Habitable Zone Dependence on Stellar Parameter Uncertainties
aa r X i v : . [ a s t r o - ph . E P ] J a n Submitted for publication in the Astrophysical Journal
Preprint typeset using L A TEX style emulateapj v. 5/2/11
HABITABLE ZONE DEPENDENCE ON STELLAR PARAMETER UNCERTAINTIES
Stephen R. Kane
Department of Physics & Astronomy, San Francisco State University, 1600 Holloway Avenue, San Francisco, CA 94132, USA
Submitted for publication in the Astrophysical Journal
ABSTRACTAn important property of exoplanetary systems is the extent of the Habitable Zone (HZ), definedas that region where water can exist in a liquid state on the surface of a planet with sufficientatmospheric pressure. Both ground and space-based observations have revealed a plethora of confirmedexoplanets and exoplanetary candidates, most notably from the Kepler mission using the transitdetection technique. Many of these detected planets lie within the predicted HZ of their host star.However, as is the case with the derived properties of the planets themselves, the HZ boundariesdepend on how well we understand the host star. Here we quantify the uncertainties of HZ boundarieson the parameter uncertainties of the host star. We examine the distribution of stellar parameteruncertainties from confirmed exoplanet hosts and Kepler candidate hosts and translate these into HZboundary uncertainties. We apply this to several known systems with a HZ planet to determine theuncertainty in their HZ status.
Subject headings: astrobiology – planetary systems INTRODUCTION
The discovery of exoplanets has proceeded through aperiod of technique refinement over the past 20 yearsin order to improve sensitivity to planets of smaller sizeand larger semi-major axis. As such, the relevance ofthe Habitable Zone (HZ) boundaries for exoplanet hoststars has moved from the realm of a theoretical exer-cise to one of enormous practical application. A rigor-ous calculation of the HZ boundaries for main sequencestars have previously been provided by Kasting et al.(1993). These calculations have since been generalizedin terms of stellar luminosity ( L ⊙ ) and effective temper-ature ( T eff ) by a variety of authors (Underwood et al.2003; Selsis et al. 2007; Jones & Sleep 2010). The rela-tion of the HZ boundaries to fundamental stellar prop-erties have been recently recalculated and extended tolower mass stars by Kopparapu et al. (2013).The radial velocity (RV) technique has revealed sev-eral possible HZ terrestrial planets, such as thosein the GJ 667 system (Anglada-Escud´e et al. 2013;Feroz & Hobson 2014). Most of the RV planets in theirstars HZ are of Jovian mass however (Kane & Gelino2012), although this opens up the prospect of habitablemoons in such cases (Hinkel & Kane 2013). A signifi-cant source of planets which potentially orbit within theHZ of their host stars has been from the Kepler mis-sion. A recent example of this includes the Kepler-62system which contains two planets of ∼ . [email protected] which are used to determine the HZ boundaries andquantify the effect of stellar parameters uncertaintieson HZ calculations. Section 2 includes a summary ofthe methodology used to determine the HZ boundariesthroughout the rest of this work. In Section 3, we exam-ine each of the primary contributing stellar parametersin detail with respect to how the uncertainties translateto our understanding the extent of the HZ. These resultsare applied to two populations of exoplanet host stars inSection 4, broadly defined as the “confirmed exoplanethost stars” and the “Kepler candidate host stars”. InSection 5, we examine several specific cases of claimedHZ exoplanets to determine the validity of those claimsand provide concluding remarks in Section 6. THE HABITABLE ZONE BOUNDARIES
The HZ for a particular star is broadly defined as theregion around the star where water can exist in the liq-uid state on the surface of a planet given adequate at-mospheric pressure. These boundaries were calculated byKasting et al. (1993) based upon one-dimensional, cloud-free climate models. The most recent revision of thesecalculations by Kopparapu et al. (2013) uses updatedmodels for H O and CO in the thermal-IR and expandsthe range of stellar effective temperatures for which thecalculations are valid. A generalization of the incidentstellar flux at the HZ boundaries is as follows: S eff = S eff ⊙ + aT ⋆ + bT ⋆ + cT ⋆ + dT ⋆ (1)where T ⋆ = T eff − T eff is the stellar effec-tive temperature. Kopparapu et al. (2013) provide val-ues for the coefficients S eff ⊙ , a , b , c , and d depend-ing on assumptions on when water-loss may have oc-curred in the Venusian and Martian history. Here wewill refer to “conservative” and “optimistic” models ofthe HZ, as used by Kane et al. (2013). The conservativemodel treats the “Runaway Greenhouse” and “MaximumGreenhouse” criteria as the inner and outer HZ bound-aries respectively. The optimistic model adopts the “Re-cent Venus” and “Early Mars” criteria for these inner Stephen R. Kane Fig. 1.—
