OHMIC HEATING SUSPENDS, NOT REVERSES, THE COOLING CONTRACTION OF HOT JUPITERS

Since the discovery of the first transiting hot Jupiter (Charbonneau et al. 2000), the radii of these objects have remained puzzling. While cooling theory predicts that these planets, regardless of mass, should contract to ∼1 R_J (Jupiter radius) at the age of a few Gyrs, the observed radii for hot Jupiters range from 0.8 R_J to 2 R_J. Factors such as stellar irradiation, a varying core mass, and atmospheric metal content can at best cause ∼10%–20% variations in the radius (see, e.g., Burrows et al. 2007).

There is a trend that the observed radii rise with the amount of stellar irradiation (e.g., Enoch et al. 2011; Laughlin et al. 2011; also see data in Figure 5) though the most inflated planet discovered to date (WASP-17b; Anderson et al. 2011) is not the hottest one.

Lacking a mechanism of energy generation, isolated Jovian planets inexorably contract as they cool. The inflated hot Jupiters look much younger than their real ages suggest. Some mechanism is at work either to stall their aging or to rejuvenate them from old age (see proposals by, e.g., Bodenheimer et al. 2001; Chabrier & Baraffe 2007; Youdin & Mitchell 2010; Batygin & Stevenson 2010). The rejuvenation may be particularly relevant for hot Jupiters, as they are believed not to have formed in situ, but to have migrated inward after they formed (perhaps well after). We consider in detail the proposal of Batygin & Stevenson (2010) where a surface wind (see review by Showman et al. 2010) blowing across the planetary magnetic field acts as a battery that sends current to the interior and gives rise to Ohmic heating in the deeper layers. A number of other studies have also discussed Ohmic heating in the atmospheres of hot Jupiters (Perna et al. 2010b, 2012; Menou 2012). Our study here is similar in spirit to that of Batygin et al. (2011, hereafter B12), where they incorporate Ohmic heating (both in the atmosphere and in deeper layers) into calculations of planet thermal evolutions. They find that Ohmic heating can explain the inflated sizes of hot Jupiters, and that low-mass hot Jupiters can be inflated to become Roche lobe overflowing. In this work, we also find that Ohmic heating appears effective in stalling the cooling contraction, largely confirming results of B12, but we find that it is incapable of re-inflating the planets after they have contracted.

Our work builds on that of Batygin & Stevenson (2010) but we focus on the thermodynamical aspects; we elucidate the condition for Ohmic heating to be effective; and we investigate the interior structure of planets as affected by Ohmic heating. We are particularly interested in the question of planet rejuvenation. Our work shows that Ohmic heating, being a largely surface phenomenon, is ineffective at re-inflating planets, as this requires entropy deposition in the very center of planets.

2.1. General Criterion for Stalling Contraction

We consider the stalling of contraction in an irradiated planet. An elucidating work in this regard is Arras & Bildsten (2006). Our discussion here is similar in spirit to theirs.

The interior of the planet is largely convective due to the high opacity there. The resulting temperature profile is adiabatic, with an internal entropy S. The layer below the photosphere, however, is nearly isothermal with T ≈ T_eq, as a result of the strong stellar irradiation. We simplify the planet into a two-zone model, an upper isothermal envelope and an isentropic core (see Figure 1).⁴ We denote the depth of the transition point as z_tr. This depth determines the cooling rate of the planet.

As we now demonstrate, the value of the internal entropy (S) determines both the planet's radius (R) and its cooling luminosity (L). The first connection is straightforward—a higher S corresponds to a higher internal temperature and therefore a larger scale height. The size of the planet is mostly determined by its bottommost scale height. Thus, in a given planet, R is a monotonic function of S.

To establish the connection between S and L, it is useful to recognize that the planet's cooling is controlled exclusively by the optical depth and the temperature gradient at the transition point, z_tr (Arras & Bildsten 2006). Radiative diffusion allows an energy leakage that scales with the temperature gradient as

$$\begin{equation} F_{\rm rad} = {{ a c}\over {3 \kappa \rho }} {{dT^4}\over {dz}} = {{4 \sigma _B}\over 3}\, {{d T^4}\over {d\tau }}, \end{equation} \tag{ 1 }$$

where τ = ∫^z₀κρdz' is the optical depth measured from the surface, and σ_B the Stefan–Boltzmann constant. Below z_tr, the temperature gradient is adiabatic and the optical depth rises quickly; above z_tr, the temperature is nearly isothermal, T_tr ∼ T_eq. The radiative flux of the entire planet is therefore set by the optical depth and the (adiabatic) temperature gradient at z_tr and is roughly ∼σ_BT⁴_eq/τ_tr. Lastly, since a planet with a cooler interior has a deeper isothermal layer (larger z_tr, see Figure 1), it will therefore have a lower cooling flux.

