Fitting Formulae and Constraints for the Existence of S-type and P-type Habitable Zones
aa r X i v : . [ a s t r o - ph . E P ] O c t Version: November 6, 2018
Fitting Formulae and Constraints for the Existence ofS-type and P-type Habitable Zonesin Binary Systems
Zhaopeng Wang and Manfred Cuntz
Department of Physics, University of Texas at Arlington,Arlington, TX 76019, USA [email protected]; [email protected]
ABSTRACT
We derive fitting formulae for the quick determination of the existence ofS-type and P-type habitable zones in binary systems. Based on previous work,we consider the limits of the climatological habitable zone in binary systems(which sensitively depend on the system parameters) based on a joint constraintencompassing planetary orbital stability and a habitable region for a possiblesystem planet. Additionally, we employ updated results on planetary climatemodels obtained by Kopparapu and collaborators. Our results are applied tofour P-type systems (Kepler-34, Kepler-35, Kepler-413, and Kepler-1647) andtwo S-type systems (TrES-2 and KOI-1257). Our method allows to gauge theexistence of climatological habitable zones for these systems in a straightforwardmanner with detailed consideration of the observational uncertainties. Furtherapplications may include studies of other existing systems as well as systems tobe identified through future observational campaigns.
Subject headings: astrobiology — binaries: general — celestial mechanics —methods: statistical — planetary systems — stars: individual (Kepler-34, Kepler-35, Kepler-413, Kepler-1647, KOI-1257, TrES-2)
1. Introduction
Several decades of detailed observations revealed that stellar binary systems consti-tute a notable component of our Galactic neighborhood (e.g., Duquennoy & Mayor 1991; 2 –Patience et al. 2002; Eggenberger et al. 2004; Raghavan et al. 2006, 2010; Roell et al. 2012).An important aspect of this type of research is the plethora of discoveries of planets in manyof those systems. Generally there are two types of possible planetary orbits (e.g., Dvorak1982): planets orbiting one of the binary components are said to be in S-type orbits, whileplanets orbiting both binary components are said to be in P-type orbits. In fact, since 1989,83 planet-hosting binary systems, encompassing 63 planets in S-type obits and 20 planetsin P-type orbits, have been detected, mostly based on the radial velocity method and tran-sit method. A survey about exoplanetary systems of binary stars with stellar separationsless than 100 au was given by Bazs´o et al. (2017); it also considers the effects of secularresonances on the systems’ habitability.In 1989, HD 114762b in the constellation Coma Berenices has tentatively been identifiedas an exoplanet, thus being the first possible planet around a main-sequence star other thanthe Sun and, incidently, the first possible planet located in a binary system. In 2012, thisplanet was finally confirmed based on the radial velocity method. A more recent example isHD 87646b, a planet in a close binary system with a 22 au separation distance (Ma et al.2016). This system contains two substellar objects in S-type orbits, which makes it thefirst close binary system known to host more than one substellar companion. Other exam-ples of planets in binary systems include Kepler-413b (Kostov et al. 2014) and Kepler-453b(Welsh et al. 2015). Both Kepler-413b and Kepler-453b are in P-type orbits, also calledcircumbinary orbits. However, S-type orbits are much more frequent, and in some systemsthe planets are in orbit around quasi-single stars. For example, Kepler-432b, a hot Jupiter-type planet orbits a giant star that is part of a super-wide binary system with a separationdistance of 750 au (Ortiz et al. 2015). Some of the S-type and P-type orbits are locatedwithin the stellar habitable zones (HZs). These systems often receive special attention asthey inspire detailed studies about the planet’s long-term orbital stability and its potentialfor hosting exolife.In previous studies, focusing on habitable zones in stellar binary systems, presented by(Cuntz 2014, 2015) denoted as Paper I and II, respectively, henceforth, a joint constraintof radiative habitable zones (RHZs, based on stellar radiation) and orbital stability wasconsidered. Previous results were given by Eggl et al. (2012, 2013), Kane & Hinkel (2013),Kaltenegger & Haghighipour (2013), Haghighipour & Kaltenegger (2013), among others . We wish to draw the reader’s attention to the online calculator
BinHab (Cuntz & Bruntz 2014), hostedat The University of Texas at Arlington (UTA), which allows the calculation of habitable regions in binarysystems based on the developed method. Another online calculator with similar capacities has been givenby M¨uller & Haghighipour (2014). Zuluaga et al. (2016) pursued a comparison study between these toolsand found that their results are consistent with each other. O and CO absorption coefficients. Previous resultsabout limits of stellar habitable zones have been given by, e.g., Kasting et al. (1993) andUnderwood et al. (2003). The latter explores how HZs are impacted by stellar evolution.Our paper is structured as follows. In Section 2, we describe the theoretical approach,including general background information. The fitting procedure is outlined in Section 3.Section 4 offers applications to observed systems, encompassing systems with S-type andP-type planets. Our summary and conclusions are given in Section 4.
