mass and angular momentum loss via decretion disks

Mass and angular momentum loss via

decretion disksarXiv:1101.1732v1

Ref:arXiv:0010517v1 etc.

Basic analytic scaling for disk mass loss Numerical models Results of numerical models Radiative ablation Mass loss of the star-disk system at the

critical limit Other processes that may influence the

outer disk radius conclusions

Outline

Basic analytic scaling for disk mass loss

Presents simple analytical relations for how the presence of a disk affects the mass loss at the critical limit

Assuming a star that rotates as a rigid body

1. Basic analytic scaling for disk mass loss

criteqcrit IJRGM

IIJ

IIJ

IJ

0,/

0

3

rGMrvK /)(

MR

RR

RIM

RRRMIJ

out

out

eq

eq

eq

outeqcritcrit

)1(2

2

Mass decouples in a spherical shell, where Rout=Req :

)2(23

2eqRIM

(2)/(1):

eqout RR /23

Numerical modelsDevelops set of equations governing structure and kinematics of the disk

obtain a detailed disc structure, stationary hydrodynamic equations, cylindrical coordinates (Okazaki 2001, Lightman1974 etc.)

vr, vΦ, and the integrated disk density , depend only on radius r

1. Equation of continuity :

dz

0)()(1)(1

zr vz

vr

vrrrt

2. The stationary conservation of the r component of momentum gives

μ=0.623. The equation of conservation of the φ

component of momentum, viscosity term

peqH rRTTmkTarGMg )/(),/(,/ 0

22

~

Hdz

rvaH

Hz

K

0

2

2

0

2

),21exp(

0,21

0 pTT eff (Millar & Marlborough 1998)

Close to the star, detailed energy-balance models show:

In the outer regions: p>0p

eq rRTT )/(0

The system of hydrodynamic equations appropriate boundary conditions For obtaining vr at r=Req we use:

We have vr(Rcrit)=a to ensure the finiteness of the derivatives at this point

At the surface: vφ=vK

rrvM 2

Results of numerical models

Solves these to derive simple scaling for how thermal expansion affects the outer disk radius and disk mass loss

Stellar parameter evolved massive first star (Teff=30000 K, M=50M⊙,R=30R⊙)

Note does not significantly depend on the assumed viscosity parameter

J~ )(

21)( critKcrit RvRv

Close to the star

2)( rrvavfor rr (Okazaki 2001)

2/1 rv2/1rrvJ

constrvvr

In the supersonic region

rvr ln2

?0~ 2

rvvra

r

Result in Shakura-Sunyaev viscosity prescription, not in the supersonic regionFrom the numerical modelsIn this case,

equation

)(21)( critKcrit RvRv

)(21)(

)()(

4103

21)(

)()(

4103

221

2

~

11

2

critKeqeqK

p

eq

eqK

p

eq

eqK

eq

crit

RJRRvMRaRv

pMJ

RaRv

pRR

Factor ½ comes from the fact that the disk is not rotating as a Keplerian one at large radii

p

eq

eqK

RaRv

M

11

)()(

)2(23

2eqRIM

)1(2out

eq

eq RR

RIM

eqout RR /23

(2)/(1):

For given the minimum

~

I

JRRpeq

crit

Radiative ablationDiscusses the effects of inner-disk ablation, deriving the associated abated mass loss and its effect on the net disk angular momentum and mass loss

Stellar outflow disk, disk wind(~r)Viscous doubling is not maintained in the

supersonic windMass-loss rate of such disk wind: - the classical Castor, Abbott & Klein (1975, CAK) stellar wind mass-loss rate

GMcLe

4

x=r/R

Assuming the disk wind is not viscously coupled to the disk, then

P1(x) solid lineP1/2(x) dashed line

)(rvv K

A more detailed calculation gives:

For Rout → ∞

Maximum disk wind mass-loss rate

Maximum angular momentum loss rate

For α≈0.6, CAKdw MM 251)(

Mass loss of the star-disk system at the critical limit

Offers a specific recipe for incorporating disk mass loss rates into stellar evolution codes

The structure of disk and radiatively driven wind , radiative force

Rout→∞If net is carried away by disk outflow < > (p=0)

J

Stellar wind disk wind disk itself

J

Conclusion

The disk mass loss is set by needed to keep the rotation at or below the Ωcrit

J

A

B

C

Thank you!