Satellite Io Research Articles

Io: Heat flow from dark volcanic fields

Dark flow fields on the jovian satellite Io are evidence of current or recent volcanic activity. We have examined the darkest volcanic fields and quantified their thermal emission in order to assess their contribution to Io’s total heat flow. Loki Patera, the largest single source of heat flow on Io, is a convenient point of reference. We find that dark volcanic fields are more common in the hemisphere opposite Loki Patera and this large scale concentration is manifested as a maximum in the longitudinal distribution (near ∼200 °W), consistent with USGS global geologic mapping results. In spite of their relatively cool temperatures, dark volcanic fields contribute almost as much to Io’s heat flow as Loki Patera itself because of their larger areal extent. As a group, dark volcanic fields provide an asymmetric component of ∼5% of Io’s global heat flow or ∼5 × 10 12 W.

Polarimetry of the Galilean satellites and Jupiter near opposition

We present new polarimetric observations of the Galilean satellites Io, Europa, Ganymede, and Callisto as well as the North and South polar regions of Jupiter in the V filter within the range of phase angles −0.13−0.62°. The differences between our results and observations of other authors are discussed. It is shown that there is no second inversion point near the phase angle 0.5°. New observations fully confirmed the presence of the polarization opposition effect for the Galilean satellites Io, Europa, and Ganymede in the region of phase angles smaller than 2°. Observations of Jupiter verified a sharp turn of the angle between the observed polarization plane and the plane perpendicular to the scattering plane by 90°.

Auroral evidence of a localized magnetic anomaly in Jupiter's northern hemisphere

We analyze more than 1000 HST/Advanced Camera for Survey images of the ultraviolet auroral emissions appearing in the northern hemisphere of Jupiter. The auroral footprints of Io, Europa, and Ganymede form individual footpaths, which are fitted with three reference contours. The satellite footprints provide a convenient mapping between the northern Jovian ionosphere and the equatorial plane in the middle magnetosphere, independent of any magnetic field model. The VIP4 magnetic field model is in relatively good agreement with the observed footprint of Io. However, in the auroral kink sector, between the 80° and 150° System III meridians, the model significantly departs from the observation. One possible way to improve the agreement between the VIP4 model and the observed footprints is to include a magnetic anomaly. We suggest that this anomaly is characterized by a weakening of the surface magnetic field in the kink sector and by an added localized tilted dipole field. This dipole rotates with the planet at a depth of 0.245 RJ below the surface, and its magnitude is set to ∼1% of Jupiter's dipole moment. The anomaly has a very limited influence on the magnetic field intensity in the equatorial plane between the orbits of Io and Ganymede. However, it is sufficient to bend the field lines near the high‐latitude atmosphere and to reproduce the observed satellite ultraviolet footpaths. JUNO's in situ measurements will determine the structure of Jupiter's magnetic field in detail to expand on these results.

The rotation of Io with a liquid core

In a previous paper (Henrard, Celest. Mech. Dyn. Astron. 178, 144–153, 2005c) we have developed an analytical theory of the rotation of the Galilean satellite Io, considered as a rigid body and based on a synthetic theory of its orbital motion due to Lainey (Theorie Dynamique des Satellites Galileens. PhD dissertation, Observatoire de Paris, 2002) (see also Lainey et al., AA A&A, 427, 371–376, 2004b). One of the most important causes of departure of the actual rotation from the rigid theory is thought to be the existence of a liquid core, the size of which is unknown but would be an important piece of information concerning the structure of the interior of the satellite. In this contribution we develop the analytical theory of a liquid core contained in a cavity filled by an inviscid fluid of constant uniform density and vorticity. The theory is based on Poincare (Bull. Astron. 27, 321–356, 1910) model and is developed by a Lie transform perturbation method, very much like in our previous contribution. Our main conclusion is that the addition of a degree of freedom (the spin of the core) with a frequency close to the orbital frequency multiplies the possibility of resonances and that for some particular size of the core one may expect a (possibly small) region of chaotic behaviour in the vicinity of the Cassini state.

