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Science 9 November 2001:
Vol. 294. no. 5545, pp. 1326 - 1328
DOI: 10.1126/science.1062492

Thickness of a Europan Ice Shell from Impact Crater Simulations
E. P. Turtle and E. Pierazzo

Supplementary Material

Water-ice is characterized by an unusual and complicated phase diagram because the solid is less dense than the liquid. The slope of the solid-liquid phase boundary changes from negative for P name

2 kbar to positive for P name

2 kbar. Such an abrupt change in slope cannot be handled by ANEOS and must be avoided. This was accomplished by artificially interpolating the phase boundary above 2 kbar down to zero-pressure, effectively extending the liquid field to lower temperatures than in reality, with a freezing point of 235.6 K. However, this change involves a very small region of the pressure range involved in an impact event, and does not significantly affect the final determination of the equation of state of water.

Experimental shock data show that the solid-solid phase transitions do not seem to influence the Hugoniot of solid ice. However, these experiments also show that the Hugoniot curve of ice lies slightly below that of liquid water, which suggests that ice undergoes a solid-solid phase transition to a denser phase, most probably ice VI (1, 2). The consequence is an increase of the amount of irreversible work done by the shock wave, which increases the post-shock temperature of the ice. This behavior is similar to the case of a porous material (3) and results in lower shock melting and vaporization pressures. Unfortunately, the current version of ANEOS does not simultaneously handle solid-solid and solid-liquid phase transitions. Because in impact events the solid-liquid and solid-vapor transitions are most important for water, we chose ANEOS to represent the melt curve, ignoring the details of solid-solid transitions. Our reference model ice, therefore, has only one solid phase with density of 1.1 g/cm³, within the density range for the various phases of ice.

Experimental data exist for both liquid water and ice shocked to high pressures and temperatures (4-7). Most experimental studies use ice I between 258 K and 263 K, or liquid water at 298 K. Recently, some work has been carried out for porous ice at much lower temperatures (2), but it is limited to very low shock states (<1 GPa). We adjusted the ANEOS input parameters (Web table 1) to best fit the phase boundary curves for both the solid-liquid and liquid-vapor transitions, and to obtain a Hugoniot curve that fits the experimental data. The only differences between the ANEOS equations of state for ice and liquid water are in the coefficients for the linear relation between shock and particle velocity. Different Hugoniot curves can be obtained for the same material by varying its starting density and temperature. To compare with the experimental data, we determined separate Hugoniot curves for ice (starting density of 0.915 g/cm³, T = 258 K) and water (starting density of 0.998 g/cm³, T = 298 K).

The main disadvantage of the current version of ANEOS is its inability to correctly represent gas phases because it cannot treat the vapor phase as a molecular gas. This results in a much higher critical point and a misrepresentation of the vapor side of the liquid-vapor phase boundary. While this effect could influence the final phases of a hydrocode simulation by changing the behavior of water/ice during the expansion of the vapor plume, it does not affect the material's shock state, which is the focus of our study.

The melt regions are constructed by determining the distance from the impact point at which the shock pressures corresponding to the incipient and complete melting entropies for water were achieved. Given the overextension of the liquid region in the EoS, the ice ANEOS underestimates the melting pressures in the region where the ice is warmer. To account for this, we assume that the threshold shock pressure for complete melting of the warm ice (T>240 K) is 1 GPa.

Supplemental Table 1. ANEOS EoS parameters for ice, values for liquid water in parentheses when different from ice.

Parameter Ice
(Water) Description

Nel 2 Number of elements in the compound

EoS 4 Type of EoS (solid/gas with electronic terms and detailed treatment of liquid/vapor region)

₀ 1.1 Reference density (g/cm³)

T₀ 233.15 Reference temperature (K)

P₀ 0 Reference pressure (GPa)

C 1.6 (1.8) Coefficient of shock-material velocity linear relation (km/s)

₀ 0.3 Reference Grüneisen coefficient

₀ 522.24 Reference Debye temperature (K)

S 0.5 (1.3) Slope of shock-material velocity linear relation

3₀ 2 3x limit of name for large compressions

E_sep 6250 Zero-temperature separation energy (J/g)

T_m 235.6 Melting temperature (K)

c₅₃ 0 Parameter for low-density modification of critical point (not used)

c₅₄ 0 Same as above

H₀ 0 Thermal conduction parameter (not included)

c₄₁ 0 Same as above

_min 0 Lowest allowed solid density (g/cm³)

D₁ 0 name ₁: start of solid-solid phase transition (g/cm³)

D₂ 0 name ₂: end of solid-solid phase transition (g/cm³)

D₃ 0 Pressure at center of phase transition at zero-T (GPa)

D₄ 0 dp/d name at high-P phase transition

D₅ 0 d²p/d name ² at high-P phase transition

H_fus 230 Heat of fusion for melting (J/g)

_liq/_s 1 Liquid/solid density at melting point

U_p 0 Upper limit for cold compression (expanded states)

L₀ 0 Lower limit for cold compression (expanded states)

0.5 Parameter for liquid EoS correction to match P_vap data and boiling point

0.95 Same as above

0.99 Same as above

c₆₀ 0.4 Interpolation parameter in name model

c₆₁ 0 Same as above

c₆₂ 0.3 Interpolation parameter in free energy expression

Flag 1 Ionization model flag (1 = Thomas Fermi)

E_sh 0 Energy shift for reactive chemistry model (0 = not included)

S_sh 0 Entropy shift for reactive chemistry model (0 = not included)

Supplemental Figure 1. Galileo images of Europa's multiple ring impact structures Callanish (A) and Tyre (B). Both images have a resolution of 250 m/pixel and are shown such that the illumination is from the right. The images are (A) ~140 km and (B) ~180 km across.

Medium version | Full size version

Supplemental Figure 2. Galileo image of the crater Manannán. Despite its diameter of 21.8 km, Manannán does not have a well-defined central peak. The image resolution is 250 m/pixel and the illumination is from the right.