Ohmic Dissipation in the Atmospheres of Hot Jupiters
DD RAFT VERSION O CTOBER
24, 2018
Preprint typeset using L A TEX style emulateapj v. 11/26/04
OHMIC DISSIPATION IN THE ATMOSPHERES OF HOT JUPITERS R OSALBA P ERNA , K RISTEN M ENOU
AND E MILY R AUSCHER
Draft version October 24, 2018
ABSTRACTHot Jupiter atmospheres exhibit fast, weakly-ionized winds. The interaction of these winds with the planetarymagnetic field generates drag on the winds and leads to ohmic dissipation of the induced electric currents. Westudy the magnitude of ohmic dissipation in representative, three-dimensional atmospheric circulation modelsof the hot Jupiter HD 209458b. We find that ohmic dissipation can reach or exceed 1% of the stellar insolationpower in the deepest atmospheric layers, in models with and without dragged winds. Such power, dissipated inthe deep atmosphere, appears sufficient to slow down planetary contraction and explain the typically inflatedradii of hot Jupiters. This atmospheric scenario does not require a top insulating layer or radial currents thatpenetrate deep in the planetary interior. Circulation in the deepest atmospheric layers may actually be drivenby spatially non-uniform ohmic dissipation. A consistent treatment of magnetic drag and ohmic dissipationis required to further elucidate the consequences of magnetic effects for the atmospheres and the contractinginteriors of hot Jupiters.
Subject headings: INTRODUCTION
The discovery of the first transiting extrasolar planetHD 209458b (Charbonneau et al. 2000) has opened a newchapter for the study of planetary bodies. This planet belongsto the class of hot Jupiters, which are close-in gaseous giantplanets thought to be tidally locked to their host star. In thelast decade, a wealth of observations has allowed direct in-vestigations of the atmospheric and bulk properties of theseplanets, while much theoretical work has been aimed at theinterpretation of these data (see, e.g., Showman et al. 2010,Burrows & Orton 2010, Baraffe et al. 2010 and Seager &Deming 2010 for recent reviews). A property of hot Jupiterswhich has constituted a longstanding puzzle is their anoma-lously large radii, which have been interpreted as requiringthat extra heat be deposited in the convective regions, or alter-natively in the deep atmospheres (at pressure of tens of bars,e.g. Guillot & Showman 2002) of these planets.In recent years, progress in modeling the unusual atmo-spheric circulation regime of hot Jupiters, with permanent dayand night sides, has also been made (e.g. Cooper & Show-man 2005; Dobbs-Dixon & Lin 2008; Showman et al. 2009;Rauscher & Menou 2010; see Showman et al. 2010 for areview). These purely hydrodynamical models have gener-ally ignored the possibility that magnetic effects acting on theweakly-ionized winds could significantly influence the circu-lation pattern.Recently, Perna et al. (2010) evaluated the level of mag-netic drag on a representative hot Jupiter atmospheric flow,and argued that it is likely to provide an effective frictionalmechanism to limit the asymptotic speed of winds in theseatmospheres. Batygin & Stevenson (2010) studied the roleof ohmic dissipation in hot Jupiters, using a prescribed zonalwind profile and a one-dimensional atmospheric structuremodel, to argue that the typical amount of heat deposited deepin these planets’ convective zones is sufficient to explain their JILA and Department of Astrophysical and Planetary Sciences, Univer-sity of Colorado, Boulder, CO, 80309 Department of Astronomy, Columbia University, 550 West 120th Street,New York, NY 10027 Kavli Institute for Theoretical Physics, UCSB, Santa Barbara, CA 93106 inflated radii. In this
Letter , we build on our previous workon magnetic drag (Perna et al. 2010; hereafter PMR10) toinvestigate the magnitude of ohmic dissipation in the atmo-spheres of hot Jupiters, and its consequences for the dynam-ics and the thermal evolution of these planets. We use spe-cific three-dimensional atmospheric circulation models of theplanet HD 209458b. We compare the amount of Ohmic heat-ing expected for typical magnetic field strengths with the extraheat required in the deep atmosphere to slow down contrac-tion according to planetary evolutionary models (Guillot &Showman 2002). While our calculations are specific to thecase of HD 209458b, our overall results are expected to holdmore generally for hot Jupiters with similar gravity, irradia-tion strength, and magnetic field. ATMOSPHERIC CURRENTS AND OHMIC DISSIPATION
