J.C. Bozier et al- A New Supercritical Shock Wave Regime

8/3/2019 J.C. Bozier et al- A New Supercritical Shock Wave Regime

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THE ASTROPHYSICAL JOURNAL SUPPLEMENT SERIES, 127:253260, 2000 April2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

A NEW SUPERCRITICAL SHOCK WAVE REGIME

J. C. BOZIER, J. P. LE BRETON, T. JALINAUD, AND J. VALADONCommissariat l Energie Atomique/DRIF, 91680 Francea` Bruye` re-le-Chatel,

Received 1999 January 19; accepted 1999 December 13

ABSTRACT

If it is very difficult to dispose of a large number of experimental results to study complex astro-physical phenomena such as those involved in the explosion of supernovae, it has, however, already beenshown that well-controlled experiments performed in the laboratory can reproduce some interesting fea-tures of these phenomena. Furthermore, it would be still more interesting if these experiments could alsohighlight events that are not yet well understood. We present here such an experimental attempt usinglaser-generated plasmas. The radiation phenomena that are referred to result from the interaction of thehigh-velocity supernova debris with the surrounding ambient plasma left over from the stellar wind ofthe supernovae progenitor.

Subject headings: hydrodynamics radiative transfer shock waves

1. INTRODUCTION

The initial interaction of a supernova with its surround-ing medium gives rise to a double-shell structure bounded

by shock waves (Chevalier & Blondin 1995). The deceler-ation of the supernova gas is subject to hydrodynamicinstabilities, and proposals have been made to study theseinstabilities by means of laser experiments (Chevalier 1997;Remington 1997).

Along with hydrodynamics, the radiative heat, exchangebetween the supernova gas and the external medium is aninteresting eld that can be investigated by laser experi-ments. Laser plasmas can eectively become an appropriatetool to study the radiation emission of the supernova sur-rounding medium. It is well known (Chevalier 1997) thatthe emission of some types of supernovae (SN 1993, forexample) is dominated by radiation in their interaction witha dense stellar wind released before explosion. If it appears

difficult to reproduce in the laboratory all the thermodyna-mic characteristics of the supernova explosion, it may bepossible to learn more about some phenomenologicalaspects of supernova radiation emission.

The blast wave created by a supernova explosion has atypical velocity of 500010,000 km s~1, but it graduallydecelerates over its lifetime as it sweeps up many cubicparsecs of interstellar gas. The initial velocities mentionedabove correspond to very high radiation temperature atequilibrium (38 keV), and the shocked circumstellarmedium should be transparent or partially transparent toits own radiation emission. Correlatively, this emission isvery weak, which is the reason why the forward shock isgenerally supposed to be adiabatic (Blondin et al. 1998). Butit is only an approximation that has some consequences;one is that the small amount of energy radiated by theshock circumstellar region creates a thermal radiative pre-cursor in the unperturbed ambient medium ahead of theforward shock. As we will see below, the temperature andthe size of this thermal precursor are going to increase withtime until the deceleration of supernova gas is importantenough to decrease the radiation temperature inside theshocked medium. This precursor makes the X-ray emissionof the shocked surrounding medium more complex.

If the high velocities that can be obtained with laserplasmas (several hundred km s~1) cannot compete withthose of supernova explosions, nevertheless, these are high

enough to study the radiative precursor phenomena. Thismeans that it becomes possible to determine the conditionsrequired for its appearance and its development phases.

2. LASER EXPERIMENTSBy a laser irradiating thin plane targets, we get high-

shock velocities that can behave like the supernova sur-rounding medium. The strong shock wave induced is at rstdescribed by the Hugoniot equations, this is the adiabaticshock wave. It exists when the temperature of the shockedmaterial is so low that no radiation is emitted. If the propa-gation velocity D is high enough, the temperature of theshocked material rises, and when it reaches a critical value

given by the relation below, the radiation losses do notTCR

remain negligible. The compressed material losing energy,the shock wave is no longer adiabatic and can be describedby the radiation heat conduction approximation. In thatcase the shock is said to be supercritical (Zeldovich &Raiser 1966, II, 536):

pTCR4 \ D(T

CR)o

0e(T

CR) , (1)

where p is the Stefan-Boltzmann constant, is the initialo0

density, and e is the specic internal energy.We studied this regime and examined the supercritical

shock wave by laser accelerating an Al foil in xenon gas,and the corresponding shock velocity was as large as1.2] 107 cm s~1 (Bozier et al. 1986). This represents thecurrent knowledge on radiating shocks. But owing to newlaser experiments, we go farther.

