The importance of magnetic-field-oriented thermal conduction in the interaction of SNR shocks with interstellar clouds

0.41 MB·English

by  S. Orlando

Draftversion February2,2008
PreprinttypesetusingLATEXstyleemulateapjv.08/13/06
THE IMPORTANCE OF MAGNETIC-FIELD-ORIENTED THERMAL CONDUCTION IN THE INTERACTION
OF SNR SHOCKS WITH INTERSTELLAR CLOUDS
S. Orlando1, F. Bocchino1
INAF-OsservatorioAstronomicodiPalermo“G.S.Vaiana”,PiazzadelParlamento1,90134Palermo,Italy
F. Reale2,1, G. Peres2,1 and P. Pagano
Dip. diScienzeFisiche&Astronomiche,Univ. diPalermo,PiazzadelParlamento1,90134Palermo,Italy
Draft versionFebruary 2, 2008
8 ABSTRACT
0
We explore the importance of magnetic-field-oriented thermal conduction in the interaction of su-
0
pernova remnant (SNR) shocks with radiative gas clouds and in determining the mass and energy
2
exchange between the clouds and the hot surrounding medium. We perform 2.5D MHD simulations

n ofa shockimpacting onanisolatedgascloud,includinganisotropicthermalconductionandradiative
a cooling; we consider the representative case of a Mach 50 shock impacting on a cloud ten-fold denser
J
than the ambient medium. We consider different configurations of the ambient magnetic field and

9 compareMHDmodelswithorwithoutthe thermalconduction. Theefficiencyofthe thermalconduc-

tion in the presence of magnetic field is, in general, reduced with respect to the unmagnetized case.
]
h The reduction factor strongly depends on the initial magnetic field orientation, and it is minimum
p when the magnetic field is initially aligned with the direction of shock propagation. The thermal
- conductioncontributes tosuppresshydrodynamicinstabilities, reducingthe massmixing ofthe cloud
o and preserving the cloud from complete fragmentation. Depending on the magnetic field orientation,
r
the heat conduction may determine a significant energy exchange between the cloud and the hot sur-
t
s rounding medium which, while remaining always at levels less than those in the unmagnetized case,
a
leadsto a progressiveheatingandevaporationofthe cloud. This additionalheating maycontrastthe
[
radiative cooling of some parts of the cloud, preventing the onset of thermal instabilities.

1 Subject headings: conduction — magnetohydrodynamics — shock waves — ISM: clouds — ISM:
v
magnetic fields — ISM: supernova remnants
3
0
4 1. INTRODUCTION tion by providing an additional tension at the interface
1
betweenthecloudandthesurroundingmedium(e.gMac
. The interaction of the shock waves of supernova rem-
1 nants (SNRs) with the magnetized and inhomogeneous Low et al. 1994; Jones et al. 1996).
0 Theinteractionoftheshockwitharadiative cloudhas
interstellar medium (ISM) is responsible of the great
8 beenonlyrecentlyanalyzedindetail(e.g.Mellemaetal.
morphologicalcomplexity of SNRs and certainly plays a
0 2002; Fragile et al. 2004). 2D calculations have shown
majorroleindeterminingtheexchangeofmass,momen-
: that the effect of the radiative cooling is to break up
v tum, and energy between diffuse hot plasma and dense
i clouds or clumps. These exchanges may occur through, the clouds into numerous dense and cold fragments that
X survive for many dynamical timescales. In the case of
for example, hydrodynamic ablation and thermal con-
r ductionand,amongotherthings,leadtothecloudcrush- theinteractionbetweenmagnetizedshocksandradiative
a clouds, the magnetic field may enhance the efficiency of
ing and to the reduction of the Jeans mass causing star
the radiative cooling, influencing the size and distribu-
formation.
tion of condensed cooled fragments (Fragile et al. 2005).
The propagation of hot SNR shock fronts in the ISM
The role played by the thermal conduction during the
and their interaction with local over-dense gas clouds
shock-cloudinteractionhasbeenlessstudiedsofar. Ina
have been investigated with detailed hydrodynamic and
previouspaper,Orlandoetal.(2005)(hereafterPaperI)
MHD modeling. The most complete review of this
have addressed this point in the unmagnetized limit. In
problemin the unmagnetized, non-conducting,and non-
particular,wehaveinvestigatedtheeffectofthermalcon-
radiativelimits is providedby Kleinetal. (1994). These
ductionandradiativecoolingonthe cloudevolutionand
studieshaveshownthatthecloudisdisruptedbytheac-
onthe massandenergyexchangebetweenthe cloudand
tionofbothKelvin-Helmholtz(KH)andRayleigh-Taylor
the surrounding medium; we have selected and explored
(RT) instabilities after several crushing times, with the
twodifferentphysicalregimeschosensothateitheroneof
cloud materialexpanding and diffusing into the ambient
the processesis dominant. In the case dominatedby the
medium. An ambient magnetic field can both act as a
radiative losses, we have found that the shocked cloud
confinement mechanism of the plasma and be modified
fragments into cold, dense, and compact filaments sur-
bytheinterstellarflowandbylocalfieldstretching. Also,
roundedbyahotcoronawhichisablatedbythethermal
a strong magnetic field is known to limit hydrodynamic
conduction. Instead, in the case dominated by thermal
instabilities developing during the shock-cloud interac-
conduction, the shocked cloud evaporates in a few dy-
1ConsorzioCOMETA,viaSantaSofia64,95123Catania,Italy namical timescales. In both cases, we have found that
