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Nuclear Reactions Important in Alpha-Rich Freezeouts

Nuclear Reactions Important in Alpha-Rich Freezeouts
Nuclear Reactions Important in Alpha-Rich Freezeouts

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Nuclear Reactions Important in Alpha-Rich Freezeouts George C.Jordan,IV,Sanjib S.Gupta,Bradley S.Meyer,Lih-Sin The 1,?1Department of Physics and Astronomy,Clemson University,Clemson,SC 29634-0978(Dated:February 8,2008)Abstract The alpha-rich freezeout from equilibrium occurs during the core-collapse explosion of a massive star when the supernova shock wave passes through the Si-rich shell of the star.The nuclei are heated to high temperature and broken down into nucleons and αparticles.These subsequently reassemble as the material expands and cools,thereby producing new heavy nuclei,including a number of important supernova observables.In this paper we introduce two web-based applications.The ?rst displays the results of a reaction-rate sensitivity study of alpha-rich freezeout yields.The second allows the interested reader to run paramaterized explosive silicon burning calculations in which the user inputs his own parameters.These tools are intended to aid in the identi?cation of nuclear reaction rates important for experimental study.We then analyze several iron-group isotopes (59Ni,57Co,56Co,and 55Fe)in terms of their roles as observables and examine the reaction rates that are important in their production.PACS numbers:24.10.-i,26.30.+k,26.50.+x

I.INTRODUCTION

Progress in the science of stars and their nucleosynthetic processes relies on the con-tinued interplay of astronomical observations and astrophysical modeling.Observations of abundances of chemical species,elemental or isotopic,constrain astrophysical models while the models,in turn,provide a framework for interpreting the observations.It has long been clear,however,that uncertainities in the input physics to the models limit them and their usefulness in interpreting abundance observations.Nuclear reaction rates are key in-puts into the astrophysical models,and,though many are measured,most have not,and modelers must therefore rely on theoretical predictions of the value of these rates.Recent theoretical reaction-rate predictions have proven fairly accurate(to within a factor of a few of the actual rate value where subsequently measured–e.g.,[1]),nevertheless experimental results are usually desirable and often crucial.A third essential e?ort in nuclear astrophysics is that of nuclear experimentalists who seek to provide better input into astrophysical mod-els by measuring the rates of nuclear reactions or nuclear properties that improve theoretical estimates of the rates.Because of the cost in time,e?ort,and?nancial resources in perform-ing the necessary experiments,however,the task of the nuclear experimentalists is greatly aided if the astrophysical signi?cance of a particular nuclear reaction can be clearly demon-strated.This in turn requires demonstration of the dependence of the predicted value of an astronomical observable in an astrophysical model on the value of the reaction rate.It is clear that observers,modelers,and experimentalists should all play an important role in the planning of nuclear astrophysics experiments,in particular by ensuring that any nuclear reaction rate proposed for experimental study satisfy the following four requirements[2]:

1.An appropriate astrophysical model of a nucleosynthesis process must exist.

2.An observable from that process,usually an abundance result,is either known or

measurable.

3.The dependency of the value of the observable on the value of the nuclear cross section

is demonstrable.

4.An experimental strategy for measuring the reaction rate,or at least using experimen-

tal data to better calculate the reaction rate,should be available.

A long-term goal of our program is to aid the dialogue among the three parties(observers, modelers,and experimentalists)in the nuclear astrophysics community by making it easier to identify those nuclear reactions most in need of experimental study.In the present work,we focus on reaction rates important in the alpha-rich freezeout in core-collapse supernovae.We explore in some detail astronomical observables from this process and the nuclear reactions that govern their nucleosynthesis.In addition,we present web-based tools that allow any interested researcher to explore reaction-rate sensitivities in the alpha-rich freezeout and thereby make a case for experiments on nuclear reactions important for other observables (yet to be identi?ed)from this process.We hope that this work can serve as a template for future work(by ourselves or others)on nuclear reactions important in other nucleosynthesis processes.