To create a very high temperature (evidence the existenceof the hypercritical) shock wave, we irradiated cylindricalCH foam targets with four laser beams focused on one oftheir bases. The cylinders were 200 or 300 km in diameter,and their lengths varied from 300 km to 1 mm. The foamdensity was equal to 5] 10~2 g cm~3. The incident laserirradiances were 1.5 or 5] 1014 W cm~2 with a 1 ns pulseduration at the wavelength of 0.35 km. The propagations ofradiative phenomena were observed owing to a back-lighting diagnostic by imaging the axis of the target ontothe slit of a streak camera. A fth laser beam was focused ona gold disk to create an X-ray source behind the target inthe observation line of sight. Its radiation was transmittedthrough the target when the foam became hot enough to betransparent to the gold emission (Le Breton et al. 1996). Theexperimental scheme is shown in Figure 1.

253


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FIG. 1.Experimental scheme

FIG. 2.Time-resolved backlighting image of the CH foam irradiated at 1.5]1014 W cm~2


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SUPERCRITICAL SHOCK WAVE REGIME 255

The imaging system was an improved version of a Kirk-patrick Baez microscope, the KBA microscope (Sauneuf etal. 1997). This one is composed of two orthogonal couplesof grazing incidence mirrors giving a spatialSiO

2resolution of 5 km in a 1 mm eld of view. The magni-cation was 10, so the spatial resolution of the images wasnot deteriorated by the streak camera characteristics. Thestreak camera temporal resolution was 30 ps. The spectralresponse of the diagnostic was given by the mirrors reec-

tivity, the lters transmission (2000 Al plus 2000 CH forA Athe rst one and 2800 Al plus 0.12 km Mylar for theAsecond one) and the streak camera photocathode sensi-tivity. Taking the X-ray source emission spectra, which ismaximum at 300 eV for gold, and the heated foam transmis-sion into account, the global spectral response of the mea-surement could mainly be represented by one window at270 eV with a 25 eV FWHM.

The experimental results are presented in Figures 2 and 3respectively for the two irradiances values we used. Thelaser beams were impinging from the right. The X-ray

source is more or less visible on the right side of the picturesbecause of its possible extension beyond the target limit.The foams are opaque at the beginning and become pro-gressively transparent as they are heated by the propagatingfronts. The comparison made between the shots performedwith and without the X-ray source demonstrates that theobserved light comes either from beyond the target limit oris really due to the foam transparency.

The observed fronts correspond to phenomena that are

integrated along the foam thickness (near the target axiswithin the alignment error that is about 30 km). And as thetarget irradiation was performed so that the laser beamsfocal spot was equal to the foam diameter with a maximumvalue in the center, the propagation fronts were slightlycurved with a maximum velocity on the target axis. Thisbent shape contributed to make the observation more diffi-cult.

Despite this fact dierent fronts can be clearly distin-guished. In both gures the most luminous front is theslower one, its velocity is about 1.5] 107 cm s~1, it corre-

FIG. 3.Time-resolved backlighting image of the CH foam irradiated at 5]1014 W cm~2


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256 BOZIER ET AL. Vol. 127

FIG. 4.Calculated temperature and density proles in the 5]10~2 gcm~3 CH foam irradiated at 1.5]1014 W cm~2.

sponds to the ablation of the foam by the laser. One or two

other fronts appear depending on the laser irradiance. At1.5] 1014 W cm~2 (Fig. 2) (for a 300 km in diameter targetand a 2 mm distance between the X-ray source and thetarget) the fastest front has a nearly constant velocity of4.5] 107 cm s~1 characteristic of a foam preheat. At5]1014 W cm~2 (Fig. 3) (for a 200 km in diameter targetand a 4 mm distance between the X-ray source and thetarget) the fastest front has an increasing velocity evaluatedto 5.4] 107 cm s~1 and an intermediate front propagatesat 3.7]107 cm s~1.

3. SIMULATIONS

The experiments interpretation has been made using theFCI1 code and considering that the CH foam was transpar-

ent at its nominal density above 20 eV. The absorptioncoefficient of the foam is very difficult to estimate because ofthe two-dimensional aspect of the heat propagation insidethe target. The X-ray source emission goes in fact through acentral hot zone and through lateral colder zones the exten-sion of which is not easily evaluable.