Atmospheric models and ionization balance
The fiducial atmospheric circulation model used here wascomputed by Rauscher & Menou (2010) for HD 209458b,under the assumption of no significant magnetic drag on theatmospheric flow. The model describes the atmospheric flowin a frame that is rotating with the bulk planetary interior. Themeridional and zonal wind speeds in the atmosphere, as wellas its thermodynamic variables, are returned at each grid lo-cation in the three-dimensional model atmosphere. Locationis identified by the angular spherical coordinates ( θ, φ ) andpressure, p , for the vertical coordinate, The model bottomis located at 220 bar while the top level is set at a pressureof 1 mbar. In addition to this drag-free model, we also per-form some of our calculations for the model with strongestdrag described in PMR10. Since, apart from wind drag, thesetwo models are identical, this allows us to evaluate the con-sequences for ohmic dissipation of an atmospheric flow withsignificantly dragged winds.In our circulation models, the local heating/cooling rate (en-ergy per unit mass) is modeled as Newtonian (linear) relax-ation, Q T = ( T eq ( p , θ, φ ) − T ) /τ rad ( p ), where τ rad represents theradiative timescale on which the local temperature T relaxesto the prescribed equilibrium profile T eq . The nature of theatmospheric circulation obtained with this simplified forcingcompares well to what is obtained with more realistic forcing a r X i v : . [ a s t r o - ph . E P ] S e p (e.g., Showman et al. 2009). Following Cooper & Showman(2005), Rauscher & Menou (2010) relied on the work of Iroet al. (2005) to implement the detailed profiles of this radia-tive forcing. The atmosphere is divided into actively forcedlayers, above the 10 bar level, and “inert” layers below. Theradiative forcing is assumed to be negligible in the inert lay-ers, which corresponds to τ rad → ∞ . The active layers, on theother hand, are forced on a finite radiative timescale, τ rad ( p ).The local atmospheric temperatures obtained after model re-laxation hover at about 1800 K in the deepest levels, while inthe upper levels they range from about 500 K on the night sideto about 1500 K on the day side (see Rauscher & Menou 2010for details ).At these temperatures, the primary source of free electronsstems from thermally ionized alkali metals with low first ion-ization potentials: Na, Al, K. For simplicity here (and con-sistently with PMR10), we adopt an approximation to Saha’sequation (Balbus & Hawley 2000) which assumes potassiumto be the dominant contributing species: x e ≡ n e n n = 6 . × − (cid:16) a K − (cid:17) / (cid:18) T (cid:19) / × (cid:18) . × n n (cid:19) / exp( − / T )1 . × − . (1)Here n e and n n are respectively the electron and neutral num-ber densities (in cm − ) and a K (cid:39) − is the potassium solarabundance. As discussed in PMR10, equation (1) is a goodapproximation as long the resulting ionization fraction, x e , is (cid:28) a K ; this condition is satisfied in our atmosphere models.We assume that the gas is overall neutral, which implies anequality between the electron number density n e and the ionicone n i . The electrical conductivity and associated resistivityare given by (see also Laine et al. 2008) σ e = n e e m e n n (cid:104) σ v (cid:105) e and η = c πσ e , (2)respectively, with the collision rate between electrons andneutrals approximated as (Draine et al 1983) (cid:104) σ v (cid:105) e = 10 − (cid:18) kT π m e (cid:19) / cm s − . (3) Induced currents and Ohmic dissipation
