Issue |
A&A
Volume 510, February 2010
|
|
---|---|---|
Article Number | A71 | |
Number of page(s) | 12 | |
Section | Interstellar and circumstellar matter | |
DOI | https://doi.org/10.1051/0004-6361/200913565 | |
Published online | 12 February 2010 |
X-ray emitting MHD accretion shocks in
classical T Tauri stars![[*]](/icons/foot_motif.png)
Case for moderate to high plasma-
values
S. Orlando1 - G. G. Sacco2,1 - C. Argiroffi3,1 - F. Reale3,1 - G. Peres3,1 - A. Maggio1
1 - INAF-Osservatorio Astronomico di Palermo, Piazza del Parlamento 1,
90134 Palermo, Italy
2 - Chester F. Carlson Center for Imaging Science, Rochester Inst. of
Technology, 54 Lomb Memorial Dr., Rochester, NY, 14623, USA
3 - DSFA-Università degli Studi di Palermo, Piazza del Parlamento 1,
90134 Palermo, Italy
Received 28 October 2009 / Accepted 9 December 2009
Abstract
Context. Plasma accreting onto classical T Tauri
stars (CTTS) is believed to impact the stellar surface at free-fall
velocities, generating a shock. Current time-dependent models
describing accretion shocks in CTTSs are one-dimensional, assuming that
the plasma moves and transports energy only along magnetic field lines
(
).
Aims. We investigate the stability and dynamics of
accretion shocks in CTTSs, considering the case of
in the post-shock region. In these cases the 1D approximation is not
valid and a multi-dimensional MHD approach is necessary.
Methods. We model an accretion stream propagating
through the atmosphere of a CTTS and impacting onto its chromosphere by
performing 2D axisymmetric MHD simulations. The model takes into
account the stellar magnetic field, the gravity, the radiative cooling,
and the thermal conduction (including the effects of heat flux
saturation).
Results. The dynamics and stability of the accretion
shock strongly depend on the plasma .
In the case of shocks with
,
violent outflows of shock-heated material (and possibly
MHD waves) are generated at the base of the accretion column
and intensely perturb the surrounding stellar atmosphere and the
accretion column itself (therefore modifying the dynamics of the
shock). In shocks with
,
the post-shock region is efficiently confined by the magnetic field.
The shock oscillations induced by cooling instability are strongly
influenced by
:
for
,
the oscillations may be rapidly dumped by the magnetic field,
approaching a quasi-stationary state, or may be chaotic with no obvious
periodicity due to perturbation of the stream induced by the post-shock
plasma itself; for
the oscillations are quasi-periodic, although their amplitude is
smaller and the frequency higher than those predicted by
1D models.
Key words: accretion, accretion disks - magnetohydrodynamics (MHD) - shock waves - stars: pre-main sequence - X-rays: stars - instabilities
1 Introduction
In the last few years, high resolution (R=600) X-ray
observations of
several young stars accreting material from their circumstellar
disk (TW Hya, BP Tau, V4046 Sgr,
MP Mus and RU Lupi) suggested the
presence of X-ray emission at a temperature T=2-5
MK from plasma
denser than cm-3
(Kastner
et al. 2002; Robrade & Schmitt 2007;
Argiroffi
et al. 2007; Günther et al. 2006;
Argiroffi
et al. 2009; Schmitt et al. 2005).
The emitting plasma is too dense to be enclosed
inside coronal loop structures (Testa
et al. 2004). Several
authors therefore suggested that this soft X-ray emission component
could be produced by the material accreting onto the star surface,
flowing along the magnetic field lines of the nearly dipolar stellar
magnetosphere, and heated to temperatures of few MK by a shock at the
base of the accretion column (Lamzin 1998; Calvet
& Gullbring 1998).
The idea that an accreting flow which impacts onto a stellar surface leads to a shocked slab of material heated at millions degrees is not new. For instance, in the context of degenerate stars, several authors have studied the dynamics and energetic of accretion shocks and the effects of radiation on the formation, structure, and stability of the shocks (Imamura 1985; Chanmugam et al. 1985; Chevalier & Imamura 1982; Langer et al. 1981,1982). Hujeirat & Papaloizou (1998) have also investigated the role of the magnetic field in confining the post-shock accreting material and in determining the evolution of the accretion shock. These studies have shown that the role of thermal and dynamical instabilities is critical to our ability to model radiative shock waves.
The first attempt to interpret the evidence of soft X-ray emission from dense plasma in classical T Tauri stars (CTTSs) in terms of accretion shocks has quantitatively demonstrated that some non-coronal features of the X-ray observations of TW Hya (Günther et al. 2007) and MP Mus (Argiroffi et al. 2007) could be explained through a simplified steady-flow shock model. But, steady models are known to be an unsuitable approximation for radiative shocks, since they neglect the important effects of local thermal instabilities as well as global shock oscillations induced by radiative cooling (Sutherland & Dopita 1993; Dopita & Sutherland 1996; Sutherland et al. 2003b,a; Safier 1998; Mignone 2005).
In fact, the first time-dependent 1D models of radiative accretion shocks in CTTSs have shown quasi-periodic oscillations of the shock position induced by cooling (Sacco et al. 2008; Koldoba et al. 2008). In particular, Sacco et al. (2008) have developed a 1D hydrodynamic model including, among other important physical effects, a well-tested detailed description of the stellar chromosphere, and they also investigated the role of the chromosphere in determining the position and the thickness of the shocked region. For hydrodynamical simulations based on the parameters of MP Mus, Sacco et al. (2008) synthesized the high resolution X-ray spectrum, as it would be observed with the Reflection Grating Spectrometers (RGS) on board the XMM-Newton satellite, and found an excellent agreement with the observations.
Up to date time-dependent models of accretion shocks in CTTSs
have been
one dimensional, assuming that the plasma moves and transports energy
only
along magnetic field lines. This hypothesis is justified if the plasma
(where
pressure/magnetic pressure) in the
shock-heated material. The photospheric magnetic field magnitude of
CTTSs
is believed to be around 1 kG (e.g. Johns-Krull et al. 1999).
Such a
powerful stellar field is enough to efficiently confine accretion
shocks
with the particle density below 1013 cm-3
and the temperature
around 5 MK (
). However, recent
polarimetric measurements
indicate that in some cases the photospheric magnetic field strength
could be less than 200 G (Valenti
& Johns-Krull 2004), and the plasma
in the slab may be around 1 or even larger. In these cases,
the magnetic field configuration in the post-shock region may change,
influencing the physical structure of the material emitting in X-rays
and the stability of the accretion shock.
