The rotational normal modes of the planet Mars are computed using a numerical approach. We deduce the associated resonances in the nutations induced by the external gravitational forcing from the Sun, Phobos, and Deimos. In particular, the influence of a possible solid inner core inside a liquid core is investigated. A normal mode associated with the inner core (Free Inner Core Nutation (FICN)) is computed, as well as the associated resonance effects in the nutations. For a small inner core, before the eutectic composition of the outer core is reached, the FICN effect on nutation is negligible. For a large inner core it can be very important, and for some nutations it can greatly decrease the resonance effect of the classical Free Core Nutation. Future observations of Mars nutation resonances will therefore provide information about the inner core.
 Our present understanding of the internal structure of Mars is mainly based on interpretation of gravity data, on analyses of SNC meteorites, on extrapolation of the Earth's internal structure to the lower pressures of Mars' interior, and on the observational determination of the precession constant from Viking and Pathfinder [Sohl and Spohn, 1997; Folkner et al., 1997]. The range of the core radius compatible with the latest measurements is between 1300 and 1700 km [Folkner et al., 1997]. Recently, Yoder et al.  estimated somewhat larger core radii from Mars Global Surveyor (MGS) tracking data analysis, and concluded from their estimation of the potential Love number that at least the outer part of the core should be liquid. From the absence of a magnetic field and from the rather high percentage of sulfur in the core deduced from the SNC meteorites, it is generally believed that the core is entirely liquid (see Spohn et al.  for a review). However, the possible existence of a solid inner core cannot be definitively ruled out. The size, mineralogy and thermal state of the core are crucial parameters for understanding the accretion and internal structure of planets [Schubert and Spohn, 1990]. Their values for Mars are constrained by the moment of inertia of Mars, as shown by Sohl and Spohn  and Bertka and Fei , but there are still uncertainties so that the existence of a solid inner core cannot be excluded.
 One of the main objectives of a radioscience experiment for Mars, such as the NEtlander Ionosphere and Geodesy Experiment (NEIGE) foreseen for the network mission NetLander, is to determine the Mars orientation and rotation parameters: precession, nutations and variations of the length of day [Barriot et al., 2001]. This will be done by measuring the Doppler shifts of the radio links between the landers and an orbiter and between this orbiter and the Earth.
 Precession and nutations are variations of the orientation of the planet in space, due to the gravitational torque imposed by the sun, moons and the other planets of the solar system. The precession is the motion like a spinning top for which the Mars rotation axis describes a circle in the sky in about 170000 years. The nutations are superimposed on the precession motion and have periods from several days to several years in the inertial space. Each nutation corresponds to an elliptical motion of the rotation pole in space, characterized by its period and its amplitude. There are two ways to describe the nutations: (1) the elliptical motion is the combination of two circular motions, one prograde and one retrograde, with the same period but different amplitudes; (2) the variable inclination of the rotation axis is associated with a motion of the equator about the ecliptic, so that each nutation can also be described by an amplitude for the variation of the longitude of the equinox, and an amplitude for the variation of the obliquity of the ecliptic.
 The future observations of Mars rotation and orientation will provide information about the interior of the planet and about the seasonal mass exchange between atmosphere and ice caps. The major questions concern the core: does it contain a solid inner core, and what are the dimensions and the moments of inertia of the core and of the possible inner core. If Mars has a liquid core, the nutations driven by the gravitational force of the Sun with frequencies at multiples of the orbital frequency are influenced by a resonance effect due to the free core nutation (FCN). The FCN is a rotational normal mode related to the possibility of having the liquid core rotating around an axis different from the rotation axis of its ellipsoidal deformable container (the mantle). The observational signature of such a resonance will confirm that the core is liquid [Dehant et al., 2000b]. The FCN is in the retrograde band of the nutation spectrum, and its eigenfrequency depends on the core-mantle-boundary flattening but also on the presence and size of an inner core. These influences are examined by Dehant et al. [2000a, 2000b], Defraigne et al. , and Van Hoolst et al. [2000a, 2000b].