The dependency of the “Runaway Greenhouse” HZ boundary (see Section 2) on the percentage changes in stellar effectivetemperature (left) and surface gravity (right). Note that the dependency on surface gravity arises when used to determine the stellar radiusand thus stellar luminosity. and outer boundaries, allowing for an expanded HZ un-der the assumption that Venus and Mars may have hada longer period of retaining surface water. These criteriaare described in more detail by Kopparapu et al. (2013).The distance, d , of the boundaries for a particular HZmodel may be determined from the stellar flux in Equa-tion 1 using the following relation: d = (cid:18) L ⋆ /L ⊙ S eff (cid:19) . (2)where L ⋆ is the stellar luminosity and the distance is inunits of AU.The above calculations depend sensitively on T eff and L ⋆ . The luminosity of the nearest stars may be deter-mined through the use of stellar parallax and subsequentdistance estimates. We discuss the uncertainties associ-ated with such luminosity determinations in Section 3.Distance estimates of reasonable accuracy are in limitedsupply for many exoplanet host stars and the luminosityis often calculated based on assumptions regarding thestellar radius, R ⋆ , via the equation L ⋆ = 4 πR ⋆ σT . Thestellar radius is often calculated based on stellar modelsand the relatively accessible quantities of T eff and sur-face gravity, log g , since these influence the stellar pho-tosphere. The radius may also be estimated from thestellar mass, M ⋆ , and surface gravity. Below is an exam-ple relationship (Smalley 2005) which allows the radiusto be determined from other fundamental stellar proper-ties: log g = log (cid:18) M ⋆ M ⊙ (cid:19) − (cid:18) R ⋆ R ⊙ (cid:19) + log g ⊙ (3)where log g ⊙ = 4 . STELLAR PARAMETER UNCERTAINTIES
The two main quantities used to determine the vari-ous HZ boundaries are T eff and L ⋆ . As seen in Equation2, the boundaries have a power law dependence on L ⋆ .However, since L ⋆ is a calculated rather than measuredquantity, it depends upon other variables ( T eff , R ⋆ , and log g ), each with their own uncertainties. Here we quan-tify the HZ boundary dependencies on these parameters.The first parameter we consider is that of effective tem-perature. As seen in Equation 1, this parameter is funda-mental to calculating the stellar flux received at the HZboundaries. Fortunately, it is also usually one of the bet-ter known stellar parameters with direct measurementsfrom stellar spectra. As such, it has little dependencyupon the knowledge of other stellar parameters. Theuncertainty in T eff also filters through to the determina-tion of luminosity since that is often calculated from T eff and R ⋆ . Thus, the T eff uncertainty plays a major role indetermining the robustness of the HZ boundaries.The left panel of Figure 1 shows the dependency ofthe “Runaway Greenhouse” boundary (inner conserva-tive HZ boundary) on percentage uncertainties in T eff .This shows that uncertainties in T eff of ∼
5% can resultin ∼
10% uncertainties in the location of the HZ bound-ary. There is a variation of this dependency on spectraltype but the variation is relatively minor in nature.The dependency of HZ boundary determinations on T eff is independent of the stellar radius, which is usuallya calculated rather than measured quantity. However,the uncertainty in R ⋆ contributes significantly to the HZerror budget and is often relatively large. There is alinear dependence of the HZ distances on R ⋆ since L ⋆ ∝ R ⋆ and d ∝ √ L ⋆ (see Equation 2). Thus the uncertaintyin the location of the HZ depends linearly on the R ⋆ uncertainties.We also consider the contribution of the surface gravity(log g ) and stellar mass ( M ⋆ ) uncertainties to the errorbudget since these can be used to determine R ⋆ . Accord-ing to Equation 3, the dependency of R ⋆ on log g and M ⋆ are R ⋆ ∝ p /g and R ⋆ ∝ √ M ⋆ respectively. Thus theHZ boundaries have the same dependencies: d ∝ p /g and d ∝ √ M ⋆ . The right panel of Figure 1 plots the effectof varying log g on the HZ boundaries which shows theinverse dependence. Note that the dependencies may bemore complicated than described since a change in massfor a main sequence star will result in a change in radius,thus an associated change in g .Finally, we consider luminosity determinations fromabitable Zone Dependence 3 Fig. 2.—