This paper begins with an introduction to the alpha-rich freezeout and a description of our freezeout calculations and the attendant reaction-rate sensitivity studies in§II.In§III we introduce two web sites accessible from the main page https://www.wendangku.net/doc/8a17480451.html,/ gjordan/nucleo/.The?rst website displays data from the sensitivity calculation performed on our alpha-rich freezeout model in order to identify nuclear reactions that are important in the production of nuclei identi?ed as astrophysical observables.With a particular isotope in mind,one may then use the web site to view the e?ect of di?erent reaction rates on the yield of that isotope.The second website can be used to calculate the e?ects of a varied reaction rate on nucleosynthesis yields under conditions that di?er from the conditions used in our sensitivity survey.Control over several of the parameters in the explosive model is given to the user so that the e?ect of a particular reaction rate on the network yields can be explored over a wide variety of conditions,thereby allowing the user to strengthen his case for measuring a particular reaction.Finally,due to their signi?cance as astrophysical observables,several isotopes from the iron-group nuclei are examined§IV.

II.THE CALCULATIONS

Many processes contribute to the production of new nuclei in core-collapse supernovae, but one of the most important for astronomy is the alpha-rich freezeout from equilibrium. During the core collapse of a dying massive star,a shock wave develops as matter falls supersonically onto the collapsed stellar core.The shock,aided by a push from neutrinos

from the cooling nascent neutron star,expands out into,heats,and expels the overlying stellar matter.In the initial heating of the innermost regions of the ejecta,post-shock temperatures are su?ciently high that nuclei are broken down into nucleons andαparticles. As the material subsequently expands and cools,the nucleons andαparticles reassemble to form heavy nuclei.Because of the fast expansion of the matter,however,not all alpha particles reassemble,and,as a result,the?nal abundances freeze out with a signi?cant numberαparticles remaining,hence the name alpha-rich freezeout.A number of signi?cant astronomical observables are produced in this process including44Ti,56Co,and57Co.

In order to explore the sensitivity of alpha-rich freezeout yields to variations in reaction rates,we utilized the Clemson nucleosynthesis code[3],which we have updated to employ the NACRE[4]and NON-SMOKER[5]rate compilations.The network used(see[2])includes 376species from neutrons up to Z=35(Bromine)and2,125reactions among them.

We began the survey with calculations using reaction rates from the compilations.Guided by detailed models of such astrophysical settings(e.g.,[6]),we chose an initial tem-perature of T9=T/109K=5.5and initial densityρ0=107g/cm3in all calculations. The matter was taken to expand exponentially so that the density evolved with time t as ρ(t)=ρ0exp(?t/τex),where the density e-folding timescaleτex=446s/√

=2,125×3×2+3=12,753.

The e?ect of modifying a particular reaction can be determined from the ratio of the modi?ed yield of a particular species with the reference yield.Clearly with2,125reactions and376species,there are many combinations to consider.Rather than present several large tables,we have opted to construct interactive web-based applications to display the results.

III.INTERNET APPLICATIONS

In this section we introduce two web-based applications.Since the web sites have complete online help?les,detailed instructions are not given here.We will simply introduce the web sites and some of their relevant features,along with instructions on how to reproduce the tables appearing in the following sections.

https://www.wendangku.net/doc/8a17480451.html,work Sensitivity Data Display

The Sensitivity Data Display website is designed to examine the results of varying the reaction rates involving a selected isotope.Upon loading the page,the user selects from one of six data sets.The data sets are speci?ed byηand by whether the reaction rates are increased or decreased(by a factor of10in each case).Next,the user can generate lists sorted by increasing or decreasing modi?ed-to-standard ratios for a selected isotope.An isotope(the chosen observable)is selected by entering its atomic number and mass number in the appropriate?elds.Aβ-decay option is also available which permits the user to evolve the abundances to arbitrarily later times upon cessation of the alpha-rich freezeout process. (The numerical technique used to construct the exact solutions to the coupled di?erential equations governing theβ-decay of the abundances is described in the Appendix).

The data in Tables V,VI,and VII of§IV were collected using this web site.To recon-struct Table VII,for example,the user selects the data set corresponding toη=0.006and a multiplicative factor of0.1.For55Fe,26is entered as the Z value and55for the A value. We target an observation of the55Fe abundance made2years after the explosion,and so the user selects theβ-decay box and enters an evolution time of2years into the time?eld.