At 1.5] 1014 W cm~2, the calculated density and elec-tron temperature proles 700 ps after the beginning of thelaser pulse are shown in Figure 4. They are characteristic ofthe propagation of a supercritical shock wave inside thetarget, the ablation front position is localized on the left ofthe diagram. The temperature inside and in front of theshock are equal and take the value of 86 eV. If the shockhad been adiabatic, the 2.7] 107 cm s~1 calculated shockvelocity (not measured because of its small separation withthe ablation front) would have led to a temperature of 175eV. The supercritical shock phenomena is consequentlycharacteristic of a very large energy loose implying asmaller shock temperature.

The radiative precursor dimension is about 90 km in thecalculation, it is not measured because of the difficulty todistinguish the shock front in the image. The followingrapid temperature dropo allows us to get a good idea ofthe precursor extension in spite of the great dependence ofthe foam absorption coefficient with temperature. The foamtransmission is evaluated to 20% from experiments. Theopaque shock thickness is small (15 km from calculations)

FIG. 5.Calculated temperature and density proles in the 5]10~2 gcm~3 CH foam irradiated at 5]1014 W cm~2.

and is not visible in the picture. The precursor velocity isabout constant in the calculation and in the experiment andremains superior to the shock velocity indicating that theasymptotic regime of the supercritical shock is not reached.


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No. 2, 2000 SUPERCRITICAL SHOCK WAVE REGIME 257

FIG. 6.Three shock wave regimes

At 5] 1014 W cm~2, the calculated density and elec-tronic temperature proles are presented in Figure 5 atdierent times after the beginning of the laser pulse. Thefastest front is associated with a radiative heating of thefoam and the intermediate one to the shock front. Theobservation of these last one is here possible during the 600rst ps because of the smaller density jump, because of thehigher foam temperature, and because of the smaller targetdiameter. After 400 ps, the temperature decreases from 340

to 70 eV through the shock. The preheated zone is 40 kmlong in the calculation and in the experiment. It is the begin-ning of a new shock structure (the hypercritical

FIG. 7.Supercritical and hypercritical temperature comparison

phenomena). After 600 ps, the temperature variationthrough the shock, decreasing from 290 to 100 eV, is lowerthan previously. The preheated zone has a 100 km dimen-sion in the calculation and can be extrapolated to a value of140 km in the experiment (its observation being stoppedafter 510 ps when the end of the foam is reached). We are ina new regime. If the shock had been adiabatic, the tem-perature corresponding to the observed shock velocitywould have been 340 eV; the shock is, consequently, not so

far from been adiabatic. After 800 ps, the temperature is 190eV near the ablation region and 125 eV near the preheatedzone, these temperatures tend to become equal. The densityand electronic temperature proles are about similar to theones we get at 1.5] 1014 W cm~2 after 700 ps (Fig. 4). Theshock regime is intermediate between the new regime andthe supercritical regime.

Our interpretation is the following: when the shockvelocity is very large the shock temperature increases somuch that the shocked material becomes transparent to itsown radiation. Then, even if the radiative ux escaping fromthe shock front keeps on growing, its radiation lossesdecrease relatively to its specic internal energy.

Consequently the shock wave tends to go back to the

adiabatic regime. We called this particular kind of shockwave the hypercritical shock wave. However, if the shockvelocity is kept constant, the shocked material thicknessincreases with time so that it tends to become opaque againand is no longer adiabatic; the new regime is consequently atemporary one. Figure 6 summarizes the kind of thermody-namic conditions that are met in the three dierent shockwave regimes discussed here.