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Hence, measuring the radius directly probes the internal entropy, which in turn informs us how fast the planet is losing energy. An inflated planet has a higher entropy, and therefore must lose heat faster. The only way to stall contraction is to supply this cooling luminosity, not from the internal energy of the planet, but from some other means (e.g., Ohmic heating, wave energy, etc.). Moreover, this extra energy source must be able to replace the cooling flux at depth z ⩾ z_tr.

In our calculations, we frequently find that a new convection zone develops in the wind zone, where Ohmic dissipation is at its greatest power. This has the effect of pushing z_tr to a deeper depth, for the same interior adiabat. However, it will not change the above discussion.

2.2. Constraints on Ohmic Heating

In this paper, we focus on the Ohmic heating mechanism (Batygin & Stevenson 2010), which is one of the best laid-out mechanisms for halting contraction. In this model, if the planet has a large-scale magnetic field that is anchored in the deep interior, and if its poorly conducting surface layer is differentially rotating relative to the interior, the current generated by pulling the field line in the surface wind layer returns through the interior, producing Ohmic heating in the interior.⁵

As established above, the relevant part of heating for stalling the planet's contraction is the part that is deposited below z_tr. The general criterion for stalling heating then translates to

$$\begin{equation} \int _R^{z_{\rm tr}} \left.{{dQ}\over {dt}}\right|_{\rm Ohmic} dz \approx F_{{\rm rad}}|_{z_{\rm tr}} \approx {{\sigma _B T_{\rm eq}^4}\over {\tau _{\rm tr}}}. \end{equation} \tag{ 2 }$$

We express the planet's equilibrium temperature as σ_BT⁴_eq ∼ L_*/16πa² with L_* the stellar luminosity and a the stellar–planet separation. If an abnormal opacity source is present and causes a strong greenhouse effect, the value of T_eq will be higher than adopted here.

The volumetric rate for Ohmic heating dQ/dt = J²/σ, where J is the electric current density and σ is the electric conductivity. There are two characteristics of the profiles of Ohmic heating that pertain to this discussion, both demonstrated in Appendix A via an analytical model. First, as wind in the shallow layer pulls the poloidal magnetic field forward, a perturbed toroidal magnetic field, b_ϕ, and a largely meridional current, J ≈ J_θe_θ, are created. Since the current is divergence-free, by geometry the radial current J_r in this thin layer is roughly δ = z_wind/R ≪ 1 smaller than the meridional current, where z_wind is the depth of the wind layer (Equation (A6)). The radial current flows continuously across the bottom of the wind zone, generating an interior radial current J_r of a comparable magnitude (Equation (A7)). The interior J_θ, however, is comparable to the local J_r in this geometrically thicker zone (Equation (A9)). As a result, the current density drops at the interface from J ≈ J_θ ∼ J_r R/z_wind above it to J_r below. The local heating rate, J²/σ, drops by a large factor of the order of (R/z_wind)² across the bottom of the wind zone. The bulk of the integrated Ohmic heating is deposited inside the wind layer, with only a fraction of the order of (z_wind/R) deposited below it (Figure 3). Our wind layer has a constant wind speed, but this discussion applies when, e.g., the wind speed varies sinusoidally with depth as in Batygin & Stevenson (2010), or decreases linearly with logarithmic pressure.

Second, since the interior current density J ≈ J_r is roughly constant below the wind layer, the interior volumetric heating rate decays as 1/σ. Since σ rises rapidly with depth, most of the interior heating occurs right below the wind zone.

Following Batygin & Stevenson (2010), we adopt a heating efficiency parameter that relates the total Ohmic heating (dominated by that in the wind layer) to the stellar insolation impinging on the planet,

$$\begin{equation} \int _R^{0} \, {{dQ}\over {dt}} dz \approx {{J_{\rm wind}^2}\over {\sigma _{\rm wind}}} z_{\rm wind} \approx \epsilon {{L_*}\over {16 \pi a^2}}\,, \end{equation} \tag{ 3 }$$

where J_wind and σ_wind are evaluated in the wind layer. Now consider the part of Ohmic heating that occurs below z_tr. We express the current there as J_tr ≈ J_wind(z_wind/R) and find

$$\begin{equation} \int _R^{z_{\rm tr}}\, {{dQ}\over {dt}} dz \approx {{J_{\rm tr}^2}\over {\sigma _{\rm tr}}} z_{\rm tr} \sim \epsilon {{L_*}\over {16 \pi a^2}} \left({{z_{\rm wind}}\over R}\right)^2 {{z_{\rm tr}}\over {z_{\rm wind}}} \left({{\sigma _{\rm wind}}\over {\sigma _{\rm tr}}}\right). \end{equation} \tag{ 4 }$$

