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Polar motion of Titan forced by the atmosphere

2011; American Geophysical Union; Volume: 116; Issue: E5 Linguagem: Inglês

10.1029/2010je003758

2156-2202

Tetsuya Tokano, Tim Van Hoolst, Özgür Karatekin,

Solar and Space Plasma Dynamics

Journal of Geophysical Research: PlanetsVolume 116, Issue E5 Free Access Polar motion of Titan forced by the atmosphere Tetsuya Tokano, Tetsuya Tokano [email protected] Institut für Geophysik und Meteorologie, Universität zu Köln, Cologne, GermanySearch for more papers by this authorTim Van Hoolst, Tim Van Hoolst Royal Observatory of Belgium, Brussels, BelgiumSearch for more papers by this authorÖzgür Karatekin, Özgür Karatekin Royal Observatory of Belgium, Brussels, BelgiumSearch for more papers by this author Tetsuya Tokano, Tetsuya Tokano [email protected] Institut für Geophysik und Meteorologie, Universität zu Köln, Cologne, GermanySearch for more papers by this authorTim Van Hoolst, Tim Van Hoolst Royal Observatory of Belgium, Brussels, BelgiumSearch for more papers by this authorÖzgür Karatekin, Özgür Karatekin Royal Observatory of Belgium, Brussels, BelgiumSearch for more papers by this author First published: 13 May 2011 https://doi.org/10.1029/2010JE003758Citations: 11AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract [1] Titan's atmosphere possesses an equatorial component of angular momentum, which can be transferred to the surface and can excite polar motion of Titan. The atmospheric excitation of Titan's polar motion is calculated using the wind and pressure data prediction from a general circulation model. The polar motion equation is solved considering Titan's triaxial shape and different hypothetical interior models. Titan's polar motion basically consists of a superposition of small diurnal wobbles and larger semiannual and annual wobbles caused by seasonal redistribution of wind and pressure pattern. If the entire interior of Titan is solid, the polar motion has total amplitudes of a few meters, but the diurnal and seasonal wobble may be intermingled due to preferential damping of the seasonal wobble by Saturn's gravitational torque. If instead there is a subsurface ocean underneath the crust, the wobble amplitude could be larger by an order of magnitude. If the crust is thin, a resonance between the seasonal and Chandler wobble further increases the polar motion amplitude and makes the polar motion path elliptical. The seasonal wobble of a crust lying on a subsurface ocean experiences damping by either gravitational and pressure torque or elastic torque, but the relative reduction of the polar motion amplitude by these torques is likely to be smaller than that of the length-of-day variations. Key Points Titan's atmosphere excites a polar motion The shape and size of the polar motion depend on the assumed interior structure The polar motion is elliptical because of Titan's triaxial shape 1. Introduction [2] Saturn's largest moon Titan is covered by a dense atmosphere of ∼1.5 bars. Such a dense atmosphere could appreciably change the rotation rate or length of day (LOD) of Titan due to a periodical exchange of angular momentum between the surface and atmosphere [Tokano and Neubauer, 2005; Friedson et al., 2009]. Observations of Titan's rotational state by Cassini have been used in combination with model predictions [Tokano and Neubauer, 2005] to constrain the interior structure of Titan, particularly the presence or absence of an internal ocean [Lorenz et al., 2008]. Subsequent theoretical studies [Karatekin et al., 2008; Van Hoolst et al., 2009; Mitchell, 2009; Goldreich and Mitchell, 2010] discussed how various additional torques would suppress the large LOD variation or cause a seasonal shift with respect to the prediction by Tokano and Neubauer [2005]. [3] All these studies considered only the relationship between the axial component of the atmospheric angular momentum (AAM) and Titan's rotation rate. However, the AAM is a three-dimensional vector, which can contain equatorial components in addition to the axial (polar) component [Barnes et al., 1983]. Cassini observations of Titan's atmosphere revealed a 4° tilt of the symmetry axis of the stratospheric atmospheric circulation from the polar axis and its westward rotation [Achterberg et al., 2008; Roman et al., 2009; Teanby et al., 2010]. This is direct evidence of an equatorial component of the AAM on Titan. Subsequently, the presence of a nonzero equatorial AAM and its westward rotation (precession) was also recognized in the output of the Titan general circulation model (GCM) of Tokano [2010]. This was ascribed to thermal tides and the same GCM also predicted a seasonal variation in the tilt angle or, equivalently, the amount of the equatorial AAM. [4] An important effect of the time variation of the equatorial AAM is the polar motion or wobble of the