In order to display only the reactions that change the55Fe yield by more than20%, the ratio cut-o?parameter is set to0.20.The“Descending List”button is then clicked to

generate the ratio list for55Fe.The?rst table generated produced gives the mass fractions of all A=55species in the network at the end of theη=0.006reference alpha-rich freezeout calculation together with their beta-decay or electron-capture lifetimes.Isotopes highlighted in black are stable while those in pink are unstable againstβ+or electron capture.Isotopes highlighted in blue are subject toβ?-decay while green denotes instability via both channels.

In the second table,the column on the far right of the output contains the modi?ed-to-standard ratios after an interval of two years.The reactions which,when their rate is decreased by a factor of ten,result in a minimum increase in the yield of55Fe of20%are entered in the table along with the respective ratios.Similarly,those reactions which,when their rate is decreased by a factor of ten,result in at least a20%decrease in the55Fe yield were also entered in the table along with their ratios.The process was then repeated with a rate factor of10andηof0.006.The data for Tables Vi and VI were created in a similar manner,but for the isotopes57Co and59Ni.

Other options on the web site include(a)a table of the isotopes used in network calcula-tions,(b)a list of reaction rates incluced in the calculations and(c)a tool for the user to plot or make a table of the standard reaction rates used in the calculations.The user can use the reaction number from the reaction list and then input that reaction into the reaction-rate engine to obtain either a plot or table of the rate as a function of temperature.(Note that the rate we present may include a stellar enhancement factor to correct for the possibility of excited states in the target nucleus.The user must account for this factor when comparing his rate with the one we used.)Other links on the page are to various related topics in the help?le and to the NACRE and NON-SMOKER rate compilation web sites.

B.Explosive Nuclear Burning

The Explosive Nuclear Burning web site simulates parameterized explosive nuclear burn-ing over the web using the same reaction network as described above.Among the variable parameters are T9(initial temperature in109K),ρ(initial density in g/cm3),τex(the ex-pansion time scale of the explosion),and the initial mass fraction of each species.The user can also alter any reaction rate in the network by a multiplicative factor as was done in§II.

The data output from the web site are the?nal mass fractions from the calculation.If a reaction rate is modi?ed,the web site performs two calculations,one with the modi?ed

reaction rate,and one with the standard reaction rates for comparison.The data output in this case are the?nal mass fractions from both calculations along with their modi?ed-to-standard ratios.The web site o?ers several options for further analysis,including aβ-decay option to evolve the?nal mass fractions in the output.

This web site was used to produce the data from Tables I,II,III,and IV.To recon-struct these tables,the user performs the following steps.First,the user selects the default parameters,which correspond to an alpha-rich freezeout of silicon-dominated matter with anηof0.006(the parameters the sensitivity calculations in§II).When the calculation is executed,the appropriate mass fractions are inserted into the“t=0years”column of the Table.Theβ-decay box is then selected from the“Final Mass Fraction Data List Controls”table in the left hand frame and a time of2years entered.The“Create List”button is then clicked.The web site returns a table with the value of all of the?nal mass fractions after2 years.The?nal mass fractions should agree with those in Tables I,II,III,and IV.

The data for Table VIII was also generated with this web site.To reconstruct this table, the user performs the following steps.First,the web site is reloaded into the browser.Then the“Edit”button in the“Reaction Rates”row of the“Parameter Controls”table in the left hand frame is clicked.This brings up an interface that allows the user to select a reaction rate to modify based on the type of reaction preferred.Since the reaction of interest in Table VIII is55Co(p,γ)56Ni(a two nuclear species to one nuclear species reaction),the radio button at the bottom of the table entitled“i+j→k”is clicked.Starting at the left hand side of this table,the consituents of the reaction are chosen?eld by?eld.When all of the ?elds in the table are selected,the“Save Changes”button at top(or bottom)of the page is clicked.The user is then prompted to input a reaction rate multiplier.A value of100 is entered and the“Save Changes”button clicked.The web site then presents a summary of the changes and the calculation is begun by clicking the“Run Nucleosynthesis Code”button.The outputted data are then evolved for two years as above and the appropriate modi?ed-to-standard ratio is entered into Table VIII.The web site is then reloaded and the same procedure is carried out except0.01is entered as the multiplicative factor on the 55Co(p,γ)56Ni reaction rate.These steps should reproduce the data in Table VIII.