2. Methodology2.1. Theoretical Background
Based on the radiative energy fluxes received by system planets from the two binarycomponents, the habitable limits could, in principle, be defined similarly to those withinthe Solar System, amounting to the concept of the RHZs; see Section 1. Following previouswork the RHZs can be calculated based on L S rel , l d + L S rel , l d = L ⊙ s l (1)with d and d denoting the distances from to the binary components (see Fig. 1), L and L indicating the stellar luminosities, and s l standing for one of the solar habitability limits (seeTable 1). S rel ,il with i = 1 , d and d can be represented by a functionof z , the distance from the center of binary system, a quartic equation for z , can be obtainedafter algebraic transformations.Hence, the RHZ, an annulus around each star (S-type) or both stars (P-type), is thus This subsection is merely intended as supplementary information; it summarizes materials previouslygiven in Paper I and II. R ( z, ϕ )) | s l, out − Max( R ( z, ϕ )) | s l, in (2)Here R ( z, ϕ ) describe the borders of the RHZs, with z and ϕ denoting the polar coordinates.Additionally, s l, in and s l, out describe the parameters tagging respectively the inner and outerlimits of the stellar RHZ; see Table 1.If a planet is assumed to stay in the HZ for timespan of astrobiological significance, astable orbit is required. Using the fitting equations developed by Holman & Wiegert (1999),the planetary orbital stability limits are obtained. They convey an upper limit as thedistance from the stellar primary for S-type orbits, and a lower limit measured from themass center of the binary system for P-type orbits. Additionally, following the terminologyof Paper I, ST-type and PT-type HZs denote the cases when the widths of the HZs areimpacted by the orbital stability limits and, therefore, the corresponding RHZs are truncated.Consequently, the width of the P/PT-type HZ (if existing) is given byWidth ( P/P T ) = RHZ out − Max (cid:0)
RHZ in , a cr (cid:1) , (3)and the the width of the S/ST-type HZ (if existing) is given byWidth ( S/ST ) = Min (cid:0)
RHZ out , a cr (cid:1) − RHZ in . (4)Here RHZ in and RHZ out denote the inner and outer limits of the RHZs, respectively,and a cr denotes the orbital stability limit. Equations (3) and (4) are relevant for devisingthe fitting formulae for the existence of P-type and S-type HZs, the main focus of this study. Various sets of binary systems, encompassing both systems of equal and non-equalmasses, have been studied to examine the existence of their HZs based on the radiativecriterion, as described by the stellar luminosities, as well as the orbital stability criterion forsystem planets. Information on the adopted stellar parameters, chosen for cases of theoretical The formulae of orbital stability by Holman & Wiegert (1999) are based on 10 binary periods, a timescale significantly shorter than required for the installment of astrobiology. However, more recent studiesby Pilat-Lohinger & Dvorak (2002) for S-type systems based on the Fast Lyapunov Integrator indicate thatthe orbital stability limits of Holman & Wiegert (1999) are also valid for notably longer time scales such as10 binary periods especially for systems with planets in nearly circular orbits. Nevertheless, improvementsof the Holman & Wiegert formulae for general systems for long time scales, ideally encompassing billions ofyears, should be considered a topic of high priority due to their significance for future astrobiological studies. M and M of 0.50 M ⊙ ,0.75 M ⊙ , 1.00 M ⊙ , and 1.25 M ⊙ ; see Figures 2 to 5 for details.For systems of masses M = M = 1 . M ⊙ , in case of e b = 0, the semi-major axis a bin is required to be smaller than 0.97 au for the P/PT-type GHZ to exist and smaller than1.03 au for the P/PT-type RVEM to exist. Regarding S/ST-type HZs, a bin needs to be largerthan 3.72 au and larger than 2.93 au for the GHZ and RVEM to exist, respectively. Largereccentricities barely affect the existence of P/PT-type HZs; however, they notably affect theexistence of the S/ST-type HZs. For e b = 0.50, a bin is required to be larger than 8.44 au andlarger than 6.66 au for S/ST-type GHZ and RVEM to exist, respectively.Different values for the existence of P/PT and S/ST-type HZs are obtained for otherkinds of equal-mass systems. For systems with masses M = M = 0 . M ⊙ , in case of e b =0, a bin is required to be smaller than 0.22 au for the P/PT-type GHZ to exist and smallerthan 0.23 au for the P/PT-type RVEM to exist. Regarding S/ST-type HZs, a bin needs to belarger than 0.76 au and larger than 0.60 au for the GHZ and RVEM to exist, respectively.Again, larger eccentricities barely affect the existence of P/PT-type HZs; however, theyimpact the existence of the S/ST-type HZs, as expected. For e b = 0.50, a bin is requiredto be larger than 1.74 au and larger than 1.37 au for S/ST-type GHZ and RVEM to exist,respectively.We also investigated non-equal mass systems, which generally are considered more sig-nificant than equal-mass systems. In systems of M = 1 . M ⊙ and M = 0 . M ⊙ , a bin isrequired to be smaller than 0.81 au and smaller than 0.86 au for the P/PT-type GHZ andRVEM, respectively, in case of e b = 0. Furthermore, a bin needs to be larger than 2.83 au andlarger than 2.23 au to allow for the existence of the S/ST-type GHZ and RVEM, respectively.Again, high eccentricities barely affect the existence of P/PT-type HZs; however, they im-pact the existence of the S/ST-type HZs as already discussed for equal-mass systems. For e b = 0.50, a bin is required to be larger than 6.78 au and larger than 5.35 au for S/ST-typeGHZ and RVEM to exist, respectively.Moreover, we also considered systems of M = 1 . M ⊙ and M = 0 . M ⊙ . The case of e b = 0 requires a bin to be smaller than 1.20 au and smaller than 1.27 au for the P/PT-typeGHZ and RVEM, respectively. Regarding S/ST-type HZs, a bin is required to be larger than4.36 au for the GHZ and larger than 3.41 au for the RVEM regarding e b = 0. Furthermore, a bin is required to be larger than 10.17 au for the GHZ and larger than 8.03 au for the RVEMand e b = 0.50. Even higher values of a bin are needed for both the GHZ and RVEM in caseof eccentricities beyond 0.50. 6 –In summary, the existence of P/PT-type HZs is barely affected by the eccentricityof the stellar system and solely controlled by M and M (or, say, L and L ). However,relatively large semi-major axes a bin are required for the existence of S/ST-type HZs in highlyeccentric systems. Large values of a bin always ensure S/ST-type HZs, as in this case, thestellar habitable environments are in essence those of single stars. In non-equal mass systemswith M + M considered as fixed, the P/PT-type HZs are barely impacted compared toequal-mass systems, but higher values for a bin are mandated for the existence of S/ST-typeHZs especially for systems with high eccentricities for the binary components.
