The Equilibrium Temperature of Planets in Elliptical Orbits
MM´endez, A. & Rivera-Valent´ın, E. G. (2017)
ApJL , 837, L1
The Equilibrium Temperature of Planets in Elliptical Orbits
Abel M´endez Planetary Habitability Laboratory, University of Puerto Rico at AreciboPO Box 4010, Arecibo, PR 00614 [email protected]
Edgard G. Rivera-Valent´ın
Arecibo Observatory, Universities Space Research AssociationHC 3 Box 53995, Arecibo, PR 00612 [email protected]
ABSTRACT
There exists a positive correlation between orbital eccentricity and the av-erage stellar flux that planets receive from their parent star. Often, though,it is assumed that the average equilibrium temperature would correspondinglyincrease with eccentricity. Here we test this assumption by calculating and com-paring analytic solutions for both the spatial and temporal averages of orbitaldistance, stellar flux, and equilibrium temperature. Our solutions show that theaverage equilibrium temperature of a planet, with a constant albedo, slowly de-creases with eccentricity until converging to a value 90% that of a circular orbit.This might be the case for many types of planets ( e.g. , hot-jupiters); however,the actual equilibrium and surface temperature of planets also depend on orbitalvariations of albedo and greenhouse. Our results also have implications in un-derstanding the climate, habitability and the occurrence of potential Earth-likeplanets. For instance, it helps explain why the limits of the habitable zone forplanets in highly elliptical orbits are wider than expected from the mean fluxapproximation, as shown by climate models.
Subject headings: stars: planetary systems, planets and satellites: fundamen-tal parameters, planetary habitability, equilibrium temperature, habitable zone,eccentricity. Corresponding Author a r X i v : . [ a s t r o - ph . E P ] F e b
1. Introduction
The climate and potential habitability of planets depend on a complex interaction ofmany stellar and planetary properties (Schulze-Makuch et al. 2011). For example, Earthexperiences a small annual change of 4 K in its average global surface temperature mostlydue to its obliquity and distribution of land and ocean areas and not its low eccentricity( i.e., e > . e.g., stability of liquid water between seasonal extremes). This is avery important effect to consider since 75% of all exoplanets with known orbits have orbitaleccentricities larger than Earth .The effects of orbital eccentricity on planetary habitability is a complex problem thathas been explored by one- and three-dimensional climate models ( e.g. , Williams & Pollard2002; Dressing et al. 2010; Dobrovolskis 2013; Wang et al. 2014; Armstrong et al. 2014;Dobrovolskis 2015; Linsenmeier et al. 2015; Bolmont et al. 2016; Way & Georgakarakos2017). The suitability for planets to maintain surface liquid water can be constrained bytheir orbital location as defined by the habitable zone (HZ), the circular region around astar in which liquid water could exist on the surface of a rocky planet (Huang 1959; Dole1964; Kasting et al. 1993). Estimates on the occurrence of Earth-like worlds around stars η ⊕ depend on recognizing those in the HZ (Traub 2015).The conservative inner edge of the HZ is given by the runaway of liquid water ( i.e., surface temperatures above ∼
340 K), corresponding to a rapid increase in stratospheric watervapor content, which leads to loss of an Earth-like ocean inventory within four to five billionyears. The conservative outer edge is given by the maximum greenhouse necessary to keeptemperatures above freezing ( i.e., surface temperature above 273 K). These limits assume anEarth-like planet with N -CO -H O atmospheres and supported by a carbonate-silicate cycle(Kasting et al. 1993). Further developments on the limits of the HZ for main-sequence starsare discussed elsewhere ( e.g. , Pierrehumbert & Gaidos 2011; Joshi & Haberle 2012; Yanget al. 2013; Shields et al. 2013; Kopparapu et al. 2013, 2014, 2016; Kitzmann 2016, 2017).The HZ for circular orbits can be defined either in terms of orbital distance, stellar flux, orequilibrium temperature. Usually, a planetary albedo of 0.3 (similar to Earth or Jupiter) isused for the calculation of the equilibrium temperature as a basis for comparisons.The equilibrium temperature is useful for comparing the temperature regimes of plan-ets, but is not sufficient to draw conclusions on their climate (Leconte et al. 2013). Surface Berkeley Earth: http://berkeleyearth.org/data/ NASA Exoplanet Archive: http://exoplanetarchive.ipac.caltech.edu/ T eq of the planet results from a balance between the inci-dent stellar flux F (cid:63) from the host star and that absorbed by the planet (Selsis et al. 2007),where F (cid:63) and T eq are given from the Stefan-Boltzmann law F (cid:63) = L (cid:63) πd = σR (cid:63) T (cid:63) d , (1) T eq = (cid:20) (1 − A ) F (cid:63) β(cid:15)σ (cid:21) = (cid:20) (1 − A ) L (cid:63) β(cid:15)σπd (cid:21) = T (cid:63) (cid:114) R (cid:63) d (cid:18) − Aβ(cid:15) (cid:19) , (2)where L (cid:63) , R (cid:63) , T (cid:63) are the luminosity, radius, and the effective temperature of the star, respec-tively, d is the distance of the planet from the star, (cid:15) is the broadband thermal emissivity(usually (cid:15) ≈ σ is the Stefan-Boltzmann constant. The planetary albedo or bondalbedo A is the fraction of incident radiation, over all wavelengths, which is scattered by thecombined effect of the surface and atmosphere of the planet. The factor β is the fraction ofthe planet surface that re-radiates the absorbed flux, β = 1 for fast rotators and β ≈ . r = d/a ⊕ , are in astronomicalunits (AU), and the stellar flux, L = L (cid:63) /L (cid:12) , in solar units such that F and T eq become F = Lr , (3) T eq = T o (cid:20) (1 − A ) Lβ(cid:15)r (cid:21) , (4)where T o = [ L (cid:12) / (16 πσa ⊕ )] = 278.5 K ( i.e. , the equilibrium temperature of Earth for zeroalbedo). The derivations presented in this paper use solar and terrestrial units from now onas shown in equations 3 and 4.Surface temperatures, though, depend on the equilibrium temperature and greenhouseeffect of the planets. The normalized greenhouse effect g can be used to connect the equi-librium and surface temperatures as g = GσT s = 1 − (cid:18) T eq T s (cid:19) , (5) 4 –where G is the greenhouse effect or forcing (W/m ), and T s is the surface temperature (Raval& Ramanathan 1989). The normalized greenhouse is very convenient since it summarizes acomplex planetary property in a unitless quantity, as the planetary albedo does. Togetherequations 4 and 5 combine to give the surface temperature of a planet as T s = T o (cid:20) (1 − A ) Lβ(cid:15) (1 − g ) r (cid:21) . (6)The final surface temperature of a planet is controlled by its heat distribution β , emissivity (cid:15) , bond albedo A , and greenhouse g ; all properties that depend on combined surface andatmospheric properties. It is also important to clarify that the average of equation 6 is in factthe effective surface temperature (cid:104) T se (cid:105) and not necessarily the actual surface temperature (cid:104) T s (cid:105) , depending on the spatial and temporal scales considered ( i.e. , global, daily, or annualaverages). For example, the annual global average surface temperature of Mars is (cid:104) T s (cid:105) ≈
202 K, but its effective surface temperature is (cid:104) T se (cid:105) ≈
214 K (Haberle 2013). This is animportant distinction necessary to calculate greenhouse effects.Equations 3 and 4 are more complicated for elliptical orbits. It is often assumed that theaverage equilibrium temperature (cid:104) T eq (cid:105) of a planet in an elliptic orbit can be simply calculatedfrom its average stellar flux (cid:104) F (cid:105) or distance (cid:104) r (cid:105) . Leconte et al. (2013) recognized that thetemporal average of the equilibrium temperature is lower than the equilibrium temperaturecomputed from the temporally averaged flux, although they did not calculate it. Thus, theexpressions (cid:104) T eq (cid:105) (cid:54) = T o (cid:20) (1 − A ) (cid:104) F (cid:105) β(cid:15) (cid:21) , (7) (cid:104) T eq (cid:105) (cid:54) = T o (cid:20) (1 − A ) Lβ(cid:15) (cid:104) r (cid:105) (cid:21) , (8)are incorrect interpretations that have been used extensively in the literature. For example,Kaltenegger & Sasselov (2011) (equation 6 in their paper) and Cowan & Agol (2011) (equa-tion 9 in their paper) suggested a formulation equivalent to equation 7. Also Hinkel & Kane(2013) (table 2 in their paper) used this interpretation to calculate the equilibrium tempera-ture for planets in elliptical orbits. Formulations similar to equations 7 or 8 might introducelarge errors in estimates of the average equilibrium temperature of planets, especially forlarge eccentricities.In this paper, we computed analytical solutions for the average orbital distance, stel-lar flux, and equilibrium temperature of planets in elliptical orbits. Orbital averages werecomputed with respect to spatial ( i.e. , geometric interpretation) and temporal ( i.e. , physicalinterpretation) coordinates for comparison purposes. In § §
3, we show correct formu-lations using both spatial and temporal averages for elliptical orbits, respectively. To our 5 –knowledge, we determined a new analytical solution for the average equilibrium temperatureof planets in elliptical orbits. In §
4, we apply our result to define an effective orbit for theHZ. The implications of our results are discussed in §
5. Our study is motivated by planetsin the HZ, but it is applicable to any planet in elliptical orbit with nearly constant albedo( e.g. , Hot-Jupiters).
