Element synthesis calculations for stellar explosions: robust uncertainties,sensitivities, and radioactive ion beam measurements

Michael S. Smitha, W. Raphael Hixb,a, Suzanne Parete-Koonb, Luc Dessieuxb, ZhanwenMab,a, Sumner Starrfieldc, Daniel W. Bardayana, Michael W. Guidryb,a, Donald L.Smithd, Jeffery C. Blackmona, Anthony Mezzacappaa

aPhysics Division, Oak Ridge National Laboratory, Oak Ridge, TN

bDept. of Physics and Astronomy, Univ. of Tennessee, Knoxville, TN

cDepartment of Physics & Astronomy, Arizona State Univ., Tempe, AZ

dNuclear Engineering Division, Argonne National Laboratory, Argonne, IL

We utilize multiple-zone, post-processing element synthesis calculations to determinethe impact of recent ORNL radioactive ion beam measurements on predictions of novaeand X-ray burst simulations. We also assess the correlations between all relevant reactionrates and all synthesized isotopes, and translate nuclear reaction rate uncertainties intoabundance prediction uncertainties, via a unique Monte Carlo technique.

1. MOTIVATION AND BACKGROUND

Novae and X-ray bursts are violent thermonuclear explosions that occur at very hightemperatures and densities - greater than 108 K and 104 g cm−3 for novae, and 109 Kand 106 g cm−3 for X-ray bursts, respectively. These conditions enable unstable nucleiproduced by capture reactions [like (p,γ)] to undergo further reactions before they decay,and sequences of such nuclear reactions on proton-rich unstable isotopes power these out-bursts [1]. The energy generation rate can be up to 100 times greater than in hydrostaticstars, and the synthesized abundances differ markedly from solar. Novae, occurring onthe surface of White Dwarf (WD) stars, are thought to produce overabundances of 13C,15N, and 17O, as well as Ne, Na, and heavier elements in some explosions. Long-livedradioactive nuclei such as 18F, 22Na, and 26Al are also synthesized and ejected, and theirobservation may provide stringent constraints on nova models [2]. X-ray bursts, occurringon the surface of neutron stars, may synthesize nuclei up to mass 100 and beyond [3],which fall back to the surface of the neutron star and alter its structure and evolution.

Progress in understanding these explosions require improved determinations of the ratesof numerous reactions on proton-rich radioactive nuclei with relatively short (<∼ 1 minute)lifetimes. Experimental determinations of some such rates are now possible with the avail-ability of beams of radioactive nuclei at a number of facilities worldwide [4]. Since thesebeams are difficult to produce, it is crucial to determine which nuclear reactions have thelargest influence on the explosion model predictions, and to determine the astrophysical

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impact of specific recent (or planned) nuclear measurements.

2. ELEMENT SYNTHESIS CALCULATIONS

To interpret lab measurements and guide future work, we utilized “post-processing”,multiple zone element synthesis calculations where the nucleosynthesis is decoupled fromthe hydrodynamics of the burst. Our method is described in detail in [5]. For novae, anuclear reaction network [6] containing 169 isotopes (1H to 54Cr) was used with nuclearreaction rates from the REACLIB [7] database. We simulated the explosion by extractinghydrodynamic trajectories – time histories of the temperature and density – from one-dimensional hydrodynamic calculations with a limited reaction rate network (similar toRef. [8]) for outbursts on a 1.00 M� CO WD, and on 1.25 M� and 1.35 M� ONeMgWDs. The first two are representative of the most prevalent classes of novae, while thethird represents a more energetic outburst. Since our reaction variations did not appre-ciably change the nuclear energy generation nor, therefore, the temperature and densityhistory of the explosion, the decoupling of nuclear and hydrodynamical effects is valid.Different mass elements (“zones”) of the stellar envelope at different radii generate uniquethermodynamic histories. In our simulations, the ejected material of each nova model wasdivided into ∼30 zones. Separate reaction network calculations with the full complementof nuclei and reactions were carried out to study the nucleosynthesis details within eachzone assuming no mixing. The total predicted abundances in the explosion ejecta werefound by summing each zone’s contribution. These calculations are more realistic thanprevious post-processing nucleosynthesis calculations, which used constant temperaturesand densities, or those that considered only the hottest zones of the explosion. For X-raybursts, a similar approach was used, but with a reaction network extending up to 136Xewith only one temperature and density trajectory (resembling the one-zone model in [3]).

3. IMPACT OF NEW REACTION RATES

We have used our element synthesis calculations to determine the astrophysical impactof several recent radioactive beam measurements at ORNL’s Holifield Radioactive IonBeam Facility (HRIBF). For example, a new rate for the 17F(p,γ)18Ne reaction – basedon an HRIBF measurement of 1H(17F,p)17F [9] – is up to a factor of 30 slower than thewidely-used rate found in REACLIB. When used in our element synthesis code, the newrate changes the abundances of 17,18F and 17,18O synthesized in the hottest zones of a1.25 M� WD nova by up to a factor of 600 compared to some previous estimates, andproduced changes by up to a factor of 2.1 in the abundances of 17,18,19F, 17,18O, 14,15N, and12,13C when averaged over the entire exploding envelope. The changes are even larger (upto a factor of 14,000 in the hot zone and 3.7 overall for 17O) for the 1.35 M�WD nova,but almost negligible for the cooler 1.0 M�WD nova [5]. Another study examined theimpact of a new direct measurement of the 18F(p,α)15O reaction at HRIBF [10]. Here,it was found that our new reaction rate, which is lower by a factor of ∼ 2 than previousestimates, increases the amount of the long lived radionuclide 18F in a nova outburst on a1.35 M� ONeMg WD by up to a factor of two in the hottest zone of the explosion. A thirdstudy [11] examined the influence of the 14O(α,p)17F reaction, and a possible competingreaction 14O(α,2p)16O, that have been studied indirectly by HRIBF measurements. We

