Formation of chondrules in radiative shock waves I. First results, spherical dust particles, stationary shocks
aa r X i v : . [ a s t r o - ph . S R ] A p r Astronomy&Astrophysicsmanuscript no. CH-V-E˙latest c (cid:13)
ESO 2018June 24, 2018
Formation of chondrules in radiative shock waves
I. First results, spherical dust particles, stationary shocks
H. Joham and E. A. Dorfi Institute for Astronomy, University of Vienna, T¨urkenschanzstraße 17, A-1180 Wiene-mail: [email protected] Institute for Astronomy, University of Vienna, T¨urkenschanzstraße 17, A-1180 Wiene-mail: [email protected]
Received 2012; Accepted 2012
ABSTRACT
Context.
The formation of chondrules in the protoplanetary nebulae causes many questions concerning the formation process, thesource of energy for melting the rims, and the composition of the origin material.
Aims.
The aim of this work is to explore the heating of the chondrule in a single precursor as is typical for radiation hydrodynamicalshock waves. We take into account the gas-particle friction for the duration of the shock transition and calculate the heat conductioninto the chondrules. These processes are located in the protoplanetary nebulae at a region around 2 . Methods.
We calculated the shock waves using one-dimensional, time-independent equations of radiation hydrodynamics (RHD)involving realistic gas- and dust opacities and gas-particle friction. The evolution of spherical chondrules was followed by solving theheat conduction equation on an adaptive grid.
Results.
The results for the shock-heating event are consistent with the cosmochemical constraints of chondrule properties. The calcu-lations yield a relative narrow range for density or temperature to meet the requested heating rates of R > K h − as extracted fromcosmochemical constraints. Molecular gas, opacities with dust, and a protoplanetary nebula with accretion are necessary requirementsfor a fast heating process. The thermal structure in the far post-shock region is not fully consistent with experimental constraints onchondrule formation since the models do not include additional molecular cooling processes. Key words. meteorites, meteors, meteoroids – interplanetary medium – protoplanetary disks – shock waves
1. Introduction
Chondrules (Greek
Chondros , grain) or chondren are silicateglobules with sizes of 0 . > K h − , whichstrongly favours melting of the origin material. A subsequentrapid cooling within a few hours is also required to produce theobserved mineralogical features (see, e.g. Hood & Kring (1996)for a detail discussion of these basic properties). The analysisof various chondrules proves that multiple heating events havebeen present to generate their final structure. Based on these in-vestigations the origin of the chondrules lies within regions ofthe early protoplanetary nebulae and their age is considered tobe between 1 and 4 million years after the formation of the solarsystem (e.g. Amelin et al. 2002; Kita et al. 2005).There exists a large number of models for chondrule for-mation. All of them contain more or less plausible assump-tions. Owing to the aforementioned structural characteristics ofthe chondrules, the duration, temperature, and pressure at theheating event have to occur within a spatially localised zone ofthe protoplanetary nebulae as summarised, e.g. by Boss (1996).Shock waves appear to meet most constraints and this scenario is currently the most plausible model, as discussed, e.g. byConnolly & Love (1998), Desch et al. (2005). Since the mod-els based on heating in shock waves make fewer assumptionson the physical environment, we will focus on these events,which have been investigated by a large amount of detailedmodelling, e.g. Hood & Horanyi (1991, 1993), Ruzmaikina & Ip(1994), Hood (1998), Desch & Connolly (2002), Miura et al.(2002), and Miura & Nakamoto (2005, 2006). Ciesla & Hood(2002) and Ciesla et al. (2004) developed gas-particle suspen-sion models with significant heating via radiation from the otherchondrule-sized particles. In the models of Desch & Connolly(2002) the heating process is caused either by frictional heat-ing between gas and dust particles when traversing an adia-batic shock wave or by thermal conduction from the shock-heated gas to the cooler dust particles. Desch and Connollyalso took significant heating by radiation from other chondrulesinto account. Nevertheless, without detailed modelling of thephysical processes within the protoplanetary disc the origin ofthese shock waves remains unanswered. In the literature severalmechanisms have been suggested to explain the shock waves,i.e. accretion shocks on the surface of the protoplanetary disc(Ruzmaikina & Ip 1994), infall of clumps of gas onto the neb-ulae (Tanaka et al. 1998), bow shocks produced by planetesi-mals on eccentric orbits (Weidenschilling et al. 1998), and X-ray flares from the young Sun (Nakamoto et al. 2005). Clearly,all these scenarios give only a simplified picture of the numerous
1. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves interactions between radiation field, and gas, and dust particles.In particular the treatment of adiabatic shock waves is mostlymade with a simple equation of state and based on the equationsof hydrodynamics without calculating a consistent radiative pre-cursor region.The recent work of Morris & Desch (2010) is based on ahydrodynamical shock model including a complete molecularline cooling due to H O, a treatment of the radiation field, andimproved opacities of the solids. These authors found the ef-fects of molecular cooling to be minimal because the combi-nation of high column densities of water, hydrogen recombina-tion / dissociation and radiation from the upstream regions reducethe rapid cooling times of chondrules in the post-shock region.In this work we investigate the reaction of dust particlesto the structure of radiative shock waves. Based on full one-dimensional time-independent equations of radiation hydrody-namics (RHD) (e.g. Mihalas & Mihalas 1984), we follow theevolution of a spherical dust particle that is swept through theseshock transitions, which are parameterised by the Mach number.In this first study we include the e ff ects of gas-particle drift butthe chondrules are still assumed to preserve their spherical shapeand are not directly coupled through their radiative properties tothe equations of RHD. Their temperature structure is obtained bysolving a heat conduction equation allowing a thermal expansionof the dust particles. Thanks to the full system of RHD, the over-all structure of the radiative shock waves is calculated correctlytogether with the radiative precursor that heats the dust particlesbefore they enter the shock jump. All models presented are lo-cated in a protoplanetary nebula at a typical distance of 2 .
2. Basic equations
The equations of radiation hydrodynamic (RHD) treat the in-terplay of matter and radiation by including the momentumand energy exchange between the two components. A detailedderivation of these equations can be found in Landau & Lifschitz(1981) and Mihalas & Mihalas (1984). To investigate the physi-cal behaviour of dust particles we restricted the problem to time-independent planar shock waves in a protoplanetary nebula inthis exploratory study. We can neglect self-gravity of the gas, andthe resulting system of ODEs can be solved without the need ofartificial viscosity to broaden shock waves. We furthermore as-sumed the Eddington-approximation as closure condition for theradiation field as well as an ideal molecular gas with a constantadiabatic index of γ = / ρ denotes the gas density and u the gas velocity. Webegin with the equation of continuity ddx ( ρ u ) = , (1) and the equation of motion with gas pressure P and radiative(Eddington) flux Hddx (cid:16) ρ u (cid:17) + dPdx − π c ρκ R H = , (2)where κ R is the so-called Rosseland-mean of the opacity, c is thelight speed. The third equation is the gas energy equation writtenas ddx Pu γ − ! + P dudx − πρκ P ( J − S ) = , (3)containing the adiabatic index γ , the Planck-mean opacity κ P , theEddington radiation energy density J and the source function S .The next two equations are needed to describe the radiationfield by writing down the Eddington moments of the radiationtransport equation. Starting with the 0 th -moment we obtain theequation of the radiation energy density J c ddx ( Ju ) + dHdx + c K dudx + ρκ P ( J − S ) = , (4)including the Eddington radiation pressure K . The 1 st -momentleads to the equation for the radiation flux density H ,1 c ddx ( Hu ) + dKdx + c H dudx + ρκ R H = . (5)This system of ODEs has to be solved for radiative shockwaves together with additional closure conditions, and boundaryconditions. Hence, we