This paper presents a numerical approach to model the electrical properties of Titan's atmosphere. The finite difference time domain technique is applied to model the atmosphere of Saturn's satellite in order to determine Schumann resonant frequencies and electromagnetic field distributions at the extremely low frequency range. Spherical coordinates are employed, and periodic boundary conditions are implemented in order to exploit the symmetry in rotation of the celestial body. Results are compared with a previous model using the transmission line matrix method up to 180 km altitude. For the first time a numerical FDTD model up to 800 km altitude is carried out, and we report lower frequencies than other previous models. The Schumann resonances of Titan were measured by the NASA–European Space Agency Cassini-Huygens mission in January 2005. Our mathematical model can be modified to change the conductivity profiles to explain the observed experimental data.
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 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 CH4, CO, and N2 atmosphere. The existence of lightning discharges on Titan is discussed by Borucki et al.  and Tokano et al. , 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.
 Direct ionization of N2, CH4, Ar, H2, and CO by cosmic rays would produce magnetospheric electrons and an ion distribution with CH5+, HCO+, HCNH+, and CH4+ [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.
 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].
 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.
 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  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.
 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.
 Several profiles for the conductivity are provided by Morente et al.  and Molina-Cuberos , 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.
 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.
 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.
2. Numerical Model
 Since Titan has an almost spherical symmetry, the spherical coordinate system is used to model the atmosphere cavity. The spherical coordinates are curvilinear coordinates that require the use of the integral expressions of the Maxwell curl equations in free source regions [Navarro et al., 1994; Stratton, 1941]:
 In the above equations, S represents the integration surface and L is a closed integration path which delimits the surface S. The continuous electric and magnetic fields are discretized to obtain the FDTD equations. The discretization of the radial component of the electric field is
where the parameter RT is Titan's radius and i, j, and k are integers corresponding to the r, θ, and ϕ directions, respectively.
 Discrete equations are obtained, taking into account the position of the electric and magnetic field components in the spherical nodes, as illustrated in Figures 1 and 2:
 The magnetic field component in the radial direction is updated with the above equation at each n + 1/2 time step. Similar expressions are derived to update the other field components. The coefficient Shr in (3) represents the integration surface element corresponding to each cell; it is obtained by integrating the differential surface element in the radial direction dSr. Similar coefficients are evaluated in the θ and ϕ directions,
 The coefficients leθ(i) = (RT + i △r) △θ, leϕ(i, j) = (RT + i △r) sin (j △θ ) △ϕ correspond to the length of the integration path in each direction and describe the geometrical properties of the mesh.
 Our mesh does not include the origin; only the discontinuity along the north-south axis (N-S axis) needs to be addressed [Fusco et al., 1991]: the zenithal angles θ = 0 and θ = π. To overcome this discontinuity, an alternative circular path for the Hϕ, surrounding the discontinuity in the N-S axis, is used to update Er, (see Figure 3). The following integral equation was solved to update Er along the discontinuity:
 There is a special treatment for the electric field along the N-S axis, with the special variables ENorthn(i) in the Northern Hemisphere and ESouthn(i) in the Southern Hemisphere. Soriano et al.  develop the special updating equations for ENorthn(i) and ESouthn(i):
 The updating equation for the ENorth points is
 The coefficient Ai, introduced in (7), depends on the discretization and electrical properties. It includes all geometrical and electrical coefficients involved in the updating equation for the electrical field along the N-S axis:
 Analogous equations are derived for the ESouthn special field components.
 A part of the mesh used in the simulation is shown in Figure 4. The periodic boundary conditions are implemented to consider that electric and magnetic fields are ϕ independent. The following equations enforce the periodic boundary conditions for the radial components:
 The complete implementation of the periodic boundary conditions requires analogous expressions for the θ components of the electric and magnetic fields. These conditions are introduced to enforce no-field variations along the ϕ direction.