3. Fitting Procedure
The main aspect of our work concerns the derivation of fitting formulae for the existenceof P/PT-type HZs and S/ST-type HZs for binary systems consisting of main-sequence stars.Through applying the least-squared method, fitting is done in two steps: first, fitting a bin versus e b by assuming fixed masses as reference (aimed at catching the sets of parameterswhere the HZs cease to exist) and, second, fitting the coefficients with stellar masses to allowthe expansion of the formulae for general binary systems. In the first step, the Bayesianinformation criterion (BIC) and mean absolute percentage error are taken into account. TheBIC is used in the second step as well for the mass fitting determination.For P/PT-type cases, the a bin versus e b fitting is done using a polynomial equation,which is a bin = α + α e b + α e b . (5)For S/ST-type cases, the a bin versus e b fitting is done using a cubic equation, placed asexponent, which reads a bin = e β + β e b + β e b + β e b . (6)The coefficients for selected systems are shown in Table 3. For e b = 0, systems withmasses of M = M = 1 . M ⊙ have 0.960 au for the fitting results and 0.97 au regardingthe data for the P/PT-type GHZ, and 1.016 au in the fitting results and 1.03 au regardingthe data for the P/PT-type RVEM to exist. For S/ST-type HZs, the fit yields 3.597 auand 2.773 au for the GHZ and RVEM, respectively, with data noted as 3.72 au and 2.93 au,respectively.Keep e b to be zero, systems with M = M = 0 . M ⊙ , render 0.215 au, 0.230 au,0.720 au, and 0.568 au in the fitting of P-GHZ, P-RVEM, S-GHZ, and S-RVEM, respectively.Conversely, the data based on the method as given in Sect. 2.1 are given as 0.22 au, 0.23 au,0.76 au, and 0.60 au, respectively. In systems with M = 1 . M ⊙ , M = 0 . M ⊙ , the 7 –fitting results read 0.805 au, 0.850 au, 2.702 au, and 2.109 au, respectively with e b as zerofor P-GHZ, P-RVEM, S-GHZ, and S-RVEM, respectively. Furthermore, 0.81 au, 0.86 au,2.83 au, and 2.23 au are the values for corresponding data. For the case of M = 1 . M ⊙ and M = 0 . M ⊙ with e b as zero, the results for the fitting of P-GHZ, P-RVEM, S-GHZ,and S-RVEM are given as 1.185 au, 1.253 au, 4.229 au, and 3.287 au, respectively. Herethe data are 1.20 au, 1.27 au, 4.36 au, and 3.41 au, respectively, again showing very closeagreement. The various fitting coefficients are listed in Table 4.To enhance the universal applicability of the fitting formulae, we also explored therelation between the coefficents in a bin versus e b fitting, and the stellar masses of the binarysystems. For the coefficents in the P-type equation, α is represented by α i = A i + A i M + A i M (7)Furthermore, for the coefficents in the S-type equation, β is represented by β i = B i + B i M + B i M + B i M + B i M (8)The BIC for the cases of mass fitting are listed in Table 5. By adding terms to theequation only containing constant and linear terms of M and M , the BIC varies andindicates that P-type cases favor adding nothing, whereas S-type cases prefer adding M and M , as done as part of the process. The general fitting coefficients for P-type andS-type HZs are listed in Table 6 and 7, respectively.Applying the calculated coefficients from stellar masses to the a bin versus e b equations,fitting results are plotted as well as the data for comparison (see Fig. 3). Most of the fitsare virtually indistinguishable from the data. Percent errors of the fits for selected cases areprovided in Tables 8 to 10. Cases not shown here reveal similar results. In Table 10, thecoefficients of determination measuring the goodness of the fit, are given for reference. Thepercentage errors are calculated asPercentage Error = (cid:12)(cid:12)(cid:12)(cid:12) data − fittingdata (cid:12)(cid:12)(cid:12)(cid:12) (9)In summary, through employing a two-step fitting procedure, fully acceptable resultsare obtained for the fitting equations in response to the existence of GHZ and RVEM HZsdepending on the system parameters (i.e., a bin , e b , M , and M ).Using the fitting formulae given above (see Eqs. 5 to 8), several observed binary systems 8 –have been studied in more detail to inquire on the existence of the stellar HZs, encompassingboth GHZs and RVEM HZs (see Sect. 4). Generally, the minimum a bin for S-type HZs toexist would increase as either stellar mass increases. As for the P-type cases, the maximum a bin is decreased with either stellar mass decreased. Thus, the maximum masses of binarycomponents were considered for S-type HZs based on their errors for their existence, andminimum masses have been taken into account for the non-existence of HZs. In contrast,P-type HZs consider stellar minimum masses (as defined by the respective observationaluncertainties) for their existence, and maximum masses for the non-existence of the HZs.As for the eccentricity, the largest eccentricity (as set by observational constraints) shouldbe considered for the study of both S-type and P-type HZs, as it corresponds to the mostadverse outcome.