2INAF,VialedelParcoMellini84,00136Roma,Italy
2 Orlando et al.
the thermal conduction is very effective in suppressing
thehydrodynamicinstabilitiesthatwoulddevelopatthe ∂B
+∇·(uB−Bu)=−∇×(η∇×B) , (4)
cloudboundaries,preservingthecloudfromcompletede- ∂t
struction. Orlando et al. (2006) and Miceli et al. (2006)
where
havestudiedtheobservableeffectsofthermalconduction
onthe evolutionofthe shockedcloudinthe X-rayband. B2 1 1B2
P =P + , E =ǫ+ u2+ ,
Here, we extend the previous studies by investigat- ∗
2 2 2 ρ
ing the effect of the thermal conduction in a magne-
tized medium, unexplored so far. Of special interest to are the total pressure,and the total gasenergy (internal
us is to investigate the role of anisotropic thermal con- energy, ǫ, kinetic energy, and magnetic energy) respec-
duction - funneled by locally organized magnetic fields - tively, t is the time, ρ = µmHnH is the mass density,
in the mass and energy exchange between ISM phases. µ = 1.26 is the mean atomic mass (assuming cosmic
In particular, we aim at addressing the following ques- abundances), mH is the mass of the hydrogen atom, nH
tions: Howandunderwhichphysicalconditionsdoesthe is the hydrogen number density, u is the gas velocity, T
magnetic-field-orientedthermalconductioninfluence the isthetemperature,Bisthemagneticfield,η istheresis-
evolutionoftheshockedcloud? Howdothemassmixing tivity according to Spitzer (1962), Fc is the conductive
of the cloud material and the energy exchange between flux, and Λ(T) represents the radiative losses per unit
the cloud and the surrounding medium depend on the emission measure (e.g. Raymond & Smith 1977; Mewe
orientationandstrengthofthemagneticfieldandonthe etal.1985;Kaastra&Mewe2000). We usetheidealgas
efficiency of the thermal conduction? law, P =(γ−1)ρǫ.
To answer these questions, we take as representative In order to track the original cloud material, we use
the model case of a shock with Mach number M = 50 a tracer that is passively advected in the same manner
(corresponding to a post-shock temperature T ≈ 4.7× as the density. We define Ccl the mass fraction of the
106 K for an unperturbed medium with T = 104 K) cloud inside the computational cell. The cloud material
impacting on an isolated cloud ten-fold denser than the is initialized with Ccl = 1, while Ccl = 0 in the ambient
ambient medium. Paper I has shown that, in this case, medium4. During the shock-cloud evolution, the cloud
the thermal conduction dominates the evolution of the andtheambientmediummixtogether,leadingtoregions
shocked cloud in the absence of magnetic field. Around with 0 < C < 1. At any time t the density of cloud
cl
this basic configuration,we perform a set of MHD simu- material in a fluid cell is given by ρ =ρC .
cl cl
lations, with different interstellar magnetic field orienta- The thermal conductivity in an organized magnetic
tions, and compare models calculated with thermal con- field is known to be highly anisotropic and it can be ex-
duction turned either “on” or “off” in order to identify traordinarily reduced in the direction transverse to the
its effects on the cloud evolution. field. Thethermalflux,therefore,islocallysplitintotwo
The paper is organized as follows: in Sect. 2 we de- components, along and across the magnetic field lines,
scribethe MHDmodelandthe numericalsetup; inSect. F =F i+F j, where
c k ⊥
3 we discuss the results; and finally in Sect. 4 we draw
our conclusions. 1 1 −1
F = + ,
k (cid:18)[q ] [q ] (cid:19)
2. THEMODEL spi k sat k
(5)
We model the impact of a planar supernova shock 1 1 −1
front onto an isolated gas cloud. The shock propagates F = + ,
⊥
through a magnetized ambient medium and the cloud is (cid:18)[qspi]⊥ [qsat]⊥(cid:19)
assumed to be small compared to the curvature radius
to allow for a smooth transition between the classical
of the shock3. The fluid is assumed to be fully ionized
and saturated conduction regime. In Eqs. 5, [q ] and
with a ratio of specific heats γ = 5/3. The model in- spi k
[q ] represent the classical conductive flux along and
cludes radiative cooling, thermal conduction (including spi ⊥
across the magnetic field lines (Spitzer 1962)
the effects ofheatflux saturation)andresistivityeffects.
The shock-cloud interaction is modeled by solving nu-
merically the time-dependent non-ideal MHD equations [q ] =−κ [∇T] ≈−5.6×10−7T5/2 [∇T]
spi k k k k
(written in non-dimensional conservative form):
n2
∂ρ [q ] =−κ [∇T] ≈−3.3×10−16 H [∇T]
+∇·(ρu)=0 , (1) spi ⊥ ⊥ ⊥ T1/2B2 ⊥
∂t (6)
where [∇T] and [∇T] are the thermal gradients along
∂ρu k ⊥
∂t +∇·(ρuu−BB)+∇P∗ =0 , (2) and across the magnetic field, and κk and κ⊥ (in units
of erg s−1 K−1 cm−1) are the thermalconduction coeffi-
cients along and across the magnetic field lines5, respec-
∂ρE
+∇·[u(ρE+P )−B(u·B)]= tively. The saturatedflux alongandacrossthe magnetic
∗
∂t fieldlines,[q ] and[q ] ,are(Cowie &McKee 1977)
sat k sat ⊥
∇·[B×(η∇×B)]−∇·F −n n Λ(T)(3,)
c e H
4Wecheckedthattheusednumericalschemeensuresthatalways
3 In the case of a small cloud, the SNR does not evolve signif- 0≤Ccl≤1.
icantly duringthe shock-cloud interaction, and the assumption of 5 ForthevaluesofT,nHandBusedhere,κk/κ⊥≈1016 atthe
aplanarshockisjustified(seealsoKleinetal.1994). beginningoftheshock-cloudinteraction.