 In addition to periodic boundary conditions, a special treatment of the θ = 0, π axis is required to reduce equations (5)–(8) to consider a single cell along the ϕ direction. In order to reduce computational requirements, no variation of the electromagnetic field is assumed along the ϕ axis. Thus Hϕ(i, 0, k) remains constant along the ϕ direction (k-index). Then the updating equation for ENorthn(i) in this particular symmetry is
 Losses in the ionosphere are introduced by means of a finite conductivity σ, which in our model is dependent on the radial distance to the Titan surface, σ = σ(r). The profile for the ionosphere conductivity for our numerical model was obtained from Morente et al. :
 The linear temporal differentiation is replaced by a first-order exponential scheme [Holland et al., 1980] that is more appropriated when high losses are present.
 The discrete updating equation for Er in the modeling of the layered structures with high losses of Titan's ionosphere is
where the conductivity profile in ionosphere is stratified, σ = σ(r). Similar equations to (12) are obtained for the rest of the electric field components, whereas the updating equations for the magnetic field components are the same for both a lossless and a lossy cavity.
3. Numerical Results
 The above proposed technique is used in the numerical calculation of Titan's Schumann resonances. The lowest resonant modes are extremely low frequency modes (ELF), as their characteristic wavelength must be on the order of magnitude of Titan's radius, RT = 2575 km, that is, lower than 150 Hz. The conductivity profile of Titan's atmosphere was predicted to have a maximum around 1000 km [Molina-Cuberos, 2004], the value at 180–200 km altitude was estimated around σ = 10−7, and the conductivity sharply increases from 500 km (σ = 2 S/m) up to 2000 km, to achieve values around 102 S/m at 2000 km altitudes. Therefore around 100 km the ionosphere was expected to behave as a quasi-conductor because for the expected ELF Schumann frequencies, the ratio σ/με is between 90 at 20 Hz and 6 at 300 Hz. Near 200 km altitude we have a conducting atmosphere with ratios σ/με higher than 100 at ELF frequencies. This justifies our previous work [Morente et al., 2003], in which we assumed that the TLM analysis could be accurate enough up to 180 km altitude; for these frequencies, a reflection coefficient was defined at the end of the TLM mesh, and the conduction current was predominant (J = σE) over the displacement current.
 Titan's surface is modeled as a perfect conductor (σ = ∞), and the ionosphere layer is modeled like a good conductor with the conductivity profile of Figure 5; this is the Earth-like model. The conductivity profile at Titan's ionosphere takes into account the present knowledge of Titan's aeronomy to calculate the concentration of electrons and ions, that is, the nominal profile of Molina-Cuberos  and Morente et al. , plotted in Figure 5.
 First, a 180 km thick atmosphere layer is simulated. Titan's atmosphere from the conducting surface to 180 km altitude is simulated using a 120 × 120 mesh, resulting in spatial and angular discretizations △r = 1.5 km and △θ = π/120 rad. As mentioned before, only one cell was computed along the ϕ direction; the angular cell size is △ϕ = 2π/50 rad. The discretizations along the r and θ directions (△r and △θ, respectively) are chosen depending on geometric parameters. The length of the cell along the ϕ direction is less important to the results than the other two. The criterion we used to determine this value was a balance to get cells as squared as possible within the equator and the poles. Because the cell length in the ϕ direction depends on the latitude, lϕ is shorter in the poles than in the equator; in our 120 × 120 mesh, lϕ was in the interval 8.47–323.58 km. The dimensions of the cells along θ directions varied between 67.4 and 72.1 km. In the radial direction, the mesh finishes with five additional cells with perfect matched layers (PML), backed by a perfect conductor. If the number of layers in the PML is increased, the accuracy in the prediction of Schumann resonances does not improve significantly, and the simulation takes a longer time.
 We tried different meshes, and we found convergency with the 120 × 120 mesh which was optimal to calculate the Schumann frequencies up to 100 Hz. In using a mesh with twice the density, we did not get a significant improvement but consumed more computer resources. Our program runs in a Pentium IV personal computer with 512 MB RAM.