underlying body [e.g., Munk and MacDonald, 1960; Barnes et al., 1983]. The polar motion is the motion of the planetary rotation axis across its surface and has been observed on Earth [Munk and MacDonald, 1960] and Mars [Konopliv et al., 2006]. Considering the large AAM of Titan, a similar effect can also be expected on Titan as tentatively suggested by Bills and Nimmo [2008]. This effect, however, has not yet been quantified. It fundamentally differs from the secular wobble forced by the spin-orbit synchronization calculated by Noyelles et al. [2008] and Noyelles [2008]. [5] This study aims at numerically quantifying the polar motion of Titan using the output of the Titan GCM of Tokano [2010] and taking into account Titan's shape and various models of Titan's interior [e.g., Sotin et al., 2009]. This is the logical next step once the relationship between the axial AAM and LOD variation is addressed and the equatorial AAM is quantified. We expect that this calculation is not only important in its own right, but can contribute to the investigation of Titan's interior structure in combination with observational data. Our approach is analogous to the calculation of Mars' polar motion by Van den Acker et al. [2002] and Sanchez et al. [2004]. [6] Section 2 describes the methods of our study, i.e., the description of the physics, of the mathematical treatment and of the model used. Section 3 presents the various results of the calculation including the atmospheric forcing, parameters of Titan's interior relevant for the polar motion and the polar motion itself. Section 4 discusses how the polar motion could be modified by additional effects. 2. Methods Chandler Wobble [7] In general, the polar motion of a planet/moon is a combination of the Chandler wobble and wobbles forced by geophysical fluids such as atmosphere or ocean. The Chandler wobble is the free wobble of a nonrigid planet/moon, which owes its existence to the misalignment of the rotation axis and the figure axis. Its period/frequency is solely determined by the shape and interior structure of the body and is fully independent of the atmospheric or oceanic forcing. [8] The angular location of the instantaneous spin pole with respect to the principal axes of Titan can be represented by the complex function m(t) = mx(t) + imy(t). By analogy with the geodetic definition for the Earth, we define the x component of m as being directed toward Saturn (0° longitude) at Titan's pericenter and the y component toward the equator at 90°E. [9] Following this convention, the three principal momenta of inertia (MOI) are defined as C (axial), A (semimajor equatorial, along the x axis) and B (semiminor equatorial, along the y axis). The equatorial principal MOI, A and B, are determined as where M = 1.345 × 1023 kg is Titan's mass and R = 2575 km is Titan's mean radius. The gravity coefficients of Titan (J2, C22 and S22) have been determined by the Cassini radio science [Iess et al., 2010] and are also listed in Table 1. Table 1. Parameters of Titan's Interior Calculated With the Gravity Data of Iess et al. [2010] Parameter Symbol Value Iess et al. [2010] Mean radius (m) R 2.575 × 106 Degree 2 order 0 gravity coefficient J2 3.18 × 10−5 Degree 2 order 2 gravity coefficient C22 9.98 × 10−6 Degree 2 order 2 gravity coefficient S22 0.22 × 10−6 Axial MOI factor C/(MR2) 0.34 This Study Axial MOI (kg m2) C 3.03218 × 1035 Semimajor equatorial MOI (kg m2) A 3.03172 × 1035 Semiminor equatorial MOI (kg m2) B 3.03208 × 1035 Polar flattening α 9.355 × 10−5 Load Love number (no ocean) k′2 −0.03 Load Love number (with ocean) k′2 −0.83 Tidal potential Love number (no ocean) k2 0.03 Tidal potential Love number (with ocean) k2 0.28–0.4 Chandler frequency (no ocean) (Hz) σCW 3.25132 × 10−10 Chandler period (no ocean) (years) PCW 612.4 [10] The equations of the Chandler wobble mx and my of Titan considering the triaxial shape can be derived from the Liouville equations, which describe the change in angular momentum of a deformable planet/moon due to an applied torque [Van Hoolst, 2007], as where Ω = 4.56 × 10−6 s−1 is Titan's angular velocity, k2 is the degree 2 tidal potential Love number describing the resistance of the body to deformation (see section 3.1) and G is the universal gravitational constant. [11] The Chandler wobble frequency is the eigenfrequency of equations (3) and (4): where describe the equatorial MOI modified by a mass redistribution of deformable Titan due to the variable centrifugal potential. [12] Equation (5) extends the classical expression for the Euler frequency for a triaxial body (for which A < B < C) by including the effect of deformation and is equivalent to the expression of Van Hoolst and Dehant [2002] up to the order in the flattenings considered by them. Equation (5) applies if Titan has no decoupling subsurface ocean. [13] If there is a subsurface ocean and the overlying crust