Combining Equation (2) and the definition for T_eq, we obtain a constraint on if Ohmic heating is to be capable of stalling contraction,

$$\begin{equation} \epsilon \ge {1\over {\tau _{\rm tr} }} \left({R\over {z_{\rm wind}}}\right)^2 {{z_{\rm wind}}\over {z_{\rm tr}}} \left({{\sigma _{\rm tr}}\over {\sigma _{\rm wind}}}\right). \end{equation} \tag{ 5 }$$

A larger is required to sustain a larger (higher entropy) planet, primarily because a higher entropy planet means a shallower z_tr and a markedly smaller τ_tr. If the wind penetrates only to a shallower depth z_wind, a higher is required to sustain the interior cooling. Note that this expression does not contain explicit dependence on the distance to the star. However, the conductivity in the wind zone (σ_wind) drops with a smaller T_eq, in which case it becomes harder to sustain the cooling using Ohmic heating.

2.3. Numerical Results

To study the contraction of a planet under Ohmic heating, we construct a sequence of planet models that are in hydrodynamical and thermal equilibrium. We include the effects of stellar insolation and Ohmic heating; Ohmic heating is calculated self-consistently with an assumed efficiency and wind depth z_wind. In this section, we are interested in how Ohmic heating stalls the cooling contraction. Thus, planets are assumed to be losing entropy and contracting in time. In Section 3, we adopt the MESA code (Paxton et al. 2011) to study the problem of re-inflating planets after they have contracted.

Let us present a few details on the model building. We interpolate the tables of equation of state (Saumon et al. 1995) and opacity (Alexander & Ferguson 1994; Ferguson et al. 2005; Cassisi et al. 2007) relevant for planetary mass objects. The center boundary condition is a dense solid core with a size of 100 km, a density of 13 g cm⁻³, and hence a small core with a mass of ∼5 × 10²² g. The outer boundary condition is that at the photosphere p = (2/3)g/κ and T = T_eq = (L_*/16πσ_Ba²)^1/4. We calculate the electron fraction for a solar composition gas (Anders & Grevesse 1989), taking into account the thermal and pressure ionization of hydrogen, helium, and three metals (Na, K, and Al, these are the elements with the lowest first ionization potential as well as high cosmic abundances). Metal contributions dominate near the surface while hydrogen is important in deeper regions. Electron conductivity is calculated using the electron–neutral collision cross-section from Draine et al. (1983). The depth of the wind layer is typically set to be z_wind = 2.5 × 10⁸ cm (near pressure p ∼ 10 bar),⁶ the surface dipole field strength is 5 G. The wind velocity is adjusted so that the total Ohmic heating efficiency reaches the preset value, typically 3%. The electric current is solved as a boundary value problem (see Appendix A), with J_r = 0 at the surface and the electric potential Φ∝r² near the center. Heat is carried out by radiative diffusion, or convection where the temperature gradient is super-adiabatic. The planet model is iterated until the profile of Ohmic heating and the thermal structure are mutually compatible.

For each model, we must assume an intrinsic cooling luminosity L_cool. This is the net energy that is lost from the planet interior. The planet's internal entropy, as well as its size, are related to the total luminosity flowing through z_tr. We align models with different values of internal entropy into a time sequence, using the relation of ∫T(dS/dt)d³x = L_cool. The Ohmic heating does not contribute to the right-hand side as it is an external energy source that hardly affects the bulk entropy of the planet.

We have also compared the results from our equilibrium models with those using MESA which integrates the temporal thermal evolution. We confirm that when we use the same Ohmic heating profile in MESA as in our equilibrium models, we obtain very similar results.

Figure 2 shows results for the planet TrEs4-b. The cooling curves in three different scenarios are plotted. If TrEs4-b was born and evolved in isolation, it would have cooled to its current measured radii (1.7 R_J) in less than 10 Myr; being irradiated by its host star at its current orbit raises this figure to ∼10⁸ yr; and having Ohmic heating on top of the irradiation prolongs it further to ∼10⁹ yr. Ohmic heating can be effective in stalling contraction.

