Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP ...astro.ru.nl/~achterb/CosmMagn/Cosmic...
Transcript of Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP ...astro.ru.nl/~achterb/CosmMagn/Cosmic...
Prof. dr. A. Achterberg, Astronomical Dept. , IMAPP, Radboud Universiteit
A way to describe fully ionized plasmas using a fluid approximation;
Uses a simplified set of equations for electrodynamic processes
Can be formulated entirely in terms of density, velocity, pressure and magnetic field
A quick recap of Hydrodynamics
• 1. Equation of motion for a fluid:
Total comoving derivative
Pressure force
Gravitational force
2. Continuity equation and equation of state (ideal gas):
Mass density ρ must adjust to flow in order to conserve mass!
Compression raises pressure and temperature, expansion lowers it. Ideal mono-atomic gas has γ = 5/3
3. Conservative form of the equations:
Mass Conservation
Momentum Conservation
Energy Conservation
Details can be found at:
http://astro.ru.nl/~achterb/Gasdynamica_2016/
Cosmic plasmas and magnetic fields
• Ionized gases (plasmas) can carry free electric charges, leading to:
- Electrostatic effects - Electromagnetic effects
• In the astrophysical context one is always dealing with highly conducting plasma’s: - Currents and the magnetic fields they generate are
long-lived: the realm of Magnetohydrodynamics
How can you detect Cosmic magnetic fields?
• In stars: Zeeman splitting of some spectral lines;
• In diffuse gas: synchrotron radiation;
• In diffuse gas with dust: polarization of scattered starlight.
• In diffuse gas: Faraday rotation and Rotation measure of
EM radiation (both are index of refraction effects)
Sun in UV light (TRACE satellite)
Synchrotron sources:
Supernova Remnants AGNs/Quasars
Typical field strengths
Object Field strength in Gauss
Earth’s dipole field at N/S Pole 0.6
Solar Dipole field at N/S Pole 10
Sunspot 2500
Solar Wind at 1 AU 2 10-6
Magnetic A Stars 10,000
Pulsar (Neutron Star) 1011-1013
Supernova Remnant 10-4-10-5
Interstellar Medium 3 10-6
Intra-Cluster Gas 10-5-10-6
Intergalactic Medium (“Voids”) < 10-9 for random component
< 10-11 for uniform component
Introduction to plasma physics
Simplest model: two-fluid model consisting of ions and electrons
Extra physics: the electromagnetic coupling between charges (ions and electrons) through the Lorentz Force
Extra equations: Maxwell’s equations that describe how: - free charges generate electric fields - currents generate magnetic fields
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a gravitational force density
q nq
P nm
nm P nmt c
ρ
ρ
≡
−
∂ × + • = − + + + ∂
g
V BV V V E g
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a gravitational force density
q nq
P nm
nm P nmt c
ρ
ρ
≡
−
∂ × + • = − + + + ∂
g
V BV V V E g
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a gravitational force density
q nq
P nm
nm P nmt c
ρ
ρ
≡
−
∂ × + • = − + + + ∂
g
V BV V V E g
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a gravitational force density
q nq
P nm
nm P nmt c
ρ
ρ
≡
−
∂ × + • = − + + + ∂
g
V BV V V E g
NOTATION CHANGE!!!
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a gravitational force density
q nq
P nm
nm P nmt c
ρ
ρ
≡
−
∂ × + • = − + + + ∂
g
V BV V V E g
Step 1: Equation of motion for a charged fluid
( )
Equation of motion for a charge .
Step 1: replace mass by mass density ( number density charges)
Step 2: re-interpret time derivative:
Step 3: replace ch
dm q qdt c
m nm n
ddt t
× = +
=
∂⇒ + •
∂
v v BE
V
( )
arge by a charge density
Step 4: add a pressure force and (where needed) a grav
itational force density
nm P nmt
q nq
P nm
cρ
ρ
∂ × + • = − + + + ∂
≡
−
V V BV E g
g
V
Glorious outcome!!
Two-fluid equations for a hydrogen plasma
( )
( )
Ions are protons with mass and charge :
Electrons: mass and charge :
p
i ii p i i i i i p
e
e ee e e e e e e e
m e
n m P n e n mt c
m e
n m P n e n mt c
+
∂ × + • = − + + + ∂
−
∂ × + • = − − + + ∂
V V BV V E g
V V BV V E g
Both species feel the same electromagnetic fields, which they generate together!
The two fluids together generate the EM fields:
( )
( )
( )2
4 4
1
0
4 4 4 4
If self-gravity is important equation for gravitational potential :
4
i p e
i e
i i e e
e
c t
c c t c c
e
t
G n m n
n n
e n
m
n
π π
π π π π
π
ρ −
−
• = =
∂× = −
∂
• =
∂ ∂× = =
∂ ∂
Φ
∇ Φ = +
J
E
BE
B
E V+ +V EB
There are additional equations needed
( )
= i
00
e,
1. Particle number conservation: 0;
2. Ideal gas law:
3. Maxwell's Equations:
4 4 (Coulomb's law)
1 (Faraday's Equation)
0
b
n nt
nP nk T P
t
q n
n
c
γ
α αα
π πρ
∂+ • =
∂
= =
• = =
∂× = −
∂
• =
∑
V
E
BE
B
=e,i
(No magnetic monopoles!)