 The Schumann resonances of the Earth-like Titan's ionosphere cavity are obtained by performing the fast Fourier transform (FFT) over the time domain fields. Since the Schumann resonances are in the ELF range, a large simulation time (N △t) is required to achieve the desired FFT resolution, given by △F = (N △t)−1. The Courant stability criterion restricts the time step to △t = 3.5 10−6 s. Therefore a large number of time iterations is required in order to achieve enough sensitivity for the FFT. In our simulations we have a frequency resolution △F = 0.1 Hz. A quasi-continuum soft spectrum is obtained because of the losses. This is because of the overlapping of contiguous modes, caused by the broadening of the resonant response after the introduction of losses. This spectral response is typical from a low-Q structure, where losses are present. To estimate the resonant frequencies of this low-Q structure, we used the multiple signal classification (MUSIC) libraries of MATLABTM and a single time series from the time domain fields.
Table 1 summarizes the results obtained with the FDTD technique in the above model. These are compared against transmission line matrix (TLM) results of our previous TLM model [Morente et al., 2003]. Resonant frequencies obtained with the TLM method and FDTD are very similar; differences range from 0.2% deviation of the fifth mode to 8% deviation for the fourth mode. The similarity of the numerical resonances obtained with both numerical methods serves as validation for the proposed FDTD model. Differences are inherent to the numerical scheme, including the numerical absorbing boundary conditions. Both TLM and FDTD are similar numerical methods; the main epistemological difference between TLM and FDTD is that the first method uses analogous transmission line circuits whose voltages and currents define the electromagnetic field, while the FDTD technique directly evaluates the electromagnetic field by solving Maxwell equations through a finite difference scheme. The FDTD model was previously validated in the analysis of the Earth [Soriano et al., 2005], and now it is validated again for the Earth-like model (up to 180 km) of Titan by comparing with TLM results.
Table 1. Numerical Resonant Frequencies
n = 1
n = 2
n = 3
n = 4
n = 5
Model has 250 km deep dielectric surface εr = 4.5.
 Our model can be easily modified to extend the mesh to cover any altitude or to include the possibility of a nonconducting surface. Therefore the model was modified to extend the calculations up to 800 km altitude, using the nominal profile for the conductivity from Molina-Cuberos . The used mesh was 500 × 100, resulting in spatial and angular discretizations △r = 1.6 km and △θ = π/100 rad. In the 500 × 100 mesh, lϕ was in the interval 10.16–323.58 km. The dimensions of the cells along θ directions varied between 80.9 and 106.0 km, and the time step was △t = 3.74 × 10−6 s. The results from this model were quite interesting: The resonances obtained with the Earth-like 180 km model were 13.5, 24.5, 35.7, 47.1, and 58.8 Hz, and the results with the Earth-like 800 km model were 8.8, 17.4, 26.7, 36.8, and 47.6 Hz. All results are presented in Table 1. The shift in frequency for the first mode was 35%; deviations for the higher-order modes were 29%, 25%, 22%, and 19%, respectively.
 To have a better understanding of the mode structure, we calculate the Fourier transform of the time domain fields to obtain the spatial distribution of the amplitude of the field components. Specifically, we analyze the first resonance for the two models. The Earth-like 180 km model provides the plots of Figure 6, which shows the average electric field versus altitude, averaged at each layer r = constant. The Earth-like 800 km model provides the plots of Figure 7. Although field magnitude was considered negligible in the first model, because at the end of the mesh it was around 10−2–10−4 a.u. (arbitrary units), the truncation of the mesh at 180 km altitude distorts the spatial shape of the resonant modes. The first model up to 180 km was not large enough to include the spatial field distribution of the modes. The observed frequency shift decreases for the higher-order modes. A higher-order mode has a more pronounced attenuation in the conductive layer; then it is less sensitive to the truncation, and the attenuation is more pronounced when the conductivity profile changes. This is illustrated in Figure 8; the plots are the electric field magnitude versus altitude for the first five resonances, averaged at each layer r = constant. From the above results, we could argue that a proper model for Titan should include the ionosphere up to 800 km altitude, proposing our Earth-like 800 km model for Titan's atmosphere.