Histograms of the stellar distances (left panel) and their associated uncertainties (right panel) for the confirmed exoplanet hoststars as determined from their stellar parallax. Each histogram uses a total of 30 bins. stellar distances derived from their parallax measure-ments. The parallax measurements are typically thoseprovided by the Hipparcos mission (Perryman et al.1997) and the revised reduction of those data(van Leeuwen 2007). This is a common method for lumi-nosity determination for the confirmed exoplanets whosediscovery resulted from the RV technique. In these casesthe stars tend to be preferentially bright and thus closeenough for reasonable parallax measurements. Figure 2shows histograms of distance measurements and their as-sociated uncertainties for the confirmed exoplanet hoststars. These data were extracted from the ExoplanetData Explorer (see Section 4). The vast majority of RVhost stars are closer than 100 pcs where distances aredetermined to ± . In the followingsections, we restrict ourselves to using the stellar param-eters noted above in order to provide a direct compari-son between different types of exoplanet host stars. Thecaveat to note is that some of the confirmed exoplanethost stars may have overestimated HZ boundary uncer-tainties where their distances are suffiently well known. APPLICATION TO KNOWN EXOPLANET HOST STARS
Here we apply the above described stellar parameteruncertainty effects to the HZ calculations for the knownexoplanet host stars. We divide these into two broadgroups: the host stars of confirmed exoplanets and thehost stars of Kepler candidates. We extracted the stel-lar parameters stored in the Exoplanet Data Explorer (Wright et al. 2011). The data are current as of 26thNovember 2013. The data are also available from theNASA Exoplanet Database (Akeson et al. 2013). The http://sci.esa.int/gaia/ http://exoplanets.org/ http://exoplanetarchive.ipac.caltech.edu/ data utilized here includes all of the host stars with thenecessary stellar parameter information to perform thisanalysis. This main criteria are that the host stars haveavailable values for the effective temperatures, radii, andsurface gravities.An inherent assumption in the following analysis isthat the uncertainties associated with stellar parame-ters are gaussian in nature and can mapped to 1 σ un-certainties in HZ boundary locations. There are numer-ous individual cases where this will not be true due tonon-gaussian posteriors and correlated errors which areinherent in the analysis of spectra (for example). Thisdoes not have a significant effect on the result since wedraw upon a large distribution to show the impact ofthese stellar uncertainties. Confirmed Exoplanet Host Stars
The number of host stars in the confirmed exoplanetsgroup of host stars which meet the criteria of availablestellar parameters is 507. This group includes host starsfor which their planets have been detected by either theRV or transit techniques. Many of the stellar proper-ties have been compiled by such sources as Butler et al.(2006) and Takeda et al. (2007). Figure 3 summarizesthe stellar properties of T eff , R ⋆ , and log g . The his-tograms show the distribution of the parameters andtheir associated uncertainties. Each of the histogramsare divided into 30 bins and have the same y-axis scalefor ease of comparison.The distribution of T eff lies mostly between 4000–6500 K; a reflection of the F–G–K target selection whichdominates RV exoplanet host stars. The distribution ofthe T eff uncertainties peaks at ∼
50 K which is ∼
1% ofthe typical T eff value.As described in Section 3, the stellar radius plays akey role in determining the uncertainty in the HZ bound-aries. The stellar radii histogram shows that dwarf starsdominate as hosts of the confirmed exoplanets with adistribution peaking at ∼ R ⊙ . The uncertainty distri-bution peaks at ∼ . R ⊙ and thus the typical radiusuncertainty for this group of host stars is ∼ g values shown in Figure 3 isalso indicative of the dominance of dwarf host stars in the Stephen R. Kane Fig. 3.—
Histograms of the stellar parameters (left column) and their associated uncertainties (right column) for the confirmed exoplanetsgroup. Included parameters are effective temperature (top row), stellar radius (middle row), and surface gravity (bottom row). Eachhistogram uses a total of 30 bins with data for 507 host stars. Note that the stellar radius histograms have a logarithmic scale. abitable Zone Dependence 5
Fig. 4.—
Histogram representations of the HZ distributions forthe confirmed exoplanets group. Top: Histogram of the calculatedHZ width (AU). Bottom: Histogram of the width of the HZ uncer-tainty region, calculated as a percentage of the HZ width shown inthe top panel. sample since it peaks between 4–5. The uncertainty inlog g peaks at ∼ .
