Stellar Magnetospheres

Stan OwockiBartol Research InstituteUniversity of DelawareNewark, Delaware USA

Collaborators (Bartol/UDel)• Rich Townsend• Asif ud-Doula • also: David Cohen, Detlef Groote, Marc Gagné

“Magnetosphere”• Space around magnetized planet or astron.body

• coined in late ‘50’s– satellites discovery of Earth’s “radiation belts”– trapped, high-energy plasma

• also detected in other planets with magnetic field– e.g. Jupiter (biggest), Saturn etc. (but not Mars, Venus)

• concept since extended to Sun and other stars– sun has “magnetized corona”– stretched out by solar wind

• but study has some key differences...

Planetary vs. Stellar Magnetospheres

• outside-in compression

• Space physics• local in-situ meas.• plasma parameters• plasma physics models

• inside-out expansion• Astrophysics• global, remote obs.• gas & stellar parameters

• hydro or MHD models

Solar coronal expansion

Interplanetary medium

Solar astrophysics includes both approaches

Earth’s Magnetosphere

Sun-Earth Connection

Magnetic loops on Solar limb

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Solar Flare

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Sun’s Active “Magnetosphere”

Solar Activity: Coronal Mass Ejections

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Processes for Solar Activity

• Convection + Rotation Dynamo– Complex Magnetic Structure– Magnetic Reconnection– Coronal Heating– Solar Wind Expansion + CME

Solar Corona in EUV & X-rays Composite EUV image from EIT/SOHO X-ray Corona from SOHO

Magnetic Effects on Solar Coronal ExpansionCoronal streamers

Corona during Solar Eclipse

Latitudinal variation of solar wind speed

Coronal streamers

Pneuman and Kopp (1971)

MHD model for base dipolewith Bo=1 G

Iterative scheme

MHD Simulation

Fully dynamic, time dependent

MHD model for magnetized plasmas

•ions & electrons coupled in “single fluid”– “effective” collisions =>– Maxwellian distributions

•large scale in length & time– >> Larmour radius, mfp, Debye, skin– >> ion gyration period

•ideal MHD => infinite conductivity

Magnetohydrodynamic (MHD) Equations

DρDt

+ ρ∇⋅v=0

Mass Momentum

mag Induction Divergence free B

€

P = ρa2 = (γ −1)e

De

Dt+γe∇⋅v=H −C

€

ρ DvDt

= −∇p+1

4π∇ × B( ) × B + ρ (gext − ggrav )

€

∂B∂t

=∇ × v × B( )

€

∇• B = 0

Energy Ideal Gas E.O.S.

€

∂e∂t

+∇ ⋅ev = −P∇ ⋅v+ H −C

Maxwell’s equations

Induction

no mag. monopoles

€

∇×E = −1

c

∂B

∂t

€

∇• B = 0€

∇• E = 4πρ e

Coulomb

€

∇×B =4π

cJ +

1

c

∂E

∂tAmpere

€

Ov

c

⎛

⎝ ⎜

⎞

⎠ ⎟2

<<1

Magnetic Induction & Diffusion

€

J =σ (E + v × B /c)

magnetic induction

Combine Maxwell’s eqns. with Ohm’s law:

∂B∂t

= ∇ × v × B( ) +c2

4πσ∇2B

Eliminate E and J to obtain:

magnetic diffusion

Magnetic Reynold’s number

€

Re ≅4πσLv

c 2

∂B∂t

= ∇ × v × B( ) +c2

4πσ∇2B

induction diffusion

€

≅induction

diffusion>>1

€

O1

Re

⎛

⎝ ⎜

⎞

⎠ ⎟<<1“Ideal” MHD

e.g., 1010!

Frozen Flux theorum

€

∂B∂t

=∇ × v × B( )Ideal MHD induction eqn.:

€

F ≡ B • dAσ∫

implies flux F through any material surface

€

dF

dt= 0

does not change in time, i.e. is “frozen”:

Magnetic loops on Solar limb

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Magnetic Reconnection

If/when/where B field varies over a small enough scale, L << R, i.e. such that local Re~1,then frozen flux breaks down,

€

dF

dt≠ 0

The associated magnetic reconnection can occur suddenly, leading to dramatic plasma heating through dissipation of magnetic energy B2/8.

