Tutorial models of the climate and habitability of Proxima Centauri b: a thin atmosphere is sufficient to distribute heat given low stellar flux
DDraft version July 31, 2018
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TUTORIAL MODELS OF THE CLIMATE AND HABITABILITY OF PROXIMA CENTAURI B:A THIN ATMOSPHERE IS SUFFICIENT TO DISTRIBUTE HEAT GIVEN LOW STELLAR FLUX
Colin Goldblatt
School of Earth and Ocean Sciences,University of Victoria,PO Box 1700 STN CSC, Victoria, British Columbia, V8W 2Y2, Canada.
ABSTRACTProxima Centauri b, an Earth-size planet in the habitable zone of our nearest stellar neighbour,has just been discovered. A theoretical framework of synchronously rotating planets, in which therisk of a runaway greenhouse on the sunlight side and atmospheric collapse on the reverse side aremutually ameliorated via heat transport is discussed. This is developed via simple (tutorial) modelsof the climate. These show that lower incident stellar flux means that less heat transport, so lessatmospheric mass, is required. The incident stellar flux at Proxima Centauri b is indeed low, whichmay help enhance habitability if it has suffered some atmospheric loss or began with a low volatileinventory.
Keywords: astrobiology — radiative transfer —planets and satellites: atmospheres — stars: low-mass— Earth INTRODUCTIONProxima b is a recently announced exoplanet (Anglada-Escud´e et al. 2016), presented as the first Earth-size planetto be found in the habitable zone. Proxima Centauri is the nearest star to Earth, 4.2 light years away. It is of M5.5V,with an effective temperature of 3050K and radius 0 . R Sun . Proxima b has m sin i = 1 . m Earth . It is on a 11.2 dayorbit, so receives 65% as much energy from its star as Earth does, and is thus in the “habitable zone” (Anglada-Escud´eet al. 2016). The importance of this discovery needs no further introduction.Many aspects of the habitability of Proxima b have been addressed by Ribas et al. (2016) and Turbet et al. (2016)(posted to the ArXiv contemporaneously with publication of the discovery) the latter focussing on climate. Turbetet al. (2016) use a General Circulation Model (GCM) to model potential climates. In this contribution, I take thecontrasting but complementary approach of using simple, tutorial, climate models to examine the climate of Proximab. This allows me to focus on intuition building for fundamental aspects of the climate system and to elucidate some(perhaps) important qualitative aspects relating to its habitability.I begin this paper with a discussion of the theory applicable to synchronously rotating planets around M-stars. Someof this is review, but I will also introduce some new theoretical framework, where I consider the risk of a runawaygreenhouse on the sunlight side and the consequences of low stellar flux.Thereafter, I will use a simple model to show that, given the low stellar flux, much less redistribution of energy byatmospheric motion is required to maintain habitable conditions. Given that any atmosphere may be at risk of lossdue to high XUV flux, this may substantially increase the chances that Proxima b is indeed habitable. ON THE HABITABILITY OF SYNCHRONOUSLY ROTATING PLANETS AROUND M-STARSTo have surface liquid water, a planet should receive neither too much nor too little energy from the star—so forms thebasis of the “circumstellar habitable zone”. Translating this into quantitative estimates of energy fluxes or circumstellardistances, boundaries outside which the existence of surface liquid water may be excluded, is thus a canonical problemin planetary climatology. The inner edge may be described by either a runaway water vapor greenhouse, which wouldbake the planet, or by massive loss of H (from water) to space, which would dessicate it. Venus, for example, probablyexperienced a runaway greenhouse followed by water loss. At the outer edge the fundamental limit is condensation ofCO , assumed to be the dominant greenhouse gas other than water. In the atmosphere, CO clouds would increasealbedo (a positive feedback on cooling). If condensate were stable at the surface, atmospheric collapse would ensue. a r X i v : . [ a s t r o - ph . E P ] A ug Given a low stellar flux and cool temperatures, the onset of this would likely be hastened by ice-albedo feedbackcausing global glaciation (e.g. Abe 1993; Kasting et al. 1993).The runaway greenhouse is described further, as some theoretical development will follow. The amount of water inthe atmosphere depends on temperature, but water is a greenhouse gas, so there is a positive feedback. With a deepcolumn of water, the atmosphere becomes optically thick across the thermal infrared region. Only the atmosphere,not the surface, may then radiate to space and the temperature of the emission level asymptotes to a constant set bythe thermodynamic and radiative properties of water. Consequently, there is a maximum amount of radiation that amoist atmosphere can emit (282 W m − , Goldblatt et al. 2013). If more sunlight than this is absorbed by an isolatedcolumn, or by the planet on average, then runaway surface warming occurs (e.g. Simpson 1927; Nakajima et al. 1992;Goldblatt & Watson 