Investigating the origins of the Jovian decimetric emission's variability

The long‐term variability of the Jovian radiation observed at 13‐cm wavelength over 30 years is investigated using a physical model of the radiation belts. The model enables us to quantify the effect of the geometric parameter DE on the synchrotron emission. When a full Jupiter rotation is taken into account in the computations, the variability of the total radio flux with DE is only ≃1% of that measured. Such a dependence is not detectable. In contrast, radio fluctuations observed at a central meridian longitude (CML) and caused by DE changes can be very important, up to 10% of that measured. This strong dependence of the total radio flux on the geometric factor can result in difficulties for examining the origins of the Jovian synchrotron emission's fluctuations during periods where times of observation are not covering large CML range values. The numerical computations reveal that the variability either of the populations injected near the outer boundary of our simulations (Io's orbit) or the inward radial transport can, independently or in combined action, conduct variability in the decimetric emission. The theoretical fluctuations required in our computations for reproducing the radio measurements do not confirm any direct correlation between the long‐term changes in the radio data and the radial transport driven by solar wind conditions or solar radio fluxes. Nevertheless, a linear relationship between the total radio flux and solar wind ram pressure for the period of 1970 to 2002 is established. The maximum correlation is reached when the Psw data are shifted by ≃2.7 years prior to the radio observations. Using this time‐lag in the computations, the simulation results show that (Psw)γ can be associated with the variations of the particle injections and the radial transport. The correlation coefficient, associated with the fit between radio data and simulated radio flux densities, is 0.59 for the period of 1971–2002, 0.93 for the period of 1971–1972, and 0.83 for the periods of 1975–1989 and 1992–1995. The index γ is set to 0.35 when only particle injections are fluctuating with time, to 0.50 when temporal variations are driving the radial transport, and to 0.15–0.20 when the particles injected in the inner magnetosphere and their transport are simultaneously following the variations of the solar wind ram pressure. The model suggests that the solar wind ram pressure fluctuations can be related to variations of the Jovian decimetric emission on timescales of months, particularly to the enhancements in total radio flux observed in 1987, 1988, 1990 and 1994. Our results thus support the idea that the increase in radio emission in July 1994 was due to the impact of Comet Shoemaker‐Levy 9 into Jupiter, in addition to the response of the Jovian magnetosphere to steep changes in solar wind conditions. The divergence between observations and simulations indicates that mechanisms other than those discussed in the modeling must govern the radiation belts dynamics for the periods of 1972–1975 and 1996–2002.

Jupiter's Nightside Airglow and Aurora

Observations of Jupiter's nightside airglow (nightglow) and aurora obtained during the flyby of the New Horizons spacecraft show an unexpected lack of ultraviolet nightglow emissions, in contrast to the case during the Voyager flybys in 1979. The flux and average energy of precipitating electrons generally decrease with increasing local time across the nightside, consistent with a possible source region along the dusk flank of Jupiter's magnetosphere. Visible emissions associated with the interaction of Jupiter and its satellite Io extend to a surprisingly high altitude, indicating localized low-energy electron precipitation. These results indicate that the interaction between Jupiter's upper atmosphere and near-space environment is variable and poorly understood; extensive observations of the day side are no guide to what goes on at night.

Satellite control of Jovian 2–6 MHz radio emission using Voyager data

Jupiter's satellite Io no longer has the only known effect in controlling the low‐frequency Jovian radio emissions. Menietti et al. (1998, 2001) used a long and contiguous set of data in the range of 2.0–5.6 MHz from the Galileo spacecraft and found correlations with the orbital phase of Ganymede and Callisto. Recently, Higgins et al. (2006) used all of the Galileo orbiter data and found that Europa's phase shows a minor but statistically significant effect on the lower range of decametric emissions. In this paper, results are found that confirm previous work for all four Galilean satellites by combining Voyager 1 and Voyager 2 flyby data in the frequency range of 2.1–5.8 MHz. The data were analyzed using the unique occurrence probability data compilation technique that includes 10‐hour averaging and satellite bias removal. A statistical significance Z‐test was computed to compare sample populations for each enhanced region of occurrence against the background. We find significance values greater than the 95% confidence level (Z > 2σ) for Io, Europa, Ganymede, and Callisto. Least squares fits to histograms of relative occurrence probability also show similar results. The cause of these correlations is believed to be Alfvénic interactions between the individual satellite and Jupiter.

