The thickness of the solid icy layer of Europa, which is coupled to the existence of a subsurface ocean, has been poorly constrained because of limitations in the amount of observational data. It remains unclear as to whether thin-crust or thick-crust models are appropriate. In the present paper, we propose a new method by which to determine the thickness of the icy layer using coherent electromagnetic emissions induced by high-energy neutrinos in outer space. Through a Monte-Carlo simulation of the interaction between neutrinos and ice, we evaluated the number of observable emissions, which depends on the thickness of the Europan ice up to 8 km. This method, if used on future missions to Europa, can provide a new constraint for the icy layer and can resolve the debate as to whether the thin-crust or thick-crust model is applicable to Europa.
 Europa, which is the second closest Galilean satellite orbiting Jupiter, may have a subsurface ocean. The state of this ocean is a key consideration not only for the thermal evolution of satellites but also for astrobiology in the solar system. Although gravitational measurements by the Galileo probe constrain the thickness of the H2O layer to be 80 to 170 km [Anderson et al., 1998], these investigations have poor resolution in terms of the phase present, i.e., water or ice. Thus, accurate determination of the thickness of the solid icy layer is important in order to unambiguously confirm the existence of a subsurface ocean, which was indicated by magnetic field measurements [Khurana et al., 1998]. Several researchers have attempted to deduce the thickness of the solid icy layer based on surface morphology [Carr et al., 1998; Pappalardo et al., 1998], induced magnetic field [Zimmer et al., 2000], and tidal stress generated by Jupiter [Hussmann et al., 2002; Tobie et al., 2003]. The results of these studies vary widely and can be classified as fitting one of two models, the thin-crust model, in which the thickness of the solid ice layer is considered to be a few kilometers [Carr et al., 1998; Greenberg et al., 1998; Hand and Chyba, 2007], and the thick-crust model, in which it is considered to be several tens of kilometers [Pappalardo et al., 1998; Hussmann et al., 2002; Tobie et al., 2003]. The ice thickness depends strongly on the magnitude and location of heat generation by tidal dissipation in Europa. As a result of the lack of observational data, there is a large uncertainty with respect to ice thickness. Therefore, new observational data are necessary in order to further constrain the state of the subsurface ocean and the surface icy layer, e.g., measurement of tidal deformation by laser altimetry, in which the amplitude at sub- and anti-Jovian points is estimated to be 29 m if an ocean exists and a few tens of centimeters if an ocean does not exist [Hussmann et al., 2010].
 In the present study, we propose a new method by which to detect the subsurface ocean by measurement of radio waves induced by cosmic neutrinos. In outer space, there exist neutrinos with very high energies, e.g., over 1018 eV (1 EeV), which are referred to as ultra-high-energy (UHE) neutrinos. UHE neutrinos are considered to be created by high-energy cosmic rays interacting with the cosmic microwave background (GZK neutrinos) [Greisen, 1966], photons from sources such as active galactic nuclei (AGN) [Nellen et al., 1993] or gamma ray bursts (GRBs) [Vietri, 1998]. When a neutrino traverses a medium, neutral or charged current interactions can occur with its nucleons, giving rise to the production of hadrons such as pions (νl + N → νl or l + X), where l, N, and X represent lepton, nucleon, and hadron, respectively. This interaction induces an electromagnetic shower. Since scattering effects such as Compton scattering induce more electrons from the medium and annihilation decreases the number of positrons, the total charge of the shower becomes asymmetric, leading to the emission of Cherenkov light. Cherenkov light with a wavelength longer than the scale of the shower becomes coherent. This process of coherent emission is referred to as the Askaryan effect [Askar'yan, 1962], which has been confirmed in silica sand [Saltzberg et al., 2001] and ice [Gorham et al., 2007]. The transverse scale of the shower depends on the density of the medium. In the case of H2O, the transverse shower scale is a few centimeters. Thus, emissions at wavelength longer than 10 cm become coherent in H2O. In the case of UHE neutrinos, coherent radio emission has a higher intensity than visible Cherenkov emission.
 The distance travelled by the radio waves depends on the state of the H2O because liquid water has a shorter attenuation length than ice. Waves emitted in the solid icy layer can maintain a sufficiently large amplitude to be detected by an antenna orbiting Europa, whereas emissions in a liquid water layer attenuate easily. Thus, the number of emissions that can be detected above Europa depends on the amount of solid ice present. Thus, if such emissions could be detected by a probe, this would provide a means of determining the thickness of the icy layer. Radio emissions form a Cherenkov cone and can be detected by a probe within this cone. Since radio emissions with a large nadir angle to the surface are reflected at the surface and cannot leave the interior, only those with a small nadir angle can be used to determine the thickness of the ice [Gorham et al., 2009].
 A radar sounder has been proposed as a viable method by which to evaluate the ice thickness [Chyba et al., 1998; Moore, 2000]. The advantage of the proposed method over a radar sounder is that no electromagnetic source is needed. All that is required is a suitable radio antenna to detect electromagnetic waves coming from Europa, which results in low power operation.
