MAGNETIC SHIELDING OF EXOMOONS BEYOND THE CIRCUMPLANETARY HABITABLE EDGE

M dwarf stars are known to show strong magnetic bursts, eventually coupled with the emission of XUV radiation as well as other high-energy particles (Gurzadian 1970; Silvestri et al. 2005). Moreover, stars with masses below 0.2–0.5 solar masses (M_☉) cannot possibly host habitable moons, since stellar perturbations excite hazardous tidal heating in the satellites (Heller 2012). We thus concentrate on stars more similar to the Sun. G dwarfs, however, are likely too bright and too massive to allow for exomoon detections in the near future. As a compromise, we choose a 0.7 M_☉ K dwarf star with solar metallicity Z = 0.0152 and derive its radius (R_⋆) and effective temperature (T_{eff, ⋆}) at an age of 100 Myr (Bressan et al. 2012): R_⋆ = 0.597 R_☉ (R_☉ being the solar radius), T_{eff, ⋆} = 4270 K. These values are nearly constant over the next couple of Gyr.

We use planetary evolution models of Fortney et al. (2007) to explore two extreme scenarios, between which we expect most giant planets: (1) mostly gaseous with a core mass M_c = 10 Earth masses (M_⊕) and (2) planets with comparatively massive cores. Class (1) corresponds to larger planets for given planetary mass (M_p). Models for suite (2) are constructed in the following way. For M_p < 0.3 M_Jup, we interpolate between radii of planets with core masses M_c = 10, 25, 50, and 100 M_⊕ to construct Neptune-like worlds with a total amount of 10% hydrogen (H) and helium (He) by mass. For M_p > 0.3 M_Jup, we take the precomputed M_c = 100 M_⊕ grid of models. These massive-core planets (2) yield an estimate of the minimum radius for given M_p. To account for irradiation effects on planetary evolution, we apply the Fortney et al. (2007) models for planets at 1 AU from the Sun.

As it is desirable to compare our scaling laws for the magnetic properties of giant exoplanets with known magnetic dipole moments of solar system worlds, we start out by considering a Neptune-, a Saturn-, and a Jupiter-class host planet. For the sake of consistency, we attribute total masses of 0.05, 0.3, and 1 M_Jup, as well as M_c = 10, 25, and again 10 M_⊕, respectively.

Kepler has been shown capable of detecting moons as small as 0.2 M_⊕ combining measurements of the planet's transit timing variation and transit duration variation (Kipping et al. 2009). The detection of a planet as small as 0.3 Earth radii (R_⊕), almost half the radius of Mars (Barclay et al. 2013), around a K star suggests that direct transit measurements of Mars-sized moons may be possible with current or near-future technology (Sartoretti & Schneider 1999; Szabó et al. 2006; Kipping 2011). Yet, in situ formation of satellites is restricted to a few times 10⁻⁴ M_p at most (Canup & Ward 2006; Sasaki et al. 2010; Ogihara & Ida 2012). For a Jupiter-mass planet, this estimate yields a satellite of about 0.03 M_⊕ ≈ 0.3 times the mass of Mars. Alternatively, moons can form via a range of other mechanisms (for a review, see Section 2.1 in Heller & Barnes 2013b). We therefore suspect moons of roughly the mass and size of Mars to exist and to be detectable in the near future.

Following Fortney et al. (2007) and assuming an Earth-like rock-to-mass fraction of 68%, we derive a radius of 0.94 Mars radii or 0.5 R_⊕ for a Mars-mass exomoon. Tidal heating in the moon is calculated using the model of Leconte et al. (2010) and assuming an Earth-like time lag of the moon's tidal bulge τ_s = 638 s as well as a second degree tidal Love number k_{2, s} = 0.3 (Heller et al. 2011).

We investigate a range of planet–moon binaries located in the center of the stellar HZ. Therefore, we compute the arithmetic mean of the inner HZ edge (given by the moist greenhouse effect) and the outer HZ edge (given by the maximum greenhouse) around a K dwarf star (Section 2.1) using the model of Kopparapu et al. (2013). For this particular star, we localize the center of the HZ at 0.56 AU.

The tighter a moon's orbit around its planet, the more intense the illumination it receives from the planet and the stronger tidal heating. Ultimately, there exists a minimum circumplanetary orbital distance, at which the moon becomes uninhabitable, called the "habitable edge" (HE; Heller & Barnes 2013b). As tidal heating depends strongly on the orbital eccentricity e_ps, amongst others, the radius of the "HE" also depends on e_ps.