2012). Earth’s tropics absorb more sunlight than the radiation limit. A runaway greenhouse isavoided for three reasons: (1) Heat is exported to the extratropics (2) Columns of unsaturated air allow more radiationout to space (Pierrehumbert 1995) (3) deep convention initiates at around the surface temperature which correspondsto the radiation limit, increasing cloudiness and albedo.Synchronously rotating planets were first though of as non-habitable as it was inferred that the atmosphere wouldcollapse on the dark side and trap all the volatiles there. However, pioneering work with an energy balance model(Haberle et al. 1996) and later a general circulation models (GCM) (Joshi et al. 1997) showed that energy redistributionby atmospheric circulation could prevent atmospheric collapse. Atmospheric heat transport is, to first approximation,proportional to surface pressure (atmospheric mass). This has subsequently been elaborated with further GCM studies,usually focusing on planets receiving the same stellar flux as the modern Earth (Joshi 2003; Edson et al. 2011; Yanget al. 2013; Turbet et al. 2016). An atmosphere with a surface pressure of 100 hPa is though to be needed.The flip-side of the climate coin is that the sunlit hemisphere would be susceptible to a runaway greenhouse, butthis has received less theoretical attention. The Haberle et al. (1996) energy balance model had a fixed greenhouseeffect, independent of temperature, so necessarily avoided a runaway greenhouse: this likely helped frame the problemin terms of dark-side collapse only. Nonetheless, heat transport that prevents collapse would play the dominant role inavoiding a runaway greenhouse. Further, some of the most capable GCMs have shown near total cloud cover aroundthe substellar point (Yang et al. 2013).Atmospheric thickness also pertains to habitability via Rayleigh scattering, pressure broadening and dilution ofwater. More atmosphere, so more Rayleigh scattering, cools a planet by increasing albedo at the blue end of the solarspectrum (Rayleigh scattering cross section is proportional to the reciprocal forth power of wavelength). Conversely,more atmosphere pressure broadens the absorption lines of greenhouse gases and warms the planet. The latter effectdominates for pressures of up to a few bars (Goldblatt et al. 2009). Lastly, more atmosphere will dilute water vapour,making hydrogen escape less likely.The overall stellar flux received by Proxima b, 65% of the modern solar constant, is lower than Earth received evenat the start of the Sun’s main sequence. The canonical problem of deep palaeoclimate is the Faint Young Sun Paradox(Sagan & Mullen 1972): despite low solar constant, how did Earth avoid ice-albedo feedback and persistent globalglaciation? (Geologic evidence is that early Earth was ice free more often than not (e.g. Nisbet 1987), though there isno agreement on the climate forcings that allowed for that). But would Proxima b not be pan-glacial?The spectrum of M-stars is shifted far to the red relative to the Sun, affecting climate and habitability. Snow andice are near white around the Wein peak of sunlight, but dark at the Wein peak of M-star emission, so the ice-albedofeedback disappears (Joshi & Haberle 2012) (also, on a synchronous rotator, snow would fall on the unilluminatedside). Little blue light is received, so Rayleigh scattering is of little importance for pressures of up to a few bar. Bothof these effects bode well for weakly illuminated Proxima b: cooling mechanisms which endanger a planet near theouter edge of the habitable zone do not apply.However, Proxima Centurai is a flare star, with a ratio of XUV flux to bolometric luminosity ∼
60 times higher thanthe Sun (Ribas et al. 2016). Whereas the overall energy flux the planet receives determines climate the XUV flux willheat the thermosphere and drive atmospheric escape. The risk of substantive atmospheric loss is a real (see extensivediscussion in Ribas et al. 2016, , and references therein).Also pertaining to atmospheric mass, the initial volatile inventory of M-star habitable zone planets may be low(Lissauer 2007, though again see discussion in Ribas et al. (2016)).The climate and habitability problem for Proxima b may be summarised. The planet may well be synchronouslyrotating, so is exposed to twin perils of a runaway greenhouse of the sunlight side and atmospheric collapse on thedark side. These can be avoided by advective heat transport from front to back, which depends on atmospheric mass.However, Proxima b may have had a low initial volatile inventory or suffered atmospheric loss. Overall stellar flux islow. The question may be posed as: how little atmosphere would be sufficient to transfer enough energy—and howdoes this depend on incident stellar flux? CLIMATE WITH NO GREENHOUSE EFFECT OR HEAT TRANSPORTThe simplest case is to consider a planet with no atmosphere (or an atmosphere with no absorbers, and no heattransport), which is an elementary problem in climate modelling. This provides insight into the climate forcing ofM-star planets under different insolation, and additionally provides a reference case to compare subsequent modelswith an atmosphere. 3.1.
