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AN APPROXIMATION FOR THE rp-PROCESS

Felix Rembges, Christian Freiburghaus, Thomas Rauscher, Friedrich-KarlThielemann

Departement für Physik und Astronomie, Universität Basel, CH-4056 Basel, Switzerland;http://quasar.physik.unibas.ch/

Hendrik Schatz, Michael WiescherDepartment of Physics, University of Notre Dame, Notre Dame, IN 46556;

hschatz@sulfur.helios.nd.edu, wiescher@nd.edu

ABSTRACT

Hot (explosive) hydrogen burning or the Rapid Proton Capture Process (rp-process)occurs in a number of astrophysical environments. Novae and X-ray bursts are the mostprominent ones, but accretion disks around black holes and other sites are candidatesas well. The expensive and often multidimensional hydro calculations for such eventsrequire an accurate prediction of the thermonuclear energy generation, while avoidingfull nucleosynthesis network calculations.

In the present investigation we present an approximation scheme which leads to anaccuracy of more than 15 per cent for the energy generation in hot hydrogen burningfrom 108–1.5×109 K, which covers the whole range of all presently known astrophysicalsites. It is based on the concept of slowly varying hydrogen and helium abundancesand assumes a kind of local steady flow by requiring that all reactions entering andleaving a nucleus add up to a zero flux. This scheme can adapt itself automatically andcovers situations at low temperatures, characterized by a steady flow of reactions, aswell as high temperature regimes where a (p, γ)–(γ, p)-equilibrium is established, whileβ+-decays or (α, p)-reactions feed the population of the next isotonic line of nuclei.

In addition to a gain of a factor of 15 in computational speed over a full networkcalculation, and an energy generation accurate to more than 15 per cent, this schemealso allows to predict correctly individual isotopic abundances. Thus, it delivers allfeatures of a full network at a highly reduced cost and can easily be implemented inhydro calculations.

Subject headings : nuclear reactions, nucleosynthesis, abundances — stars: novae —

X-rays: bursts

1

http://arxiv.org/abs/astro-ph/9701217v1http://quasar.physik.unibas.ch/

1. INTRODUCTION

Close binary stellar systems can exchangemass, when at least one of the stars fills itsRoche Lobe. After a critical mass ∆M ofunburned transferred matter is accumulatedon the surface of the accreting star, igni-tion sets in, typically under degenerate con-ditions when the accreting object is a whitedwarf or neutron star. Degenerate condi-tions for which the pressure is not a functionof temperature, prevent temperature adjust-ment via pressure increase and expansion andcause a thermonuclear runaway and explosiveburning. For white dwarfs, a layer of 10−5–10−4 M⊙ forms, before pycnonuclear ignitionof hydrogen burning sets in (e.g. Sugimoto& Nomoto 1980; Starrfield et al. 1993; Cocet al. 1995 ) and causes a nova event (explo-sive H-burning on white dwarfs) with maxi-mum temperatures of ≈(2–3)×108 K and atotal energy release of 1046–1047 ergs. Theburning takes 100–1000 s before the partiallyburned hydrogen envelope is ejected.

X-ray bursts (for an observational overviewsee Lewin, van Paradijs, & Taam 1993) in-volve accreting neutron stars with an un-burned, hydrogen-rich surface layer. Thecritical size of the hydrogen layer before ig-nition is as small as 10−12 M⊙. Temper-atures of the order (1-2)×109 K and den-sities ρ≈106–107 g cm−3 are attained (seee.g. Wallace & Woosley 1981; Ayasli & Joss1982; Hanawa, Sugimoto, & Hashimoto 1983;Woosley, Taam, & Weaver 1986; Taam et al.1993; Taam, Woosley, & Lamb 1996). Thisexplosive burning with rise times of about 1–10 s leads to the release of 1039–1040 ergs.Many of the observed features and generalcharacteristics are understood, however, thereis still a lack of a quantitative understand-ing of the detailed observational data. An-

other aspect is that the explosion energies aresmaller than the gravitational binding energyof the accreted hydrogen envelope. An evenlydistributed explosion energy would not un-bind and eject matter. It remains to be seenwhether an interesting amount of matter canescape the neutron star. Super-Eddington X-ray bursts (Taam et al. 1996) are the bestcandidates for this behavior.

A description of hot (explosive) hydrogenburning has been given by Arnould et al.(1980), Wallace & Woosley (1981), Ayasli &Joss (1982), Hanawa et al. (1983), Wiescheret al. (1986), van Wormer et al. (1994),Thielemann et al. (1994) [see also Biehle(1991), Biehle (1994), Cannon et al. (1992),and Cannon (1993) for Thorne-Zytkow ob-jects, but see the recent results by Fryer,Benz, & Herant (1996) which puts doubtson their existence]. The burning is describedby proton captures, β+-decays, and possiblyalpha-induced reactions on unstable proton-rich nuclei, usually referred to as the rp-process (Rapid Proton Capture Process). Crosssections can either be obtained from the bestavailable application of present experimen-tal knowledge, e.g. the determination ofresonance properties from mirror nuclei andtransfer and/or charge exchange reaction stud-ies, or actual cross section measurements, likefor 13N(p, γ)14O, the first reaction cross sec-tion analysed with a radioactive ion beam fa-cility (see Champagne & Wiescher 1992 andreferences therein). There exist two majormotivations in nuclear astrophysics (a), tounderstand the reaction flow to a necessarydegree, in order to predict the correct en-ergy generation required for hydrodynamic,astrophysical studies, and (b), to predict adetailed isotopic composition which helps tounderstand the contribution of the process in

