A Note on Planet Size and Cooling Rate
AA Note on Planet Size and Cooling Rate
Johnny Seales and Adrian Lenardic
Abstract
Variation in the balance of forces that drive and resist tectonic plate motions allows small terres-trial planet to cooler slower than larger ones. Given that interior cooling affects surface environment,through volcanic/geologic activity, this indicates that small planets should not be down-weighted inthe search for life beyond Earth.
Keywords: planetary cooling, habitability, exoplanets
The idea that small planets cool faster than larger ones stems from an area to volume argument. For aplanet of radius, R p , heat flow scales with surface area while heat produced within its interior scales withvolume. Taking the ratio, cooling scales as 1 /R p . For scaling between planets, this assumes that planetshave the same internal heat source concentrations, valid for planets of similar chemical composition. Italso assumes equivalent surface heat flux for a given internal temperature.It has been noted that the relationship between heat flux and internal temperature depends on thetectonic mode of a planet. Planets with plate tectonics will have a different cooling efficiency than singleplate planets [Stevenson, 2003]. Although that potential has been acknowledged, there is still the thoughtthat surface to volume arguments remain valid for planets with the same tectonic modes - in particular,plate tectonics. This assumption has not been called out to date, and it too is invalid.Plate tectonics is a kinematic theory [McKenzie and Parker, 1967, Morgan, 1991, Le Pichon, 1968].Connecting plate tectonics to interior cooling is a dynamic problem and a dynamic theory of platetectonics is not agreed upon at present. It is an active research problem with no deficiency of hypotheses.Different assumptions regarding the balance between the forces driving and resisting plate motions leadto different scaling relationships between heat flux and internal temperature [Tozer, 1972, Christensen,1985, Conrad and Hager, 1999b, Crowley and O’Connell, 2012]. Such relationships have been used in1 a r X i v : . [ a s t r o - ph . E P ] F e b odels that track the evolution of the Earth’s internal temperature over time (thermal history models).Proponents of different hypotheses regarding the cooling efficiency of plate tectonics have argued thattheir models can match thermal history constraints. However, using an agnostic approach that accountedfor model and observational uncertainties, Seales et al. [2019], Seales and Lenardic [2020] showed thatmultiple hypotheses remain viable. More critically for planetary studies, no observational data demandsequivalency between the cooling efficiency of planets within a tectonic mode akin to Earth and the Earthitself. In this note we explore how variances in plate tectonic cooling efficiencies couple with variableplanetary size to determine cooling rates. Plate tectonics, on Earth and potentially other terrestrial planets, is a surface manifestation of thermalconvection within a planet’s rocky interior layer (i.e., its mantle). Thermal history models predict mantlecooling trajectories based on how internal heat sources ( H ) and convective heat flux ( q conv ) evolve withtime. A large class of such models use a global energy balance that determines the spherically averagedtemperature of the mantle, T , according to ρc p V ˙ T = H − Aq conv (1)where rho , c p , and ˙ T are the density, heat capacitance and the time derivative of mantle temperature.The volume of the convecting mantle is V = π (cid:0) R p − R c (cid:1) and its surface area is A = 4 πR p , where R c is the radius of the iron core of a terrestrial planet/moon. Radiogenic decay produces heat within themantle according to H = V H o exp ( − λt ) (2)where H o is a scaling constant representing the heat produced per unit time per unit volume, λ is thedecay constant, and t is time. The heat flux through the surface depends on convective vigor in themantle. It is typically parameterized using a scaling equation given by Schubert et al. [1979, 1980]: N u = aRa β (3)where N u is the Nusselt number (a measure of surface heat flux), Ra is the Rayleigh number (a measureof convective vigor), a is scaling constant that accounts for geometric effects (e.g., the wavelength ofconvection), and β is a scaling exponent that encapsulates the efficiency of convective cooling. Thevalue of β varies between different hypotheses for the dynamics of plate tectonics. We will return to2his issue after we develop the final model equations. The Nusselt number is the convective heat flux, q conv , normalized by the amount of heat that would be conducted through the