Lecture 9 Prominences and Filaments
Filaments are formed in magnetic loops that hold relatively cool, dense gas suspended above
the surface of the Sun," explains David Hathaway, a solar physicist at the NASA Marshall Space
Flight Center. "When you look down on top of them they appear dark because the gas inside is
cool compared to the hot photosphere below. But when we see a filament in profile against the
dark sky it looks like a giant glowing loop -- these are called prominences and they can be
spectacular.
September 23, 1999
SoHO-EIT
H
(http://spaceweather.com/glossary/filaments.html)
Physics 777 Week 12
2004
Physics 777 Week 12
2004
Lecture 9
Prominences
Filaments  Disk
Prominences  Limb
Quiescent: high
Active Region: low
Quiescent prominence is a huge, almost vertical sheet of dense cool plasma, surrounded by a hotter and
rarer coronal environment.
T: 5,000 ~ 8,000 K
H: 60,000 ~ 600,000 km
: 1016 ~ 1017 m-3
Height: 15,000 ~ 100,000 km
Formation of Filament
Consider a hot plasma, with T0 , 0 and thermal equilibrium under a balance between heat h and
radiation  0 : 0 = h -  0
Perturbation from its equilibrium :
T
kN 2 T
cp
h
t
t2
m 0
kT
linearize :
T T0 T1 ,
0
1
T1
kN 2 T1
0
cp
T1
T
1
t
s2
Assume :
T
wt
2
i
s
l
perturbation vanishes at loop ends
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
w
Prominences
kH 4 2
2
0L
0
c p T0
If conduction is absent, w>0, plasma is thermally unstable
Presence of conduction stabilizes the plasma, provided
L
Lm
2
c p K T0
1
2
0
Formation in a loop:
Active Region prominence energy equation:
d
ds
k0 T5
2
dT
ds
T
h ,
mp
kT
If  or L is large, h is small, state of thermally non-equilibrium ensures, loops cool down to a new
equilibrium of prominence temperature.
S=L
n1 , T1
L
d
s
rd
S=0
T0, n0
D
Use force equation to derive T,  structure
dp
dR
R
d
B 2 Bz2
dR
2
2LB
twist
R Bz
B
2
R
Solutions are shown in Fig. 11.1 -------- formation of a cool
core, , T droops quicker in the core.
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
Formation in a coronal Arcade
When coronal pressure becomes too great, force-free equilibrium ceases to exist and plasma
cools to form a quiescent prominence
The arcade is in equilibrium under force balance:
0
j
B
dp
dz
p
0
Energy Equation:
j
B
jz
 to field
g
// To field
d
ds
k
dT
ds
k
B
dB dT
ds ds
2
z
L
x
a
B0 Cos
Linear field solution:
a
L
L2
x
By
1
B0 Cos
2 a2
L
z
x
a
Bz
B0 Sin
L
inclination
Sec 1
a L
T
h , k
k0 T5
2
Bx
Boundary condition:
n
2m
n0
5
1014 m
3
& T
dT
0 at the summit z
H,
ds
summit height H
ln Cos
x0
T0
z
a
106 K at base z
0
L
Modeling depends on 5 parameters 0, T0, h, L, . It is found that when 0 exceeds a
critical value ~ 1015 m-3, the plasma can not have a hot equilibrium --- cool down to form
prominence.
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
c decreases as L or  increases
Neglecting heating term, energy balance equation becomes:
0 T1
5 2
1
Solution has the form:
g
f T1
T1
T1
1 T1
H2
0
f T1
T0
T0
Exp
T1
5 2
T0
,
dp
dz
H T0
,
0 T1
kT
mg
0
2
0
T1 , g
g
2
2 T0
0,
H
T1
Fig. 11.4 shows the solution.
Formation in a current sheet:
For a T & , characteristic of lower corona, a neutral sheet becomes thermally unstable when
L > 100,000 Km. Horizontal force balance and thermal equilibrium:
p20
d
dy
B2
, p20
1
2
5 2 dT
2
0T
dy
k
m
T
20
T20
h
0
Physics 777 Week 12
2004
Lecture 9
Prominences
If heating balances radiation outside the sheet,
h
0 T20
5 2
h0 T205
T 1 , conduction term
1 x1
T1
T20
2
20 L
20
T20
1
1
T1
T1
2
T20
L2
1
0
Equations 11.18, 11.20, & 11.21 determine 20, B20, T20 in term of L and B. Fig. 11.7a shows that when L > Lmax, a
hot equilibrium condition does not exist, plasma cools down along a dotted line to a new equilibrium at
prominence temperature. E.g., at B = 1 G, Lmax = 50, 000 km --- height of quiescent prominence.
Colling time :
pressure balance: p2
p1
B2
, p2
2
Time dependent energy equation:
Cp
k
2T
m
T2
t
1
T1
2
T2
k0 T25
2
T1
T2
2
L2
Assume L = Lmax ( 1 +
), solution is shown in Fig. 11.7b.
T decreases slowly first, then drops suddenly.
, cooling time decreases. E.g.,
= 10-2.  ~ 105 sec ( 1 day )
Line – Tying :
During the condensation of plasma in a vertical current sheet, lorentz force will tend to oppose the transverse
motions because the magnetic field lines are anchored in the dense photosphere. The effect of line-tying is to
favour the formation of thin wedges.