2

question to nucleosynthesis in general. Ourmain motivation in this paper is (a), to pro-vide a fast energy generation network as atool for nova and X-ray burst studies. Thismight be underlined by the fact that no-vae and X-ray bursts seem not to be ma-jor contributors to nucleosynthesis due to thesmall ejected masses involved or the questionwhether gravitational binding can be over-come at all. [However, novae can be impor-tant for nuclei like 7Li, 15N, 22Na, 26Al, andeven Si and S; and super-Eddington X-raybursts (Taam et al. 1996) will be able to ejectsome matter, probably containing some lightp-process elements.] It will turn out at theend that our efficient approximation and en-ergy generation scheme can also be used topredict abundances accurately. The nucle-osynthesis for nuclei above Kr will, however,be discussed in a second paper (Schatz et al.1996).

In order to understand the energy gener-ation correctly, we have to be able to under-stand the main reaction fluxes. In the past weperformed a series of rp-process studies (Wi-escher et al. 1986 to Ar, and van Wormeret al. 1994, extending up to Kr). The reac-tion rate predictions were based on resonanceand direct capture contributions for proton-rich nuclei, making use of the most recent ex-perimental data. Several (p, γ)-reaction ratesbelow mass A=44 have been recalculated byHerndl et al. (1995) in the framework of ashell model description for the level structureof the compound nucleus. A preliminary anal-ysis of the major aspects was given by Thiele-mann et al. (1994). In the following section2 we will present this in more detail and dis-cuss the constraints which an approximationscheme has to fulfill in the whole range oftemperatures occurring in explosive hydrogen

burning environments, i.e. 108–1.5 × 109 K.The approximation scheme will be presentedin section 3, its application and comparisonwith full network calculations in section 4, fol-lowed by concluding remarks in section 5.

2. REACTION FLOWS IN EXPLO-

SIVE HYDROGEN BURNING

At low temperatures, the rp-process is dom-inated by cycles of two successive proton cap-tures, starting out from an even-even nucleus,a β+-decay, a further proton capture (into aneven-Z nucleus), another β+-decay, and a fi-nal (p, α)-reaction close to stability (similarto the hot CNO). The final closing of the cy-cle via a (p, α)-reaction occurs because pro-ton capture Q-values increase with increas-ing neutron number N in an isotopic chainfor a given Z, while α-capture Q-values or -thresholds show a very weak dependence onN . Thus, a cycle closure occurs at the mostproton-rich compound nucleus, which has alower alpha than proton-threshold. Theseare the even-Z and especially well bound“alpha”-nuclei with isospin Tz=(N−Z)/2=0,like 16O, 20Ne, 24Mg, 28Si, 32S, 36Ar, 40Ca.Therefore, (p, α)-reactions can operate on odd-Z targets with Tz=+1/2:

15N, 19F, 23Na,27Al, 31P, 35Cl, 39K. The hydrogen burningcycles are connected at these nuclei, due to apossible competition between the (p, γ)- and(p, α)-reaction. An exception to the rule isthe Ne isotopic chain, where the transitionhappens already for the compound nucleus19Ne. Thus, 18F(p, α)15O is a possible reac-tion, bypassing the OF(Ne)-cycle. Therefore,only the extended CN(O)-, the NeNa-, MgAl-, SiP-, SCl-cycles etc. exist. This is displayedin Figure 1.

The progress of burning towards heaviernuclei depends on the leakage ratio (p, γ)/(p, α)

3

Fig. 1.— The hot hydrogen burning cycles,established at T9=0.3 and ρ=10

4 g cm −3.

into the next cycle, which makes a good ex-perimental determination of these reactionsimportant and is possible as we deal here withstable targets. On the other hand, the excita-tion energies in the corresponding compoundnuclei are of the order 8.5–12 MeV and makestatistical model approaches also reliable (seee.g. the experiments by Iliadis et al. 1993,1994 and Ross et al. 1995 for 31P(p, γ), and35Cl(p, γ) and the later discussion).

Increasing temperatures allow to overcomeCoulomb barriers and extend the cycles tomore proton-rich nuclei, which permit addi-tional leakage via proton captures, compet-ing with long β-decays. The cycles open firstat the last β-decay before the (p, α)-reactionvia a competing proton capture, i.e. at theeven-Z Tz=–1/2 nuclei like

23Mg, 27Si, 31S,35Ar, and 39Ca, excluding 15O (because 16Fis particle unstable) and 19Ne, which is by-passed by the stronger 18F(p, α)-rate. Thesenuclei are only one unit away from stabilityand have small decay Q-values and long half-lives. The situation is illustrated in Figure

Fig. 2.— Q-values of (p, γ)-reactions between20Ne and 44Ti.