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The radial profile of Ohmic heating is displayed in Figure 3, when TrEs-4b is at 1.6 R_J. Less than a percent of the total Ohmic heating is deposited below z_tr yet this amount is sufficient for suspending the contraction. Figure 2 also demonstrates that when the wind layer is much shallower, the required is much larger (Equation (5)).

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Lastly, the radial profiles of electrical conductivity and gas temperature are displayed in Figure 4. Compared to the models shown in Figure 2 of B12, this planet model has a larger radius, a higher internal entropy, and a shallower z_tr. In particular, the intense Ohmic heating near the surface causes the region above ∼10 bar to become convective. A similar model without Ohmic heating will have an isothermal atmosphere (Figure 1). The surface convection zone can significantly alter the local hydrodynamics.

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The electrical conductivity we obtain differs from that shown in B12, but compares well against that in Huang & Cumming (2012), a paper that appeared after the submission of this work. Given our assumption of constant total heating efficiency, the overall magnitude of σ does not affect the results. But the radial profile (especially that near the surface) will. This may partly explain (see Section 4) the difference between results here and those in B12.

We proceed to calculate the expected radii at 1 Gyr for the list of transiting planets, as compiled by Southworth (2011; see http://www.astro.keele.ac.uk/jkt/tepcat), who has updated some of the values from previous publications through his homogeneous analysis. These are shown in Figure 5 for which we assume that both insolation and Ohmic heating have acted on each planet since birth. We find that the expected radius excess can be roughly fitted by the following expression:

$$\begin{equation} {{R - R_J}\over {R_J}} = {{\Delta R}\over {R_J}} \approx 0.5 \left({{T_{\rm eq}}\over {1500\, {\rm K}}}\right)^{1.2}\,, \end{equation} \tag{ 6 }$$

for planets at ∼1 M_J. This results from the increasing conductivity and the larger irradiation luminosity when T_eq is raised. This scaling is obtained for a constant heating efficiency of 3% (see Section 4 for discussions). This scaling is compatible with the ΔR∝T^{1.4 ± 0.6}_eq scaling obtained from the observed sample by Laughlin et al. (2011). For different planet masses, we find the excess decreases with mass: T_eq ≈ 1500 K, the excess scales approximately as (R − R_J)/R_J ∼ 0.5(M/M_J)^−1/2. More massive planets have smaller pressure scale heights for the same entropy.

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The theoretical radii exceed those of almost all known hot Jupiters, with the exceptions of a handful of planets at T_eq ∼ 1700 K. Among these, WASP-17b (Anderson et al. 2011) and HAT-P-32b (Hartman et al. 2011) are larger than the theoretical values by more than 2σ. In fact, these two planets are so large their interiors are almost fully adiabatic. For planets with such a high entropy, z_tr is almost at the photosphere, and hence an efficiency of Ohmic heating ⩾ 1 (Equation (5)) is required to stop the loss of entropy.

We turn to consider the scenario where the planet has contracted significantly before it is subjected to intense stellar irradiation and Ohmic heating. This is possible in a variety of migration theories (e.g., Wu & Murray 2003; Fabrycky & Tremaine 2007; Wu & Lithwick 2011; Nagasawa et al. 2008), where the hot Jupiters arrive at their current locations a few million to a few billion years after their formation.

We focus on WASP-17b (0.49 M_J, T_eq = 1770 K; Anderson et al. 2010, 2011), one of the lowest mass hot Jupiters that is also heavily irradiated. These two factors should combine to produce the most pronounced re-inflation.⁷ Since this is a non-equilibrium phenomenon, we employ the MESA package (Paxton et al. 2011), which has the option of a user-supplied routine for additional heating sources. The heating source we adopt has an energy generation rate per unit mass that scales quadratically with pressure above z_wind, dropping by a factor of (R/z_wind)² at z_wind, and scales as p^−1.3 below the wind zone until p = 10¹⁰ dyn cm⁻², below which the high conductivity leads to a diminishingly small heating that scales as p⁻⁴. Such a profile (the integrated heating is shown in Figure 3) closely approximates the actual Ohmic heating profile we calculate for WASP-17b and agrees with our analytical model in Appendix A. The results on radius expansion are shown in Figure 6 where planet models are exposed to both Ohmic heating (3%) and surface irradiation (with T = T_eq), after having cooled in isolation for a period ranging from 10⁶ to 10⁹ yr.

