AS 4002 Star Formation & Plasma Astrophysics

Angular momentum conservation: 1

• Azimuthal component x R gives torque balance:

ρ u ⋅∇( )u = −∇p +1

μ 0

∇ × B( ) × B −ρGM

r2 ˆ r

ρ u ⋅∇( ) Ru( )[ ]φ =R

μ 0

∇ × B( ) × B ⎡

⎣ ⎢

⎤

⎦ ⎥φ

⇒ ρu ⋅∇ Ruφ( ) =R

μ 0

− ∇ × B( )R Bz + ∇ × B( )z BR[ ]

=R

μ 0

Bz

R∂∂z

RBφ( ) +BR

R∂∂R

RBφ( ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Angular momentum transported by flow

Couple exerted by Lorentzforce on unit volume

AS 4002 Star Formation & Plasma Astrophysics

Angular momentum conservation: 2

• Equation of motion becomes:

∇ Ruφ( ) has no φ - component, so:

u ⋅∇ Ruφ( ) = up ⋅∇ Ruφ( ) = κBp ⋅∇ Ruφ( )

ρκB ⋅∇ Ruφ( ) = B ⋅∇RBφ

μ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

⇒ B ⋅∇ ρκRuφ −RBφ

μ 0

⎛

⎝ ⎜

⎞

⎠ ⎟ = 0.

constant

AS 4002 Star Formation & Plasma Astrophysics

Net angular momentum flux

• Integrate azimuthal equation of motion to get:

ρκRuφ −RBφ

μ 0

≡ ρκL = constant.Constant of integration on eachfield-streamline. L is thenet angular momentum perunit mass carried in the plasmamotion and the magnetic stresses

• Wind carries angular momentum away from star.

• Lorentz force transmits torque to stellar surface.

• Star spins down.

AS 4002 Star Formation & Plasma Astrophysics

The story so far:

up =κBp uφR−κBφ

R=Ω=constant.

ρup

Bp

=ρκ ≡η =constant.

ρκRuφ −RBφμ0

≡ρκL =constant.

Induction

Mass continuity

Torque balance

1

2

3

AS 4002 Star Formation & Plasma Astrophysics

The Alfvénic point

• Use (1) - μ0κ/R2 (3) to eliminate B:

uφ

R−μ0ρκ

2uφR

=Ω−μ0ρκ

2LR2

κ 2 =up

2

Bp2 and uA

2 =Bp

2

μ 0ρ,• Substitute:

• to get:uφ

R1−

up2

uA2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ =Ω−

up2L

uA2 R2 .

• u is singular at Alfvenic point up=uA unless:

RA2Ω−L =0.

AS 4002 Star Formation & Plasma Astrophysics

Weber-Davis radial-field model

• Radial field lines close to star (“split monopole”).

• Spherical Alfvénic surface, radius rA.

• Isotropic mass loss.

• Angular momentum flux across area ds is:

ρAuA( ). ΩrA2 cos2 θ( ). 2πrA cosθ . rAdθ( )

Mass flux Net specific angularmomentum transportedalong field line anchored at latitude

Area of surface element ds

⇒ −˙ J = ρ AuArA2( ). ΩrA

2( )2π cos3 θ−π /2

π /2

∫ dθ

=4/3

AS 4002 Star Formation & Plasma Astrophysics

Net spindown torque on star

• Density at Alfven radius:

• Net torque on star:

ρA =BA

2

μ 0uA2

⇒ −˙ J =8π3μ 0

BArA2( )

2 ΩuA

=8π3μ 0

B0r*2( )

2 ΩuA

Radial or dipole field

• For thermal driving, uA~ 2 to 3 cs, indep. of Ω.

• For linear dynamo law, B0 ~ Ω.

• If stellar moment of inertia is constant:

J =k2M*r*2Ω⇒

dΩdt

∝−Ω3

AS 4002 Star Formation & Plasma Astrophysics

General braking laws

• Asymptotically:

• Hence for p = 3,

dΩdt

=−c0Ωp ⇒

11−p

Ω1−p −Ω01−p

( ) =−c0 t−t0( )

⇒ Ω = Ω01−p +c0 p−1( ) t−t0( )[ ]

1/(1−p)

Ω → c0 p−1( ) t−t0( )1/(1−p)

for t−t0 >>Ω0

1−p

c0 p−1( )

Ω(t) → t−1/2

cf. Skumanich.