4. EXISTENCE CONDITIONS OF THE HYPERCRITICALSHOCK WAVE

The conditions for the existence of the hypercritical shockwave and most of its fundamental properties can be


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258 BOZIER ET AL. Vol. 127

deduced from a qualitative description. At rst the shockedmaterial must be transparent to its own radiation, and con-sequently the Planck mean free path has to be large inj

Pcomparison with the thickness of the compressed material.This can be described by the following inequality:

jP

Ac[ 12

BUP

t , (2)

where is the particle velocity and c is the isentropicUPexponent.Second, the radiation energy must be small in compari-

son with the energy received by the shocked material. Thiscondition can also be stated by an inequality:

(c[ 1)

(c] 1)

pTC3 t

2o0

CVjP

> 1 , (3)

where is the shock temperature and is the specicTC

CV

heat at constant volume.Third, we can dene a life duration for the hypercriti-t

hccal shock wave, which depends on the degree of adiabaticityof the shock wave. This characteristic which we might callthe degree of adiabaticity , can be measured by the fol-lowing ratio:

thc

\Ac] 1c[ 1

B 2o0

CVjP

pTc3

*Tc

Tc

, (4)

with

TC

\UP2

2CV

, (5)

where is the variation of the shock wave temperature*TC

due to the fact the shocked material is not perfectly adia-batic.

Fourth, the propagation law of the radiative precursor

associated with the hypercritical shock wave can bededuced from the basic laws governing the energy radiatedby a transparent material, so that its radiation ux can'

Cbe written as (Zeldovich & Raiser 1966):

'C

\Dt

jP

pTC4 , (6)

we can also write with'C

\'0

t,

'0

\c[ 1

25

pUP9

jP

CV4

, (7)

but will be usually too small to make the material ahead'C

of the shock wave transparent, we consequently have to use

the diusion approximation to describe the propagation ofthe radiative precursor through the material at rest. We alsowill not have to take the albedo of the precursor intoaccount because of the important gap existing between theshock temperature and the temperature of the precursor.This is due to the fact that the energy received by the shockwave is much smaller than the energy it loses by radiation.So we can set the following relation:

'0

t \16

3pj

RT3

LT

Lx, (8)

with the Rosseland mean free path and T the precursorjR

temperature at the x abscissa.

The position of the thermal wave front can be inferredXF

from simple dimensional considerations, being approx-jR

imated by a power function (usually m[ 1) :jR

\ ATm

XFB

C 16Ap3(m ] 4)

D1@(m`5)'

0(m`3)@(m`5)

]

1

(o0

CV

)(m`4)@(m`5)t(2m`7)@(m`5) . (9)

This relation is not valid when t is close to 0, but the termmust be added to relation (9) to obtain an approximateU

Pt

solution of the thermal wave expansion at its beginning.

5. COMPARISON BETWEEN THE SUPERCRITICAL ANDHYPERCRITICAL SHOCK WAVES

The thermodynamic considerations we have pointed outat the beginning allow us to compare the supercritical andthe hypercritical thermal precursors. The relation (9) showsthat the velocity of the thermal wave increases with time aslong as the shock wave remains hypercritical becausem[ 0. On the contrary, for the supercritical shock wave, ascan be deduced from the relation (10) we established some

years ago (Bozier et al. 1986), the thermal wave velocitydecreases with time:

XFB

C 16Apo0

3(m ] 4)

D1@(m`5)CUP3(c] 1)

4

D2@(m`5)C tCV

D(m`4)@(m`5)(10)

the characteristic temperature is given byTSC*

TSC* \

C3(c] 1)64

m ] 4

po

0UP3D0.25

. (11)

We can compare the characteristic temperature of aTSC*

supercritical thermal wave and the characteristic tem-

perature of the hypercritical thermal wave.THC*

THC* \

C3(m ] 4)(c[ 1)UP

t

32jP

D0.25TC

, (12)

with

TC

\UP2

2CV

. (13)

For a piston propagating at a given velocity in two dierentmaterials having the same density, and if the shock wave issupercritical in the rst one of these materials and hyper-critical in the other one, we can set the following inequality:

THC

*

TSC* \ 1 . (14)

This one shows that the supercritical shock wave precursoris hotter than the hypercritical one; the correspondingcurves in the velocity domain of concern are shown inFigure 7.

The dierences between the two kinds of thermal precur-sors can also be seen considering the waves time evolutions.

The piston velocity starts from 0 and goes up to itsmaximum value (we shall suppose that this maximum valueis high enough to make the shock hypercritical). If the timeneeded to reach the maximum piston velocity is of the sameorder of magnitude as the lifetime (given by eq. [2]) of the


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No. 2, 2000 SUPERCRITICAL SHOCK WAVE REGIME 259

FIG. 8.Radiative astrophysical shock wave description

hypercritical shock we want to establish, we shall suc-cessively obtain an adiabatic shock wave, a supercriticalshock wave, then the hypercritical shock wave. But at theend of its lifetime, as the hypercritical shock wave willthicken it will go back to the supercritical regime. With

'0\c[ 1

25

pUP9

jP

CV4 ,

the relation (9) shows the spatial expansion of the thermalprecursor depends only weakly on the time it takes for thepiston to reach its maximum velocity and tells us the expan-sion of the thermal precursor is quite fast compared withthe expansion of the supercritical thermal precursor, as canbe seen from relation (10).

