s M (cid:63) .The distance from the SMBH at which 50% of planets fall outof the HZ of their host star depends on the AGN luminosity,thus M (cid:63) through the relation log r AGN (kpc) = − . + . × log M (cid:63) (M (cid:12) ). We then add the spherical volume defined by thisradius to the value of V irr given in Eq. 8. Combining all the elements described in the previous sections,we define the habitability h G of a galaxy at redshift z as the ratioof the number of main sequence stars of mass ≤ . (cid:12) , age ≤ t min = z . In a slightly shortened notation h G = (cid:82) t z − t min (1 − V irr ( t )) Ψ ( t ) (cid:82) . m min m φ ( m , t ) H ( t MS ( m ) − t ) w h d m d t (cid:82) t z Ψ ( t ) (cid:82) m max m min m φ ( m , t ) H ( t MS ( m ) − t )d m d t (9)for SF galaxies. The habitability h G is here a function of the time t z after the onset of star formation and, through the SFH Ψ , theM (cid:63) of the galaxy at z . In the case of passive galaxies, both sidesof the fraction are a combination of di ff erent age populations asdescribed in Eq. 4.
3. Results
Figure 3 shows the galaxy habitability h G , computed in the massrange 10 < M < × M (cid:12) , as a function of M (cid:63) at z = z =
0, between 0.65% and 0.8% of all stars host potentiallyhabitable planets and h G is only weakly dependent on M (cid:63) , witha maximum around 4 × M (cid:12) (consistent with Zackrissonet al. 2016). The habitability of ETGs is ∼ h G are, in order of importance: • metallicity : lower mass galaxies also have lower metallicityand therefore a higher fraction of stars below the threshold forterrestrial planets, while for higher mass galaxies the decreasein h G stems from f T being inversely proportional to metallicity.In contrast, the habitability for Case 1 increases monotonicallywith M (cid:63) and is comparable in behavior to the model of Dayalet al. (2015). The M (cid:63) − Z (cid:63) relation is well defined for both SFand passive galaxies, with an intrinsic scatter whose value isnot precisely known, although it is believed to be small. Usinga value of 0.1 dex close to the observed scatter would add adispersion of 0.03 dex and 0.008 dex at the low- and high-massend of Fig. 3, respectively, for Case 2, and a constant dispersionof 0.2 dex for Case 1. • IMF : while metallicity drives the mass dependence of h G ,the IMF determines its average value because the number ofhabitable planets in a stellar population depends on its fractionof low-mass stars. For example, using a Salpeter IMF decreasesthe habitability by ∼ (cid:46) (cid:12) stars compared to the IGIMF, while the variation of h G with M (cid:63) is similar in both cases. The negative slope of theIMF in the range 0 . −
100 M (cid:12) implies that most terrestrialplanets will be hosted by subsolar mass stars, which havetighter HZs than solar mass stars. As a consequence, h G israther sensitive to the slope of the distribution of orbital periodsand an error of 0.1 in β P (Cumming et al. 2008) would o ff set h G by 0.2 dex, making this parameter as important as f T , indetermining its normalization. • supernovae : since our model assumes a small lethal radius,the e ff ect of SN is very weak. It is slightly stronger at highermasses and for ETGs, which have SFHs that peak higher thandisks of identical mass and at earlier times, when galaxies aredenser. On the other hand, while we have ruled out large lethalradii in our model, the 8 pc value of Gehrels et al. (2003) mightstill be relevant if we were to consider a more restrictive criterionfor planetary habitability (see Sec. 3.1). As shown in Fig. 3, theimpact of SN would in this case lead to significant di ff erences inthe h G of passive and SF galaxies at high masses. Active galacticnuclei behave in a similar fashion, fractionally increasing theirradiated volume in high-mass galaxies. The larger irradiatedvolume and fraction of stars below the metallicity thresholdmake these two e ff ects stronger at higher redshift as well.Other sources of uncertainty may include the GMF andthe various parameters describing the distribution of terrestrialplanets in Sec. 2.2. They are, however, inconsequential com-pared to the ones discussed above: the uncertainty generatedon the h G of passive galaxies and its evolution by the errors on Article number, page 6 of 10obat & Hong: The habitability of galaxies
Fig. 3. (A):
Galaxy habitability at z = f T . To illustrate the e ff ect of SN on the mass-dependency of h G , the dottedlines show the IGIMF case with the lethal radius of Gehrels et al. (2003) r II = r Ia = t rec > t H . (B): Galaxy habitability at z = ff erent cases of terrestrial planet incidence (Case 1 and 2; dashed and solidlines, respectively). The filled gray circle shows the ratio of stars with terrestrial planet candidates in the habitable zone to the total number ofstars with at least one planet candidate, taken from the NASA Exoplanet Archive using the same criteria (star mass, illuminance, planetary radius)as described in Sect. 2.2. The error bar assumes Poisson uncertainties. (C): Galaxy habitability integrated over the range of galaxy masses, as afunction of redshift and IMF (IGIMF and Salpeter; solid and dashed lines, respectively), for SF (blue) and passive (red) galaxies. As in (A), thispanel assumes a Case 2 f T . The dotted lines show the evolution of habitability in the IGIMF case if we use the larger (8 pc) SN lethal radius ofGehrels et al. (2003). (D): As in (C), evolution of galaxy habitability with redshift as a function of the metallicity dependence of f T . In both panels,the black lines show the evolution of habitability averaged over the whole galaxy population (SF and passives). the GMF parameters is negligible, while the errors on f HJ and α P (0.038 and ∼
