Thermal forces and the coronal helium abundance
V.H. Hansteen∗ , Ø. Lie-Svendsen∗† and E. Leer∗
∗
Institute of Theoretical Astrophysics, P.O. Box 1029, Blindern, NO–0315 Oslo, Norway
†
Norwegian Defence Research Establishment, P.O. Box 25, NO–2027 Kjeller, Norway
Abstract. The interaction between protons and minor ions in the chromosphere-corona transition region produces an upward
force on the minor ions and an enhanced coronal abundance. In this presentation we compare a “classical” hydrodynamical
model of a hydrogen – helium solar wind and a model based on a 16-moment fluid description where the heat flux is treated
in a self-consistent manner.
BACKGROUND
as noted earlier, the electric field is unimportant as long
as the electron temperature is on the order of 1MK and
because the frictional coupling between protons and α particles is very small when the ion temperatures are as
high.
Even so a Helium abundances on the order of 50%
would challenge our ideas of the solar corona and revisiting the problem is certainly justified. The Helium abundace increase in the corona in the models mentioned is
due the thermal force which in turn is a consequence of
the heat flux in the ion fluids. The classical description of
the ion heat flux has recently been shown to be incorrect
for conditions in the lower corona (e.g. [5]). In addition
it seems likely that the deposition of energy may be in
the moment perpendicular to the magnetic field, through
for example cyclotron resonance. This will decrease the
amount of ion heat flux that is brought down into the transition region. Both of these phenomena may have consequences for the thermal force between Hydrogen and
Helium in the transition region. This paper is a first attempt at a study of the behaviour of Helium in models
in light of a "better than classical" description of the ion
heat flux.
Another finding in the [4] paper was that the frictional
coupling between neutral Hydrogen and Helium in the
chromsophere had to be greater than nominal in order to
be able to pull any Helium into the corona/solar wind at
all. We will also revisit this problem.
The observed abundance variations of Helium, as
well as other elements, between the photosphere and
corona/solar wind have raised considerable interest.
A study of the separation mechanisms could possibly
reveal insight into processes controlling the presumably
magnetically open fast solar wind streams as well as the
partially closed source regions of the slow solar wind.
Helium is a special case in that it alone may not be
regarded as a minority speicies. Indeed previous studies
[1, 2] showed that a Helium rich corona could play an
important role in the solar wind acceleration mechanism
through the altered electric field set up the α -particles as
well as by the frictional coupling between protons and
α -particles. Later studies by [3], showed that the corona
could in certain cases become Helium rich. These studies
invoked the large thermal force between Helium and
Hydrogen in the transition region to transport Helium
into the corona.
In a follow up study the entire chromosphere, transition region, corona, solar wind system was considered
consistently ([4]). From these models it was concluded
that, indeed, it was often the case that the thermal force
produced coronae with a high Helium abundance. However, it was also found that the models which reproduced
the observations best were those that were characterized
by a low electron temperature and hence an insignificant
electric field. These "best fit" models derived their acceleration from the large pressure gradients that the direct
heating of ions results in. As such these models predicted
the high ion temperatures that were later observed by the
UVCS instrument on board SOHO but also that the role
Helium plays in the accelaration of the largely proton
wind was insignificant, even when Helium abundances
greater than 50% were found in the corona. This because,
NEW SIMULATIONS
In these new simulations we use the gyrotropic approximation to solve for the particle density, velocity, temperature and heat flux moments parallell and perpendicular to the magnetic field. The gyrotropic approximation
CP679, Solar Wind Ten: Proceedings of the Tenth International Solar Wind Conference,
edited by M. Velli, R. Bruno, and F. Malara
© 2003 American Institute of Physics 0-7354-0148-9/03/$20.00
620
to the 16-moment transport equations has been adopted
from [6]. We solve the equations for a plasma containing
eletrons, protons, singly ionized helium and α -particles
as well as neutral hydrogen and helium. We require
charge (quasi) neutrality and no current. The electron-ion
plasma is created dynamically, in part through photoionization of the neutral hydrogen and helium gas, in part
through collisional ionization in the transition region. For
each particle species s we get six conservation equations
for density, momentum, parallel and perpendicular temperature, and heat flux densities along the magnetic field
of parallel and perpendicular thermal motion:
Particle conservation:
∂ ns
∂t
δ ns
1 ∂
−
(ns us A) +
A ∂r
δt
=
Here k is Boltzmann’s constant, G is Newton’s gravitational constant, MS is the solar mass, ms and es are
the mass and charge of the particle (thus for electrons
ee = −e and for protons e p = e, where e is the elementary
charge), and r is the heliocentric distance. A is the area
of the radial flow tube (A ∝ 1/B where B is the strength
of the radial magnetic field) and E is the radial electric
field.
