Automated Formulation of Reactions and Reaction Chains in Nuclear Astrophysics

Automated Formulation of Reactions and Reaction Chains in Nuclear Astrophysics
Sakir Kocabas
Department of Space Sciences and Technology
ITU, 80626 Maslak, Istanbul, TURKEY

Pat Langley
(LANGLEY @ NEWATLANTIS.ISLE.ORG)
Institute for the Study of Learning and Expertise
2164 Staunton Court, Palo Alto, CA 94306 USA

Abstract
In this paper we describe ASTRA, a computational research aid for the formulation and analysis of process explanations in nuclear astrophysics. The system operates in two independent modules. The first module generates fusion and decay reactions for the light elements from hydrogen to oxygen by using knowledge of quantum theory, and from these reactions, the second module constructs all theoretically possible reaction chains as process explanations for the nucleosynthesis of helium, carbon and oxygen. ASTRA has found apparently novel reactions that involve proton, electron and neutron capture. Currently, there is a small number of reactions and pathways proposed by astrophysicists to explain the synthesis of these elements and their relative abundance in stellar systems. ASTRA also produces many alternative reaction pathways, some of which are of interest to scientists working in this domain.

1 Introduction
Computational modeling of scientific discovery has been a primary concern of a small number of research groups in artificial intelligence, and has made considerable advances in its short history. A number of models have been developed in the last two decades to simulate discoveries in fields such as mathematics, physics, chemistry, and biology. These models addressed different aspects of discovery in formal and experimental sciences, such as mathematical theory formation (Lenat, 1979), searching for quantitative relationships and hypothesis formation (Langley, Simon, Bradshaw & Zytkow, 1987), theory development through the discovery of componential models (Zytkow & Simon, 1986; Rose & Langley, 1986), scientific problem formulation and experiment design (Kulkarni & Simon, 1990), theory formation and theory revision (Kocabas, 1991), and theory formation (Valdes-Perez, 1994).

In recent years however, interest increased towards the computational discovery of new scientific knowledge by means of new models. One of the earliest computational tools used in producing new scientific knowledge was DENDRAL (Feigenbaum, Buchanan & Lederberg, 1971), which helped analytical chemists to build correct 2-d models of some complex chemical substances. Two recent examples are Hendrickson's (1995) SYNGEN which designs the synthesis of some organic compounds from initial and intermediate compounds, and Valdes-Perez's (1995; 1997) MECHEM which has found new reaction pathways in physical chemistry.

This paper focuses on the results of ASTRA, an astrophysical research aid designed to support scientists in explaining the nucleosynthesis of elements and their relative abundance in stars. The program is a successor of BR-4 (Kocabas & Langley, 1995) which was developed as an integrated model for studying the role of predictions in particle physics. The behavior and the results of ASTRA is described with an emphasis on the system's abilities as a research tool in astrophysics.

2. Research Problems in Astrophysics
Astrophysics is a curious field of study related with the tiniest and the largest objects in the universe, the elementary particles, and stars and galaxies. One of its subfields, nuclear astrophysics, mainly concerns with the formation of chemical elements from hydrogen (H) and helium (4He), thought to have emerged in the early history of the universe, through a series of fusion and decay reactions in stars. Another important problem concerns the relative abundances of elements, in particular the abundance carbon (12C), nitrogen (14N) and oxygen (16O) relative to lighter elements lile lithium (7Li), beryllium (9Be) and boron (11B).

According to the current astrophysical theories, stars go through several stages in their lifetimes. The first stage, which follows the star. s initial formation by the condensation of cosmic clouds and hydrogen gas, involves . hydrogen burning. . During this stage, stars radiate energy emitted by a series of exothermic fusion reactions in which hydrogen is transformed into helium. Astrophysicists propose three different pathways (Audouze & Vauclair, 1980, p. 52; Williams, 1991, p. 351) to account for hydrogen burning in stars the size of the sun and smaller. Later stages consist of more complex reactions, typically involving heavier elements.

In their attempt to explain nucleosyntheses, the theorists first select a stellar model in thermal equilibrium which makes certain assumptions about the mass, temperature, density, and the element distribution in the stellar plasma. They then identify the particle and nuclear reactions consistent with quantum physics by calculation. Finally, they calculate the rates of these reactions, by using experimental and theoretical knowledge about nuclear cross-sections and reactant abundances. In this way they obtain a set of valid reactions with their rate coefficients. Scientists use the reactions with high rates to construct the pathways, either by working forward from lighter elements to the final element, or backward from the final element, until reaching to the existing lighter elements.