1; Wright et al. 2012; Fischer & Valenti 2005;Gonzalez 2014) induce an uncertainty of at most 0.005 dex and0.003 dex, respectively, across the mass range we consider. Theuncertainties associated with planetary distribution functionsarise mostly from instrumental limitations and small samplesizes. We can thus expect them to be mitigated in the nearfuture as complementary surveys will extend our coverage ofparameter space and increase existing samples by orders ofmagnitude (e.g.,
Gaia for long-period Jovians and PLATO forintermediate-period terrestrials; Perryman et al. 2014; Rauer etal. 2014).We compare our estimates with observations from the
Kepler mission using the NASA Exoplanet Archive . If the Kepler sample is mostly unbiased and terrestrial planets havethe same distribution of orbital inclinations as giant ones, theratio of stars with planet candidates within the HZ to the totalnumber of stars with planet candidates should represent an http: // exoplanetarchive.ipac.caltech.edu estimate of h G (if all stars host at least one planet) or an upperlimit to it. We first select all stars observed by Kepler with atleast one confirmed or candidate planet and T e ff ≤ < . (cid:12) to the total and assuming a Kroupa (2001)IMF for simplicity. We repeat the procedure adding conditionsfor the planet’s radius ( < R ⊕ ) and insolation (between 0.22 and1.13 times that of Earth) consistent with Sec. 2.2, and find 44candidates. This corresponds to a ratio of ∼ . × M (cid:12) for the MW disk; Licquia & Newman2015). This is not unexpected, since our model uses parametersderived from the Kepler sample. It would however tend tovalidate the use of Case 2 and our assumptions for galaxyevolution, as well as our chosen combination of f T , and β P . Onthe other hand, the Case 1 prediction lies significantly below theobserved ratio, and can be reconciled only if f T , =
1, i.e., if allstars of solar metallicity or greater host a terrestrial planet, in
Article number, page 7 of 10 & A proofs: manuscript no. ms conflict with observations.In panels C and D of Fig. 3 we show the redshift depen-dence of h G , integrated over the mass range 10 − × M (cid:12) .The evolution of the integrated galactic habitability is monotonicwith time for both SF and passive galaxies, and in both cases canbe broadly divided into an early linear rise followed by a plateauin which h G increases by no more than 0 . z > . z < . h G ) galaxies to high-mass, high-metallicity (andhigh- h G ) ones. Therefore, the apparent decrease in habitabilityat z < . ff erent GMFs for z ∼ z > . ∼ z ∼ z ∼
1, i.e., in the last 7-8 Gyr. In the localuniverse, most of the stellar mass is held in large passive galax-ies that formed at z >
1. This suggests that most of the habitableplanets in the present epoch should belong to stars older than ourown. Indeed, as shown in Fig. 4, the median age of stars hostinghabitable planets at z = ∼ ∼ ∼ ∼
70% of stars in disks that host habitable planetsat z =
0, in agreement with the earlier estimate of Lineweaver(2001). Unlike the work presented here, the studies referencedabove did not consider stellar death. However, the shape of theIMF implies that most planets orbit long-lived subsolar massstars, greatly mitigating its e ff ect. On the other hand, the im-pact of stellar death would be more important when consideringtimescales longer than the Hubble time or non-standard IMFs.Finally, we note that we might be overestimating the number ofpresently habitable planets formed near the peak of SF in pas-sive galaxies ( z ∼
2, or ∼
10 Gyr ago) as a result of the simplisticassumptions on galaxy structure that our model uses, as stated inSec. 2.3.1.
In the previous sections we used a rather broad definition of plan-etary habitability, which likely includes environments that are atbest marginal to life (e.g., planets that spend only a short amountof time in the HZ). However, we as a culture seem to be most fas-cinated by the possibility of existence of other worlds and intel-ligences similar to our own. The latter in particular has become arecurrent part of both our folklore and scientific discourse (e.g.,Dyson 1960; Hewish et al. 1968; Bowyer et al. 1982; Wright etal. 2014). Complex surface ecosystems such as the ones presenton Earth today, let alone intelligent species, require time to ariseby random evolution and are more sensitive to outside catastro-phes (e.g., SN) than life in general. If we simply assume thatthe emergence of complex and / or intelligent life on a habitableplanet is not contingent upon the type and metallicity of its host Fig. 4.