6. DISCUSSION

A particular question comes from the inuence in theinterpretation of the radiation created at the critical surface,corresponding to the limit of the laser energy absorption

zone, and at the ablation front. These can be consideredcomparing the relative intensities of the X-ray emitted fromthe dierent parts of the foam.

For an optically thin material, like the heated CH foamused in the hypercritical experiments, the radiative emis-sion rate (energy emitted per gram) varies proportionally toT4 but inversely to the Planckian mean free path whichj

p,

in CH at low density o, varies like T3.8/o1.9. So, in CH at aconstant density the shock radiative emission rate varieso

0,

like and will slightly increase, like T0.2, with thepT4/o0jp

temperature Tof the shock wave.In the heated foam, between the critical density up to the

beginning of the shocked material, the relative decrease ofthe density is much more important than the relativeincrease of the temperature. In the domain of temperatureand density we are dealing with, the Planckian mean freepath varies inversely with density o, like o1.8 (see above). Asa consequence of this, the radiative emission rate of theablated part of the target is going to decrease from theablation front up to the critical density. As a result, the


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260 BOZIER ET AL.

radiative net ux will be oriented from the ablation fronttoward the critical density. This radiative ux will induce acooling wave propagating in the opposite direction. Thisphenomenon will be more or less severe depending on theoptical depth of the piston (the ablated mass of the targetin the hypercritical shock experiment we did in CH).

So, in the case of a low-density and low-Z foam where thehypercriticity condition is valid, the radiative emission ofthe ablated part of the foam is smaller than the radiative

emission of the shocked material.A second question arises from the hot electrons propaga-

tion. An evaluation of the heat ux due to electronic con-duction in the hypercritical shock shows that it is orders ofmagnitude lower than the radiative ux. As for the hotelectrons generated by resonant absorption, their number isquite low at 0.35 km laser wavelength, so that electrons arefar from being the main source of preheat.

7. CONCLUSION

We have evidenced and characterized a new shockregime, which we have named the hypercritical regime. It

has been obtained in a CH foam target and its temperaturewas about 90 eV, that is, 5 times larger than the one we getin the supercritical experiment.

Its analytical description as well as simulations conrmedthe experimental results. This regime is characterized by anincreasing propagation velocity of the radiative precursorand by a large temperature jump between the shock and theprecursor which disappears when the energy feed decreases.

This new shock regime and its connection to the super-

critical one should be applicable to the description of radi-ative astrophysical shocks. These processes should occurwith a more-or-less important intensity during the earlytimes of the interaction of the supernovae gas with its sur-rounding medium. We think the usual hydrodynamic andradiative description might be usefully completed asdescribed in Figure 8. The radiative shocked gas is quiteisothermal and transparent to its own radiation during the hypercritical phase and opaque and nonisothermalduring its supercritical phase, but both phases are charac-terized by a thermal precursor ahead of the forward shockwave.

REFERENCES

Chevalier, R. A., & Blondin, J. M. 1995, ApJ, 444, 312Remington, B. A., et al. 1997, Phys. Plasmas 4, 1994Chevalier, R. A. 1997, Science, 276, 1374Blondin, J. M., Wright, E. B., Borkowski, K. J., & Reynolds, S. P. 1998,

ApJ, 500, 342Bozier, J. C., Thiell, G., Le Breton, J. P., Azra, S., Decroisette, M., &

Schirmann, D. 1986, Phys. Rev. Lett., 57, 1304

Le Breton, J. P., Bozier, J. C., Jalinaud, T., & Valadon, C. 1996, APSMeeting

Sauneuf, R., Dalmasso, J. M., Jalinaud, T., Le Breton, J. P., Schirmann, D.,Marioge, J. P., Bridou, F., Tissot, G., & Clotaire, J. Y. 1997, Rev. Sci.Instrum. 68 (9)

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J.C. Bozier et al- A New Supercritical Shock Wave Regime