Stellar Magnetospheres Stan Owocki Bartol Research Institute University of Delaware Newark, Delaware...
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Transcript of Stellar Magnetospheres Stan Owocki Bartol Research Institute University of Delaware Newark, Delaware...
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
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
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 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.
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
MHD model for coronal expansion vs. solar dipole
Pneumann & Kopp 1971Iterative solution MHD Simulation