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The MESA calculations show that re-inflation is ineffective: the planet can at best be re-inflated by ∼10% if it has already cooled (Figure 6). In Figure 7, we illustrate in detail why this is so. Ohmic heating is intrinsically shallow and most of the energy is deposited at p < 10 bar =10⁷ dyn cm⁻², whereupon most of it is quickly transported outward by the newly generated surface convection zone. The interior of the planet experiences the heating only when a temperature inversion is built up and heat diffuses gradually inward.⁸ By the time heat has diffused downward to p = 10¹¹ dyn cm⁻², the temperature inversion is minimal and the radiative diffusion has nearly ground to a halt. As a result, only layers shallower than this experience appreciable heating. Since the pressure scale height is ∼c²_s/g ∼ 4%R_J(T/7000 K)^1/2, the planet can at most expand a few times this value. This result has significance for the survival of hot Jupiters.

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In contrast to our result here, B12 find that low-mass planets can be reinflated by Ohmic heating to become Roche Lobe overflowing. In fact, B12 typically find a more drastic effect of Ohmic heating than that obtained here: our TrEs-4b model reaches an equilibrium radius of 1.6 R_J (Figure 2), while a similar model in B12 yields 1.9 R_J (Figure 5 in B12). Detailed differences in, e.g., radial profiles of conductivity and wind, treatment of radiative atmosphere, between our models may help to explain this difference. However, re-inflation by an order of unity requires raising temperature in the planetary center (bottom scale height). We find that radiative diffusion could not accomplish this (Figure 7).

Figure 7 also demonstrates that heating a cold planet from the outside produces a much more pronounced radiative region (compare with Figure 4). This may have consequences for tidal dissipation, elemental diffusion, and magnetic dynamo in the planet.

In conclusion, we find that it is difficult to re-inflate a planet that has contracted by a substantial amount.

We clarify two issues here. Our assumption of a fixed Ohmic efficiency is ad hoc. It allows us to focus on the thermodynamics of Ohmic heating, but obscures the atmospheric physics that determines the actual wind speed, as well as any dependence on planet temperature or magnetic field strength. Menou (2012) and B12 have argued that in the weather layer (∼0.1 bar) where stellar insolation produces a large horizontal pressure gradient, the wind is driven by the thermal gradient but dragged by the Lorentz force. Assuming the two forces balance each other, Menou (2012) obtains an equilibrium wind speed and an Ohmic heating efficiency (in the weather layer) that peaks at a few percent when the surface temperature is ≈1500 K (B ∼ 10 G). However, this argument may be incomplete. The biggest Lorentz drag is applied not at the weather layer, but deeper down at the bottom of the wind zone where conductivity is the highest (see, e.g., Liu et al. 2008). Thus, the actual wind speed depends on how deep the wind zone extends. This in turns depends on how well the weather layer couples dynamically to the deeper atmosphere, a process currently not well understood (Perna et al. 2010a; Rauscher & Menou 2010; Lian & Showman 2008; Liu & Schneider 2010). An added complication is that Ohmic heating itself can modify the local temperature gradient and stratification (see below). Further study is necessary.

An additional question is whether heat deposited above z_tr can affect the radius evolution. This is the majority of Ohmic heating and can be characterized by GCM simulations. We argue (Section 2) that this fraction of heat cannot directly supplant the internal entropy loss and halt the contraction. But it raises the local temperature gradient (Figure 4)⁹ and exerts an indirect effect on the radius evolution, as is observed in our numerical results. At a given interior adiabat, an atmosphere with such a superficial heating has a deeper z_tr than an atmosphere that is only irradiated and isothermal. Deeper z_tr implies that the amount of energy necessary to suspend internal cooling, the one that is deposited below z_tr), is reduced (see Section 2). The rising temperature in the wind zone brings about a greater conductivity, a larger magnetic drag, and likely, a smaller wind speed. This could lead to saturation of Ohmic heating.

We study the radius evolution of hot Jupiters under Ohmic heating and stellar insolation. We consider initial conditions in which either (1) the planet starts with an high entropy, or (2) it has previously cooled and contracted. In the former case, we find that planets can be sustained at large radii for billions of years (Figure 5). For our example case of TrEs-4b, the planet may remain in a state of perpetual youth, at a cooling age of 10 Myr (Figure 2). Less massive planets can be suspended at larger radii than the more massive ones in the same radiation environment. We place a constraint on the Ohmic heating if it is to suspend the cooling contraction: for an efficiency of a few percent, the wind layer must extend to a depth of the order of 10 bar; a shallower layer would require a higher heating efficiency (Equation (5)).

