The excitation of 5-min oscillations in the solar corona

0.19 MB·English

by  T. V. Zaqarashvili

Astronomy & Astrophysics manuscript no. manuscript (cid:13)c ESO 2011
January 21, 2011
5
The excitation of -min oscillations in the solar corona
T.V. Zaqarashvili,1,4 K. Murawski,2 M.K. Khodachenko,1 D. Lee3
1 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, 8042 Graz, Austria
e-mail: [email protected], [email protected]
2 Group of Astrophysics, Instituteof Physics, UMCS, ul. Radziszewskiego 10, 20-031 Lublin, Poland
e-mail: [email protected]
3 ASC/Flash Center, TheUniversity of Chicago, 5640 S. Ellis Ave,Chicago, IL 60637, USA
e-mail: [email protected]
4 AbastumaniAstrophysical Observatory at Ilia StateUniversity,Kazbegi ave. 2a, Tbilisi, Georgia
1
1 received / accepted
0
2 ABSTRACT

n Aims. Weaim to studyexcitation of theobserved ∼ 5-min oscillations in thesolar corona bylocalized pulses that are
a
launched in thephotosphere.
J Methods. We solve the full set of nonlinear one-dimensional Euler equations numerically for the velocity pulse propa-

0 gating in thesolar atmosphere that is determined bythe realistic temperature profile.
2 Results.Numerical simulations showthat aninitial velocity pulsequicklysteepensintoaleading shock,while thenon-
linearwakeinthechromosphereleadstotheformation ofconsecutivepulses.Thetimeintervalbetweenarrivalsoftwo

] neighboring pulses to a detection point in the corona is approximately 5 min. Therefore, the consecutive pulses may
R result in the ∼ 5-min oscillations that are observed in thesolar corona.
S Conclusions.The∼5-minoscillations observedinthesolarcoronacanbeexplainedintermsofconsecutiveshocksthat
. result from impulsive triggers launched within the solar photosphereby granulation and/or reconnection.
h
p Key words.Sun:atmosphere – Sun:oscillations
-
o
r
st 1. Introduction impulsive (e.g. granulation and/or explosive events due to
a magnetic reconnection). Both types of drivers may be re-
[ Propagating acoustic waves are frequently seen in the so- sponsible for the observed dynamical phenomena in upper

1 lar corona as periodic variations of spectral line intensity atmospheric regions.
(De Moortel et al. 2000, 2002, Marsh et al. 2003, Lin et
v As a result of the rapid decrease of mass density with
al. 2005, 2006, Srivastava et al. 2008, Wang et al. 2009).
2 height, finite-amplitude high-frequency photospheric oscil-
As these waves are often observed within the frequency
1 lations can quickly grow in their amplitudes and steepen
9 rangecorrespondingtotheacousticwavesinthesolarpho-
into shocks, which by energy dissipation can lead to the
3 tosphere/chromosphere, this logically leads to the idea of
chromosphericheating(Narain&Ulmschneider1990,1996,
. penetration of the photospheric acoustic oscillations into
1 Carlsson & Stein 1997, Ruderman 2006). Lower-frequency
the corona. However, the photospheric 5-min oscillations
0 waves, those with ∼5 min period, are not good candidates
are evanescent in the gravitationally stratified solar atmo-
1 for the chromospheric heating (Narain & Ulmschneider
1 sphereastheirfrequencyislowerthanthecut-offfrequency
1990).
: (Lamb 1908, Roberts 2004, Musielak et al. 2006). Bel &
v Leroy (1977) suggested that the cut-off frequency of the It was found by Hollweg (1982) that a localized pulse
i
X magneticfield-freeatmosphereislowerforwavespropagat- that is launched initially sets up a nonlinear wake which
ing obliquely to the vertical direction. De Pontieu et al. results in a trail of consecutive shocks. Such shocks were
r
a (2005) proposed that p-modes may be channeled into the called rebound shocks by Hollweg (1982).