4. Applications to Observed Systems4.1. P-type Systems
Welsh et al. (2012) have reported transiting circumbinary planets both regarding Kepler-34 and Kepler-35; their study also conveys detailed information about the system’s data (seeTable 11). Kepler-34 has two Sun-like stars revolving around each in 27 . +0 . − . d,with stellar masses to be 1 . +0 . − . M ⊙ and 1.0208 ± M ⊙ , respectively. The systempossesses a 0 . +0 . − . M J circumbinary gas giant, i.e., somewhat less massive than Saturn,with a 1 . ± . . +0 . − . eccentricity. By measuring theeffective temperature and metallicity of both stars, an age between 5 and 6 Gyr has been de-duced (Yi et al. 2001), based on Yonsei–Yale theoretical models of stellar evolution, and thusthe stars should still be in their main-sequence stages. The semi-major axis of the binaryis 0 . +0 . − . , and the eccentricity is 0 . +0 . − . . Considering the smallest possiblestellar masses and largest possible eccentricity, both P-type GHZ and P-type RVEM HZsare expected to exist, noting that those should require for a bin to be less than 0.683 au andless than 0.722 au, respectively. These criteria are fulfilled based on the observational data(see Fig. 6). Welsh et al. (2012) also pointed out that the circumbinary planet is locatedinterior to the HZ. 9 – Kepler-35 is known to have a 0 . +0 . − . M J circumbinary gas giant orbiting hostingstarts on a nearly circular orbit ( e b = 0 . +0 . − . ); see Welsh et al. (2012) for details. Theplanet, which has a semi-major axis to be 0 . +0 . − . au, is within the P-type HZs.The primary star of Kepler-35 has a mass to be 0 . +0 . − . M ⊙ , and the secondary being0 . +0 . − . M ⊙ , with an orbital period of 20 . +0 . − . days. Based on the Yonsei–Yale theoretical models of stellar evolution (Yi et al. 2001), the age of this system is about8 Gyr to 12 Gyr. This is larger than the solar age; however, based on the masses of the twostellar components, this system is still considered to be composed of main-sequence stars.The binary system has a semi-separation of 0 . +0 . − . au, and the eccentricity is givenas 0 . +0 . − . . The system’s semi-major axis is clearly less than the requirements for P-type HZs to exist, which are 0.674 au and 0.712 au for the GHZ and RVEM (see Fig. 6).Welsh et al. (2012) pointed out that the circumbinary planet is located interior to the HZ. Following Kostov et al. (2014), Kepler-413 has a 0 . +0 . − . M ⊙ K dwarf as primary, anda 0 . +0 . − . M ⊙ M dwarf as secondary (see Table 11). The two stars orbit each other on anearly circular orbit, which has an eccentricity of 0 . +0 . − . . The orbital period is givenas 10 . +0 . − . d. Kepler-413b, a 67 +22 − M ⊕ circumbinary planet, orbiting both starswith 0 . +0 . − . au as semi-major axis and 0 . +0 . − . as eccentricity, is slightly outsideof the GHZ, but insight of the RVEM. Fitting with the minimum possible stellar massesand maximum possible binary eccentricity (the most adverse choices for the existence of thecircumstellar HZs), the P-type GHZ and RVEM require a bin to be smaller than 0.572 au and0.605 au to allow their existence. The semi-major axis of the system is 0 . +0 . − . au,which satisfies the requirements for circumbinary HZs to exist (see Fig. 6). Following Kostov et al. (2016), Kepler-1647b has been identified a 483 ± M ⊕ gasgiant in the eclipsing binary system Kepler-1647. The semi-major axis of the planet is2.7205 ± M = 1 . ± . M ⊙ . The secondary star is similar to theSun; it has a mass of M = 0 . ± . M ⊙ (see Table 11). The estimated age of thesystem, identified as approximately 4.4 Gyr, corresponds to mid-age main-sequence. The 10 –binary orbital period is given as 11.2588179 ± ± ± a bin to be less than 1.037 au for P-type GHZ to exist, and 1.096 au for P-typeRVEM to exist. Fitting results for this system are shown in Figure 6, which clearly showthat both the GHZ and RVEM are able to exist in this system. Following Daemgen et al. (2009), TrES-2, also known as Kepler-1, consists of a planet-hosting G0V primary star with a mass of 1.05 M ⊙ . The 1 . ± . M J planet orbits thestellar primary based on a 0 . ± . M ⊙ ; it is a zero-age main-sequence star. The binary separation of thesystem, which is estimated to be 232 ±
12 au, ensure that effects by the secondary star onthe primary’s habitable environment are largely negligible; therefore, resulting in conditionsakin to a single star. Thus, S-type HZs should exist around the primary star. Although thebinary eccentricity is unknown, the semi-major-axis of the binary system is larger than therequired value as 23.679 au for S-type GHZ and 19.303 au for S-type RVEM (see Fig. 7).