layer of thickness D .The conductive flux is given by Fourier’s Law, q cond = k ∆ TD . The values k and ∆ T are the thermalconductivity and the difference between surface and interior temperatures. Ra is defined as Ra = ρgα ∆ T D κη ( T ) (4)where ρ , g , α and κ are density, gravity, thermal expansivity and thermal diffusivity. The temperature-dependent mantle viscosity is defined as η ( T ) = η ref exp (cid:20) A e R (cid:18) T − T ref (cid:19)(cid:21) (5)where A e is the activation energy and R the universal gas constant, and eta ref and T ref are referencevalues [Karato and Wu, 1993]. Conduction becomes unstable when Ra exceeds a critical threshold value, Ra c . Taking this into account, the convective heat flux is q conv = ak ∆ TD (cid:18) RaRa c (cid:19) β (6)Combining the above we arrive at˙ T = 1 ρc p (cid:34) H o exp ( − λt ) − AV ak ∆ TD (cid:18) RaRa c (cid:19) β (cid:35) . (7)If we assume that all values in Equation (4) are constant except T and η ( T ), then combining Equations(3), (4), and the definition of N u leads to q conv = a (cid:48) T β η ( T ) β (8) a (cid:48) = akD (cid:18) ρgαD κ (cid:19) β (9)where all constants have now been combined into a (cid:48) . The material constants can be determined usingexperimental values. The geometric constant, a , can be determined from laboratory and/or numericalconvection experiments in combination with boundary layer theory [Davies, 1980, Schubert et al., 1980].We will refer to that approach as a classic thermal history model (CTM). An alternative approach, thatwe refer to as a scaled thermal history model (STM), sets the constant a (cid:48) to a particular heat flow, q o ,at a scaling temperature, T o , and viscosity, η o [Christensen, 1985]. In doing so, we have an alternative3ormulation given by ˙ T = 1 ρc p (cid:34) H o exp ( − λt ) − Aq o V (cid:18) TT o (cid:19) β (cid:18) η o η ( T ) (cid:19) β (cid:35) . (10)CTMs integrate forwards in time from an initial mantle temperature value. STMs have historicallybuilt in Earth’s present day heat flux, temperature, and viscosity directly into the model formulation(akin to a data assimilation approach). Following this rationale, STMs have integrated backwards intime to model past mantle temperatures starting from present day values. That is not conducive tomodeling exoplanets, but the STM approach can be adapted for integrating forwards in time [Sealeset al., 2019, Seales and Lenardic, 2020]. For completeness, we evaluated how variable planetary mass/sizeand tectonic cooling efficiency (i.e., different β values) affected thermal histories by evolving model pathsof both CTMs and STMs forwards in time.Table 1: Model constants, scaling values and parameter rangesSymbol Parameter Value Units H o Initial radiogenic concentration 1.25e-7
W m − λ Decay constant 0.34
Gyr − α Thermal expansivity 2e-5 K − κ Thermal diffusivity 1e-6 m s − T s Surface Temperature 273 Kη ref Reference viscosity 1 e P a ∗ sA e Activation energy 3e5
Jmol − R Universal gas constant 8.314 J ( K ∗ mol ) − T ref Reference temperature 1855
KRa c Critical Rayleigh number 1100 - c p Heat capacitance 1400 J ( kg ∗ K ) − k Thermal conductivity 4.2 W ( m × K ) − T o Scaling temperature 1600 Kq o Scaling convective heat flow 0.069
W m − η o Scaling viscosity 4.45e19
P a ∗ sM ⊕ Mass of Earth 5.97e24 kgR ⊕ Radius of Earth 6371 kmG
Gravitational constant 6.67408e-11
N m kg − β Tectonic cooling efficiency constant 0-0.33 - M p Mass of Planet 0.1-5 M ⊕ - R p Planet radius Calculated kmR c Core radius Calculated kmρ
Mantle density Calculated kgm − g Surface gravity Calculated ms − The cooling efficiency of plate tectonics remains a matter of debate. For this reason, thermal historymodels have assumed different values of β . Given that different β values represent different physicalassumptions regarding the dynamics of plate tectonics, and by association Earth cooling, it follows thatdifferent values of β represent different hypotheses. The earliest thermal history models used a β valueof 0.33 [Schubert et al., 1980, Spohn and Schubert, 1982, Jackson and Pollack, 1984]. This assumes thatmantle viscosity dominantly resists convective motion [Tozer, 1972]. Gurnis [1989] incorporated analogues4o tectonic plates and showed this scaling could be recovered provided that weak plate boundaries werealso incorporated. Moresi and Solomatov [1998] allowed weak plate boundaries to develop dynamically,which lead to a scaling exponent of 0.30. If plate boundaries are not assumed to be so weak that energydissipation along them can be neglected and/or if plate strength offers resistance to convective motion,then the scaling exponent will be lower, with a range between 0 < = β < = 0 .
