Tidal heating of Earth-like exoplanets around M stars: Thermal, magnetic, and orbital evolutions
TTidal heating of Earth-like exoplanets around M stars:Thermal, magnetic, and orbital evolutions
Driscoll, P. E. ∗ and Barnes, R. Astronomy Department, University of Washington, Seattle, WA NASA Astrobiology Institute, Virtual Planet Laboratory Lead Team
Compiled September 25, 2015Accepted inAstrobiology, Volume 15, Number 9, 2015DOI: 10.1089/ast.2015.1325Note: Small deviations exist between this versionand the published version.
Abstract
The internal thermal and magnetic evolution of rocky exoplanets is critical to theirhabitability. We focus on the thermal-orbital evolution of Earth-mass planets aroundlow mass M stars whose radiative habitable zone overlaps with the “tidal zone”, wheretidal dissipation is expected to be a significant heat source in the interior. We develop athermal-orbital evolution model calibrated to Earth that couples tidal dissipation, witha temperature-dependent Maxwell rheology, to orbital circularization and migration. ∗ [email protected], (206) 543-0777 a r X i v : . [ a s t r o - ph . E P ] S e p e illustrate thermal-orbital steady states where surface heat flow is balanced by tidaldissipation and cooling can be stalled for billions of years until circularization occurs.Orbital energy dissipated as tidal heat in the interior drives both inward migrationand circularization, with a circularization time that is inversely proportional to thedissipation rate. We identify a peak in the internal dissipation rate as the mantle passesthrough a visco-elastic state at mantle temperatures near 1800 K. Planets orbiting a0.1 solar-mass star within 0 .
07 AU circularize before 10 Gyr, independent of initialeccentricity. Once circular, these planets cool monotonically and maintain dynamossimilar to Earth. Planets forced into eccentric orbits can experience a super-coolingof the core and rapid core solidification, inhibiting dynamo action for planets in thehabitable zone. We find that tidal heating is insignificant in the habitable zone around0 .
45 (or larger) solar mass stars because tidal dissipation is a stronger function oforbital distance than stellar mass, and the habitable zone is further from larger stars.Suppression of the planetary magnetic field exposes the atmosphere to stellar winderosion and the surface to harmful radiation. In addition to weak magnetic fields,massive melt eruption rates and prolonged magma oceans may render eccentric planetsin the habitable zone of low mass stars inhospitable for life.
Keywords: tidal dissipation; thermal history; planetary interiors; magnetic field.
Gravitational tides are common in the Solar System, from the Moon, responsible fordriving the principle diurnal tides in Earth’s oceans and atmosphere, to Io, the mostvolcanically active body in the Solar System. Tidal dissipation as a heat source in thesolid Earth is weak at present and often neglected from thermal history calculationsof its interior. However, rocky exoplanets with eccentric orbits close to their starare expected to experience significant tides (Dole, 1964; Rasio et al., 1996; Jacksonet al., 2009; Barnes et al., 2010) that likely influence their thermal, orbital, and even tmospheric evolution (Barnes et al., 2013; Luger et al., 2015). Recent progresses inmodeling tidal dissipation in a visco-elastic mantle (Bˇehounkov´a et al., 2010, 2011;Henning et al., 2009; Henning and Hurford, 2014) have advocated using Maxwell-typetemperature-pressure dependent rheology and emphasized the limited applicability ofa constant tidal quality factor “ Q ” model. These more complicated dissipation modelsare necessary to better characterize the tidal and orbital states of rocky exoplanetsover a range of internal temperatures.The search for habitable Earth–like exoplanets commonly targets planets in orbitaround low mass M–type stars to maximize the number of small mass planets found(Mayor et al., 2014; Seager, 2013). Targeting low mass stars is beneficial for at leastthree reasons: (1) the habitable zone around M stars is much closer to the star (Koppa-rapu et al., 2013), making an Earth–mass planet in the habitable zone an easier targetfor both transit and radial velocity detection, (2) low mass M stars are more abundantin the nearby solar neighborhood, and (3) M stars have longer main sequence times.On the other hand, there are several reasons why targeting M stars may be risky.M stars are intrinsically faint making most observations low signal to noise, and theirflux peaks at wavelengths close to those absorbed by Earth’s atmosphere. M stars aremore active, especially early on, which may induce massive amounts of atmospheric loss(Luger et al., 2015) and biologically hazardous levels of radiation at the surface. Earth–mass planets in the habitable zones of M stars likely experience larger gravitationaltides associated with star-planet and planet-planet interactions, especially consideringthat most exoplanet systems are dynamically full (Barnes and Quinn, 2004; Barnesand Greenberg, 2006; Van Laerhoven et al., 2014). However, the implications of thesetides on the thermal evolution of the interior have not yet been explored.The thermal history of a planet is critical to its habitability. Mantle temperaturedetermines the rates of melting, degassing, and tectonics, while the thermal state ofthe core is critical to the maintenance of a planetary magnetic field that shields thesurface from high energy radiation. The thermal evolution of the Earth and terrestrial lanets involves solving the time evolution of mantle and core temperature through abalance of heat sources and sinks. The thermal history of the Earth, although betterconstrained than any other planet, is still subject to significant uncertainties. However,avoiding both the thermal catastrophe in the mantle (Korenaga, 2006) and the newcore paradox (Olson et al., 2013) add significant constraints that predict a monotoniccooling of the mantle and an active geodynamo over the history of the planet (Driscolland Bercovici, 2014). Previous models of the thermal evolution of rocky exoplanets(e.g. Gaidos et al., 2010; Driscoll and Olson, 2011; Tachinami et al., 2011; Summerenet al., 2013; Zuluaga et al., 2013) have focused on the influence of planet size on thermalevolution, but neglected tidal dissipation as an internal heat source, and therefore likelyover estimate magnetic field strength and lifetime in the HZ around low mass stars.We improve the thermal and magnetic evolution model used in Driscoll and Bercovici(2014) by adding tidal heating as a internal heat source, and couple this to orbitalevolution.In this paper we focus on the influence of tidal dissipation in the solid mantle ofEarth–like exoplanets in the habitable zone around M stars. Tidal dissipation depositsheat in the planetary interior while simultaneously extracting energy from the orbit,which can lead to circularization and migration. We couple the thermal and orbitalevolution equations into a single model to identify the conditions and timescales forEarth–like geophysical and magnetic evolution. Section § § § §
5. The possibility of aninternally driven runaway greenhouse is addressed in §
6. The influence of tides on theinner edge of the habitable zone over a range of stellar masses is explored in §
7. Asummary and discussion are in § Model Description
In this section we describe the details of the thermal-orbital evolution model. § § § Q ’s refer to heat flows (units of [W]), q ’srefer to heat fluxes (units of [W m − ]), and the script symbol Q refers to the tidalquality factor (dimensionless), also known as the specific dissipation function. The gravitational perturbation experienced by a secondary body (the planet) in orbitabout a primary body (the star) is approximated by the lowest order term in thepotential expansion, which is the semidiurnal tide of degree 2 (Kaula, 1964). The powerdissipated by tidal strain associated with this term in the secondary with synchronousrotation is (Segatz et al., 1988), Q tidal = −