2. Spatial Averages for Elliptical Orbits
Spatial averages for orbital distance r , stellar flux F , and equilibrium temperature T eq do not have a practical observational application, but were computed here for comparisonpurposes. They could be calculated with respect to the eccentric anomaly E or true anomaly f (orbital longitude). Equations 3 and 4 were integrated over the true anomaly (making thesubstitution r = a (1 − e ) / (1 + e cos f )) for a full orbit to obtain the corresponding averages, r = 12 π (cid:90) π r d f = a √ − e , (9) F = 12 π (cid:90) π F d f = L (2 + e )2 a (1 − e ) , (10) T eq = 12 π (cid:90) π T eq d f = T o (cid:20) (1 − A ) Lβ(cid:15)a (cid:21) π √ − e E (cid:32)(cid:114) e e (cid:33) (11) ≈ T o (cid:20) (1 − A ) Lβ(cid:15)a (cid:21) (cid:2) e + e + O ( e ) (cid:3) , (12)where E is the complete elliptic integral of the second kind (Weisstein 2016; GSL 2016) and a is the semi-major axis. Equation 9 is sometimes used as the average distance of planets inelliptical orbits. This is a geometric average and the more physically meaningful quantity, thetemporal average (see §
3. Temporal Averages for Elliptical Orbits
Temporal averages for orbital distance (cid:104) r (cid:105) , stellar flux (cid:104) F (cid:105) , and equilibrium temperature (cid:104) T eq (cid:105) were computed. They could be calculated with respect to time t or the mean anomaly 6 – M . Equations 3 and 4 were integrated over time (making the substitutions r = a (1 − e cos E ), M = E − e sin E , and M = (2 π/T ) t ) for a full orbital period T to obtain the correspondingaverages (cid:104) r (cid:105) = 1 T (cid:90) T r d t = a (cid:18) e (cid:19) , (13) (cid:104) F (cid:105) = 1 T (cid:90) T F d t = La √ − e , (14) (cid:104) T eq (cid:105) = 1 T (cid:90) T T eq d t = T o (cid:20) (1 − A ) Lβ(cid:15)a (cid:21) √ eπ E (cid:32)(cid:114) e e (cid:33) (15) ≈ T o (cid:20) (1 − A ) Lβ(cid:15)a (cid:21) (cid:2) − e − e + O ( e ) (cid:3) , (16)where E is the complete elliptic integral of the second kind (Weisstein 2016; GSL 2016)and a is the semi-major axis. Equation 13 is the correct physical average distance of planets,but equation 9 is sometimes used instead. Equation 14 is a well known expression for theaverage stellar flux of elliptical orbits (Williams & Pollard 2002). Equation 15 is a new resultof this study. These equations show that the average distance ( a ≤ (cid:104) r (cid:105) < a ) and stellar flux( F | e =0 ≤ (cid:104) F (cid:105) < ∞ ) increases with eccentricity while the average equilibrium temperature( T eq | e =0 ≤ (cid:104) T eq (cid:105) < √ π T eq | e =0 ) slowly decreases until converging to ∼
90% of the equilibriumtemperature for circular orbits (figure 2).