Anisotropic thermal conduction in SNRs 3
TABLE 1 TABLE 2
Summaryof theinitial physicalparameters SummaryoftheMHDsimulations. Inallrunstheshock
characterizingtheMHDsimulations. Machnumberis M=50, thedensity contrast isχ=10,
andthecloud crushingtimeisτcc≈5.4×103 yr.
Temperature Density Velocity
ISM 104 K 0.1cm−3 0.0 Run µ|BG| β0 CFoimeldp. TChoenrdm.. LRoassde.s Res.a
Cloud 103 K 1.0cm−3 0.0
Post-shock medium: 4.7×106 K 0.4cm−3 430kms−1
NN 0 ∞ − no no 132
NR 0 ∞ − no yes 132
TN 0 ∞ − yes no 132
TR 0 ∞ − yes yes 132
[qsat]k =−sign [∇T]k 5φρc3s, NN-Bx4 1.31 4 Bx no no 132
(cid:0) (cid:1) (7) NN-By4 1.31 4 By no no 132
[qsat]⊥ =−sign([∇T]⊥) 5φρc3s, TNNN--BBxz44 11..3311 44 BBxz yneos nnoo 113322
TN-By4 1.31 4 By yes no 132
where cs is the isothermal sound speed, and φ is a num- TN-Bz4 1.31 4 Bz yes no 132
ber of the order of unity; we set φ = 0.3 according TR-Bx1 2.63 1 Bx yes yes 132
to the values suggested for a fully ionized cosmic gas: TR-By1 2.63 1 By yes yes 132
0.24 < φ < 0.35 (Giuliani 1984; Borkowski et al. 1989; TR-Bz1 2.63 1 Bz yes yes 132
TR-Bx4 1.31 4 Bx yes yes 132
Fadeyev et al. 2002, and references therein). As dis- TR-By4 1.31 4 By yes yes 132
cussed in Paper I, this choice implies that no thermal TR-Bz4 1.31 4 Bz yes yes 132
precursordevelopsduringtheshockpropagation,consis- TR-Bx100 0.26 100 Bx yes yes 132
TR-By100 0.26 100 By yes yes 132
tentwiththe factthatnoprecursorisobservedinyoung TR-Bz100 0.26 100 Bz yes yes 132
and middle aged SNRs.
The initial unperturbed ambient medium is magne- TR-Bz4-hr 1.31 4 Bz yes yes 264
tized, isothermal (with temperature T = 104 K, cor- TR-Bz4-hr2 1.31 4 Bz yes yes 528
ism
responding to an isothermal sound speed c = 11.5
km s−1), and uniform (with hydrogen numbisemr density aInitialnumberofzonespercloudradius
n =0.1 cm−3; see Table 1). The gas cloud is in pres-
ism
extendinginfinitelyalongthez axisperpendiculartothe
sure equilibrium with its surrounding and has a circular
(x,y) plane. The primary shock propagates along the y
cross-section with radius r = 1 pc; its radial density
cl
axis. In this geometry, we consider three different field
distribution is given by
orientations: 1) parallel to the planar shock and per-
n −n pendicular to the cylindrical cloud, 2) perpendicular to
cl0 ism
n (r)=n + , (8)
cl ism cosh[σ(r/r )σ] both the shock front and the cloud, and 3) parallel to
cl
both the shock and the cloud. The magnetic field com-
where ncl0 is the hydrogen number density at the cloud ponents along the x and the z axis are enhanced by a
center,r is the radialdistancefromthe cloudcenterand factor (γ +1)/(γ −1) (where γ is the ratio of specific
σ = 10. The above distribution describes a thin tran- heats)inthepost-shockregion(inthestrongshocklimit;
sition layer (∼ 0.3 rcl) around the cloud that smoothly Zel’dovich&Raizer1966),whereasthecomponentalong
brings the cloud density to the value of the surrounding theyaxisiscontinuousacrosstheshock. Weincluderuns
medium6. Theinitialdensitycontrastbetweenthecloud in the strong and weak magnetic field limits, consider-
center and the ambient medium is χ = ncl0/nism = 10. ing initial field strengths of |B| = 2.63, 1.31, 0.26, 0 µG
Thecloudtemperatureisdeterminedbythepressurebal- in the unperturbed ambient medium7, corresponding to
ance across the cloud boundary. β = 1, 4, 100, ∞, where β = P/(B2/8π) is the ratio
0 0
The SNR shock front propagates with a velocity w = of thermal to magnetic pressure in the pre-shock region.