The low-
approximation was challenged by recent findings of
Drake et al. (2009).
In the case of
,
the radiative
shock instability is expected to lead to a detectable periodic
modulation
of the X-ray emission from the shock-heated plasma, if the density and
velocity of the accretion stream do not change over the time interval
considered (in agreement with predictions of 1D models). But, the
analysis of soft X-ray observations of the CTTS TW Hya (whose
emission
is believed to arise predominantly from accretion shocks) produced no
evident periodic variations (Drake
et al. 2009). These authors
therefore concluded that 1D models may be too simple to describe the
multi-dimensional structure of the shock, and that the magnetic field
may
play an important role through the generation and damping of
MHD waves.
In this paper, we investigate the stability and dynamics of
accretion
shocks for cases where the low-
approximation cannot be
applied. We analyze the role of the stellar magnetic field in the
dynamics and confinement of the slab of shock-heated material. To
this end, we model an accretion stream which propagates through the
magnetized atmosphere of a CTTS and impacts onto its chromosphere,
using
2D axisymmetric MHD simulations and, therefore, an explicit description
of the ambient magnetic field. We investigate cases of
in the
post-shock region for which the deviations from 1D models are
expected to be the largest. In Sect. 2 we describe the
MHD model
and the numerical setup; in Sect. 3 we describe the
results;
in Sect. 4
we discuss the implications of our results and
finally we draw our conclusions in Sect. 5.
2 MHD modeling
The model describes an accretion stream impacting onto the surface of a CTTS. We assume that the accretion occurs along magnetic field lines that link the circumstellar disk to the star and consider only the portion of the stellar atmosphere close to the star.
The fluid is assumed to be fully ionized with a ratio of
specific heats
.
The model takes into account the stellar magnetic field,
the gravity, the radiative cooling, and the thermal conduction
(including
the effects of heat flux saturation). Since the magnetic Reynolds
number
1
considering the typical velocity (107 cm s-1)
and
length scale (109 cm) of the system,
the flow is treated as an ideal
MHD plasma. The impact of the accretion stream is modeled by
numerically
solving the time-dependent MHD equations (written in
non-dimensional
conservative form):
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
where

are the total pressure, and the total gas energy (internal energy,









![[*]](/icons/foot_motif.png)

![]() |
Figure 1: Radiative losses for an optically thin plasma from the APED V1.3 atomic line database (Smith et al. 2001), assuming the metal abundances of 0.5 of the solar values (Anders & Grevesse 1989). |
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The thermal conductivity in an organized magnetic field is known to
be highly anisotropic, and it can be highly reduced in the direction
transverse to the field. The thermal flux is therefore locally
split into two components, along and across the magnetic field lines,
,
where (see,
for instance, Orlando
et al. 2008)
to allow for a smooth transition between the classical and saturated conduction regime. In Eq. (5),
![$[q_{\rm spi}]_{\parallel}$](/articles/aa/full_html/2010/02/aa13565-09/img32.png)
![$[q_{\rm spi}]_{\perp}$](/articles/aa/full_html/2010/02/aa13565-09/img33.png)
where
![$[\nabla T]_{\parallel}$](/articles/aa/full_html/2010/02/aa13565-09/img35.png)
![$[\nabla T]_{\perp}$](/articles/aa/full_html/2010/02/aa13565-09/img36.png)


![$[q_{\rm sat}]_{\parallel}$](/articles/aa/full_html/2010/02/aa13565-09/img39.png)
![$[q_{\rm sat}]_{\perp}$](/articles/aa/full_html/2010/02/aa13565-09/img40.png)
where



![]() |
Figure 2:
Initial geometry of the system in cylindrical coordinates. The stellar
surface lies on the r axis, and the
unperturbed stellar magnetic field is uniform and oriented along
the z axis (vertical lines). The accretion
stream propagates downwards through the stellar corona with a velocity |
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We solve the MHD equations using cylindrical coordinates in the plane
(r,z), assuming axisymmetry and
the stellar surface lying on the r axis
(see Fig. 2).
The initial unperturbed stellar atmosphere
is assumed to be magneto-static and to consist of a hot (maximum
temperature 106 K)
and tenuous (
cm-3)
corona linked through a steep transition region to
an isothermal chromosphere
that
is uniformly at temperature 104 K and
is
cm thick
(see Fig. 3).
The choice of an isothermal chromosphere is
adopted for ease of modeling, and different choices of more detailed
chromospheric models have been shown not to lead to significantly
different results (Sacco et al. 2010, in preparation). The
unperturbed
stellar magnetic field is uniform, oriented along the z axis
and perpendicular to the stellar surface. The gravity is calculated
assuming the star mass
and the star radius
,
which is appropriate for the CTTS MP Mus (see
Argiroffi et al. 2007).
Different choices of stellar parameters
should not lead to different results.
![]() |
Figure 3: Initial hydrogen number density (dashed line) and temperature (dotted line) as a function of height above the stellar surface z for the unperturbed stellar atmosphere. |
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The accretion stream is assumed to be constant and propagates along
the z axis. Initially the stream is in
pressure equilibrium with the
stellar corona and has a circular cross-section with a radius cm;
its radial density distribution is given by
![]() |
(8) |
where





The velocity of the stream along the z axis
also has a radial
profile:
![]() |
(9) |
where

The simulations presented here describe the evolution of the
system
for a time interval of about 3000 s. We used the accretion
parameters
(velocity and density) that match the soft X-ray emission of
MP Mus
(Argiroffi et al.
2007), namely cm-3and
km s-1
at a height
cm
above the stellar surface
.
The initial field strengths of
G
in the unperturbed stellar atmosphere
(see Table 1)
maintain
in the post-shock region,
where
is the ratio of thermal to magnetic
pressure. For instance,
G
corresponds to
ranging
between
104,
at the base of the chromosphere, and
10-2
up in corona. We also performed a 1D hydrodynamic
simulation to provide a baseline for the 2D calculations. This
simulation corresponds to the strong magnetic field limit (
), in which the plasma moves
and transports energy exclusively along
the magnetic field lines (see Sacco
et al. 2008).
Table 1: Relevant parameters of the simulations.