We also consider Ohmic heating on planet models that have previously contracted. Ohmic heating, being very superficial, has a limited capacity for re-inflating these planets. We illustrate this using WASP-17b, a low-mass planet that has one of the highest irradiation fluxes, yet still cannot be significantly inflated after cooling (Figure 6). As such, Ohmic heating is best called "Ohmic suspension," rather than "Ohmic inflation."

The interior structure of the planet is affected by Ohmic heating: the planet has an adiabatic interior and a largely isothermal atmosphere in the case of no Ohmic heating, but the wind zone tends to become convective when Ohmic heating is applied. This may impact the physics of global circulation, magnetic dynamo, tidal dissipation, and element settling.

While a few planets have observed radii comparable to or even larger than predictions using our fiducial Ohmic heating parameters (WASP-17b and HAT-P-32b, in particular), the vast majority fall below these predictions. In fact, many are consistent with cooling without any Ohmic heating, and some are too small for pure gaseous spheres. There could be a number of explanations for these. They could have lower Ohmic heating efficiencies than stipulated here, or they could have massive solid cores and/or heavy-metal enriched envelopes (see, e.g., Baraffe et al. 2008), or, alternatively, they may have a different migration history, as we now describe. Consider the scenario where the planet is migrated to its current location well after formation. Before the migration, it contracts and loses entropy in isolation. If during the migration there is no significant entropy injection into the deep interior of the planet, the planet will then follow one of the trajectories in Figure 6 and end up having a different radius depending on its time of migration. If this scenario is correct, it would mean that the present-day planet radii retain the memory of their past dynamical history.

There is a caveat to our above scenario. During the migration process, there is likely a large amount of tidal energy deposited inside the planet and this may have rejuvenated the planet. However, there are substantial theoretical uncertainties regarding the mechanism and location of tidal heating. We address tidal rejuvenation in an upcoming publication.

The super-sizes of WASP-17b and HAT-P-32b are surprising: if Ohmic heating is responsible for their sizes, it must be both very efficient and remarkably continuous. For instance, if the dynamo field weakens for, say, 10⁸ years, the internal entropy would be mostly drained away and the planet would have contracted.

We thank the referee, Kristen Menou, for suggestions that improved the clarity of this paper. Y.W. acknowledges useful conversations with P. Goldreich, T. Guillot, K. Batygin, P. Arras, and especially thanks B. Paxton and other MESA creators for this wonderful package. Y.L. acknowledges support from NSF grant AST-1109776.

Here we describe our model for Ohmic heating. We assume a fixed wind profile near the surface of the planet, and a dipolar magnetic field threading the planet. We ignore here the effect of the Lorentz force on the fluid velocity, but return to discuss it in Appendix B. The electric current is obtained by solving

$$\begin{eqnarray} \mbox{\boldmath $J$\unboldmath } &=& \sigma \left(-{\bf \nabla }\Phi + {{{\bf v_{\rm wind}}\times {\bf B_{\rm dipole}}}\over c}\right) \end{eqnarray} \tag{ A1 }$$

$$\begin{eqnarray} \mbox{\boldmath $\nabla \cdot J$\unboldmath }&=&0, \end{eqnarray} \tag{ A2 }$$

subject to the boundary conditions that J_r = 0 at the planet's surface (r = R) and center (r = 0). We have used that the steady-state electric field is the gradient of a potential, $\mbox{\boldmath $E$\unboldmath }=-\mbox{\boldmath $\nabla $\unboldmath }\Phi$. The dipolar field is

$$\begin{eqnarray} &&\mbox{\boldmath $B_{\rm dipole}$\unboldmath }=-{\bf \nabla }\left({\bf M}\cdot {\bf r}\over r^3\right)=M\frac{(2\cos \theta {\bf e}_r+\sin \theta {\bf e}_\theta)}{r^3}{,}\quad \end{eqnarray} \tag{ A3 }$$

in spherical co-ordinates, and we assume that the wind has a constant angular velocity ω in a zone that extends to a depth δR, i.e.,

$$\begin{eqnarray} &&\mbox{\boldmath $v_{\rm wind}$\unboldmath }= \omega r \sin \theta \mbox{\boldmath $e$\unboldmath }_\phi \ \ \ \ {\rm for\ } 1-\delta < r/R <1 \end{eqnarray} \tag{ A4 }$$