can be assumed to be decoupled from the deeper interior, the Chandler wobble frequency inversely scales with the effective MOI, so equation (5) has to be multiplied by C/Cc, where Cc is the polar MOI only of the outer crust. Cc is a function of the crustal thickness, which is a free parameter in the model, and is calculated as where ρc = 917 kg m−3 is the density of the outer crust (ice I), cc is the inner radius of the crust (radius of the crust-ocean interface), i.e., R − cc is the crustal thickness, and α is the dynamical polar flattening. Equation (8) is modified from equation (8) of Sohl et al. [2003]. The factor 2/3α accounts for the relative difference between the mean and polar MOI. [14] The dynamical polar flattening is defined as Atmospheric Excitation of Polar Motion [15] Besides the Chandler wobble, the atmosphere also excites wobbles with periods characterized by the various timescales occurring in the atmosphere such as seasonal or diurnal cycle. The equation of polar motion (wobble) predicts the change in the instantaneous pole position m in response to perturbations of inertia products due to mass redistribution and motion. It can be deduced from the linearized Liouville equations, which can be expressed as by extending the method of Barnes et al. [1983] for a biaxial planet to a triaxial body (see Van Hoolst [2007] for the derivation for a triaxial body). The difference of equations (10) and (11) with those for a biaxial body arises solely from the inequality of and . Here, k′2 is the degree 2 load Love number (see section 3.1). The atmospheric excitation manifests itself in the relative AAM (wind term of the AAM), hx and hy, and the incremental inertia elements (pressure term of the AAM divided by Ω), c13 and c23, which are both explained in section 2.3. [16] The solutions of equations (10) and (11) yield the instantaneous position of the pole as a function of time, which are given by integrating the polar motion at a given frequency over the entire frequency spectrum If Titan is entirely solid, the x and y component of m(σ) are given by [17] Here, the terms on the right-hand side including hx, hy, c13 and c23 are complex quantities depending on σ. This solution shows that polar motion in both x and y components is resonant at the Chandler wobble frequency as for the classical polar motion solution for a biaxial planet, to which it reduces when A is set equal to B. [18] If Titan's interior harbors a subsurface ocean underneath the outer crust and no coupling between the outer crust and mantle/core is assumed, only the outer crust undergoes polar motion. Therefore, all the momenta of inertia in equations (13) and (14) have to be replaced by those of the outer crust, except the MOI differences in the first term: Here, the internal parameters (k′2, σCW) for the ocean case have to be used. The MOI of the deformed crust, and , are calculated from and , respectively, analogously to equation (8). In equations (15) and (16) the MOI differences C − and C − are not replaced by the differences of the MOI of the crust because full decoupling implies a spherically symmetric interior. [19] The variable m(t) is expressed in radians, which can be converted to meters by multiplying by Titan's radius. Equation (12) is solved numerically with a spectral resolution of Δσ = 1.69 × 10−9 Hz. [20] In our baseline calculation we assume that there is no interaction between Titan's polar motion and Saturn or between the outer crust and the subsurface ocean and mantle/core. This is not correct since there are additional torques that would counteract the polar motion. Depending on the assumed interior structure the additional torques comprise Saturn's gravitational torque (external coupling) on Titan [Van Hoolst et al., 2009], the gravitational and pressure torque between the outer crust above a subsurface ocean (internal coupling) [Karatekin et al., 2008; Van Hoolst et al., 2009] or the elastic torque acting on the crust [Goldreich and Mitchell, 2010]. We discuss separately in section 4 how these additional torques may affect our model results. Atmospheric Forcing [21] In this work only the forcing of Titan's polar motion by the atmosphere is considered, although we do not a priori rule out the presence of other forcing mechanisms on Titan. We apply the angular momentum approach, which means that the atmospheric torque acting on Titan is not explicitly calculated but derived from the time derivative of the equatorial AAM predicted by a GCM. The alternative, torque approach is known to be more delicate than the angular momentum approach [de Viron and Dehant, 1999]. [22] We use the time series of the equatorial AAM predicted by the three-dimensional Titan GCM of Tokano [2010]. The time series covers a full Titan year and is sampled 24 times per Titan day in order to resolve the diurnal cycle. This GCM does not contain topography and no