The calculations described in this paper were performed using PLUTO
(Mignone et al.
2007), a modular, Godunov-type code for
astrophysical plasmas. The code provides a multiphysics, multialgorithm
modular environment particularly oriented toward the treatment of
astrophysical flows in the presence of discontinuities as in the
case treated here. The code was designed to make efficient
use of massive parallel computers using the message-passing interface
(MPI) library for interprocessor communications. The MHD equations are
solved using the MHD module available in PLUTO
which is based on the
Harten-Lax-van Leer Discontinuities (HLLD) approximate Riemann solver
(Miyoshi & Kusano
2005). Miyoshi
& Kusano (2005) have shown
that the HLLD algorithm can exactly solve isolated discontinuities
formed
in the MHD system; consequently the adopted scheme is particularly
appropriate to describe accretion shocks. The evolution of the
magnetic field is carried out using the constrained transport method of
Balsara & Spicer
(1999) that maintains the solenoidal condition at
machine accuracy. PLUTO includes optically
thin radiative losses in a
fractional step formalism (Mignone
et al. 2007), which preserves the
time accuracy, as the advection and source steps are at least
of the
order accurate; the radiative losses
values are
computed at the temperature of interest using a table
lookup/interpolation
method. The thermal conduction is solved with an explicit scheme that
adds the parabolic contributions to the upwind fluxes computed at cell
interfaces (Mignone
et al. 2007). Such a scheme is subject to
a rather restrictive stability condition (i.e.
,
where
is the maximum diffusion coefficient),
as the thermal conduction timescale is generally shorter than the
dynamical one (e.g. Orlando et al. 2008;
Hujeirat
& Camenzind 2000; Orlando et al. 2005;
Hujeirat
2005).
The symmetry of the problem allows us to solve the MHD
equations in
half of the spatial domain with the stream axis coincident with the z axis.
The 2D cylindrical (r,z)
mesh extends between 0 and cm
in the r direction and between 0 and
cm
in the z direction and consists of a
uniform grid with
grid
points. Additional runs were done with setups identical to
those used for runs By-01, By-10, and By-50, but with a higher
resolution
(
grid points; see
Table 1)
to evaluate the
effect of the spatial resolution (see Sect. 3.4); the simulations
with higher resolution cover a time interval of about 1000 s.
We use fixed boundary conditions at the lower (z
= 0) boundary,
imposing zero material and heat flux across the boundary. With
this condition, matter may progressively accumulate at the base of the
chromosphere; we have estimated that this effect may become significant
on timescales a factor of 100 longer than than those explored by our
simulations. Axisymmetric boundary conditions
are imposed at r = 0 (i.e. along the symmetry axis
of the problem),
a constant inflow at the upper boundary (
cm)
for
,
and free outflow
elsewhere.
3 Results
3.1 One-dimensional reference model
The results of our 1D reference hydrodynamic simulation HD-1D are
analogous to those discussed by Sacco
et al. (2008): the impact of
the accretion stream onto the stellar chromosphere generates a reverse
shock, which propagates through the accretion column and produces a hot
slab. According to Sacco et al., the expected temperature and the
maximum thickness
of the slab are in the strong
shock limit (Zel'Dovich
& Raizer 1967):
where

Figure 4
shows the time-space plot of the temperature evolution
for run HD-1D. The base of the hot slab penetrates the chromosphere
(the dashed line in Fig. 4
marks the initial position of the
transition region between the chromosphere and the corona) down to
the position at which the ram pressure, ,
of the post-shock plasma equals the thermal pressure
of the chromosphere (Sacco
et al. 2008). As is evident from
the figure, the shock front is not steady, and the amplitude and period
of
the pulses rapidly reach stationary values as the initial transient
disappears. The shock position oscillates with a period of
600 s
due to intense radiative cooling at the base of the slab, which
robs the post-shock plasma of pressure support, causing the material
above the cooled layer to collapse back before the slab expands
again (see Sacco
et al. 2008, for a detailed description
of the system evolution). The maximum thickness of the slab is
cm,
in agreement with the prediction
(see Eq. (11)).
The post-shock plasma reaches the temperature
MK
during the expansion phase, and
MK
during the cooling phase.
As already mentioned, the evolution of the accretion shock
described
by the 1D reference model is appropriate if the magnetic field
lines
can be considered to be rigid to any force exerted by the accreting
plasma
flowing along the lines (
). In case of plasma
values
around 1 or even higher, the hot slab is expected to be only
partially
confined by the magnetic field, and flow can occur sideways because of
the strong pressure of the post-shock plasma and may ultimately perturb
the dynamics of the shock itself. Moreover, 2D or
3D structures leading
to fast cooling zones are expected to develop in the post-shock
plasma if the latter is characterized by
.
In this case,
Sutherland et al.
(2003a) showed that the evolution of 1D
and 2D radiative shocks may be significantly different.
![]() |
Figure 4: Time-space plot of the temperature evolution for run HD-1D. The spatial extent of the shock lies in the vertical direction at any instant in time. The dashed line marks the initial position of the transition region between the chromosphere and the corona. |
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3.2 Two-dimensional radiative MHD shock model
The cases of
are described by our 2D model. The main
differences of the evolution from the 1D model are expected to occur
at the border of the stream where the shock-heated plasma may not be
efficiently confined by the magnetic field; so we focus our attention
there.
Figure 5
shows the spatial distribution of temperature (on the
left) and plasma
(on the right) for runs By-01, By-10, and By-50
at time t=530 s (at early stage of
evolution). Movies showing the
complete evolution of 2D spatial distributions of mass density
(on the
left) and temperature (on the right) in log scale for runs By-01,
By-10,
and By-50 are provided as on-line material. The accretion flow follows
the magnetic field lines and impacts onto the chromosphere, forming a
hot
slab at the base of the stream with temperatures
5 MK
and
.
In all cases, the slab is rooted in the chromosphere,
where the thermal pressure equals the ram pressure, and part (
1/3) of the
shock column is buried under a column of optically thick
material and may suffer significant absorption. In runs By-01 and
By-10,
the dense hot plasma behind the shock front causes a pressure-driven
flow
parallel to the stellar surface, expelling accreted material sideways
(see upper and middle panels in Fig. 5; see also on-line
movies
for runs By-01 and By-10). This outflow strongly perturbs the shock
dynamics and is absent in the 1D shock model. As expected, this feature
determines the main differences between these 2D and the
1D simulations.