need an equation of state and we adoptedin this first study an ideal gas P = R µ ρ T . (6)The source function S is given by S = σπ T (7)for LTE, which we assumed throughout. T is the temperatureand σ the Stefan-Boltzmann constant. The corresponding radia-tion temperature T rad can be obtained from the radiation energydensity by J = σπ T . (8)Since the above equations contain three moments, J , K , and H ofthe grey specific radiation intensity, we need an additional clo-sure condition for the radiation field. Within the limit of isotropicintensity distributions we can use the well-known Eddington ap-proximation f = KJ = , (9)relating the second to the zeroth moment. For the radiating shockwaves in a dusty environment such as the solar nebula with itshigh optical depths this approximation is usually valid. According to the previous section our five basic variables aregiven by the gas density ρ , the gas velocity u , the gas pres-sure P , the radiation energy density J , and the radiation flux H .Hence, we have to specify five far up-stream values at x → −∞ .Denoting this gas density by ρ and the gas velocity by u , the
2. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves gas pressure P also fixes the gas temperature T through theequation of state (6). From the equation of motion (2) we seethat only H = J = S and togetherwith the source functions we have T = T rad , , i.e the radiationtemperature is equal to the gas temperature for the far up-streamregion. The same conditions hold for the far down-stream re-gion x → + ∞ for stationary problems (e.g. Zel’dovich & Raizer1967). As already discussed in the literature, these conditionsappear to be not fulfilled in all shock computations used to studythe formation of chondrules. Because the radiative flux H is zeroat both boundaries, a maximum (or minimum) has to occur in be-tween and any physical possible solution has a maximum locatedat the shock position x =
0. Ignoring for this discussion the dif-ference between the Rosseland- and Planck mean, we introducethe optical depth τ by d τ = κρ dx (10)and can transform the system of stationary RHD equations into dd τ ( ρ u ) = dd τ (cid:16) ρ u + P (cid:17) − π c H = dd τ Pu γ − ! + P dud τ − π ( J − S ) = dd τ (cid:18) Juc + H (cid:19) + Kc dud τ + ( J − S ) = dd τ (cid:18) Huc + K (cid:19) + Hc dud τ + H = . (15)We see that the optical depth defines through ( κρ ) − a lengthscale and hence ( κρ u ) − yields a typical time scale availablefor heating of particles in the precursor region.The properties of the shock waves are parameterised by theMach number M in the up-stream region; M is given for an adi-abatic sound velocity a by M = u a with a = γ P ρ = γ R µ T . (16)A detailed analytical treatment of RHD shock waves can befound in Zel’dovich & Raizer (1967). We used the inverse com-pression ratio η = ρ /ρ , and a value of η = + ( γ − M ( γ + M (17)has to be reached in the far down-stream region behind the shockwave.A close view on the aforementioned stationary RHD-equations reveals that the model assumes a single shock wavewith a precursor for the melting of the chondrules. However,the complexity observed in meteorites favours formation regionswhere many precursors may exist, e.g. due to the interactionwith bow shocks of moving planetesimals and to accretion pro-cesses. A more definite answer to the evolution of chondrulestherefore requires a more elaborate treatment of the various in-teraction processes between the gas, dust particles and radiationfields within time-dependent flows. In particular, in another stepit will be necessary to consider also the momentum and energyexchange between gas and particles because this study only takesinto account the heating of the dust particles embedded in thegas without additional energy and momentum exchange. Hence,the cooling rates in the post shock region can disagree with datacoming from furnace experiments (e.g. Hewins et al. 2005). As mentioned before, we neglected the backreaction of the dustparticles on the gas, e.g. the evolution of the dust particle’s sizedistribution function during the shock transition, which also in-fluences the transparency of the gas-dust mixture. The opaci-ties adopted in our computations were computed for evolvedplanetary discs with a chemical composition like our solar sys-tem, i.e. also a solid-to-gas ration of about 0 .