solar corona along inclined magnetic field lines as a result The time interval between consecutive shocks is close
of the decrease of the acoustic cut-off frequency. McIntosh to the period of the nonlinear wake. In the linear case,
& Jefferies (2006) found the observational justification of the wake oscillates with the acoustic cut-off frequency
the modification of the cut-off frequency by inclined mag- of the stratified atmosphere. Nonlinearity modifies the
netic field. As the magnetic field of active region loops is wave period of the wake, with many features of spicules
predominantly vertical in the photosphere/chromosphere, which exhibit the periodicity of about 5-min (Murawski &
it is unclear how p-modes may penetrate into the coronal Zaqarashvili 2010). Then, these quasi-periodic shocks may
regions. leadtotheoscillatorydynamicsofcoronalplasma,whichis
There are two different types of drivers in the highly observed as intensity oscillations in coronal spectral lines.
dynamic solar photosphere: oscillatory (e.g. p-modes) and The aim of this paper is to study the role of rebound
shocks,whichare formed by an impulsive perturbation, on
Send offprint requests to: T. Zaqarashvili e-mail: the observed∼5-min oscillations in the solar corona.Here
[email protected] we consider the simplest hydrodynamic case, which can be
2 T.V. Zaqarashvili et al.: 5-min oscillations in thesolar corona
developed to more realistic magnetohydrodynamic model 2.3. Linear approximation
in future studies.
Neglecting all nonlinear terms in Eqs. (1)-(3) leads to the
This paper is organized as follows. The basic equations
classical Klein-Gordon equation (Rae and Roberts 1982,
and the atmospheric model are described in Sect. 2. The
Roberts 2004)
numerical model and results of numerical simulations of
impulsivephotosphericdriverarediscussedinSect.3.This ∂2Q ∂2Q
paper is concluded by a summary of the main results in −c2 +Ω2Q=0, (6)
∂t2 s ∂y2 c
Sect. 4.
where Q(t,y)=V(t,y) ̺0(0)c2s(0)/̺0(y)c2s(y) and
2. Basic model
c2 ∂Λp
2.1. Hydrodynamic equations Ω2(y)= s 1+2 . (7)
c
4Λ ∂y
(cid:18) (cid:19)
Our model system is taken to be composed of a
gravitationally-stratifiedsolaratmospherethatisdescribed Inthe caseofanisothermalatmosphere,the Klein-Gordon
by one-dimensional (1D) Euler equations: equation implements a natural frequency of the stratified
medium,Ωc =cs/2Λ,calledthecut-offfrequencyforacous-
∂̺ ∂(̺V)
+ =0, (1) ticwaves.Aphysicalinterpretationofthecut-offfrequency
∂t ∂y is that waves of lower (higher) frequencies than Ωc are
∂V ∂V ∂p evanescent (propagating). According to the theory devel-
̺ +̺V =− −̺g, (2)
oped for the linear Klein-Gordon equation the initially
∂t ∂y ∂y
launched pulse results in a leading pulse which propagates
∂p ∂(pV) ∂V
+ =(1−γ)p . (3) with the sound speed. This pulse is followed by the wake,
∂t ∂y ∂y which oscillates with Ωc and gradually decays as time pro-
Here ̺ denotes the mass density, V is a vertical compo- gresses (Lamb 1908, Rae and Roberts 1982). However, sit-
nent of the flow velocity, p = kB̺T/m is the gas pressure, uation is changed when one considers nonlinear terms in
γ =5/3 is the adiabatic index, g =272 m s−2 is the gravi- Eqs.(1)-(3).Then,alarge-amplitudeinitialpulsemaylead
tational acceleration, T is the temperature, m is the mean to consecutive shocks due to nonlinearity (Hollweg 1982).
particle mass and kB is Boltzmann’s constant. In the nonlinear regime, the wake may oscillate with the
period, which differs from the linear cut-off period. Then
theintervalbetweenconsecutiveshocksmaydependonthe
2.2. The equilibrium
amplitude of the initial pulse (Murawski and Zaqarashvili
We assume that at the equilibrium the solar atmosphere 2010). We solve the fully nonlinear equations for a realis-
is settled in a static (V = 0) environment in which the tic atmospheric model numerically and the results are pre-
pressure gradient force is balanced by the gravity, that is sented in the next section.