![]() |
Figure 5:
Temperature ( left panels) and plasma |
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In run By-01, the magnetic field is too weak to confine the post-shock
plasma (
at the border of the slab), and a
conspicuous amount of material continuously escapes from the border of
the accretion column at its base, where the flow impacts onto the
stellar
surface. The maximum escape velocity is comparable to the free-fall
velocity
,
and this outflow acts as an additional cooling
mechanism. The resulting outflow advects and stretches the magnetic
field
lines (see upper panels in Fig. 5), taking the
material away from
the accretion column and strongly perturbing the stellar atmosphere
even
at several stream radii. As a result of the outflow, a large component
of
perpendicular to the stream velocity (
)
appears in the post-shock region (see upper panels
in Fig. 5).
As discussed by Toth
& Draine (1993), the
presence of even a small magnetic field perpendicular to the flow can
stabilize the overstable oscillations. In fact, at variance with the
force due to gas pressure, the Lorentz force is not affected by cooling
processes (see also Hujeirat
& Papaloizou 1998), and this mechanism may
contribute to stabilize the shock oscillations (see the on-line movie).
![]() |
Figure 6: Temperature ( left panel) and mass density ( right panel) distributions in the (r,z) plane in log scale in the simulation By-10, at a time t=1800 s. The white lines mark magnetic field lines. |
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In the intermediate run By-10, the magnetic field is trapped at the
head
of the escaped material, leading to a continuous increase of the
magnetic
pressure and field tension there. As a result, the escaped material is
kept close to the accretion column by the magnetic field (at variance
with the By-01 case) and may eventually perturb the stream
itself. In
fact, the escaped material accumulates around the accretion column,
forming a growing sheath of turbulent material gradually enveloping
the stream. The magnetic pressure and field tension increase at the
interface between the ejected material and the surrounding medium,
pushing on the material and forcing it to plunge into the stream after
1.3 ks.
Figure 6
shows the temperature and mass density
distributions at a time t=1.8 ks, when the
expelled material has already
entered into the stream and has deeply perturbed the accretion column
(see also the on-line movie to follow the complete evolution). As
a result of the stream perturbation, the hot slab may temporarily
disappear altogether, as shown in Fig. 6. In this phase, the
region of impact of the stream onto the chromosphere is characterized
by a rather complex structure with knots and filaments of material and
may involve possible mixing of plasma of the surrounding corona with
accretion material.
In the By-50 case, the post-shock plasma is confined
efficiently by
the magnetic field, and no outflow of accreted material forms (see
lower panels in Fig. 5).
In particular, the plasma is
10-4
and the magnetic pressure is
at the stream border, where
is the ram
pressure. The 2D shock therefore evolves similarly to the 1D overstable
shock simulation, with alternating phases of expansion and collapse of
the post-shock region (see the on-line movie). The maximum thickness of
the slab is
cm,
i.e. less than in the
1D case by a factor of
1.8.
This result is analogous to that
described by Sutherland
et al. (2003a) for the case of
2D hydrodynamic
radiative shocks and is explained there as due to the formation of
denser
knots of more rapidly cooling gas. Note that, at variance with our
MHD simulations, the hydrodynamic model of Sutherland et al.
(2003a) does
not include the thermal conduction. In our simulations, the thermal
conduction acts as an additional cooling mechanism of the hot slab,
draining energy from the shock-heated plasma to the chromosphere, and
partially contrasts the radiative cooling. Depending on the temperature
of the shock and on the plasma
,
the thermal conduction may
therefore lead to significant differences between our results and those
of Sutherland
et al. (2003a).
Figure 7
shows snapshots of the evolution of temperature
distribution (in linear scale) in run By-50. The post-shock region gets
bigger (t=1.05 ks), reaches the maximum
extension (t= 1.15 ks),
and collapses (t= 1.25 ks); the cycle then
repeats and the slab
begins to get larger again (t= 1.35 ks).
Although the magnetic
field is powerful enough to confine the post-shock plasma, the
plasma
is slightly larger than 1 inside the slab, and the
1D approximation is not valid there. As in the
2D hydrodynamic simulations
(Sutherland et al.
2003a) therefore, complex 2D cooling structures,
including knots and filaments of dense material, form there due to
the thermal instability of the post-shock plasma (see t=1.15 ks
in
Fig. 7).
As discussed by Sutherland
et al. (2003a), these
2D complex structures lead to the zones cooling more efficiently than
those in 1D models (by virtue of the increased cold-hot gas boundary)
and consequently the amplitude of the oscillations is expected to be
reduced. It is worth noting that the similarity of our results with
those
of the hydrodynamic models is due to the fact that, because
in the slab in run By-50, 2D cooling structures can form. In the case
of
everywhere in the slab, we expect that the stream can be
considered as a bundle of independent fibrils (each of them describable
in terms of 1D models), and the 2D MHD simulations would produce the
same results as 1D models.
![]() |
Figure 7: Temperature distribution in the (r,z) plane in linear scale in the simulation By-50 at the labeled times. The white lines mark magnetic field lines. |
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We also studied the global time evolution of 2D shocks by
deriving
time-space plots of the temperature evolution analogous to that derived
for the 1D reference simulation (to compare directly
our 1D and 2D results). From the 2D spatial
distributions of temperature and mass
density, we first derive profiles of temperature along the z-axis
by
averaging the emission-measure-weighted temperature along the r-axis
for
cm (i.e. 80% of the
stream radius); then we derive the
time-space plots of the average temperature evolution from these
profiles.
![]() |
Figure 8: As in Fig. 4 for the 2D simulations By-01 ( top), By-10 ( middle) and By-50 ( bottom). The blue curve in the bottom panel marks the shock position derived in the 1D reference model, HD-1D. |
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Figure 8
shows the results for runs By-01, By-10, and By-50, and
can be compared with Fig. 4.
In all the cases, the amplitude
of the shock oscillations is smaller than the one observed in the 1D
reference model. In runs By-01 and By-10, the evolution of the 2D shock
is markedly different from that of the 1D shock (compare Fig. 4
with top and middle panels in Fig. 8): in run By-01, the
oscillations of the shock are stabilized after 2.5 ks and the
solution
approaches a quasi-stationary state; in run By-10, the oscillations
appear chaotic without an evident periodicity (at least in the time
lapse explored here). As discussed before, the stabilization of the
shock oscillations in the former case is due to a magnetic field
component perpendicular to the flow (Toth
& Draine 1993),
whereas the chaotic oscillations in the latter case are due to the
continuous stream perturbation by the material ejected sideways at the
stream base (see Fig. 6).