2, which shows the possible break-out pointsand the Q-values of the proton capture reac-tions. Since the reaction proceeds towards theproton drip-line and into an odd-odd nucleus,the capture Q-values of these (p, γ)-reactionsare small (generally less than 2 MeV). Thisimplies a too small density of excited statesfor employing statistical model cross sections(see discussion in the following paragraphs).At about 3×108 K essentially all such cyclesare open and a complete rp-pattern of protoncaptures and β-decays is established, with theexception of the extended CNO-cycle. Theprocess time is determined by the sum ofβ-decay and proton capture time scales, τβand τp,γ, along the rp-path, which does gener-ally not extend more proton-rich than Tz=–1for even-Z nuclei. The reaction time scalesare dominated by the Tz=–1/2 even-Z nucleiclose to stability, listed above.

At 4×108 K the CNO-cycle breaks openvia 15O(α, γ)19Ne. 14O(α, p)17F can also leadto a successful break-out when followed by aproton capture and (α, p)-reaction on 18Ne.

4

Due to higher energies, Coulomb barriers canbe more easily overcome by heavier projec-tiles (like alpha captures on CNO-targets),and proton captures can occur further awayfrom stability. However, in all cases the decayof Tz=–1 even-Z nuclei (and at high densi-ties the competition with (α, p)-reactions) isa prominent part of the reaction path. Alpha-reactions are not important for Z>20, due toincreasing Coulomb barriers and decreasingQ-values. For heavier nuclei beyond Ca, thereaction pattern seems complicated. How-ever, temperatures approaching 109 K causestrong photodisintegrations for proton cap-ture Q-values smaller than 25kT≈3 MeV,which is comparable to Q-values found foreven-Z Tz=–1 nuclei. [25-30kT≥Q is a roughcriterion to find out whether photodisinte-grations can counterbalance capture reactionsand lead to an equilibrium situation.] Thus,reactions breaking out of the cycles discussedabove (up to the proton drip line) come intoa (p, γ)–(γ, p)-equilibrium along isotonic lineswith equal N . Under such circumstances thenuclear connection boils down to the nec-essary knowledge of nuclear masses and β-decay half-lives, similar to the r-process ex-periencing an (n, γ)–(γ, n)-equilibrium in iso-topic chains (see Cowan, Thielemann, & Tru-ran 1991; Kratz et al. 1993; Meyer 1994).

The general situation is explained in Fig-ure 3, which contains all possible reactionsfrom the original cycles over proton capturesbreaking out of the cycles, the accompany-ing beta decays and (α, p)-reactions bridgingwaiting points with long decay half-lives. Bycomparing with Figure 2, which also lists theQ-values involved, Figures 4a and 4b give anindication for the minimum temperatures re-quired to permit statistical model calculationsfor proton and alpha-induced reactions. The

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Fig. 3.— Recurrent structure in the rp-process network.

reason is that a high density of compound nu-clear states is necessary for statistical modelcalculations to become reliable. The energyof the Gamow peak, where the integrand ina thermonuclear cross section integral (crosssection times Boltzmann distribution) has amaximum, increases with increasing temper-ature. Because higher level densities are en-countered at higher excitation energies corre-sponding to higher temperatures of the Boltz-mann distribution, a critical temperature ex-ists beyond which such calculations are reli-able (for more details see Rauscher, Thiele-mann, & Kratz 1996). We can analyse theexpected behavior: Reactions j, k, and lof Figure 3 have reaction Q-values less than≈3 MeV and for conditions close to 109 Kwe have 25kT≈Q which permits a (p, γ)–(γ, p)-equilibrium. Thus, although the crosssections are highly uncertain and statisticalmodel approaches are not valid (see Figure

5

N

Z

Applicability of the Statistical Model (p)

10 15 20 25 30 35 40

10

15

20

25

30

35

Fig. 4a.— Applicability of the StatisticalModel for (p, γ)- and (p, α)-reactions.

4a), only the Q-values, i.e. masses, are neededfor these nuclei. Reactions b, c, d, and e [(p, γ)and (p, α) reactions] have Q-values in excessof 5 MeV and in most cases high enoughlevel densities to ensure the usage of statis-tical model cross sections for the appropriatetemperature regime. Reaction f , an (α, p)-reaction, provides also a sufficiently high leveldensity to apply statistical model cross sec-tions (see Figure 4b, for 18Ne(α, p) see Görres& Wiescher 1995). Reaction g is crucial forthe break-out, a statistical model approachis clearly not permitted, and a (p, γ)–(γ, p)-equilibrium is not valid at the lower appropri-ate temperatures of about 3-4×108 K. Thus,all reactions of type g ask strongly for experi-mental determination, possibly with radioac-tive ion beams. Reactions h, i and all otherconnecting β-decays are required, preferablyfrom experiments, otherwise from theoreticalQRPA predictions.