212 Im( k ) GM ∗ R p ωe a (1)where G is the gravitational constant, M ∗ is stellar mass, R p is planet radius, ω isorbital frequency, e is orbital eccentricity, a is orbital semi-major axis, and Im( k ) isthe imaginary part of the complex second order love number k . If planetary rotation issynchronous then the tidal frequency is equal to the mean motion ω = n = (cid:112) GM ∗ /a ,and the tidal power reduces to Q tidal = −
212 Im( k ) G / M / ∗ R p e a / . (2)This expression for tidal dissipation is the product of three physical components: (1)tidal efficiency ( − Im( k )), (2) star-planet size ( M / ∗ R p ), and (3) orbit ( e /a / ).For illustrative purposes it is helpful to compare the radiative “habitable” zone HZ) to the “tidal” zone, the orbital distance at which tidal dissipation is likely todominate the internal heat budget of the planet. For this comparison we compute thetidal heat flow using (2) for an Earth-sized planet with e = 0 . − Im( k ) = 3 × − ,similar to present-day Earth. Figure 1 shows that for M star < . M ∗ the HZ overlapswith the tidal zone, defined as when tidal heating is the dominant term in the heatbudget. This diagram implies that Earth-mass planets in the HZ of low mass starscould experience extreme tidal heating and a rapid resurfacing rate that may renderthe surface uninhabitable. The HZ and tidal zone intersect because stellar radiationflux is more sensitive to stellar mass than the gravitational tidal potential. We notethat larger eccentricity or tidal dissipation efficiency ( − Im( k )) would push the tidalzone limits out to larger orbital distances, rendering the HZ tidally dominated for largermass stars.The one-dimensional dissipation model in (2) assumes a homogeneous body withuniform stiffness and viscosity. To derive the dissipation efficiency ( − Im( k )) we firstdefine the love number, k = 3 /
21 + µβ st (3)where µ is shear modulus and β st is effective stiffness (Peale and Cassen, 1978). Writingshear modulus as a complex number and using the constitutive relation for a Maxwellbody (Henning et al., 2009), one can derive the dissipation efficiency in (2) as − Im( k ) = 57 ηω β st (cid:18) (cid:104)(cid:16) µ β st (cid:17) ηωµ (cid:105) (cid:19) (4)where η is dynamic viscosity. We note that this model does not involve a tidal Q factor, rather the rheological response of the mantle is described entirely by Im( k ).For comparison with other models, one can compute the standard tidal Q factor of theMaxwell model as Q = ηωµ . (5) he common approximation is then − Im( k ) ≈ Re( k ) / Q (e.g. Goldreich and Soter,1966; Jackson et al., 2009; Barnes et al., 2013). Although the tidal Q factor does notappear explicitly in the calculations below, it will be used to calibrate this model toEarth and is useful in comparing it to other models (also see Appendix A).Following previous studies (Sotin et al., 2009; Bˇehounkov´a et al., 2010, 2011), weensure that the model reproduces the observed tidal dissipation in the solid Earth bycalibrating the effective material properties in (4) appropriately. This calibration allowsus to approximate the total tidal dissipation over the whole mantle by a single volumeaveraged dissipation function. The Q factor of the solid Earth has been estimatedempirically to be Q E ≈
100 (Ray and Egbert, 2012). Effective viscosity follows anArrhenius Law form, ν = ν ref exp (cid:18) E ν R g T m (cid:19) /(cid:15) phase (6)where ν = η/ρ m is kinematic viscosity, ρ m is mantle density, ν ref is a reference viscosity, E ν is the viscosity activation energy, R g is the gas constant, T m is average mantletemperature, and (cid:15) phase accounts for the effect of the solid to liquid phase change (seeTable 1 for a list of constants). Shear modulus is similarly described, µ = µ ref exp (cid:18) E µ R g T m (cid:19) /(cid:15) phase . (7)This model predicts the rapid drop in shear modulus with melt fraction demonstratedexperimentally by Jackson et al. (2004). The reference shear modulus µ ref = 6 . × Pa and effective stiffness β st = 1 . × GPa are calibrated by k = 0 . Q = 100for the present-day mantle.We model the influence of melt fraction φ on viscosity following the parameteriza-tion of Costa et al. (2009), (cid:15) phase ( φ ) = 1 + Φ δ ph [1 − F ] Bφ ∗ (8) F = (1 − ξ )erf (cid:20) √ π − ξ ) Φ(1 + Φ γ ph ) (cid:21) (9) here Φ = φ/φ ∗ , and φ ∗ , ξ , γ ph , and δ ph are empirical constants (Table 1).The functional form of tidal shear modulus at high temperature–pressure is notwell known, so we investigate the influence of shear modulus activation energy A µ onthe model. Contoured in Figure 2 are tidal power using (2-4), tidal power using theapproximation − Im ( k ) = k / Q , love number k , and tidal factor Q as functions ofshear modulus µ and viscosity ν . Evolution paths of Q tidal as a function of T m in therange 1500 − A µ = 0, 2 × , and 4 × J mol − . For the nominal activation energyof A µ = 2 × J mol − the mantle cooling path passes through a maximum tidaldissipation around ν = 10 Pa s and µ = 10 Pa, corresponding to T m ≈ § Q tidal they do not passthrough the global maximum (dark red), which has been invoked for Io (Segatz et al.,1988). The main influence of increasing A µ is to shift the dissipation peak to lowertemperatures. The nominal value of A µ = 2 × J mol − produces a dissipation peakwhen melt fraction is about 50%.The tidal dissipation factor Q (Figure 2d) and tidal power using Q (Figure 2b)change by about one order of magnitude over the entire temperature and activationenergy range (Figure 2d). However, tidal power using the Maxwell model in (2) fluc-tuates by 10 TW over the same range (Figure 2a). This comparison emphasizes that,although Q is not far from constant, dissipation using the Maxwell model is signifi-cantly different than the Q approximation (also see Appendix A). The Love number k increases monotonically with T m for all cases (Figure 2c) up to the limit of k = 3 / µ/β << /
19 in (3).We note that dissipation in this model is a lower bound as dissipation in the liquidis not included, which can occur by resonant dissipation (e.g. Tyler, 2014; Matsuyama,2014). Dissipation in the liquid is not likely to be a major heat source, but could drive echanical flows in the core (Zimmerman et al., 2014; Le Bars et al., 2015) and amplifydynamo action there (Dwyer et al., 2011; McWilliams, 2012). The thermal evolution of the interior solves the balance of heat sources and sinks inthe mantle and core. The thermal evolution is modeled as in Driscoll and Bercovici(2014), with an updated mantle solidus and inclusion of latent heat release due tomagma ocean solidification (see Appendix B.3). The conservation of energy in themantle is, Q surf = Q conv + Q melt = Q rad + Q cmb + Q man + Q tidal + Q L,man (10)where Q surf is the total mantle surface heat flow (in W), Q conv is heat conductedthrough the lithospheric thermal boundary layer that is supplied by mantle convection, Q melt is heat loss due to the eruption of upwelling mantle melt at the surface, Q rad isheat generated by radioactive decay in the mantle, Q cmb is heat lost from the core acrossthe core-mantle boundary (CMB), Q man is the secular heat lost from the mantle, Q tidal is heat generated in the mantle by tidal dissipation, and Q L,man is latent heat releasedby the solidification of the mantle. Crustal heat sources have been excluded becausethey do not contribute to the mantle heat budget. Note that heat can be released fromthe mantle in two ways: via conduction through the upper mantle thermal boundarylayer ( Q conv ) and by melt eruption ( Q melt ). Detailed expressions for heat flows andtemperature profiles as functions of mantle and core properties are given in AppendixB. The