01. However, thechondrules may be formed in a dust-enriched environment (e.g.Ciesla & Hood 2002) where this solid-to-gas ratios can be 100 oreven 1000 times the solar ratio. In these zones the particle den-sities are much higher than the adopted values and the passageof a shock wave through this particle-gas suspensions requiresa more detailed treatment of the opacities (Desch et al. 2005).Increasing the dust-to-gas ratio increases the total opacity, andtherefore all length scales defined through d τ = κρ dx will be-come smaller, in particular the time t p a chondrule needs to passthe radiative precursors (cf. Fig. 2) is shortened.Since we use frequency-integrated moments of the radiationfield, the frequency-integrated gas and dust opacities that enterRHD equations, in particular the frequency-dependent opacity κ ν , have to be integrated to obtain either the Planck-mean κ P forthe optically thin case or the Rosseland-mean κ R in the opticalthick case. This use of opacity tables for a dust-gas mixture canonly be a first approximation to a more realistic situation wherethe dust particles are processed by the interaction with largerbodies.In particular, we used the Rosseland- and Planck-mean dustopacities for protoplanetary discs from Semenov et al. (2003),which are based on optical constants for spherical dust withporous aggregate particles and normal silicates. When the gas passes the shock wave, the velocity changes inthe precursor region, the adiabatic shock, and the subsequentpostshock cooling region. A particle changes its velocity throughcollisions with the moving gas and will therefore need some timeto relaxate to zero relative velocity. The speed of a chondrule u C will be higher compared to the gas since it maintains its initialup-stream velocities in the absence of friction. Consequently, theexposition of dust grains in the hot shock zone is shorter com-pared to the gas. To account for this e ff ect, we solved the fol-lowing system of equations for a moving chondrule at a position x C u C = dx C dt or u C dtdx C − = . (18)The equation of motion for a chondrule can be written as F C = m C du C dt or F C dtdx C − m C du C dx C = . (19)The simple formulae for the frictional force F C is related to therelative or drift-velocity u D by F C − π r C ρ C D u D = , with u D = u C − u , (20)where m C is the mass of the particle and r C its radius. C D is thedrag coe ffi cient given for the numerical calculations accordingto (Sandin & H¨ofner 2003) C D = C D , di ff + C D , com . (21)
3. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves
Fig. 1.
Gas and particle velocities as a function of the distancefrom the shock wave in [km] for a Mach number M = . r C = C D , di ff = π (1 − ε ) S D (cid:18) T C T (cid:19) (22) C D , com = sign ( S D ) S D + S D − . S D + . | S D | + (23)with S D = u D v mp , v mp = s k B T µ m H . (24) ε defines the type of collision, i.e. di ff usive collisions lead to ε = ε = T C denotes the tem-perature of the particles, T is the temperature of the gas and v mp is the thermal speed of the Maxwellian velocity distribution.Figure 1 shows the velocity di ff erence between gas and particlesand also the force on the particles plotted for a shock wave withMach number M = .
1, a preshock temperature T =
300 K,a preshock density ρ = . · − kg m − and an adiabatic in-dex γ = /
5. The radius of the particle is set to r C = . · kg m − assuming forsteritMg SiO .The stopping length l s of a typical chondrule with radius r C and density ρ C can be estimated by equating the mass of a dustparticle with the gas mass contained in a cylinder of length l s and a cross section corresponding to the dust particle. This simpleapproximation leads roughly to l s ≃ r C ρ C ρ (25)and for the above values (also used in Fig. 1) we derived l s ≃ ± As a first step we studied the heating of a spherical dust particleby heat conduction where the temperature changes because ofthe passage through the shock at the particle surface. In the sim-plest approximation we assumed a spherical shape of the particleas an initial condition in the upstream region. The particle is alsoassumed to be in thermal equilibrium with the surrounding gas,to be located many optical depths away from the radiating shockwave, and to move with the gas, i.e. no drift velocity u D = T C the spherical heat equationreads ∂ T C ∂ t = r ∂∂ r r D ∂ T C ∂ r ! , (26)where D specifies the heat conduction coe ffi cient. This coe ffi -cient can be written as D = λρ C c s , (27) λ is the heat conduction e ffi ciency, ρ C is the chondrule densityand c s its specific heat.For the numerical solution of this PDE we used a standardimplicit di ff erence scheme (e.g. Richtmyer 1967), which leadsto a tridiagonal system of equations for the temperature distribu-tion T C ( r ) at the di ff erent particle radii. In this formulation thetotal radius can change in time due to a thermal expansion of theparticle. The initial condition is specified by a constant temper-ature within the particle T ( r , = T ( r ) for 0 ≤ r ≤ r C , and lateron we assumed an outer boundary condition of T ( r C , t ) = T ( x C )as well as dT C / dr = x C isdetermined by integrating Eq. (18). For the numerical solution of the RHD system (Eqs. 1-7) to-gether with the opacity tables we adopted a standard packageSLGA2 (Raith, Schoenauer & Glotz 1984) for ODEs, written inFORTRAN. We sought solutions where H = dH / d τ > T is not possiblebut the radiation energy density J and the radiation temperature T rad cross the shock wave without discontinuity.
4. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves
On top of the RHD shock solutions we followed the path ofa dust particle by integrating Eqs. 18-24. Numerical experimentsshow that the type of gas-particle collision is negligible for ourcalculations and we adopted ε = . T C is not yet known, but shouldbe in between radiation temperature T rad and gas temperature T ,and for these computations we considered the gas temperatureas the appropriate outer boundary condition for the chondrules,i.e. T C ( x , r = r C ) = T ( x C ).The calculation of the heat conduction in the dust particleswas made by numerically solving the tridiagonal equation sys-tem, which we obtained from discretisation of the heat conduc-tion equation in spherical symmetry (u.v. 2.5). We also includedthe thermal expansion and estimated the amount of radial in-crease by taking a linear expansion coe ffi cient of α = . · − K − together with a temperature change of ∆ T = ∆ r C = . · − mm or about 1.5%for a mm-sized chondrule. Clearly, this e ff ect is negligible butwe adopted a finite volume discrete version of Eq. (26) for sub-sequent applications, studying also the temporal evolution fromflu ff y to spherical particles.
3. Results
For all RHD shock wave models listed in Table 1 we considereda spherical dust particle with a radius of r = γ = / T = − γ = /
3. However, to study the e ff ectswithin the radiative precursor as well as the radiative coolinge ff ects behind the shock waves, we tried to keep the equationof state as simple as possible for the moment. The chondruleswere assumed to consist of forsterit (Mg SiO ) with the fol-lowing particle characteristics. The particle density was set to ρ = . · kg m − , the heat conductivity λ = .
33 W / m K andthe specific heat c s = .
01 kJ / kg K were taken from the litera-ture (e.g. Krishnaiah, Singh & Jadha 2004; Yingwei 2004). Thelinear heat expansion coe ffi cient is α = . · − K − .The protoplanetary disc with accretion has been modelledby Bell et al. (1997), where an angular momentum transportcoe ffi cient α = − leads to a mass accretion rate of ˙ M = · − M ⊙ y − . Outside of 5 AU these discs are gravitationallyunstable. Taking a typical solar distance of 2 . ρ ∼ − kg m − in the mid plane at a temperature T ∼
300 K. According to Hood & Kring (1996), the temperaturein the chondrule precursor region must not exceed T =
650 Kso that in chondrules the existence of chemical compounds suchas FeS can be explained. Furthermore, the melting time has tobe in the range of about 100 s, which requires heating rates of R > K h − and the subsequent temperature decline has tooccur over a period of an hour or more (Hewins et al. 2005).In Figure 2 a schematic curve illustrates the basic temper-ature structure as required from the cosmochemical evidencefound within chondrules. Table 2 lists several models with amaximum temperature between T peak = T peak = ρ and the tem-perature T in the up-stream region ( x p , t p ) correspond to dif-ferent Mach numbers M of the shock transitions. The adoptednumbers vary around values suggested by simple models of theprotoplanetary nebulae. Fig. 2.