Mcism in the ambient medium, where M is the shock This range of β0 includes typical values inferred for the
Mach number, and cism is the sound speed in the inter- diffuseregionsoftheISM(e.g.MacLow&Klessen2004)
stellar medium; we consider a shock propagating with and for shock-cloud interaction regions in evolved SNR
M = 50, i.e. a shock velocity w ≈ 570 km s−1 and a shells (e.g Bocchino et al. 2000). There is no magnetic
temperature Tpsh ≈ 4.7×106 K. As discussed in Paper field component exclusively associated to the cloud.
I, in this case (for a cloud with rcl = 1 pc and χ = 10) Wefollowtheshock-cloudinteractionfor3.5τcc,where
theclouddynamicswouldbedominatedbythermalcon- τ ≈χ1/2r /w is the cloud crushing time, i.e. the char-
cc cl
ductionintheabsenceofmagneticfield. Thepost-shock acteristic time of the shock transmission through the
conditions of the ambient medium well before the im- cloud; for the conditions considered here (χ = 10 and
pact onto the cloud are given by the strong shock limit M=50),τ ≈5.4×103yr. Eachsimulationisrepeated
cc
(Zel’dovich & Raizer 1966). either with or without thermal conduction for each field
Startingfromthisbasicconfiguration,weconsideraset orientation. Table2liststherunsandtheinitialphysical
of simulations with different initial magnetic field orien- parameters of the simulations.
tations. We adopt a 2.5D Cartesian coordinate system We solve numerically the set of MHD equations us-
(x,y), implying that the simulated clouds are cylinders
7 Theunmagnetized case(i.e. |B|=0)describedhereisanalo-
6Afinitetransitionlayer,ingeneral,isexpectedinrealinterstel- gous to the one studied in Paper I except for the fact that in the
lar clouds due, for instance, to thermal conduction (Balbus 1986; present case the cloud is a cylinder rather than a sphere and has
seealsoNakamuraetal.2006). smoothboundaries.
4 Orlando et al.
Fig. 1.— Mass density distribution(gm cm−3) inthe (x,y) plane, inlog scale, inthe simulations NN (left half panels) and TR (right
halfpanels),sampledatthelabeledtimesinunitsofτcc. Thecontour enclosesthecloudmaterial.
ing flash (Fryxell et al. 2000), a multiphysics code in- Bz4, TR-Bx4, TR-By4, TR-Bz4). The left (right) half
cluding the paramesh library (MacNeice et al. 2000) panels showthe result ofmodels without (with) thermal
for the adaptive mesh refinement. The MHD equations conduction and radiative losses.
aresolvedusing the flash implementationofthe HLLE From Fig. 1, we note that the thermal conduction
scheme (Einfeldt 1988). The code has been extended drives the cloud evolution in the unmagnetized case
with additional computational modules to handle the (β = ∞; run TR): after the initial compression due
0
radiative losses and the anisotropic thermal conduction to the primary shock, the cloud expands and gradually
(see Paganoetal.2007, forthe details ofthe implemen- evaporatesduetotheheatingdrivenbythethermalcon-
tation). ductioninafewdynamicaltimescales(seerighthalfpan-
The2.5DCartesian(x,y)gridextendsbetween−4and elsinFig.1). Theheatconductionstronglycontraststhe
4pcinthexdirectionandbetween−1.4and6.6pcinthe radiativecoolingof some parts ofthe cloud andno ther-
ydirection. Initiallythecloudislocatedat(x,y)=(0,0) mal and hydrodynamic instabilities (visible in run NN;
and the primary shock front propagates in the direction see left-panels in Fig. 1) develops during the cloud evo-
of the y axis. At the coarsest resolution, the adaptive lution, making the cloud more stable and longer-living
mesh algorithm used in the flash code uniformly cov- (the mass mixing is strongly reduced; see Paper I for
ers the 2.5D computational domain with a mesh of 42 more details).
blocks,eachwith82 cells. We allowfor5levelsofrefine- Wenowdiscusstheeffectofthemagnetic-field-oriented
ment,withresolutionincreasingtwiceateachrefinement (anisotropic)thermalconductionontheshock-cloudcol-
level. The refinement criterion adopted (L¨ohner 1987) lision when an ambient magnetic field permeates the
follows the changes of the density and of the tempera- ISM. We first summarize the expected evolution in the
ture. This grid configuration yields an effective resolu- presence of an ambient magnetic field, according to the
tion of≈7.6×10−3 pc atthe finest level, corresponding well-establishedresults of previousmodels without ther-
to ≈132 cells per cloud radius. In Sect. 3.5, we discuss mal conduction. We distinguish between fields perpen-
theeffectofspatialresolutiononourresults,considering dicular to the cylindrical clouds (i.e. with only B and
x
the additional runs TR-Bz4-hr and TR-Bz4-hr2 which B components; referredto as “external”fields by Frag-
y
use an identical setup to run TR-Bz4, but with higher ileetal.2005)andfieldsparalleltothecylindricalclouds
resolution (≈ 264 and ≈ 528 cells per cloud radius, re- (i.e. with only the B component; referred to as “inter-
z
spectively; see Table 2). nal” fields). In the former case, the magnetic field plays
We use a constant inflow boundary condition for the a dominant role along the cloud surface and in the wake
post-shock gas at the lower boundary, with free outflow of the cloud where it reaches its highest strength (and
elsewhere. For runs with zero magnetic field (β = ∞), the plasmaβ its lowestvalues; e.g. Mac Lowetal.1994;
0
weusereflectingboundaryconditionsatx=0alongthe Jones et al. 1996). In the case of B , the magnetic field
x
symmetryaxisoftheproblemandonlyevolvehalfofthe is trapped at the nose of the cloud, leading to a contin-
grid. uous increase of the magnetic pressure and field tension
there (see upper panels in Fig. 2); in the case of B , the
y
3. RESULTS cloud expansion leads to the increase of magnetic pres-
3.1. Dynamical evolution sure and field tension laterally to the cloud (see middle
panels in Fig. 2). In the case of B (internal field), the
Figs.1and2showtheevolutionofthemassdensityin z
magnetic field, being parallel to the cylindrical cloud,
the(x,y)planeinthesimulationswithβ =∞(runsNN
0 modifies only the total effective pressure of the plasma
and TR) and with β = 4 (runs NN-Bx4, NN-By4, NN-
0
Anisotropic thermal conduction in SNRs 5
Fig. 2.—AsinFig.1forthesimulationswith β0=4andthe magnetic fieldoriented along x(upper panels), y (middlepanels), and z
(lowerpanels). Thefigureshowsthedistributioninmodelseitherwithout(lefthalfpanels)orwith(righthalfpanels)thermalconduction
andradiativelosses. ForrunsNN-Bx4,TR-Bx4,NN-By4andTR-By4,weplotthemagneticfieldlines;forrunsNN-Bz4andTR-Bz4,we
includecontours oflog(B2/8π).