Model description
The surface of the planet is in energy balance, with absorbed stellar radiation and a geothermal heat flux balancedby thermal emission, which is taken to be a black body. Thus: F s + F g = σT , (1)with F s = S (1 − α ) cos φ for φ < π φ ≥ π (2)where F s is the stellar flux incident per unit surface area, solar (or stellar) constant S is the flux at the top of theatmosphere at the substellar point (modern Earth, S = 1368 W m − ), α is the bond albedo, φ is the angular separationfrom the substellar point, σ = 5 . × − W m − K − is Stefan’s constant and T eff is the effective temperature.Geothermal heat flux F g may be set via comparative planetology; F g, Earth = 0 .
08 W m − , dominated by radiogenicheat production in the mantle and internal cooling, F g, Io = 2 W m − , dominated by tidal heating. A conservativevalue of F g = 0 . − is used.Note that surface temperature for the airless case is equivalent to effective temperature T eff . The meteorologicaldefinition is required; effective temperature is the emission temperature required for a column to be in energy balancewith F s , noting that planetary albedo must be considered .3.2. Results
Incident sunlight and surface temperature for the airless planet are shown in Figure 1 for the stellar flux at Proximab, S = 0 . S and the reference value S = S . Representative albedos used are 0.15 (bare rock), 0.3 (patch waterclouds, as modern Earth) and 0.6 (complete water cloud coverage).Incident stellar flux exceeds the runaway greenhouse threshold over much of the sunlit hemisphere. This illustratesone bound on habitable climate for synchronously rotating planets; with surface water, and were the light and darkhemispheres thermally isolated, the illuminated hemisphere would heat beyond the stability of liquid water. Thedifference between S = 0 . S and S = S is immediately apparent. With modern Earth insolation, most of the sunlithemisphere would be unable to radiate the absorbed sunlight locally, so prodigious heat export to the dark hemispherewould be required. With the stellar flux at Proxima b, the excess sunlight is much smaller. The implication is clear:given that stellar flux at Proxima b is low, less energy redistribution is required and, by inference, less atmosphere isneeded.Surface temperature for the airless case is instructive too: there is a band which is both sunlight and temperate.One could imagine locally stable soil moisture and life within the soil. CLIMATE WITH AN ATMOSPHERE4.1.