and is zero for r/R < 1 − δ. The above equations combine to yield

$$\begin{eqnarray} \mbox{\boldmath $J$\unboldmath } &=& {-}\sigma \mbox{\boldmath $\nabla $\unboldmath }\left(\Phi +\frac{2\omega M}{3cr}\left(P_2-P_0\right) \right)\nonumber\\ \ms\ms &&\quad\quad{\rm for}\;\; 1-\delta < r/R <1 \end{eqnarray} \tag{ A5 }$$

and $\mbox{\boldmath $J$\unboldmath }=-\sigma \mbox{\boldmath $\nabla $\unboldmath }\Phi$ below the wind zone. We discard the uninteresting term P₀ and assume all variables follow a P₂ dependence, adopting the form Φ = Φ₂(r)P₂ for the potential. In the body of the paper, we numerically solve the ODE for Φ₂ that results from setting $\mbox{\boldmath $\nabla \cdot J$\unboldmath }=0$ to obtain the Ohmic heating rate, which is the volume-integrated J²/σ.

In the remainder of this Appendix, we solve analytically a simple example that illustrates two important features of Ohmic heating: first, the rapid drop in the heating profile by the factor ∼δ² just below the wind, and second, the inward decay in the heating profile in the interior (see numerical results in Figure 3). We take σ to be a piecewise function of radius, with σ = σ_shallow through the wind zone to a depth ΔR, where Δ > δ; and σ = σ_deep below that. There are then three shells of interest. In the outer one (1 − δ < r/R < 1), there is a wind and σ = σ_shallow; in the middle one (1 − Δ < r/R < 1 − δ), there is no wind and σ = σ_shallow; and in the inner one (r/R < 1 − Δ), there is no wind and σ = σ_deep. The ODE for Φ₂ is easily solved analytically in each shell, and between the shells one requires that Φ and J_r are continuous. Omitting the algebra, we find that in the limit δ < Δ ≪ 1 and σ_shallow ≪ σ_deep the current in each of the three shells is given by

$$\begin{eqnarray} {\rm Outer\ shell \ (wind\ zone)}:\ \ \ J_r &\approx & {-}6 \left(1-{r\over R} \right) \cdot J_0 P_2 \ ,\nonumber\\ J_\theta &\approx & J_0 {dP_2/ d\theta }. \end{eqnarray} \tag{ A6 }$$

$$\begin{eqnarray} {\rm Middle\ shell}:\ \ \ J_r &\approx & {-}6\delta \cdot J_0 P_2, \nonumber\\ J_\theta &\approx & 6\delta \left(1-{r\over R}-\Delta \right) \cdot J_0 {dP_2/d\theta }.\quad\quad \end{eqnarray} \tag{ A7 }$$

$$\begin{eqnarray} {\rm Inner\ shell}:\ \ \ J_r &\approx & {-}6{r\over R}\delta \cdot J_0 P_2 ,\nonumber\\ J_\theta &\approx & {-}3{r\over R}\delta \cdot J_0 dP_2/d\theta , \end{eqnarray} \tag{ A8 }$$

where the constant

$$\begin{eqnarray} &&J_0 \equiv {2\omega M\sigma _{{\rm shallow}}\over 3 c R^2}\,. \end{eqnarray} \tag{ A9 }$$

Hence the total current in the wind zone is dominated by J_θ which is ∼1/δ greater than the radial current J_r. Below the wind zone, J_r is continuous while J_θ drops discontinuously to less than J_r. This causes a large drop in the rate of Ohmic heating (J²/σ) below the wind zone, of order δ² = (z_wind/R)². Moreover, the higher conductivity in the core ensures that there is little Ohmic dissipation there. Substituting values of δ = 0.001, and Δ = 0.005 and σ_deep/σ_shallow = 10⁷, the fractions of the total Ohmic dissipation taking place in the outer, middle, and inner shells are 0.999855, 0.000144851,  and 9.04343 × 10⁻¹⁰, respectively We argue in the text that only heating in the middle shell can contribute to halting the planet's contraction.

While wind is forced in the outer shell, there is no external forcing in the inner shells. A simple estimate for the Lorentz force (J × B/c) shows that the inner layers could be spun up in a short timescale. We investigate the dynamical response of the planet to the surface battery by looking for steady-state solution to the following equations,

$$\begin{eqnarray} \rho {{\partial {\bf v}}\over {\partial t}} & = & {{{\bf J}\times {\bf B}}\over c} \end{eqnarray} \tag{ B1 }$$

$$\begin{eqnarray} {1\over c} {{\partial {\bf b}}\over {\partial t}} & = & {-} {\bf \nabla }\times {\bf E}\,. \end{eqnarray} \tag{ B2 }$$

Here, b denotes the perturbed field line, while B the background field. At steady state, J = ∇ × b.