07. The range of uncertainty values issufficient to determine a luminosity class, but can resultin substantial ambiguity in derived stellar radius whenEquation 3 is employed.From the values of T eff and R ⋆ described above, wecalculated the HZ boundaries for the conservative model(see Section 2). The width of the conservative HZ wasthen determined for each of the stars. The distributionof the HZ widths is represented by the histogram shownin the top panel of Figure 4. This distribution appearsto be symmetric but is shown on a logarithmic scale andpeaks at a width of ∼ . σ parameteruncertainties for the inner and outer boundaries respec-tively. This results in an uncertainty region which islarger than the calculated width of the HZ. The distribu-tion of this HZ uncertainty region is shown in the bottompanel of Figure 4 as a percentage of the HZ width. Thus,an uncertainty region close to 100% means that the stel-lar parameter uncertainties are relatively small and thelocation of the HZ is well constrained. An uncertaintyregion of 200% means that any planet in the HZ of sucha system may not be in the HZ at the 1 σ level. The dis- tribution shown in Figure 4 shows that typical HZ un-certainty regions are only 20% larger than the calculatedHZ width for the confirmed exoplanet group. Kepler Candidate Host Stars
There have been several releases of Kepler candidateswhich is an ever-growing list of likely transiting exo-planets detected by the Kepler mission (Borucki et al.2011a,b; Batalha et al. 2013). With the number ofKepler candidate host stars numbering in their thou-sands, determining accurate stellar parameters is adaunting task. Stellar parameters for these stars havebeen measured and estimated using a combination ofphotometric calibration, spectroscopy, and astroseis-mology (Brown et al. 2011; Silva Aguirre et al. 2012;Everett et al. 2013; Huber et al. 2013). There are nu-merous difficulties in determining these stellar proper-ties which can result in systematic offsets in parameterssuch as temperature, matallicity, radius, etc. These aredescribed in detail by Huber et al. (2014) and referencestherein. This highlights the need for accurate stellar pa-rameters in order to decrease the HZ boundary uncer-tainties discussed here.The number of Kepler candidate host stars which meetour criteria of available stellar parameters is 1852. Thedistributions of the T eff , R ⋆ , and log g and their associ-ated uncertainties are shown in the histograms of Figure5. As with the confirmed exoplanets group (see Figure 3)each of the histograms are divided into 30 bins. However,the y-axes of the surface gravities have a different scaleto the other histograms due to their substantially differ-ent distributions. A general difference that can be seenbetween the confirmed exoplanet and Kepler candidatehost stars is that the uncertainty distributions for theKepler candidate host stars are skewed towards higheruncertainties. This is not unexpected since the Keplerstars are systematically fainter than those monitored bymost ground-based RV and transit programs. This doeshowever have a significant effect on HZ calculations aswe will soon show.The distribution of T eff for the Kepler candidate hoststars is similar to that for the confirmed exoplanet hoststars but with a strong emphasis on solar (G-type) stars.The uncertainty distribution peaks at ∼
80 K, almosttwice that of the confirmed exoplanet host stars.The stellar radii distribution shown in Figure 5 is like-wise similar to that shown in Figure 3 with relatively fewgiant stars in the sample. However, the uncertainty dis-tribution is significantly worse with a major peak in thebimodal distribution of ∼ . R ⊙ , an order of magnitudehigher than that of the confirmed exoplanet host stars.As stated earlier, large efforts have been made to bettercharacterize the Kepler stars, particularly the exoplanetcandidate host stars, resulting in a minor distributionwith radius uncertainties less than ∼ R ⊙ .Determining log g values for the relatively faint Ke-pler stars is a difficult task, and several efforts havebeen made to do so from photometric calibrations(Claret & Bloemen 2011; Creevey et al. 2013). The his-togram of log g values shown in Figure 5 is consistentwith most of the Kepler candidate host stars being se-lected because of their dwarf classification and thus con-sistent with the T eff and R ⋆ distributions. The uncertain-ties for log g are particularly unreliable with most hav- Stephen R. Kane Fig. 5.—