Beyond Ca and Ti, this scheme simpli-fies into one dominated by proton capturesand beta-decays, with the proton captures in

N

Z

10 15 20 25 30 35 40

10

15

20

25

30

35

Fig. 4b.— Applicability of the StatisticalModel for (α, γ)- and (α, p)-reactions.

a (p, γ)–(γ, p)-equilibrium. The proton dripline develops a zig-zag shape and can separatetwo even-Z proton-stable nuclei on the sameisotone by an unstable odd-Z nucleus. Görreset al. (1995) have shown that such gaps canbe bridged by two-proton capture reactions,proceeding through the unstable nucleus, sim-ilar to the 3α-reaction passing through 8Be.This can extend the (p, γ)–(γ, p)-equilibriumbeyond the drip line to proton-stable ”penin-sulas”, thus avoiding long beta-decay half-lives. While alpha-induced reactions are notimportant for nuclei beyond Ca and Ti, smallQ-values for A > 56 can permit (γ, α)-photo-disintegrations and cause a back flow, de-creasing the build-up of heavy nuclei.

After having analized the reaction patternsin the previous discussion, we have now toimplement these findings into an approxima-tion which can handle and describe all of thepatterns. Considering the hydrogen and he-lium abundances as slowly varying, and as-suming a kind of local steady flow by requir-ing that all reactions entering and leaving a

6

nucleus add up to a zero flux, we will devisea scheme which can adapt itself to the wholerange of conditions of interest in hot hydro-gen burning. It can automatically cover situa-tions at low temperatures, characterized by asteady flow through all connected hot CNO-type cycles and branchings feeding the nextcycle, to high temperature regimes where a(p, γ)–(γ, p)-equilibrium is established, whileβ-decays or (α, p)-reactions feed the popula-tion of the next isotonic line. The details arediscussed in the following section.

3. APPROXIMATION SCHEME

3.1. Reaction Rates and Quasi-Decay

Constants

The thermonuclear reaction rates used inour calculations have been discussed in detailby van Wormer et al. (1994) (cf. the ap-pendix therein).1 The temporal change of theisotopic abundances Yi=

XiAi

can be calculatedby all depleting and producing reactions

Yi =∑

j

iαj Yj +∑

j,k

iαj,k Yj Yk

+∑

j,k,l

iαj,k,lYj Yk Yl . (1)

The first term in (1) includes all one-particlereactions (decays or photodisintegrations) ofnuclei j producing (i6=j) or destroying (i=j)nucleus i. The αj are either decay con-stants or effective temperature dependent de-cay constants in the case of photodisintegra-tions. The second sum represents all two-particle reactions between nuclei j, k with

1The parameterized reaction rates can be foundfor instance in the database REACLIB located athttp://csa5.lbl.gov/ f̃chu/astro.html/.

He

Li

Be

B

C

6

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structure extended to

53

Ni

N

O

F

Ne

Na

Mg

Al

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Fig. 5a.— rp-process network being used inthe approximation. The structure in the boxis eight times repeated up to 50Mn. Bold ar-rows correspond to large, normal arrows tosmall decay constants.

iαj,k=icj,k ρNA

i〈σv〉j,k. Similarly, the thirdterm represents three particle processes withiαj,k =

icj,k,lρ2N2A

i〈σv〉j,k,l (Fowler, Caughlan,& Zimmermann 1967). The coefficients c in-clude the statistical factors for avoiding dou-ble counting of reactions of identical particles.

van Wormer et al. (1994) investigatednumerically which reactions must be consid-ered in the rp-process. Apart from the tem-perature insensitive β+-decays, the only one-particle reactions which must be especiallyconsidered at high temperatures are photo-disintegrations. The two-body reactions dom-inating the rp-process are (p, γ)-, (p, α)- and,(α, p)-reactions. Three body reactions can beneglected in explosive hydrogen burning withthe exception of the 3α-reaction which playsa central role in this context. Taking into ac-count individual reactions, equation (1) can

7

http://csa5.lbl.gov/

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6

Z

Fig. 5b.— The proton capture sequencesare connected to each other by (α, p)- and(β+, ν)-breakout reactions respectively.

be rewritten as

Ẏi =∑

j

iλβ+

j Yj +∑

j

iλγ,p

j Yj

+∑

j

ρNAi〈σv〉j,pYjYp

+∑

j

ρNAi〈σv〉j,αYjYα . (2)

If the proton and helium abundances do notchange greatly during relevant rp-process timescales, we are able to describe proton cap-tures and (α, p)-reactions with quasi-decayconstants, provided that temperature and den-sity do not change greatly during the processtime scale of interest:

iλp,γ

j := ρNAi〈σv〉j,pYp = const (3a)

iλp,α

j := ρNAi〈σv〉j,pYp = const (3b)

iλα,γ

j := ρNAi〈σv〉j,αYα = const (3c)

iλα,p

j := ρNAi〈σv〉j,αYα = const . (3d)

Thus the system of nonlinear differential equa-tions can be written in a linearized form

