**FULL TITLE**
ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION**
**NAMES OF EDITORS**
Dynamic hydrogen ionization in simulations of the solar
chromosphere.
Jorrit Leenaarts & Sven Wedemeyer-Böhm
Sterrekundig Instituut, Postbus 80 000, NL–3508 TA Utrecht, The
Netherlands
Kiepenheuer Institut für Sonnenphysik,Schöneckstrasse 6, D–79104
Freiburg, Germany
Abstract.
Since the assumption of statistical equilibrium does not hold under
the conditions of the dynamical solar chromosphere, the time dependence of the
rate equations has to be taken into account when calculating ionization stages of
elements. We present a method based on the work by Sollum (1999) to calculate
the dynamic hydrogen ionization degree and electron density in the 3D radiationhydrodynamics code CO5BOLD. In our model chromosphere, both quantities
are more constant over time and horizontal position than LTE theory predicts.
We compare synthetic brightness temperature images at λ = 1 mm calculated
with LTE and time-dependent NLTE electron densities. Both formation height
and average brightness temperature change significantly compared to LTE when
using time-dependent electron densities.
1.
Introduction
Carlsson & Stein (2002) showed in 1D simulations that the hydrogen ionization
degree in the chromosphere is far from equilibrium and found that it is relatively
constant at a level set by the temperatures in the shock waves. Any realistic
chromospheric model should take this into account as the ionization degree has
an effect on the hydrodynamics and the radiative cooling of the atmosphere. We
apply an approximate method to calculate the ionization degree to 2D and 3D
simulations of the solar chromosphere.
2.
The method
We solve the time-dependent rate equations
l
l
X
X
∂ni
+ ∇ · (ni~v ) =
nj Pji − ni
Pij ,
∂t
j6=i
j6=i
n
n
(1)
where ni is the number density of level i of the hydrogen atom, ~v the flow velocity
and Pij = Cij +Rij is the sum of the collisional and radiative rates from level i to
j. The radiative rates are fixed, calibrated such that they approximate the full
solution in dynamical 1D radiation-hydrodynamics simulations (Sollum 1999).
The collisional rates depend on the local temperature and electron density. We
allow no radiative transitions from the ground level into any other level.
1
2
Leenaarts and Wedemeyer-Böhm
Figure 1. Effect of time dependence in the upper part of the 2D simulation.
First row: LTE ionization degree (left) and time-dependent NLTE ionization
degree (right). Second row: LTE electron density (left) and time-dependent
NLTE electron density (right). Third row: gas temperature, which is the
same for the LTE and the time-dependent case.
The temperature and mass density come from the CO5 BOLD code (Wedemeyer et al. 2004). Currently there is no back-coupling to the hydrodynamics,
but we include advection of the hydrogen populations and the electron density.
We performed a 2D computation that ran for 1800 s of solar time and a 3D
computation that ran for 3900 s of solar time. Both simulations extended up
to 1500 km above average optical depth unity. The hydrogen population and
electron density were initialized with LTE values. After about 600 s of solar
time the simulations reach a statistically stable state.
3.
Results
Figure 1 shows a snapshot from the 2D simulation after 1640 s. Clearly, in the
time-dependent case the hydrogen ionization degree and the electron density are
much more constant than in LTE. The high temperature shocks, that show up
as streaks of increased electron density and ionization degree in LTE are barely
visible in time-dependent NLTE. The ionization degree tends to be at a level set
by the temperature in the shocks. On average the ionization degree is 1.1% at
1.5 Mm above τ500 = 1. In Table 1 we compare some statistical parameters of
the ionization degree in LTE and time-dependent NLTE computations.
Figure 2 shows brightness temperature images computed from a 3D snapshot at a wavelength of 1 mm. At this wavelength the extinction is dominated
by H I free-free processes which are strongly dependent on the electron density
Dynamic hydrogen ionization
3
Figure 2. Brightness temperature at 1 mm computed with the RH code
(Uitenbroek 2001) for a snapshot from the 3D simulation. Left: Simulation
snapshot with LTE electron densities. Right: same snapshot, but with timedependent NLTE electron densities. The black line indicates the position of
the cut in Fig. 3.
Table 1. Comparison of statistical properties of the ionization degree
np /nHtot at z = 1.5 Mm in the 2D simulation.
np /nHtot
average
minimum
maximum
std. dev.
LTE
td-NLTE
0.06
0.011
1.2 × 10−26
3.2 × 10−4
0.99
0.03
0.17
0.005
and thus on the ionization degree of hydrogen. The effect of the NLTE electron
densities are clearly visible. The peak brightness tends to be lower and the dark
areas are slightly darker in the time-dependent case. The average brightness
temperature is 4086 K (standard deviation 916 K) for the LTE and 3661 K
(standard deviation 725 K) for the time-dependent case.
Figure 3 shows the formation mechanism of the intensity in a cut through
Fig. 2. Compared to LTE the time-dependent extinction coefficient is lower
in shocks and higher in cool areas. Owing to the slow rates, the hydrogen
populations lag behind the fast fluctuations in temperature. As a result, the
horizontal variation in extinction coefficient is mainly set by the density. This
gives rise to a smaller variation in formation height, and sometimes to a doublepeaked contribution function.
4.
Conclusions & Outlook
We have implemented a fast approximate way of computing time-dependent
hydrogen ionization in a radiation-hydrodynamics code. We have shown that
4
Leenaarts and Wedemeyer-Böhm
Figure 3. Vertical cut through the snapshot shown in Fig. 2. Top left:
intensity at 1 mm in time-dependent NLTE (solid) and LTE (dashed). Top
right: Gas temperature with average formation heights in time-dependent
NLTE (solid) and LTE (dashed). Middle row: contribution function in LTE
(left) and NLTE (right), with average formation heights overplotted. The
brightness scale has been reversed and clipped to enhance visibility. Bottom row: extinction coefficient in LTE (left) and NLTE (right), with average
formation heights overplotted.
Dynamic hydrogen ionization
5
the time-dependent ionization degree and the electron density in our simulated
chromosphere are much more constant than in LTE. As an application we have
computed the the brightness temperature of the emergent millimeter radiation in
our simulation. In the time dependent case, the average brightness temperature
is 10% lower than in LTE. In future work we will take back-coupling of the
ionization to the hydrodynamics into account.
Acknowledgments. The authors would like to thank O. Steiner and J. Bruls
for illuminating discussions. Jorrit Leenaarts recognizes travel support from the
Deutscher Akademischer Austauschdienst and hospitality at the Kiepenheuer
Institut für Sonnenphysik. Sven Wedemeyer-Böhm acknowledges support by
the Deutsche Forschungsgemeinschaft, project STE G15/5
References
Carlsson, M. & Stein, R. F. 2002, ApJ, 572, 626
Sollum, E. 1999, Master’s thesis, University of Oslo
Uitenbroek, H. 2001, ApJ, 557, 389
Wedemeyer, S., Freytag, B., Steffen, M., Ludwig, H.-G., & Holweger, H. 2004, A&A,
414, 1121