In the run By-50, the shock evolves
somewhat similarly to the 1D overstable shock simulation, showing
several
complete cycles with expansion and collapse of the post-shock region
(see
lower panel in Fig. 8).
However, as already discussed before,
some important differences arise: by comparing By-50 with HD-1D, the
oscillations are reduced in amplitude by a factor of 1.8 and
occur
at higher frequency (period
s).
3.3 Effect of stream parameters
The details of the shock evolution described in this paper depend on
the model parameters adopted. In particular, the temperature of the
shock-heated plasma is determined by the free-fall velocity with which
the
plasma impacts onto the star (Eq. (10)); the
stand-off height of
the hot slab generated by the impact depends on the velocity and
density
of the stream (see Eq. (11)).
Assuming a typical value for the
free-fall velocity of 400-500 km s-1
(leading to temperatures of
3-5 MK,
as deduced from observations), the maximum thickness of
the slab is determined only by the density of the stream: the heavier
the
stream, the thinner is the expected slab. Also the sinking of the
stream
in the chromosphere depends on the stream density and velocity at
impact. In fact, we find that the slab penetrates the chromosphere to
the position at which the ram pressure
of
the post-shock plasma equals the thermal pressure of the chromosphere
(see also Sacco
et al. 2008). As a result, heavier streams
are expected to sink more deeply into the chromosphere.
In the case of ,
the shock evolution is expected to be
influenced also by the stream radius
.
In fact, as shown
by our simulations, the complex plasma dynamics close to the border
of the stream (e.g. the generation of outflows of accreted plasma;
see runs By-01 and By-10) may affect the shock evolution in the inner
portion of the slab. The shock evolution is expected to be modified in
the whole slab if the oscillation period of the shock,
,
is
larger than the dynamical response time of the post-shock region, which
can be approximated as the sound-crossing time of half slab:
,
where
is the isothermal sound
speed. In the cases discussed in this paper, the oscillation period
is
s,
and the sound-crossing time of the slab is
s;
as shown by our simulations, the shock
evolution is modified in the whole slab. Assuming an accretion stream
with twice the radius considered here, we find
s,
and the region at the center of the slab should not be affected
by the plasma dynamics at the stream border. In this case, we expect
quasi-periodic shock oscillations (analogous to those described by run
By-50) at the center of the stream and a strongly perturbed shock at
the stream border.
3.4 Effect of spatial resolution
In problems which involve radiative cooling, the spatial resolution and the numerical diffusion play an important role to determine the accuracy with which the dynamics of the system is described. In particular, in the case of radiative shocks, we expect that the details of the plasma radiative cooling depend on the numerical resolution: a higher resolution may lead to different peak density and hence influence the cooling efficiency of the gas, and therefore also the amplitude and frequency of shock oscillations. In the simulations presented here, the thermal conduction partially contrasts the radiative cooling and consequently alleviates the problem of numerical resolution (see, for instance, Orlando et al. 2008).
To check if our adopted resolution is sufficient to capture
the basic shock evolution over the time interval considered, we
repeated simulations By-01, By-10, and By-50, but with twice the
spatial
resolution (runs By-01-HR, By-10-HR,
and By-50-HR). The efficiency of
radiative cooling is expected to be the largest in the model with
G,
because there is no loss of material through
sideway outflows. In fact, quasi-periodic oscillations of the accretion
shock occur (see Fig. 8)
and make this case adequate for a
comparison of different spatial resolutions. Since this case is one of
the most demanding for resolution, it can be considered a worst case
comparison of convergence.
![]() |
Figure 9: Evolution of the averaged shock-front location for runs By-50 (solid blue line) and By-50-HR (dashed red line). The dotted line marks the initial position of the transition region between the chromosphere and the corona. |
Open with DEXTER |
Figure 9
compares the evolution of the averaged shock-front
location for runs By-50 and By-50-HR. The main differences between the
simulations appear in the first bounce, which starts earlier and is
larger by a factor of 2
in By-50-HR than in By-50. The first
expansion of the hot slab occurs after that the stream has penetrated
the
chromosphere to the position where the ram pressure of the shock-heated
material equals the thermal pressure of the chromosphere. The first
bounce is a transient, therefore, and in fact its amplitude and width
are different from those of subsequent bounces in all the simulations
examined. On the other hand, Fig. 9 clearly shows that
apart from
the first bounce the results of the two simulations agree in general
quite
well, showing differences of <
.
We expect therefore that, with an
increase of the spatial resolution, the results of our simulations may
slightly change quantitatively (e.g. the amplitude of oscillations),
but not qualitatively.
4 Discussion
4.1 Distribution of emission measure vs. temperature of the shock-heated plasma
As discussed in the introduction, there is a growing consensus that two distinct plasma components contribute to the X-ray emission of CTTSs: the stellar corona and the accretion shocks. This idea has been challenged recently by Argiroffi et al. (2009), who compared the distributions of emission measure EM(T) of two CTTSs with evidence of X-ray emitting dense plasma (MP Mus and TW Hya) with that of a star (TWA 5) with no evidence of accretion (i.e. only the coronal component is present) and with the EM(T) derived from a 1D hydrodynamic model of accretion shocks (i.e. only the shock-heated plasma component is present; Sacco et al. 2008). They proved that the EM(T) of MP Mus and TW Hya can be naturally interpreted as due to a coronal component, that dominates at temperatures of T> 5 MK, plus a shock-heated plasma component, that dominates at T < 5 MK.
In case of shocks with ,
our simulations show that the
distributions of temperature and density at the base of the accretion
column can be rather complex (see, for instance, Fig. 7). It is
interesting therefore to investigate how the EM(T)
of the shock-heated
plasma changes with
and whether it is possible to derive a
diagnostic of the plasma-
in the post-shock region.
In order to derive the EM(T) distribution
of the accretion region from
the models, we first recover the 3D spatial distributions of density
and temperature by rotating the corresponding 2D distributions
around
the symmetry z axis (r=0).
The emission measure in the jth
domain cell is calculated as ,
where
is the particle number density in the cell, and Vj
is the cell volume. The EM(T) distribution is then
derived
by binning the emission measure values into slots of temperature; the
range of temperature [
]
is divided into 15 bins, all equal on a logarithmic scale (
).