4. The Habitable Zone
The HZ can be defined either in terms of distance, stellar flux, or equilibrium tem-perature for circular orbits, using equations 3 and 4 to convert from one another. This isnot trivial for elliptical orbits because these quantities diverge differently with eccentricity.The usual approach is to compare the average stellar flux of the planet in an elliptical orbit(equation 14) with the stellar flux limits of the HZ, the so called ‘mean flux approximation’(Bolmont et al. 2016). Equivalently, Barnes et al. (2008) suggested an HZ definition for el-liptical orbits based on the mean flux approximation, but in terms of orbital distance. Theireccentric habitable zone (EHZ) is the range of orbits for which a planet receives as muchflux over one orbit as a planet on a circular orbit in the HZ. This consists of comparing thesemi-major axis of the planet with the HZ limits for circular orbits scaled by (1 − e ) − / .Here we prefer to avoid redefining the limits of the HZ, which adds a complication formultiplanetary systems ( i.e. , each planet having a different HZ). Instead, we suggest to define 7 –some effective distance for planets in elliptical orbits, not necessarily equal to their averageorbital distance as given by equations 9 or 13. Based on the mean flux approximation, theeffective flux distance r F , the equivalent circular orbit with the same average stellar flux (cid:104) F (cid:105) as the elliptic orbit is given by r F = a (1 − e ) / . (17)Alternatively, using equation 15 we define a new effective thermal distance r T , the equivalentcircular orbit with the same average equilibrium temperature (cid:104) T eq (cid:105) as the elliptic orbit, givenby r T = a (cid:34) √ eπ E (cid:32)(cid:114) e e (cid:33)(cid:35) − (18) ≈ a (cid:2) e + e + O ( e ) (cid:3) . (19)The effective distance based on the mean flux approximation (equation 17) and our mean thermal approximation (equation 18) are not equivalent, one decreasing while theother increasing with eccentricity, respectively. The difference between these two solutionsare less than 5% for eccentricities below 0.7, so our solution is a small correction that ismore significant for high eccentric orbits. Nevertheless, we argue that the mean thermalapproximation provides a better characterization of the limits of the HZ than the mean fluxapproximation for elliptical orbits since the HZ is constrained by the thermal regime of theplanet. Therefore, we suggest that equation 18 is better to determine whether planets inelliptical orbits are within the HZ limits .
5. Discussion
Our temporal solution shows that the average equilibrium temperature of planets slightlydecreases with eccentricity until converging to a value about 10% less than for circular orbits.This result might seem contradictory because the average stellar flux strongly increases witheccentricity; however, the average distance also increases with increasing eccentricity. Thenet effect is a small decrease in the equilibrium temperature with eccentricity as long as theplanetary albedo stays constant throughout the orbit. That might be the case for bodieswithout atmospheres; however, the bond albedo of planets with condensables ( e.g. , oceans) inelliptic orbits might change significantly from the hot periastron ( e.g. , due to the absorptionfrom water vapor) to the cold apastron ( e.g. , due to the reflectivity of ice). VPL’s HZ Calculator: http://depts.washington.edu/naivpl/content/hz-calculator gsl sf ellint Ecomp (GSL 2016) and the Mathematica function
EllipticE (note that theMathematica implementation of this function requires the square of the argument). Our re-sults were also validated with numerical solutions. The equations 3 and 4 were independentlynumerically integrated for a full orbit in IDL using the
RK4 procedure, which is a fourth-orderRunge-Kutta method. Both the analytic and numerical solutions agree as shown in figures1 and 2.As discussed in the introduction §
1, there are different and incorrect interpretations inthe literature of the equilibrium temperature of planets in elliptical orbits (equations 7 and8). They might introduce large errors especially for large eccentricities (figure 3). The mostcommon interpretation is one based on the average stellar flux (equation 7), also knownas the ‘mean flux approximation’. This interpretation gives errors larger than 5% for theaverage equilibrium temperature of planets with eccentricities higher than 0.48, and over10% for e > .