Ẏi =∑

j

iλx

jYj , (4)

where x represents one of the following re-actions: (γ, p), (p, γ), (α, p), (p, α), (α, γ),(γ, α), (β+ν). When we choose Y0=(Y1, . . . , Yi,. . . , Yn) to represent the initial abundance dis-tribution, the system of equations (4) can bewritten as

Ẏ = AY , A (N ×N)-matrix . (5)

Such systems of linear differential equa-tions with constant coefficients can be solvedby Jordan-normalizing matrix A. Its trans-formation

J = SAS−1 , S ǫGLn(RR) (6)

yields theN -dimensional set of eigenfunctions

ui;j(t) = eλmi

tpi;j(t) . (7)

Here, λmi represents eigenvalue i with degen-eracy m, ui;j eigenvector j to eigenvalue i(1≤j ≤m) and pi;j(t) a polynomial of de-gree (j–1). However, the transformationsu′i;j=S ui;j are in general difficult to surveyand to handle. Furthermore, the elements ofA become slowly varying functions when theabundances of hydrogen and helium changeslightly over a certain process time scale. Thisis for instance the case, when we consider hothydrogen burning. Only numeric simulationswill be feasible for these two reasons.

We will show in the following subsectionhow certain sequences of a thermonuclear re-action network can be regarded as quasi-

8

Fig. 6a.— Comparison of the approximatedabundances of 53Ni and 55Ni with results fromfull network calculations.

nuclei. In such sequences a steady flow ma-terializes and the ratios between the differentabundances are constant and are only deter-mined by the values of the quasi-decay con-stants. A procedure of this kind provides uswith a system of equations with a smaller di-mension, which can be solved numerically.

3.2. The Approximation

In section 2, we discussed the burning cy-cles with the same structure as the hot CNO-cycle (see also Figures 1–3). It is well un-derstood how a steady state equilibrium isestablished in hot CNO-like reaction chainsat temperatures T9 ≃ 0.2 and on time scalesgoverned by the longest β+-decay in the cy-cle. Then the abundance Yi,j of nucleus (i) ina cycle (j) can be calculated from the equi-librium condition

Ẏi,j!= 0. (8)

This condition can be used for simplifyingthe system of equations which determines thethermonuclear reaction flows. However, anapproximation of this kind is no longer jus-

Fig. 6b.— Same as Figure 6a but for 64Geand 72Kr.

tified, when we consider time scales shorterthan the half-lives of the β+-unstable nuclei ortemperatures greater than T9=1 where pho-todisintegrations become so important thata (p, γ)–(γ, p)-equilibrium is attained ratherthan a steady flow (see Figure 3). Then,proton-induced reactions can be associatedwith large quasi-decay constants and, in com-parison to β-decays, small half-lives. Thus,all the abundances of isotonic nuclei whichare connected by proton capture reactionscan be considered in a quasistatic equilibriumon time scales exceeding the half-lives associ-ated with the quasi-decay constants. Such se-quences shall be now defined as quasi-nuclei.

In the following paragraphs, the abundancesof nuclei in proton capture sequences shall becalculated under the assumption of a quasi-static equilibrium. Consider the jth of m se-quences which consists of n−1 proton cap-tures between n nuclei (cf. Figure 5b: Here,two proton capture sequences are shown withn=4 and n=5). Each nucleus (i) in a sequence(j) is involved in k+2 reactions including pro-ton captures and photodisintegrations. Fur-thermore, we assume for simplicity that the

9

abundances of helium and hydrogen stay con-stant over the time scale of interest and thatnucleus (i, j) is not fed by reactions from an-other chain j′. Using the equilibrium condi-tion (8), the temporal change of the abun-dance of a nucleus (i, j) can be written as

Ẏi,j = λp,γi−1,jYi−1,j −

∑

k

λxi,j,kYi,j

+λγ,pi+1,jYi+1,j = 0 . (9)

The terms summed over k are leaks out ofchain (j) via nucleus (i, j) The extreme casesof flux equilibria, like steady flows on the onehand (λp,γ ≫ λγ,p, λβ

+

) and (p, γ)–(γ, p)-equilibria on the other hand (λp,γ, λγ,p ≫λβ

+

) can be discussed on the basis of thisequation. Here, the general case shall be dis-cussed.

For a chain of n nuclei, n−1 such equa-tions permit to relate the abundance Yi−1,j toYi,j starting with a “known” Yn,j, the abun-dance of the last nucleus in a sequence. Thuswe obtain for each sequence (j) an (n−1)-dimensional system of linear equations whichallows us to express each abundance Yi,j interms of Yn,j:

Yi,j = vi,j(λx1,j . . . λ

xn,j)Yn,j . (10)

The vi,j depend only on the constant or slowlyvarying quasi-decay constants λi,j. The abun-dance of a sequence can be treated as theabundance of a quasi-nucleus by Yj:=

∑

i Yi,j.When Yj is already determined at a point intime t=ts, each abundance Yi,j can be writtenas

Yi,j(ts) =vi,jvj

Yj(ts) , (11)

with vj=∑

i vi,j. Hence, when the abundancesof the quasi-nuclei Yj are given at a time t,

Fig. 7a.— Comparison of the energy genera-tion rate ǫ̇ for for nova-like conditions.

the abundances Yi,j(t) in each sequence canbe determined simultaneously.