Histograms of the stellar parameters (left column) and their associated uncertainties (right column) for the Kepler candidatesgroup. Included parameters are effective temperature (top row), stellar radius (middle row), and surface gravity (bottom row). Eachhistogram uses a total of 30 bins with data for 1852 host stars. Note that the stellar radius histograms have a logarithmic scale. abitable Zone Dependence 7
Fig. 6.—
Histogram representations of the HZ distributions forthe Kepler candidates group. Top: Histogram of the calculated HZwidth (AU). Bottom: Histogram of the width of the HZ uncertaintyregion, calculated as a percentage of the HZ width shown in thetop panel. ing been assigned a default value of 0.3 dex (Brown et al.2011). This does not affect the HZ calculations presentedhere since we use the R ⋆ values and uncertainties due totheir availability.The widths of the conservative HZ for all of the Keplercandidate host stars are shown in the top panel of Fig-ure 6. These were calculated in the same way as for theconfirmed exoplanet host stars. An important differencebetween the two is that the distribution of HZ widths forthe Kepler candidate host stars peaks at ∼ . ∼ . T eff distribution is similar be-tween the two groups so it is the slightly smaller radiidistribution of the Kepler candidate host stars that re-sults in the overall reduction in HZ widths. The maindifference between the two groups can be seen in the bot-tom panel of Figure 6. The slight bimodal distribution inthe stellar radii results in a similar bimodal distributionin the width of the HZ uncertainty region. However, themajority of the Kepler stars have a HZ uncertainty regionwhich is between 200–400% larger than the width of theHZ; significantly worse than for the confirmed exoplanethost stars. The repercussion of this is that Kepler candi-date exoplanets whose orbits are supposed to lie within Fig. 7.—
The calculated extent of the conservative (light-gray)and optimistic (dark-gray) HZ for the GJ 581 system. The Keple-rian orbits of the planets are shown as solid lines. The 1 σ uncer-tainty boundaries for the conservative HZ are indicated by dashedlines. the HZ of their host star are unlikely to fall within theHZ at all. Thus statistics of Kepler HZ candidates mustbe individually examined to determine if they reasonablyqualify to retain their HZ status. SPECIFIC HABITABLE ZONE SYSTEMS
Here we consider some specific systems to study theextent of the HZ both with and without stellar parameteruncertainties accounted for.The GJ 581 system has been of particular interestwith regards to the HZ boundaries as there have beenclaims that planets within the system have HZ status(Vogt et al. 2010). The host star is especially well char-acterized due in no small part to the long-baseline inter-ferometric observations carried out by von Braun et al.(2011). These measurements resulted in determining thefundamental stellar parameters of T eff = 3498 ±
56 K and R ⋆ = 0 . ± . R ⊙ .Figure 7 shows a top-down view of the GJ 581 systemwith the calculated HZ regions. The light gray representsthe conservative HZ model and the dark gray regionsrepresent the extensions to the HZ from the optimisticmodel. The orbits of the GJ 581 planets are overlaid onthe plot (solid lines) where we have adopted the four-planet model described by Forveille et al. (2011). Thedashed lines indicate the 1 σ extensions of the conserva-tive model boundaries due to the stellar parameter uncer-tainties. In this case, these 1 σ uncertainties are almostnegligible in size and the HZ uncertainty region (see Fig-ures 4 and 6) is 102% of the HZ width of 0.104 AU. Thus,the location of the HZ is well-defined for this system.A prominent confirmed multi-planet system detectedby the Kepler mission is that of Kepler-62 Borucki et al.(2013). This five-planet system includes two which earlyestimates showed are in the HZ of the host star. Thepublished stellar parameters are T eff = 4925 ±
