1 S. Davis, April 2004 A Beta-Viscosity Model for the Evolving Solar Nebula Sanford S Davis Workshop...

1S. 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

13-17 April, 2004Ringberg, Baveria, Germany


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


Hot Nebula (t ~ 102 yrs)

Cool Nebula (t ~ 106 yrs)

The Gaseous Nebula Evolves and Cools


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


Turbulence Model Characteristics

is proportional to the product of a length and velocity scale (H,c) or (H,Uk)

H and r related: H ~ 5% r

c and Uk are problematic

c: random energy; Uk 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)


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)


The Basic Dynamic equation

Evolution depends on choice of kinematic viscosity

1/ 21/ 2( , ) 3 ( , )

( ) 0r t r t r

rt r r r

2 /cH c Conventional viscosity model:

1/ 2kU r GM r viscosity model


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

= 6.3 10-6

(r,t) T(r,t)

=.01

Match M0 andJ0 at t = 0


5 10 50 100 500radius, AU

10

100

1000

10000

ecafruS

ytisneD

,mgmc2

5 10 50 100 500radius, AU

40

20

0

20

40

laidaR

yticoleV

,mcces 104

107

104

107 r-1/2

r,t) Vradr,t)

Outflow

Inflow Stagnationradius

Viscosity Disk EvolutionM0 = .23 Msun, J0 = 5 Jsun

• Analytical formulas for surface density and radial accretion,• Independent of opacity


Global Mass Accretion Rates

M0=.111 Msun

J0= 49.8 Jsun

Data fromCalvet et al.(2000)Excess IR emissions fromClassical T Tauri stars (cTTS)


Viscosity Mass Accretion Rates

Ruden & Pollack (1991)=.01

1 / 4

0

1/ 3

0 0 0

0( ) (1 / )

( ) (1 3 / )

M t M t t

M t M M t M

Accretion startsat 1000 yrs

Heavy Disk

Light Disk

Analytical

ConventionalPower Law Model


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 = 2r r to obtain (r) for the ground-up planets

Application of the Evolution Equation


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

CumGas/Dust Mass,

Earth masses Mercury 0.3871 0.0553 0.0553 27 27. Venus 0.7233 0.815 0.8703 235 262. Earth 1 1 1.8702 320 582. Mars 1.5327 0.1074 1.9776 27 609. Jupiter 5.2028 317.894 319.87 12000 12609. Saturn 9.5388 95.185 415.05 6000 18609. Uranus 19.1914 14.537 429.59 2000 20609. Neptune 30.0611 17.132 446.72 2000 22609. Pluto 39.5294 0.0022 446.72 0.129 22609.


Nebula Surface Density total lifetime ~ 106-7 years

Note: slope ~ -1/2


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.

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 = (Tv

4 + Tcs4 )1/4


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 and T (from eqn)

pXvap is the vapor pressure of species X at a given T (from tables)

At equilibrium, pXgas = pX

vap, solve for eqTeq and the corresponding radius req.

gas vapX X

d Ice(X)=C (p -p )

dt


Phase Equilibrium Nomograph

XH2O = 10-4


Condensation Front Evolution


Conclusions

Characterization of the dynamic field is important forChemistry: outer region hot at early timesInter-radial transfer processes: space-time regime of inflow/outflow

The viscosity can be a useful tool in addressing multiphysics problems

1 S. Davis, April 2004 A Beta-Viscosity Model for the Evolving Solar Nebula Sanford S Davis Workshop...

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