 The dynamical flattening of the core-mantle interface and the deformation of this interface are parameters that influence strongly the value of the FCN period. For an initial hydrostatic equilibrium state and a core radius between 1250 and 1700 km, the resonance period lies between 230 and 280 Martian days depending on the dimension of the core [see Van Hoolst et al., 2000a; Dehant et al., 2000b, Figure 2]. However, Defraigne et al.  have shown that the flattening of the core-mantle boundary can additionally be largely increased by non-hydrostatic effects associated with mantle convection, considering the geoid perturbation above Tharsis as a result of a hot plume underneath. The difference between the equatorial radius and the polar radius can increase by up to 5.2 km (extreme case), and consequently the values of the FCN period can decrease down to 120 days, depending on the mantle convection model used. The consideration of non-hydrostatic equilibrium thus has important impacts on the nutation transfer function in the retrograde frequency band as shown by Dehant et al. [2000a], because it moves the resonance that can be very close to some of the solar-induced nutations. From this, the future nutation observations will allow the determination of the FCN period, which will then give insight on the core flattening, and hence on mantle convection parameters (mantle viscosity profile, density anomalies, presence of phase transitions). In particular, Defraigne et al.  have shown that a very short FCN period (less than 200 days) indicates the absence of a perovskite layer at the bottom of the mantle, while a FCN period close to the hydrostatic value cannot be interpreted in terms of such a phase transition inside Mars.
 Van Hoolst et al. [2000a] have studied the effect on the FCN of an inner core varying in size. The consequences on the FCN period and thus on the resonance in the nutation transfer function are important when the inner core is large. In that case, the FCN period can decrease by more than 100 days. In addition to affecting the FCN, the existence of a solid inner core leads to an extra rotational mode, the Free Inner Core Nutation (FICN), describing a relative rotation of the inner core with respect to the outer core and mantle. This mode is in the prograde frequency band of the nutation spectrum, and induces a second resonance in the nutations, at a frequency different from the FCN resonance frequency. The present paper computes the nutation resonances for different inner core sizes and evolution stages, and investigates the possibility to detect the FICN in the future NEIGE observations, either as a resonance in the nutations or as a modification of the FCN resonance.
 The paper is organized as follows. The procedure for computing nutations is described in section 2. Section 3 presents the modeling of an inner core in Mars interior. The numerical results for nutation transfer functions are given in section 4, and the nutation amplitudes as well as the time evolution of the orientation parameters are given in section 5.
2. Numerical Computation
 The theoretical nutations of Mars are computed using a convolution of a transfer function with the nutation amplitudes of the rigid planet deduced from the gravitational torque implied by the Sun, Phobos and Deimos. The transfer function corresponds to the response (nutation in our case) of the planet to a unit forcing. It is frequency dependent and accounts for the non-rigidity of the planet. It can be expressed as the sum of rotational normal mode resonances [Wahr, 1981a, 1981b]. The modes as well as the transfer function are computed using a numerical integration of the equation of motion, Poisson's equation for the gravitational potential, and by using stress-strain relations corresponding to the rheologies modeled for the different layers (mantle, liquid core and possibly solid core), with appropriate boundary conditions at the internal and external boundaries [Smith, 1974; Wahr, 1981a, 1981b; Dehant and Defraigne, 1997]. The code initially written for the Earth has been adapted to the planet Mars [Dehant et al., 2000a]. The solutions of the integration are the stress field, the potential readjustment and the displacement field everywhere inside the planet, as a response to a unit tidal gravitational potential introduced as forcing. The toroidal displacement of degree 1, order 1 in the spherical harmonic development of the displacement field corresponds to a rotation about an axis in the equator, i.e., a nutation.
 The FICN has a very small influence on the Earth nutations. The reason is that its eigenperiod is not very close to a main nutation term, and furthermore its resonance strength is very small. For Mars, depending on the properties of the inner core, the FICN eigenperiod could be very close to the period of a nutation, inducing large resonance effects. It could therefore lead to an observable signature in the future NEIGE data. In order to determine the Martian FICN period and resonance strength for different inner core dimensions, we modified the coding of the boundary conditions at the Inner Core Boundary (ICB), as well as the initial solution close to the center, for which the integration did not converge for an arbitrary small inner core. For uniformity reasons we also changed the variables used in the liquid core, using a Lagrangian representation as in the solid layers, rather than the Eulerian representation used before. This last modification does not change the numerical results.
3. Inner Core Modeling
 The equations which have to be solved to obtain the nutations contain geophysical parameters like the density, the shear and bulk moduli, the initial gravity, and the flattening everywhere inside the planet. For Mars' interior, we used the mean spherical density Model A of Sohl and Spohn , constrained by the Martian mass, radius, and polar moment of inertia. This model does not contain an inner core because the melting temperature of the iron alloy with 14.2 wt % sulfur, which is based on analyses of the SNC meteorites, is below the calculated temperature at the center. However, the uncertainties on the temperature profile in Mars and on the melting properties for core alloys are such that an inner core with 14.2 wt % of sulfur cannot be excluded.