Schaefer et al.  briefly considered this method although a quantitative evaluation was not presented. Through a 3-D global Monte-Carlo simulation, we estimate the number of radio signals detectable by an antenna. We report the intensity distributions of the emissions is a key in this method and we describe the practicality of this method.
2.1. Intensity of Radio Waves
 The energy of Cherenkov emission in the frequency range νmin to νmax from a single charged particle propagating a distance L can be represented as follows:
where h, c, and α are Plank's constant, the speed of light, and the fine structure constant (1/137), respectively [Gorham et al., 2009]. In addition, n represents the refractive index of the medium. Here, we use L = 6 m and n = 1.8, which are typical values for Antarctic ice on Earth [Jelly, 1996]. In the above equation, β is the ratio of the speed of the charged particle to the speed of light. Charged particles induced by high-energy neutrino traverse Europa at nearly the speed of light. Thus, we can consider β ∼ 1. The total number of charged particles N induced by neutrinos can be estimated as follows:
where Eν is the energy of the neutrino, and y represents the Bjorken inelasticity, which is 0.2 in the case a UHE neutrino [Gorham et al., 2009]. In the ice layer, the number of excess electrons giving rise to Cherenkov light emission is approximately 20 percent of the total number N [Gorham et al., 2009]. Thus, the number of charged particles that results in coherent Cherenkov emission is 0.2N. By the Askaryan effect, these emissions become coherent, and the total energy of the emission is amplified to wtot = 0.04 × N2w. This is the total energy of Cherenkov emissions caused by a single interaction between a neutrino and Europa.
 Cherenkov light is emitted as a cone with an apex half-angle of θ, which can be expressed as cosθ = 1/(nβ). The spread of the Cherenkov cone is expressed as Δθ = c sinθ/(νL), where ν is the mean frequency of emission. The solid angle of the Cherenkov emission can be determined as Ω = 2πΔθ sinθ. The intensity of emission at the antenna is wtot/(r2ΩνΔt) Jy, where r is the length of the emission path from the shower to the antenna, and 1 Jy = 10−26 W m−2Hz−1. Δt is the mean duration of the emission, which is detected by the antenna, and can be expressed as Δt ∼ 1/Δν, where Δν is the dispersion of the frequency from ν. The intensity of the emission increases with the mean frequency ν and the frequency band Δν. Since radiation with a wavelength of less than 10 cm does not become coherent, the intensity becomes maximum at a few gigahertz. Here, we use the frequency band between 1 and 3 GHz, i.e., νmin, νmax, ν, and Δν are 1 GHz, 3 GHz, 2 GHz, and 1 GHz, respectively.
 In this method, attenuation of the emission strongly affects the limit of the measurable thickness of ice. The attenuation length l is defined as the length an electromagnetic wave can travel until its amplitude attenuates to 1/e, as follows:
where ω, μ, and ε are the angular frequency of the wave, the permeability, and the permittivity, respectively. In this equation, tan δ = σ/(ωε), where σ is the electric conductivity. In the case of pure water, σ ∼ 10−3 S/m and ε ∼ 7.1 × 10−10 F/m. The electric conductivity of ice is smaller than that of liquid water. Thus, tan δ ≪ 1 at over 1 MHz. In this case, the attenuation length l can be approximated as l ∼ (2/σ) and does not depend on the frequency. This length changes depending on the conductivity hence the temperature of the ice. Chyba et al.  and Moore  analyzed the attenuation in the megahertz region by considering several models of ice composition and the temperature profile. Particularly they discussed importance of the temperature profile in the ice layer in the radar sounding, which should have a similar effect in this case. Here, for simplicity a single value of the attenuation length is used. The attenuation length of Europan ice is from 540 to 2,400 m if the ice is mixed with a small amount of rock or chloride. We therefore assume the attenuation length of the Europan ice to be 2 km, which is consistent with the attenuation level estimated by Moore . This value is considered to be efficient in the gigahertz region because, based on the approximation of l, the attenuation length does not depend on the frequency over the megahertz region.
 The attenuation length of liquid water is shorter than that of solid ice because the conductivity of water is larger than that of ice. If the water of Europa contains compounds such as salt, then the attenuation length becomes even shorter because of the higher conductivity. Model compositions of the Europan ocean are proposed by Kargel et al. . The lower limit of electrical conductivity of the Europan ocean is estimated to be ∼0.02 S/m [Hand and Chyba, 2007]. This value is still much larger than that for ice. Based on equation (3), using this value, we set the attenuation length of the water layer to be 2 m. The resulting strong attenuation would cause emissions from the water layer to be too weak to reach the antenna. Based on this principle, the number of observed emissions depends on the thickness of the ice.