Following Santerne et al. (2014), KOI-1257 consists of two main-sequence stars withstellar masses of 0.99 ± M ⊙ and 0.70 ± M ⊙ , respectively (see Table 11). A 1.45 ± M J planet is in an S-type orbit around the primary star with 0.772 ± ± . +0 . − . , and its semi-major axis is 5.3 au with an uncertainty of 1.3 au. In this system,the existence of S-type HZs strongly depends on the binary parameters, which are subjectto notable uncertainties. At e b = 0 .
68, the semi-major axis is required to be larger than13.919 au for an S-type GHZ and 11.019 au for RVEM, which means that for this valueof e b , there are no S-type HZs. However, for the smallest value of e b , given as 0.10, therequirements for the semi-major axis now read 3.899 and 3.085 au for the GHZ and RVEM,respectively, indicating the existence for both S-type types of HZs (see Fig. 7). If disregardingthe observational uncertainties, the semi-major axis is given as 5.3 au. This value is largerthan 5.386 au for the S-type GHZ to exist, and larger than 4.280 au for the S-type RVEM 11 –to exist. We also have compared some of our results with previous work that is based on moder-ately different methods for the calculations of the HZ limits. For S-type and P-type HZs, thiswork has been given by Haghighipour & Kaltenegger (2013) and Kaltenegger & Haghighipour(2013), respectively. Figure 8 shows the comparisons for the systems of Kepler-34, Kepler-35,Kepler-413, and KOI-1257. Comparisons between results based on GHZ and EVEM climatemodels are shown examples, with the observational uncertainties for the eccentricities takeninto account. The percent differences are calculated as difference between habitability limitsdivided by the average.Regarding Kepler-34, the luminosity of the primary is given as 1 . L ⊙ and for thesecondary it is 1 . L ⊙ . The GHZ inner limit has a percent difference between 3.04% and3.14%, whereas the outer limit’s difference varies between 0.777% and 0.780%. In caseof RVEM, the inner limit has a minimum difference of 0.319% and a maximum differenceof 0.321%. The values for the outer limit are given as 0.390% and 0.391%, respectively.Regarding Kepler-35, the luminosity of the primary is given as 0 . L ⊙ and for the secondaryit is 0 . L ⊙ . The GHZ inner and outer limits have minimum percent differences of 2.94%and 0.703%, and maximum percent differences of 3.08% and 0.706%, respectively. For theRVEM climate models, the percent difference for the inner limit and outer limit are close to0.3% and 0.5%, respectively. Regarding Kepler-413, the luminosity of the primary is givenas 0 . L ⊙ and for the secondary it is 0 . L ⊙ . The inner and outer GHZ limits have percentdifferences ranging from 2.82% to 3.01% and from 0.562% and 0.570%. RVEM has percentdifference between 1.12% and 1.22% for the inner limits, and between 0.518% and 0.519%for the outer limits.KOI-1257, considered as an S-type system, consists of two stars with luminosities of1 . L ⊙ and 0 . L ⊙ , respectively. The minimum and maximum percent differences of theGHZ inner limit are 3.22% and 3.73%, respectively. The value for the GHZ outer limit isapproximately 0.75%. The RVEM inner limit is noted for having percent differences between0.053% and 0.102%, whereas the RVEM out limit has a percentage difference of 0.43%with virtually no variation regarding the assumed eccentricity of the binary components.Therefore, in conclusion, our results on obtaining fitting formulae for the existence of S-type and P-type HZs are unaffected by the choice of habitability limits as available in theliterature. 12 –
5. Summary and Conclusions
The aim of this study was the evaluation of the mathematical constraints for the possi-bility of HZs in stellar binary systems — an effort of interest irrespectively of hitherto planetdetections in those systems. This allowed us to deduce fitting formulae that permit — inthe framework of the adopted model — a straightforward “yes/no” answer whether HZsexist. The underlying mathematical concept is based on the work of Paper I and II, whichfollows a comprehensive approach for the computation of habitable zones in binary systems.The latter includes (1) the consideration of a joint constraint including orbital stability anda habitable region for a putative system planet through the stellar radiative energy fluxes(RHZ) needs to be met; (2) the treatment of different types of HZs as defined for the SolarSystem and beyond; (3) the provision of a combined formalism for the assessment of both S-type and P-type HZs based on detailed mathematical criteria — in particular, mathematicalcriteria are presented for which kind of system S-type and P-type habitability is realized; and(4) applications to stellar systems in either circular or elliptical orbits. Note that previousless sophisticated fitting procedures for the existence of HZs in binary systems were givenby Wang & Cuntz (2016).The adopted planetary climate models follow the previous work by Kopparapu et al.(2013, 2014), which allowed us to define the HZs referred to as GHZ and RVEM, includingthe definitions of the respective inner and outer limits. The inspection of the planetaryorbital stability limits follows the work by Holman & Wiegert (1999), which expands onprevious results including work by Dvorak (1986) and Rabl & Dvorak (1988). Results onthe formation and dynamics of planets in dual stellar systems compared to single stars havebeen given by, e.g., Haghighipour (2008) and subsequent work. The fitting formulae for theexistence of the HZs, both regarding P-type and S-type HZs, obtained in our study relatethe axes a bin between the stellar binary components and an algebraic expansion for theirorbital eccentricities; they target the limits where the respective HZs ceases to exist. ForP-type cases, the attained algebraic expansion is of second order, and for S-type cases, it isof third order, but written as an exponential exponent. The various coefficients also dependon M and M , the masses of the stellar components, which ensures the comprehensiveapplicability of the fitting formulae. The current version of our methods is aimed at systemsof main-sequence stars.The stellar masses are assumed to range between 0.50 and 1.25 M ⊙ , i.e., between spec-tral type M0V and F6V (e.g., Gray 2005; Mann et al. 2013). Thus, the respective stellarluminosities range between 0.036 and 2.15 L ⊙ . Therefore, notwithstanding late-type reddwarfs, the kind of stars for which our approach is applicable comprises more than 95% ofmain-sequence stars (e.g., Kroupa 2001, 2002; Chabrier 2003). Furthermore, detailed tests 13 –for our fitting formulae demonstrate that their accuracy compared to the exact results basedon the solutions for the underlying quartic equations (see Papers I and II) is better than 5%in most cases, noting that the least accurate results are found for systems of high eccentricity,i.e., e b ∼ > .