Numerical simulation of fine structure in the Io plasma torus produced by the centrifugal interchange instability

The Io plasma torus as a whole has a radial width scale ∼1 RJ, much larger than the width of the localized Io plasma source (∼1 RIo ≈ RJ/39). One of the most prominent features of the Io torus observed by the Voyager spacecraft and Earth‐based instruments is the “ribbon” structure near Io's orbit. The stability properties of this narrower ribbon structure embedded within the larger torus have been investigated by the Rice Convection Model for Jupiter. Four initial plasma distributions having different radial widths are each represented by 82 longitudinally symmetric edges establishing 41 levels of the flux tube mass content η with the peak η value at Io's orbit. The same initial perturbation is put on each of these edges and is subjected to centrifugal interchange. Our simulations produce regularly spaced long, thin fingers moving outward from the outer edges. It is shown that the azimuthal width of the interchange convection cells (the distance between outflowing fingers in the nonlinear stage of development) is proportional to the radial width scale of the initial distribution that produced them. The constant of proportionality is ≈0.5. Since the exponential growth rate is essentially proportional to the azimuthal wave number of the disturbance and hence is inversely proportional to its azimuthal width, the ribbon‐scale interchange structures grow faster than torus‐scale interchange structures.

Dependence of the Io-related decametric radio emission of Jupiter on the central meridian longitude and Io's “active” longitudes

Context.The statistical analysis of the Io-related decametric radio emission of Jupiter shows that this emission depends precisely on the central meridian longitude. This dependence is the result of the existence of Io's “active” longitudes, i.e. particular regions of Io's orbit, which are fixed with respect to the Jovian magnetic field and at which Io-related emission occurs more often.

Magnetic field induced by elliptical instability in a rotating spheroid

The tidal or the elliptical instability of the rotating fluid flows is generated by the resonant interaction of the inertial waves. In a slightly elliptically deformed rotating sphere, the most unstable linear mode is called the spin-over mode, and is a solid body rotation versus an axis aligned with the maximum strain direction. In the non-viscous case, this instability corresponds to the median moment of the inertial instability of the solid rotating bodies. This analogy is furthermore illustrated by an elliptical top experiment, which shows the expected inviscid heteroclinic behaviour. In geophysics, the elliptical instability may appear in the molten liquid cores of the rotating planets, which are slightly deformed by the tidal gravitational effects of the close bodies. It may then participate in the general outer core dynamics and possibly the geodynamo process. In this context, Kerswell and Malkus (Kerswell, R.R. and Malkus, W.V.R., Tidal instability as the source for Io's magnetic signature. Geophys. Res. Lett., 1998, 25, 603–606) showed that the puzzling magnetic field of the Jovian satellite Io may indeed be induced by the elliptically unstable motions of its liquid core that deflect the Jupiter's magnetic field. Our magnetohydrodynamics (MHD) experiment is a toy-experiment of this geophysical situation and demonstrates for the first time the possibility of an induction of a magnetic field by the flow motions due to the elliptical instability. A full analytical calculation of the magnetic dipole induced by the spin-over is presented. Finally, exponential growths of this induced magnetic field in a slightly deformed rotating sphere filled with galinstan liquid metal are measured for different rotating rates. Their growth rates compare well with the theoretical predictions in the limit of a vanishing Lorentz force.

Models, figures, and gravitational moments of Jupiter’s satellites Io and Europa

Two types of trial three-layer models have been constructed for the satellites Io and Europa. In the models of the first type (Io1 and E1), the cores are assumed to consist of eutectic Fe-FeS melt with the densities ρ1 = 5.15 g cm−3 (Io1) and 5.2 g cm−3 (E1). In the models of the second type (Io3 and E3), the cores consist of FeS with an admixture of nickel and have the density ρ1 = 4.6 g cm−3. The approach used here differs from that used previously both in chosen model chemical composition of these satellites and in boundary conditions imposed on the models. The most important question to be answered by modeling the internal structure of the Galilean satellites is that of the condensate composition at the formation epoch of Jupiter’s system. Jupiter’s core and the Galilean satellites were formed from the condensate. Ganymede and Callisto were formed fairly far from Jupiter in zones with temperatures below the water condensation temperature, water was entirely incorporated into their bodies, and their modeling showed the mass ratio of the icy (I) component to the rock (R) component in them to be I/R ∼ 1. The R composition must be clarified by modeling Io and Europa. The models of the second type (Io3 and E3), in which the satellite cores consist of FeS, yield 25.2 (Io3) and 22.8 (E3) for the core masses (in weight %). In discussing the R composition, we note that, theoretically, the material of which the FeS+Ni core can consist in the R accounts for ∼25.4% of the satellite mass. In this case, such an important parameter as the mantle silicate iron saturation is Fe# = 0.265. The Io3 and E3 models agree well with this theoretical prediction. The models of the first and second types differ markedly in core radius; thus, in principle, the R composition in the formation zone of Jupiter’s system can be clarified by geophysical studies. Another problem studied here is that of the error made in modeling Io and Europa using the Radau-Darvin formula when passing from the Love number k2 to the nondimensional polar moment of inertia \(\bar C\). For Io, the Radau-Darvin formula underestimates the true value of \(\bar C\) by one and a half units in the third decimal digit. For Europa, this effect is approximately a factor of 3 smaller, which roughly corresponds to a ratio of the small parameters for the satellites under consideration αIo/αEuropa ∼ 3.4. In modeling the internal structure of the satellites, the core radius depends strongly on both the mean moment of inertia I* and k2. Therefore, the above discrepancy in \(\bar C\) for Io is appreciable.