 So, we used the mantle model of Sohl and Spohn  and modified their core modeling in order to include an inner core. The inner core formation can be separated in two stages (see Figure 1). The first stage corresponds to the precipitation of solid iron particles from the fluid core. Consequently, the concentration of sulfur in the liquid core increases until it reaches the eutectic composition of iron-sulfur alloy. At this point the second stage begins. During this stage the inner core growth is related to the freezing of the fluid alloy due to the temperature decrease. Before the eutectic composition is reached in the fluid core the density of the inner core corresponds to the density of pure solid iron at the pressure and temperature expected in Mars' core. For an inner core within a liquid core that has reached the eutectic composition the inner core is stratified, with a first layer of solid iron, and a second layer of a solid alloy of iron and sulfur with the eutectic composition.
 In the present study, we used densities of Fe and FeS corresponding to those expected for the mean core pressures and temperatures of Mars. For the solid phase we have ρFe = 8386 kg/m3 and ρFeS = 5704 kg/m3, and for the liquid phase the densities are taken to be 1% smaller. For an initial weight percentage of sulfur equal to 14.2%, we then determined the mean density of the inner core and the outer core as a function of inner core radius [see Van Hoolst and Jacobs, 2003]. We furthermore introduced a density gradient in the core by adapting the one of model A of Sohl and Spohn . We slightly modified the outer core radius in order to conserve the mass of the planet. This modification corresponds to an increase of the core radius of 12 to 20 km depending on the inner core radius. Figure 2 shows the density profiles so-obtained for some selected inner core sizes.
4. Transfer Functions for Different Inner Core Stages
 We computed the transfer function for nutation for all inner core sizes, from no inner core up to a fully solidified core. We calculated the ellipticities of the surfaces of equi-rheological properties as well as of the boundaries by assuming our interior models to be in hydrostatic equilibrium. Figure 3 represents the evolution of the frequency dependent amplification factor (i.e., the transfer function) as a function of the inner core radius. The locus for which the resonance effects are largest correspond to the eigenperiods of the rotational normal modes. Therefore this figure also shows the evolution of the FCN and FICN eigenperiods. The third resonance which appears at the period of 0 day is actually not a resonance, but rather a linear increase due to the term proportional to the frequency in the transfer function [see, e.g., Mathews et al., 2002] inducing an infinite value for infinite frequencies.
 Before the eutectic composition is reached, i.e., in our case, for inner core radii smaller than 965 km, the FCN period is nearly constant, at a period of −250 Martian days, and the FICN period increases slowly from about 580 to about 700 days. However, the resonance strength is very small for a small inner core, the effect on nutations of a very small inner core is therefore nearly negligible. But, the FICN period could be very close to the annual nutation (corresponding to 669 Martian sidereal days) and in that case induce a very large resonance. For our models, this corresponds to an inner core radius of 790 km. The detectability of this resonance in the frame of a geodetic experiment such as the NEIGE experiment will be discussed in the next section.
 The right-hand part of Figure 3 corresponds to an inner core with two layers, i.e., models in an evolution stage where the outer core has the eutectic composition. Here, the FCN as well as FICN periods strongly depend on the inner core size, and tend to zero for a completely solid core. To understand these changes physically, Dehant et al.  have used an analytical model to calculate the rotational normal modes and the nutations. They show that the FICN period is determined by two competing terms: one is related to the difference between solid and liquid core densities, and the other is proportional to the ratio between the moments of inertia of the inner and outer cores. The relative importance of these two terms varies with increasing inner core radius. In particular after the eutectic the density jump decreases as the inner core becomes richer in sulfur, so that the term related to the moments of inertia dominates, giving rise to the FICN period evolution seen in Figure 3. The decrease of the FCN period for very large inner core is due to the increasing ratio between the moments of inertia of the solid and liquid core.
 Due to this important change of the FCN and FICN periods, we can have inner core radii for which one of these modes is very close to a principal nutation (annual, semiannual, ter-annual, etc., both prograde and retrograde). In particular, for our models, the FICN period is very close to the semiannual prograde nutation for an inner core radius of about 1120 km. We also observe that the resonance strength of the FICN increases with increasing inner core size (the resonance zone is wider around the eigenperiod), so that this mode can efficiently enhance a particular nutation even when its period is not exactly this nutation period.