However, we also have to take care of in-coming fluxes into a sequence (j) not jet con-sidered in equation (9). As a consequence, thecoefficients vi,j are no longer constant becausethey become explicitly abundance-dependentfor the following reason: in contrast to theoutgoing fluxes, the incoming fluxes into a se-quence j are not proportional to the abun-dances Yi,j of the nuclei in this sequence.Therefore, abundances can no longer be de-termined simultaneously.

Incoming fluxes can nevertheless be consid-ered as long as the abundances do not changetoo fast during a time step. We can addto equation (9) all incoming fluxes from nu-clei (l) of different chains (m) proportional totheir abundances Yl,m(ts−1), m6=j (when notime t is indicated, the quantities are given att=ts),

Ẏi,j = −Yi,j∑

k

λxi,j,k + λγ,pi+1Yi+1,j + λ

p,γi−1,jYi−1,j

10

Fig. 7b.— Comparison of the energy genera-tion ǫ for nova-like conditions.

+∑

l,m

m6=j

λxl,mYl,m(ts−1) = 0 . (12)

This can be solved for Yi−1,j, leading to

Yi−1,j =1

λp,γi−1,j

(

Yi,j∑

k

λxi,j,k − λγ,pi+1Yi+1,j

)

︸ ︷︷ ︸

vi−1,j Yn,j

−1

λp,γi−1,j

∑

l,m

m6=j

λxl,mYl,m(ts−1)

︸ ︷︷ ︸

wi−1,j(λxl,m,Yl,m(ts−1))

, (13)

or more generally

Yi,j = vi,j Yn,j − wi,j(ts−1) . (14)

When a slowly varying function v′j(ts−1) isgiven so that we can approximate Yn,j by

Yn,j(ts) ≈1

v′j(ts−1)Yj(ts) , (15)

we can extend wi,j in (14) as follows

Fig. 7c.— With this approximation, energygeneration for nova-like conditions can be re-produced with an accuracy of 15-5 per cent.

Yi,j = vi,jYn,j −v′jYj

wi,jYn,j , (16)

and finally obtain

Yi,j = v′

i,jYn,j (17)

v′i,j = vi,j −v′jYj

wi,j , (18)

with v′j=∑

i v′

i,j . Setting v′

i,j(t0=0)=vi,j(t0=0)as initial values, the existence of the functionsv′j is guaranteed.

So far we have made two approximations.(a): including reactions within a chain andleaks out of a chain described by equation(9). In that case all abundances could be de-scribed simultaneously. (b): When also con-sidering incoming fluxes [see equation (12)],we become dependent on abundances in otherchains. Approximating their abundances att=ts by abundances at t

′=ts−1 permits to staywithin the same approximation scheme. Butwe have to be aware that this is only valid

11

Fig. 8a.— Same as Figure 7a but for X-rayburst conditions.

for slowly varying abundances or equivalentlysmall time steps.

We obtained this approximation under theassumption that the abundances of hydrogenand helium stay constant. Yp and Yα areslowly varying functions in realistic environ-ments. We can adjust for the change of Ypand Yα and obtain their new values applyingrelation (11)

Ẏp =∑

i,j

(λγ,pi,j + λα,pi,j − λ

p,αi,j

−λp,γi,j )Yi,j(ts) (19a)

Ẏα =∑

i,j

(λp,αi,j − λα,pi,j + λ

γ,αi,j

−λα,γi,j )Yi,j . (19b)

As a consequence, the quasi-decay con-stants must be recalculated after each timestep, making use of the updated proton andalpha abundances considering them constantfor one time step.

Thus it is possible to express simultane-ously every abundance Yi,j in a sequence with

Fig. 8b.— Same as Figure 7b but for X-rayburst conditions.

the sequence abundance Yj. With an approx-imation scheme of this kind, we reduced ourfull network described by

∑mj=1 n(j) equations

to a system of m equations. In the followingwe want to check the accuracy of such an ap-proximation.

4. RESULTS

The calculations were done with a reactionnetwork which consists mainly of proton cap-ture sequences, each starting out from a Tz=0nucleus. The number of nuclei in a sequenceis determined by the condition that the quasidecay constant i+1λ

p,γ

i of a nucleus (i) intonucleus (i+1) is larger than a certain λmin.When this condition is fulfilled, nucleus (i+1)is considered to be in the sequence. For thefollowing calculations λmin was set to 10

3 s−1

because we wanted to reproduce the energygeneration of explosive hydrogen burning ontime scales τ=λ−1

min> 10−2−10−1 s. As a con-

sequence, a sequence contains not more than5 nuclei for temperatures T9≤1.5 and densi-ties ρ≤106 g cm−3.

In addition to the proton capture sequences,

12

Fig. 8c.— Same as Figure 7c but for X-rayburst conditions.