2.2. Geometry and Condition of Simulation
 We assume a three-layered (surface solid ice, liquid water, and rock core) spherical model for Europa. We assume the thickness of the surface H2O layer to be 150 km, which is within the gravitational constraint of 80 to 170 km [Anderson et al., 1998]. Although Athar et al.  showed that the ratio of the three flavors of cosmic neutrino is 1:1:1, for simplicity, we consider only 1019 eV mu-neutrinos. In the simulation, 10 million neutrinos were shot into the Europa model. This amount is equivalent to the neutrinos arriving at Europa over a period of approximately seven days if we consider the energy range from 1018.5 eV to 1019.5 eV of the UHE neutrino flux reported by Engel et al. . In this energy range, we did not consider the effect of neutrinos from AGN or GRBs because the flux of these neutrinos is small [Stecker et al., 1991; Alvarez-Muñiz et al., 2000]. The incident position and direction of the neutrinos were given using random numbers. Calculations of neutrino propagation were performed using high-energy lepton calculation software called JULIeT [Yoshida et al., 2004]. We calculated the intensity of Cherenkov emissions that reach an antenna orbiting Europa for different antenna altitudes and ice thickness.
 The intensity of the thermal noise of the antenna, ΔS, can be estimated as follows:
where kB is the Boltzmann constant, Tsys is the temperature of the detector, and Aeff is the effective area of the detector [Gorham et al., 2009]. Here, we assume that Tsys is the surface temperature of Europa (∼100 K). If we use detector antennas of approximately 1 m2 in size, the magnitude of thermal noise is approximately 7 × 104 Jy. Based on these results, we evaluated radiation greater than 105 Jy as a detectable signal event.
3. Determination of Thickness
 The number of detectable emissions is shown in Figure 1 as a function of the thickness of the icy layer and the altitude of the antenna. The highest number of emissions is expected at an altitude of approximately 350 km. As the altitude is increased up to 350 km, emissions from a larger volume of the ice can participate. When the altitude of the antenna exceeds 350 km, the number of detectable events begins to decrease due to attenuation during propagation over long distances. At every altitude, the number of detectable events increases with ice thickness up to 8 km. This is because more emissions could reach the antenna before attenuation when the volume of ice was increased. However, at thicknesses greater than 6 km, the rate of increase becomes smaller, and no further increase occurs at thicknesses greater than 8 km. Even though the attenuation length at the icy layer is much longer than that at the water layer, emissions from deeper than 8 km would be attenuated and not reach the antenna.
 The subtended volume V, which is defined as the ice volume constrained by the angle of view of the antenna over Europa, can be roughly estimated as follows:
where Rs, Rw, and ha are the mean radius of the orbit of the satellite (1,560 km), the radius from the center to the liquid water layer (Rs minus the thickness of the ice), and the altitude of the antenna above the surface, respectively. Figure 2 shows the number of signal events and the subtended volume of ice as a function of ice thickness when the altitude is 200 km. The cross symbols indicate the number of detectable signal events. The line indicates the subtended volume of ice calculated by equation (5) at ha = 200 km. At this altitude, the difference in the number of events can be used to determine the thickness of ice with a resolution of 1 km up to 6 km. Up to 5 km, the number of events increases proportionally with ice thickness. However, above 5 km, attenuation affects the emission, and the number of events saturates. Based on the difference in the number of signal events, we were able to determine the thickness of the icy layer with a resolution of 1 km up to 6 km.
 The total number of events depends on the absolute neutrino flux, which is quite difficult to estimate in the vicinity of Jupiter. Instead, we propose a supplementary method to specify the thickness based on the intensity distributions of the signals. Figure 3 shows the distributions of intensity when the icy thickness is 4 km, 6 km, and 8 km. The altitude of the antenna is 200 km in all cases. The number of emissions with intensities of 105 − 5 × 106 Jy and 105 − 106 Jy increases with thickness from 4 to 6 km and from 6 to 8 km respectively. Since the effect of attenuation becomes significant with increasing ice thickness, there should be an increase in the number of lower intensity signals. By limiting the intensity range to 105 − 106 Jy at (b) and (c), the number of detectable emissions increases by approximately 20 percent. This intensity is approximately 10 times the amplitude of the thermal noise. Thus, we can determine the thickness of the ice in this range above 6 km. Using the intensity distributions, the limit of thickness determined by this method could be expanded. Since the total number of signals is saturated above 8 km (Figure 1), we could not determine thicknesses above 8 km. The limit of thickness depends largely on the attenuation length of the Europan ice, which remains unknown and must be determined in order to perform further examinations.
 By measuring the number of emissions detected by an antenna orbiting Europa, the ice thickness could be determined with a resolution of 1 km for thicknesses of up to 6 km. If the intensity distribution of the emissions is available, then the thickness is further constrained up to 8 km. In this simulation, we could not determine thicknesses above 8 km. However, this method can at least determine whether the thin-crust model or the thick-crust model provides an appropriate approximation of the Europan ice layer. Although further studies are necessary in order to take into account the existence of three flavors of neutrino, and other effects such as radio emissions from Jupiter, this is an effective method by which to provide a new constraint on the thickness of the Europan ice and to detect the existence of a subsurface ocean.
 We thank K. Hoshina and Chiba IceCube Group for introducing the simulation software JULIeT. ANITA project team opens the outline of their mission, which is helpful to our research.
 The Editor thanks the two anonymous reviewers for their assistance in evaluating this paper.