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found little influence of variations of either reaction on element synthesis in novae, andin X-ray bursts we found that the mass flows of nuclei through 14O(α,p)17F and through14O(α,2p)16O(p,γ)17F are fairly independent of these rates unless they are changed by 3-5orders of magnitude.

4. MONTE CARLO SIMULATIONS

To determine the correlation between all relevant reaction rates and all synthesized iso-topes in an explosion, and to translate nuclear reaction rate uncertainties into abundanceprediction uncertainties, we use a Monte Carlo technique which simultaneously assignsa different, uncorrelated, random enhancement factor to each reaction rate in the simu-lation. The nucleosynthesis is calculated with these modified reaction rates, the resultsstored, and the process repeated with different enhancement factors. After 10000 itera-tions, the mean values and 90% confidence limits are determined from the distributionof abundance predictions. Monte Carlo methods have been employed with great successin the analysis of Big Bang nucleosynthesis [12], but have not previously been appliedto other thermonuclear burning scenarios. Since there is no tabulation of uncertaintiesfor all the rates we use, we employed the following prescription. We assigned uncertain-ties of ∼ 50% to rates whose measurement would require radioactive ion beams, anduncertainties of a factor of 2 for rates calculated by Hauser-Feshbach methods. Betadecay rates are given their tabulated values [13]. For all other rates we assign a 20%uncertainty. We have deliberately used modest uncertainties to ensure that our resultingMonte Carlo uncertainties are not unduly inflated. The reaction rate enhancement factorsare distributed according to the log-normal distribution, which is the correct uncertaintydistribution for quantities like reaction rates which are manifestly positive [14]. Our useof the log-normal distribution for the reaction rates represents a significant improvementover previous Monte Carlo calculations.

In this paper, we discuss results from the innermost zone of a 1.25 M� WD nova model,though the method can examine the impact on entire outburst models. A representativeresult: our Monte Carlo calculations give an upper and lower 90% confidence limits thatdiffer by a factor of 3.6 for the important radionuclide 22Na. This is typical: even withour conservatively chosen rate uncertainties, many of the abundant metals have 90% con-fidence limits with a width of a factor of 2 or larger, such as 16O (2.7), 17O (2.7), 18O(3.3),and 30Si (5.6). The predicted abundances of radionuclides also have large uncertainties:for 18F [26Al], the 90% confidence level spans a factor of 3.3 [3.6]. These confidence limitsare the first statistically robust uncertainties determined for nova nucleosynthesis. Theyhave important implications for determining the sensitivity of orbital observatories (e.g.,INTEGRAL) for detection of gamma rays from novae.

We also determine the correlation between small variations of all relevant reactionrates and all synthesized isotopes in the outburst. Figure 1 shows a representative result- the distribution of the predicted 22Na abundance with the variation in two reactionrates. A linear fit for the 22Na(p,γ)23Mg rate determines that there is a negative cor-relation – as the capture rate increases, the resulting 22Na abundance decreases – andthat the correlation is statistically significant – the slope is more than a few standarddeviations different from zero. The figure also shows a positive, statistically significant

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Rate Variation22Na(p,γ)23Mg

1 2 50.5

slope = - 0.904 0.003

10-5

10-4

22N

a A

bund

ance

1 20.5

Rate Variation20Ne(p,γ)21Na

slope = 0.589 0.020

Figure 1. Distribution of the predicted 22Na abundance with the variation in two reactionrates, along with linear fits (solid lines) and their slopes.

correlation with the 20Ne(p,γ)21Na reaction. We have used our analysis to determine aprioritized list of reactions that most influence the production of radioisotopes that maybe observable tracers of novae. For 18F, the critical reactions are (in order of importance)17O(p,γ)18F, 17F(p,γ)18Ne, 16O(p,γ)17F, 18F(p,α)15O, and 17O(p,α)14N. For 22Na, the mostimportant reactions are 22Na(p,γ)23Mg, 20Ne(p,γ)21Na, 23Na(p,α)20Ne, 23Na(p,γ)24Mg,and 21Na(p,γ)22Mg. For 26Al, they are 23Mg(p,γ)24Al, 26Al(p,γ)27Si, 23Na(p,γ)24Mg,23Na(p,α)20Ne, 20Ne(p,γ)21Na, 25Mg(p,γ)26Al, and 25Al(p,γ)26Si. Our calculations canbe used to prioritize measurements of these reactions with radioactive and stable beams.

ORNL is managed by UT-Battelle, LLC, for the U.S. Dept. of Energy under contractDE-AC05-00OR22725.

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