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A Beta-Viscosity Model for the Evolving Solar Nebula  Sanford S Davis

A Beta-Viscosity Model for the Evolving Solar Nebula  Sanford S Davis Workshop on Modeling the Structure, Chemistry, and Appearance of Protoplanetary Disks 13-17 April, 2004 Ringberg, Baveria, Germany. Outline of Talk. Review of the  viscosity model

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A Beta-Viscosity Model for the Evolving Solar Nebula  Sanford S Davis

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  1. A Beta-Viscosity Model for the Evolving Solar Nebula  Sanford S Davis Workshop on Modeling the Structure, Chemistry, and Appearance of Protoplanetary Disks 13-17 April, 2004 Ringberg, Baveria, Germany S. Davis, April 2004

  2. Outline of Talk • Review of the  viscosity model • Global behavior of  and  turbulence models l Unsteady surface density model applied to a Solar Nebula • l Condensation front migration in an early Solar Nebula S. Davis, April 2004

  3. The Gaseous Nebula Evolves and Cools Hot Nebula (t ~ 102 yrs) Cool Nebula (t ~ 106 yrs) S. Davis, April 2004

  4. Thin disk nebula model S r • Keplerian rotation curve with S(r,t) to be determined from the evolution equation • T(r,t) found from energy equation • Generally coupled to one another in a viscosity model T S. Davis, April 2004

  5. Turbulence Model Characteristics • n is proportional to the product of a length and velocity scale (H,c) or (H,Uk) • lH and r related: H ~ 5% r • c and Ukare problematic • c: random energy; Ukdirected energy; turbulence velocity scale is in between • The factors a and b reflect choice of scales. a model used since 1970s. b model based on scaling of hydrodynamic sources of turbulence (Richard & Zahn 1999) S. Davis, April 2004

  6. Why use a β model? • Exclude thermodynamics from the evolution equation (opacity model is not a factor) • Turbulence modeling is historically an incompressible hydrodynamic problem • Temperature follows from radiation transfer (energy equations) • As a vehicle for moving to multiphysics problems • Described in Davis (2003, ApJ) S. Davis, April 2004

  7. The Basic Dynamic equation Evolution depends on choice of kinematic viscosity Conventional a viscosity model: b viscosity model S. Davis, April 2004

  8.  = 6.3 10-6 S(r,t) T(r,t) a=.01 Match M0 and J0 at t = 0 Comparison with Ruden-Lin (1986) Numerical Simulation • Analytical formulas for surface density compared with numerical soln (coupled momentum, energy) • Central plane temperature is not smooth using both approaches S. Davis, April 2004

  9. Vrad(r,t) S(r,t) 104 104 Outflow 107 Inflow Stagnation radius 107 r-1/2 • Viscosity Disk Evolution • M0 = .23 Msun, J0 = 5 Jsun • Analytical formulas for surface density and radial accretion, • Independent of opacity S. Davis, April 2004

  10. Global Mass Accretion Rates M0=.111 Msun J0= 49.8 Jsun Data from Calvet et al.(2000) Excess IR emissions from Classical T Tauri stars (cTTS) S. Davis, April 2004

  11.   Viscosity Mass Accretion Rates Analytical Conventional Power Law Model Heavy Disk Ruden & Pollack (1991) a=.01 b= Accretion starts at 1000 yrs Light Disk S. Davis, April 2004

  12. Application of the Evolution Equation • What is an appropriate M0, J0, and ? • How well can it predict the early evolution of our Solar System? Procedure: • Fit an analytical curve (tan-1) to the total mass vs r distribution. This is the monotonic cumulative mass distribution, M(r). • Divide the incremental mass M = dM/dr r by the incremental area A = 2r r to obtain (r) for the ground-up planets S. Davis, April 2004

  13. Application of the Evolution Equation • Convert current-day planetary masses • to a smooth nebula of dust and gas S. Davis, April 2004

  14. Nebula Surface Density total lifetime ~ 106-7 years Note: slope ~ -1/2 S. Davis, April 2004

  15. Evolution of a Condensation Front • Recent work shows that radial drift across H2O condensation front at 5 AU may enhance water vapor content and contribute to Jupiter’s growth. • l Sweep of condensation front across the nebula may help in solidifying moderately volatile species for subsequent planetary formation. • The b viscosity formulation can be a useful tool in this interdisciplinary field • Use a quasi steady model with Mdot variable • Includes viscous heating and central star luminosity so that T = (Tv4 + Tcs4 )1/4 S. Davis, April 2004

  16. Application of the Evolution Equation: Gas/Solid Sublimation Fronts Rate of increase of a solid species (Water ice, Ammonia ice, Carbon Dioxide ice) is governed by the Hertz-Knudsen relation pXgas is the partial pressure of species X at a given S and T (from eqn) pXvap is the vapor pressure of species X at a given T (from tables) At equilibrium, pXgas = pXvap, solve for SeqTeq and the corresponding radius req. S. Davis, April 2004

  17. Phase Equilibrium Nomograph XH2O = 10-4 S. Davis, April 2004

  18. Condensation Front Evolution S. Davis, April 2004

  19. Conclusions • Characterization of the dynamic field is important for Chemistry: outer region hot at early times Inter-radial transfer processes: space-time regime of inflow/outflow • The b viscosity can be a useful tool in addressing multiphysics problems S. Davis, April 2004

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