This web site has many other useful tools that are not described above but are documented in the on-line help?les.As shown,this web site allows the user to examine the e?ects of a reaction on nucleosynthesis yields under a wide variety of circumstances.Our hope is that

this easily accessible data will provide additional insight into the e?ects that uncertainties in reaction rates have on the yields of the alpha-rich freezeout and other nucleosynthesis processes.

IV.NUCLEAR REACTIONS GOVERNING THE SYNTHESIS OF IRON-GROUP OBSER V ABLES

In this section,we present some results of our calculations drawn from the web sites described in§III.For brevity we limit the discussion to theη=0.006calculations.The purpose is to illustrate how one may identify relevant observables and identify their governing reactions using the web sites.

At least three types of isotopic observables are relevant for stellar nucleosynthesis:

1.The bulk yields are important for understanding galactic chemical evolution and solar

system abundances.

2.Radioactive species such as26Al and44Ti can be observed from space telescopes,and

consequently provide important constraints on their production sites.

3.Isotopic abundances in presolar meteoritic grains carry important information about

the nucleosynthesis environments in which their isotopes formed and the astrophysical settings in which the dust grains condensed(e.g.,[7,8]).

In the subsections to follow,we present several isotopes from the iron-group nuclei for which we used the web-based applications in§III to obtain data.These four isotopes are directly or indirectly linked to all three types of observables listed above.We have chosen to analyze57Co and56Co because of their prominent role asγ-ray observables and59Ni and55Fe for their potential as future X-ray observables.In addition,the abundances of the daughter isotopes of all four of these species may eventually be measured in presolar grains. We do not analyze44Ti,a key alpha-rich freezeout observable,because this has been done previously[2].

For each species,we list any reaction rate that produces at least a20%increase or decrease in the alpha-rich freezeout yield due to a factor of10change in the reaction rate(except for 57Co as explained below).Since we primarily consider theγ-ray or X-ray observations of

these species,we chose a time of analysis2years after the freezeout.By this time,overlying supernova should have become transparent to these radiations.Often,several years pass,if not thousands,before data from astronomical observables can be collected.It is,therefore, up to the user to make his own analyses using appropriately chosen times in the web tools described in§III.For each observable considered,we list the important reactions,and for two particularly interesting reactions we explore in some detail the reason for their e?ect on the chosen observable.

A.56Co

This species is produced primarily as the radioactive parent56Ni(τ1/2=6.075days) which decays through56Co(τ1/2=77.2days)to56Fe(see Table I).The overlying supernova largely obscuresγ-rays from the decay of56Ni,but the2.598Mevγ-rays(for example)from the56Co can escape and be detected[9],making the latter species a valuable alpha-rich freezeout observable[21].Theγ-rays from56Co decay can be used to determine the total yield of the parent(56Ni)as well as the opacities of the outer envelope of the supernova(for a review ofγ-ray observables see[10]).

From the web sites one?nds that the abundance of56Co after two years is dependent on the yield of56Ni from the supernova.We?nd that the yield of56Ni immediately after theη=0.006alpha-rich freezeout is quite insensitive to factor-of-ten changes in the reaction rates. The largest e?ect was due to changes in the triple-αreaction which produced about a6% change in the yield.This6%change propagates as a6%change in the yield of56Co after the decay of56Ni.These are not signi?cant changes and we conclude that the calculated yields of56Ni(and thus the yields of56Co)are quite robust against any reaction rate uncertainties. This result is expected since the production of56Ni is dominated by the equilibrium phase of the expansion in which the abundances are set by binding energies and nuclear partition functions(not by individual reaction rates).