Schematic temperature curve as required by chondrulecharacteristics with preheating wave T =
300 K up to T − = t V =
400 s ( R > K h − ). The shock front heats thegas to T + = T = x p , t p ) and the relaxation zone ( x r , t r ) T ( x ) ≃ T − e −| x | / x p and T ( x ) ≃ T + e −| x | / x r , (28)which also defines the flow time through these regions by t p = Z − x p dxu ( x ) and t r = Z x r dxu ( x ) . (29) x denotes the distance from the shock wave located at x = ff erence between the surface and the cen-ter of a chondrule is relative small, as seen from the values of ∆ T max . Figure 5 shows the maximum temperature in a sphericalbody in dependence of its radius with T peak =
4. Discussion
For model 1 with initial temperature T =
300 K and peak tem-perature T peak = ρ = . · − kg m − was chosen so that the requirement of a heating rate R > K h − is fulfilled. Model 2 shows that at initial density of ρ = . · − kg m − the necessary heating rate is not achieved.A higher initial density ρ , like in model 3, leads to significantlyhigher heating rates. Models 4 and 5 show variations of the ini-tial temperature T , based on model 1. A temperature decreaseleads to higher heating rates (model 5). The necessary heatingrates cannot be reached any more by model 4. Models 1 to 5 arealmost subcritical shock waves, models 6 to 10 with peak tem-peratures T peak = ff ect on the duration of the preheating wave and thereby on theheating rate. The increase of temperature in supercritical shockwaves at the beginning of the preheating waves fulfills the nec-essary heating rates. The particles stay for a long time in thepreheating wave until the shock occurs.
5. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves
Table 1.
Parameter of RHD shock waves with γ = / Model 1 2 3 4 5 ρ [10 − kg m − ] 2.0 1.0 3.0 2.0 2.0 T [K] 300 500 100 T + [K] 1700 M x p [km] 83 3600 370 1100 980 x r [km] 175 225 132 220 131 t p [s] 179 780 79 240 199 t r [s] 43 55 32 60 30 R [10 K h − ] ∼ . ∼ . ∼ . ∼ . ∼ . ∆ T max [K] 13 . . . . . ρ [10 − kg m − ] 5.83 5.7 6.0 5.83 T [K] 300 400 200 T + [K] 2100 M x p [km] 1150 5600 310 76000 106 x r [km] 122 122 122 125 119 t p [s] 232 1220 59 17100 20 t r [s] 36 36 36 38 32 R [10 K h − ] ∼ . ∼ . ∼ . ∼ . ∼ . ∆ T max [K] 26.8 26.2 28.8 20.7 33.7 Investigating the range of possible initial conditions as con-strained by models of protoplanetary discs, we found that thegas can be heated up to T peak = γ = /
3, opacities without dust (Alexander opacities ofGrevesse & Noels (1993)) and a disc models without accretion(e.g. the minimum mass nebulae of Black & Metthews (1985))reveal that the particles are propagating within the preheatingwave for days to years, which does not match the required heat-ing rates. Molecular gas and the dust-enriched opacities resultin better agreements with the observational facts of chondruleresearch. The models of a protoplanetary disc including accre-tion (Bell et al. 1997) have much higher gas densities in the midplane around 2 . ρ together with a decrease ofthe temperature T makes molecules and therefore γ = / ff ects reduce the du-ration of the dust particle in the preheating wave and increasethe heating rates.The calculated temperature di ff erences between surface andcore of the chondrules (Fig. 5) are correct as long as the chon-drules remain solid particles. But up to now we have not incorpo-rated phase transitions in this chondrule model. Clearly, strongshock waves with high Mach numbers M as well as shock waveswith smaller preheating waves will produce larger temperaturedi ff erences but are unlikely to ensure the stringent cosmochem-ical constraints. As depicted in model 10, the preheating wavevanishes completely and the sudden increase of the temperatureat the outer boundary yields the largest temperature di ff erenceswithin chondrules. For these conditions it is easier to explainpartial or shell melting of the particles.Immediately after the shock passage we observe a rapidcooling on a time scale of minutes owing to radiative losses de-termined by the opacities. This e ff ect is followed by a longer Fig. 3.
