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Heat and Gas Diffusion in Comet Nuclei

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Heat and Gas Diffusion in Comet Nuclei HEAT AND GAS DIFFUSION IN COMET NUCLEI Walter F. Huebner Southwest Research Institute, USA Johannes Benkhoff1 Institute for Planetary Research, DLR-Berlin, Germany Maria-Teresa Capria, Angioletta Coradini, Christina De Sanctis, Roberto Orosei Istituto di Astrofisica Spaziale, Italy Dina Prialnik Tel Aviv University, Israel August 8, 2006 1Presently with Research and Scientific Support Department, ESA-ESTEC, The Netherlands ii Contents Foreword xv Preface xvii 1 Introduction { Observational Overview 1 2 The Structure of Comet Nuclei 9 2.1 Size and Composition . 9 2.2 Some Physical Properties . 15 2.3 Comet { Asteroid Transitions . 19 2.4 Laboratory Simulations . 21 2.4.1 KOSI Experiments . 21 2.4.2 Cometary Materials: the Effects of Bombardment with Energetic Charged Particles . 23 2.4.3 Experiments on Gas Trapping . 24 2.4.4 Measurements of Thermal Conductivity . 25 2.4.5 Other Laboratory Measurements . 28 3 Physical Processes in Comet Nuclei 31 3.1 Sublimation of Ices . 32 3.2 The Phase Transition of Amorphous Ice . 33 3.3 Gas Diffusion in Pores . 35 3.3.1 Comments on Porosity . 35 3.3.2 The Surface-to-Volume Ratio . 37 3.3.3 Gas Flow . 39 3.4 The Coma/Nucleus Boundary Layer . 41 3.4.1 The Knudsen Layer in the Coma . 41 3.4.2 Effects of Nucleus Surface Topography . 46 3.5 Dust Entrainment and Dust Mantling . 48 3.5.1 The Critical Dust Particle Size . 49 3.5.2 Models of Dust Mantle Formation . 50 3.5.3 Porosity of the Dust Mantle . 51 3.6 Fracturing, Splitting, and Outbursts . 53 4 Basic Equations 57 4.1 Mass Balance . 57 4.2 Energy Balance . 59 4.3 Momentum Balance . 61 iii iv Contents 4.4 Boundary Conditions . 61 4.5 Initial Structure and Parameters . 63 4.6 Flow Regimes and their Transitions . 66 4.7 Dust Flow and Mantling . 68 4.8 Sublimation and Condensation in Pores . 72 4.9 Effective Thermal Conductivity . 73 5 Analytical Considerations 79 5.1 Early Models . 79 5.2 Characteristic Properties of the Nucleus . 84 5.3 Characteristic Timescales . 86 5.3.1 The Surface Temperature . 87 5.3.2 The Onset of Crystallization . 89 5.3.3 Fracture Instability . 90 5.3.4 The Effect of Radioactivity . 90 5.4 An Analytical Model for Crystallization and its Implications . 91 5.4.1 A Two-Zone Model . 92 5.4.2 Implications for the Onset of Cometary Activity . 95 5.4.3 The Intermittent Progress of Crystallization . 97 6 Numerical Methods 99 6.1 1-D Difference Schemes . 99 6.2 Treatment of Boundary Conditions . 103 6.3 From 1-D to Multi-Dimensions . 107 6.4 Simultaneous Solution for Transfer of Heat and Mass . 109 6.5 Stability Problems . 111 7 Comparison of Algorithms 115 7.1 Rationale . 115 7.2 Thermal Algorithm: Different Formulations . 118 7.3 The Models . 120 7.4 Results of Different Algorithms for Various Models . 121 7.5 Conclusions . 131 8 Orbital Effects 135 8.1 Inward Heat Flux . 135 8.2 Short-Period vs. Long-Period Comets . 137 8.2.1 Dynamical Evolution . 137 8.2.2 Differences between Long- and Short-Period Comets . 140 8.3 Changing Orbits . 144 8.4 Multistage or Direct Injection . 146 8.5 Sungrazing Comets . 147 v 9 Spin Effects 151 9.1 Diurnal Evolution . 151 9.2 Gas Emission . 151 9.3 Day { Night Temperature Difference . 153 9.4 Effect of Spin Axis Inclination . 155 9.4.1 Uneven Distribution of Dust Mantles . 155 9.4.2 Uneven Erosion . 158 9.5 Effect of Spin Rate . 160 10 Comparison of Models with Observations 165 10.1 Modeling Guided by Observations . 165 10.1.1 Example 1: Comet Hale-Bopp (C/1995 O1) . 165 10.1.2 Example 2: Comet 46P/Wirtanen . 174 10.2 Conclusions Based on Multiple Simulations . 181 10.3 Comet Outbursts . 184 10.3.1 Distant Outbursts of Comet 1P/Halley . 186 10.3.2 Pre-Perihelion Activity of 2060 Chiron . 186 10.3.3 Erratic Activity of 29P/Schwassmann-Wachmann 1 . 187 10.3.4 Distant Activity of Comet Hale-Bopp (C/1995 O1) . 188 10.4 Coma Versus Nucleus Abundances . 188 10.4.1 Multi-Volatile Model of Comet 67P/Churyumov-Gerasimenko . 188 10.4.2 Volatile Production Rates Compared with Nucleus Composition . 192 11 Internal Properties of Comet Nuclei 197 11.1 Temperature Profiles . 197 11.2 Stratification of Composition . 198 11.3 Dust Mantle Thickness . 201 12 Conclusions 205 12.1 Numerical Algorithms . 206 12.2 Goals of Comet Nucleus Modeling . 206 12.2.1 Derivation of Internal Properties . 207 12.2.2 Identification of Internal Processes . 208 12.3 General Characteristics of Comet Nuclei . 208 12.4 General Behaviour Patterns . 