75. In fact, for the vast majority of cases the accuracy of the fitting formulae isfound to be about 1% or 2%.Our method is particularly useful for the quick assessment of observed systems withrelatively large (or poorly known) uncertainties in a bin , e b , M , and M , while noting thatthe latter can play a decisive role regarding whether or not S-type or P-type HZs exist. Todemonstrate the applicability of our method, we explored the existence of P-type HZs forKepler-34, Kepler-35, Kepler-413, and Kepler-1647 and S-type HZs for TrES-2 and KOI-1257. Observational uncertainties of the various system parameters have been consideredas well, which can be relevant for the outcome. A good example is KOI-1257, where theexistence of the S-type HZ is strongly affected by the values for both the semi-major axisand the eccentricity of the stellar motion, which are somewhat uncertain. On the otherhand, we found that all P-type systems considered possess P-type HZs irrespectively of theuncertainties in the relevant observational parameters. These results are also unaffected bythe planetary climate models.Generally, the likelihood for the existence of HZs is relatively high for low values of the e b , but relatively small, or virtually non-existing, for high values of the e b . Moreover, thelikelihood if a HZs can exist is also increased if RVEM-type HZs are considered rather thanGHZ-type HZs, as expected. The fact that high values for e b decisively reduce the possibilityof S-type habitability has previously been pointed out by, e.g., Cuntz (2015). Our futurework will also consider stellar systems of stars other than main-sequence components andalso take into account future advances about planetary climate models, including the impactof planetary masses and atmospheric structures.This work has been supported by the Department of Physics, University of Texas atArlington (UTA). We also appreciate comments by Elke Pilat-Lohinger about research onorbital stability limits in binary systems. Moreover, we wish to thank the anonymous refereefor her/his useful suggestions allowing us to improve the manuscript. 14 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.0.
17 – α ϕ a S2S1 d d z α ϕ S1 S22 ad d z Fig. 1.—: Mathematical set-up of S-type (top) and P-type (bottom) habitable zones ofbinary systems as given by the stellar radiative fluxes. Here 2 a denotes the separationdistance between the stellar binary components, corresponding to the semi-major axis a bin (as used by the observational community) of the binary system. It is not necessary for thestars S1 and S2 being identical (adopted from Paper I). 18 – e b a b i n ( A U ) HZs for M =0.50M ⊙ , M =0.50M ⊙ S/ST−TypeP/PT−Type
S−GHZS−RVEMP−RVEMP−GHZ e b a b i n ( A U ) HZs for M =1.00M ⊙ , M =0.50M ⊙ S/ST−TypeP/PT−Type e b a b i n ( A U ) HZs for M =1.00M ⊙ , M =1.00M ⊙ S/ST−TypeP/PT−Type e b a b i n ( A U ) HZs for M =1.25M ⊙ , M =0.75M ⊙ S/ST−TypeP/PT−Type
Fig. 2.—: Required a bin and e b for the GHZ and RVEM to exist regarding selected theoreticalbinary systems. The GHZ can exist when the system parameters are within the gray region.System parameters fall in either gray or light gray region would allow RVEM to exist. Themagenta and green curves show the critical pairs of values for the GHZ and RVEM to existcorrespondingly. 19 – e b a b i n ( A U ) HZs for M =0.50M ⊙ , M =0.50M ⊙ S/ST−TypeP/PT−Type
S−GHZS−RVEMP−RVEMP−GHZFitting 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.802468101214161820 e b a b i n ( A U ) HZs for M =1.00M ⊙ , M =0.50M ⊙ S/ST−Type P/PT−Type
S−GHZS−RVEMP−RVEMP−GHZFitting0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.802468101214161820 e b a b i n ( A U ) HZs for M =1.00M ⊙ , M =1.00M ⊙ S/ST−TypeP/PT−Type
S−GHZS−RVEMP−RVEMP−GHZFitting 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.802468101214161820 e b a b i n ( A U ) HZs for M =1.25M ⊙ , M =0.75M ⊙ S/ST−TypeP/PT−Type
S−GHZS−RVEMP−RVEMP−GHZFitting
Fig. 3.—: Fitting of the data for selected theoretical main-sequence stars. The magenta andgreen lines represent the boundaries for the GHZ and the RVEM to exist, respectively. Ineach subfigure, the areas beyond the magenta and green curves (top) identify the existenceof the S/ST-type HZs, whereas the areas below the magenta and green curves (bottom)identify the existence of the P/PT-type HZs. The thin black curves depict the fitting resultsfor the curve nearby, and they are virtually indistinguishable from the data curves. 20 – e b a b i n ( A U ) P-GHZ for M =1.25M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) P-GHZ for M =0.50M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) P-RVEM for M =1.25M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) P-RVEM for M =0.50M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ Fig. 4.—: Results for P-type GHZ and RVEM. P-type HZs are realized beneath the respec-tive curve. 21 – e b a b i n ( A U ) S-GHZ for M =1.25M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) S-GHZ for M =0.50M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) S-RVEM for M =1.25M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ e b a b i n ( A U ) S-RVEM for M =0.50M ⊙ M =1.25M ⊙ M =1.00M ⊙ M =0.75M ⊙ M =0.50M ⊙ Fig. 5.—: Results for S-type GHZ and RVEM. S-type HZs are realized above the respectivecurve. 22 – e b a b i n ( A U ) HZs for Kepler-34