64. In fact, the interpretation based on the average distance (equation 8) ismore consistent with our result ( i.e. , also decreases with eccentricity). In any case, averagevalues should always be between the corresponding minimum orbital distance at periastron r p = a (1 − e ) and maximum at apastron r a = a (1 + e ).A similar, but incorrect analytic solution to our average equilibrium temperature fromequation 15 was proposed by Brasser et al. (2014) (equation 13 in their paper). Theirsolution suggests that the equilibrium temperature increases instead with eccentricity, but itis not self-consistent and for eccentricities below 0.8 gives values larger than the equilibriumtemperature at periastron ( i.e. , the maximum value). For example, the solution of Brasser etal. (2014) gives 230 K for the average equilibrium temperature for Mars based on its orbitalparameters and assuming a 0.25 bond albedo. This average value is outside the possiblerange of 200 K to 220 K between apastron and periastron, respectively. Our solution gives210 K, which is consistent with the NASA Ames Mars General Circulation Model (Haberle2013). Our calculated values are also in agreement from those in the literature for otherSolar System objects (table 1).Unfortunately, previous studies on the climate of planets in elliptical orbits did not in-clude equilibrium temperature estimates to directly compare with our results. Nevertheless,our results might explain previous conflicting studies, using surface temperature as a proxy.There are different ways to study the effect of eccentricity on the climate of planets ( e.g. , atconstant semi-major axis, stellar flux, etc.). Bolmont et al. (2016) studied the effect of ec-centricity on the climate of tidally-locked ocean planets using a Global Climate Model LMDz 9 –under constant stellar flux. They showed that the higher the eccentricity of the planet andthe higher the luminosity of the star, the less reliable is the mean flux approximation. Asimilar conclusion was previously obtained by Dressing et al. (2010) for Earth-like planetsin high eccentric orbits with a one-dimensional energy balance climate model (EBM). Ourequilibrium temperature estimates positively correlate with the surface temperature resultsof Bolmont et al. (2016), which show that temperature decreases with eccentricity, contraryto the mean flux approximation (table 2). In this case, our results characterize the decreasingthermal response with eccentricity and corresponding increase in effective orbital distance.Our results are more relevant for planets in very eccentric orbits e > . e.g. , hot-jupiters). In any case, they seem better to describe the thermal regimeof planets than the mean flux approximation (figure 3). Temperatures could increase witheccentricity as also shown by climate models (Williams & Pollard 2002; Dressing et al.2010). Planets with condensable species ( e.g. , Earth-like) could experience strong variationson their albedo and heat redistribution during their elliptical orbit. Kasting et al. (1993)showed that the planetary albedo not only depends on the surface and atmospheric propertiesof a planet, but it is also affected by both the stellar flux and the spectrum of the star. Forexample, it could go below 0.2 for Earth-like planets receiving over 20% the stellar flux thatEarth receives around a Sun-like or cooler star. Heat redistribution is also more efficient inlonger period tidally-locked planets (Bolmont et al. 2016). Future studies will consider theseimportant dynamical corrections.
6. Conclusion
Here we determined analytic solutions of the spatial and temporal averages of orbitaldistance, stellar flux, and equilibrium temperature for planets in elliptical orbits and constantalbedo. The solutions were validated (see §
5) by reproducing known solutions with thesame approach, obtaining equal results with numerical solutions, comparing with some SolarSystem bodies, and producing results consistent with previous climate models for planets inelliptic orbits. In particular, we determined that:1. The average equilibrium temperature (cid:104) T eq (cid:105) of planets in elliptical orbits slowly de-creases with eccentricity e until a converging value ∼
90% ( √ π ) the equilibrium tem-perature for circular orbits, assuming a constant albedo A and heat redistribution β .The temperature is given by (cid:104) T eq (cid:105) = T o (cid:20) (1 − A ) Lβ(cid:15)a (cid:21) √ eπ E (cid:32)(cid:114) e e (cid:33) , (20) 10 –where T o = 278.5 K, L is the star luminosity (solar units), a is the semi-major axis(AU), (cid:15) is the emissivity, and E is the complete elliptic integral of the second kind(Weisstein 2016; GSL 2016).2. The potential stability of surface liquid water on rocky planets in elliptical orbits isbetter characterized by a mean thermal approximation rather than the mean fluxapproximation. Using observational data, a planet is inside the HZ if their effectivethermal distance r T , given by r T = a (cid:34) √ eπ E (cid:32)(cid:114) e