Figure 10
shows the EM(T) distributions averaged over
3 ks
for runs By-01, By-10, and By-50 together with the average EM(T)derived
from our 1D reference model HD-1D (blue dashed lines). The
figure shows the EM(T) distributions of plasma with
a density
of cm-3
(black) and
cm-3
(red). The 2D MHD models have been normalized to
have an X-ray luminosity
erg
in the
band [0.5-8.0] keV,
in agreement with the luminosity derived from the
low-temperature (
)
portion of the EM(T) distribution
of MP Mus (Argiroffi
et al. 2009). This is obtained assuming
that
10
accretion streams similar to that modeled here are
present simultaneously. The 1D model has been normalized to match
the EM peak in By-50. Inspecting Fig. 10 we note that: i)
in all the cases, the EM(T) has a peak at
5 MK
and a shape
compatible with those observed in MP Mus and TW Hya
and attributed to
shock-heated material (e.g. Argiroffi
et al. 2009); ii) most of
the X-ray emission arises from shock-heated plasma with a density of
>1011 cm-3
regardless of the
(i.e. the most dense
component of the post-shock region dominates); iii) the slope of the
ascending branch of the EM(T) distribution is
comparable in runs
By-50 and HD-1D, and gets steeper for decreasing values of
.
The
time-averaged X-ray luminosity
derived from these EM(T) distributions ranges
between
erg
(run By-10) and
erg
(run By-01), with only a weak dependence on the plasma
.
![]() |
Figure 10:
Distributions of emission measure vs. temperature, averaged over
3 ks, for runs By-01, By-10, and By-50. Black (red) lines mark
the average EM(T) distributions of plasma with
density |
Open with DEXTER |
Our model consequently supports the idea that dense shock-heated plasma
may
contribute significantly to the low temperature portion of the EM(T)
distributions of CTTSs regardless of the value. We also
suggest
that the shape of the EM(T) could be used as a
diagnostic of
in
the post-shock region, if the coronal contribution to the low
temperature
tail can be neglected.
4.2 Mass accretion rates
Time-dependent models of radiative accretion shocks provide a
convincing
theoretical support to the hypothesis that soft X-ray emission from
CTTSs
arises from shocks due to the impact of the accretion columns onto the
stellar surface. But, several points remain still unclear. Among
these, the most puzzling is probably the fact that the mass accretion
rates derived from X-rays, M,
are consistently lower by one
or more orders of magnitude than the corresponding M
values derived
from UV/optical/NIR observations (e.g. Günther et al. 2007;
Argiroffi
et al. 2009; Schmitt et al. 2005;
Drake
2005, Curran et al. 2010,
in preparation). We have here the opportunity to discuss the problem
of the accretion rate, in particular focusing on the cases with
in the post-shock region.
The model parameters adopted in this paper describe a stream
with an
accretion rate of
yr-1.
According with the discussion in Sect. 4.1,
10 streams
are needed to match the soft X-ray luminosity of MP Mus. In
this case
the accretion rate is
yr-1,
which agrees nicely with that deduced from observations,
namely M
yr-1(Argiroffi et al. 2007).
Alternatively, the observed M
may be
reproduced by our model if the accretion stream has a larger cross
section (with a radius of
cm).
In this case,
we expect some changes to the dynamics of the shock-heated plasma as
described in Sect. 3.3.
On the other hand, the mass accretion rate of MP Mus,
as deduced
from optical observations, is M
yr-1
(Argiroffi et al.
2009), which
exceeds by more than one order of magnitude the value obtained from
X-rays. Similar discrepancies are found in all CTTSs for which it
is possible to derive M
(see also Curran et al. 2010,
in preparation). The discrepancy might be reconciled if M
values are
underestimated due for instance to absorption from optically
thick plasma. But, as we explain below, even assuming that the
absorption
can account for the observed M discrepancy, the
idea that the
same streams determine both M
and M
has to be
discarded. Indeed, assuming the accretion parameters adopted here, we
derive that
300
streams must be present to match M
or the stream should have a
cross section
cm2,
implying that in both cases
20% of the stellar
surface would have to be involved in accretion. None of these
hypotheses
is realistic, as the surface filling factor of hot spots due to
accretion is up to a few percent (e.g. Gullbring 1994; Hartmann
& Kenyon 1990; Bouvier & Bertout 1989;
Bouvier
et al. 1993; Drake et al. 2009;
Kenyon
et al. 1994 and references therein).
The above inconsistency can be removed if the mass density of
accretion
streams is higher than assumed here; for instance, for a density of
the stream cm-3
(a factor 50 higher than modeled), the mass accretion rate is
yr-1,
and few streams are needed to match
M
.
In this case, however, the model cannot explain the
lower values of density derived from X-ray observations (
cm-3;
Argiroffi et al. 2007).
A possible solution to remove the discrepancy in the M
values
(and in the values)
is that few accretion streams
characterized by different mass density values are present at the same
time: those with a density of
1011 cm-3
would produce
shocks that are visible in the X-ray band, leading to the observed
M
;
those with higher densities would produce shocks not
visible in the X-ray band and leading to the observed M
(being the dominant component
in the UV/optical/NIR bands). The reason
of the different visibility could be due to local absorption of the
X-ray emission, as explained below.
Our simulations show that the accretion stream penetrates
the chromosphere to the position at which the ram pressure of the
post-shock plasma equals the thermal pressure of the chromosphere (see
also Sacco
et al. 2008). Part of the shock-heated plasma is
buried in the chromosphere and is expected to be obscured by
significant
absorption from optically thick plasma (see also Sacco et al.
2010, in
preparation). In the simulations presented here, this portion is 1/3 of
the hot slab (
cm;
see Fig. 8),
and most of the post-shock plasma (above the chromosphere) is expected
to be visible with minimum absorption. Instead, in denser streams,
the shock column is buried more deeply in the chromosphere, due to the
larger ram pressure, and its maximum thickness is smaller, according
to Eq. (11)
(see Sect. 3.3).
Assuming the
accretion parameters adopted in this paper, but with
cm-3,
Eq. (11)
gives
cm,
that is much smaller than the expected sinking
of the stream in the chromosphere (
cm).
In this case, the post-shock column is buried under a hydrogen
column density
cm-2,
and the photoelectric absorption of the O VII
triplet
(i.e. the lines commonly used to trace the accretion in the X-ray band)
is given by
.