6 Orlando et al.
(Jones et al. 1996); in the case of radiating shocks, the runTR-Bx4. Oneofthiscoldanddensestructuresisev-
additional magnetic pressure may play a crucial role in identinFig.2(upperpanels)andislocatedatthecloud
theshockedcloud,preventingfurthercompressionofthe boundary near the nose of the cloud (at x ≈ 0.4 pc and
cloud material (Fragile et al. 2005). y ≈3.0 pc) at t=3 τ .
cc
IntheB case,theinitialfielddirectionismostlymain-
y
3.1.1. External magnetic fields tained in the cloud core during the evolution, allowing
efficient thermal exchange between the core and the hot
In the case of predominantly external magnetic fields,
mediumupwindofthecloud(seecenterpanelsinFig.3):
MacLowetal.(1994)andJonesetal.(1996)haveshown
the core is gradually heated and evaporates in few dy-
that the hydrodynamic instabilities can be suppressed
namical timescales. This is illustrated by run TR-By4
even in models neglecting the thermal conduction due
in Fig. 2. On the other hand, the cloud is thermally in-
to the tension of the magnetic field lines which maintain
sulated laterally where the magnetic field lines prevent
a more laminar flow around the cloud surface (see also
thermalexchangebetweenthecloudandthesurrounding
Fragile et al. 2005): for a γ = 5/3 gas, the KH instabil-
ities are suppressed if β < 2/M2, whereas RT instabili- medium. Also, a strong magnetic field component along
ties are suppressed if β < (2/γ)(χ/M)2 (see also Chan- the x axis developsin the wake ofthe cloud andinhibits
thermal conduction with the medium downwind of the
drasekhar 1961). However, for the parameters used in
cloud. Thethermalinsulationatthesideofthecloudde-
thispaper(χ=10andM=50),themagneticfieldcan-
terminesthegrowthofthermalinstabilitieswhereshocks
notsuppressKHinstabilitiesinanyofourruns,whereas
transmitted into the cloud collide (see middle panels in
theRTinstabilitiesaresuppressedonlyinrunsthatlead
Fig. 2).
to locally very strong field (β <0.05). This can be seen
In both external field configurations, elongated struc-
in model NN-Bx4 (upper panels in Fig. 2), presenting a
tures of strong field concentration are produced on the
large field increase at the cloud boundary, compared to
axis downwind of the cloud due to the focalization of
model NN (Fig. 1): in the latter case the growth of KH
the magnetized fluid flows there (see upper and middle
andRTinstabilitiesatthecloudboundaryismuchmore
panels of Fig. 2, and lower panels in Fig. 3). These fil-
evident than in NN-Bx4. On the other hand, the hy-
amentary structures, identified as “flux ropes” by Mac
drodynamic instabilities are suppressed more efficiently
Low et al. (1994), are formed by magnetic field lines
in models including the thermal conduction (runs TR-
stretched around the cloud shape and do not carry a
Bx4 and TR-By4) even in cases with low field increase
significant amount of cloud material (as shown by the
(for instance in our B case) as it is evident in Fig. 2
y tracer C ) although the plasma there moves with the
by comparingmodels NN-Bx4 andNN-By4 with models cl
cloud (see also Gregori et al. 2000).
TR-Bx4 and TR-By4, respectively.