 The numerical values given here correspond to the model we have used, with a core consisting of iron and 14.2 wt % of sulfur. For a different percentage of sulfur in the core, the eutectic would be reached at another inner core radius, but the general shape of the curve of Figure 4 would be similar to what we have obtained (see Dehant et al.  for more details). Another important parameter for the FICN period is the ICB flattening, in the same way as the CMB flattening is important for the FCN period, as discussed in section 1. Convection in the inner core could lead to some deviation from the hydrostatic value for the ICB flattening, but this convection is unlikely. Indeed, during the evolution of the inner core, lighter material is added on top of a denser layer which tends to stabilize the inner core against convection. Also, the gravitational effect due to a possible CMB deformation could lead only to minor changes of the inner core flattening as shown for the Earth by Defraigne et al. .
 The nutation series for Mars considered as rigid have been computed from the VSOP87, ESAPHO and ESADE ephemerides of the Bureau des Longitudes by two different teams, at the Observatoire de Paris [Bouquillon and Souchay, 1999], and at the Royal Observatory of Belgium [Roosbeek, 2000]. Both computations are in very good agreement at the level of 1 milliarcsecond (mas). The only limitation of these computations is that the observed precession, and hence the dynamical flattening of Mars, is not very accurate; it is deduced from Viking and Pathfinder data [Yoder and Standish, 1997; Folkner et al., 1997] and has a precision of one percent. Observations by future geodetic experiments such as the NEIGE experiment of the NetLander mission will allow the refinement of this value.
 Both models give nutation amplitudes as periodic variations of the longitude of the Martian equinox and of the obliquity of the Martian ecliptic. We give in Table 1 the amplitudes for the prograde and retrograde components of each principal nutation. The largest nutation is by far the prograde semiannual nutation. Although this nutation is far from the FCN period, its rigid-Mars amplitude is amplified by at least 5 mas if the core is liquid [Dehant et al. 2000b].
 When we add an inner core to the model, the FCN resonance is modified and an additional (FICN) resonance appears. Therefore the amplification or decrease of the nutation amplitudes that we have for a fully liquid core is modified. This is particularly important for the prograde semiannual nutation, which can be very close to the FICN for certain inner core radii. We present in Figure 4 the amplification of the main nutations with respect to the rigid amplitudes as a function of the inner core radius, and we compare them with two possible levels of precision in the frequency domain for the future NEIGE experiment: 5 mas (blue lines) and 2 mas (green lines). These precisions have been deduced by [Yseboodt et al., 2003] considering a 0.1 mm/s precision on the Doppler measurements [Barriot et al., 2001]; the 5 mas was obtained using two landers and 9 month of observations only, and the 2 mas was obtained using four landers and one Martian year of observations.
 For our internal models with a small inner core the amplification of the semiannual prograde nutation is about 7 mas, well above the nominal precision of 2 mas obtained after one terrestrial year of observations. This amplification is due to the FCN and its observation will lead to the conclusion that the core is liquid. The curves for all nutations are nearly constant up to the radius corresponding to the eutectic (965 km) except for the annual prograde which shows a resonance to the FICN at 790 km. The constant values of the curves are equal to the amplification obtained for no inner core (inner core radius equal to zero in the figure). Therefore if the Mars' inner core is small, and did not yet reach the eutectic composition, it will not be possible to observe the FICN in the upcoming observations of nutations, and hence not possible to detect the presence of the inner core, except if the FICN is very close to the annual period.
 For a larger inner core, some km after the eutectic, the resonances of the prograde nutations become observable with amplification higher than the precision levels, so that it is possible to detect the FICN and hence the presence of a solid inner core. However, for inner core radius larger than 1150 km in our model, the semiannual prograde nutation resonance to the FCN is largely cancelled by the resonance to the FICN. As a consequence, the observation of no amplification of the semiannual prograde nutation could mean either a fully solidified core, or a very large inner core with still a liquid part in the core. In other words, the inner core will be detected, but it will be difficult to know if there is still a liquid layer around it. The other nutations which are affected by the FICN have smaller rigid amplitudes, so that their resonance could be observed only if the FICN frequency is very close to them, i.e., if the inner core radius is in a certain range as shown by the curves. When comparing the curves of Figure 4 to the precision levels of 5 and 2 mas, we come to the conclusion that only for a very large inner core (larger than 1300 km for our models) the state of the core and the inner core will be difficult to determine. The green and blue regions correspond to inner core radii for which it is not possible, from the NEIGE observations alone and with the precision of 5 mas (blue) or 2 mas (green), to distinguish between liquid core with large inner core and fully solid core.