We will study a cylindrical analog to the planet where the cylindrical z-direction approximates that of the spherical θ direction, and cylindrical radius that of the spherical radius. The dipole field is now B = B_re_r with B_r = B₀R/r. We include only the components b = b_ϕe_ϕ and j = j_ze_z. We further assume a harmonic z dependence $\propto e^{i k_z z}$.

In the wind zone, an external forcing, F = Fe_ϕ maintains the current. The time-independent equations are now

$$\begin{eqnarray} {{B_r}\over c} {1\over r} {\partial \over {\partial r}} \left(r b_\phi \right) + F & = & 0\nonumber \\ - {{k_z^2}\over {\sigma }}b_{\phi } + {\partial \over {\partial r}} \left({1\over {\sigma r}} {\partial \over {\partial r}} (r b_\phi )\right) - {{B_r}\over c} {{\partial v_{\phi }}\over {\partial r}} & = & 0\,. \end{eqnarray} \tag{ B3 }$$

The boundary condition is that the radial current vanishes at r = R, or b_ϕ = 0 at R. Taking r ≈ R, and a constant forcing F, we find the solution to be

$$\begin{eqnarray} b_\phi & = & {{F c R}\over {B_0}} (1-r/R)\nonumber \\ v_\phi & \approx & v_{\rm wind} = {\rm const.} \end{eqnarray} \tag{ B4 }$$

since we take the wind zone be of thickness δ ≪ 1 ∼ 1/(Rk_z).

At the bottom of the wind zone, b_ϕ behaves continuously (radial current has to be continuous), but a jump in the tangential velocity (Δv) is necessary to ensure that the tangential electric field is continuous. The velocity there is

$$\begin{equation} v_\phi (r = R-\delta)= v_{\rm wind}- \Delta v = v_{\rm wind} - {{F c^2}\over {B_r^2 \sigma }}. \end{equation} \tag{ B5 }$$

Further below, both the velocity and b_ϕ decay inward as

$$\begin{eqnarray} b_\phi & = & {{F c R \delta }\over {B_0 r}} e^{i k_z z}\nonumber \\ {{\partial v_\phi }\over {\partial r}} & = & {-} {{F c^2 k_z^2 \delta }\over {B_0^2 \sigma }} e^{i k_z z}. \end{eqnarray} \tag{ B6 }$$

So the region of lowest σ is also where v_ϕ decreases most steeply. Compared to Δv, the total shear across the no-forcing zone is much smaller,

$$\begin{equation} {v_\phi (r = R-\delta) - v_\phi (r=0)} \sim k_z^2 \delta ^2 \, \Delta v \ll \Delta v\,. \end{equation} \tag{ B7 }$$

This result indicates that at steady state the interior is able to maintain a largely uniform rotation with v_ϕ ≈ v_ϕ(r = R − δ). Contrary to our initial expectation, the interior can rotate at a different rate than that in the wind zone, despite the large Lorentz torque. It is not spun up.

Let us present a few words on the energetics. Stellar insolation drives the wind which suffers both turbulent dissipation and Ohmic loss. We take the timescale for turbulent dissipation of the wind to be the wind traveling time across the planet, R/v_wind

$$\begin{equation} \left({{dE}\over {dt}}\right)_{\rm turb} \approx 2 \pi r \delta {{\rho v_{\rm wind}^3}\over R}\,, \end{equation} \tag{ B8 }$$

and the rate for Ohmic loss is

$$\begin{equation} \left({{dE}\over {dt}}\right)_{\rm Ohmic} \approx 2 \pi r \delta F v_{\rm wind} = 2 \pi r \delta v_{\rm wind}^2 {{B_r^2 \sigma }\over {c^2}}\,. \end{equation} \tag{ B9 }$$

We find the ratio between the two rates to be

$$\begin{equation} {{\left({{dE}/{dt}}\right)_{\rm turb} }\over {\left({{dE}/{dt}}\right)_{\rm Ohmic} }} \approx {{\rho v_{\rm wind} c^2}\over {R B_r^2 \sigma }}\,. \end{equation} \tag{ B10 }$$

Adopting parameters ρ = 10⁻⁴ g cm⁻³, σ = 10¹⁰ s⁻¹, B_r = 5 G, and v_wind = 10⁵ cm s⁻¹, this ratio is ≈4, or Ohmic loss is non-negligible. This suggests that the Ohmic efficiency may be higher than the canonical 3% we adopt in the text.