lecture 9 prominences and filaments filaments are formed in magnetic loops that hold relatively...

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.

Physics 777 Week 12Physics 777 Week 12 2004 2004

September 23, 1999 SoHO-EIT H

(http://spaceweather.com/glossary/filaments.html)

Lecture 9 ProminencesFilaments 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 :

cp T

t h

kN

2 T

t2

m 0

k Tlinearize :

T T0 T1, 0 1

cp T1

t

0

T1

T1 kN

2 T1

s2

Assume :

T wt 2 i sl perturbation vanishes at loop ends

Lecture 9 Prominences



If conduction is absent, w>0, plasma is thermally unstable

Presence of conduction stabilizes the plasma, provided


w 0

cp T0

kH 4 2

0 L2

L Lm 2 cp KT0

0

12Formation in a loop:

Active Region prominence energy equation:dds

k0 T52 dTds

T h , m pkT

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 = 0T0, n0

d

D

rd

s

Ln1 , T1

S = LUse force equation to derive T, structured pd R

dd R

B2 Bz2

2 B2

R

R 2 L BR Bztwist

Solutions are shown in Fig. 11.1 -------- formation of a cool core, , T droops quicker in the core.



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 p j

B j z

j

B 0d pd z

g

to field

// To field

Energy Equation:ddskdTds k

BdBds

dTds

2 T h , k k0 T52

Linear field solution: Bx L a

B0 CosxL

za

By 1

L2

2 a2B0 Cos x

L za

Bz B0SinxL

za

inclination Sec1 aLBoundary condition: n

2m n0 5 1014 m3 & T T0 106K at basez 0

dTds

0atthesummitz H,

summitheight H lnCos x0LModeling 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.



c decreases as L or increases

Neglecting heating term, energy balance equation becomes:

Solution has the form:

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:

0T152 T0 T1

H2

1T1

,dpdz

g

1 0T0T1

Exp HT00 T1, 0 kTm g

fT1 gfT1 T1

52 T0 T1, g 022 2T00,

HT1

p20 1 B2

2 , p20

km20T20

ddy

0T52dTdy 2

T h 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:

Time dependent energy equation:

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.

If heating balances radiation outside the sheet,

h 1x1T1, conductionterm h0T20

52T1 T20L2

0T2052T1 T20

20L2 20 T20

11T1

1 0

p2 p1 B2

2 , p2

k 2Tm

Cp T2t

1T1

2T2

k0T252 T1 T2

2L2



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:

Assume Bx, By are uniform, Bz is a function of x. x, z direction equations:

Boundary conditions:

Solutions:

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°

0 p g z B22

B B, m p

k T

0 d

dxp

B2

2

, 0 g Bx

dBz

dx

x , p 0, Bz Bzx 0, Bz 0

Bz Bz tanhBz x

Bxp

Bz 2

2 Sech2

Bz x

BxHomework: derive theseHomework: derive these



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.

The problem is to solve .

with boundary conditions: Bz =

Solution. Averaged lorenz Force Bx = g(z) x = 0, 0 z H

FL = J Bx0. J = 2 Bzd / . Current flowing through prominence

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:

Current-free:

Bz with x, for stable configuration, fields must be concave upwards.

2 B 0 0 x 0, z H

0 z 0, x a

fxz 0, 0 x a

JdBxdz

0 w 0 Bz x

Bx z

, so Bx z

Bz x

0

X

coronaZ

Z = H

Prominence



Helical structure

B has uniform Bx0, By0 and a pure azimuthal pinch field.

resulting field lines depend on the value of

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 the force of repulsion between two line current,

This force supports a prominence of mass m = R2 Balance between them:

Support in a horizontal Field

B field has the form . Prominence has a radius of R0 and its axis is located at

x = 0, z = h

B

Bx0 x By0y

B

B I R

2 a2R a

I

2 RR a

R x2 z2

C I2 a Bx0 I2

4 h, I 2 B R

I2

4 h m g, so,

B2

h g

10 10 kg m3, h 10, 000 m,B 6 G

B

A

z, By,

A

x



Outside prominence, magmetic field is potential,

Boundary conditions:

BR, B continuous at ( y, z )

Solution:

Inside prominence:

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.

2A 0

z , Bx B0, Bz 0z 0, Bz 0

A F0 B0z

F0 F lnz a2 x2z a2 x2

2A JR, J

4 F1 2 02R2R021 02R2 2 02R Cos 2

0 h h2 R0212

R02



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.

force balance:

magnetic pressure tension 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:

Baz2

Rc

Bl2

Rc

n m G M

R2

mdR2

dt2 Fr

m MG

R2,

Fris lorentz force 2

r

r1L



At certain twist = c, CME speed is constant,

> c, acceleration

< c, deceleration

Untwisted loop Models

Conservation of flux:

`` of mass :

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.

dr12

dt2

B2

D

B2

Rc

G M

r2

Rc r tag

1 tan

B B0D02

D2

0D02 r02

D2rBz Bz0r0

2r2

lecture 9 prominences and filaments filaments are formed in magnetic loops that hold relatively...

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