Naturally, there are many possible reactions, and a great number of reaction pathways even for a small number of reactions to start with. Astrophysicists deal with this problem by focusing their attention on only a small set of reactions, relying on heuristics to constrain the generation of explanatory hypotheses. In this process, there seems to be at least two places where automation could be used: In the formulation of all possible reactions within any selected energy band, and in the construction of pathways from any selected set of reactions.

In developing ASTRA our main objective was to see its results on several important research topics in nuclear astrophysics. These were: 1) hydrogen-burning processes, 2) helium burning processes, 3) formation of heavier elements carbon, nitrogen and oxygen through hydrogen and helium burning, and other fusion chains, 4) the role of neutrons in such processes, and 5) the anomaly in the relative abundance of the light elements.

We have examined a number of books and journal papers on nuclear astrophysics, notably the following work : Audouze & Vauclair (1980); Clayton (1983); Fowler (1986); Fowler, et al., 1967; Fowler et al., 1975; Harris & Fowler, et al., 1983; Cujec & Fowler, 1980; Kippenhahn & Weigert (1994); Lang (1974); and Williams (1991).

In the next section, we describe ASTRA in terms of its inputs, outputs and operations. Section 3 describes the experimental results of ASTRA, and Section 4 discusses its results. The paper ends with a summary of the conclusions.

3. System Description of ASTRA
Before we describe our application of ASTRA to nuclear astrophysics, we should first describe its inputs, outputs and procedures, which include two main stages. The first generates all theoretically valid reactions, and the second produces reaction chains as process explanations for the nucleosynthesis of elements.

3.1 Generating Reactions
The first stage of ASTRA takes as input descriptions for a set of elements and isotopes. Each entity is characterized in terms of five quantum properties: rest mass (in MeV/c2), electric charge, spin counts, lepton counts, and baryon counts. We also give ASTRA theoretical knowledge about conservation relations over these quantum properties that hold in reactions among the elements and isotopes. Finally, we constrain the system to consider only the exothermic reactions, assuming that endothermic reactions play a relatively minor role in stellar nucleosynthesis.

Based on this information, ASTRA systematically generates all fusion and decay reactions among these elements that obey the conservation laws, together with their energy emissions, or Q-values, in terms of mega electron volts (MeV). The reactions generated by the program are in the form: Rm Pn , m = 1,2; n = 1,2,3 where Rm and Pn are the sets of the reacting and resulting elements respectively, and m and n are the number of elements in the sets. (For m = 1, the formula represents decay reactions). An example of the output of this module for the H + 6Li reactions is as follows:*

-------
* The reaction formulations of ASTRA are based on neutral atoms. For this reason, there appear minor differences with textbook notations, such as in the second reaction above whose textbook version is H + 6Li 7Li + n, instead of H + 6Li 7Li + /e + n.


reaction( [h,li6], [be7], 5.68 ),
reaction( [h,li6], [li7, nu], 6.48 ),
reaction( [h,li6], [he4, he3], 4.08 ).


where the first list shows the reacting elements, the second the resulting elements, and the figures before the right parantheses, the total Q-values.
For the runs described in this paper, we provided ASTRA with the elements from hydrogen to oxygen, their isotopes and a few elementary particles like the electron, proton, neutron and the neutrino with their antiparticles, giving a total of 36 distinct entities. From these, the system generated some 400 different reactions, but some were minor variations on one another such as 3He + 9Be 12C + e + /e and 3He + 9Be 12C + n + /n. We eliminated such near repetitions manually, leaving 276 reactions that included 262 fusion reactions and 14 decays.

3.2 Generating Reaction Chains
ASTRA. s second stage takes as input these primitive reactions, along with an element E whose syntheses we want explained and the basic elements/isotopes (E) that we assume as given (typically hydrogen and deuterium). In response, the system generates all reaction chains that lead from the starting elements to the final element through the various reactions identified in the first stage. The system uses a depth-first, backward chaining search to construct the reaction chains. On the first step, ASTRA finds those reactions that give as an output the final element E. Upon selecting one of these reactions, R, it recursively finds those reactions that give as an output one of more R. s input elements. The algorithm continues this process, halting its recursion when it finds a reaction chain for which all the reacting elements are in (E), or when it cannot find a reaction off which to chain. ASTRA generates all possible reaction chains in this systematic manner.

ASTRA produces all possible exothermic fusion reactions and decays including the ones given in the astrophysics literature that we have looked at. The program constructs a large number of reaction chains, most of which would be ruled out by physicists on grounds of low reaction rates. However, some of the pathways produced by the program seem to be viable alternatives to the currently proposed mechanisms, both on the account of the energy emissions and the existence of the elements in stars.