∂p0
− −̺0g =0. (4)
∂y 3. Numerical simulations for impulsively generated
With the use of the equation of state we obtain the equi- waves
librium gas pressure and mass density as
Equations(1)-(3)aresolvedwithauseofthe code FLASH
y dy′ p0(y) (Dubey et al. 2009). We set the simulation region as −0.5
p0(y)=p00 exp − Λ(y′) , ̺0(y)= gΛ(y). (5) Mm≤y ≤29.5Mmandatthenumericalboundarieswefix
Zyr ! intime allplasmaquantitiestotheirequilibriumvalues.In
Here Λ(y)=kBT(y)/(mg) is the pressure scale-height and ourstudiesweuseanadaptivemeshrefinement(AMR)grid
p00 is the gaspressureatthe referencelevel,chosenhereat with a minimum (maximum) level of refinement blocks set
yr =10 Mm. to 3 (9). In the simulations, we excite waves by launching
WeadoptarealistictemperatureprofileT(y)fortheso- at t=0 s the initial velocity pulse of a Gaussian profile
lar atmosphere (Vernazza et al. 1981). In this profile T at-
tainsavalueofabout5700Katthetopofthephotosphere (y−y0)2
whichcorrespondstoy =0.5Mm.AthigheraltitudesT(y) V(y,t=0)=Avexp − w2 (8)
(cid:20) (cid:21)
falls off until it reaches its minimum of 4350 K at y ≃0.95
Mm.HigherupT(y)growsgraduallywithheightupto the with Av the amplitude of the initial pulse, y0 its initial
transition region which is located at y ≃ 2.7 Mm. Here position and w its width. We set and hold fixed y0 = 0.5
T(y) experiences a sudden growth up to the coronal value Mm and w = 0.1 Mm, but allow Av to vary in different
of 1.5 MK at y = 10 Mm. Having specified T(y) we can simulations.
obtain mass density and gas pressure profiles with the use Attheinitialstageofthewaveevolutiontheinitialpulse
of Eq. (5). splits into counter-propagating waves. As a result of the
Equations(1)-(3)aretobesolvednumericallyforanim- rapiddecreaseofthe equilibrium massdensity, the upward
pulsiveperturbationlaunchedinitiallywithinthesolarpho- propagatingpulsegrowsinitsamplitudeandquicklysteep-
tosphere. The numerical simulations for a harmonic driver ens into a shock. Figure 1 illustrates the spatial profile of
were already reported elsewhere (e.g. Erd´elyi et al. 2007, V(y)att=250sforAv =1kms−1.Atthistime,thepulse
Fedun et al. 2009 and references therein). Therefore, it is reaches the level y = 23.5 Mm, while the secondary shock
justifiabletolimitourstudytothe caseofimpulsivelygen- beginstoformfromthenonlinearwakeaty =2Mm.Later
erated waves. on, this secondary shock propagates upwards, and then it
T.V. Zaqarashvili et al.: 5-min oscillations in thesolar corona 3
Fig.1. Velocity (in units of 1 km s−1) profile vs height y
(in units of 1 Mm) at t = 250 s for Av = 1 km s−1. Here
y = 0 Mm (y ≃ 2.7 Mm) corresponds to the base of the
photosphere (the transition region).
is followed by the next shock. This process repeats in time
until the perturbation energy finally subsides to zero.
Figure2displaysthe temporaldynamicsofthevelocity
that is collected at y = 20 Mm for the initial pulse ampli-
tude of Av = 1 km s−1 (top panel) and Av = 2 km s−1
(bottom panel). In the top panel, the arrival of the lead-
ing shock to the detection point occurs at t ≃ 250 s and
the second pulse reaches the detection point at t ≃ 570 s
i.e. after ∼320 s. Thus, the nonlinear wake in the chromo-
sphere excites quasi-periodic upward propagating pulses,
which enter the corona and may cause quasi-periodic in-
Fig.2. Time-signatures of V (in units of km s−1) collected at
tensityoscillationswhichareobservedintheimagingdata.
For Av = 1 km s−1, which is close to the typical granular Ayv==220kMmms−f1or(btohtetocmasepaonfeAl).vT=im1e iksmexsp−r1es(steodpinpaunneitl)s oafnd1
velocity,theintervalbetweenarrivalsofneighboringshocks
s.
is near 5-min (Fig. 2, top). This means that the nonlinear
wake in the realistic atmosphere and for the initial pertur-
bations, corresponding to the solar granular velocity, ex-
hibits about a 5-min wave period, i.e. it is longer than in should be θ ∼ 500 (cosθ ≈ 3/5). Therefore, the leakage of
the linear isothermal case (∼ 3-min in the chromosphere). p-modes may take place only in particular regions of the
−1
Time-signature for Av = 2 km s shows that the leading solar atmosphere, where the magnetic field is significantly
shock arrives to the detection point at t≃190 s, while the inclined in the chromosphere.
secondshockreachesthispointatt≃630s,i.e.after∼440
In this paper, we suggest an alternative mechanism to
s (Fig. 2, bottom panel). It is clearly seen that the interval
explainthe observedoscillationsin the solar corona,which
between arrival times of two consecutive shocks is longer
is based on the rebound shock model of Hollweg (1982).
for a larger amplitude of the initial pulse.