Model description
I modify the energy balance model of Haberle et al. (1996) by trivial addition of a geothermal heat flux andsubstantive addition of a temperature dependent greenhouse effect, corresponding to the water vapour feedback onclimate and ultimately a runaway greenhouse. The original model solves for energy balance at the surface and in asingle-layer atmosphere on the both the light and dark sides of the planet. The atmosphere is transparent to sunlightand grey to thermal radiation. The two sides are connected by an atmospheric heat flux (see Figure 1 of Haberle S t e ll a r f l u x ( W m - ) , = 0.15 , = 0.30 , = 0.60 Angluar separtation from substellar point E ff e c t i v e t e m pe r a t ue ( K ) Figure 1 . Incident stellar flux (top) and effective temperature / surface temperature for airless case (bottom). Solid lines for S = 0 . S , dashed line for S = S . Horizontal lines are reference fluxes and temperatures: solid red is the radiation limit forrunaway greenhouses, dotted green is the diurnal average insolation at Earth’s equator, blue is the triple point temperature forwater. et al. 1996). The equations (not written out in the original paper) are:Light , surface : 0 = S (1 − α ) + εσT la − σT ls (3)Light , atmos . : 0 = − εσT la + εσT ls − A ( T la − T da ) (4)Dark , surface : 0 = εσT da − εσT ds (5)Dark , atmos . : 0 = − εσT da + εσT ds + A ( T la − T da ) (6) T is temperature, with subscripts l for light, d for dark, s for surface and a for atmosphere. Emissivity 0 < ε < A = p s c p gt adv (7)with advective timescale t adv = LU . (8)Using modern Earth values for all variable in A (surface pressure p s = 10 Pa, specific heat capacity of air c p =1000 J kg − s − , gravity g = 9 . − , length scale L = π r , planet radius r = 6 . × m and windspeed U = 10 m s − )gives A = 10 W m − K − .My modified equations are:Light , surface : 0 = S (1 − α ) + ε l σT la − σT ls + F g (9)Light , atmos . : 0 = − ε l σ ( T la + T lr ) + ε l σT ls − A ( T la − T da ) (10)Dark , surface : 0 = ε d σT da − σT ds + F g (11)Dark , atmos . : 0 = − ε d σ ( T da + T dr ) + ε d σT ds + A ( T la − T da ) (12)where F g is a geothermal heat flux and T r is the temperature of atmospheric radiation to space (defined below).Emissivity is no longer a free parameter, but set as ε = 1 − e − τ (13)where optical depth comprises contributions from water vapour and other greenhouse gases τ = τ H2O + τ dry . (14)I take τ dry = 0 .
5, somewhat similar to Earth’s atmosphere. Optical depth from water is directly proportional to arepresentative saturation vapor pressure; τ H2O = kp sat (15)with absorption coefficient k = 0 . p sat = p sat , exp (cid:18) L RT (cid:19) (16)with p sat , = 1 . × Pa, L = 43655 J mol − and R = 8 .
14 J mol − K − . For the light side, T ls is appropriate forcalculation of p sat , for this will represent the column water vapour. For the night side, however, there is commonlya temperature inversion (the atmosphere is warmer than the surface), so the maximum of p sat ( T ds ) or 0 . p sat ( T da ) isused (the factor of 0.9 is based on an assumption that the inversion level is at nine-tenths of the surface pressure).Specification of atmospheric radiative temperature, T r , is key to representing the runaway greenhouse. When theatmosphere is not optically thick from water, the radiative temperature will be the bulk atmospheric temperature T a .When the atmosphere is optically thick with water, radiation is emitted only from the level where τ H2O ∼ T r = (1 − ε H2O ) T a + ε H2O T limit , for T a < T x (1 − ε H2O ) T a + ε H2O T limit + 0 . T a − T x ) , for T a ≥ T x (17)where T limit = 265 K is the blackbody temperature corresponding to a limiting flux of 282 W m − (Goldblatt et al.2013). This parameterization, with parameters tuned as above, well represents modern radiative transfer model output(Goldblatt et al. 2013).At very high surface temperatures ( T > µ m water vapor window, allowing a hot dry equilibrium temperature to be found (Goldblatt 2015). Thisis difficult to represent well with a single layer atmosphere; it is crudely parametrized by setting some maximumatmospheric temperature T x = 600 K, above which the radiative temperature again increases.With k = 0, my modified model simplifies to the original (Haberle et al. 1996) model.4.2. Results
Surface and atmosphere temperatures for light and dark side of a synchronously rotating planet, for S = S and S = 0 . S , are shown in Figure 2. These serve as the basis for assessing how much atmosphere is required forhabitability via three criteria: (1) moderate surface temperatures on the light side, for mesophile life. (2) water vapournot the dominant atmospheric constituent, to avoid rapid water loss. (3) Sufficiently warm temperatures on the nightside to avoid atmospheric collapse.Moderate temperatures are sketched with a representative temperature range of 273 < T s <