Solar Flare

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Magnetic Lorentz force:magnetic tension and

pressure

€

fLor =J × B

c

magnetictension

Lorentz force

€

fLor =B • ∇B

4π−∇

B2

8π

⎛

⎝ ⎜

⎞

⎠ ⎟

Use BAC-CAB + no. div. to rewrite:

magnetic pressure

€

=1

4π∇ × B( ) × B

Alfven speed

€

VA ≡B

4πρ

Alfven speed

€

ρVA2

2=B2

8π

Note:

€

=Pmag

€

=Emag

€

a ≡Pgasρ

Compare sound speed

€

Pgas = ρa2

Plasma “beta”

low“beta” plasma strongly magnetic: VA > a

€

=2a

VA

⎛

⎝ ⎜

⎞

⎠ ⎟

2

€

β ≡PgasPmag

€

=Pgas

B2 /8π

high “beta” plasma weakly magnetic: VA < a

Key MHD concepts •Reynolds no. Re >> 1•Re->inf => “ideal” => frozen flux

•breaks down at small scales: reconnection

•Lorentz force ~ mag. pressure + tension

• Alfven speed VA~B/sqrt(ρ)

•plasma beta ~ Pgas/Pmag ~ (a/VA)2

Magnetohydrodynamic (MHD) Equations

€

Dρ

Dt+ ρ∇ • v = 0

Mass Momentum

mag Induction Divergence free B

€

P = ρa2 = (γ −1)e

€

∂e∂t

+∇ ⋅ev = −P∇ ⋅v+ H −C€

ρ DvDt

= −∇p+1

4π∇ × B( ) × B + ρ (gext − ggrav )

€

∂B∂t

=∇ × v × B( )

€

∇• B = 0

Energy Ideal Gas E.O.S.

Apply to solar corona & wind

•Solar corona–high T => high Pgas

–scale height H ~< R–breakdown of hydrostatic equilibrium–pressure-driven solar wind expansion

•How does magnetic field alter this?–closed loops => magnetic confinement–open field => coronal holes –source of high speed solar wind

Hydrostatic Scale Height

€

≈T6

14Scale Height:

€

H

R=a2R

GM €

P = ρa2

€

≡a2

H

€

H

R≈

1

2000 solar photosphere:

€

T6 = 0.006

€

−GM

r2=

1

ρ

dP

dr

Hydrostaticequilibrium:

€

H

R≈

1

7 solar corona:

€

T6 = 2

Failure of hydrostatic equilibrium for hot, isothermal corona

€

0 = −GM

r2−a2

P

dP

dr

hydrostaticequilibrium:

€

P(r)

Po= exp −

R

H1−

R

r

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

€

logPTRPISM

⎛

⎝ ⎜

⎞

⎠ ⎟=12observations for

TR vs. ISM:

# decades of pressure decline:

€

logPoP∞

⎛

⎝ ⎜

⎞

⎠ ⎟=R

Hloge ≈

6

T6€

→ exp −R

H

⎡ ⎣ ⎢

⎤ ⎦ ⎥ for r → ∞

Solar corona T6 ~ 2 => must expand!

Spherical Expansion of Isothermal Solar Wind

Momentum and Mass Conservation:

€

vdv

dr= −

GM

r2−a2

ρ

dρ

dr

€

d ρvr2( )

dr= 0

Combine to eliminate density:

€

1-a2

v2

⎛

⎝ ⎜

⎞

⎠ ⎟vdv

dr=

2a2

r−GM

r2

RHS=0 at “critical” radius:

€

rc =GM

2a2

€

v2

a2- ln

v2

a2= 4 ln

r

rc+

4rcr

+CIntegrate for transcendental soln:

€

v rc( ) ≡ a

€

C = −3=> Transonic soln:

€

rc = rs sonic radius

Solution topology for isothermal wind

C=-3

Corona during Solar Eclipse

Kopp-Holzer Non-Radial Expansion

Mach number topology

Latitudinal variation of solar wind speed

Coronal streamers

MHD model for coronal expansion vs. solar dipole

Pneumann & Kopp 1971Iterative solution MHD Simulation

Wind Magnetic Confinement

€

η(r) ≡B2 /8π

ρv 2 /2

Ratio of magnetic to kinetic energy density:

e.g, for dipole field, q=3; η ~ 1/r 4

€

=B2r2

M•

v=B*

2R*2

M•

v∞

(r /R*)2−2q

(1− R* /r)β

η*

for solar wind with Bdipole ~ 1 G: η*~ 40

η* >>1 => strong magnetic confinement of wind