thermal evolution model assumes a mobile-lid style of mantle heat loss wherethe mantle thermal boundary layers maintain Rayleigh numbers that are critical forconvection. In contrast, a stagnant lid mantle parameterization would have a lowerheat flow than a mobile-lid at the same temperature (e.g. Solomatov and Moresi, 2000).However, a stagnant lid mantle that erupts melt efficiently to the surface can lose heat s efficiently as a mobile-lid mantle with no melt heat loss (Moore and Webb, 2013;Driscoll and Bercovici, 2014). Io is an example of this style of mantle cooling (O’Reillyand Davies, 1981; Moore et al., 2007).Similarly, the thermal evolution of the core is governed by the conservation of energyin the core, Q cmb = Q core + Q icb + Q rad,core (11)where Q core is core secular cooling, Q rad,core is radiogenic heat production in the core,and heat released by the solidification of the inner core is Q icb = ˙ M ic ( L icb + E icb ),where ˙ M ic is the change in inner core mass M ic , and L icb and E icb are the latent andgravitational energy released per unit mass at the inner-core boundary (ICB).The internal thermal evolution equations are derived by using the secular coolingequation Q i = − c i M i ˙ T i , where c is specific heat and i refers to either mantle or core,in equations (10) and (11). Solving for ˙ T m and ˙ T c gives the mantle and core thermalevolution equations,˙ T m = ( Q cmb + Q rad + Q tidal + Q L,man − Q conv − Q melt ) /M m c m (12)˙ T c = − ( Q cmb − Q rad,c ) M c c c − A ic ρ ic (cid:15) c dR ic dT cmb ( L F e + E G ) (13)where the denominator of (13) is the sum of core specific heat and heat released bythe inner core growth, A ic is inner core surface area, ρ ic is inner core density, (cid:15) c is aconstant that relates average core temperature to CMB temperature, dR ic /dT cmb isthe rate of inner core growth as a function of CMB temperature, and L F e and E G arethe latent and gravitational energy released at the ICB per unit mass (Table 1). SeeAppendix B and Driscoll and Bercovici (2014) for more details. The orbital evolution of the planet’s semi-major axis a and eccentricity e , assumingno dissipation in the primary body (the star), is (Goldreich and Soter, 1966; Jackson t al., 2009; Ferraz-Mello et al., 2008)˙ e = 212 Im ( k ) M ∗ M p (cid:18) R p a (cid:19) ne (14)and ˙ a = 2 ea ˙ e. (15)Mean motion can be replaced by using n = GM ∗ /a ,˙ e = 212 Im ( k ) M / ∗ G / R p M p ea / . (16)The differential equations for thermal evolution (12, 13) and orbital evolution (16,15) are solved simultaneously to compute coupled thermal-orbital evolutions. Before exploring the full model it is useful to highlight the influence of tidal heating onthe thermal evolution in a steady state sense by comparing heat flows as functions ofmantle temperature. Figure 3 shows the tidal heat flow (a) and orbital circularizationtime (b), t circ = e/ ˙ e (17)as a function of mantle temperature for a range of initial orbital distances and eccen-tricity of e = 0 .
1. Figure 3a shows that a peak in dissipation occurs when the mantle isin a partially liquid visco-elastic state ( T m ≈ eflects the preferred cooling rate of the interior. Conceptually, a tidal steady stateis achieved as the planet cools down from an initially hot state ( T m > − a = 0 .
02 AU), circularization occurs in less than ∼ t T = M m c m ( T m (0) − T m ) /Q surf , which is thetime required for the mantle to cool from T m (0) = 2500 K to T m . This shows that ittakes the mantle ∼ This section presents full thermal-orbital evolutions with a single Earth-mass planet inorbit around a 0 . . § ave A µ = 2 × J mol − , T m (0) = 2400 K, and T c (0) = 6000 K. The results areindependent of initial mantle and core temperatures up to approximately ±
500 K.
First, we investigate the evolution of three models that start with e (0) = 0 . a = 0 .
01 AU, (2) within theHZ at a = 0 .
02 AU, and (3) outer edge of HZ at a = 0 .
05 AU.Figure 4 compares the tidal heat flow and eccentricity of these three models as afunction of T m . The inner case with a (0) = 0 .
01 AU begins with a rather high initialtidal heat flow ( Q tidal ∼ . a (0) = 0 .
02 AU begins with a lower tidal heat flow becauseit orbits farther from the star. As the mantle cools and solidifies, tidal dissipationevolves through the visco-elastic peak at T m ≈ ∼ Q tidal ∼
100 TW occurs. This increase in dissipation drives arapid circularization (Figure 4b), which then decreases the dissipation as the mantlecools further.The outer case at a (0) = 0 .
05 AU experiences the lowest initial tidal heat flowof Q tidal (0) ∼ − TW due to it being farthest from the star. Dissipation remainslow and the orbit remains eccentric until the mantle cools to T m ∼ Q tidal ∼
100 TW and T m ∼ e ∼ . nd a tidal heat flow of Q tidal ∼
10 TW. This shows that at the outer edge tidaldissipation can linger longer due to slower circularization times.A detailed comparison of these three models over time is shown in Figure 5. Rela-tively small differences in their temperature histories (Figure 5a) are driven by smalldifferences in mantle and core cooling rates (Figure 5b). Circularization of the innermodel occurs in the first Myr and by 100 Myr for the middle model, while the outermodel retains a small eccentricity of e = 2 × − after 10 Gyr (Figure 5d). Thesecircularization times are reflected in the tidal heat flow peaks (Figure 5b), which occuraround 0.1 Myr for the inner case, 10 Myr for the mid case, and 1 Gyr for the outercase. Inward migration by 10 −
20% also accompanies this circularization (Figure 5c).The thermal evolutions are mainly controlled by secular cooling and radiogenicdecay, with tidal dissipation as a temporary energy source. Mantle heating due tolatent heat released during the solidification of the mantle is of order 10 TW until ∼ ∼ . −
10 Myr.Mantle solidification, and the drop in mantle heat flow, occurs slightly later for theinner model for two reasons: (1) the surface heat flow is lower than the other modelsin the first Myr because the surface is hot, decreasing the upper mantle temperaturejump; (2) tidal heating is initially moderate ( ∼ . eratures in order to accommodate the same cooling rates (e.g. Solomatov and Moresi,2000; Driscoll and Bercovici, 2014). Therefore, we expect mobile-lid planets to coolfaster, dissipate tidal energy more efficiently, and circularize faster than stagnant-lidplanets. Stagnant lid planets that are strongly tidally heated likely rely on meltingrather than conduction to remove heat from the interior, as Io demonstrates today.Core cooling rates are similar at these three orbital distances, which results insimilar magnetic moment histories and inner core nucleation times (Figure 5e). Innercore nucleation induces a kink in the core compositional buoyancy flux and magneticmoment around 4 Gyr, similar to predictions for the Earth (Driscoll and Bercovici,2014). Surface melt eruption rate is determined by the mantle cooling rate throughthe upper mantle geothermal gradient, so that the eruption rates at these three orbitaldistances are similar and follow the mantle heat flow history (Figure 5f). After 6 Gyrthe middle and outer planet’s mantles are completely solid so that melt eruption ends,while melt eruption continues longer for the inner case due to a slightly hotter mantle.In summary, Earth-like planets near the inner edge of the HZ around 0 . M ∗ starscircularize rapidly (within a few Myr), allowing internal cooling and core dynamo ac-tion to proceed similar to Earth. On the outer edge of the HZ orbital circularizationis slower, leading to a prolonged period of tidal dissipation that is accentuated by thecooling of the mantle through a visco-elastic state after ∼ e (0) = 0 . In this section we compare the final states (after 10 Gyr) of orbital-thermal evolutionfor 132 models that span a range of initial orbital distances of 0 . − .
