Wind-Fed Magnetospheres:Mass Budget & Spindown

Stan OwockiUniversity of DelawareNewark, Delaware USA

Collaborators• Asif ud-Doula • Rich Townsend

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Questions

• What processes limit the build-up of mass in a wind-fed magnetosphere?– how/where does it leak?– how does total mass scale with B, vrot/vcrit, Mdot, etc.

• How does angular momentum loss & spindown scale with B*, Mdot, etc.?– can we explain slow rotators w/ magnetic spindown?– how to generalize for higher multipoles?

Rigid Field - Hydro Model

QuickTime™ and a decompressor

are needed to see this picture.

Wind Magnetic Confinement

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η(r) ≡B2 /8π

ρv 2 /2Ratio of magnetic to kinetic energy density:

e.g, for dipole

q=3; η ~ 1/r 4 =B*

2R*2

M•

v∞

(r / R*)2−2q

(1− R* / r)β

η*

Alfven Radius: η( RA) ≡ 1

For dipole (q=3) with beta=0: RA = η*1/4 R*

Magnetic confinement vs. Wind + Rotation

RA =η*1/4R*

Alfven radius

RK =W−2/3R*

Kepler radius

W ≡Vrot

GM / R*

Rotation vs. critical

η∗≡B*

2R*2

M.

V∞

Wind mag. confinement

Field-aligned rotation

η*=100RA=3.2R*

Vrot/Vorb=1/2RK=1.6 R* QuickTime™ and a

BMP decompressorare needed to see this picture.

RK RA

Strong Field + Rapid rotationη*=100 W=1/2

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RK RA

Radial Mass Distribution

dme(r, t)

dt≡2πr2 ρ(r,θ,t)

π /2−Δθ /2

π /2+Δθ /2

∫ sinθ dθ

Time evolution of Radial distribution of equatorial disk mass

η* = 100 & Vrot/Vcrit =1/2

RK

RA

Time (ksec)

Rad

ius

(R*)

Corotation Parameter

η* = 100 & Vrot/Vcrit =1/2

RK

RA

Time (ksec)

Rad

ius

(R*)

Temporal evolution of radial distribution of equatorial disk mass

Stronger Magnetic Confinement --->More

Rap

id R

ota

tion

--

->

r=1-5

t=0-3 Msec

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are needed to see this picture.

Strongest MHD sim

η*=1000W=1/2

RK RA

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Breakout time & Asymptotic mass

tb (r) ≈η* tff

R*RK3

r(r3 −RK3 )

but for sigOriE, eta* ~106,

tb (2RK ) ≈100 yr

M∞ ≈10−7 Me

for MHD sims with eta* =100,

tb (2RK ) ≈106 sec~10 day ≈3 ×10−9M e

B32R12

2

g4

Independent of Mdot!

M∞ ≈0.2M

g

η* tff(for W=1/2)

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Angular Momentum Loss & Spindown

J•= 2

3 M•ΩRA

2

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Weber & Davis 1967

j =Vφr −BφBrrρVr

j =ΩRA2

At r= RA, MA=1 implies

Tot. equat. Ang. mom/mass jgas ≡Vφr =

jMA2 −Ωr2

MA2 −1

=>

Bφ

Br

=Ωr −Vφ

Vr &

Frozen flux

gas field

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Time Variation of Angular Momentum Loss

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Time variation of totalAngular Momentum Loss

Gas Field Total

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Angular Momentum Loss vs.latitude & time

+

+ =

=

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Magnetic confinement parameter =>

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Dipole spindown times

τ spin ≈ τ mass32 k

η*

≈11Myrk−1

BkG

M*

R*

V8

M•

−9

≈32 kM*

BeqR*

V∞

M•

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Spindown age

P =Poe−t/τspin Po ≈ fPc

Pc =0.21d R RM ⇒ Po ~ day

e.g. HD191612, with Po = 0.5 to 1 day:

τ age ≈ 6.3 → 6.9 τ spin ≈ 2.5 → 2.9 Myr

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Extrapolated spindown law for higher order multipoles?

p=2 monopole =3 dipole =4 quadrapole... etc.

=> Spindown weaker for more complex fields?

If so, hard to explain tau Sco by spindown??

Need 3D MHD sims to test this!

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Summary• Wind feeding of magnetosphere

– balanced by inner & outer “leakage”?– observations should estimate Mtot

– breakout analysis predicts Mtot indep of Mdot!

• Wind Magnetic Spindown– tspin ~ tmass/Sqrt[eta*] for aligned dipole– complex field => slower spindown?– need 3D sims to confirm!

Alfven speed

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VA ≡B

4πρ

Alfven speed

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ρVA2

2=B2

8πNote:

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=Pmag

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=Emag

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a ≡Pgasρ

Compare sound speed

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Pgas = ρa2