This paper continues our effort to under-stand the condensation of carbon solids in a gas of pure C and O atoms when these exist within the interior of an expanding young supernova. This setting has a sufﬁciently large number of energetic electrons that the CO molecule is disrupted with a lifetime of a few months, causing the abundance of CO to be no greater than 1% or so of the C abundance. As a consequence, the CO molecule cannot consume the available C atoms, even when the O abundance exceeds that of 

Clayton, Donald D

Deneault, E

Meyer, B S

The, L.-S.

English

Clemson University: TigerPrints

Clemson UniversityTigerPrintsPublications Physics and Astronomy2001Abundances of Linear Carbon-Chain Molecules inSupernovaeDonald D. ClaytonClemson University, claydonald@gmail.comE DeneaultClemson UniversityB S. MeyerClemson UniversityL.-S. TheClemson UniversityFollow this and additional works at: https://tigerprints.clemson.edu/physastro_pubsThis is brought to you for free and open access by the Physics and Astronomy at TigerPrints. It has been accepted for inclusion in Publications by anauthorized administrator of TigerPrints. For more information, please contact kokeefe@clemson.edu.Recommended CitationPlease use publisher's recommended citation.ABUNDANCES OF LINEAR CARBON-CHAIN MOLECULES IN SUPERNOVAE. D.D. Clayton, E. Deneault, B. S. Meyer,L.-S. The,Department of Physics and Astronomy, Clemson University, Clemson SC 29634-0978.Introduction. This paper continues our effort to under-stand the condensation of carbon solids in a gas of pure C andO atoms when these exist within the interior of an expandingyoung supernova. This setting has a sufficiently large numberof energetic electrons that the CO molecule is disrupted with alifetime of a few months, causing the abundance of CO to be nogreater than 1% or so of the C abundance. As a consequence,the CO molecule cannot consume the available C atoms, evenwhen the O abundance exceeds that of C. The maintenanceof free C atoms is the driver for condensation of carbon (1,2).Clayton et al. (2) presented a sweeping picture of the chemi-cal sequence that enables large micron-sized carbon grains tocondense even when C<O. We adopt this same sequence: (1)carbon association reactions balanced approximately by rapiddestruction reactions determine the abundance of linear Cnmolecules within the gas; (2) given sufficient length, the chainmolecules isomerize to ringed molecules, for which further Cassociations become more rapid than oxidations; (3) mono-cyclic rings grow to bicyclic and multicyclic sizes, includingfullerenes; (4) multicyclic rings seed the inexorable growth tomacroscopic particles owing to very fast C association. In thispicture, the large Cn linear chains and their ringed isomers con-stitute seed nucleations for the growth of macroscopic grains ofcarbon; therefore the key chemistry governs the abundances ofthose linear Cn molecules. Because the destruction reactions(oxidation, C association, or C-induced fission) for Cn occurvery rapidly (about 1 sec when N(C) = N(O) =1010cm−3 withdestruction rate coefficient k =10−10cm3s−1), destructive re-actions limit the abundance of linear Cn to a small steady statevalue. Steady state is achieved more rapidly than environmen-tal changes (density and temperature), which occur only overmonth timescales after one year, when condensation mostlyhappens. For that reason, we evaluate in this work the station-ary abundances of Cn and seek to understand what chemicalrates are of importance to that abundance. Our approach maybe likened to that of classical nucleation theory, except thatthe nucleation abundances are determined by the balance ofkinetic rates for creation and destruction rather than by themore restrictive assumption of chemical equiibrium (the usualassumption of nucleation theory). Indeed, the abundances ofCn molecules are vastly greater than would obtain in thermalequilibrium, for in that case CO molecule formation wouldconsume the carbon.Reaction Rates. We take as our standard set of kineticrates those given by (2) in their Table 1. The destruction ratefor even linear molecules C2n reaches the maximum kineticrate k =10−10cm3s−1 by both oxidationO + Cn −→ CO + Cn−1 (1)and by association-induced fissionC + Cn−1 −→ Cm + Cn−m (2)The odd chains C2n+1 are also destroyed at the maximumkinetic rate, but instead by C association reactionsC + Cn−1 −→ Cn + hν (3)whereas their oxidation is slower kO = 10−13cm3s−1 and fis-sion is taken to be even slower yet. Association rates arefast because the larger chains do not have to radiate imme-diately to conserve energy, but instead simply exist in statesof high excitation to be stabilized eventually by photon emis-sion. This greatly increases the asociation rate kC above thevalue required if radiative stabilization must be prompt. TheC association rate kC is slower for even-n chains owing to thecompetition by the fast C-induced fission (2). Using this basicset of rates for reference calculation we solve for the steadystate abundances. The reference calculation takes the numberdensities to be1010cm−3 for both C and O; but the dependenceon the C/O ratio is determined by numerical solutions.Setting the rate of change of each molecular abundance tozero yields a matrix equation for the abundance vector [Nn]= [C2, C3, C4,...C24]. With [M] being the matrix of reactionrates, the abundance equation can be written [M]× [Nn] = 0except in its first row, which equals instead the production ofC2 from C atoms (which lie outside the elements of the vec-tor [N]). The solution is straightforward and will be describedmore completely in a work in progress (3).Results. Figure 1 displays stationary abundances of thelinear molecules as fractional abundances N(Cn)/N(C) rela-tive to free C atoms, which prior to condensation constituteroughly 99% of the total density of C atoms. The standard cal-culation has ambient ratio C/O =1; but results for C/O = 5, 1/5and 1/25 are also shown for comparison. For the range shown(C8 to C18 only) the fractional abundance drops steeply fromabout10−18 for C8 to about10−31 for C18. To appreciatesuch small numbers consider that a fractional abundance near10−20 is needed for the final micron-sized graphites if eachcontains1015 C atoms (say) and if they