10 AU andinitial eccentricities of 0 − .
5. The results are displayed as contours in initial orbital − e space (Figures 7-13). Here we consider models whose orbits evolve (left panels ofFigures 7-13), while § e/a / in (16). In other words, orbital circularization isa stronger function of orbital distance than eccentricity. Circularization also causesthe orbit to migrate inwards (Figure 6a), although this results in a maximum inwardsmigration of only 22% of the initial distance. The evolution of orbital distance, whichproduces mainly horizontal iso-contours (Figure 6a), is controlled by initial eccentricitybecause migration is proportional to e /a / by (15); hence migration ( ˙ a ) is a strongerfunction of eccentricity than circularization ( ˙ e ).Figure 7a contours tidal heat flow for these models. We identify the tidal heatflow boundaries defined by Barnes and Heller (2013) as an Earth Twin for Q tidal < < Q tidal < Q tidal > e inFigure 6b. Beyond a ∼ .
07 AU tidal dissipation is too weak to result in any signifiantcircularization. Therefore, there are gradients in Q tidal on both sides of this boundaryat a ∼ .
07. There is also a decrease in Q tidal with initial e because models with lowinitial e circularize earlier. The combination of these effects results in a peak in tidaldissipation around a ∼ .
07 AU and e ∼ . a). A second maximum in core heat flow occurs at the innermost orbits due to thehigh surface temperature insulating the mantle, which keeps mantle temperature high(Figure 10a) and thins the lower mantle thermal boundary layer. Core temperature islow where core heat flow is high (Figure 11a) due to secular cooling of the core.After 10 Gyr of significant core cooling all models have a large solid inner core. Thesize of the inner core (also contoured in (Figure 11) is proportional to Q cmb (Figure9a) and is between 80 − .
07 AU. A secondary peak in melt mass flux at close-in orbits (upper left corner ofFigure 13a) is caused by a slightly higher mantle temperature associated with a hotter,insulating surface (Figure 10a).
In this section we consider planets whose orbits are fixed ( ˙ e = ˙ a = 0). This includeseccentric orbits, which could be fixed, for example, through interactions with a plane-tary companion (Van Laerhoven et al., 2014). Figures 7b-13b show contours in orbitalspace, similar to those discussed above except with fixed orbits. Figure 14 shows the time evolution of a specific case with a fixed orbit of a = 0 . e = 0 . §
3. Tidal heating initially starts low ( ∼ − TW) before increasing as the mantlecools for the first 10 Myr, until the mantle reaches a steady state temperature of m ≈ Q surf ≈ q surf ≈ − ),implying that this planet might be better characterized as a super-Io than Earth-like(e.g. Barnes et al., 2010). The tidal steady state still allows the core to cool slowlybecause a significant temperature difference between the mantle and core persists. Corecooling drives a core dynamo for all 10 Gyr, although the magnetic moment rapidlydeclines as the core is nearly entirely solid by 10 Gyr (Figure 14c). These fixed orbit models, in contrast with the evolving models in §
4, have tidal heatflows that are mainly determined by the orbital state rather than the cooling history.Specifically, Q tidal increases with e and decreases with a , producing a maximum inthe upper left corner of Figure 7b. Mantle heat flow (Figure 8b) and temperature(Figure 10b) increase with tidal heating due to the positive feedback between mantletemperature and surface heat flow.Core heat flow peaks in models at moderate orbital distances where tidal heatflow is similar in magnitude to the sum of all other mantle heat sources; i.e. Q tidal ∼ Q cmb + Q man + Q rad (Figure 9b). This peak can be understood by considering how Q cmb behaves at the two tidal extremes: (1) where tidal heating is strong (upper leftcorner of Figure 7b) the mantle is forced into a hot steady state so that surface heatflow can accommodate all heat sources, which thins the lower mantle thermal boundarylayer and allows a moderate core heat flow of Q cmb ∼
10 TW; (2) where tidal heatingis weak (lower right corner of Figure 7b) the mantle and core are free to cool similarto Earth, so that Q cmb decreases monotonically over time. In between these limitstidal dissipation heats the mantle slightly, increasing the surface heat flow, but doesnot dominate the heat budget, which allows the interior to cool. Note that evenwhen mantle temperature is high ( ∼ ore ( ∼ Q cmb as the super-cooling of the core.The influence of this peak in core cooling rate on the dynamics of the core isdramatic. Intuitively, core temperature is lowest where core cooling is high (center ofFigure 11b), and the coldest models with T cmb ≈ R ic = R core ). A fully solid core prevents fluid motion and therefore dynamo action.These models lose their dynamos at ∼ . kg yr − correspond to a global basalt layer resurfacingrate of 7 m kyr − . For reference, the Siberian traps, one of the largest igneous provinceson Earth and thought to be responsible for the Permian mass extinction event, isestimated to have produced a basalt layer at a rate of 1 m kyr − over the area of thetraps (Reichow et al., 2002). Therefore, continuous eruption rates of ∼ kg yr − re likely to prevent such planets from being habitable.We also compute the same range of models but only fixing eccentricity, allowing a to evolve. This might occur if a neighboring planet forces the eccentricity but allowsinward migration. In these cases we find that all models with initial orbits of a < . a < .
02 AU) and e > . e > .
1) migrate into the central star within 10Gyr, and most by 5 Gyr.
