Prometheus Lava-Frost Interaction Robert R. Howell University of Wyoming

Overview SO 2 gas generation rates –HST SO 2 column density (Jessup et al. 2004) plus numerical plume dynamics (Zhang et al. 2004) plume dynamics –Confirms updated ~10 4 kg/s from Ingersoll type models SO 2 erosion rates (m/s as function of time) – From modified lava-flow model or simple energetics Integration of rates over various age flows –Also modified from lava flow model Combine above with Galileo SSI 5 m 2 /s lava on frost spreading rate (Milazzo et al. 2001) to get –Constraints on frost or lava thickness –Comparison of IR and gas power (GW)

Original Calculation – Ingersoll model Presented in Jessup et al. (2004) with correction in Lellouch et al. (2006) (Jupiter book) Analytical model where excess pressure (above vapor pressure) drives excess collisions with surface. –P surface (or column density), T  collision rate with surface (per unit area per unit time). –  (Sticking coef.), P vapor  excess collision rate (above equil.) –Excess rate  area of plume  kg/s Updated geometry from Geissler? Updated value 10 4 kg/s of SO 2

Zhang et al Cross-Section of Plume density Numerical model follows expansion and condensation of gas (and small plume particles which follow gas) Shape of plume constrained by Voyager on-the-limb brightness contours (which sense dust). Model assumes gas characteristics at vent – but are not well constrained by Voyager observations since dust/gas ratio isn’t very certain. Density of gas in plume proportional to assumed vent density. Can use HST SO 2 column density to constrain Zhang model and therefore obtain “independent” estimate of SO 2 generation rate.

Zhang et al Prometheus SO 2 density profile Zhang et al. (2004) SO 2 number density normalized to 5  m -3. Assumes vent conditions equivalent to 1.6  10 3 kg/s but poorly constrained

Column density derived from density profile Column density falls rapidly beyond 15 km radius, with long tail HST observations have ~300 km spatial resolution (i.e. ~150 km radius) –Horizontal line represents HST value and spatial resolution Ratio of observed to averaged model is ~5 (so assumed vent gas density too low) Supply rate = 5 x 1.6  10 3 kg/s = 8.0  10 3, close to “Ingersoll model” 10 4 kg/s

Lava – Frost interaction overview Assume lava crust in contact with frost (or perhaps liquid) SO 2 –Assume lava maintains coherent crust Same assumption as in Milazzo et al Ignores possibility of violent mixing Calculate heat out of lava flow using modified “flow model” Assume all that heat is used to vaporize SO 2 –So vaporization rate  heat flow / (latent heat of sublimation) Vaporization rate will vary with time as lava crust thickens and heat flow drops Mathematics for averaging vaporization rate over different age flows is exactly analogous to mathematics for averaging infrared emission –Final results different because different power laws involved in rates

Heat flow from lava flow Published Howell (1997) model uses “Stefan solution” to find heat flow and temperature from cooling and solidifying lava flow. Stefan solution is “exact” if surface of lava is clamped at some T 0 –Gives heat flow q(t) through surface, plus interior temperature of lava crust –Flow model makes initial estimate of T 0 then uses it to calculate q(t) Then uses q(t) and radiate boundary condition to refine estimate of surface T(t) –Approximations are even better for lava crust quenched by contact with frost Only requires minor modification of initial T 0 guess (or  T  T melt -T 0 ) Use T 0  198K = triple point temperature of SO 2 so  T  1200 K –Lava flow model assumed T 0 ~ 400K so  T  1000 K Insensitive to exact temperature of boundary since that has little effect on  T.

Erosion rates from heat flow SO 2 mass flux from surface (F m ) has similar functional form Velocity of vaporization wave is just F m /  frost Integrate velocity to get erosion depth of SO 2

Others’ erosion rates Difference from Milazzo et al estimate –This work: –Milazzo et al times less Factor of 2 could be material constants, rest unexplained Test of model with Kieffer et al estimate –“Back-of-the-envelope” calculation of thickness of silicate crust, then heat flux through it gives erosion rate after 200 minutes of 6.4  m/s –Previous page’s eqn gives 6.1  m/s, in very good agreement

Averaging over different ages Equations analogous to spectral average Assume new flow created at constant areal rate R a Assume flow started time t a ago, ended time t b ago –t b =0 for ongoing flow

Implications of SO2 and flow rate known Milazzo et al using I24 vs. I27 images measured R A =5m 2 /s of new dark flow over bright (SO2) areas Substitution into previous equation gives t A ~ 1 day. If flow continued for months, what else could cut off vaporization after one day? –Depth of SO 2 snow field 4 meters eroded after one day –Solidification then cooling of lava 0.5 meters solid after one day – but cooling would continue vaporization Energetics of cooling imply 0.2m lava depth will work More complicated geometries –Lava burrows under SO 2 (but should still vaporize snow above by radiation) –Other more complicated geometries?

Comparison of Powers from IR, Vaporization 10 4 kg/s vaporization requires 5 GW Observed infrared power GW Factor of difference simply implies that only 1/20 or 1/30 of newly created lava flows interact with SO2. –The rest are presumably new lava flows on top of “dry” slightly older flows

Summary Gas rate 10 4 kg/s confirmed (if geometry OK) Adapted lava flow models show how SO 2 rate depends upon material constants, areal rate, and time parameters If 5 m 2 /s rate and 10 4 rate are right, something cuts off vaporization after 1 day –4 meter depth of snow field? –0.2 meter depth of lava? –More complicated geometry?