e (cid:33)(cid:35) − , (21)is within the HZ limits for circular orbits as defined by Kasting et al. (1993). In general,planets move farther outside the HZ ( i.e. , become colder) with increasing eccentricityif all other conditions stay constant; however, orbital variations of albedo, greenhouse,and heat redistribution might accentuate or invert this trend.3. All climate models of planets should include at least global calculations of albedo A ,normalized greenhouse g , equilibrium T eq , and surface temperatures T s as a function oforbital time, not just orbital longitude. The distinction between the use of the surfacetemperature or the effective surface temperature should also be clear (Haberle 2013).These global parameters help to compare different climate models and to translatethem to simple parametric functions that are necessary to determine whether planetsare in the HZ from observational data.The results of this paper might be used to reevaluate the effective location in the HZ ofmany exoplanets with known eccentricity (using equation 21), especially those small enoughto be considered potentially habitable. For example, the planet Proxima Cen b is probablyin a circular orbit (Anglada-Escud´e et al. 2016), but the outer two planets in the red-dwarfstar Wolf 1061 are in eccentric orbits crossing the HZ (Wright et al. 2016; Kane et al. 2016).Also, many giant planets with elliptical orbits that cross the HZ might support habitableexomoons (Forgan & Kipping 2013; Hinkel & Kane 2013). Results for individual exoplanetsare available and updated in the PHL’s Exoplanet Orbital Catalog .This work was supported by the Planetary Habitability Laboratory (PHL) and theCenter for Research and Creative Endeavors (CIC) of the University of Puerto Rico at Exoplanet Orbital Catalog: http://phl.upr.edu/projects/habitable-exoplanets-catalog/catalog
11 –Arecibo (UPR Arecibo). Part of this study was originally presented in a poster at the2014 STScI Spring Symposium:
Habitable Worlds Across Space And Time (April 28 - May1, 2014). This research has made use of the NASA Exoplanet Archive, which is operatedby the California Institute of Technology, under contract with the National Aeronautics andSpace Administration under the Exoplanet Exploration Program. We gratefully acknowledgethe anonymous referee, whose comments improved the quality of the paper.
REFERENCES
This preprint was prepared with the AAS L A TEX macros v5.2.
14 –Table 1: Calculated temporal averages of orbital distance (cid:104) r (cid:105) , stellar flux (cid:104) F (cid:105) , andequilibrium temperature (cid:104) T eq (cid:105) for some Solar System bodies.Literature Values a Temporal Averages b Name a (AU) e A T eq (K) (cid:104) r (cid:105) (AU) (cid:104) F (cid:105) (cid:104) T eq (cid:105) (K)Mercury 0.38709927 0.20563593 0.068 439.6 0.395 6.819 439Venus 0.72333566 0.00677672 0.77 226.6 0.723 1.911 227Earth 1.00000261 0.01671123 0.306 254.0 1.000 1.000 254Moon 1.00000261 0.01671123 0.11 270.4 1.000 1.000 271Mars 1.52371034 0.0933941 0.250 209.8 1.530 0.433 210Pluto 39.48211675 0.2488273 0.4 37.5 40.704 0.001 39 a Literature values are from the orbital elements data of NASA Solar System Dynamic:http://ssd.jpl.nasa.gov/?planets, and albedo and equilibrium temperature data of the NASAPlanetary Fact Sheets: http://nssdc.gsfc.nasa.gov/planetary/planetfact.html. b Average values calculated from the literature values and the temporal average equations13, 14, and 15, respectively.
15 –Table 2: Estimated average equilibrium temperature (cid:104) T eq (cid:105) and effective thermal distance r T for a modeledocean planet with increasing eccentricity, but under aconstant average stellar flux equal to Earth.Planet Model a Estimated Values e a (AU) A (cid:104) T s (cid:105) (K) (cid:104) T eq (cid:105) b (K) r T c (AU)0.0 1.000 0.250 267 259 1.0000.2 1.003 0.295 263 253 1.0080.4 1.045 0.350 255 242 1.067 a Global Climate Model LMDz for a tidally-locked oceanplanet in an elliptical orbit around a Sun-like star with L =1 and (cid:104) F (cid:105) = 1 (Bolmont et al. 2016). b Average equilibrium temperature from equation 15 with (cid:15) = 1 and β = 1. c Effective thermal distance from equation 18.
16 –Fig. 1.— Comparison of spatial averages of orbital distance, stellar flux, and equilibriumtemperature as a function of eccentricity (normalized with respect to the circular orbitsvalues). Both analytic (color lines) and numerical solutions (black lines within the colorlines) are shown. 17 –Fig. 2.— Comparison of temporal averages of orbital distance, stellar flux, and equilibriumtemperature as a function of eccentricity (normalized with respect to the circular orbitsvalues). Both analytic (color lines) and numerical solutions (black lines within the colorlines) are shown. 18 –Fig. 3.— Comparison of different averages for orbital equilibrium temperature as a functionof eccentricity (normalized with respect to the circular orbits values). Averages based onstellar flux (red line) ( i.e.i.e.