As described by Barnes et al. (2013), if interior heat flux exceeds the limit at whichenergy can be radiated from the top of the atmosphere then runaway heating of thesurface occurs, evaporating the ocean, and leading to rapid water loss (Goldblatt andWatson, 2012). Figure 15 shows the time spent in an internally driven runaway green-house, defined as the period of time when the surface heat flux exceeds the threshold q runaway = 300 W m − .For both evolving (Figure 15a) and fixed (Figure 15a) orbital models the runawaygreenhouse period is shorter at closer orbital distances, almost independent of eccen-tricity. This implies that tides, which depend strongly on eccentricity, play a minorrole in the length of the runaway greenhouse state. The runaway greenhouse stateis shorter for close-in planets because they have higher effective surface temperaturescloser to the star, which insulates the mantle and decreases the initial surface heatflow. With lower initial surface heat flows, these inner planets drop below the runawayheat flow threshold earlier (Figure 5b). A second trend in Figure 15a towards evenshorter times spent in a runaway greenhouse is found for the inner-most, high eccen-tricity planets. This drop in surface heat flow at around 50 −
100 kyr occurs during thecircularization of the inner planets’ orbits, when the tidal heat flow rapidly declines(Figure 5b). Circularization causes a small dip in the surface heat flow as the interiortemperatures and heat flows adjust to the smaller internal (tidal) heat source. Thisadjustment to lower heat flows, although seemingly minor, actually shortens the time pent above the runaway threshold (Figure 5b).Interestingly, when the heat flow is high enough to drive a runaway greenhouseduring the first few hundred Myr the mantle is so hot that tidal dissipation is inefficient.Typically tidal dissipation is not a major heat source until the mantle solidifies andcools down to ∼ ∼
100 kyr) is shorter than the typical magma ocean solidificationtime ( ∼
10 Myr), a period when the surface is likely uninhabitable anyway. Thesemodels assume mobile lid cooling at all times, however Foley et al. (2012) proposedthat a runaway greenhouse could induce a transition from mobile to stagnant lid, whichwould also slow internal cooling and would be detrimental to habitability. In § The above calculations assumed a stellar mass of 0 . M sun . In this section we explorethe influence of stellar mass, in the range 0 . − . M sun , at the inner edge of thehabitable zone. Similar to the contours in § − . Q tidal is 50% or more of the total surface heat flow Q surf ; (b) the time to reach Earth’s resent-day surface heat flow of Q ∗ surf = 40 TW.The island-like shapes of these time contours can be explained by a combinationof three physical effects. First, planets with initially low eccentricity ( e (0) < .
1) ex-perience weak tides and spend little time, if any, in the tidally dominated regime. Athigher eccentricity tides become stronger, so that more eccentric planets are tidallydominated longer (Figure 16a). Second, as stellar mass increases the habitable zonemoves to larger orbital distances and the tidal dissipation decreases because tidal dis-sipation in (2) is a stronger function of orbital distance ( ∝ a − / ) than stellar mass( ∝ M +5 / ∗ ). The net result is a decrease in tidal dissipation within the habitablezone for increasing stellar mass, and shorter time spent in the tidally dominated state(Figure 16a). This effect produces contour boundaries with positive slope in Figure16. Third, models with high initial eccentricity ( e (0) > .
2) and close-in initial orbitsaround low mass stars ( M star < .
12) experience extreme early tides that drive rapidorbital circularization. This leads to short times spent in the tidally dominated state.Figure 16b, similar to Figure 16a, shows that eccentric planets on the inner edgearound 0 . − . M sun stars maintain surface heat flows in excess of Q ∗ surf for 10Gyr due to strong tidal dissipation. Interestingly, Figure 16b shows that planets thatexperience only a temporary period of tidal heating actually cool to an Earth-likeheat flow before 4.5 Gyr. These planets cool faster than Earth because their thermaladjustment timescale is longer than their circularization (or tidal heating) timescale,so they are still adjusting to the new heat balance with a lower tidal heat source. Inother words, the surface heat flow, that was increased during the tidal heating phase,is still slightly larger than it would have been with no tidal heating. This super-coolingeffect was also discussed in § . M sun stars with eccentricity of 0 . anion. We find a threshold at a stellar mass of 0 . M sun , above which the habitablezone is not tidally dominated. These stars would be favorable targets in the search forgeologically habitable Earth-like planets as they are not overwhelmed by strong tides. In summary, we have investigated the influence of tidal dissipation on the thermal-orbital evolution of Earth-like planets around M-stars with masses 0 . − . M sun . Athermal-orbital steady state is illustrated where, under certain conditions, heat fromtidal dissipation is balanced by surface heat flow. We find that mantle temperaturesin this balance are hotter for planets with shorter orbital distances and larger eccen-tricities. Orbital energy dissipated as tidal heat in the interior drives both inwardmigration and circularization, with a circularization time that is inversely proportionalto the dissipation rate. The cooling of an eccentric planet in the habitable zone leadsto a peak in the dissipation rate as the mantle passes through a visco-elastic rheologystate. Planets around 0 . a < .
07 AU circular-ize before 10 Gyr, independent of initial eccentricity. Once circular, these planet coolmonotonically and maintain dynamos similar to Earth. Generally, we find that tidaldissipation plays a minor role on the dynamo history if the orbit is free to evolve intime.When the orbit is fixed the planet cools until a tidal steady state balance betweentidal dissipation and surface cooling is reached. In the habitable zone this steady statecan produce a super-cooling of the core when tidal heating is strong enough to heat themantle and decrease its viscosity and low enough to not dominate the surface heat flow.This rapid cooling leads to complete core solidification, prohibiting dynamo action formost models in the habitable zone with e > .
05 by 10 Gyr. In addition to weakmagnetic fields, massive melt eruption rates in the habitable zone may render thesefixed orbit planets uninhabitable.Commonly the term “habitability” refers to the influx of radiation necessary to aintain surface liquid water. However, the full habitability of a planet must involvethe dynamics of the interior and its interaction with the surface environment. We findthat tidal heating of a planetary mantle can influence surface habitability in severalimportant ways:1. Prolonged magma ocean stage. Close-in planets with a high eccentricity ( e (cid:38) . ∼ ∼ ∼ ∼ kg yr − ).These extreme eruption rates can lead to rapid global resurfacing and degassingthat render the surface environment a violent and potentially toxic place forlife. Volcanically dominated atmospheres could be significantly different fromthe modern-day Earth’s and are potentially detectable with future space- andground-based telescopes (Misra et al., 2015).3. Lack of magnetic field. Planetary magnetic fields are often invoked as shields nec-essary to maintain life. Magnetic fields can protect the atmosphere from stellarwind erosion (Driscoll and Bercovici, 2013) and the surface from harmful radia-tion (Dartnell, 2011; Griessmeier et al., 2005). Super-cooling of the core, whichcan solidify the entire core and kill the dynamo, occurs in the habitable zone after ∼ lanets may have magnetic fields that are too weak to hold the stellar wind abovethe atmosphere or surface. In either case, the lack of a strong magnetic shieldwill be detrimental to life.4. Tidally driven runaway greenhouse. In § ∼ TW (Figure 2a). A significant amountof tidal energy can also be dissipated in the liquid portions of the planet (Tyler,2014), which is beyond the scope of this study.With growing interest in the habitability of Earth-like exoplanets, the developmentof geophysical evolution models will be necessary to predict whether these planets haveall the components that are conducive for life. This paper focused on a single planetmass, but the mathematical equations can be developed to model the evolution of otherrocky planet/star mass ratios, including large rocky satellites around giant planets.However, significant uncertainties make the application to super-Earths particularlychallenging. The fundamental physical mechanisms underpinning plate tectonics, bothin terms of its generation and maintenance over time, are not fully understood, mak-ing extrapolation to larger planets questionable. Perhaps most importantly, materialproperties, such as viscosity, melting point, solubility, and conductivity, are poorly con-strained at pressures and temperatures more extreme than Earth’s lower mantle andcore. This uncertainty prevails in our own Solar System where the divergent evolutionof Earth and Venus from similar initial conditions to dramatically different present-daystates remains elusive.Future thermal-orbital modeling improvements should include coupling the evolu- ion of the interior to the surface through volatile cycling and atmosphere stability.Advancing the orbital model to include gravitational interactions with additional plan-etary companions would allow for tidal resonances, variable rotation rates, and othertime-dependent orbital forcings. In addition to the eccentricity tide explored here, anobliquity tide could also be important. Further improvements could include dissipa-tion in oceans or internal liquid layers, variable internal compositions, structures, andradiogenic heating rates, core light element depression, continental crust formation,and eventually a direct coupling of first-principles numerical simulations. Acknowledgements
The authors thank W. Henning and T. Hurford for helpful discussions. This workwas supported by the NASA Astrobiology Institutes Virtual Planet Laboratory underCooperative Agreement solicitation NNH05ZDA001C.