 Finally, we modify the Earth-like 800 km model to include a dielectric in Titan's surface. Titan's surface remained a mystery, in part because it was not directly observable because of the clouds. Different hypotheses have been put forward about its composition: water, ice cleaned by methane rains for bright observed regions, also solid ammonia, carbon dioxide, and other elements [Hamelin et al., 2000; Coustenis and Taylor, 1999]. The data from the SSP and HASI accelerometer are consistent with a soft substrate material similar to sand or a fluid component analogous to a wet sand or a textured tar/wet clay [Zarneckil et al., 2005].
 The model is modified following the scheme of Morente et al.  to consider a dielectric layer with a thickness 250 km and εr = 4.5. This approach was based on the reported measurements of the complex permittivity of frozen aqueous ammonia solutions at liquid nitrogen temperatures. The real part of the dielectric constant of 30% ammonia ice is around εr = 4.5 at near-ELF frequencies. The new data about Titan do not reveal yet a detailed composition of the surface; however, in the present FDTD model, an averaged εr must be introduced because a singe cell is used in the ϕ axis, and the variation of εr is not known along the surface in the r axis. The new model is named Titan 800 km model and includes in the surface a 250 km deep layer with no losses and εr = 4.5 and the atmosphere up to 800 km altitude. The used mesh was 600 × 100, resulting in spatial and angular discretizations △r = 1.75 km and △θ = π/100 rad. In the 500 × 100 mesh, lϕ was in the interval 9.18–292.17 km. The dimensions of the cells along θ directions varied between 73.0 and 106.0 km, and the time step was △t = 4.09 × 10−6 s. The obtained results for the first five Schumann resonances are shown in Table 1, and a plot of the spectral response of the atmosphere is in Figure 9.
 There was an attempt to model Titan's surface in the work of Morente et al. ; however, the assumptions about the composition of Titan's surface should be revised under the new knowledge provided by the Cassini-Huygens mission [Fulchignoni et al., 2005; Zarneckil et al., 2005]. These previous models from our research team now are improved with the addition of the present FDTD model up to 800 km altitude, which provides a better understanding of the electrical activity of Saturn's satellite. Our present Titan 800 km model demonstrates a magnetic diffusion of the ELF fields up to 800 km, having an influence in the numerical value of the frequencies, as well as the deep dielectric layer in the surface. A clear representation of the field diffusion is presented in Figure 10, which shows a picture of the magnetic field intensity for the entire r – θ plane from 250 km under the surface to 800 km over the surface of Titan, obtained with our FDTD Titan 800 km model. This magnetic diffusion is not clear in the representation of the electric field of Figure 11; however, it has a strong influence in the propagation of the Schumann modes and its frequencies.
 To summarize, despite the different methods used to evaluate Schumann frequencies in Titan, TLM or FDTD, the results depend on the precise definition of the conductivity profile, related to the ion distribution and electron number density. These also depend on the electrical properties of its surface related to its composition and layers. Our FDTD Earth model [Soriano et al., 2005] was exported to Titan, and three models were built: an Earth-like 180 km model, an Earth-like 800 km model, and a Titan 800 km model. There are frequency variations in the obtained resonances, depending on the model. The last two models are considered the more realistic because of the magnetic diffusion up to 800 km altitude.
 The FDTD numerical method has been used to model Titan's atmosphere in order to determine the Schumann resonant frequencies. The present paper describes the numerical procedure and results following three FDTD models of Saturn's moon, named the Earth-like 180 km model, the Earth-like 800 km model, and the Titan 800 km model, changing the altitude and the properties of the surface. These resonances would be very smooth, and the FDTD results have shown a fundamental mode between 8.8 and 10.9 Hz, depending on the electrical properties of the surface. The results demonstrate that a reasonable model for Titan at ELF frequencies should include the ionosphere up 800 km altitude.
 The electrical characteristics of Titan's environment have already been measured by the NASA-ESA Cassini-Huygens mission in January 2005. The presented models are going to be very useful in the analysis of the data and also in developing new models for the composition of Titan's atmosphere and surface in combination with the measurements of the Cassini-Huygens mission.
 This work has been supported by the Spanish government projects TIC2002-04096-C03-01, FIS2004-03273, and GV2005-254 and grant AP2001-3741.