We consider two thresholds for a transition into an uninhabitable state. (1) When the moon's tidal heating reaches a surface flux similar to that observed on Jupiter's moon Io, that is 2 W m⁻², then enhanced tectonic activity as well as hazardous volcanism may occur. Such a scenario could still allow for a substantial area of the moon to be habitable, as tidal heat leaves the surface through hot spots, (see Io and Enceladus; Ojakangas & Stevenson 1986; Spencer et al. 2006; Tobie et al. 2008), and there may still exist habitable regions on the surface. Hence, we consider this "Io-limit HE" (Io HE) as a pessimistic approach. (2) When the moon's global energy flux exceeds the critical flux to become a runaway greenhouse (RG), then any liquid surface water reservoirs can be lost due to photodissociation into hydrogen and oxygen in the high atmosphere (Kasting 1988). Eventually, hydrogen escapes into space and the moon will be desiccated forever. We apply the semi-analytic RG model of Pierrehumbert (2010; see Equation (1) in Heller & Barnes 2013b) to constrain the innermost circumplanetary orbit at which a moon with given eccentricity would just be habitable. For our prototype moon, this model predicts a limit of 269 W m⁻² above which the moon would transition to an RG state. We call the corresponding critical semimajor axis the "runaway greenhouse HE" (RG HE) and consider it as an optimistic approach.

We calculate the Io and RG HEs for e_ps ∈ {0.1, 0.01, 0.001} using Equation (22) from Heller & Barnes (2013b) and introducing two modifications. First, the planetary surface temperature (T_p) depends on the equilibrium temperature ($T_\mathrm{p}^\mathrm{eq}$) due to absorbed stellar light and on an additional component (T_int) from internal heating: $T_\mathrm{p}=([T_\mathrm{p}^\mathrm{eq}]^4+T_\mathrm{int}^4)^{1/4}$. Second, we use a Bond albedo α_opt = 0.3 for the stellar illumination absorbed by the moon and α_IR = 0.05 for the light absorbed from the relatively cool planet (Heller & Barnes 2013a).

We apply scaling laws for the magnetic field strength (Olson & Christensen 2006) that consider convection in a spherical conducting shell inside a giant planet and a convective power Q_conv (Equation (28) in Zuluaga et al. 2013), provided by the Fortney et al. (2007) models. The ratio between inertial forces to Coriolis forces is crucial in determining the field regime—whether it is dipolar- or multipolar-dominated—for the dipole field strength on the planetary surface. We scale the ratio between dipolar and total field strengths following Zuluaga & Cuartas (2012).

The core density (ρ_c) is estimated by solving a polytropic model of index 1 (Hubbard 1984). Radius and extent of the convective region are estimated by applying a semi-empirical scaling relationship for the dynamo region (Grießmeier 2006). Thermal diffusivity κ is assumed equal to 10⁻⁶ for all planets (Guillot 2005). Electrical conductivity σ is assumed to be 6 × 10⁴ for planets rich in H and He, and σ = 1.8 × 10⁴ for the ice-rich giants (Olson & Christensen 2006).

We have verified that our model predicts dynamo regions that are similar to results obtained by more sophisticated analyses. For Neptune, our model predicts a dynamo radius R_c = 0.77 planetary radii (R_p) and ρ_c = 3000 kg m⁻³, in good agreement to Kaspi et al. (2013). We also ascertained the dynamo scaling laws to reasonably reproduce the planetary dipole moments of Ganymede, Earth, Uranus, Neptune, Saturn, and Jupiter (J. I. Zuluaga et al., in preparation). For the mass range considered here, the predicted dipole moments agree within a factor of two to six. Discrepancies of this magnitude are sufficient for our estimation of magnetospheric properties, because they scale with $M_\mathrm{dip}^{1/3}$. Significant underestimations arise for Neptune and Uranus, which have strongly non-dipolar surface fields.

The shape of the planet's magnetosphere can be approximated as a combination of a semi-sphere with radius $R_\mathcal {M}$ and a cylinder representing the tail region (Figure 1). The planet is at a distance $R_\mathcal {M}-R_\mathcal {S}$ off the sphere's center, with

$$\begin{equation} R_\mathcal {S} = \left(\frac{\mu _0 f_0^2 }{8 \pi ^2}\right)^{1/6} \mathcal {M}^{1/3} P_{\rm sw}^{-1/6} \end{equation} \tag{ 1 }$$

being the standoff distance, μ₀ = 4π × 10⁻⁷ N A⁻² the vacuum magnetic permeability, f₀ = 1.3 a geometric factor, $\mathcal {M}$ planetary magnetic dipole moment, and P_sw∝n_swv_sw the dynamical pressure of the stellar wind. Number densities n_sw and velocities v_sw of the stellar wind are calculated using a hydrodynamical model (Parker 1958). Both quantities evolve as the star ages. Thus, we use empirical formulae (Grießmeier et al. 2007) to parameterize their time dependence.

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Observations of stellar winds from young stars are challenging, and thus the models only cover stars older than 700 Myr. We extrapolate these models back to 100 Myr, although there are indications that stellar winds "saturate" when going back in time (J. Linsky 2012, private communication).