Tz=12nuclei had to be considered in the net-

work. They are the most neutron-rich nu-clei of the hot CNO-type cycles, where aftera proton capture, the alpha-threshold in thecompound nucleus is lower than the proton-threshold, and (p, α)-reactions can occur whichcompete with the still possible (p, γ)-transitions(see Figure 5b). Since the (p, α)- and (p, γ)-reactions occur on the same time scale, theseTz=

12nuclei can not affiliated to a proton cap-

ture chain but have to be considered sepa-rately.

Between 12C and 50Mn, the network canbe composed of proton capture reaction se-quences and Tz=

12nuclei with one exception.

The transition occurs already at compoundnucleus 19Ne in the Ne isotopic chain. A(p, α)-reaction is possible on the Tz=0 nucleus18F and a proton capture reaction sequencestarts out from 19Ne with isospin Tz=−

12.

The low alpha-threshold of 19Ne, whichcaused an exception of the rule and preventedan OFNe-cycle, permits on the other hand the15O(α, γ)-reaction and triggers the break-outfrom the CNO-cycles. The network described

Fig. 9.— Same as Figure 7a but for varyingdensity and temperature starting with novaepeak conditions. The approximation is validas long as the temperature does not fall con-siderably below T9 = 0.1.

so far is shown in the Figure 5a and 5b.

At temperatures exceeding T9=0.8 and ρ ≥106 g cm−3, the network must be completedwith the 3α-reaction. Furthermore, it is verylikely that fluxes up to 72Kr may influencethe energy generation. In the region between51Mn and 72Kr, the β+-reactions of the forthe rp-process relevant nuclei are, with theexception of 55Ni, 64Ge and 72Kr, consider-ably smaller than the typical time scales ofnovae and X-ray bursts. Consequently, onlythe above mentioned nuclei were consideredexplicitly in the approximation scheme. Theapproximated abundances of 53Ni and thesethree waiting point nuclei are compared withfull network calculation results in Figure 6aand 6b.

The nuclei used in the approximation schemeare listed in Table 1 according to their af-filiation to a certain proton capture reactionchain. Nuclei which had to be considered sep-arately (p, 4He, Tz=

12nuclei, 55Ni, 64Ge and

72Kr) are given an own sequence number.

13

Fig. 10.— Same as Figure 9 but for X-rayburst peak condition.

The waiting point abundances obtained withthis approximation are compared with fullnetwork calculation results in Figure 6a and6b.

Approximating the network according tosection 3.2, we calculated the energy gener-ation rate ǫ̇ and the integrated energy gen-eration ǫ for nova-like conditions (T9=0.2,ρ=105 g cm−3) and X-ray burst peak condi-tions (T9=1.5, ρ=10

6 g cm−3) and comparedthe results to full size network calculationswhich take into account each nucleus sepa-rately. In each case we assumed a solar abun-dance distribution for the accreted matter.The results for small temperatures are shownin Figure 7a–7c and those for X-ray burst con-ditions in Figure 8a–8c. Considering the timescales important for novae (100–1000 s) andX-ray bursts (1–10 s), we see that the approx-imation scheme devised in section 3 agreesalways to a high degree for the astrophysi-cal applications considered here. The overallresults show only deviations smaller than 15per cent or even less (down to 5 per cent)at a gain in computational speed by a fac-tor of 15. Several calculations at intermediate

temperatures and densities, i. e. 104≤ρ≤106,0.2≤T9≤1.5 confirmed the above mentionedefficiency and accuracy of the approximationscheme.

Such an approximation would not be veryuseful if it was only valid for constant tem-peratures and densities. In order to check itsaccuracy for varying temperatures and den-sities we assumed after peak conditions wereobtained in the ignition an exponentially de-creasing temperature and an adiabatically ex-panding accretion layer with a radiation dom-inated equation of state (γ = 4

3):

T = T0e−

tτ (20a)

ρ = ρ0

(T

T0

) 43

. (20b)

Using typical peak temperatures T0, densi-ties ρ0 and time scales τ for X-ray bursts andnovae, we again calculated the energy genera-tion (Figure 9 and 10). The results show thatthe approximation is valid for changing tem-peratures and densities as long as they arein the range of 104≤ρ≤106, 0.1≤T9≤1.5. Nosignificant change in computational speed wasobserved.

The differences at time scales smaller thant=10−2 s are due to the very fast reactionflows within the proton capture reaction chains:They occur in full scale network calculationsuntil equilibrium is reached according to equa-tion (8).

Table 1:

Proton Capture Sequences1

14

Sequence Number NucleiSequence 1 . . . . . . . pSequence 2 . . . . . . . 4HeSequence 3 . . . . . . . 12C