We numerically solved the full set of Euler equations for
the realisticVAL-C temperatureprofileandforaGaussian
velocity pulse launched within the photosphere. We found
4. Discussion and summary
that velocity pulses, originating from granules or magnetic
Frequentlyobserved∼5-minoscillationsinthesolarcorona reconnectioninthelowerregions,leadtodifferentresponses
are often explained by leakage of photospheric p-modes of the chromosphere/transition region than the periodic
alonginclinedmagneticfield.Verticallypropagatingacous- acoustic waves resulting from p-modes. It must be men-
ticwaveswith5-minperiodareevanescentduetothestrat- tioned, that the energy of granular motions is higher than
ificationofthe solaratmosphereas chromosphericacoustic the energy of p-modes, therefore the impulsively triggered
cut-off period is ∼ 3 min (Roberts 2004). But, acoustic- waves should have more power than those triggered by a
gravity waves have a smaller cut-off frequency when they periodic driver.
propagate with the angle about the vertical. This may al- The numerical simulations show that as a result of the
low the photospheric 5-min oscillations to channel along a rapiddecreaseoftheequilibriummassdensitytheinitialve-
stratified chromosphere and penetrate into the corona (De locity pulse quickly steepens into a shock.The shockprop-
Pontieuet al. 2005, Erd´elyiet al. 2007, Fedun et al. 2009). agates into the corona, while the nonlinear wake is formed
In order to increase the cut-off period from 3-min up to 5- in the chromosphere due to the atmospheric stratification.
min, the propagation angle (or magnetic field inclination) Thisnonlinearwakeleadstoconsecutiveshocksaswasfirst
4 T.V. Zaqarashvili et al.: 5-min oscillations in thesolar corona
shown by Hollweg (1982). The interval between the arrival forstrongerinitialpulsesbeing∼5-minfor1kms−1 inthe
times of two consecutive shocks depends on the amplitude photosphere. Then the rebound shock model predicts the
of the initial pulse; a stronger pulse leads to longer inter- longer period oscillations above the regions of strong gran-
vals. The initial pulse with the granular velocity of 1 km ular power. The granulation is suppressed in strong mag-
s−1 leads to ∼ 5-min intervals between consecutive shocks. neticfieldregionsofthephotosphere(e.g.sunspots),there-
Therefore, the quasi-periodic arrival of consecutive shocks foretheinitialpulsesshouldhavesmalleramplitudesthere,
inthesolarcoronamaycausetheintensityoscillationswith leading to the oscillation at near cut-off frequency. Indeed,
a period close to the interval between the shocks. the strong magnetic field regions (sunspots, magnetic net-
We implemented a simple 1D analytical model in order work cores) show predominantly 3-min oscillations in the
to avoid the propagation of acoustic oscillations with the chromosphere. Note that some observations also show the
angletothevertical.Weshowedthatpurelyverticallyprop- 3-min oscillations in the solar corona above sunspots (De
agating pulses may lead to quasi 5-min oscillations in the Moortel et al. 2002) and other magnetic structures (Lin
corona due to consecutive shocks. Therefore, it is not nec- et al. 2005). These oscillations can be excited as a conse-
essary to invoke the propagation along inclined magnetic quence of consecutive shocks due to chromospheric 3-min
field in order to explain the observed 5-min periodicity in oscillations.
the corona.
