DETECTION OF EXOMOONS THROUGH OBSERVATION OF RADIO EMISSIONS

Io is an intensely volcanic moon orbiting inside Jupiter's magnetosphere. The volcanic activity creates a light atmosphere of SO₂ around Io, which ionizes to create an ionosphere (Lopes & Spencer 2007). This ionosphere then injects ions into Jupiter's magnetosphere to create a plasma torus, which orbits Jupiter's magnetic equator at an angle of 96 from the rotational equator and co-rotates with the magnetic field at a speed of 74 km s⁻¹ (Su 2009). Io orbits Jupiter at a linear speed of 17 km s⁻¹, so Jupiter's magnetic field passes Io at a speed of 57 km s⁻¹. The speed difference gives rise to a unipolar inductor (Grießmeier et al. 2007), which induces a current across Io's atmosphere of a few million amps. The current then accelerates the electrons that produce the characteristic radio emissions (Crary 1997). It must be noted that while volcanism is essential to the formation of a dense ionosphere around Io, such a process might not be required for larger moons, since moons like Titan are already large enough to sustain a thick atmosphere, which in turn can give rise to an ionosphere.

Furthermore, the interaction between Io and the plasma torus gives rise to Alfvén waves (Belcher 1987). The precise mechanism by which Alfvén waves interact with the torus is complex, and several analytical and numerical models have been proposed. In these models, Alfvén waves produce electric fields parallel to Jupiter's magnetic field lines, which then transport and accelerate electrons toward Jupiter's magnetic poles (Su 2009; Saur et al. 1999; Crary 1997; Neubauer 1980 and references therein). The electrons traveling through the field lines create a cyclotron maser, which then emits radio waves whose existence in the Jupiter–Io system has been observationally verified (Crary 1997; Mauk et al. 2001).

Our studies of exoplanet–moon interactions are based on an extrapolation of both Io's plasma environment and Jupiter's magnetic field to different scenarios that could potentially be encountered in newly discovered planetary systems. We begin by finding an expression for the maximum intensity of the radio emissions, then proceed to address the magnetic field and plasma properties, and finally we analyze the dependence of the radio emissions on these parameters.

Assuming a simple current distribution around Io and that the magnetic field lines are approximately perpendicular to the plasma velocity, Neubauer (1980) found that the maximum Joule dissipation of the system is given by

$$\begin{equation} P_{T}=\frac{\pi R_{{\rm Io}}^{2}V_{0}^{2}B_{{\rm Io}}^{2}}{\mu _{0}\sqrt{V_{A}^{2}+ V_{0}^{2}}}\,, \end{equation} \tag{ 1 }$$

where R_Io is Io's radius, B_Io is the magnetic field at Io's position, V₀ is the plasma speed relative to Io, μ₀ is the permeability of free space, and V_A is the Alfvén velocity. The Alfvén velocity depends on the magnetic field, B_Io, and the plasma density, ρ_Io, through the relationship $V_{A}=B_{{\rm Io}}/\sqrt{\mu _{0}\rho _{{\rm Io}}}$.

Since only a small fraction of the maximum Joule dissipation, P_T, is converted to radio waves, we will introduce the efficiency coefficient β_Io, which, based on previous studies, is ≈1% for the Jupiter–Io system (Zarka et al. 2001). There is little information on the variability of this parameter, so we assume that other exoplanet–moon systems have similar efficiency coefficients. Hence, the maximum radio emission intensity, P_rad, from these systems is given as

$$\begin{equation} P_{{\rm rad}}=\frac{\pi \beta _{S}R_{S}^{2}V_{0}^{2}B_{S}^{2}}{\mu _{0}\sqrt{B_{S}^{2} /(\mu _{0}\rho _{S})+V_{0}^{2}}}\,, \end{equation} \tag{ 2 }$$

where the subscript "Io" was switched to "S" to denote that these variables now belong to a generic exomoon or "satellite." The plasma speed, V₀, is computed assuming it corotates with the planet's magnetic field, as in the Jupiter–Io case. Explicitly, if the moon orbits at a distance of r_S from the planet, then $V_{0}=\omega _{P}r_{S}-\sqrt{GM_{P}/r_{S}}$, where G is the gravitational constant, M_P is the planet's mass, and ω_P is the planet's angular velocity. The other parameters in Equation (2) are explored in later sections.