Thethermalexchangesbetweenthecloudandthesur-
3.1.2. Internal magnetic fields
roundingmediumstronglydependontheinitialfieldori-
entation. Fig. 3 shows the heat flux and magnetic field Predominantly internal magnetic fields strongly sup-
strength distributions in the (x,y) plane in runs TR- press the heat conduction, providing an efficient ther-
Bx4, TR-By4, and TR-Bz4, at time t = 2 τ . In our mal insulation of the cloud material (see right panels in
cc
B case(upper panels inFig.2), the magnetic fieldlines Fig. 3). In the realistic configuration of an elongated
x
gradually envelope the cloud, reducing the heat conduc- cloud with finite length L along the z axis, some heat
tionthroughthe cloudsurface (see left panels in Fig.3): would be conducted along the magnetic field lines. The
thermal exchanges between the cloud and the surround- characteristic timescales for the conduction along mag-
ingmediumarechannelledthroughsmallregionslocated netic field lines is (see Paper I)
at the side of the cloud. The cloud expansion and evap-
orationarestronglylimitedbytheconfiningeffectofthe n L2
τ ≈2.6×10−9 H . (10)
magnetic field (cf. the unmagnetized case TR in Fig. 1 cond T5/2
with model TR-Bx4 in Fig. 2) that becomes up to 30
We estimate that the cloud would thermalize in τ >
times stronger just outside the cloud than inside it (see, cond
3.5 τ (i.e. the physical time covered by our simula-
also,thelowerleftpanelinFig.3). Theconsequentther- cc
tions), if the length scale of the cloud along the z axis
malinsulationinduces the radiativecoolingandconden-
is L>3 pc. In this case, hydrodynamic instabilities de-
sation of the plasma into the cloud during the phase of
velop at the cloud boundary, being both the magnetic
cloud compression (t < τ ). At the end of this phase,
cc
the cloud material has temperature T ≈105 K and den- field and the thermal conduction not able to suppress
sity n ≈ 10 cm−3 where primary and reverse shocks them. The growth of these instabilities is clearly seen
H
in Fig. 2 (lower panels). The combined effect of hy-
transmitted into the cloud are colliding; for these values
drodynamicinstabilitiesandshockstransmittedintothe
ofT andn ,the Fieldlengthscale (Begelman& McKee
H
cloudleadsto unstable high-densityregionsatthe cloud
1990) derived from the ratio of cooling timescale over
boundaries that trigger the development of thermal in-
conduction timescale (see Paper I for details) is
stabilities there(see lowerpanels inFig.2). However,as
T2 discussedby Fragileetal.(2005),internalmagnetic field
l≈106 ≈3.2×10−4 pc . (9) lines are expected to resist compression in the shocked
n
H cloud, thus reducing the cooling efficiency. In fact, in
The radiative cooling dominates over the effects of the our run TR-Bz4, the cloud material is prevented from
thermal conduction in cold and dense regions with di- cooling below T ≈ 103 K. Since the thermal conduction
mensions larger than l. At variance with our unmagne- does not play any significant role in the shock-cloud in-
tized case TR, therefore, thermal instabilities develop in teraction, our B case leads to results similar to those
z
Anisotropic thermal conduction in SNRs 7
Fig. 3.—Heatflux(upperpanels)andmagneticfieldstrength(lowerpanels)distributionsinthe(x,y)planeinthesimulationsTR-Bx4
(left panels), TR-By4 (center), and TR-Bz4 (right), at time t = 2 τcc. The arrows in the upper panels describe the heat flux and scale
linearlywithrespecttothereferencevalueshownintheupperrightcornerofeachpanel. Thescaleofthemagneticfieldstrengthislinear
andisgivenbythebarontheright,inunitsof10µG.Theredcontour encloses thecloudmaterial.
obtained by Fragile et al. (2005) and we do not discuss
further this case.
C ρT da
cl
Z
hTi = A(Ccl>0.9) (11)
cl
C ρ da
cl
3.2. Role of thermal conduction ZA(Ccl>0.9)
Inthissection,westudymorequantitativelytheeffect
C B da
ofthermalconductiononthecloudevolutionand,inpar- Z cl
ticular, on the cloud compression and on the magnetic hBicl = A(Ccl>0.9) (12)
field increase. To this end, we use the tracer defined in C da
cl
Z
Sect. 2 to identify zones whose content is the original A(Ccl>0.9)
cloud material by more than 90%. Then, we define the where we integrate on zones with C > 0.9. Note that
cl
cross-sectional area of cloud material, A (t), as the to- our choice of considering cells with the value of the pas-
cl
tal area in the (x,y) plane occupied by these zones. We sive tracer C >0.9 is arbitrary. To determine how sen-
cl
define the cloud compression(or expansion)as A /A , sitive the results are to this value and, in particular, to
cl cl0
whereA isthe initialcross-sectionalarea. We alsode- smallchangesinit,wederiveourresultsalsoconsidering
cl0
fine an average mass-weighted temperature of the cloud thevaluesC >0.85andC >0.95. Inallthecases,we
cl cl
and anaveragemagnetic field strengthassociatedto the findthattheresultsderivedwiththedifferentthresholds
cloud as show the same trend with differences lower than 10%.
8 Orlando et al.
Fig. 4.—Evolutionofcloudcompression(upperpanels),ofaveragetemperature(middlepanels),andofaveragemagneticfieldstrength
(lowerpanels)ofthecloudforrunswhichneglect thethermalconduction andtheradiation(dot-dashed lines;leftpanels), forrunswhich
includethethermal conduction but neglect theradiation(solid;leftpanels), forruns whichincludethe radiationbut neglect thethermal
conduction(dotted;rightpanels)andforrunswhichincludebothphysicaleffects(dashed;rightpanels). Themagnetizedcaseswithβ0=4
aremarkedwithred(initialmagneticfieldalongx),green(initialBalongy)andblue(initialBalongz)lines;theunmagnetizedcasesare
marked with black lines. The light yellow regions markthe location of solutions which have thermodynamical characteristics inbetween
the cases of maximum efficiency of the thermal conduction (models TN and TR) and the cases without thermal conduction (models NN
and NR). By comparing the position of the magnetized models curves inside the yellow region, it is possible to quantitatively assess the
degreeofsuppressionoftheeffects ofthethermalconduction bythemagneticfields.