209 12.5 Input Data Required from Observations and Experiments . 210 12.5.1 Recommended Advances for Numerical Modeling . 213 12.5.2 Physical Processes . 213 12.5.3 Modeling the Evolution of Comet Nuclei . 213 vi Contents Appendix A: Orbital Parameters and Sizes of Comet Nuclei 215 Appendix B: Thermodynamic Properties 221 B.1 Vapour Pressures and Changes in Enthalpy of Sublimation . 221 B.2 Specific Heat . 223 B.3 Thermal conductivity . 223 B.4 Phase Transitions . 224 Glossary 225 Bibliography 227 Index 253 List of Figures 1.1 Four comet nuclei visited by spacecraft. Top left: The nucleus of Comet 1P/Halley is about 15:5 8:5 8 km in size. The × × best spatial resolution (2 pixels) is about 100 m at the top part of the image as obtained by the Halley Multicolour Cam- era on the Giotto spacecraft (Courtesy H.U. Keller; copyright 1986 MPAE). Top right: The nucleus of Comet 19P/Borrelly is about 16 8 8 km in size. Spatial resolution (2 pix- × × els) over most of the image is about 90 m. The image was obtained with the Miniature Imaging Camera and Spectrom- eter (Courtesy L. Soderblom). Bottom left: The nucleus of Comet 81P/Wild is about 5:5 4:0 3:3 km in size. The × × spatial resolution (2 pixels) is about 20 m as obtained by the Stardust mission (Courtesy R. Newburn). Bottom right: A composite of many images from the Impactor Targeting Sensor of the Deep Impact mission on approach to the nu- cleus of 9P/Tempel 1. The nucleus is about 6:2 4:6 km in × size. Highest resolution, approximately 2-3 metres, is in the area near the impact site, where small, sub-frame, close-up images were obtained. The resolution gradually degrades to- ward the edges of the frame (Courtesy Deep Impact Project. Image processing by A. Delamere and D. Stern). 3 3.1 Hexagonal packing of spheres . 37 4.1 Thermal conductivity correction formulae: parallel combina- tion (par), series combination (ser), geometrical mean (geo), Maxwell upper limit (MaxU ) and lower limit (MaxL). Re- sults are given for two ice to pore ratios: solid lines for the higher ratio(s), and dotted lines for the lower one (d). The green line represents the Monte Carlo fractal model. 78 vii viii List of Figures 5.1 Change in enthalpy of sublimation of water ice. The black dashed curve is the change in enthalpy of sublimation of wa- ter ice under equilibrium conditions (Gibbins, 1990). The solid red curve presents this correct change in enthalpy of sublimation into vacuum. The green squares represent the change in enthalpy for sublimation into vacuum as obtained by Delsemme and Miller (1971) based on data from Washburn (1928). Note that two points are outside the limits of validity of the Washburn data, indicated by blue triangles. The blue dashed curve is the fit by Cowan and A'Hearn (1979) to the data of Delsemme and Miller. The black dotted curve has been corrected twice for sublimation into vacuum using the data of Gibbins. The similarity between the blue and black dotted curves leads us to believe that Delsemme and Miller made the correction twice. 81 5.2 Net flux and surface temperature for sublimation of H2O ice, in terms of the molecular flux, ZH2O = QH2O=mH2O, where mH2O is the mass of a water molecule. 83 5.3 Timescales of different evolutionary processes (see text) as a function of temperature for different cases: (top) 1 m at 1 AU; (middle) 10 m at 10 AU; (bottom) 1000 m at 1000 AU. 88 5.4 Schematic representation of the two-zone model . 93 5.5 Regions of dominance of the two energy sources: insolation and crystallization of amorphous ice, as a function of initial comet temperature T0 and heliocentric distance, rH; is the emissivity. 96 6.1 Numerical grid. Tc is the central temperature, which cor- responds to T0 in the text. The Fi are the fluxes crossing boundaries. 100 6.2 Daily variation of surface temperature at aphelion as a func- tion of the phase angle from the local meridian, for the tem- perature surface boundary conditions Eq. (4.25) (solid line) and Eq. (6.20) (dashed line). 105 6.3 Daily variation of surface temperature at aphelion as a func- tion of the phase angle from the local meridian, for the tem- perature surface boundary conditions Eq. (4.25) (solid line) and Eq. (6.20) (dashed line) and for a surface layer thickness of 5 mm. The curves of Fig. 6.2 are shown as dotted lines for reference. 107 ix 6.4 Schematic representation of numerical grids for a spinning nucleus, commonly used in model calculations. Dots indicate radial directions along which heat conduction is computed; only in the 2.5-D model is lateral conduction included, and only along the meridian, as shown.
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