The first step towards the computation of currents involvesan estimate of the importance of various non-ideal MHDterms in the full induction equation for the weakly-ionizedmedium. For the atmospheric models under considerationhere, PMR10 found that, for a surface magnetic field strengthof 3 G, the Hall term is completely negligible throughoutthe flow, while the resistive term largely dominates over theambipolar diffusion term everywhere except possibly in themodel uppermost levels. Therefore, to a good approximation,the induction equation can be considered as purely resistive. While the simulations by Rauscher & Menou (2010) adopt an approx-imate treatment for radiative transfer, other simulations by Showman et al.(2009) with explicit radiative transfer find rather similar results for the day-night temperature gradient and wind pattern under comparable physical con-ditions. However, it should be recognized that all these models are subject tosome uncertainties for the development of the deep atmospheric winds due tothe very long integration times needed for spin-up. F IG . 1.— Cylindrical maps of Joule heating times (log τ J , in sec), at fourpressure levels in our fiducial, drag-free atmospheric model. The sub-stellarpoint is centered at longitude and latitude zero. From top to bottom, Jouleheating times are shown at 1 mbar, 50 mbar, 2 bar, 90 bar. Ohmic dissipationis not spatially uniform and it can dominate over heating by stellar insolationat deep enough pressure levels ( p ∼ > B = 3 G,but note that ohmic dissipation increases steeply with magnetic field strength( τ J ∝ B − ). Assuming that zonal winds are dominant, one needs only toconsider the toroidal component of the induction equation ∂ B φ ∂ t = r sin θ (cid:20) ∂ Ω ∂ r B r + r ∂ Ω ∂θ B θ (cid:21) + r ∂∂ r (cid:20) η ∂∂ r ( rB φ ) (cid:21) + r ∂∂θ (cid:20) η sin θ ∂∂θ (sin θ B φ ) (cid:21) , (4)where Ω = v φ ( r sin θ ) − is the local angular velocity of theflow, r is the radial spherical coordinate, and the magneticfield is assumed to be an axisymmetric dipole.In PMR10, we also argued that, to leading order, the lati-tudinal component of the current induced by the zonal flowshould be dominant. This assumption could be verified inmore general versions of our models. Since the magneticReynolds number is R m (cid:28) B r , B θ ) is maintained by currents inthe interior of the planet, while the zonal flows separately in-duce a B φ component from the dipolar one in the superficialatmospheric layers. Under these conditions (Liu et al. 2008),the resulting steady-state latitudinal current can be computedas: j θ ( r , θ, φ ) = − c sin θ π r η ( r , θ, φ ) (cid:90) Rr dr (cid:48) r (cid:48) (cid:18) ∂ Ω ∂ r (cid:48) B r + r (cid:48) ∂ Ω ∂θ B θ (cid:19) + R η ( R , θ, φ ) r η ( r , θ, φ ) j θ ( R , θ, φ ) , (5)where the last term includes a boundary current in the upper-most modeled level, j θ ( R , θ, φ ). Lacking information aboutthe nature of currents possibly flowing from regions above themodeled atmospheric layers, we set this boundary current tozero for simplicity. As discussed in PMR10, this unknownboundary current represents an important source of uncer-tainty in our modeling, but, unless near cancellations occur,additional boundary currents could in principle contribute toeven stronger ohmic dissipation than estimated here.The ohmic power per unit volume dissipated locally is read-ily computed as (e.g., Liu et al. 2008) Q J ( r , θ, φ ) = [ j θ ( r , θ, φ )] σ e ( r , θ, φ ) . (6) Joule-driven Circulation
We first examine the possibility that spatially non-uniformohmic dissipation drives a circulation in regions of the atmo-sphere where it is comparable or stronger than heating dueto stellar insolation. To permit a direct comparison betweenthe two heating sources, we find it convenient to define a typ-ical local timescale associated with Joule heating, which isdeduced from the energy equation ρ C p dTdt = Q J , (7)so that the Joule heating time is τ J ∼ ρ C p TQ J . (8)The specific heat of the gas in our model atmospheres is C p =1 . × erg g − K − .Fig. 1 shows cylindrical maps of Joule heating times fora nominal surface dipolar field of 3 G, at four depths in the atmospheric flow of our fiducial, drag-free model. In the up-permost levels, the Joule heating times span almost 20 ordersof magnitude, with the shortest times found on the day side.This wide span reflects the large variations in resistivity, η ,between the day and the night sides. On the day side, whichhas higher temperatures, η is considerably smaller than onthe night side, yielding larger currents, which in turn resultin larger Q J and correspondingly smaller τ J values. On theother hand, deep in the inert layers, which are little affectedby stellar irradiation, Joule heating times span a more mod-est range of values, ∼ − s over a large fraction of theflow. Longer timescales along the equator in the deeper levelsare mostly the result of geometric effects: in Eq.