Author Disclosure Statement
No competing financial interests exist. Q surf ≈
80 TW, leftcurve) or Earth ( Q surf = 40 TW, right curve). Tidal heat flow is calculated by (2) assuming e = 0 . − Im( k ) = 3 × − ( k = 0 . Q = 100). The gray shaded region denotes thezone where the planet is predicted to be radiatively “habitable” but tidally dominated, andtherefore possibly not habitable. 27a) (b)(c) (d)Figure 2: Comparison of tidal properties as a function of viscosity and shear modulus forthree shear modulus activation energies A µ = 0 J mol − (blue line), 2 × J mol − (blackline), and 4 × J mol − (red line). Lines show tracks of ν ( T m ) and µ ( T m ) for mantletemperatures in the range 1500 − Q tidal . (b) Contourof tidal power using the approximation − Im( k ) = k / Q . (c) Contour of Love number k . (d)Contour of tidal dissipation factor Q . Calculations use M ∗ = 0 . M sun , A ν = 3 × J mol − , e = 0 .
1. 28a) (b)Figure 3: Tidal dissipation properties as a function of mantle temperature T m for M ∗ =0 . M sun and e = 0 .
1. (a) Comparison of tidal heat flow from the Maxwell model (curves)with the constant Q model ( Q = 100, k = 0 .
3) at four orbital distances (see legend).Also shown in (a) is the mantle surface heat flow Q surf as a function of temperature (solidblack) and constant runaway greenhouse threshold (dashed). (b) Timescales for orbitalcircularization using the Maxwell model (same colors as in (a)) and mantle cooling (solidblack).Figure 4: Thermal-orbital evolution for three models with initial orbits of e (0) = 0 . a (0) = 0 .
01 (red), 0 .
02 (black), and 0 .
05 (blue). The temperature that corresponds to 50%melt fraction is denoted by the dashed line. 29a) (b)(c) (d)(e) (f)Figure 5: Time evolution of models with initial orbits of e = 0 . a = 0 .
01 (red), 0 . .
05 (blue). (a) Temperature in the mantle (solid) and core (dashed). (b) Heatflow at the surface Q surf (solid), tidal dissipation Q tidal (dash-dot), mantle radiogenic heating(dotted), mantle latent heat (dash-dot-dot), and core heat flow Q cmb (dashed). The runawaygreenhouse heat flow threshold (1 . × TW) is label as a solid grey line. (c) Orbitaldistance. (d) Eccentricity. (e) Magnetic moment of core dynamo (solid) and inner coreradius (dashed). Inner core radius axis has been scaled by core radius so the top correspondsto a completely solid core. For reference, Earth’s present day magnetic moment is about 80ZAm . (f) Melt mass flux to the surface. Melt eruption fluxes for present-day mid-oceanridges (10 kg yr − ) and the Siberian traps (10 kg yr − ) shown for reference (grey dashed).30a) (b)Figure 6: Contour of orbital evolution after 10 Gyr for a range of initial orbital distancesand eccentricities. (a) Change in orbital distance: ( a − a ) /a . (b) Change in eccentricity:( e − e ) /e . Orbits are free to evolve in both panels. The habitable zone in denoted byvertical dashed white lines.(a) (b)Figure 7: Contour of (log) tidal heat flow after 10 Gyr for a range of initial orbital distancesand eccentricities. (a) Orbit evolves. (b) Orbit is fixed. The tidal heat flow boundariesdefined by Barnes and Heller (2013) are shown for Earth Twins Q tidal <
20 TW, TidalEarths 20 < Q tidal < Q tidal > Q ∗ surf = 40 TW).(a) (b)Figure 9: Contour of core heat flow after 10 Gyr for a range of initial orbital distances andeccentricities. (a) Orbit evolves. (b) Orbit is fixed.32a) (b)Figure 10: Contour of mantle temperature after 10 Gyr for a range of initial orbital distancesand eccentricities. (a) Orbit evolves. (b) Orbit is fixed.(a) (b)Figure 11: Contour of core temperature after 10 Gyr for a range of initial orbital distancesand eccentricities. Line contours show solid core fraction R ic /R c as a percentage (i.e. 100%corresponds to a completely solid core). (a) Orbit evolves. (b) Orbit is fixed.33a) (b)Figure 12: Contour of magnetic moment after 10 Gyr for a range of initial orbital distancesand eccentricities. (a) Orbit evolves. (b) Orbit is fixed. For reference, Earth’s present daymagnetic moment is about 80 ZAm .(a) (b)Figure 13: Contour of surface melt mass flux after 10 Gyr for a range of initial orbitaldistances and eccentricities. (a) Orbit evolves. (b) Orbit is fixed. White line contourdenotes Earth’s approximate present-day mid-ocean ridge melt flux (10 kg yr − ). Notecolor scales in (a) and (b) are different. 34a) (b)(c) (d)Figure 14: Time evolution of a model with fixed orbit of e = 0 . a = 0 .
