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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 expansionInterplanetary medium
Solar astrophysics includes both approaches
Earth’s Magnetosphere
Sun-Earth Connection
Magnetic loops on Solar limb
Solar Flare
Sun’s Active “Magnetosphere”
Solar Activity: Coronal Mass Ejections
Processes for Solar Activity
• Convection + Rotation Dynamo– Complex Magnetic Structure– Magnetic Reconnection– Coronal Heating– Solar Wind Expansion + CME
Solar Corona in EUV & X-raysComposite 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
Coronalstreamers
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 = 0Mass Momentum
mag InductionDivergence free B
€
P = ρa2 = (γ −1)e
€
∂e∂ t
+∇ ⋅ ev = −P∇ ⋅ v + H −C€
ρDvDt
= −∇p +14π
∇ × B( ) × B + ρ(gext − ggrav )
€
∂B∂t
=∇ × v × B( )
€
∇ •B = 0
Energy Ideal Gas E.O.S.
Maxwell’s equations
Induction
no mag. monopoles
€
∇ × E = −1c∂B∂t
€
∇ •B = 0€
∇ • E = 4πρe
Coulomb
€
∇ × B =4πcJ +
1c∂E∂t
Ampere
€
O vc
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( ) +c 2
4σ∇2B
Eliminate E and J to obtain:
magneticdiffusion
Magnetic Reynold’s number
€
Re ≅ 4πσLvc 2
induction
€
∂B∂t
=∇ × v × B( ) +c 2
4σ∇2B
diffusion
€
≅inductiondiffusion
>>1
€
O 1Re
<<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 σ,
€
dFdt
= 0
does not change in time, i.e. is “frozen”:
Magnetic loops on Solar limb
Magnetic ReconnectionIf/when/where B field varies over a small enough scale, L << R, i.e. such that local Re~1,then frozen flux breaks down,
€
dFdt
≠ 0
The associated magnetic reconnection can occur suddenly, leading to dramatic plasma heating through dissipation of magnetic energy B2/8π.
Solar Flare
Magnetic ReconnectionIf/when/where B field varies over a small enough scale, L << R, i.e. such that local Re~1,then frozen flux breaks down,
€
dFdt
≠ 0
The associated magnetic reconnection can occur suddenly, leading to dramatic plasma heating through dissipation of magnetic energy B2/8π.
Magnetic Lorentz force:magnetic tension and pressure
€
fLor =J × Bc
magnetictension
Lorentz force
€
fLor =B •∇B4π
−∇B2
8π
Use BAC-CAB + no. div. to rewrite:
magneticpressure
€
=14π
∇ × B( ) × B
Alfven speed
€
VA ≡B4πρ
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
€
= 2 aVA
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
€
DρDt
+ ρ∇ •v = 0Mass Momentum
mag InductionDivergence free B
€
P = ρa2 = (γ −1)e
€
∂e∂ t
+∇ ⋅ ev = −P∇ ⋅ v + H −C€
ρDvDt
= −∇p +14π
∇ × 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
€
≈T614
Scale Height:
€
HR
=a2RGM €
P = ρa2
€
≡a2
H
€
HR≈
12000
solar photosphere:
€
T6 = 0.006
€
−GMr2
=1ρdPdr
Hydrostaticequilibrium:
€
HR≈17
solar corona:
€
T6 = 2
Failure of hydrostatic equilibriumfor hot, isothermal corona
€
0 = −GMr2
−a2
PdPdr
hydrostaticequilibrium:
€
P(r)Po
= exp − RH1− R
r
€
log PTRPISM
=12
observations for TR vs. ISM:
# decades of pressure decline:
€
log PoP∞
=
RHloge ≈ 6
T6€
→ exp − RH
for r→∞
Solar corona T6 ~ 2 => must expand!
Spherical Expansion of Isothermal Solar WindMomentum and Mass Conservation:
€
v dvdr
= −GMr2
−a2
ρdρdr
€
d ρvr2( )dr
= 0
Combine to eliminate density:
€
1- a2
v2
vdvdr
=2a2
r−GMr2
RHS=0 at “critical” radius:
€
rc =GM2a2
€
v2
a2- ln v
2
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-HolzerNon-Radial Expansion
Mach number topology
Latitudinal variation of solar wind speed
Coronalstreamers
MHD model for coronal expansion vs. solar dipole
Pneumann & Kopp 1971Iterative solution MHD Simulation
Wind Magnetic Confinement
€
η(r) ≡ B2 /8πρv 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