Equation (2) does not depend on the properties of the host star and therefore the exoplanet–moon system does not have to be close to its star (or even have a host star) to be detectable. In fact, exomoons around exoplanets with small orbits might be undetectable because stellar winds can also induce radio emissions, which increase with decreasing planetary semi-major axis (Zarka 1998). Furthermore, moon-induced radio emissions do not solely occur along the orbital plane, which means the system's bodies do not have to orbit parallel to our line of sight.

The dependence of Equation (2) on $R_{S}^{2}$ clearly favors large exomoons. Although there has been no observational evidence, the possibility of detecting such exomoons is still plausible. Nevertheless, given that mass grows as the cube of the radius, the long-term stability of such systems can also be called into question; however, orbital stability analysis is beyond the scope of this study and it will not be discussed any further.

All the giant planets and several rocky bodies in the solar system have magnetic fields and extended magnetospheres, of which Jupiter's is the largest. Consequently, one can expect exoplanets to have similar magnetic fields and extended magnetospheres. To model an exoplanet's magnetic field, we begin by assuming that the field is mostly dipolar, as is the case for all magnetized bodies in the solar system, and that its angle of inclination with respect to the exoplanet's axis of rotation is small enough to be neglected. If the exomoon orbits close to the rotational equator of the exoplanet, then the magnetic field affecting the exomoon at its location is given by B_S = $(\mu _0/4\pi)(m/{r_S^{3}})$, where m is the exoplanet's magnetic dipole moment.

To approximate the magnetic moment of an exoplanet, we adopt the approximation introduced by Durand-Manterola (2009) who found that the magnetic moment of a planetary body can be expressed as m = 10⁻⁵(σM_P/T_P)^K, where the exponent K is experimentally determined to be 1.10 ± 0.13 (we use 1.15 to better fit the giant planets), T_P is the planet's rotation period, and σ is the conductivity of the liquid at its core, which is responsible for creating the magnetic field. In the case of giant planets, the liquid that creates their magnetic field is metallic hydrogen, which is estimated to have a conductivity of about 2 ± 0.5× 10⁵ S m⁻¹ (Shvets 2007). The expression for the magnetic field affecting the exomoon is thus

$$\begin{equation} B_{S}=\frac{10^{-12}}{r_{S}^{3}}\left(\frac{M_{P}\sigma }{T_{P}}\right)^{K}. \end{equation} \tag{ 3 }$$

It must be mentioned that there are other approximations that we could have chosen to calculate m, but we chose Durand–Manterola's formula because of its simplicity and because it can be at least partially justified using standard electrodynamics. If the exomoon is sufficiently large, it could also have its own magnetic field, but here we assume its effects to be negligible. However, based on the interaction between planets and the solar wind, we hypothesize that the exomoon's magnetic field creates a bow shock where the exomoon's magnetic field pressure equals the plasma torus pressure and its net effect is to increase the apparent cross-sectional area of the exomoon, thereby increasing the power of its radio emissions. This effect will be quantitatively discussed in later publications.

The intensity of exomoon-induced radio emissions depends on many parameters, thus a thorough clarification of how each of these parameters are treated is crucial. A very important parameter in our calculation of a radio signal's flux is T_P, but it is also very difficult to measure or predict. Therefore, it is typically assumed to be equal to Jupiter's rotational period, T_J, or that the planet is tidally locked if it is closer than 0.1 AU to its host star (Lazio et al. 2004 and references therein). Since we mostly consider exoplanets with large orbits, we assume T_P = T_J throughout the calculations.