Fig. 4 shows the cloud compression, A /A , the av- the unmagnetized cases (see Table 2).
cl cl0
erage temperature of the cloud, hTi , normalized to In all the NNs models either with (NN-Bx4, NN-By4,
cl
the post-shock temperature of the surrounding medium and NN-Bz4) or without (NN) the magnetic field, the
(T = 4.7 × 106 K), and the average magnetic field evolution of the cloud compression and of the average
psh
strength associated to the cloud, hBi , normalized to cloudtemperatureis roughlythesame(seeleftpanelsin
cl
the initial field strength (B = 1.31 µG, corresponding Fig.4). Thecloudisinitiallycompressedoveratimescale
0
to β = 4) as a function of time for models neglecting t ≈ τ due to the ambient post-shock pressure; during
0 cc
thermal conduction and radiation (hereafter NNs mod- this phase hTi rapidly increases. After t ≈ τ , the
cl cc
els),formodelsincludingconductionbutneglectingradi- cloudpartially reexpands, leading to a decrease of hTi .
cl
ation (TNs models), and for models including both con- In the last phase (t > 2.0 τ ), the cloud is compressed
cc
ductionandradiation(TRs models); wealsoinclude the again by the interaction with the “Mach stem” formed
results derived from the unmagnetized case NR with ra- during the reflection of the primary shock at the sym-
diative cooling and without thermal conduction. The metry axis, and hTi increases; later A /A continues
cl cl cl0
figure shows both the magnetized cases with β =4 and to decrease, because of the mixing of the cloud material
0
Anisotropic thermal conduction in SNRs 9
with the ambient medium (see Sect. 3.3; see also Paper atedtothe cloud,being the fieldlockedwithinthe cloud
I), while hTi stabilizes at ≈0.17T . material.
cl psh
Thefieldincreaseinthecloudmaterialdependsonthe In our unmagnetized case TR (including thermal con-
initial configurationof B (see lower left panel in Fig. 4). duction and radiative cooling), the thermal conduction
Inthecaseofexternalfields(B andB components),B prevents the onset of thermal instabilities, and the evo-
x y
is mainly intensified due to stretching of field lines due lution of the shocked cloud is the same as found in the
to sheared motion. In the B case, the magnetic field TNmodel. AtvariancewithourunmagnetizedcaseTR,
x
undergoes the greatest increase and hBi keeps increas- Fig. 4 shows that thermal instabilities develop in all our
cl
ingduringthewholeevolution. Infactthefieldismainly magnetized TRs runs, being the effects of thermal con-
intensifiedatthenoseofthecloudwherethebackground duction reduced by the magnetic field. The effects of
flow continues to stretchthe field lines during the evolu- radiative cooling are very strong for internal fields (our
tion(seeupperpanelsinFig.2). IntheB case,thefield B case; see run TR-Bz4 in Fig. 4). In this case, the
y z
increaseoccursmainlyatthe sideofthecloudwherethe heat conduction is totally suppressed by the magnetic
field lines are stretched along the cloud surface. In the field and the evolution of A /A and of hTi are the
cl cl0 cl
case of internal fields (B component), the field increase same as those found in the unmagnetized case with ra-
z
is due to squeezing of field lines through compression. diative cooling and without thermal conduction (model
hBi , therefore, follows the changes in A /A , since NR); at t = 3.5 τ , run TR-Bz4 (and NR) shows the
cl cl cl0 cc
the field is locked within the cloud material. Thus the largest cloud compression (A /A ≈ 0.1) and the low-
cl cl0
greatest field increase occurs at t≈τ when the shocks est cloud average temperature (hTi ≈ 0.12 T ). In
cc cl psh
transmitted into the cloud collide. the case of external fields (runs TR-Bx4 and TR-By4),
The effects of thermal conduction are greatest in the the effects of heat conduction are reduced but not sup-
unmagnetized model (TN) which can be considered an pressed and the results are intermediate between those
extreme limit case (see left panels in Fig. 4). During derivedfor runs NR and TR(i.e. within the lightyellow
the first stage of evolution (t < 0.8 τ ), the cloud is region in the right panels in Fig. 4). The cooling effi-
cc
heated efficiently by the thermal conduction and its av- ciency is largely reduced in our B case (run TR-By4),
y
erage temperature increases rapidly to ∼ 0.5 T . As namely that with the magnetic field configuration that
psh
a consequence, the pressure inside the cloud increases allows the most effective thermal conduction.
and the cloud reexpands earlier than in model NN. Af-
terwards, the average cloud temperature, hTi , keeps 3.3. Mass mixing and energy exchange
cl
increasing up to ∼0.9 Tpsh at t=3.5τcc. Weusethetracertoderivethecloudmass,Mcl,asthe
In the case of predominantly external magnetic fields total mass in zones whose content is the original cloud
(models TN-Bx4 and TN-By4), the thermal conduction material by more than 90%,
still plays a significant role in the cloud evolution, al-
thoughitseffectsarenotaslargeasintheunmagnetized
M =L C ρ da , (13)
case (TN). During the initial compression, the thermal cl cl
Z
conduction contributes to the cloud heating: the aver- A(Ccl>0.9)
age temperature of the cloud reaches values larger than where L is the cloud length along the z axis, and the
in models neglecting the conduction (compare TNs with integral is done on zones with Ccl >0.9. We investigate
NNs models in the left panels of Fig. 4). This effect is the mixing of cloud material with the ambient medium
greatestinthe By casewhichistheconfigurationoffield bydefiningtheremainingcloudmassasMcl/Mcl0,where
lines that allowsthe most efficient thermal exchange be- Mcl0 is the initial cloud mass.