geographical variation in the surface properties (albedo, thermal inertia etc.). The only mechanism to cause longitudinal anisotropy is Saturn's gravitational tide. It was shown by Tokano [2010] that the tilt of the AAM vector in the stratosphere predicted by the GCM roughly agrees with Cassini data [Achterberg et al., 2008], while the axial AAM in the stratosphere is underestimated. Since the tilt angle is given by tan−1 of the ratio of the equatorial to the axial AAM this could mean that the equatorial AAM in the stratosphere is underestimated as well. However, the vast majority of the equatorial AAM is contributed by tropospheric winds, which are not underestimated by this GCM. [23] The two equatorial components of the relative AAM are given [Tokano, 2010] by The two equatorial components of incremental inertia elements are given by Here, g = 1.354 m s−2 is Titan's gravitational acceleration, ps is the surface air pressure, u is the zonal wind, v is the meridional wind, ϕ is the latitude, λ is the longitude and p is the air pressure. [24] The time derivative of hx and hy corresponds to the surface friction torque as a result of tangential stress placed upon the surface by the surface wind. In the absence of topography in the GCM we used, there is no mountain torque associated with the atmospheric pressure on the topography. Therefore, we note that the atmospheric torque in our model may be underestimated to some extent. 3. Results Parameters of Titan's Interior and Chandler Wobble [25] The polar motion of a planet/moon strongly depends on the parameters that describe the planetary interior. In the case of Titan, some of the parameters are reasonably well known, while others are uncertain or even totally unknown. [26] The axial MOI factor of Titan was determined by gravity measurements of the Cassini radio science [Iess et al., 2010]. An axial MOI factor of 0.34 was derived, so the axial moment of inertia amounts to C = 0.34M R2 = 3.03218 × 1035 kg m2. With Titan's gravity coefficients J2, S22 and C22 determined by Iess et al. [2010], equations (1), (2), and (9) yield equatorial MOI of A = 3.03172 × 1035 kg m2, B = 3.03208 × 1035 kg m2 and polar flattening of α = 9.355 × 10−5. [27] In the case of an internal ocean the unknown thickness of the outer crust controls several model parameters. Among various models of Titan's interior, the model of Grindrod et al. [2008] predicts the largest crustal thickness (176 km). Medium crustal thicknesses of 90 km and 100 km are suggested by Mitri and Showman [2008] and Nimmo and Bills [2010], respectively. Older models preferred a crustal thickness around 70 km [Grasset et al., 2000; Sohl et al., 2003; Tobie et al., 2005]. The smallest nominal crustal thickness (10–50 km) was suggested by Béghin et al. [2009] based on measurements of the Schumann resonance by the Huygens probe. [28] Figure 1a shows how the ratio C/Cc depends on the crustal thickness. Should Titan have a decoupling subsurface ocean, the polar motion amplitude would amplify by at least a factor of 5 compared to Titan without such an ocean. As long as the crust has a relatively large thickness this ratio almost linearly increases with decreasing thickness. However, if the crustal thickness approaches the lower limit, C/Cc rapidly grows. Figure 1Open in figure viewerPowerPoint Parameters of Titan's interior relevant for the polar motion as a function of the crustal thickness (R − cc) above a putative subsurface ocean. (a) Ratio of momentum of inertia of whole Titan to that of the outer crust only. The amplitude of the polar motion amplifies by this factor if Titan has a decoupling internal ocean. Cc is obtained from equation (8). (b and c) Chandler wobble frequency and period. [29] The available gravity data of Cassini do not yet allow a reliable estimation of the degree 2 tidal potential Love number k2 [Iess et al., 2010], which is required for the Chandler frequency. Therefore, we adopt the theoretically calculated value after Sohl et al. [2003]. If there is a subsurface ocean, k2 increases with decreasing crustal thickness from 0.28 (200 km crustal thickness) to 0.4 (20 km crustal thickness). If there is no internal ocean, k2 = 0.03. [30] We adopt the load Love number (mass load coefficient) k′2 from the theoretical work of Sohl et al. [1995]. In the absence of a subsurface ocean k′2 = −0.03 and in the presence of an ocean k′2 = −0.83. The latter number should somewhat depend on the crustal thickness, but there are other uncertainties that affect the number as well. Therefore, we refrain from varying k′2 with the crustal thickness. [31] In the presence of a subsurface ocean, mass redistribution of the crust is facilitated, so it largely compensates the atmospheric load, which is given by the pressure term. This is expressed by a load Love number close to −1, so 1 + k′2 is close to zero. If instead no ocean is present, the