Warwick PX420 Solar MHD 2015-2016: Prominences
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Prominences
1. Phenomenology
A quiescent prominence is a cool, dense sheet of plasma in the corona. It is seen on
the disk as a thin dark filament lying above a magnetic polarity inversion line on the
photosphere. Being cool and dense it should not be in the hot and rarified coronal plasma
and it must be supported against the force of gravity by some mechanism. The magnetic
field is believed to be responsible for the support of prominences.
Typical prominence properties are compared with typical coronal properties in table:
Prominence properties
Coronal properties
Length L ≈ 200 Mm
Height h ≈ 50 Mm
Width w ≈ 6 Mm
T ≈ 5, 000 − 10, 000 K
T ≈ 1 − 3 × 106 K
ρ0 ≈ 2 × 10−10 kg m−3
ρ0 ≈ 2 × 10−12 kg m−3
pressure scale height Λ ≈ 0.6Mm
Λ ≈ 50Mm
Warwick PX420 Solar MHD 2015-2016: Prominences
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2. Support of Prominences
The simplest model of the internal structure of a prominence was suggested by Kippenhahn and Schlüter (1957). The temperature is assumed to be a constant T = T0 .
Since the width of a prominence is much shorter than the height and length, we assume
that inside the prominence we may neglect variations in the vertical direction and only
consider variations in the horizontal direction across the prominence. Thus,
B = (Bx0 , By0 , Bz (x)) ,
p = p(x),
ρ = ρ(x),
(1)
where Bx0 and By0 are constants. These function must satisfy the magnetostatic equations:
0 = −∇p + j × B + ρg,
(2)
coupled with
∇ · B = 0,
(3)
µj = ∇ × B,
(4)
p=
ρRT
,
µ̃
(5)
and T satisfies an energy equation.
The horizontal and vertical components of the force balance equation (2) are
Bz dBz
dp
=−
,
dx
µ dx
(6)
Bx0 dBz
= ρg.
µ dx
(7)
Since the temperature is constant we may use the gas law, (5) to eliminate the density in
favour of the pressure to obtain from (7)
Bx0 dBz
p
= ,
µ dx
Λ
(8)
where the pressure scale height, Λ = RT0 /µ̃g. Solving (6) we get
p+
Bz2
= constant.
2µ
(9)
To determine the constant we apply the boundary conditions that the pressure and density
tend to zero as we move away from the prominence and that Bz tends to a constant value
Bz0 . Thus,
Warwick PX420 Solar MHD 2015-2016: Prominences
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p→0
as |x| → ∞
|Bz | → Bz0 as |x| → ∞
(10)
(11)
(12)
Thus,
p=
1 2
Bz0 − Bz2 .
2µ
(13)
Substituting (13) into (8) gives
Bx0 dBz
1 2
=
Bz0 − Bz2 ,
µ dx
2µΛ
Z
dBz
x
=
+ constant,
2
2
Bz0 − Bz
2ΛBx0
Bz
x
1
tanh−1
+ C,
=
Bz0
Bz0
2ΛBx0
Bz0 x
Bz = Bz0 tanh
+C .
2Bx0 Λ
From symmetry at x = 0 we must have Bz (0) = 0 and this gives C = 0. Therefore,
Bz = Bz0 tanh
Bz0 x
.
2Bx0 Λ
(14)
and the pressure from (13) is
2
Bz0
Bz0 x
.
sech2
2µ
2Bx0 Λ
p=
(15)
Since the temperature is constant the density is given by the gas law as
2
µ̃ Bz0
Bz0 x
ρ=
sech2
.
RT0 2µ
2Bx0 Λ
The profiles of Bz and ρ are shown in figures:
(16)
Warwick PX420 Solar MHD 2015-2016: Prominences
Equation of the field lines for the Kippenhahn-Schlüter prominence model is
dx
dz
=
,
Bx
Bz
⇒
Z
Bz0
Bz0 x
tanh
dx = z + c,
Bx0
2Bx0 Λ
and so integrating we obtain
Bz0 x
2Λ log cosh
2Bx0 Λ
= z + c.
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Warwick PX420 Solar MHD 2015-2016: Prominences
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Note how the magnetic field lines are bent and the magnetic tension force opposes the
force due to gravity. In addition, the magnetic pressure is higher away from the centre
of the prominence and so there is a magnetic pressure acting towards the centre that
compresses the plasma and opposes the outward pressure gradient.
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3. Thermodynamics of Prominences
3.1 Radiatively-Driven Thermal Instability
Consider a plasma which is in equilibrium with temperature T0 and density ρ0 , under a
balance between
• mechanical heating of amount H = hρ per unit volume (where h is constant, and
• optically thin radiation of amount Lr = χρ2 T α (where χ and α are constant).
Also, we assume that the thermal conduction and the ohmic dissipation are negligible.
So, the energy loss function is
L = Lr − H = χρ20 T0α − hρ0 = 0,
(17)
(It is equal to zero because of the equilibrium condition).
For a perturbation at constant pressure, the energy equation becomes
cp
∂T
= h − χρT α ,
∂t
(18)
Also, we can express the density through the temperature to close the equation,
ρ=
mp0
.
kb T
(19)
So, Equation (22) of Lecture 2 becomes
∂T
T α−1
α
= χρ0 T0 1 − α−1 .
cp
∂t
T0
!
(20)
Thus, if α < 1,
a small decrease in temperature (T < T0 ) makes the right-hand side negative, so that
∂T /∂t and the perturbation continues:
this is the thermal instability.
α > 1:
α < 1:
Warwick PX420 Solar MHD 2015-2016: Prominences
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1
0.8
0.8
0.6
7
0.6
T/T_0
T/T_0
0.4
0.4
0.2
0.2
0
1
2 t/t_rad 3
0
5
4
1
2 t/t_rad 3
4
5
The time-scale (growth-time) of the instability is
τrad =
cp
.
χρ0 T0α−1
(21)
The growth-time (s) for thermal instability in a plasma of number density n0 (m−3 ) and
temperature T0 (K):
n0
1014
1015
1016
T0
105 5 × 105
440
2200
44
220
4.4
22
106
3.2 × 104
3.2 × 103
320
2 × 106
1.3 × 105
1.3 × 104
1.3 × 103
107
3.2 × 106
3.2 × 105
3.2 × 104
The thermal instability can be prevented by the efficient heat conduction (along magnetic
field lines).
Taking into account this effect, Eq. (18) becomes
∂T
1
= h − χρT α + ∇ · (κ|| ∇T ),
∂t
ρ0
where κ|| is the coefficient of thermal conduction parallel to the field,
cp
κ|| = κ0 T 5/2 .
(22)
(23)
Thus, if the length of a field line is L, the thermal conduction time is
τcon =
L2 ρ0 cp
5/2
κ0 T0
.
(24)
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When the length L is so small that
τcon < τrad ,
(25)
the plasma is stable. This condition can be rewritten as
7/2−α 1/2
κ
T
0
0


χρ20

L < Lmax =

.
(26)
Thus, when the length of magnetic field lines (say, in a coronal loop) exceeds the threshold
value Lmax , the plasma filling the loop becomes thermally unstable and cools down until
a new equilibrium is reached, with cooler temperature and, consequently, higher density:
condensation takes place.