Figure 2 visualizes the evolution of $R_\mathcal {S}$ (thick blue line) as a snail curling around the planet, indicated by a dark circle in the center. Stellar age (t_⋆) is denoted in units of Gyr along the snail, starting at 0.1 Gyr at "noon" and ending after 4.6 Gyr at "midnight." $R_\mathcal {S}$, as well as the RG and Io HEs, are given in units of R_p. At t_⋆ = 0.1 Gyr (t_⋆ = 4.6 Gyr), R_p = 0.385, 1.023, and 1.195 R_Jup (0.329, 0.862, and 1.056 R_Jup) for the Neptune-, Saturn-, and Jupiter-like planets, respectively.

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In panel (a) for the Neptune-like host, $R_\mathcal {S}$ starts very close to the planet and even inside the RG HE for the e_ps = 0.001 case (thin black snail), at roughly 2 R_p from the planetary center. This means that at an age of 0.1 Gyr, any Mars-like satellite with e_ps = 0.001 would need to orbit beyond 2 R_p to avoid transition into a RG state, where it would not be affected by the planet's magnetosphere. As the system ages, wider orbits are enshrouded by the planetary magnetic field, until after 4.6 Gyr, $R_\mathcal {S}$ reaches as far as the Io HE for e_ps = 0.1 (thick dashed line).³ In summary, moons in low-eccentricity orbits around Neptune-like planets can be close the planet and be habitable from an illumination and tidal heating point of view, but it will take at least 300 Myr in our specific case until they get coated by the planet's magnetosphere. Exomoons on more eccentric orbits will either be uninhabitable or affected by the planet's magnetosphere after more than 300 Myr.

Moving on to Figure 2(b) and the Saturn-like host, we find that $R_\mathcal {S}$ reaches the e_ps = 0.001 RG HE after roughly 200 Myr, but it transitions all the other HEs substantially later than in the case of a Neptune-like host. After ≈1 Gyr, the e_ps = 0.01 RG HE and the Io HE for e_ps = 0.01 are transversed. After 4.6 Gyr, all orbits except for the Io HE at an eccentricity of 0.1 are covered by the planet's magnetic field. In conclusion, close-in Mars-sized moons around Saturn-like planets can be magnetically affected early on and be habitable from a RG or tidal heating point of view, if their orbital eccentricities are small.

Finally, Figure 2(c) shows that exomoons at the RG HE of Jovian planets for e_ps = 0.001 will be bathed in the planetary magnetosphere at stellar ages as young as 100 Myr. After roughly 1.3 Gyr, $R_\mathcal {S}$ transitions the RG HE for e_ps = 0.1 and the Io HE for e_ps = 0.01. Even the Io HE with an eccentricity of e_ps = 0.1 is covered at around 3.1 Gyr. After 4.6 Gyr, the planet's magnetic shield reaches as far as 21 R_p. Clearly, exomoons about Jupiter-like planets face the greatest prospects of interference with the planetary magnetosphere, even in orbits that are sufficiently wide to ensure negligible tidal heating.

Looking at the standoff radii for the three cases in Figure 2, we wonder how strong the magnetic dipole moment $\mathcal {M}_\mathrm{dip}$ would need to be after 0.5 Gyr, when the atmospheric buildup should have mostly ceased, in order to magnetically enwrap the moon at a given HE. This question is answered in Figure 3.

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While planets with relatively massive cores (brown solid line) shield a wider range of orbits for given M_p below roughly 1 M_Jup, the low-mass core model (blue solid line) catches up for more massive giants. What is more, our model tracks for the predicted M_dip, which we expect to be located between the thick brown and thick blue line, "overtake" the RG and Io-like HEs for a range of eccentricities. HE contours that fall within the shaded area are magnetically protected, while moons above a certain HE are habitable from a tidal and illumination point of view.

We finally examine temporal aspects of magnetic shielding in Figure 4. The question answered in this plot is "how long would it take a planet to magnetically coat its moon beyond the HEs?" Again, planetary mass is along the abscissa, but now stellar lifetime t_⋆ is along the ordinate. Clearly, the magnetic standoff radius of lower-mass planets requires more time to expand out to the respective HEs. While low-mass giants with low-mass cores (blue lines) require up to 3.5 Gyr to reach the RG HE for e_ps = 0.01, equally mass planets but with a high-mass core (thin brown line) could require as few as 650 Myr.

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Planets more massive than roughly 0.3 M_Jup will coat their moons at the RG HE for e_ps = 0.01 as early as 1 Gyr after formation. For lower eccentricities, timescales decrease. As the RG HE is more inward to the planet than the Io HE, it is coated earlier than the Io limit for given e_ps. RG and Io HEs for e_ps = 0.001 are not shown as they are covered earlier than ≈300 Myr in all cases.