(5), the term( ∂ Ω /∂ r ) B r generally dominates over 1 / r ( ∂ Ω /∂θ ) B θ . Since Ω ∝ sin − θ and B r ∝ cos θ , the net result is j ∝ cos θ , andhence τ J ∝ cos − θ . For a non-axisymmetric dipolar field, theanisotropy pattern would be different.To evaluate the possibility of Joule-driven circulation, wecompare local Joule heating times with a representative heat-ing time associated with stellar irradiation, τ irr . For con-sistency with the Newtonian forcing scheme used in ouratmospheric models, we adopt a simple downward insola-tion flux that approximately matches the absorption prop-erties of the one-dimensional models computed by Iro etal. (2005) at a few bars level. The flux is taken to obey F ( p ) = F exp[ − . p / (2bar)], where F = 2 . × erg cm − s − is the incident flux at the model top for HD 209458b, aftergeometric dilution by a factor 1 / τ irr , is then computed following Eq. (8),with a heating rate taken as the vertical divergence of the stel-lar irradiation flux, Q irr = − dF ( r ) / dr . This allows us to eval-uate τ irr simply as a function of the local pressure, pressurescale height and density. For example, at p = 2 bar, using ρ ∼ . × − g cm − , we estimate τ irr ∼ × s. At p = 10bar, where the stellar insolation flux has been very strongly at-tenuated already (Iro et al. 2005), we estimate τ irr ∼ × sfor ρ ∼ × − g cm − . Deeper in the atmosphere, the inso-lation flux is further attenuated (exponentially so) and heatingby insolation becomes largely inconsequential.A comparison between values for τ irr and the detailed mapsof τ J values in Fig. 1 shows that, for the nominal B = 3 G usedin our τ J calculation, irradiation dominates over Joule heating( τ irr (cid:28) τ J ) everywhere on the day side of the atmosphere atpressure levels above a few bars. In these upper regions, weexpect the atmospheric thermal structure to be largely unaf-fected by the extra Ohmic heating. Deeper than a few bars,however, Joule heating times start becoming comparable toor shorter than the typical heating time associated with stellarinsolation, over significant regions of the atmospheric flow.Eventually, at levels deeper than about 10 bar, ohmic dissipa-tion easily dominates over stellar insolation over much of theatmosphere. Since this extra source of heating in the deeperregions of the atmosphere is spatially non-uniform (Fig. 1),it should lead to some form of Joule-driven circulation inthe “inert” layers located well below the radiatively-forced“weather” layers. This conclusion is largely independent ofthe specific stellar insolation model adopted here since theexistence of radiatively inactive layers at levels deeper than afew bars is a rather generic property of irradiated hot Jupiteratmospheres (e.g., Hansen 2008). However, it remains to beseen what the nature of such Joule-driven circulation mightbe given the uneven heating pattern shown in Fig. 1, the non-uniformity of stellar insolation from the day-side to the night-side and the presence of an additional, spatially uniform netflux emerging from the deep planetary interior. Finally, it isworth remembering that, since τ J ∝ B − , a stronger field couldproduce a substantially more dominant ohmic dissipation thanestimated above for B = 3 G . Inflating Hot Jupiters