pressure term is barely compensated. [32] If there is no subsurface ocean, the Chandler frequency is 3.2513 × 10−10 Hz after equation (5), corresponding to a Chandler period of 612.4 years, which is 20 times longer than one Titan year (29.5 years, i.e., Saturn's orbital period). As long as the AAM variation is characterized by seasonal or subseasonal oscillations, a resonance between the atmospherically excited wobble and the Chandler wobble is unlikely in the no-ocean case. [33] We can quantify how the triaxiality and deformation of Titan change the free wobble period of Titan. If Titan were a biaxial, rigid body, the free wobble period (Euler period) would be where = (A + B)/2. PE amounts to 472.8 years. [34] The free wobble period of a triaxial, rigid body is given by equation (5), where and are replaced by A and B, respectively. This amounts to 599.6 years, i.e., the triaxiality lengthens the free wobble period by 126.8 years. The deformation of Titan lengthens the free wobble period by another 12.8 years. As a whole the deformation in Titan's interior and the triaxial shape lengthen the free wobble period by ∼30%. [35] However, if there is a subsurface ocean, the Chandler wobble of the crust accelerates because the crust has a smaller MOI than entire Titan. Generally, a subsurface ocean reduces the Chandler period by an order of magnitude (Figure 1c). The Chandler period decreases with decreasing crustal thickness. It turns out that there is a certain thickness at which the Chandler period coincides with the annual period of Titan. This coincidence is found at a crustal thickness of ∼35 km. In such a case a substantial modification of the polar motion by the Chandler wobble can be expected, as shown below. Atmospheric Forcing [36] In our study the temporal variation in the equatorial AAM is the only forcing mechanism of Titan's polar motion. The time series of the relative AAM used in this study were presented in Figures 1 and 3 of Tokano [2010]. The time series of c13 and c23 times Ω were presented in Figure 4 of Tokano [2010]. hx and hy undergo a diurnal oscillation due to thermal tides. The amplitude of the diurnal oscillation varies semiannually and becomes maximal twice per Titan year between solstice and subsequent equinox. [37] Figure 2 shows the atmospheric forcing in the frequency domain used in equations (13) to (16). It elucidates how much the atmosphere forces polar motion at which frequency. The wind term exhibits the largest peak at the diurnal period (4.56 × 10−6 Hz frequency). Secondary, broad peaks are seen in the low-frequency band. They reflect atmospheric waves present in the model, which are neither thermal nor gravitational tides. The most distinct peaks in the low-frequency band are found at the semiannual (σ = 1.35 × 10−8 Hz) and annual (6.76 × 10−9 Hz) period. The semiannual variation of the amount of the equatorial AAM of Titan is mainly caused by the reversal of the Hadley circulation twice per Titan year that covers both the troposphere and stratosphere [Tokano, 2010]. Figure 2Open in figure viewerPowerPoint (left) Equatorial components of the AAM in the frequency domain. (a and b) The wind term of the AAM (hx and hy). (c and d) The pressure term of the AAM (c13Ω and c23Ω). The term σ = 4.56 × 10−6 Hz corresponds to the diurnal frequency, σ ∼ 1.35 × 10−8 Hz to the semiannual frequency and 6.76 × 10−9 Hz to the annual frequency on Titan, latter of which is marked by an arrow. Positive and negative σ causes prograde (anticlockwise) and retrograde (clockwise) rotation of the rotation axis around the geographical reference pole. (right) A zoom of the low-frequency band. [38] The pressure term is 2 orders of magnitude smaller than the wind term, so it contributes much less to the polar motion of Titan. In the presence of a subsurface ocean the pressure contribution to the polar motion is further reduced by the load Love number k′2 close to −1. The seasonal forcing (at low frequencies) is as strong as the diurnal forcing and there is another broad peak around 10 Titan days. While negative frequencies are negligible in the wind term, they are present in the pressure term. They are caused by the fact that gravitational atmospheric tides contain both westward and eastward propagating modes. However, because of the overall weakness of the pressure term forcing on Titan a retrograde wobble does not become apparent. Polar Motion [39] In this section we present and discuss the polar motion predicted under the hypothetical assumption that besides the atmospheric torque no external and internal torques and dissipation exist. This serves as a baseline model to which further studies on additional torques can be referred. Some discussion on additional torques follows in section 3.4. Therefore, the model prediction in this section should not be directly used for the interpretation of possible observations of