## Thermal And Rheological Structure

The tremendous heat flow measured at the surface of Io is the result of silicate volcanism, with typical inferred eruption temperatures of 1,200-1,400 K, and extreme temperatures over 1,800 K (McEwen et al., 1998). It is likely that the sulfur-based volcanism observed by Voyager is also driven by silicate magma. Io is the most volcanically active body in the Solar System. The surface is so active that not a single impact crater has been identified in the images of the satellite. In this section we explore the implications of such activity for the thermal and rheological structure of the interior.

Despite the intensity of the volcanic activity on Io and the inferred high temperatures of its interior, Io supports some of the highest and steepest topography of any planetary body. Mountains up to 17 km high dot the surface (Jaeger et al., 2003) and oddly, these are not volcanoes. Instead, volcanic centers are generally depressions, while the mountains appear to be fault-bounded tectonic uplifts. The presence of such large topography seems incompatible with a lithosphere that must allow the transport of - 1014W of heat.

A solution to this problem was found by O'Reilly and Davies (1981), who proposed a heat-pipe mechanism for heat transport through Io's lithosphere accomplished by melt transport through fissures. The most important implication of this model is that, as successive eruptions spread across the surface and cool, they bury the flows of previous eruptions, resulting in an advective transport of cold material from the surface downward.

In order to match the observed heat flow, Io must be resurfaced globally by silicate magma to a depth of 1-2 cm every year. This is then the rate at which material is advected downward within the lithosphere. The equation for the temperature T in the lithosphere is then:

where k is the thermal conductivity, p is the density, cP is the specific heat, z is the depth from the surface, v is the downward velocity (resurfacing rate), and H is the volumetric heat production. The temperature must match the surface temperature Ts at the surface and the melting temperature Tm at the base of the lithosphere. The solution in the case of no heat production is:

where £ = z/D is the depth normalized by the thickness of the lithosphere D, and the dimensionless parameter A = DvpcP/k is the advective velocity normalized by the conductive velocity scale. This can be related to the heat flux carried by melting F as follows:

pep dz2 dz pep

DePF

1200

Temperature [K]

Figure 5.3. Temperature as a function of normalized depth for different values of A, the normalized advective velocity defined in the text.

1200

1500

Temperature [K]

Figure 5.3. Temperature as a function of normalized depth for different values of A, the normalized advective velocity defined in the text.

where Lf is the latent heat of fusion of the silicate rocks. For a heat flux of 2.5 Wm-2 and a 30 km thick lithosphere, A is about 10. Solutions for different values of A are shown in Figure 5.3. For A of 3, more than 80% of the thickness of the lithosphere is below 900 K, and for A of 10, more than 95% of the lithosphere is cold enough to sustain elastic stresses for very long periods of time. This is how Io's lithosphere can support huge mountains while at the same time allowing a heat flux of 2.5 Wm-2 to pass through.

The very steep gradient at the base of the lithosphere and the continuous flux of material through it means that any chemical layering in the lithosphere (i.e., crust) must be very closely linked to the rheological structure. That is, the crust and lithosphere are essentially the same. Schenk et al. (2001) pointed out that the lithosphere must be at least as thick as the tallest mountains (~15 km) if the mountains form as thrust blocks. In order to form the mountains by compressive stresses due to the global resurfacing, a thickness of at least 12 km is required (Jaeger et al., 2003). As an upper limit, the tidal deformation will be restricted by a thick, strong lithosphere, so we can determine the maximum thickness elastic lithosphere (shear modulus 1011 Pa) that will allow Io to dissipate the observed heat flow. This limit turns out to not be very useful, since a 500-km elastic lithosphere is required to reduce the maximum dissipation below 1014W.

What sort of melt fractions does the heat-pipe mechanism imply for the astheno-sphere? It is straightforward to calculate the melt segregation velocity due to Darcy flow driven by the buoyancy of the melt (e.g., Scott and Stevenson, 1986):

where fi is the melt volume fraction (porosity), g is the gravitational acceleration, Ap is the difference between the solid and melt densities (^500 kg m-3), is the melt viscosity (^ 1,000Pas), and kfi is the permeability, which is related to the porosity by a function of the form:

where b is a typical grain size (~1 cm), and n and r are constants which are functions of the geometry of the melt. The dependence of kfi on grain size is overly simplified in this model, since real systems may have broad grain size distributions. Using the 1 cm yr-1 resurfacing velocity (which is actually the melt flux fiv) and inserting experimentally determined values for the constants n (3) and r (200) (Wark and Watson, 1998; Liang et al., 2001) results in an estimate for fi between 10 and 20% (Moore, 2001). Melt fractions exceeding this will transport too much heat (the heat transported goes as kfi), thus cooling the asthenosphere and bringing the melt fraction back to the equilibrium value.

Based on petrological models (using L- and LL-chondrite compositions, above), it has been pointed out by Keszthelyi et al. (1999, 2004) that the highest temperatures (up to 1,870 K) observed from surface eruptions on Io (McEwen et al., 1998) require very high melt fractions (above 50%) in Io's interior. The apparent low viscosity of Ionian magmas also suggests ultramafic compositions and high melt fractions. Generalizing this to a global "mushy magma ocean'' Keszthelyi et al. (2004) arrive at a model for Io's interior that is partially molten throughout, decreasing from 60% melt at the base of the lithosphere to 10-20% at the base of the mantle. Though attractive for its explanation of the very highest magma temperatures observed, it has not been shown how such a melt distribution can be maintained against melt segregation and the resulting heat loss (as described above), or how such a rheological structure can allow sufficient tidal heat generation (Moore, 2001). The temperature of most eruptions observed on Io (1,200-1,400 K) seem to be consistent with the melt fractions (10-20%) required for thermal equilibrium (Moore, 2001), so perhaps there are local processes that account for the high-temperature outliers.

A rough estimate for the thickness of the asthenosphere may be obtained by extrapolating along the adiabat from the temperature at the top of the melt zone to the solidus:

where ATm = fiLf /cP is the temperature excess required to reach a melt fraction fi, and the denominator is the difference between the slopes of the solidus 0.8 K km-1) and the adiabat 0.1Kkm-1). Using the melt fractions estimated above, the asthenosphere is 60-120 km thick.

The physical state of Io's core cannot be determined from the Galileo spacecraft's observations of Io's permanent tidal deformation. The density difference between solid and liquid core material is too small to be resolvable from these data. If Io had a magnetic field, it would be possible to conclude that at least part of the core would have to be liquid. However, multiple fly-bys ofIo by the Galileo spacecraft have shown that Io does not possess an internal magnetic field (Kivelson et al., 2004). All that can be concluded from the absence of a magnetic field is that there is insufficient convective activity in Io's core to support a dynamo. The core could be completely solid or liquid; it could even be partially solidified, although the lack of dynamo action would be more difficult to understand if there were a growing solid inner core in Io. Because Io's mantle is so intensely heated, it seems most likely that Io has no magnetic field because it has a completely liquid core that is kept from cooling and convecting by the surrounding hot mantle (Weinbruch and Spohn, 1995).

The state of the core is also not determined by the amount of tidal dissipation in Io, which is determined almost entirely by the viscosity of the mantle. Early work using parameterized (Q-model) dissipation (Peale et al., 1979; Cassen et al., 1982) suggested that a liquid core was required in order to allow the mantle to dissipate sufficient heat, but this is not supported by the results of more detailed calculations using the solutions to the equations of motion for viscoelastic bodies (Segatz et al., 1988; Moore, 2003). The difference in dissipation between fluid cores and elastic cores in such models is less than a factor of 3.

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