Another potentially important consequence of ohmic dis-sipation is the possibility that it contributes to the tendencyfor hot Jupiters to have inflated radii (Batygin & Stevenson2010). To evaluate the magnitude of this effect on the basis ofour three-dimensional atmospheric models, we compute thecumulative ohmic power dissipated in successive atmosphericshells, from the top level at a pressure p t of 1 mbar, down toa pressure p ( r ) within the atmosphere; this is readily obtainedby integration of Eq.(6), P J ( p ) = (cid:90) r ( p t ) r ( p ) dr (cid:48) r (cid:48) (cid:90) π d φ (cid:90) π d θ [ j θ ( r (cid:48) , θ, φ )] σ e ( r (cid:48) , θ, φ ) . (9)Note that the power as computed above includes a horizontalaverage over the day and night sides. This averaging washesout the horizontal features in Joule heating shown in Fig. 1, anissue deserving further attention in future multi-dimensionalmodels. Fig. 2 displays this ohmic power as a function ofpressure for our fiducial, drag-free atmospheric model and forthe model with strongest drag described in PMR10, with zonalwind speeds typically reduced by about 30 − B = 3 G, and B = 10 G. This allows us to sepa-rately explore the effects of drag on the zonal winds and of themagnetic field strength on the resulting magnitude of ohmicdissipation. This is a useful exercise until models with a self-consistent treatment of magnetic drag and ohmic dissipationbecome available.Fig. 2 shows that much of the dissipated ohmic powerbuilds up at pressure levels from a few bars to several tensof bars. The power scales as B and, relative to the drag-freemodel, it is reduced by a factor ∼ ∼ × W, Fig. 2 shows that the total ohmic power dis-sipated in these model atmospheres typically approaches orexceeds 1% of the stellar insolation power. Notice that, whilethe Ohmic power in Fig.2 is integrated down to the lowestgrid points ( ∼
200 bar) available in the model of Rauscher& Menou (2010), it is not yet seen to saturate at those lev-els. This is because, while the zonal velocities decrease withdepth, they are still non-zero in the deepest, ’inert’ model lay-ers. It is in fact not a priori clear where exactly the transitionto the convection zone should be located in hot Jupiters andthis constitutes an uncertainty for Joule heating models of thedeepest atmospheric layers.Overall, our results on the dissipated ohmic power areconsistent with the estimates made by Batygin & Stevenson The magnetic field strengths of hot Jupiters are unconstrained from anobservational point of view. In the case of Jupiter, the measured field is 14 Gat the pole and 4.2 G around the equator. Arguments (e.g. Christensen et al.2009) suggest that the field strength should scale with the square root of thedensity and the spin frequency, which could make the B field in hot Jupiterssomewhat weaker. Calculations of the internal structure and convective mo-tions of giant planets (Sanchez-Lavega 2004) yield surface magnetic fields ∼ − IG . 2.— Cumulative ohmic power dissipated above pressure level p , infour different models for HD 209458b. The solid lines correspond to the drag-free model described in Rauscher & Menou (2010) while the dashed linescorrespond to the model with strongest drag, and reduced winds, describedin Perna et al. (2010). Curves are labeled with the value of the magnetic fieldadopted for the ohmic dissipation calculation. The ohmic power can reachor exceed 1% of the stellar insolation power, ∼ × W, in the deepestatmospheric layers, in models with and without dragged winds. (2010) on the basis of a more idealized, one-dimensional at-mosphere model, using a parametrized zonal wind profile.Like them, we find that a large fraction of the ohmic power isdissipated in the deeper atmospheric levels, below ∼
10 bar.Contrary to Batygin & Stevenson (2010), however, we do notsolve for radial currents in our models, only meridional ones(Eq. 5), and our results are thus largely independent of thepresence or absence of an insulating layer high in the atmo-sphere. In fact, we find that an insulating layer is present onthe nightside in our three-dimensional models, but not neces-sarily on the much hotter, less resistive dayside. This raisesthe possibility that, in more detailed three-dimensional MHDmodels, current loops could actually close high in the atmo-sphere, rather than deep in the planetary interior as conjec-tured by Batygin & Stevenson (2010). While our results, re-lying on leading-order meridional