Composition of jovian dust stream particles

The Cassini spacecraft encountered Jupiter in late 2000. Within more than 1 AU of the gas giant the Cosmic Dust Analyser onboard the spacecraft recorded the first ever mass spectra of jovian stream particles. To determine the chemical composition of particles, a comprehensive statistical analysis of the dataset was performed. Our results imply that the vast majority (>95%) of the observed stream particles originate from the volcanic active jovian satellite Io from where they are sprinkled out far into the Solar System. Sodium chloride (NaCl) was identified as the major particle constituent, accompanied by sulphurous as well as potassium bearing components. This is in contrast to observations of gas in the ionian atmosphere, its co-rotating plasma torus, and the neutral cloud, where sulphur species are dominant while alkali and chlorine species are only minor components. Io has the largest active volcanoes of the Solar System with plumes reaching heights of more than 400 km above the moons surface. Our in situ measurements indicate that alkaline salt condensation of volcanic gases inside those plumes could be the dominant formation process for particles reaching the ionian exosphere.

Europa's atmosphere, gas tori, and magnetospheric implications

A two-dimensional kinetic model calculation for the water group species (H 2O, H 2, O 2, OH, O, H) in Europa's atmosphere is undertaken to determine its basic compositional structure, gas escape rates, and velocity distribution information to initialize neutral cloud model calculations for the most important gas tori. The dominant atmospheric species is O 2 at low altitudes and H 2 at higher altitudes with average day–night column densities of 4.5 × 10 14 and 7.7 × 10 13 cm −2 , respectively. H 2 forms the most important gas torus with an escape rate of ∼ 2 × 10 27 s −1 followed by O with an escape rate of ∼ 5 × 10 26 s −1 , created primarily as exothermic O products from O 2 dissociation by magnetospheric electrons. The circumplanetary distributions of H 2 and O are highly peaked about the satellite location and asymmetrically distributed near Europa's orbit about Jupiter, have substantial forward clouds extending radially inward to Io's orbit, and have spatially integrated cloud populations of 4.2 × 10 33 molecules for H 2 and 4.0 × 10 32 atoms for O that are larger than their corresponding populations in Europa's local atmosphere by a factor of ∼200 and ∼1000, respectively. The cloud population for H 2 is a factor of ∼3 times larger than that for the combined cloud population of Io's O and S neutral clouds and provides the dominant neutral population beyond the so-called ramp region at 7.4–7.8 R J in the plasma torus. The calculated brightness of Europa's O cloud on the sky plane is very dim at the sub-Rayleigh level. The H 2 and O tori provide a new source of europagenic molecular and atomic pickup ions for the thermal plasma and introduce a neutral barrier in which new plasma sinks are created for the cooler iogenic plasma as it is transported radially outward and in which new sinks are created to alter the population and pitch angle distribution of the energetic plasma as it is transported radially inward. The europagenic instantaneous pickup ion rates are peaked at Europa's orbit, dominate the iogenic pickup ion rates beyond the ramp region, and introduce new secondary plasma source peaks in the solution of the plasma transport problem. The H 2 torus is identified as the unknown Europa gas torus that creates both the observed loss of energetic H + ions at Europa's orbit and the corresponding measured ENA production rate for H.