tweenthecloudandthehotenvironment(seeSect. 3.1). The tracer allows us to investigate also the energy ex-
At t=3.5τ , hTi in TNs models reachesvalues larger change between the cloud and the surrounding medium;
cc cl
than in NNs models (≈ 0.21 Tpsh in the Bx case and wederivetheinternalenergy,Icl,andthekineticenergy,
≈0.33Tpsh intheBy case). Forinternalmagneticfields, Kcl of the cloud as
the thermal conduction plays no role in the evolution of
the shocked cloud, being strongly ineffective due to B I =L C ρǫ da , (14)
cl cl
(see Sect. 3.1). As a consequence, the TN-Bz4 model Z
A(Ccl>0.9)
leads to the same results as NNs models.
L
In general, therefore, the effects of the thermal con- K = C ρ|u|2 da , (15)
cl cl
duction in the presence of an ambient magnetic field 2 Z
A(Ccl>0.9)
are reduced with respect to the corresponding unmag-
where again L is the cloud length along the z axis, and
netized case, but not entirely suppressed. This can be
the integral is done on zones with C > 0.9. We also
seen in Fig. 4, where we have markedin light yellow the cl
define the total energy of the cloud as
region between the fully conductive unmagnetized case
(TN) and the case without thermal conduction (NN).
E =I +K . (16)
The magnetized TNs models are always within this re- cl cl cl
gion,meaning that the effects of the thermalconduction Fig.5showstheevolutionofthecloudmass,M /M ,
cl cl0
are never as large as in the unmagnetized case (TN) but for NNs, TNs, and TRs models; again we also include
not completely suppressed as in the model NN. the unmagnetized case with radiative cooling and with-
We also note that the thermal conduction influences outthermalconduction(modelNR).Bothunmagnetized
indirectly the magneticfieldincrease. The mainchanges cases and magnetized cases with β = 4 are shown. In
0
are in the By case and are due to the larger expansion models without thermal conduction and radiation (NNs
of the cloud that reduces the increase of the field associ- models), the hydrodynamic instabilities drive the mass
10 Orlando et al.
Fig. 5.—PresentationasinFig.4fortheevolutionofthecloudmass(upperpanels),oftheinternalenergyofthecloud(middlepanels),
andofthekineticenergyofthecloud(lowerpanels).
mixing of the cloud8. The mass loss rate of the cloud, conduction suppresses most of the hydrodynamic insta-
m˙ , increases significantly after 1.5 τ (i.e. after the bilities and the mass loss mainly comes from the cloud
cl cc
hydrodynamic instabilities have fully developed at the evaporation driven by the thermal conduction rather
cloud boundary), with m˙ ≈ 1.5×10−6L M yr−1, thanfromhydrodynamicablation. Notethatourunmag-
cl pc ⊙
whereL is thecloudlengthalongthez axisinunits of netized TN model is an extreme limit case in which the
pc
pc: ∼20%ofthecloudmassiscontainedinmixedzones hydrodynamic instabilities are totally suppressed by the
att=3.5τ . TheonlyexceptionisrunNN-Bx4(∼15% thermalconductionwhichdrivesthecloudmixing;inthis
cc
ofthe cloudmassis inmixed zonesatt=3.5τ ), being case,themasslossrateism˙ ≈1.5×10−7L M yr−1
cc cl pc ⊙
in this case RT instabilities partially suppressed by the (∼5%ofthecloudmassisinmixedzonesatt=3.5τ ).
cc
magnetic field (compare run NN-Bx4 with runs NN-By4 In magnetized TRs models, the onset of thermal in-
and NN-Bz4 in Fig. 2). stabilities increases the mass loss rate of the cloud with
In TNs models with external magnetic fields (TN- respect to the unmagnetized case (m˙ ranges between
cl
Bx4 and TN-By4), the mass loss rate of the cloud 1.5×10−6L M yr−1 and 4×10−6L M yr−1)
pc ⊙ pc ⊙
is less efficient than in NNs models with m˙ ≈ 6 × due to the fragmentation of the cloud in dense and
cl
10−7L M yr−1 (∼10%ofthe cloudmassisinmixed cold cloudlets. We expect, therefore, that the larger
pc ⊙
zones at t = 3.5τ ). In fact, in these cases the thermal the amount of cloud mass mixed with the surrounding
cc
mediumattheendoftheevolution,themorelimitedthe
8 This is alsotrueinour magnetized cases because, forthe pa- thermal exchange between the cloud and the hot ambi-
rameters used in this paper (M = 50 and χ = 10), the hydrody- ent medium (and, therefore, the greater the efficiency of
namic instabilities are partially suppressed by the magnetic field
radiativecooling). Infact,theupperrightpanelinFig.5
onlyinrunsevolvingtostrongfields(seeSect. 3.1).