As a result, we expect that ∼ 5-min oscillations can
One dimensional propagation for acoustic waves is jus- be detected above the regions where the granular energy
tified for purely verticalmagnetic field. In this case,acous- is significant i.e. the quiet Sun and surroundings of active
tic waves are in fact slow magneto-acoustic waves for low regions/magneticnetwork.This is consistentwith observa-
plasma β ∼ c2s/vA2 < 1 and fast magneto-acoustic waves tions.Ontheotherhand,thedetectionof5-minoscillations
for high plasma β >1. Here vA is the Alfv´en speed. In the in the quiet corona is not an easy task due to a relatively
solar photosphere β is larger than unity, but it rapidly de- weak intensity of coronal lines. However,a careful analysis
creasesduetothemassdensityfalloffwithheight(andcon- still can be performed. But, one should keep in mind that
sequent increase of the Alfv´en speed). It becomes smaller the dynamicsofacousticwavesinthefield-freeregionsand
than unity in the chromosphere (Gary 2001) and tends to along the vertical magnetic field could be quite different
unitysomewherebetweenthephotosphereandthechromo- as it was discussed above. An initial acoustic pulse may
sphere and this surface should be thinner than the width be spread horizontally in field-free regions, while almost
of the chromosphere. The linear fast and slow magneto- the whole energy would be guided along the field lines in
acoustic waves are coupled near the level of β ∼ 1 when magnetic structures. Therefore, the amplitude of intensity
they propagate obliquely to the magnetic field (Rosenthal oscillations could be smaller in the quiet corona than near
et al. 2002, Bogdan et al. 2003). However, these waves re- active regions and chromospheric network cores. However,
main purely acoustic for parallel propagation unless the thedivergenceofmagneticfieldandincreasedthermalcon-
tube dispersive effects are taken into account. Therefore, duction may significantly weaken the amplitude of coronal
a pulse propagating along the vertical magnetic field may oscillations, which seem to be quite strong and non-linear
not feel the β ∼1 surface.On the other hand, the acoustic onFigs.1and2.Then,thestrongslowwavepulsesmaybe-
wake,whichisformedbehindthepulseandoscillatesalong comealmostlinearoncetheypenetrateintothecoronaasit
themagneticfield,mayleadtothenon-linearenergytrans- isseenbyobservations.Ournumericalmodelthenrequires
fer into Alfv´en waves near the β ∼ 1 region (Zaqarashvili the inclusion of magnetic field and the thermal conduction
and Roberts 2006, Kuridze and Zaqarashvili 2008). This (at least, in the coronal part of the atmosphere). This will
effect may be revealed as far as one includes the magnetic be done in future studies.
field into numericalsimulations.Then, a part of oscillation
Itshouldbenotedthatthegranularvelocitiesmaytake
energymaybetransformedintotransverseoscillationsnear
valuesbetween0.5-2kms−1 withpeakon1kms−1.Asthe
this region, but most of energy will remain in longitudi-
interval between consecutive shocks strongly depends on
nal oscillations due to thinness of β ∼1 region. Therefore,
theamplitudeoftheinitialpulse,thentheresultedcoronal
β ∼ 1 region may not significantly affect the formation
oscillationsmaytakevaluesbetween4-7minwithapeakon
of rebound shocks in the chromosphere and consequently
5-min. Indeed, the wavelet analysis of coronal line images
quasi-periodic acoustic oscillations in the lower corona.
obtainedby Hinode/EISshowthatthe oscillationpowerof
However, a two-dimensional consideration and inclusion of
coronaloscillationsisconcentratedattheperiodinarange
magneticfieldarenecessaryforthecompleteunderstanding
of4-6min(Wangetal.2009),whichisfullyconsistentwith
oftheproposedscenario.Thefirststepinthisdirectionwas
our theory.