02 AU. (a)Temperature in the mantle (solid) and core (dashed). (b) Heat flow at the top of the mantle Q conv (solid), tidal Q tidal (dash-dot), mantle radiogenic heating (dotted), and core heat flow Q cmb (dashed). (c) Magnetic moment of core dynamo (solid) and inner core radius (dashed).Inner core radius axis goes from zero to total core radius. For reference, Earth’s present daymagnetic moment is about 80 ZAm . (d) Melt mass flux to the surface. Melt eruption fluxesfor present-day mid-ocean ridges (10 kg yr − ) and the Siberian traps (10 kg yr − ) shownfor reference (grey dashed). 35a) (b)Figure 15: Contour of time spent in an internally driven runaway greenhouse, defined aswhen surface heat flow exceeds the threshold for a runaway greenhouse (300 W m − , or1 . × W), for a range of initial orbital distances and eccentricities. (a) Orbit evolves.(b) Orbit is fixed.(a) (b)Figure 16: Contour of (a) time spent in a tidally dominated state (i.e. Q tidal /Q total ≥ . Q surf = 40 TW). In (b) a whitecontour line is shown at 4 . ppendixA Tidal Dissipation Model This section demonstrates the dependence of the tidal dissipation model and materialproperties on mantle temperature. Figure B.1 shows several parameters related tothe tidal dissipation rate as a function of mantle temperature for the nominal shearmodulus activation energy of A µ = 2 × J mol − . The Maxwell model that usesthe full form of − Im ( k ) in (4) differs from the common approximation of − Im ( k ) ≈ k / Q for mantles hotter than the present-day ( T m > Q ∝ η/µ ,which is constant, whereas the Maxwell model predicts a sharp drop in tidal dissipationwith viscosity when µ/β << / B Thermal History Model
B.1 Geotherm
The mantle temperature profile is assumed to be adiabatic everywhere except in thethermal boundary layers where it is conductive. The adiabatic temperature profile inthe well mixed region of the mantle is approximated to be linear in radius, which is agood approximation considering that mantle thickness D = 2891 km is much less thanthe adiabatic scale height H = c p /αg ≈ T ad = T UM + γ ad ( R − r − δ UM ) , (18) here the adiabatic gradient is γ ad ≈ . T UM erf (cid:20) R − rδ UM (cid:21) + T s , Upper mantle (19)∆ T LM erf (cid:20) R c − rδ LM (cid:21) + T cmb , Lower mantle (20)replace the adiabat. Thermal boundary layer temperature jumps are ∆ T UM = T UM − T g and ∆ T LM = T cmb − T LM , and thermal boundary layer depth is δ . Figure B.2shows an example whole planet geotherm T ( r ) at four times in the evolution. Surfacetemperature T g is assumed to be equal to the equilibrium temperature, T eq = (cid:18) L ∗ πσa (cid:19) / , (21)where L ∗ is stellar luminosity and σ is the Stefan-Boltzmann constant.The core temperature profile is assumed to be adiabatic throughout the entirecore, i.e. the thermal boundary layers within the core are ignored. This is a goodapproximation because the low viscosity and high thermal conductivity of liquid ironproduce very small thermal boundary layers that are insignificant on the scale of thewhole planet. The core adiabatic profile is approximated by T c ( r ) = T cmb exp (cid:18) R c − r D N (cid:19) , (22)where D N ≈ T F e = T F e, exp (cid:20) − (cid:18) − γ c (cid:19) r D F e (cid:21) , (23)where T F e, = 5600 K, γ c is the core Gruneisen parameter, and D F e = 7000 km is aconstant length scale (Labrosse et al., 2001). This simple treatment of the core solidusdoes not account for volatile depression of the solidus, which has been demonstrated xperimentally (Hirose et al., 2013), and would act to slow inner core growth. Innercore radius can then be solved for by finding the intersection of (22) and (23). Fordetails see Driscoll and Bercovici (2014). B.2 Mantle and Core Heat Flows
In this section we define the remaining heat flows that appear in the mantle (10) andcore (11) energy balance.The convective cooling of the mantle Q conv is proportional to the temperaturegradient in the upper mantle thermal boundary layer, Q conv = Ak UM ∆ T UM δ UM , (24)where A is surface area and k UM is upper mantle thermal conductivity. Q conv is writtenin terms of T m and the thermal boundary layer thickness δ UM by requiring that theRayleigh number of the boundary layer Ra UM be equal to the critical Rayleigh numberfor thermal convection Ra c ≈
660 (Howard, 1966; Solomatov, 1995; Sotin and Labrosse,1999; Driscoll and Bercovici, 2014). This constraint gives, Q conv = Ak UM (cid:18) αgRa c κ (cid:19) β ( (cid:15) UM ∆ T m ) β +1 ( ν UM ) − β , (25)where the thermal boundary layer temperature jump ∆ T UM has been replaced by∆ T UM ≈ (cid:15) UM ∆ T m , (cid:15) UM = exp( − ( R UM − R m ) αg/c p ) ≈ . T m = T m − T g , and the mantle cooling exponent is β = 1 / U , U , T h , and K , which is approximated in the mantle by, Q rad ( t ) = Q rad, exp( − t/τ rad ) , (26) here Q rad, is the initial radiogenic heat production rate at t = 0 and τ rad is theradioactive decay time scale that approximates the decay of the four major isotopes.The precise bulk silicate Earth radiogenic heat production rate is somewhat uncertain,so we use a nominal value of Q rad ( t = 4 . Q cmb = A c k LM ∆ T LM δ LM , (27)where A c is core surface area and k LM is lower mantle thermal conductivity. The lowermantle and CMB temperatures, T LM and T cmb , are extrapolations along the mantleand core adiabats: T LM = (cid:15) LM T m and T cmb = (cid:15) c T c , where (cid:15) LM = exp( − ( R LM − R m ) αg/c p ) ≈ . (cid:15) c ≈ .
8. The lower mantle thermal boundary layer thicknessis also derived by assuming the boundary layer Rayleigh number is critical and that ν LM = 2 ν UM , which was found by Driscoll and Bercovici (2014) to produce a nominalEarth model.Core secular cooling is Q core = − M c c c ˙ T c , (28)where M c is core mass, c c is core specific heat, and ˙ T c is the rate of change of theaverage core temperature T c .Radiogenic heat in the core is produced primarily by the decay of K (Gessmannand Wood, 2002; Murthy et al., 2003; Corgne et al., 2007). Its time dependence istreated the same as mantle radiogenic heat in (26), but with a radioactive decay timescale of τ rad,c = 1 . K in the core that correspondsto 2 TW of heat production after 4.5 Gyr. .3 Melting The mantle solidus is approximated by a third-order polynomial (Elkins-Tanton, 2008), T sol ( r ) = A sol r + B sol r + C sol r + D sol , (29)where the coefficients are constants (see Table 1). This solidus is calibrated to fit thefollowing constraints: solidus temperature of 1450 K at the surface, solidus temperatureof 4150 K at the CMB (Andrault et al., 2011), and present-day upwelling melt fractionof f melt = 8%. The liquidus is assumed to be hotter by a constant offset ∆ T liq = 500K, so T liq ( r ) = T sol ( r ) + ∆ T liq .Mantle melt heat loss (or advective heat flow) is modeled as, Q melt = (cid:15) erupt ˙ M melt ( L melt + c m ∆ T melt ) , (30)where (cid:15) erupt = 0 . M melt is melt mass flux (see below), L melt is latent heat of the melt, c m is specific heat of the melt, and ∆ T melt is the excesstemperature of the melt at the surface (see below). This formulation of heat loss issimilar to the ”heat pipe” mechanism invoked for Io (O’Reilly and Davies, 1981; Moore,2003), where melt is a significant source of heat loss. We note that this mechanismis more important for stagnant lid planets where the normal conductive heat flow islower (Driscoll and Bercovici, 2014).The melt mass flux ˙ M melt is the product of the upwelling solid mass flux times themelt mass fraction f melt , ˙ M melt = ˙ V up ρ solid f melt ( z UM ) , (31)where solid density is ρ solid , volumetric upwelling rate is ˙ V up = 1 . κA p /δ UM , z UM = − δ UM , and melt fraction is f melt ( z ) = T m ( z ) − T sol T liq − T sol . (32)This model predicts a ridge melt production of ˙ M melt = 2 . × kg s − for δ UM = 80km and f melt = 0 .