P−RVEMP−GHZ e b a b i n ( A U ) HZs for Kepler-35
P−RVEMP−GHZ e b a b i n ( A U ) HZs for Kepler-413
P−RVEMP−GHZ 0.14 0.15 0.16 0.17 0.180.00.20.40.60.81.01.2 e b a b i n ( A U ) HZs for Kepler-1647
P−RVEMP−GHZ
Fig. 6.—: The red and blue curves are the fitting results that show the maximum a bin forP-type HZs to exist; i.e., P-type HZs are possible below these curves. The dashed lines arethe results considering the uncertainties in the stellar masses. The gray domains indicatethe indicated a bin and e b values for the respective stellar systems with the observationaluncertainties taken into account (the purple ellipses are placed to enhance the domains’visibility). 23 – e b l og ( a b i n ) ( A U ) HZs for TrES-2
S−GHZS−RVEM 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8024681012 e b a b i n ( A U ) HZs for KOI-1257
S−GHZS−RVEM
Fig. 7.—: The red and blue curves are the fitting results that show the minimum a bin forS-type HZs to exist; i.e., S-type HZs are possible above these curves. The dashed lines arethe results considering the uncertainties in the stellar masses. The gray domains indicatethe indicated a bin and e b values for the respective stellar systems with the observationaluncertainties taken into account. A logarithmic scale y -scale is used for TrES-2 because ofthe system’s very large semi-major axis. 24 – e b H ab i t ab l e Z one s ( A U ) Kepler−34
HaKa−GHZHaKa−RVEMWaCu−GHZWaCu−RVEMStability e b H ab i t ab l e Z one s ( A U ) Kepler−35
HaKa−GHZHaKa−RVEMWaCu−GHZWaCu−RVEMStability0.034 0.036 0.038 0.0400.00.51.01.52.0 e b H ab i t ab l e Z one s ( A U ) Kepler−413
HaKa−GHZHaKa−RVEMWaCu−GHZWaCu−RVEMStability e b H ab i t ab l e Z one s ( A U ) KOI−1257
KaHa−GHZKaHa−RVEMWaCu−GHZWaCu−RVEMStability
Fig. 8.—: Comparisons of our results (WaCu) for Kepler-34, Kepler-35, Kepler-413, andKOI-1257 regarding the existence of HZs with previous work pertaining to the calculationsof HZ limits. That work has been given by Haghighipour & Kaltenegger (2013) for S-typeHZs and by Kaltenegger & Haghighipour (2013) for P-type HZs, thus denoted as HaKa andKaHa, respectively. Results are given for GHZ and RVEM climate models. 25 –Table 1. Habitability Limits for the Solar SystemDescription Indices Models This work... l Kas93 Kop1314 ...... ... 5700 K 5780 K 5780 K ...... ... (au) (au) (au) ...Recent Venus 1 0.75 0.77 0.750 RVEM Inner LimitRunaway greenhouse effect 2 0.84 0.86 0.950 GHZ Inner LimitMoist greenhouse effect 3 0.95 0.97 0.993 ...Earth-equivalent position 0 0.993 ≡ ≡ condensation 4 1.37 1.40 ... ...Maximum greenhouse effect 5 1.67 1.71 1.676 GHZ Outer LimitEarly Mars 6 1.77 1.81 1.768 RVEM Outer LimitNote. — This table depicts the various values of s ℓ (see Eq. 1), as previously obtained inthe literature. Here Kas93 denotes the work by Kasting et al. (1993), and Kop1314 denotesthe combined work by Kopparapu et al. (2013) and Kopparapu et al. (2014). 26 –Table 2. Stellar Parameters M ∗ Spectral Type T ∗ R ∗ L ∗ ( M ⊙ ) ... (K) ( R ⊙ ) ( L ⊙ )1.25 ∼ F6V 6257 1.253 2.1541.00 ∼ G2V 5780 1.000 1.00000.75 ∼ K2V 5104 0.766 0.35680.50 ∼ M0V 3664 0.472 0.03593Note. — Adopted from Paper I and II. 27 –Table 3. BIC Values for a bin versus e b FittingBIC Linear Quadratic Cubic QuarticP − GHZ − − − − − RVEM − − − − − GHZ − − − − − RVEM − − − − M = M = 1 . M ⊙ is givenas an example for the determination of the a bin versus e b fitting. For all S-type results, the logarithm of a bin isapplied. The mean absolute percentage error (MAPE)is calculated as well, and a 2% threshold is used. TheBICs are found to decrease as the orders of the equa-tions increase from 1 to 3 for all cases indicating thatit is acceptable to have cubic equations. Lowest orderequations satisfy the MAPE requirement are chosen forless complexity. 28 –Table 4. Fitting CoefficientsModel Coefficient Case 1 Case 2 Case 3 Case 4P-GHZ α α − − − − α α α − − − − α β − β β − − − − β β − β β − − − − β M = M = 0 . M ⊙ ; Case 2: M =1 . M ⊙ , M = 0 . M ⊙ ; Case 3: M = M = 1 . M ⊙ ;Case 4: M = 1 . M ⊙ , M = 0 . M ⊙ . Table 5. BIC Values for Mass Fitting
Model BIC Linear to M and M Adding M and M Adding M M and M Adding M M and M Adding M and M Adding M and M P − GHZ Constant − − − − − − e b term − − − − − − e b term − − − − − − − RVEM Constant − − − − − − e b term − − − − − − e b term − − − − − − − − − − − − − GHZ Constant − − − − − − e b term − − − − − − e b term − − − − − − e b term − − − − − − − RVEM Constant − − − − − − e b term − − − − − − e b term − − − − − − e b term − − − − − − − − − − − − a bin versus e b fitting are further fitted based on an equation linear in the stellar masses. Additional terms are added by checking the BIC. Thesmallest BIC in each line is given in bold font. The total BIC values in each column are compared. For P-type, adding nothing is preferred, whereas for S-type, adding M and M turns out to be the best choice.