13N14O

Sequence 4 . . . . . . . 14N15O

Sequence 5 . . . . . . . 15N

Table 1 — Continued

Sequence Number NucleiSequence 6 . . . . . . . 19Ne

20Na21Mg

Sequence 7 . . . . . . . 16O17F

18NeSequence 8 . . . . . . . 20Ne

21Na22Mg23Al24Si

Sequence 9 . . . . . . . 22Na23Mg24Al25Si

Sequence 10 . . . . . . 23NaSequence 11 . . . . . . 24Mg

25Al26Si27P28S

Sequence 12 . . . . . . 26Al27Si28P29S

Sequence 13 . . . . . . 27AlSequence 14 . . . . . . 28Si

29P30S

31Cl32Ar

Sequence 15 . . . . . . 30P31S

32Cl33Ar

Sequence 16 . . . . . . 31P

15

Table 1 — Continued

Sequence Number NucleiSequence 17 . . . . . . 32S

33Cl34Ar35K

36CaSequence 18 . . . . . . 34Cl

35Ar36K

37CaSequence 19 . . . . . . 35ClSequence 20 . . . . . . 36Ar

37K38Ca39Sc40Ti

Sequence 21 . . . . . . 38K39Ca40Sc41Ti

Sequence 22 . . . . . . 39KSequence 23 . . . . . . 40Ca

41Sc42Ti43V

44CrSequence 24 . . . . . . 42Sc

43Ti44V

45CrSequence 25 . . . . . . 43ScSequence 26 . . . . . . 44Ti

45V46Cr

47Mn48Fe

Table 1 — Continued

Sequence Number NucleiSequence 27 . . . . . . 46V

47Cr48Mn49Fe

Sequence 28 . . . . . . 47VSequence 29 . . . . . . 48Cr

49Mn50Fe51Co52Ni

Sequence 30 . . . . . . 50Mn51Fe52Co53Ni

Sequence 31 . . . . . . 51MnSequence 32 . . . . . . 55NiSequence 33 . . . . . . 64GeSequence 34 . . . . . . 72Kr

1The nuclei used in the approximation scheme

are listed according to their affiliation to a

certain proton capture reaction chain. Nuclei

which had to be considered separately (p, 4He,

Tz=1

2nuclei, 55Ni, 64Ge and 72Kr) are given

an own sequence number.

5. CONCLUSION

The main motivation for the present in-vestigation was to find a fast and efficientapproximation scheme which permits to pre-dict the energy generation in explosive hydro-gen burning for a large range of conditions,i.e. temperatures (0.2≤T9≤1.5) and densities(104 g cm−3≤ρ≤ 106 g cm−3) respectively, forapplications in hydro calculations which can-not afford full nuclear network. This goal hasbeen achieved. The energy generation of fullnetwork calculations (150 nuclei) could be re-produced with a high accuracy, resulting in

16

Fig. 11.— Comparison of the 33Ar-abundances calculated by a full network cal-culation and the approximation scheme.

deviations of 5 to a maximum of 15 per centwhile gaining a factor of about 15 in compu-tational speed.

The approximation scheme discussed inthis paper is therefore well suited for realistichydro calculations of novae or X-ray burstsand other possible sites of explosive hydrogenburning.

The additional advantage of this method isthat it also permits to predict isotopic abun-dances with a similar accuracy and can thuseven replace full network calculations for thispurpose (see Figure 11).

We were also able to determine key nu-clear properties, which directly enter the pre-cision of calculations for explosive hydrogenburning, its energy generation and abundancedetermination. We have identified the even-Z Tz=–1/2 nuclei like

23Mg, 27Si, 31S, 35Arand 39Ca, as essential targets for proton cap-tures which are in competition with β-decays.As their small reaction Q-values (less than2 MeV) do not permit the application of sta-

tistical model cross sections, they have tobe determined experimentally. Thus, theseresults are an ideal guidance for future ex-perimental investigations with radioactive ionbeams. Nuclei closer to stability permit theapplication of statistical model cross sections.Nuclei more proton-rich than Tz=-1 are onlypopulated for conditions when a (p, γ)–(γ, p)-equilibrium is established.

Thus, only their masses, spins and half-lives enter the calculation. Connecting (α, p)-reactions have high enough Q-values, evenclose to the drip-line, that the level densi-ties are sufficient for statistical model crosssections (this is true at least up to Ca andTi, where alpha-induced reactions can playan important role). Therefore, the remain-ing experimental properties which should bedetermined and enter the calculations in acrucial way are β+-decay half-lives and nu-clear masses. The latter control the abun-dances in a (p, γ)–(γ, p)-equilibrium. BeyondSe, the details of the proton drip-line are ac-tually not that well known yet, but can influ-ence the endpoint of the rp-process. First cal-culations which include 2p-capture reactions(Schatz et al. 1996) indicate that it seemspossible to produce nuclei with A=90-100 inthe rp-process under X-ray burst conditions,depending on the choice of mass formulae andhalf-life predictions (similar to r-process stud-ies). Especially a region of strong deformationand small α-capture Q-values around A=80seems to be predicted with large variationsamong different mass models, and the even-even N=Z nuclei between A=68 and 100 playa dominant role.

The work of F. R., Ch. F., and F.-K. Twas supported by the Swiss National Sci-ence Foundation grant 20-47252.96. T. R.

17

acknowledges support by an APART fellow-ship from the Austrian Academy of Sciences.H. S. was supported by the German AcademicExchange Service (DAAD) with a “Doktoran-denstipendium aus Mitteln des zweiten Hochschul-sonderprogramms”. M. W. was supported bythe DOE grant DE-FG02-95-ER40934.

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