currents in the atmosphericregion alone, are presumably not strongly sensitive to theseconditions affecting radial currents, it remains to be seen howcurrents would flow in more realistic non-axisymmetric andthree-dimensional models.Various authors have evaluated the extra power that needsto be continuously dissipated deep in the convective interiorof hot Jupiters to explain their inflated radii (e.g., Gu, Boden-heimer & Lin 2004, Burrows et al. 2007, Ibgui et al. 2009).While Batygin & Stevenson (2010) emphasize such a deep de-position scenario, our own ohmic dissipation models say littleabout this scenario since we only calculate currents and ohmicdissipation in the superficial atmospheric region of the planet.However, another means by which hot Jupiter radii can be in-flated is by slowing down their rate of contraction, throughmodifications to the thermal structure of their overlaying at-mospheres which act as boundary conditions for the coolingisentropic interiors. Indeed, Guillot & Showman (2002) ar-gue that, if ∼
1% of the stellar insolation flux were depositedat pressures of tens of bars deep in the atmosphere, this wouldslow down cooling sufficiently to explain the inflated radii ofhot Jupiters. While Guillot & Showman (2002) suggested thata downward flux of kinetic energy could in principle achievesuch energy deposition, our Fig. 2 indicates that this is in factnaturally achieved by ohmic dissipation in our atmosphericmodels, with or without drag, for a magnetic field strength B ∼ >
10 G.More specifically, Guillot & Showman (2002) describe anevolutionary model in which 2 . × W are deposited ata location centered around 21 bar in the atmosphere of HD209458b, which can explain its observed inflated radius. Thisis achieved by the two ohmic dissipation models representedby the upper dashed and solid lines in our Fig. 2. In fact, sinceeven more ohmic power is dissipated deeper in, we anticipatethat models with fields even weaker than 10 G, possibly as lowas 3 G, will be able to meet the inflated radius requirement forHD 209458b. This leads us to conclude that ohmic dissipa-tion deep in the atmospheres of hot Jupiters, which indirectlytaps into the kinetic energy of dragged winds driven by stel-lar insolation higher up, is a promising scenario to explain theinflated radii of hot Jupiters. SUMMARY AND CONCLUSIONS
In this
Letter , we have computed the rate of ohmic dissipa-tion in representative, three-dimensional atmospheric circu-lation models of the hot Jupiter HD 209458b. We find that,for a fiducial magnetic field strength of 3 G, ohmic dissipa- tion starts dominating over stellar insolation heating at levelsdeeper than a few bars. The spatial non-uniformity of this ex-tra source of heat could induce Joule-driven circulation in thedeep layers traditionally considered as “inert”. For a magneticfield strength ∼ >
10 G, our models also indicate that enoughheat is deposited at pressures of several tens of bars to slowdown cooling sufficiently that the inflated radii of hot Jupiterscan be explained.Our results hence suggest that magnetic interactions in hotJupiter atmospheres play a fundamental coupling role for thedynamics and the thermal evolution of these planets. As such,our work calls for the problem to be treated more consistently:magnetic drag affects wind speeds and induces currents, whilethe ohmic dissipation of these currents can generate a deepatmospheric circulation which could, in turn, feedback on thecirculation higher up. Indeed, the extra heat source might alsoenhance convection in the night side, which might in turn en-hance cooling. These various ingredients will have to be in-corporated consistently in circulation models for a better as-sessment of their influence on the structure and the evolutionof hot Jupiters. Some diversity may naturally arise from vari-ations in the magnetic field strength and geometry of differentplanets.We thank Tristan Guillot for useful discussions, and AdamBurrows, Jeremy Goodman and the referee Douglas Lin forhelpful comments on our manuscript. This work was sup-ported in part by the National Science Foundation under GrantNo. PHY05-51164.