Titan's polar motion. [40] We calculated the polar motion for a period of 1 Titan year for various models of Titan's interior (no subsurface ocean, with subsurface ocean, three different crustal thicknesses). Polar motion with a period of longer than 1 Titan year is neglected since the atmospheric data themselves cover only 1 Titan year, i.e., the lower limit of σ in equation (12) is 6.76 × 10−9 Hz. The upper limit is σ = 1.09 × 10−4 Hz, corresponding to 1/24 Titan day. We do not predict the possible interannual variability in the polar motion since the significance of interannual variability in the atmospheric circulation of Titan is yet unclear. [41] The solution of the polar motion equation is an initial value problem with a possible nonzero phase with respect to σt in equation (12). In other words, it is not possible to unambiguously determine the absolute coordinates of x and y as a function of season unless the initial position is known. In the lack of observational data we started the simulation arbitrarily from LS = 0° at t = 0. If the simulation is started from an another season, the starting point would be at another location on the displayed polar motion path. [42] The calculation shows that the polar motion depends both qualitatively and quantitatively on the interior structure assumed. Figure 3 shows the time series of the polar motion. A superposition of high-frequency and low-frequency oscillations can be readily recognized in each Titan model, although the low-frequency oscillations do not necessarily exhibit a simple sinusoidal shape. In order to better understand the shape of the polar motion and its dependence on the crustal thickness we also decomposed the polar motion time series into different frequency bands, which are shown in Figures 4 and 5. Figure 3Open in figure viewerPowerPoint Time series of the polar motion for different interior models. The curves show the instantaneous position of the rotation axis (deviation from Titan's figure axis in meters) as a function of time. The left and right column show the x and y component, respectively. The first row shows the result for Titan without a subsurface ocean, the second through fifth rows show the results for Titan with a subsurface ocean of different crustal thickness. The polar motion amplitude has been converted from radians to meters for convenience. The corresponding paths of the polar motion are depicted in Figures 6 to 8. Figure 4Open in figure viewerPowerPoint Zoom of Figure 3 at the season LS = 90° (northern summer solstice) for 2 consecutive Titan days showing the diurnal wobble. The solid and dashed line show the x and y component, respectively. Figure 5Open in figure viewerPowerPoint Time series of the polar motion in different frequency bands. (left) The medium-frequency band (10−8 Hz < σ < 4.56 × 10−7 Hz) covering forcing periods between 10 Titan days and 456 Titan days (0.676 Titan year). (right) The low-frequency band (σ < 10−8 Hz) covering forcing periods longer than 456 Titan days. The solid and dashed line show the x and y component, respectively. [43] Figure 4 elucidates that the high-frequency oscillation is a diurnal wobble caused by thermal tides described by Tokano [2010]. In the absence of a subsurface ocean the amplitude of the diurnal wobble is of the order of a meter. However, the amplitude (polar motion radius) of the diurnal wobble is modulated seasonally as with the tilt angle of the AAM vector [Tokano, 2010]. If Titan has a decoupling subsurface ocean, the amplitude of the diurnal wobble is 1 order of magnitude larger because only the outer crust responds immediately to the atmospheric forcing. The amplitude gradually grows from ∼5 m at a crustal thickness of 100 km to ∼20 m at a crustal thickness of 30 km. This amplitude increase mostly reflects the increase of C/Cc shown in Figure 1a. The tidal potential Love number has little influence since c13 and c23 are small on Titan. Also the Chandler wobble has no influence on the diurnal wobble because the periods of both wobbles differ by several orders of magnitude from each other. [44] The medium-frequency band (Figure 5, left) is dominated by the semiannual wobble (two waves per Titan year) associated with the periodical reversal of the Hadley circulation, which occurs twice per Titan year around equinoxes. The low-frequency band almost exclusively consists of the annual wobble with one wave per Titan year. This analysis shows that the semiannual and annual wobble have comparable amplitudes although the relative importance of each wobble depends on the interior structure as discussed below. In these two frequency bands the wobble has an amplitude larger than the diurnal wobble although the forcing is weaker. The reason for the larger wobble is the closer proximity of the forcing frequency σ to the Chandler frequency σCW. [45] The overall shape in Figure 3 can be basically