THE LENSE–THIRRING EFFECT IN THE JOVIAN SYSTEM OF THE GALILEAN SATELLITES AND ITS MEASURABILITY

We investigate the possibility of measuring the post-Newtonian general relativistic gravitomagnetic Lense–Thirring effect in the Jovian system of its Galilean satellites Io, Europa, Ganymede and Callisto in view of recent developments in processing and modeling their optical observations spanning a large time interval (125 years). The present day best observations have an accuracy between several kilometers to few tens of kilometers, which is just the order of magnitude of the Lense–Thirring shifts of the orbits of the Galilean satellites over almost a century. From a comparison between analytical development and numerical integration it turns out that, unfortunately, most of the secular component of the gravitomagnetic signature is removed in the process of fitting the initial conditions. Indeed, an estimation of the magnitude of the Lense–Thirring effect in the ephemerides residuals is given; the resulting residuals have a maximum magnitude of 20 meters only (over 125 years).

Monitoring active volcanism with the Autonomous Sciencecraft Experiment on EO-1

The ability to monitor and rapidly react to remote detection of volcanic activity has been greatly improved through use of the Autonomous Sciencecraft Experiment (ASE), an advanced software application installed on a spacecraft in Earth orbit. ASE is a NASA New Millennium Program experiment demonstrating science-driven autonomous command and control of a spacecraft. Flying on the Earth Observing-1 ( EO-1) spacecraft, ASE successfully detected thermal emission from the Mt. Erebus lava lake on 7 May 2004, having analyzed a Hyperion hyperspectral data product on board the spacecraft. EO-1 was re-tasked by ASE to obtain a follow-up observation 7 h later and sent a notification of detection of volcanic activity to the ground. The entire process was carried out autonomously. Initial acquisition to receipt on the ground of the positive detection took less than 3 h, a process that without ASE would have taken weeks. The ASE Thermal Classifier has detected several styles of effusive volcanic activity: active lava lakes, pahoehoe flow fields, open channel flows and lava domes. ASE successfully demonstrated that science-driven spacecraft operation greatly enhances science return per returned byte through the identification of the most valuable data, allowing prioritization of downlink products and the discarding of null data sets. This technology has applications on missions elsewhere in the solar system. Modified thermal classifiers can be used for detecting and monitoring active volcanism on the jovian satellite Io, the neptunian moon Triton, and searching for active volcanism on Mars and icy satellites. The success of ASE is an incentive for future instrument and mission designers to consider on-board data-processing requirements (especially data storage capacity, number of processors and processor speed, and RAM) in order to take advantage of this flight-proven technology.

Dynamic features of Io's extended sodium distributions

We report on two-dimensional imaging observations of D-line emissions from the extended distribution of iogenic sodium atoms with two fields of view (±20 R J (narrow FOV) and ±400 R J (wide FOV)) simultaneously by using a portable small telescope or camera lens. We derived dynamic feature of the band-shaped and spray-shaped distributions near Io's orbit by means of continuous observation. The observations confirm the phenomenological behavior of the sodium cloud on two spatial scales, as previously observed by Pilcher et al. [Pilcher, C.B., Smyth, W.H., Combi, M.R., Fertel, J.H., 1984. Astrophys. J. 287, 427–444], Schneider et al. [Schneider, N.M., Trauger, J.T., Wilson, J.K., Brown, D.I., Evans, R.W., Shemansky, D.E., 1991. Science 253, 1394–1397], and Mendillo et al. [Mendillo, M., Baumgartner, J., Flynn, B., Hughes, W.S., 1990. Nature 348, 312–314]. We also confirm an elongated oval emission distribution of the sodium nebula and derivation of its detailed east–west asymmetry depending on Io's phase angle, which was first noted by Flynn et al. [Flynn, B., Mendillo, M., Baumgartner, J., 1994. J. Geophys. Res. 99, 8403–8409]. We then did model analyses to investigate the source process for sodium atoms and the dynamics behind their distribution. We conclude that the essential of molecular ion mechanisms to the band-shaped distribution is in agreement with Wilson and Schneider [Wilson, J.K., Schneider, N.M., 1999. J. Geophys. Res. 104, 16567–16583]. We differ from Wilson et al. [Wilson, J.K., Mendillo, M., Baumgartner, J., Schneider, N.M., Trauger, J.T., Flynn, B., 2002. Icarus 157, 476–489] in finding that charge exchange process contributes more to the spray-shaped distribution and sodium nebula than sputtering does. These results derived the double-peaked velocity distribution of released sodium atoms, and re-confirmed the source rates in agreement with past studies.