The atmospheric plasma density of an exomoon (or exoplanet) is difficult to determine if the environmental properties of the body are not already known. Even though we cannot predict the plasma density of a hypothetical exomoon, we can find a reasonable estimate based on what we know about the solar system. The plasma density depends not only on the number of ions present, but also on the molecular weight of the ions that constitute it. For example, on Io, one can find O⁺, S⁺, SO⁺, etc., because Io's atmosphere is made from the SO₂ emitted by its volcanoes (Su 2009). Io's mean plasma density is ∼4.2 × 10⁴ amu cm⁻³, or ∼7 × 10⁻¹⁷ kg m⁻³ (Kivelson 2004). Earth's ion number density is typically on the order of 10⁵ cm⁻³, and the dominant ion is O⁺, which gives us a plasma density of ∼1.7 × 10⁶ amu cm⁻³, or ∼2.7 × 10⁻¹⁵ kg m⁻³ (Schunk & Nagy 2009). Venus and Mars also have similar plasma densities to Earth, with O$_{2}^{+}$, and O⁺ ions from CO₂ being the dominant species in their ionospheres (Schunk & Nagy 2009). Given the large variability of this parameter, we chose to work with three different values: 10⁴ amu cm⁻³, 10⁵ amu cm⁻³, and 10⁶ amu cm⁻³, which cover most of the range of plasma densities that can lead to a detectable exomoon with reasonable size (as explained in the next section). We can see the dependence of P_rad on the plasma density, ρ_S, by rearranging Equation (2) to get

$$\begin{equation} P_{{\rm rad}}=\frac{\pi \beta _{S}R_{S}^{2}B_{S}^{2}V_{0}}{\mu _{0}}\sqrt{\frac{\rho _{S}}{\rho _{S}+\frac{1}{\mu _{0}}\left(\frac{B_{S}}{V_{0}}\right)^{2}}}. \end{equation} \tag{ 4 }$$

Equation (4) allows us to define a critical density $\rho _{C}=\mu _{0}^{-1}(B_{S}/V_{0})^{2}$, which in turn allows us to characterize the three limiting cases:

$$\begin{eqnarray} P_{-} &\equiv & P_{{\rm rad}}(\rho _{S} \ll \ \rho _{C}) = \pi \beta _{S}R_{S}^{2}V_{0}^{2}B_{S}\sqrt{\frac{\rho _{S}}{\mu _{0}}}, \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray} P_{C} &\equiv & P_{{\rm rad}}(\rho _{S} = \rho _{C}) = \frac{1}{\sqrt{2}}\pi \mu _{0}^{-1}\beta _{S}R_{S}^{2}B_{S}^{2}V_{0}, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} P_{+} &\equiv & P_{{\rm rad}}(\rho _{S} \gg \ \rho _{C}) = \pi \mu _{0}^{-1}\beta _{S}R_{S}^{2}B_{S}^{2}V_{0}. \end{eqnarray} \tag{ 7 }$$

The fact that $P_{C} = ({1}/{\sqrt{2}}) P_{+} \sim 71\% P_{+}$ tells us that the emitted radio power is much less dependent on ρ_S for plasma densities larger than ρ_C. Furthermore, since P₋ decreases with decreasing ρ_S, then systems with higher plasma densities are more likely to be observable.

Regarding the orbital radius r_S, the only real physical constraint on the exomoon's orbit is that it must be close enough to the exoplanet to be well inside the magnetosphere and gravitationally stable, but farther than the Roche limit to avoid structural instability. However, we can also make use of Equation (2) to find orbits that might favor radio emissions, since exomoons in these orbits are the most likely to be detected.

The function P_rad(r_S) is plotted in Figure 1 for various parameter combinations. The purpose of Figure 1 is to demonstrate how changing a single parameter in P_rad affects its properties. Clearly, P_rad always has a single maximum in the orbits larger than the synchronous orbit. The only parameter that does not affect the orbital distance at which this maximum occurs is R_S; however, ρ_S only affects the point's position weakly and T_P is always held constant, so M_P is the dominant parameter when finding P_rad's maximum. The region around the maximum is also relatively flat; therefore, it seems reasonable to assume that an exomoon could be close to this maximum. In fact, Io and four Saturnian moons (including the large moon Enceladus) have orbits well within an area that would allow the moons to output at least 95% of the maximum predicted radio power. Therefore, we set r_S to be the value that gives the maximum radio power and rename it r_Opt from this point forward to avoid confusion. We avoid the treatment of orbits smaller than the synchronous orbit due to their proximity to the Roche limit.

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It is noteworthy that the optimal orbital radius for each planetary mass needs to be found numerically, but it approximately follows a power law. For example, using ρ_S = ρ_Io we get $r_{{\rm Opt}}=5.4\,M_{P}^{0.32}$, where M_P and r_Opt are given in units of Jupiter's mass and radius.