B.57Co

A second important isotope is57Co.This isotope is similar to56Co in that its stable daughter57Fe owes a signi?cant portion of its synthesis to alpha-rich freezeouts.From

Table II we see that57Co is primarily produced as the parent isotope57Ni.Gamma-rays from the decay of57Co are observable with space detectors,and they also power the supernova light curve at somewhat later times than56Co[11].As shown,factors that a?ect the yield of57Ni thus have an e?ect on the yield of57Co.

The web sites show that the yield of57Ni immediately after the alpha-rich freezeout is (like56Ni)largely insensitive to the value of any particular reaction rate because both are primarily equilibrium products.The reaction with the most signi?cant e?ect(forη=0.006) is57Ni(n,p)57Co.We have chosen to list this reaction even though it does not produce more than a20%change because of the importance of57Co as an observable.From Table V we see that this reaction produces at most a17.7%change in the yield of57Ni for a factor of ten change in the rate.The primary reaction rate governing the yield of57Co forη=0.006 is57Ni(n,p)57Co and a factor of a few uncertainty in the reaction rate results in a several percent uncertainty in the observable.

The reason57Ni(n,p)57Co has an e?ect on the yield of57Ni(and thus the yield of57Co) is that it governs when57Ni falls out of quasi-statistical equilibrium(QSE–see,for example, [12]).As shown in?gure1,the57Ni abundance diverges from nuclear statistical equilibrium (NSE)early but remains in line with QSE expectations until it freezes out below T9≈

3.5.The QSE favors57Ni early,but as the material cools below T9≈

4.3,the QSE abun-dances shift to higher-mass nuclei;thus,the longer57Ni remains in QSE,the lower its?nal abundance.57Ni falls out of QSE when the57Ni(n,p)57Co reaction becomes too slow,which means that increasing the rate for this reaction causes57Ni to remain in QSE longer and to have a lower?nal abundance.Decreasing the rate means57Ni falls out of QSE earlier and retains more of its originally high QSE abundance.

C.59Ni

Both59Ni and55Fe have been chosen because they hold promise as detectable radioactive X-ray sources(for a prospectus see[13]).In the decay of these isotopes by electron capture, a K-shell vacancy occurs which is often?lled by an electron previously occupying a higher-level bound state.The energy loss from this event is carried away as an X-ray which,if detected,would not only provide data complementary toγ-ray observations,but also shed light on the production of radioactive isotopes that emit noγ-rays.

The observable59Ni,because of its75,000y half-life,should produce a signi?cant di?use emission from the interstellar medium similar to that observed from interstellar26Al.It may also be detectable in close,individual supernova remnants[14].From Table III we see that the majority of59Ni is produced in the alpha-rich freezeout as59Cu;thus,factors that a?ect the production of59Cu ultimately a?ect the production of the observable59Ni.From the web sites it is evident that there are several reactions which,when changed by an order of magnitude,have a moderate e?ect on the production of59Cu(see Table VI).The most in?uential of these is the59Cu(p,γ)60Zn reaction.A factor of10decrease in this reaction rate produces about a50%increase in the yield,while a factor of10increase in the reaction rate drops the yield of59Cu by about30%.These are modestly signi?cant changes and show that59Ni is sensitive to speci?c reactions in the alpha-rich freezeout.

D.55Fe

Unlike59Ni,55Fe has a relatively short half life(τ1/2=2.73yrs),making it only detectable in very young supernovae(supernova1987A remains a good target for detecting55Fe[13]). Since the detection of55Fe could become available at very early times after the explosion (assuming the ejecta is transparent to X-rays),its detection should give information on small-scale structure formation in the ejecta.Further,it may help elucidate the velocity structure of the core and the velocity structure of55Fe[14].

As can be seen from Table IV,55Fe is predominantly produced as55Co.The production of55Co in the alpha-rich freezeout withη=0.006is most sensitive to the triple-αreaction and the55Co(p,γ)56Ni reaction(Table VII).The triple-αreaction has about a factor of two e?ect on the yield when a factor of10change in the triple-αrate is made.The largest e?ect on the yield is seen in the dependence of the55Co abundance on the55Co(p,γ)56Ni reaction.