Distribution of the ages of habitable planets per unit volumeat z =
0, for SF (blue) and passive (red) galaxies and with the totaldistribution shown as a black curve. The dotted lines show the respectivemedian ages, while the dashed gray line marks the formation time of thesolar system. star, the frequency of Earth-analog “garden worlds” and that ofintelligent life (hereafter civilizations) should then be directlyproportional to the number of habitable planets, with two mod-ifications: First, we increase t min , with the added condition thatthe planets stay within the (shifting) HZ for a period at leastequal to t min (e.g., the rise of our own species corresponds to t min = . t rec =
50 Myr to account for SN-induced massextinctions.On the other hand, the probabilities of the emergence of life,of Earth-like biospheres, and of intelligent species are so far un-constrained. Consequently, we can only estimate the frequencyof civilizations with respect to some arbitrary reference point.Here we choose to normalize the number of planets estimatedwith the modified criterion to 1 at z = M (cid:12) galaxy has ∼ − . × ( t min / Gyr ) civilizations exist in the observableuniverse when accounting for look-back time. The occurrencerate of civilizations, as shown in Fig. 5, peaks 3 + . × t min Gyrafter the onset of star formation. This suggests that, if Earth isnot unique in this galaxy and the timescale of our own evolutionis typical, we exist ∼ ∼ t min = . . × planets following the above cri-terion have formed in a MW-mass disk since the onset of starformation. For us to exist before most civilizations our galaxywill produce would then imply either that the incidence of civ-ilizations per suitable planet is < × − or that, contrary towhat the timescale of our evolution suggests, the typical delaytime is >
4. Conclusion
We have used an analytic model of galaxy evolution to estimategalaxy habitability, measured by the fraction of stars withhabitable planets, as a function of galaxy type, M (cid:63) , and redshift.Our model includes passive galaxies through a simple treatment
Article number, page 8 of 10obat & Hong: The habitability of galaxies
Fig. 5.
Occurrence rate of civilizations in a MW-type disk, normalizedto 1 at z =
0, as a function of look-back time and for delay times of 2,4.5, and 6 Gyr (dashed, solid, and dotted lines, respectively). The solidline corresponds to our own case. of galaxy quenching and metal enrichment, as well as thermale ff ects of stellar evolution and supernovae on habitable zoneplanets. We summarize our findings as follows: • We consider two di ff erent types of metallicity dependencefor the frequency of terrestrial planets and find that, giventhe assumptions of the model, a weak negative metallicitydependence (Case 2) reproduces observations better. In thiscase, between 0.65% and 0.8% of stars in > M (cid:12) galaxiesare expected to host planets in their HZ, close to the < Kepler observations. The habitability of galaxiesat z = M (cid:63) = . • We estimate that the radius where the thermal e ff ects ofSN on planets become significant is 0.3-0.5 pc. The impactof SN on galaxy habitability is therefore negligible, as thefractional irradiated volume is almost always very small exceptat high M (cid:63) and redshift, where SFRs are extreme and galaxiesmore compact. The e ff ect of the central AGN is likewise limited. • The habitability of passive galaxies is slightly but sys-tematically higher than that of star-forming galaxies and hasremained mostly unchanged since z ∼ .
5. On the other hand,the habitability of SF galaxies has increased monotonically since z = • The median age of habitable planets is ∼ ∼ ∼
10 Gyr in passive galaxies. Using a more restrictive criterion,the occurrence rate of habitable planets similar to present-dayEarth (which we can assume is proportional to that of alien civi-lizations) peaked ∼ ∼ × − . Acknowledgements.
We thank E. Daddi for enlightening discussions, F. Adams,C. Park, O. Snaith, and H.S. Hwang for their helpful suggestions which helpedimprove this paper. This research has made use of the NASA Exoplanet Archive,which is operated by the California Institute of Technology, under contract withthe National Aeronautics and Space Administration under the Exoplanet Explo-ration Program.
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