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
Magnetohydrostatics of support in a simple arcade
Kippenhahn --- Schlüter model Fig. 11.9
Field lines are bowed down by dense plasma in prominence. Magnetic tension provides upward force to balance
gravity to support plasma; magnetic pressure increases with distance from z-axis to provide transverse force to
compress plasma and balance plasma pressure gradient
Force balance: 0
p
B2
g z
B
2
B
mp
,
kT
Assume Bx, By are uniform, Bz is a function of x.  x, z direction equations:
d
0
Boundary conditions:
Solutions: Bz
p
x
x
, p
0, Bz
Bz tanh
Bz
2
dx
p
B2
2
0, Bz
, 0
g
Bx dBz
dx
Bz
0
Bz x
2
Sech
Bx
2
Bz x
Homework: derive these
Bx
More complete treatment includes magnetic shear and heat transfer conclusion: prominence can not exist below
hmin. hmin  as Bx increases, so active region filaments are lower. Also, there exists a maximum share ~ 75° to 83°
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
External Fields
Fig. 11.2 shows a typical magnetic configuration of a prominence --- thin current sheet PLUS surrounding fields
which are potential in x-z plane.
0
x
0, z
H
0.
The problem is to solve 2 B
0
z
0, x
a
with boundary conditions: Bz =
f x
z
0, 0
x
a
Solution. Averaged lorenz Force Bx = g(z) x = 0, 0  z  H
FL = J Bx0. J = 2 Bzd / . Current flowing through prominence
Z
corona
Z=H
Prominence
X
FL > 0 for z > 17,000 km, can support a reasonable plasma mass of nd  1.8 x 1024 m-2.
MHD stability
Using energy principle, condition for stability: J
Current-free:
Bz
x
Bx
,
z
so
Bx
z
Bz
x
dBx
dz
0
w
0
0
Bz  with x, for stable configuration, fields must be concave upwards.
Physics 777 Week 12
2004
Lecture 9
Prominences
Helical structure
B has uniform Bx0, By0 and a pure azimuthal pinch field.
B
IR
2 a2
I
2 R
R
a
Bx0 x
x2
R
R
B
By0 y
B
z2
a
resulting field lines depend on the value of
C
I
2 a Bx0 C < 1, field has a dip; C > 1, closed field lines in x – z plane.
Support of current sheet
Fields are treated by vertical current sheet together with a current filament field ( Fig. 11.3 ) supporting force is
I2
the force of repulsion between two line current,
4
h
This force supports a prominence of mass m =  R2 
B 2
Balance between them:
I2
m g, so,
4
h
h
B
10 10 kg m3 , h
6G
, I
2B R
g
10, 000 m,
Support in a horizontal Field
B field has the form B
x = 0, z = h
A
z
, By ,
A
x
. Prominence has a radius of R0 and its axis is located at
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
Outside prominence, magmetic field is potential,
Boundary conditions: z
, Bx
B0 , Bz
z
0, Bz
2
A
0
0
0
BR, B continuous at ( y, z )
Solution: A
F0
F0
B0 z
z
F ln
2
Inside prominence:
J
z
A
R0 2
h
0
a
2
x2
a 2 x2
J R,
4F
1
1
0
h2
2
R2
R0 2
2
2
0
2
0
R2
2
R Cos
2
1 2
R0 2
A field component aling filament is necessary to produce prominence-like temperature.
Coronal Mass Ejections ( coronal transient )
mass 1015 g, energy up to 1032 ergs. Speed is 100 to 1,000 km/s
consequence: geomagnetic storms
solar energetic particles
may be related to filament eruptions and or flares.
Typical structure includes: Front, Cavity & Core.
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
They may have limb events or halo events ( Earth directed ).
A CME may produce magnetic cloud in interplanetary space.
They may cause coronal dimming.
Let’s discuss two simple models of CMEs. ( Fig. 11.15 )
Twisted loop Model
Longitudinal field Bl is surrounded by an azimuthal field Baz, speed of CME is constant.
2
Bl 2
nmGM
force balance: Baz
Rc
magnetic pressure
Rc
tension
R2
gravity
Conservation of longitudinal field: Be h2 = const.
``
of azimuthal
`` : Baz h R = const.
``
of mass
`` : n h2 R = const.
Also assume Baz / Bl = const.
Then: h ~ R, Rc ~ R, Bl ~ R-2
Background field in solar wind ~ R-3, so, CME magnetic field is dominant.
In a more general equation:
dR2
mM G
m
dt2
Fr
R2
,
2
Fr is lorentz force
r
r
1
L
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004
Lecture 9
Prominences
At certain twist  = c, CME speed is constant,
 > c, acceleration
 < c, deceleration
Untwisted loop Models
dr1 2
B2
D
dt2
Rc
B
Conservation of flux:
B2
Rc
GM
r2
r tag
1
tan
B 0 D0 2
D2
D0 2 r0 2
``
Bz
of mass
Bz0 r0
2
:
0
D2 r
2
r
This model also explain that CMEs are accelerated to a certain speed and then keeps constant speed.
More recent Models
S.T. Wu MHD model
J. Chen ejecting flux rope model
Magnetic clouds and Ace data.
Physics 777 Week 12
2004
Lecture 9
Prominences
Physics 777 Week 12
2004