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

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

3 S. Davis, April 2004 Hot Nebula (t ~ 10 2 yrs) Cool Nebula (t ~ 10 6 yrs) The Gaseous Nebula Evolves and Cools

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

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

6 S. Davis, April 2004 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)

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

8 S. Davis, April 2004 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  =  (r,t) T(r,t)  =.01 Match M 0 and J 0 at t = 0

9 S. Davis, April r -1/2  r,t) V rad  r,t) Outflow Inflow Stagnation radius  Viscosity Disk Evolution M 0 =.23 M sun, J 0 = 5 J sun Analytical formulas for surface density and radial accretion, Independent of opacity

10 S. Davis, April 2004 Global Mass Accretion Rates M 0 =.111 M sun J 0 = 49.8 J sun Data from Calvet et al.(2000) Excess IR emissions from Classical T Tauri stars (cTTS)

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

12 S. Davis, April 2004 What is an appropriate M 0, J 0, 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 Application of the Evolution Equation

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

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

15 S. Davis, April 2004 Evolution of a Condensation Front Recent work shows that radial drift across H 2 O condensation front at 5 AU may enhance water vapor content and contribute to Jupiter’s growth. Sweep of condensation front across the nebula may help in solidifying moderately volatile species for subsequent planetary formation. The  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 = (T v 4 + T cs 4 ) 1/4

16 S. Davis, April 2004 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 p X gas is the partial pressure of species X at a given  and T (from eqn) p X vap is the vapor pressure of species X at a given T (from tables) At equilibrium, p X gas = p X vap, solve for  eq  T eq and the corresponding radius r eq.

17 S. Davis, April 2004 Phase Equilibrium Nomograph X H2O = 10 -4

18 S. Davis, April 2004 Condensation Front Evolution

19 S. Davis, April 2004 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  viscosity can be a useful tool in addressing multiphysics problems