373 K (grey area).Avoidance of a moist atmosphere is sketched with as p s > p sat ( T ), equivalently water vapour mixing ratio x H2O < . S = 0 . S , the planet is in this region with A an order of magnitude lower than if S = S . Recallthat A ∝ p s ; the implication is that a reduced solar constant will allow a habitable day side with an order of magnitudeless atmosphere.Atmospheric collapse can be assessed by comparing night side surface temperatures (which are colder than theatmosphere) to the thermodynamic properties of popular atmospheric gases. For S = 0 . S , the low A end of dayside habitability corresponds to T ds = 138 K. Hydrogen and nitrogen have critical point temperatures of 33 K and126 K, so these would have no condensed phase and are absolutely safe as bulk atmospheric gases. The triple andcritical points of methane are 91 K and 191 K. At 138 K, p sat,CH = 5 . × Pa, so a thick methane atmosphereis reasonable. Comparative planetology supports this; methane is an important (though condensible) atmosphericconstituent on Titan where p s = 90 K. The triple point for carbon dioxide is 217 K, so the situation here is moredelicate. The saturation vapour pressure is plotted as a function of night side surface temperature (Figure 3). At138 K, p sat,CO = 135 Pa, several times higher than modern Earth pCO of 40 Pa. The corresponding CO mixingratio would be 3 – 20%: carbon dioxide would be stable as a minor atmospheric constituent. The triple point of water -5 -4 -3 -2 -1 Advection parameter, A (W m -2 K -1 ) T e m pe r a t u r e , T ( K ) T light,surf T light,atmos T dark,surf T dark,atmos -1 Surface pressure, p s (Pa) Figure 2 . Model temperatures (solid lines for S = 0 . S , dashed line for S = S ). Light grey shaded areas are modest surfacetemperatures. Green shaded area corresponds to water vapour mixing ratios less than 0.5. Vertical grey line is the referenceadvection parameter, A = W m − K − . Note that pressure scales linearly with A . -5 -4 -3 -2 -1 Advection parameter, A (W m -2 K -1 ) S a t u r a t i on V apo r P r e ss u r e , p s a t ( P a ) -1 Surface pressure, p s (Pa) Figure 3 . Saturation vapor pressure of CO corresponding to T ds (solid line S = 0 . S , dashed line S = S ). pCO > p sat,CO2 would cause deposition and low partial pressure implies atmospheric collapse. Grey solid line indicates a pure CO atmosphere(using p s implied by A ) and dotted grey lines mixing ratios of 0.1 to 10 − . The green dotted line is the partial pressure of CO on Earth in 2016. Note that pressure scales linearly with A . is 273 K; ice formation on the dark side is inevitable. With a very low water inventory, the risk of cold trapping allthe water exists. With a medium-size inventory, glaciers would flow to the terminator, melt and evaporate. With alot of water, there would be a global ocean like Earth’s, and water distribution would not be a problem. DISCUSSIONSimple climate models are some of the best teachers. Here, the the theoretical framework is risk of runawaygreenhouse on the sunlight hemisphere and collapse on the reverse. Models clearly show the extent of the potentialrunaway greenhouse problem depending on incident stellar flux, and the amount of atmospheric heat transport requiredto offset this decreasing as incident stellar flux decreases. For Proxima b, low stellar flux means that the requirementson the atmosphere are less stringent. To achieve the goal of a temperate climate, an atmosphere of mostly nitrogenwith minor CO seems the best choice.So far, the atmospheric composition has been set as a contrivance of the modeller. This is, of course, not so: realterrestrial planet atmospheres are controlled by a mix of geological and biological processes. Indeed, the proposalof life detection by atmospheric analysis (Lovelock 1965) arose from the observation of biological control on Earth’satmosphere, indeed the proposer of this saw Earth’s atmosphere as a biological contrivance (Lovelock 1972; Lovelocket al. 1974). Contingency in atmospheric evolution means that there is a paradox: habitability and inhabitance areinseparable (Goldblatt 2016).Theory on the control of Proxima b’s atmospheric composition must await a later contribution; differences relativeto solar system planets are expected. For example, CO will likely condense to some extent (Mars-type control), but inif there is a water ocean then carbonate deposition would be expected (Earth-style control)—what would the dynamicsof these contributions be? The most wonderful thing about Proxima b is, of course, that we will likely be able tocharacterize his atmosphere—its presence or absence, its temperature and composition—in my lifetime, and therebyprove all our theories wrong.Financial support came from an NSERC discovery grant.REFERENCESwill likely condense to some extent (Mars-type control), but inif there is a water ocean then carbonate deposition would be expected (Earth-style control)—what would the dynamicsof these contributions be? The most wonderful thing about Proxima b is, of course, that we will likely be able tocharacterize his atmosphere—its presence or absence, its temperature and composition—in my lifetime, and therebyprove all our theories wrong.Financial support came from an NSERC discovery grant.REFERENCES