are to constitute about10−5 of all C by mass in the ISM. This will require isomer-ization rates into ringed C that produce about10−20 rings perC atom during roughly the first year of expansion. This mightLunar and Planetary Science XXXII (2001) 1109.pdfLINEAR CARBON CHAINS: D. D. Clayton et al.be achieved in various ways: a chain with abundance 10−20isomerizing at the rate10−7s−1 would yield 3×10−20 ringsper year; or an abundance 10−28 isomerizing at rate 10s−1would also give 3×10−20 rings per year. These possibilitieswill be explored by (3). Note that the abundances of Fig. 1 donot include depletion by the unknown isomerization rates.Another conclusion from Fig. 1 is that different C/O ra-tios produce parallel results; but reduction of C/O by factor5 reduces the fractional abundances of Cn only by a factornear 100. Although this makes already small abundances evensmaller, it is important to note that the abundances are not zerofor C/O = 1/5. This illustrates the kinetic non-LTE nature ofthis problem. Were one to instead calculate chemical ther-mal equilibrium at C/O = 1/5, the abundance of Cn would bevanishingly small! Secondly, even if C/O = 1/5,10−20 ringsmay be produced per year from C14, say, if its isomerizationrate is near10s−1. On the other hand, a carbon-rich C/O =5 produces no great increase of abundances in the supernovaproblem whereas it is a huge facilitator in equilibrium conden-sation calculations because when C/O=5 one has free carbonin the equilibrium calculation; but carbon is maintained freeby radioactivity in the supernova problem. These ideas clarifyimmediately that graphite grains may grow even if C<O (2).Population Control. Having more abundant nucleationsdoes not imply more large particles. The nucleations must notbe too abundant or there will not exist enough C atoms for themto grow into large graphite particles. The kinetic destructionof the linear molecules controls their population, which in turnallows them to grow large before free C is depleted. Sufficientcollisions to deplete all of the carbon do occur within an ex-panding supernova interior. If the ringed nucleation moleculeswere as abundant as10−12 they could grow no larger than 1micron (1012 atoms) before depleting all of the C into graphite.They would thus be too abundant to give mass fraction10−5 to1 micron particles. We call this principle "population control",and we note that for this problem ringed-isomer abundancenear10−20 appears to be about right.Rate Survey. Because supernova condensation is con-trolled by kinetic rates it is of interest to examine which re-action rates are the most critical for the outcome. In whatfollows we increase or decrease all of the reaction rates ofspecified type. Fission rates are very important. We havetaken their rates to be as fast as oxidation in the destructionof the even-n chains. Figure 2 shows the standard referencecalculation as well as a comparison result if the fission rateis slowed from the fast kf = 10−10cm3s−1 to 10−11cm3s−1and then even more to10−12cm3s−1 with all other rates heldat the reference values (green curves). It is clear that despitethe continued rapid destruction by oxidation, the abundancesincrease greatly (about 15 powers of ten for C20) by the slow-down factor (100) in these even-n fission rates. Fission is sodestructive owing to the high probability that the smaller frag-ment may be lost altogether by oxidizing it down to C2 whereanother oxidation removes a molecule.The carbon association rate with linear C2n is also impor-tant but unknown. Because we have taken the C associationrate to be small for even C2n, only 1/1000 of the fission rate,we explored that high sensitivity as well in Fig. 2 (red curves).Increasing kC from 10−13cm3s−1 to 10−11cm3s−1 whileholding other rates fixed vastly increases the abundances. In-creasing association tenfold to kC = 10−12cm3s−1 is almostidentical to slowing the fission rate by a factor 10. This showsthat it is the branching ratio kC /kf that governs the abundancesas long as modest variations are involved. But further increaseto kC = 10−11cm3s−1 results in huge abundance increase.Even though that association rate remains smaller than bothdestruction rates of C2n by a factor ten, it produces a muchflatter abundance curve. With such a flat curve, the doublingof molecule number accomplished by each fission plays animportant new role in the steady state abundances, which be-come even greater than those that would obtain were there nofission at all. Inefficient population control resulting from thatvalue of kC creates too many particles (unless isomerizationto ringed structure is much slower than expected), resulting ina large number of small graphite grains rather than a smallernumber of large ones (as observed). We have also shown thatthe assumption of fast kC for the odd-n linear chains C2n+1is not so crucial. This is because C association dominates de-struction of odd-n molecules; decreasing all such rates wouldtherefore increase their abundances but for its also decreas-ing the creations of the even-n C2n by compensating factors.These compensate within the general decline. We find thateven two powers of ten decrease of kC leaves the same generaldecline in abundances, although it does smooth out the odd-even effect.We have taken the oxidation rate for odd-n chains to bekO = 10−13cm3s−1, which is much smaller than the C asso-ciation rate for odd-n. But that is not a critical unknown. Wefind that increasing the value of kO by even a factor 100 doesnot alter the abundance curve. By such results we propose toclarify for chemists the identity of chemical rates that governthe supernova presolar-grain problem.Acknowledgements. This work was supported by grantsfrom NASA’s Cosmochemistry and Origins of Solar SystemsPrograms and from the NSF.References. (1) Clayton, D.D. (1998)LPSC 29. (2)Clayton, D.D., Liu, W. and Dalgarno, A. (1999)Science, 283,1290-1292. (3) Clayton, D.D., Deneault, E., Meyer, B.S. andThe, L.-S. (2001)ApJ, to be submitted.Lunar and Planetary Science XXXII (2001) 1109.pdf

Abundances of Linear Carbon-Chain Molecules in Supernovae

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Abundances of Linear Carbon-Chain Molecules in Supernovae

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