## 1. Introduction

[2] Titan is the largest satellite of Saturn, and it was one of the main targets of the NASA–European Space Agency (ESA) Cassini-Huygens mission in January 2005. Several instruments on board the Cassini-Huygens mission were devoted to the detection of electromagnetic waves produced near Titan's surface, which could indicate the presence of lightning and electrical activity in a CH_{4}, CO, and N_{2} atmosphere. The existence of lightning discharges on Titan is discussed by *Borucki et al.* [1984] and *Tokano et al.* [2001], who predict that lightning takes place between the charged clouds and the ground, even though the Voyager mission could not detect this electrical activity because of the shielding produced in a hidden ionospheric layer by some meteoric ionization.

[3] Direct ionization of N_{2}, CH_{4}, Ar, H_{2}, and CO by cosmic rays would produce magnetospheric electrons and an ion distribution with CH_{5}^{+}, HCO^{+}, HCNH^{+}, and CH_{4}^{+} [*Molina-Cuberos*, 1999], providing an electron number density and an electrical conductivity profile that extends from the Titan surface up to 1400 km altitude [*Schwingenschuh et al.*, 2001]. The existence of cloud-to-ground strokes would produce electromagnetic waves over a broad frequency range that would propagate through the atmosphere and would be attenuated or reflected depending on the wave frequency and electrical conductivity of the ground and ionospheric layers. These electromagnetic fields at the extremely low frequency (ELF) range are known as Schumann resonances. Depending on the ionospheric losses, related to the electron number density, these ELF radio waves are able to circle the planet for many times before they completely attenuate. The celestial body behaves as a huge electromagnetic resonator with losses; where the electromagnetic field resonates between the ground and the layers of the ionosphere, the *Q* factor of the resonator is related to the ionospheric losses, related to the conductivity profile of the ionosphere also associated with the electron number density.

[4] The radio and plasma wave science (RPWS) instrument on board the Cassini orbiter measured the electromagnetic waves during the Titan flybys at distances greater than 900 km in January 2005. Cassini released the Huygens probe on 25 December 2004. The probe entered Titan's atmosphere and successively deployed its parachutes, taking 2 hours, 27 min to descend to the satellite's surface. The Huygens probe included the permittivity, wave, and altimeter (PWA) instrument as a part of the Huygens atmospheric structure instrument (HASI). These instruments were devoted to investigating the electric properties and electric field fluctuations during the descent [*Fulchignoni et al.*, 1997, 2005].

[5] Beforehand, the detection of the instruments was adjusted using expected results from quasi-analytical models of the atmosphere [*Nickolaenko et al.*, 2003], exploiting the results of Greifinger [*Greifinger and Greifinger*, 1978], and numerical data obtained with a transmission line matrix modeling (TLM) [*Morente et al.*, 2003]. Now these electrical measurements are partly available for the broad scientific community [*Fulchignoni et al.*, 2005]. Therefore numerical models like TLM [*Morente et al.*, 2003] and the finite difference time domain (FDTD) model presented in this manuscript are very useful for future analysis and interpretation of the measured data.

[6] The Schumann resonances in the Earth were predicted by W. O. Schumann in 1952 [*Schumann*, 1952] and were detected by Balser and Wagner in 1960 [*Balser and Wagner*, 1960]. The techniques available in the literature for the study of Schumann resonances were primarily based upon frequency domain waveguide theory [*Wait*, 1970]. Recently, *Cummer* [2000] applied a two-dimensional finite difference time domain (FDTD) technique in cylindrical coordinates to the modeling of propagation from lightning radiation in the Earth-ionosphere waveguide. Cummer showed that the FDTD technique was extremely well suited to the characterization of such a phenomenon in the very low frequency (VLF) range. In more recent papers, *Simpson and Taflove* [2002, 2004] developed a two-dimensional FDTD technique involving a mix of trapezoidal and triangular cells to map the entire surface of the Earth and described antipodal ELF propagation and Schumann resonances.

[7] In this paper we extend our electrical FDTD model of the Earth's atmosphere [*Soriano et al.*, 2005] to Titan. Our Earth-like model permits the characterization of Schumann resonances, using different expected profiles for the electrical conductivity of its ionosphere. Our computational demands are minimized by the implementation of periodic boundary conditions. A simpler model of the atmosphere considers Titan's surface and the ionosphere as perfect conductors, the gap between both conducting surfaces being around 180 km. This model provides a first approximation to derive Schumann resonances [*Morente et al.*, 2003]. However, better results are expected if electrical conductivity is inserted into the model.

[8] Several profiles for the conductivity are provided by *Morente et al.* [2003] and *Molina-Cuberos* [1999], and these were introduced in our FDTD model. We compare our Earth-like FDTD results with our previous TLM results [*Morente et al.*, 2003], obtaining a reasonable agreement for a model that considers a conducting surface and a conductivity profile up to 180 km altitude. Both TLM and FDTD models used the same conductivity profiles up to 180 km altitude. We found that the ELF field components have a negligible value for the first Schumann frequency and almost negligible values for the other resonances near the end of our simulation domain near 180 km. In the TLM model a small part of the energy escapes from the outer layer because of a large but finite conductivity; to calculate this escaping energy, a reflection coefficient is calculated at 180 km. In the FDTD model our mesh arrived to 180 km altitude to compare with the previous TLM results. The mesh of the FDTD model was extended up to 800 km using the conductivity profiles of the atmosphere from the surface (conducting) up to 800 km altitude [*Morente et al.*, 2003; *Molina-Cuberos*, 2004]. We found that Schumann resonances decreased the frequencies, providing results similar to the Earth for the fundamental mode.

[9] Our model does not include ionosphere day/night asymmetry or the anisotropy of the ionosphere; in this case it is not necessary to complete a three-dimensional (3-D) model, but the implementation of the symmetry and periodicity obviously improves the accuracy of the overall results. We look for ELF resonant fields below 100 Hz in the Saturnian moon. Because of the spherical symmetry, the resonant frequencies have no dependence in *ϕ* [*Morente et al.*, 2003]; then in our model we assume there is no *ϕ* variation for all the field values, deriving in a 2-D azimuthally symmetry. However, in our FDTD scheme we can introduce more cells along the *ϕ* direction to complete the Titan perimeter for a future analysis of day/night ionosphere asymmetry.

[10] A very simple model also efficient in terms of computer resources is presented, which will be very useful in the analysis of the electrical ELF properties of Titan, by means of simulation, analysis, and comparison with the available data [*Fulchignoni et al.*, 2005]. Our FDTD Earth-like model, validated in the analysis of the Earth [*Soriano et al.*, 2005], is also validated for Titan (180 km model) and is demonstrated as a useful tool for the analysis of the ionosphere and electrical activity at other celestial bodies.