70 K and R ⋆ = 0 . ± . R ⊙ . The HZ conservative and optimistic Stephen R. Kane Fig. 8.—
The calculated extent of the conservative (light-gray)and optimistic (dark-gray) HZ for the Kepler-62 system. The Kep-lerian orbits of the planets are shown as solid lines. The 1 σ uncer-tainty boundaries for the conservative HZ are indicated by dashedlines. Fig. 9.—
The calculated extent of the conservative (light-gray)and optimistic (dark-gray) HZ for the Kepler-27 system. The Kep-lerian orbits of the planets are shown as solid lines. The 1 σ uncer-tainty boundaries for the conservative HZ are indicated by dashedlines. regions are shown in Figure 8. As noted by Borucki et al.(2013), the Kepler-62 system is unstable with the Kep-lerian orbital parameters presented in their paper due tooverlapping orbits. We therefore adopt circular orbits forthe planets which are overlaid on the plot. The widthof the conservative HZ region is 0.37 AU. The dashedlines for the 1 σ uncertainty boundaries show that theHZ location is less well-known in this case than it is forGJ 581. The HZ uncertainty region for Kepler-62 is 120% of the HZ width. Even so, the uncertainties are smallenough such that the e and f planets are likely to be inthe optimistic and conservative HZ regions as shown. Toquantify this, we assume a normal distribution of the HZboundary uncertainties. The e planet is 1.92 σ away fromthe conservative HZ boundary and thus 94.5% likely tobe outside of this region. The f planet is 2.77 σ away fromthe conservative HZ boundary and thus 99.4% likely tobe inside of the conservative HZ region.The two planets of the Kepler-27 system were con-firmed using the Transit Timing Variation (TTV) tech-nique by Steffen et al. (2012). Although neither of theseplanets are purported to lie within the HZ, this is an in-teresting system to study as an example of one with rel-atively large stellar parameter uncertainties. For Kepler-27, these are T eff = 5400 ±
60 K and R ⋆ = 0 . ± . R ⊙ .The HZ and planets for this system are shown in Figure 9.The width of the conservative HZ is 0.38 AU and the HZuncertainty region is 201% of this width. As describedin Section 4, an uncertainty region of this size resultsin considerable doubt as to the HZ status of any planetdescribed to be in HZ in such cases. We have shownthat the vast majority of the Kepler exoplanet candidatefall into this category with the currently known stellarparameters. CONCLUSIONS
The HZ is an increasingly important property of exo-planet host stars with the ever-increasing sensitivity toexoplanets of smaller size/mass and at longer orbital pe-riods. The greatest hindrance to understanding the prop-erties of exoplanets is the difficulty in fully characterizingthe host star properties. This also results in limitationsin defining the extent of the HZ in exoplanetary systems.Here we have shown these limitations as imposed bythe specific stellar properties of T eff , R ⋆ , and log g . Theimportance of R ⋆ is in deriving the stellar luminosity andthe value of log g is utilized only when the radius is notdetermined through other means. The stellar parametersfor the confirmed exoplanet host stars are sufficiently welldetermined such that the size of the HZ uncertainty re-gion lies below 150% for most of the stars. However, theHZ uncertainty region distribution for the Kepler candi-date host stars is dominated by those in the 200-400%range where the HZ status of exoplanets is highly dubi-ous. Analysis of the stellar properties between the twogroups shows that the uncertainty in stellar radius is theprimary cause of this HZ uncertainty difference.With the continuous rate of new discoveries fromboth ground and space-based surveys, the search forterrestrial-size planets in the HZ of their host stars isa difficult but achievable goal. When such discoveriesare made, it will always be critical to quantify the extentto which we can correctly classify these discoveries as HZplanets. Without such analysis, the understanding of thefrequency of Earth-size planets in the HZ (sometimes re-ferred to as η ⊕ ) will be less secure than we suppose. ACKNOWLEDGEMENTS