already done by Murawski and Zaqarashvili (2010) in the
modelingofspiculeformation,buttheyconsideredasimple Oursimulationswereperformedforanisolatedpulsein
temperatureprofilecoveringonlythechromosphere-corona. ordertoshowclearlytheeffectofreboundshocks.However,
The plasma β was considered less than unity along the the solar photosphere is very dynamic, hence the initial
whole simulation region, therefore the effects of the β ∼ 1 pulse probably is followed by other pulses. The subse-
region were absent in these simulations. We intend to con- quent pulse coming from the photosphere may also trigger
siderthe reboundshockmodelinthecaseofmagneticfield the consecutive shocks in the chromosphere. The interac-
and realistic temperature profile in the future. tion between rebound shocks formed by different photo-
An important consequence of the rebound shock model spheric pulses may set up complex dynamics in the chro-
is that the intervalbetween consecutive shocks depends on mospheric plasma with immediate influence on the lower
the initial amplitude of pulse (see Fig. 2). The smaller am- corona.Therefore, the coronaloscillations may have broad
plitude pulses lead to shorter interval between consecutive spectrum as we already discussed in the previous para-
shockswithlowerlimitof3-min,whichisthelinearacoustic graph. On the other hand, if the mean interval between
cut-offperiodofthesolaratmosphere.Theintervalislonger subsequent initial pulses is close to the mean interval be-
T.V. Zaqarashvili et al.: 5-min oscillations in thesolar corona 5
tween consecutive shocks, then a resonance may occur in Lamb,H.,1908,Proc.LondonMath.Soc,1908,7,122
the chromosphere.This could be subject of future study. LinC.H.,Banerjee,D.,Doyle,J.G.,O’SheaE.,2005,A&A,444,585
The excitation of coronal oscillations due to leakage of LinC.H.,Banerjee,D.,O’SheaE.,Doyle,J.G.,2006,A&A,460,597
Marsh, M.S., Walsh, R.W., De Moortel, I., Ireland, J., 2003, A&A,
p-modes may occur only along significantly inclined mag-
404,L37
netic field inthe chromosphere(in orderto reduce the cut- McIntosh, S.W.,Jefferies,S.W.,2006,ApJ,647,L77
off frequency). The magnetic field lines, which are signifi- Murawski,K.,Zaqarashvili,T.,2010,A&A,519,A8
cantly inclined from the verticalin the chromosphere,may Musielak, Z.E., Musielak, D.E., Mobashi, H., 2006, Phys. Rev. E,
036612
not reach the corona at all. Therefore, the p-mode leak-
Narain,U.,Ulmschneider,P.,1990, SpaceSci.Rev.,54,377
age may have problems in real geometry of active region
Narain,U.,Ulmschneider,P.,1996, SpaceSci.Rev.,75,453
magnetic field. Onthe contrary,the reboundshockmecha- Rae,I.C.,Roberts,B.,2002,ApJ,256,761
nismmayworkinanygeometryofmagneticfieldincluding Roberts,B.,2004,InProc.SOHO13Waves,OscillationsandSmall-
the purely vertical field lines. Therefore, the excitation of Scale Transient Events in the Solar Atmosphere: A Joint View
from SOHO and TRACE, Palma de Mallorca, Spain, (ESA SP-
5-min oscillations in the solar corona by photospheric im-
547),1
pulsivedrivershaslargerareaofapplicationthanthatofby Rosenthal, C.S., Bogdan, T.J., Carlsson, M., et al. 2002, ApJ, 564,
p-modes. We believe that the future sophisticated models 508
may shed lighton the excitation ofcoronalacoustic waves. RudermanM.S.,2006,Phyl.Trans,R.Soc.A,364,485
Srivastava, A.K., Kuridze, D., Zaqarashvili, T.V., Dwivedi, B.N.,
Our conclusions are:
2008,A&A,481,L95
Vernazza,J.E.,Avrett,E.H.,Loeser,R.,1981, ApJ,45,635
(a) a velocity pulse that is initially launched at the photo-
Wang,T.J.,Ofman,L.,Davila,J.M.,2009,ApJ,696,1448
spheric level (due to granules or reconnection) quickly Zaqarashvili,T.V.,Roberts,B.,2006, A&A452,1053
steepensintoashockandcanpenetrateintothecorona,
whileanonlinearwakethatisformedbehindthisshock
leads to consecutive shocks in the chromosphere;
−1
(b) forthe initialphotosphericpulseamplitude of1kms
thetimeintervalbetweentwoconsecutiveshocksis∼5-
min;theconsecutiveshockspropagateupwardsandmay
cause the observed 5-min intensity oscillations in the
solar corona;
(c) the final conclusion is that the observed ∼ 5-min oscil-
lationsinthe solarcoronacouldbecausedbyimpulsive
photospheric perturbations (convection, reconnection)
not necessarily by p-modes.
Acknowledgements: The authors express their thanks
to the unknown referee for his/her constructive comments.
The work of TZ and MK was supported by the Austrian
Fond zur F¨orderung der Wissenschaftlichen Forschung
(project P21197-N16).The work of KM was supported by
the Polish Ministry of Science (the grant for years 2007-
2010). TZ was also supported by the Georgian National
Science Foundation grant GNSF/ST09/4-310. The soft-
ware used in this work was in part developed by the
DOE-supported ASC / Alliance Center for Astrophysical
Thermonuclear Flashes at the University of Chicago.
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