The parallel and perpendicular temperatures and heat
fluxes are related to the average temperature Ts and total
heat flux qs through
(1)
Particle momentum:
∂ us
∂t
k ∂ Ts kTs ∂ ns
∂ us
= −us
−
−
∂r
ms ∂ r
n s ms ∂ r
1 dA k
−
(T − Ts⊥ )
A dr ms s
GM
es
1 δ Ms
+ E− 2S +
ms
r
n s ms δ t
(2)
∂t
∂ Ts⊥
∂t
1 ∂ qs
∂ us
−
∂r
∂r
ns k ∂ r
1
1 dA qs 2 dA qs⊥
+
+
Q
−
A dr ns k A dr ns k ns k sm
1 δ Es
(3)
+
ns k δ t
∂T
1 dA
1 ∂ qs⊥
= −us s⊥ −
us Ts⊥ −
∂r
A dr
ns k ∂ r
1
2 dA qs⊥
+
Q
−
A dr ns k ns k sm⊥
1 δ Es⊥
(4)
+
ns k δ t
=
−us
∂ Ts
− 2Ts
∂t
∂ qs⊥
∂t
−us
∂ qs
=
(7)
(8)
In order to illustrate the type of models we expect to arrive at with this set of equations we have computed three
sets of models, all with roughly the same energy input
(coronal heating) per particle. Energy is inserted into the
temperature moment perpendicular to the magnetic field
and it is apportioned between protons and α -particles
such that protons receive 85% while α -particles receive
15% of the deposited energy. (Electrons are also heated
with 10 Wm−2 in the lower corona in all simulations).
∂ us
1 dA
− us qs
∂r
∂r
A dr
k2 ns Ts ∂ Ts δ qs +
(5)
−3
ms
∂r
δt
∂q
∂ us
1 dA
= −us s⊥ − 2qs⊥
− 2usqs⊥
∂r
∂r
A dr
k2 ns Ts ∂ T
s⊥
−
ms
∂r
δq 1 dA k2 ns Ts⊥
−
(Ts − Ts⊥ ) + s⊥ . (6)
A dr ms
δt
=
qs
1
(T + 2Ts⊥)
3 s
1
(q + 2qs⊥).
2 s
RESULTS
Particle heat flux:
∂ qs
=
The term δ ns /δ t in (1) is the rate of production (or
loss) of particles due to ionization and recombination.
The following particle destruction and creation reactions have been included: radiative recombinations from
[7], collisional ionization and three-body recombination
from [8], and photoionization which is roughly modelled
by simply assuming constant rates taken from [9].
The collision terms δ Ms /δ t, δ Es(⊥) /δ t, and
δ qs(⊥) /δ t contain both elastic and inelastic (ionization, recombination and charge exchange) collisions.
Charge exchange between neutral hydrogen and protons
is included using the coefficients found by [10]. The expressions for elastic collisions, as given by [6], are very
extensive. We therefore use much simplified expressions
for these.
Particle energy:
∂ Ts
Ts
− 4qs
•
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Models with radial (r 2 ) geometry. In these models
we insert 110 Wm−2 ; 10 Wm−2 in the inner corona
with a scale height of 14 Mm, the remaining 100
Wm−2 with a scale height of 0.5R S . The models differ in the amount of frictional coupling between
neutral hydrogen and neutral helium in the chromosphere; the three models presented have 10× nominal, 5× nominal and nominal (e.g. [11]) coupling.
The radial model with only nominal coupling in the
FIGURE 1. Rapidly expanding ( fmax = 5) geometry and 5× nominal coupling between neutral hydrogen and neutral helium.
All of the energy deposited in the corona (500 Wm−2 ) is dissipated with a large scale height (0.5RS ), in addition 25% of this
energy is put into the perpendicular moment of the α -particle temperature as opposed to 15% for all other models. The proton flux
np up = 1.75 × 1012 m−2 s−1 , while the α -particle flux nα uα = 1.03 × 1010 m−2 s−1 at 1 AU which is 5.9% of the proton flux.
a small amount, 10 Wm −2 , in the electron fluid),
but rather deposit all 500 Wm −2 with a scale height
of 0.5RS. Here, again, we vary the frictional coupling between neutral hydrogen and neutral helium
in the chromosphere using 10× nominal, 5× nominal coupling. But we also vary the percentage of energy deposited in protons and α -particles. A model
where α -particles receive 25% of the deposited energy is presented.
chromosphere is not shown, as in that model helium settled gravitationally and almost no helium
was pulled into the corona or solar wind.