We conclude
accordingly that heavy streams may produce X-ray emitting shocks that
are still hardly visible in X-rays, because they are buried too deeply
in the chromosphere.
4.3 Variability of X-ray emission from shock-heated plasma
Another important point in the study of accretion shocks in CTTSs is the periodic variability of X-ray emission due to the quasi-periodic oscillations of the shock position induced by cooling, predicted by time-dependent 1D models. Still in the only case analyzed up to date, namely TW Hya, no evidence of periodic variations of soft X-ray emission (thought to arise predominantly in an accretion shock) has been found (Drake et al. 2009). This result apparently contradicts the prediction of current 1D models, and Drake et al. suggested that these models might be too simple to explain the 3D shock structure.
On the other hand, quasi-periodic shock oscillations are expected if the accretion stream is homogeneous and constant (no variations of mass density and velocity). But, there is substantial observational evidence that the streams are clumped and inhomogeneous (e.g. Bouvier et al. 2007,2003; Safier 1998; Gullbring et al. 1996). In these conditions, periodic shock oscillations are expected to be hardly observable.
The 2D MHD simulations presented here show that, even assuming
constant
stream parameters, periodic oscillations are not expected if in the
post-shock region (runs By-01 and By-10). In these cases the
time-space plots of temperature evolution (top and middle panels
in Fig. 8)
actually predict that the oscillations may be
rapidly dumped, approaching a quasi-stationary state with no
significant
variations of the shock position, or the variability may be chaotic
(with
no obvious periodicity) due to an intense perturbation of the stream by
the
accreted material ejected sideways. In none of these cases we therefore
expect to observe periodic modulation in the X-ray emission.
At the other extreme, for shocks with ,
we expect that the
single accretion stream is structured in several fibrils, each
independent
on the others due to the strong magnetic field which prevents mass and
energy exchange across magnetic field lines. Time-dependent
1D models
describe one of these fibrils. As they are independent from each other,
the
fibrils can be characterized by both a slightly different mass density
and
velocity, which would result in different instability periods, as also
by random phases of the oscillations. Drake
et al. (2009),
assuming that 1D models describe a single stream, derived the number
of streams needed to account for the absence of a periodic variability
in TW Hya and concluded that this number contrasts with the
presence
of conspicuous rotationally modulated surface flux with a small filling
factor. Following Drake
et al. (2009), but considering the fibrils
instead of the streams, we may argue that an accretion stream
consisting
of 200-300 different fibrils with a radius of
cm
and with different instability periods and random phases would produce
a signal pulsed at a level of less than 5% as measured in
TW Hya.
The intermediate situation is for shocks with
around 1. In this
case, our run By-50 shows quasi-periodic oscillations of the shock
position (see bottom panel in Fig. 8) and predicts
periodic
variations of the X-ray emission arising from a single stream. In
fact, the magnetic field is strong enough in this case to confine
the shock-heated plasma, but it is too weak to consider as valid the
1D approximation inside the slab that cannot be described as a
bundle of
fibrils. As a result, the shock oscillates coherently in the slab. As
noted by Drake
et al. (2009), in this case it is not possible to
reproduce the absence of the periodic modulation of X-ray emission
observed
in TW Hya, and we conclude that shocks with
do not
occur in this star.
It is worth to emphasize that further investigation is needed
to
understand to which degree the intermediate case described by run By-50
is
frequent and observable: an intensive simulation campaign is needed to
assess the range of
values leading to streams with quasi-periodic
oscillations; a systematic analysis of the variability of soft
X-ray emission should also be performed, considering a complete sample
of
CTTSs with evidence of X-ray emitting accretion shocks, to assess if
and when periodic variability is observed.
4.4 Effects of accretion shocks on the surrounding stellar atmosphere
In the case of shocks with ,
our simulations predict that
the stellar atmosphere around the region of impact of the stream can
be heavily perturbed by the impact, leading to the generation of MHD
waves and plasma motion parallel to the stellar surface: the
larger the
plasma
in the post-shock region, the larger the perturbation
of the atmosphere around the shocked slab. The resulting ejected flow
also advects the weak magnetic field in such a way that the conditions
for ideal MHD may break down, magnetic reconnection may be possible
and eventually a release of the stored energy from the magnetic field
may occur (which is not described by our model however, as it does not
include resistivity effects).
A possible effect of the perturbation of the stellar
atmosphere is that
shocks with may
contribute to the stellar outflow. In fact,
Cranmer (2008)
suggested that the MHD waves and the material
ejected from the stream (as in our runs By-01 and By-10) may trigger
stellar outflow and proposed a theoretical model of accretion-driven
winds
in CTTSs (see, also, Cranmer
2009). His model originates
from a description of the coronal heating and wind acceleration in the
Sun and includes a source of wave energy driven by the impact of
accretion
streams onto the stellar surface (in addition to the convection-driven
MHD turbulence which dominates in the solar case). The author found
that
this added energy seems to be enough to produce T Tauri-like
mass loss
rates. It would be interesting to assess how the different plasma-
cases
discussed in this paper contribute to the added energy.
4.5 Limits of the model
Our simulations were carried out in 2D cylindrical geometry, implying
that all quantities are cyclic on the coordinate .
This choice
is expected to affect some details of the simulations, but not our
main conclusions. In particular, adopting a 3D Cartesian
geometry, the
simulations would provide an additional degree of freedom for
hydrodynamic
and thermal instabilities, increasing the complexity of cooling zones
and
thus the cooling efficiency of the plasma in the post-shock region. As
a
result the amplitude of the shock oscillations might be slightly
smaller
in 3D simulations and the frequency slightly higher than that
observed
in 2D simulations.
As discussed in Sect. 3.3, some details
of our simulations
depend on the choice of the model parameters. For instance, the
temperature of shock-heated plasma, the stand-off height of the hot
slab and also the sinking of the stream in the chromosphere depend
on the stream density and the velocity at impact. The accretion
parameters adopted in this paper originate from the values derived
by Argiroffi et al.
(2007) for MP Mus; the cases presented here
are representative of a regime in which the shock-heated plasma has
a temperature of 5 MK
and part of the slab is above the
chromosphere, which renders it observable with minimum absorption. The
range of the magnetic field strength considered has been chosen
in order to explore shocks with
.
Nevertheless, our
results undoubtedly show that the stability and dynamics of accretion
shocks strongly depends on
,
and that a variety of phenomena not
described by 1D models arise, including the generation of
plasma motion
parallel to the stellar surface and MHD waves.