15 having been proposed[Christensen, 1985, Giannandrea and Christensen, 1993, Conrad and Hager, 1999a,b]. H¨oink et al. [2011]and Crowley and O’Connell [2012] argued that different sized plates can have different balances betweenplate driving and resisting forces. This leads to a mixed mode scaling that allows for β values between0.15 and 0.30 [H¨oink et al., 2013]. We will consider the full range of β values cited above. As notedin the introduction, within data and model uncertainties, multiple models within that range can matchobservational constraints on the cooling of the Earths interior over time [Seales et al., 2019, Seales andLenardic, 2020].Our choice of constants, scaling values and parameter ranges are listed in Table 1. We calculatedthermal paths for planets ranging from 0.1 to 5 earth masses ( M ⊕ ). For the remainder of this paper ⊕ refers to Earth-referenced values. For scaling models with planetary mass ( M p ), we followed in the spiritof Schaefer and Sasselov [2015] in using the scalings of Valencia et al. [2007b] to determine the planetary( R p ) and core ( R c ) radii, assuming a constant core mass fraction of 0.3259. We calculated the averagemantle density ( ρ ) based on the planetary mass, mantle volume and the average gravitational acceleration( g ), which scales as GM p /R p . We ran model suites with two different sets of initial temperatures. Inone scenario, all planets began with the same average mantle temperature. The second suite of modelsstarted all planets with the same potential temperature - the temperature of the interior mantle removingthe effects of adiabatic self-compression. We used the scaling of Schaefer and Sasselov [2015] to convertbetween average mantle temperature and potential temperature. Figure 1 shows sample thermal histories of different models and different starting temperatures. Forlow β values, temperatures were considerably warmer for CTMs than STMs. This behavior was firstnoted by McNamara and Van Keken [2000]. It occurs principally because CTMs have one initial value,mantle temperature, while STMs have effectively two boundary values, temperature and heat flux. Thisdifference did not impact our principal conclusions. For a fixed tectonic cooling efficiency, small planetmodels cool faster than larger ones. Allowing for different plate tectonic cooling efficiencies producedmore nuanced results. For example, a 5 M ⊕ planet with β = 0 . a) (b)(c) (d) Figure 1: Sample thermal histories of average mantle temperature for CTMs (a and b) and STMs (c andd) that begin at the same (a and c) and different (b and d) temperatures.Figure 2 shows contoured mantle temperatures after five billion years of model time plotted in plan-etary mass and cooling efficiency space. Models on the same contour have cooled to the same tempera-ture. The contours show that differences in plate tectonic cooling efficiencies allowed planets of differentmasses/sizes to be at the same temperature, i.e., small planets can cool to the same temperature aslarger planets over time. Figures 2c and 2f demonstrate this using sample thermal paths, which arecolor-coded to the parameter space shown in Figures 2a and 2d. The CTM samples had similar coolinghistories despite an order of magnitude difference in planetary mass. Similar behavior occurred for STMmodels with the addition that less massive (smaller) planets could remain significantly warmer than moremassive ones for five billion years of model time.