(For reference we find a proton flux n p up = 4.78 ×
1012 m−2 s−1 , while the α -particle flux is n α uα =
1.3×108m−2 s−1 at 1 AU; 2.7×10 −3% of the proton
flux in that model.)
• Models with rapidly expanding geometry (expansion factor 5 as defined by [12]) and with coronal heat inputs of 500 Wm −2 ; 50 Wm−2 in the inner corona with a scale height of 14 Mm, the remaining 450 Wm −2 with a scale height of 0.5R S . A
small amount, 10 Wm −2 , is inserted into the electron fluid. Again the difference between the models
lies in the coupling factor between neutral hydrogen and neutral helium in the chromosphere, using
the same values as above.
• Models with rapidly expanding geometry where we
do not deposit energy in the inner corona (except
The various models are summarized in Table 1 while a
specific example with properties close to those observed
in high speed streams is plotted in Figure 1.
CONCLUSIONS
We can vary the behaviour of the models by changing
four parameters: The ratio of the energy dissipated to
the expansion factor, the amount of energy dissipation in
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TABLE 1. Summary of the runs shown above. Below fexm is the geometrical expansion factor of the flow tube (1 is radial), Fm0 is the total energy (including that in the
inner corona) deposited in the corona, Fm,inner the energy deposited in the inner corona.
The two next columns signify the collisional coupling between neutral hydrogen and
neutral helium in the chromosphere and the percentage of deposited energy put into the
α -particles. For comparison we have included the equivalent data for the old ‘standard’
model in the last row of the table.
fexm
Fm0
Wm−2
Fm,inner
Wm−2
H i – He ii
coupling
heating %
α -paricles
np up
m−2 s−1
nα uα
m−2 s−1
1
1
1
5
5
5
5
5
5
110
110
110
500
500
500
500
500
500
10
10
10
50
50
50
0
0
0
10×
5×
nominal
10×
5×
nominal
10×
5×
5×
15
15
15
15
15
15
15
15
25
2.99 × 1012
3.95 × 1012
4.78 × 1012
2.32 × 1012
2.43 × 1012
2.85 × 1012
1.57 × 1012
1.72 × 1012
1.75 × 1012
1.31 × 1011
1.05 × 1011
1.30 × 108
1.72 × 1011
1.63 × 1011
3.10 × 1010
1.11 × 1011
1.00 × 1011
1.03 × 1011
1
100
0
15×
15
2.02 × 1012
1.07 × 1011
The α -particle speed, but little else, is set by the
percentage of energy dissipated into said α -particles.
the inner corona, the frictional coupling between helium
and hydrogen in the chromosphere, and the percentage
of energy going into α -particles.
The first number more or less sets the total mass
flux from the model. This is modified by the amount
of heating that occurs in the inner corona: A greater
percentage of heating there leads to a greater heat flux
in the transition region and thus a denser corona and a
slower, denser wind. Likewise, if heating is concentrated
in the outer corona or if the expansion factor is large,
heat flow back to the transtion region is inhibited and a
tenuous, faster wind arises.
Due gravitational settling helium drains out of the upper chromosphere on a timescale of hours unless the frictional coupling between hydrogen and helium is (artificially) increased. On the sun some process (such as turbulent motions?) must hinder this settling. Note that the
helium flux in the models above is set by this number
more or less alone. Note also that helium is not a minority species in these models and the proton flow is modified by the prescence of helium; i.e. helium ‘holds’ the
protons back, through frictional coupling in the lower
corona. This effect is less pronounced in the rapidly expanding models with no lower corona heating and indetecable in the old standard model where coronal heating took place quite far out in the corona. Note also that
the coronal abundance of Helium found in these models varies greatly, from roughly 20% found in the model
presented in Figure 1 up to greater than 80% in the radial
model with 10× frictional coupling in the chromosphere
and with 15% of the alloted energy deposited in the coronal α -particle fluid.
ACKNOWLEDGMENTS
This work was supported in part by teh Research Council
of Norway under grants 121076/420 and 136030/431.
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