 In the case of a large inner core, information can also be obtained from resonance effects to another rotational mode, the inner core wobble (a mode similar to the planet Chandler wobble but for the solid inner core) which could have a period close to the main period of polar motion of Mars and largely enhance this signal [Dehant et al., 2003]. Observation of this effect would prove the core to be partially liquid. On the other hand, although it could be difficult in some cases to get direct evidence for an at least partially liquid core from observations of nutation and polar motion, a completely solid core for Mars is highly unlikely. A small liquid outer core would contain a high concentration of sulfur, equal to the eutectic concentration for our iron-sulfur alloy here, which would reduce the melting temperature for the outer core alloy to a value below any reasonable estimate of the present temperature at the core-mantle boundary of Mars. Moreover, Yoder et al.  have obtained positive evidence from MGS tracking data for at least the outer part of the core to be liquid. Although for a large inner core an experiment like NEIGE possibly could not give independent proof of the partial liquidity of the core, the absence of a detectable amplification of the nutations, combined with an analysis of the time variation of the orbit element of the satellite as done by Yoder et al. , could then be interpreted as evidence for the existence of a large inner core with an outer core of eutectic composition.
 The nutations driven by the Sun have a total amplitude of about 1000 mas corresponding to a displacement of the rotation axis at Mars' surface of 16 m. We have represented in Figure 5 the time variations of the longitude of the equinox of Mars. With the expected precision of the NEIGE experiment on the direct measurements (50 mas) it appears clear that if one nutation is very close to the FCN period or FICN period it will be possible to distinguish between liquid and solid core very rapidly, if the measurements begin near an extremum. For example, in the figure we show the case of a large induced resonance in the prograde semiannual nutation to the FICN. In other cases, it will be necessary to wait longer and to use determination of resonance parameters in the frequency domain as envisaged by the NEIGE strategy [Yseboodt et al., 2003]. We do not show the results for the variations of the obliquity of Mars which lead to the same conclusions.
 In this paper, we investigated the effect of a solid inner core inside Mars on the nutations of the planet. The presence of an inner core induces an additional rotational mode called the Free Inner Core Nutation (FICN), leading to additional resonance in the nutation frequency band. We computed the effects of the FICN on nutations using a particular model of Mars' interior based on Model A of Sohl and Spohn  with 14.2 wt % sulfur. Using an analytical approach for a simplified model with three homogeneous layers, Dehant et al.  have shown that a change in this model would give the same general shape of the nutation transfer function. Therefore the general conclusions we got concerning the detection of the liquid outer core and of the solid inner core remain valid for the range of possible models of Mars interior.
 Our results show that for a small inner core, before the eutectic composition is reached in the liquid outer core, the effect of the FICN on nutations is negligible, except if the FICN is very close to the annual prograde nutation. This resonance occurs for an inner core radius of 750 to 800 km for our models of Mars' interior based on 14.2 wt % sulfur. Therefore if the inner core of Mars is before the eutectic, and not at a radius implying a resonance of the annual prograde nutation, it will not be possible to observe the FICN in the future observations of nutations, and hence not possible to detect the presence of the inner core.
 For larger inner core we have shown that the resonances of the prograde nutations become observable with amplification higher than the precision levels, so that it is possible to detect the FICN and hence the presence of a solid inner core. However, for very large inner core (more than 1150 km in our model) the resonance with the FICN decreases strongly the FCN resonance effect on the semiannual prograde nutation. The amplitude of this nutation, which is by far the largest of the Martian nutations can then not be distinguished observationally from the amplitude for a rigid planet. Therefore, if the upcoming mission does not detect any semiannual prograde nutation resonance, this will give the proof that there exists an inner core in Mars, but two cases are possible. First, the planet is fully rigid, i.e., the core is totally solid, although this is very unlikely because it requires unexpected low temperatures at the CMB. Second, there is a large inner core and a liquid outer core of eutectic composition. This interpretation would be in agreement with the liquid core detected by Yoder et al.  from MGS data. As a conclusion, if the inner core is small it will not be possible to detect it using nutation observations except if the FICN is very close to the annual period. If the inner core is large, it will be detected, but in that case it will be difficult to determine if there is still a liquid layer using only nutation observations, except if the FICN period is very close to a nutation period.
 We also showed that for particular inner core radii, the FICN can induce a large resonance of a particular nutation. In that case, the direct determination of Mars orientation in space by the future geodesy experiment NEIGE could detect the existence of an inner core, without waiting a long time to perform a least square fit as foreseen for NEIGE. This fit however will be necessary in order to determine more precisely the FICN eigenfrequency, and hence the dimensions of the inner core. These results indicate therefore the necessity to include the FICN in the NEIGE strategy which consists of inverting the Doppler shifts between the landers and an orbiter and between the orbiter and the Earth in order to deduce the Mars orientation parameters and their resonances.