4 4 4 4
(Ampere's equation)c c t c c t
q nα α αα
π π π π∂ ∂× = =
∂ ∂∑E EB + +J V
Charge density ρ and current density J are determined by both the ions and the electrons
Conductivity for static fields: Ohms Law
Free charge carriers in a plasma conduct current:
electrostatic force frictional force due toon the electrons electr
Simple model: stationary ions and moving electrons:
Balance between electric force and electron-ion friction:
ee e e e e ei e
dn m en n mdt
ν= − −
V E V
on-ion collisions
2
0
Solve for electron velocity in STATIC case:
Associated current density:
=
ee ei
ee e e
e ei
em
e nenm
ν
σν
=
= −
− = ≡
EV
EJ V E
Current density J and driving electric field E are proportional!
Scalar and tensor conductivity
Our simple calculation gives a scalar conductivity:
eσ=J E
More complicated case: a tensor conductivity
(in component form: )i ij jJ Eσ= • =J E
High school version of Ohms Law is the same!
current resistance
current density area
electric field strenght length
V I R
I J
V EL
= × = ×
= × = ×
= = ×
Basic relations:
High school version of Ohms Law is the same!
current resistance
current density area
electric field strenght length
V I R
I J
V EL
= × = ×
= × = ×
= = ×
Basic relations:
Combine:
( = / )I V ELJ E L RR R
σ σ= = = ≡
The MHD condition
• Valid for plasma’s with a very high conductivity!
'c
σ σ × = = +
V BJ E E
Ohm’s law in a moving plasma:
In terms of the resistivity η=1/σ:
cη
σ×
+ = =V B JE J
Limit of infinite conductivity (zero resistivity):
0
c
c
ησ
×+ = = ↓
⇔
×
V B JE J
V BE = -
Alternative interpretation in terms of Lorentz force:
0 if L qc× = =
v BF E + v = V
Lorentz force vanishes for any charge that moves with the plasma!
Basic equations: 1. dynamics
( )
( )
Ions are protons with mass and charge :
Electrons: mass and charge :
p
i ii p i i i i i p
e
e ee e e e e e e e
m e
n m P n e n mt c
m e
n m P n e n mt c
+
∂ × + • = − + + + ∂
−
∂ × + • = − − + + ∂
V V BV V E g
V V BV V E g
Basic assumptions: 1) electrons and protons almost move together: 2) electron and ion densities are the same:
e i ≡V V V
e in n n= ≡
Add the two equations of motion:
( )
( )
i ii p i i i i i p
e ee e e e e e e e e
n m P n e n mt c
n m P n e n mt c
∂ × + • = − + + + ∂ ∂ × + • = − − + + ∂
V V BV V E g
V V BV V E g
( ) ( ) ( ) ( ) ( )0 p e i e i e i i e e
Total plasma Charge density Total cTotal mass densitypressure
1n m m P P n e n e n e n et c
∂ + + • = − + + − + − ∂
+ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -V V V E V V ( )0 p e
urrent density
n m m× +
B + g
MHD approximation: neglect charge density, define:
( )( )
0 p e e i
0 i e
mass density , pressure
current density
n m m P P P
n e
ρ = + = +
= −J V V
MHD equation of motion:
( ) Pt c
ρ ρ∂ × + • = − + ∂ V J BV V + g
( ) ( ) ( ) ( ) ( ) ( )0 p e i e i e i i e e 0 p e
Total plasma Charge density Total current densityTotal mass densitypressure
1n m m P P n e n e n e n e n m mt c
∂ + + • = − + + − + − × + ∂
V V V E V V B + g
Maxwell’s equations:
1. We do not need the equations for the electric field! 2. Ampere's equation simplifies considerably:
4 1c c tπ
∂= +×
∂B J E
MHD describes slow phenomena: neglect the displacement current!
Still missing: a dynamical equation for the magnetic field!
Faraday’s equation + Ampere’s equation + Ohms Law:
( )
( )2
2
4
4
ct
cc t
c
σ πσ
π
∂ = − × ∂ × ∂= − + ⇒ = × × + ∇ ∂= ×
B E
V B J BE V B B
J B
The Induction Equation
Summary: basic MHD equations
( )
( )
( )2
2
0
4
4
0
Pt c
t
ct
c
c
ρ ρ
ρ ρ
πσ
π
σ
∂ × + • = − + ∂
∂+ • =
∂
∂= × × + ∇
∂
× =
• =
×= − +
V J BV V + g
V
B V B B
B J
B
V B JE
Force Balance Mass conservation Induction Equation Ampères law No Monopoles!! MHD Condition
Additional relations:
00
and TP P Pγ
ρ ρµ ρ
= =
2 (and 4 for self-gravity)Gπ ρ= − Φ ∇ Φ =g
Ideal gas law and the Equation Of State
Governing equations for the gravitational field
The Lorentz Force in the MHD Approximation
( ) Pt c
ρ ρ∂ × + • = − + ∂ V J BV V + g
( )L
L 44
c
c
ππ
× = × ×
⇒ =× =
J BfB B
f
B J
The Lorentz Force in the MHD Approximation
( ) Pt c
ρ ρ∂ × + • = − + ∂ V J BV V + g
( )L
L 44
c
c
ππ
× = × ×
⇒ =× =
J BfB B
f
B J
( ) ( )4
Pt
ρ ρπ
× ×∂ + • = − + ∂
B BV V V + g
In the end, only the magnetic field appears in the eqn. of motion!