The temporal structure of the temperature within a RHDshock wave reaching a maximum of T peak = ρ , T = = const; bottom:Variations in T , ρ = · − kg m − = const; solid line: model 1,dotted line: model 2, dashed line: model 3, dash-dot line: model4, dash-dot-dot line: model 5time scale of several days where non-equilibrium processes fur-ther reduce the post shock temperature of the gas and the chon-drules. The rapid cooling rates of our models of about 10 K / hrseem to disagree with furnace experiments, which indicate val-ues of around 100 K / hr. The reason for this discrepancy is dueto di ff erent time scales involved during cooling processes in thepost shock region. The calculations of, e.g. Desch et al. (2005)include the energy and momentum exchange between the chon-drules and the gas. Therefore the temperature structure in thepost shock region is controlled over longer time scales by anadditional cooling term Λ ( x ), which includes the radiative cool-ing by H O-molecules. Consequently, the far down-stream gastemperature can be set to the far up-stream temperature T . Thisleads to overall cooling rates of 100 K / hr, which cannot be ob-tained within our physical description, which neglects this addi-tional cooling term. The boundary condition demands T = T for the gas temperature as specified, e.g. in Desch et al. (2005),and the post shock region cools down to the initial tempera-ture by radiative looses into the surrounding protosolar nebula.However, this additional cooling occurring on a longer spatialand temporal scale reduces the overall cooling rates to less than100 K / hr. From furnace experiments (Hewins et al. 2005) it hasbecome clear that the exact shape of cooling path has no influ-ence on the final cooling rate.The recent work of Morris & Desch (2010) reveals how ad-ditional physical mechanisms like molecular line cooling dueto water influence the cooling history in the post-shock region.However, these authors emphasize that the post-shock region isonly slightly changed because several e ff ects partly cancel each
6. Joham and E. A. Dorfi: Formation of chondrules in radiative shock waves
Fig. 4.
Heating to T peak = ρ , T = = const; bottom: variations in T , ρ = . · − kg m − = const; solid line: model 6, dotted line:model 7, dashed line: model 8, dash-dot line: model 9, dash-dot-dot line: model 10 Fig. 5.
Maximum relative temperature di ff erence within a spher-ical chondrule as a function of its radius in [ mm ] for shock waveswith T peak = / s to8 km / s.Several models have been developed during the past years tocalculate the behaviour of dust particles within shock waves (e.g.Desch & Connolly 2002; Miura & Nakamoto 2005). However, none of the proposed models have included the full set of RHDequations with the most recent gas and dust opacities, which areessential for the correct structure of the precursor region. Thiszone is shaped by the radiation transmitted from the shockeddown-stream gas and the dust particles are advected through thisregion towards the shock wave. As inferred from the cosmo-chemical evidence, this region provides a basic heating processand will therefore restrict the possible Mach numbers. As seenfrom the models (cf. Table 1), already moderate Mach numbersaround M ≃ T =
300 K) meet the constraints of chon-drule formation with parameters reasonably within protoplane-tary nebula models.Opacities changes due to dust-enriched environments causedby fragmentation and collisional debris (e.g. Ciesla & Hood2002) may significantly increase solid-to-gas ratios, and val-ues of 100 or even 1000 times the solar ratio can be ex-pected. Clearly, shock waves propagating in such a densely dust-populated environment requires a more detailed treatment of theopacities (Desch et al. 2005), as adopted in these simulations. Inparticular, all time- and length scales controlled by the opacityare then changed.As mentioned already, the shape and / or the mass of the par-ticles can be modified as they propagate through the shock re-gion. First, the partial or total melting could lead to a massloss by evaporation or, depending of the surface tension, alsoto a rearrangement of the surface, i.e. the particles can becomemore spherical. Secondly, drift velocities acting on the surfacecan result in an erosion and thereby decrease the particles mass.Thirdly, the radiative interactions with the dust particles togetherwith the additional heat needed to melt the particle should be in-cluded into these models of radiative shock-heated chondrules.The dissociation of molecules also demands a more realistictreatment of the equation of state. These structural e ff ects of thedust particles will be discussed and included in a forthcomingpaper in more detail. Acknowledgements.
The publication is supported by the Austrian Science Fund(FWF).
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