Polarization opposition effect for the Galilean satellites of Jupiter

We present results of polarimetric observations of the Galilean satellites Io, Europa, Ganymede, and Callisto at phase angles ranging from 0.19° to 2.22°. The observations in the UBVR filters were performed using a one-channel photoelectric polarimeter attached to 70-cm telescope of the Chuguev Observation Station (Ukraine) on November 19–December 7, 2000. We have observed the polarization opposition effect for Io, Europa, and Ganymede to be a sharp secondary spike of negative polarization with an amplitude of about −0.4% centered at phase angles of 0.2°–0.7° and superimposed on the regular negative polarization branch. Although these minima for Io, Europa, and Ganymede show many similarities, they also exhibit a number of distinctions. The polarization opposition effect appears to be wavelength-dependent, at least for Europa and Ganymede. No polarization opposition effect was found for Callisto. The results obtained are discussed within the framework of different mechanisms of light scattering.

The rotation of Io

The paper develops, in the framework of Hamiltonian mechanics, a theory of the rotation of Io, considered as a rigid body. The theory includes the perturbation due to Jupiter (considered as an oblate body) and the indirect perturbations due to the other Galilean satellites. In order to describe the orbit of Io around Jupiter, we use the synthetic theory of Lainey [2002, PhD dissertation, Observatoire de Paris], the result of a frequency analysis of a numerically integrated jovian system. The direct effects of the other Galilean satellites are found to be negligible, but their indirect effects are important. Our theory is consistent with the rigid body model and with Lainey's description of the orbit of Io, at least down to 10 −6 rad (0.2 arc-second). Of course the effects of the nonrigidity of Io and of a probable liquid core should be considered. We find a mean obliquity of 7.619 × 10 −4 rad (157 arc-second) and the period of the three free librations to be 13.25 days (free libration in longitude), 159.39 days (free libration in latitude), and 229.85 days (free wobble). Fourier series are produced describing, in the body frame, the motion of the polar axis of Jupiter, the motion of the unit vector pointing toward Jupiter, and the “motion of the pole” (the motion of the angular momentum with respect to the axis of largest inertia). Free librations (depending on three arbitrary parameters) are also computed.

Statistics of depleted flux tubes in the jovian magnetosphere

On many of its passes through the Io torus the Galileo spacecraft has detected the presence of what appear to be thin magnetic flux tubes with fields somewhat higher than their surroundings. On these flux tubes the magnetic pressure is sufficiently above the pressure of neighboring tubes that it is possible the plasma contributions to the pressure within these tubes are depleted. Due to their short duration, they are only detectable in high time-resolution magnetometer data. Herein we survey all high time-resolution data that are available over the full Galileo mission and present a final statistical study. These tubes occupy 0.32% of the torus outside the orbit of Io. None are found inside. Their strength indicates that the ratio of the thermal pressure to magnetic pressure in the outer torus is about 2%. Comparison of the observed electron density in the neighborhood of these tubes indicates that the ion temperature is in the range 30–100 eV, consistent with other estimates. The amount of magnetic flux transported by these thin tubes could supply the amount of magnetic flux mass-loaded and transported to the magnetotail if the inward velocity is about 300 times that of the outward transport. Finally, the thin flux tubes are found in clusters, as they would occur if they resulted from the breakup of larger flux tubes.

Moments of inertia and the Chandlerian period for two-and three-layer models of the Galilean satellite Io

We consider two-layer (Fe-FeS core+silicate mantle) and three-layer (Fe-FeS core+silicate mantle+crust) models of the Galilean satellite Io. Two parameters are known from observations for the equilibrium figure of the satellite, the mean density ρ0 and the Love number k2. Previously, the Radau-Darwin formula was used to determine the mean moment of inertia. Using formulas of the Figure Theory, we calculated the principal moments of inertia A, B, and C and the mean moment of inertia I for the two-and three-layer models of Io using ρ0 and k2 as the boundary conditions. We concluded that when modeling the internal structure of Io, it is better to use the observed value of k2 than the moment of inertia I derived from k2 using the Radau-Darwin formula. For the models under consideration, we calculated the Chandlerian wobble periods of Io. For the three-layer model, this period is approximately 460 days.