A factor of10decrease in this reaction rate increases the yield of55Co by526%while an increase in the reaction rate reduces the yield by almost80%.

The reason for the sensitivity of55Co to the55Co(p,γ)56Ni reaction rate is similar to that of the sensitivity of57Ni to the57Ni(n,p)57Co reaction rate,but the e?ect is much larger. The55Co remains in(p,γ)-(γ,p)equilibrium with abundant56Ni until T9drops below3.0. As the temperature drops,the equilibrium shifts abundance from55Co to higher-mass nuclei (particularly56Ni).Increasing the55Co(p,γ)56Ni reaction rate causes this abundance shift to

persist longer,which in turn leads to a lower?nal55Co yield.A lower value for the reaction rate leads to an earlier freezeout from QSE and a higher?nal55Co yield.From this we see that55Co,and thus55Fe(the relevant observable),are very sensitive to the55Co(p,γ)56Ni reaction rate.

V.CONCLUSION

The above described web sites are intended to be a starting point for identifying those nuclear reactions that govern the production of astrophysical observables from the alpha-rich freezeout.It is our hope that these web sites will help motivate future measurements of these key reactions,especially as new observables from the alpha-rich freezeout are identi?ed.

While we identi?ed and analyzed four key observables from the alpha-rich freezeout, we expect new ideas or emphases in astrophysics to make new observables available.For example,as models of Galactic chemical evolution become increasingly re?ned,better yields from massive stars will become necessary.This means that any isotope that owes a signi?cant portion of its solar system abundance to the alpha-rich freezeout,such as40Ca,48Ti,52,53Cr and the ones discussed above,will become an observable.Similarly,new technologies may also open new possibilities for observing isotopes.As a possible example,44Ca excesses found in presolar supernova silicon carbide(SiC)X grains arose from condensation of live 44Ti[15].One speculation is that much of this excess radiogenic44Ca may be concentrated in small titanium carbide(TiC)subgrains within the larger SiC X grains.Such TiC subgrains are known to exist in mainstream SiC grains that condensed in out?ows from low-mass stars[16,17];however,in two SiC X grains studied so far,there is no direct evidence for the presence of such TiC subgrains[18],though the search continues.If such subgrains do exist,they are likely to be dominantly comprised of alpha-rich freezeout material.If the new generation of secondary-ion mass spectrometers are able to measure the isotopic abundances in the subgrains,then the alpha-rich freezeout abundances of all titanium and carbon isotopes,as well as the isotopes of any element that might condense in TiC,would become observables.Those new observables will open up new governing reactions requiring experimental study.

We view the web sites presented here as a?rst step for similar,future work on other nucleosynthesis processes.The goal in those future e?orts,as in the present work,will be

to help identify nuclear reactions that govern the nucleosynthesis of astronomical observ-ables.As always,however,the challenge remains not only to identify nuclear reactions that govern the synthesis of those observables,but also to identify the key astronomical observ-ables themselves.Only by maintaining a healthy dialogue among astronomers,astrophysical modelers,and nuclear experimentalists can we face that challenge and advance the science of nuclear astrophysics.

Acknowledgments

This research was supported by NASA grant NAG5-4703,NSF grant AST-9819877,and a DOE SciDAC grant.The authors acknowledge helpful discussions with D.D.Clayton,M.

D.Leising and P.Mohr.

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APPENDIX:ANALYSIS OF THEβ-DECAY ALGORITHM

We now draw attention to the algorithm used by the Sensitivity Data Display and the Explosive Nucleosynthesis web-sites to compute alpha-rich freezeout yields at a speci?ed point in time.Traditional approaches rely on Runge-Kutta or implicit Euler techniques to integrate the coupled di?erential equations(preferably with adaptive timestep adjustments). However,we have sought a rapid,but exact solution.To do this,we exploit the elementary matrix solution to the system of linear coupled DEs that utilizes eigenvalues and eigenvectors. For any matrix equation of the form