30 –Table 6. General Fitting Coefficients, P-TypeModel Coefficient A i A i A i P-GHZ α − α − α − − α − α − − α − − B i B i B i B i B i S-GHZ β − − β − − − β − − β − − − β − − β − − − β − − β − − − e b M = M = 1 . M ⊙ M = 1 . M ⊙ , M = 0 . M ⊙ ... P-GHZ P-RVEM S-GHZ S-RVEM P-GHZ P-RVEM S-GHZ S-RVEM0.0 0.53% 0.17% 4.96% 6.36% 0.58% 0.02% 2.71% 4.09%0.1 2.28% 2.21% 0.67% 0.44% 2.25% 2.20% 2.21% 2.46%0.2 2.66% 2.46% 0.11% 0.89% 2.27% 2.33% 3.26% 4.09%0.3 2.20% 2.04% 0.70% 0.09% 1.31% 1.38% 2.14% 2.80%0.4 1.59% 1.40% 1.59% 1.52% 0.33% 0.19% 0.66% 0.86%0.5 1.27% 1.11% 1.82% 2.20% 0.07% 0.40% 0.18% 0.20%0.6 1.59% 1.50% 0.98% 1.40% 1.20% 0.16% 1.13% 0.62%0.7 2.99% 2.98% 0.67% 0.08% 4.13% 2.32% 2.27% 2.59%0.8 5.78% 5.82% ... ... 9.26% 6.34% ... 0.92% 33 –Table 9. Errors of Fitting, Continued e b M = 0 . M ⊙ , M = 0 . M ⊙ M = M = 0 . M ⊙ ... P-GHZ P-RVEM S-GHZ S-RVEM P-GHZ P-RVEM S-GHZ S-RVEM0.0 3.48% 3.47% 2.97% 3.36% 0.33% 0.00% 5.41% 6.08%0.1 1.87% 1.85% 3.77% 3.80% 1.82% 2.05% 0.58% 0.59%0.2 2.02% 1.95% 5.40% 5.52% 2.50% 2.74% 2.30% 2.26%0.3 2.96% 2.92% 4.04% 4.12% 2.37% 2.53% 1.21% 1.09%0.4 3.99% 4.14% 1.91% 2.12% 2.00% 1.94% 0.65% 0.75%0.5 4.38% 4.83% 0.87% 0.98% 1.80% 1.49% 1.60% 1.59%0.6 3.58% 4.52% 1.63% 1.63% 2.25% 1.47% 0.67% 0.65%0.7 1.29% 2.88% 3.43% 3.34% 3.60% 2.33% 1.24% 1.12%0.8 2.58% 0.45% 0.34% 0.46% 6.10% 4.56% 2.27% 2.86% 34 –Table 10. Coefficient of DeterminationSystems P-GHZ P-RVEM S-GHZ S-RVEM M = 1 . M ⊙ , M = 1 . M ⊙ M = 1 . M ⊙ , M = 1 . M ⊙ M = 1 . M ⊙ , M = 0 . M ⊙ M = 1 . M ⊙ , M = 0 . M ⊙ M = 1 . M ⊙ , M = 1 . M ⊙ M = 1 . M ⊙ , M = 0 . M ⊙ M = 1 . M ⊙ , M = 0 . M ⊙ M = 0 . M ⊙ , M = 0 . M ⊙ M = 0 . M ⊙ , M = 0 . M ⊙ M = 0 . M ⊙ , M = 0 . M ⊙ M M a bin e b Reference... ( M ⊙ ) ( M ⊙ ) (au) ... ...Kepler-34 1 . +0 . − . ± . +0 . − . . +0 . − . Welsh et al. (2012)Kepler-35 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Welsh et al. (2012)Kepler-413 0 . +0 . − . . +0 . − . . +0 . − . . +0 . − . Kostov et al. (2014)Kepler-1647 1.2207 ± ± ± ± ±
12 ... Daemgen et al. (2009)KOI-1257 0.99 ± ± ± . +0 . − .21