A Beta-Viscosity Model

Transcription

1 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 1

2 Outline of Talk! Review of the β viscosity model! Global behavior of α and β turbulence models! Unsteady surface density model applied to a Solar Nebula! Condensation front migration in an early Solar Nebula 2

3 The Gaseous Nebula Evolves and Cools Hot Nebula (t ~ 10 2 yrs) Cool Nebula (t ~ 10 6 yrs) 3

4 Thin disk nebula model! Keplerian rotation curve with Σ(r,t) to be determined from the evolution equation!t(r,t) found from energy equation! Generally coupled to one another in α viscosity model T Σ r 4

5 Turbulence Model Characteristics! ν 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 α and β reflect choice of scales. α model used since 1970s. β model based on scaling of hydrodynamic sources of turbulence (Richard & Zahn 1999) 5

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) 6

7 The Basic Dynamic equation 1/2 Σ (,) rt 3 ν Σ (,) ( r 1/2 rtr ) = 0 t r r r Evolution depends on choice of kinematic viscosity Conventional α viscosity model: αch c ν α 2 = = / Ω β viscosity model ν = βur k = β GMr 1/2 7

8 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 β = Σ(r,t) T(r,t) α=.01 Match M 0 and J 0 at t = 0 8

9 β Viscosity Disk Evolution M 0 =.23 M sun, J 0 = 5 J sun Analytical formulas for surface density and radial accretion, Independent of opacity Σ(r,t) V rad (r,t) radius,au radius,au 9 Surface D ensit y, gm m c 2 Radial V elocit y, cm c se 10 7 r -1/2 Outflow 10 7 Stagnation Inflow radius

10 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) 10

11 αβviscosity Mass Accretion Rates M() t = M (1 + 3 M & t/ M ) /3 Analytical M() t = M (1 + t / t ) 0 0 1/4 Conventional Power Law Model Heavy Disk β= Ruden & Pollack (1991) α=.01 Accretion starts at 1000 yrs Light Disk 11

12 Application of the Evolution Equation! What is an appropriate M 0, J 0, and β?! 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 = 2πr r to obtain Σ(r) for the ground-up planets 12

13 Application of the Evolution Equation Convert current-day planetary masses to a smooth nebula of dust and gas r, AU Mass, Earth masses CumMass, Earth masses Gas/Dust Mass, Earth masses Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto CumGas/Dust Mass, Earth masses 13

14 Nebula Surface Density total lifetime ~ years Note: slope ~ -1/2 14

15 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 β 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 4 v + Tcs 4 ) 1/4 15

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 d Ice(X) =C (p gas vap X -p X ) dt p X gas is the partial pressure of species X at a given Σ and T (from eqn) p X vap is the vapor pressure of species X at a given T (from tables) At equilibrium, p gas X = px vap, solve for Σeq T eq and the corresponding radius r eq. 16

17 Phase Equilibrium Nomograph X H2O =

18 Condensation Front Evolution 18

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 β viscosity can be a useful tool in addressing multiphysics problems 19