?x=Ax(A.1)

x(0)=b(A.2) the solution at all later times is given by:

x(t)= c k(Φk eωk t),(A.3)

whereΦk are the eigenvectors andωk the corresponding eigenvalues of the matrix A (constructed fromβ?decay rates and the application of mass conservation)which remain constant during the evolution.The coe?cients c k of the linear combination are the elements of the product T?1b(where T is the matrix of eigenvectors of A ordered strictly according to the index“k”)

The technical challenge here is that A is singular and thus reduction to upper Hessenberg form(the practical method of computing the spectrum of a large asymmetric system)using QR Factorization fails.We may circumvent this di?culty by employing very small perturba-tions to the rates and thus disturbing the rate matrix away from singularity,as is the norm for practical Numerical Linear Algebra applications in most?elds of engineering.However, we must be cognizant of the fact that a theorem of Watkins[20]requires the eigenvalues to be reasonably separated for the system to be insensitive to perturbations.Unfortunately for the full376×376matrix,the observed degeneracies are quite severe.

On closer inspection,however,it is realized that the cause of the degeneracies is quite simple:the matrix is composed of several non-communicating blocks.Since onlyβ?decays are considered,the vector space decomposes into the direct sum of these blocks,each cor-responding to a particular mass number.Blocks comprised of isotopes from di?erent mass numbers cannot tra?c with each other since we exclude other kinds of nuclear reactions. Several of these component submatrices are very similar in their spectra,and therefore the full space su?ers from many closely spaced eigenvalues.

The solution,however,is now obvious–we merely compute the spectrum of each block, con?dent in the knowledge that the isotopes within a particular block are not a?ected by those in any other.The complete state vector of the abundances at any given time is then a simple concatenation of the state vectors from the subspaces.Watkins’theorem is now observed to hold for each block(though the blocks themselves are singular as the full matrix was before)and abundances are output within seconds.

Obviously,the next step is to extend the perturbation technique to matrices which can-not be decomposed into non-communicating subspaces(blocks).Weakly coupled blocks may lend themselves to similar perturbation analyses.Unfortunately Watkins’theorem does not predict error bounds and the most we can do at this stage is carry out some rough numerical experiments to check the sensitivity of the system to small random perturbations.In our trials we found the e?ects to be negligible.Propagation of the perturbations caused?uctu-

ations on the order of10?15in the most ill-conditioned blocks.Abundances of order10?15 or less are quite small and for any practical purpose such small abundances are nonexistant. We conclude that the system was very robust to the small kicks we gave it.

We hope to incorporate the latest techniques in spectral computations on large asymmet-ric systems as they become available,since nucleosynthesis would be an exciting test-bed for these tools from the frontlines of research in Numerical Linear Algebra.In the interim, perturbation allows us to practically carry out an important calculation in seconds which would otherwise have been impossible if we were to insist on a spectral solution.

TABLE I:Mass Fractions for A=56withη=0.006

Isotope t=0years t=2years

57Cu00

57Ni3.967×10?20

57Co6.735×10?76.197×10?3 57Fe3.429×10?143.348×10?2

TABLE III:Mass Fractions for A=59withη=0.006

Isotope t=0years t=2years

55Ni3.306×10?100

55Co1.013×10?50

55Fe3.393×10?106.058×10?6 55Mn04.072×10?6

TABLE V:Data from theη=0.006network survey for57Co

Reaction Reaction Rate×0.1Reaction Rate×10.0

59Cu(p,γ)60Zn1.5560.615

2α(α)12C1.3240.731

59Cu(p,α)56Ni0.8481.444

TABLE VII:Data from theη=0.006network survey for55Fe Reaction Reaction Rate×0.1Reaction Rate×10.0

55Co(p,γ)56Ni34.5445.469×10?2

FIG.1:Mass fraction of57Ni versus T9for di?erent values of the reaction rate57Ni(n,p)57Co and in NSE and QSE.The value of the reaction rate near T9≈3.5governs when57Ni breaks out of

QSE.

FIG.2:Mass fraction of55Co versus T9for di?erent values of the reaction rate55Co(p,γ)56Ni and in(p,γ)-(γ,p)equilibrium.The value of the reaction rate near T9≈2.5determines when55Co breaks out of(p,γ)-(γ,p)equilibrium with56Ni.

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