The thermal history of a terrestrial planet affects its volcanic and geologic history. Volcanic/geologichistory, in turn, affects the cycling of volatiles between a planet’s interior and surface reservoirs, which is6 a) (b) (c)(d) (e) (f)
Figure 2: Contoured mantle temperatures at 5 Gyr for CTMs (a and b) and STMs (d and e) withthe same (a and c) and different (b and d) initial mantle temperatures. Sample paths from this spacedemonstrating smaller planets can cool more slowly than larger ones (c and f).a critical factor in determining whether liquid water can exist at the surface of a planet over geologicaltime [Walker et al., 1981, Berner et al., 1983, Kasting et al., 1993, Kopparapu et al., 2014]. In additionto liquid water being key for life as we know it, life forms can also use a planets internal energy as a fuelsource for their survival [Baross and Hoffman, 1985, Jannasch and Mottl, 1985]. For these reasons, thesolid body thermal evolution of a terrestrial planet has had a long standing connection to astrobiology.The discovery of terrestrial exoplanets has reinvigorated interest in that connection and in thermalhistory models. The discovery of terrestrial exoplanets larger and more massive than the Earth kick-started thinking about how differences in planetary size could affect a planets thermal history and, byassociation, life potential [Valencia et al., 2007a].A first-wave of research into planetary size effects on geological history focused on whether largerplanets would be more or less likely to have plate tectonics [e.g., Valencia et al., 2007a, O’Neill andLenardic, 2007]. The focus was on the initiation of plate tectonics. That is, would internal energyovercome rock strength such that plate margins could be generated. Although an interesting problem,the cooling efficiency of plate tectonics does not depend solely one whether a planets internal energy7ources can overcome rock strength to initiate plate subduction. It also depends on the source(s) ofresistance to plate motions after plate tectonics is established. As discussed, that remains debated forthe Earth and exoplanets with plate tectonics can have different cooling efficiencies. Allowing for thisleads to trade-offs between a planets size/mass and tectonic cooling efficiency. A principal result isthat planets smaller than the Earth, and of the same absolute age, can remain and geologically andvolcanically active.We have only considered differences in cooling efficiency for a particular tectonic mode (i.e., platetectonics). Other tectonic regimes, such as episodic and stagnant lid, will further increase the possibilitythat planets of the same size as Earth may not have the same interior temperatures and/or that planetssmaller than the Earth may have hotter interiors. Within our own solar system, it has been arguedthat Venus may have liquid magma at the base of its mantle [O’Rourke, 2020]. This suggests theinterior of Venus may be hotter than Earth, despite the two planets having similar size and mass.Mars is considerably smaller and less massive than the Earth, yet estimates of its potential temperatureare similar to Earth [Filiberto, 2017]. In addition, Ruiz et al. [2011] argued that the Martian mantleexperienced recent warming. An added effect that could allow small planets to remain geologicallyactive is tidal locking. In some cases, tidal heating may be the dominant heat source in the mantle.Rocky bodies in that setting may maintain the same interior temperature for billions of years. With alarge enough volatile inventory, this could provide steady, persistent outgassing of life essential elements[Driscoll and Barnes, 2015].By looking at a range of plate tectonic cooling efficiencies, we have shown that smaller planets cancool slower than larger ones. This implies that, to the degree that geological activity is critical forplanetary habitability, exoplanets smaller than the Earth, and of the same age or older, should not bedown weighted in target selection strategies. An added implication is that planets sharing a range ofEarth characteristics, including absolute age, can be at different times in their geological lifetimes –the time window over which a planet can remain geologically active. To the degree that variations involcanic/geologic/tectonic activity over time have influenced the evolution of life on Earth, this suggeststhat we should anticipate that Earth-like exoplanets, of the same age as Earth, need not be at the sameevolutionary stages.
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