1, similar to present-day global melt production estimates (Cogn´eand Humler, 2004).We define the magma ocean as the region of the mantle with temperature exceedingthe liquidus. Given the geotherm in (18,20) and the liquidus T liq ( r ) similar to (29),the mantle will mainly freeze from the bottom of the convecting mantle up because theliquidus gradient is steeper than the adiabat (e.g. Elkins-Tanton, 2012). However, ifthe core is hot enough a second melt region exists in the lower mantle boundary layer,where the temperature gradient exceeds the liquidus and the mantle freezes towardsthe CMB. As can be seen in Figure B.2, a basal magma ocean exists for about 4 Gyrbefore solidifying.Latent heat released from the solidification of the mantle is Q L,man = ˙ M sol L melt , (33)where L melt is the latent heat released per kg and ˙ M sol is the solid mantle growthrate. The growth rate is calculated assuming a uniform mantle density ρ m so that˙ M sol = ρ m ˙ V sol , where ˙ V sol = − ˙ V liq . The rate of change of the liquid volume of themantle is ˙ V liq = dV liq dT m ˙ T m , (34)where ˙ T m is the mantle secular cooling rate and dV liq /dT m is linearly approximated by8 × m K − , which is the change in liquid volume from a 90% liquid to a completelysolid mantle. This approximation implies that the latent heat released due to mantlesolidification is linearly proportional to the mantle secular cooling rate, and the ratioof the latent heat flow to the mantle secular cooling heat flow is Q L,man /Q sec,m ≈ . or example, a mantle solidification time of 100 Myr corresponds to an average latentheat release of Q L,man ≈
400 TW over that time.
B.4 Core Dynamo
Given the thermal cooling rate of the core, the magnetic dipole moment M is estimatedfrom the empirical scaling law, M = 4 πR c γ d (cid:112) ρ/ µ ( F c D c ) / (35)where γ d = 0 . µ =4 π × − H m − is magnetic permeability, D c = R c − R ic is the dynamo region shellthickness, R c and R ic are outer and inner core radii, respectively, and F c is the corebuoyancy flux (Olson and Christensen, 2006). We assume that the field is dipolar,ignoring the complicating influences of shell thickness and heterogeneous boundaryconditions (e.g. Heimpel et al., 2005; Driscoll and Olson, 2009; Aubert et al., 2009;Olson et al., 2014). In this formulation a positive buoyancy flux implies dynamo action,which is a reasonable approximation when the net buoyancy flux is large, but mayoverestimate the field strength at low flux. The total core buoyancy flux F c is the sumof thermal and compositional buoyancy fluxes, F c = F th + F χ (36)where the thermal and compositional buoyancy fluxes are F th = α c g c ρ c c c q c,conv (37) F χ = g i ∆ ρ χ ρ c (cid:18) R ic R c (cid:19) ˙ R ic , (38)where the subscript c refers to bulk core properties, core convective heat flux is q c,conv = q cmb − q c,ad , gravity at the ICB is approximated by g ic = g c R ic /R c , and the outer core ompositional density difference is ∆ ρ χ = ρ c − ρ χ with ρ χ the light element density.For simplicity, the expression for light element buoyancy (38) ignores buoyancy due tolatent heat release at the ICB because it is a factor of 3 . q c,ad = k c T cmb R c /D N , (39)where core thermal conductivity is approximated by the Wiedemann-Franz law, k c = σ c L c T cmb , (40)and electrical conductivity is σ c and L c is the Lorentz number. For typical values ofhigh pressure-temperature iron, σ c = 10 × Ω − m − (Pozzo et al., 2012; Gomi et al.,2013), L c = 2 . × − WΩK − , and T cmb = 4000 K, the core thermal conductivity is k c = 100 Wm − K − . References
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A turbulent,high magnetic Reynolds number experimental model of Earth’s core. Journal ofGeophysical Research: Solid Earth 119 (6), 4538–4557. uluaga, J. I., Bustamante, S., Cuartas, P. A., Hoyos, J. H., 2013. The influence ofthermal evolution in the magnetic protection of terrestrial planets. The AstrophysicalJournal 770 (1), 23. ymbol Value Units Reference A ν × J mol − Viscosity activation energy in (6) A µ × J mol − Nominal shear modulus activation energy in (7) A sol − . × − K / m Solidus coefficient in (29) (ET08) α × − K − Thermal expansivity of mantle α c × − K − Thermal expansivity of core B . B sol . × − K / m Solidus coefficient in (29), calibrated β / β st . × GPa Effective mantle stiffness, calibrated in § c m − K − Specific heat of mantle c c
840 J kg − K − Specific heat of core C sol − . × − K / m Solidus coefficient in (29), calibrated D D F e D N D sol . × K Solidus coefficient in (29), calibrated δ ph E G × J kg − Gravitational energy density release at ICB (cid:15) UM (cid:15) LM (cid:15) c φ ∗ . g UM . − Upper mantle gravity g LM . − Lower mantle gravity g c . − CMB gravity γ c . γ dip ymbol Value Units Reference γ ph k UM . − K − Upper mantle thermal conductivity k LM
10 W m − K − Lower mantle thermal conductivity κ m s − Mantle thermal diffusivity L F e
750 kJ kg − Latent heat of inner core crystallization L melt
320 kJ kg − Latent heat of mantle melting L e . × − W Ω K − Lorentz number L ∗ . × W Stellar luminosity for M ∗ = 0 . M sun (B13) M m . × kg Mantle mass M c . × kg Core mass µ ref . × Pa Reference shear modulus in (7) µ π × − H m − Magnetic permeability ν ref × m s − Reference viscosity ν LM /ν UM Q rad,
60 TW Initial mantle radiogenic heat flow (J07) R R c R m T m Ra c
660 nd Critical Rayleigh number ρ c − Core density ρ ic − Inner core density ρ m − Mantle density ρ melt − Mantle melt density ρ solid − Mantle upwelling solid density∆ ρ χ
700 kg m − Outer core compositional density difference σ c × S m − Core electrical conductivity T F e, ymbol Value Units Reference τ rad .
94 Gyr Mantle radioactive decay time scale τ rad,c . ξ × − nd Rheology phase coefficient in (8, 9)nd Rheology phase coefficient in (8, 9)