We wish to stress that our model does not include magnetic resistivity effects. The complex shock evolution described by our simulations shows that violent plasma flows may advect the magnetic field in a way that the conditions for ideal MHD may break down and magnetic reconnection may possibly occur (not described by our simulations). In particular, field lines at the stream border are squeezed close together and may reconnect, resulting in a change of the magnetic topology as well as in a release of magnetic energy. Since most of these events are expected to occur at the stream border, magnetic reconnection may play an important role in the dynamics and energetic of the ejected accreted plasma. For instance, the magnetic reconnection may trigger flares at the stream border outside the accretion column. This scenario does not contradict Reale (2009), who showed that it is unlikely that flares can be triggered in an accreting flux tube. This issue deserves further investigation in future studies.
5 Conclusion
We investigated the stability and dynamics of accretion shocks in
CTTSs,
considering the case of
in the post-shock region,
through numerical MHD simulations. To our knowledge, the simulations
presented here represent the first attempt to model
2D accretion
shocks that simultaneously include magnetic fields, radiative cooling,
and magnetic-field-oriented thermal conduction. Our findings lead to
several conclusions:
- 1.
- In all the cases, a hot slab of shock-heated material is
generated at the base of the accretion column due to the impact of the
stream with the chromosphere. In the case of shocks with
, violent outflows of shock-heated material and possibly of MHD waves are generated at the border of the hot slab, and they may perturb the surrounding stellar atmosphere. For shocks with
, the shock-heated plasma is efficiently confined by the magnetic field, and no outflow forms.
- 2.
- If the magnetic field is too weak to confine the shock-heated plasma, but is powerful enough to keep it close to the stream, the escaped accreted material may strongly perturb the accretion column, modifying the dynamics and stability of the shock itself.
- 3.
- The accretion shocks are susceptible to radiative shock
instability. The resulting shock oscillations strongly depend on the
plasma
: for
, the oscillations may be rapidly dumped by the magnetic field, approaching a quasi-stationary state, or may be chaotic with no obvious periodicity due to perturbation of the stream induced by the post-shock plasma itself; for
around 1 the oscillations are quasi-periodic with an amplitude smaller and a frequency higher than those predicted by 1D models.




We thank the referee for constructive and helpful criticism. We are grateful to Andrea Mignone and Titos Matsakos for their support in using the PLUTO code. PLUTO is developed at the Turin Astronomical Observatory in collaboration with the Department of General Physics of the Turin University. The simulations have been executed at CINECA (Bologna, Italy), and at the HPC facility (SCAN) of the Osservatorio Astronomico di Palermo. This work was supported in part by the EU Marie Curie Transfer of Knowledge program PHOENIX under contract No. MTKD-CT-2005-029768 and by Agenzia Spaziale Italiana under contract No. ASI-INAF I/088/06/0.
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Online Material
Footnotes
- ... stars
- Three movies are only available in electronic form at http://www.aanda.org
- ... measure
- Note that the radiative losses are dominated by emission lines in the temperature regime common to CTTSs and are set equal to zero for T< 104 K.
- ... chromosphere
- Note that the radiative losses are set to zero in the chromosphere to keep it in equilibrium.
- ... surface
- Note that the stream
velocity is
km s-1 when the stream impacts onto the chromosphere, due to gravity.
- ... conditions
- Variables
are symmetrized across the boundary, and both radial and angular
components of vector fields (
) change their sign.
- ... outflow
- Set zero gradients across the boundary.
- ...
resolution
- Note that in the case of simulations with higher
spatial resolution (very CPU time consuming), the time interval covered
was
1 ks instead of 3 ks to reduce the computational cost.
- ... luminosity
- The synthetic X-ray spectra in the band [0.5-8.0] keV have been derived with the PINTofALE spectral code (Kashyap & Drake 2000) with the APED V1.3 atomic line database (Smith et al. 2001), assuming the metal abundances adopted in the whole paper, namely 0.5.
All Tables
Table 1: Relevant parameters of the simulations.
All Figures
![]() |
Figure 1: Radiative losses for an optically thin plasma from the APED V1.3 atomic line database (Smith et al. 2001), assuming the metal abundances of 0.5 of the solar values (Anders & Grevesse 1989). |
Open with DEXTER | |
In the text |
![]() |
Figure 2:
Initial geometry of the system in cylindrical coordinates. The stellar
surface lies on the r axis, and the
unperturbed stellar magnetic field is uniform and oriented along
the z axis (vertical lines). The accretion
stream propagates downwards through the stellar corona with a velocity |
Open with DEXTER | |
In the text |
![]() |
Figure 3: Initial hydrogen number density (dashed line) and temperature (dotted line) as a function of height above the stellar surface z for the unperturbed stellar atmosphere. |
Open with DEXTER | |
In the text |
![]() |
Figure 4: Time-space plot of the temperature evolution for run HD-1D. The spatial extent of the shock lies in the vertical direction at any instant in time. The dashed line marks the initial position of the transition region between the chromosphere and the corona. |
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Temperature ( left panels) and plasma |
Open with DEXTER | |
In the text |
![]() |
Figure 6: Temperature ( left panel) and mass density ( right panel) distributions in the (r,z) plane in log scale in the simulation By-10, at a time t=1800 s. The white lines mark magnetic field lines. |
Open with DEXTER | |
In the text |
![]() |
Figure 7: Temperature distribution in the (r,z) plane in linear scale in the simulation By-50 at the labeled times. The white lines mark magnetic field lines. |
Open with DEXTER | |
In the text |
![]() |
Figure 8: As in Fig. 4 for the 2D simulations By-01 ( top), By-10 ( middle) and By-50 ( bottom). The blue curve in the bottom panel marks the shock position derived in the 1D reference model, HD-1D. |
Open with DEXTER | |
In the text |
![]() |
Figure 9: Evolution of the averaged shock-front location for runs By-50 (solid blue line) and By-50-HR (dashed red line). The dotted line marks the initial position of the transition region between the chromosphere and the corona. |
Open with DEXTER | |
In the text |
![]() |
Figure 10:
Distributions of emission measure vs. temperature, averaged over
3 ks, for runs By-01, By-10, and By-50. Black (red) lines mark
the average EM(T) distributions of plasma with
density |
Open with DEXTER | |
In the text |
Copyright ESO 2010
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