4. The Results of ASTRA
In this section we report the results of our tests with ASTRA, within the conceptual framework of research topics in nuclear astrophysics. We first address two broad classes of reactions that are believed to play an important role in stellar nucleosyntheses, then turn to reaction chains that explain the synthesis of heavier elements.

4.1 Proton, Electron and Neutron Captures
The synthesis of chemical elements from hydrogen and helium in successive steps in stellar systems are explained by astrophysicists by a series of fusion and decay reactions. Two main processes among these reactions are proton and neutron captures in which a nucleus reacts with a proton or a neutron, giving a heavier element or isotope. Electron captures in which an orbital electron is absorbed by the atomic nucleus with the emission of a neutrino, play a relatively minor role in the nucleosyntheses.

Proton captures are an important class of exothermic reactions that take part in hydrogen burning processes. We have found 33 examples of proton captures given in astrophysics literature (e.g., Fowler, et al., 1967, 1975, 1983) for elements from hydrogen to oxygen (16O).

ASTRA. s first stage predicts that all elements from hydrogen to nitrogen (15N), with the exception of 4He, participate in proton capture. The program produces 46 such reactions, including all 33 examples we have found in texts, but also 13 others which we have not seen in astrophysics texts that we examined. Some of these reactions are:

H + 6Li 7Be
H + 9Be 4He + 4He + D
H + 9Be 10B
H + 10B 7Be + 4He
H + 11B 12C.


Electron capture reactions are weak interactions in which an electron is absorbed by the atomic nucleus to be transformed into one with a smaller atomic number. An important example which also take place in what is called the pp2 chain given below, is (e + 7Be 7Li + n). ASTRA. s first stage produces 6 electron capture reactions of which only the one just given appears in astrophysics texts.
In fusion reactions that involve neutron capture, an element combines with a neutron to form a heavier isotope of the same element. We found 17 neutron captures for light elements in the literature, while ASTRA predicts 59 such reactions that are theoretically possible for the same elements. These include the following reactions that we did not see in the texts:

n + 6Li 7Be + n
n + 7Be 4He + 4He
n + 8Be 9Be
n + 10B 11B


The third reaction may play an important role in stellar reaction pathways, which we will consider shortly.

4.2 Neutron and Deuteron Production
Neutron capture requires a continuous supply of neutrons in the stellar plasma, so that it relies on some neutron producing reaction. Audouze & Vauclair (1980, p. 86) suggest that

D + D 3He + n ,

is the only reaction that releases neutrons in the hydrogen burning stage of main-sequence stars. Yet, ASTRA also predicts six additional reactions that produce neutrons, three of which are

D + T 4He + n
3He + 7Li 9B + n
D + 9Be 10B + n
4He + 9Be 12C + n.

The first reaction appears likely in main-sequence stars, as D and T exist in them. However, astrophysicists would ignore most of these reactions for their low reactant abundances in stellar plasma. Most of the neutron-producing reactions rely on a deuteron as one of their inputs. The best known deuteron source is the reaction:

H + H D + /e + n ,

and in astrophysics texts we have found two more reactions that produce deuterium (T + 3He 4He + D and H + 9Be 8Be + D ). However, the first stage of ASTRA predicts 15 other reactions of this sort. These include:

3He + 6Li 7Be + D
3He + 7Li 8Be + D
4He + 10B 12C + D
3He + 11B 12C + D
3He + 13C 14N + D.


The first two of these reactions should take place in main-sequence stars, as 6Li and 7Li are known to exist there, yet we have not found either reaction in the literature that we examined. Again, astrophysicists would presumably ignore most of these reactions for their low reactant abundances.

4.3 Helium Synthesis by Hyrogen Burning
The transformation of hydrogen into helium in a series of nuclear processes which take place in main sequence stars as the principal source of energy. The standard reaction chains given in astrophysics texts (e.g. Audouze & Vauclair, 1980, p. 52; Williams, 1991, p. 351) for helium synthesis in such stars are the hydrogen-burning processes called . proton-proton. or pp chains. The first of these chains, is given as:


a. H + H D + /e + n
b. D + H 3He
c. 3He + 3He 4He + H + H.


The net effect of this reaction chain when reaction (a) occurs twice, is 4 H 4He + 2 n + 26.72 MeV. Another pathway hypothesized, is called . alpha-catalyzed chain. , is

d. 3He + 4He 7Be
e. 7Be + e 7Li + n
f. H + 7Li 8Be
g. 8Be 4He + 4He


in which reactions b and c provide both the 3He and the 4He needed by reaction d. An alternative pathway, which also appears in texts, replaces reaction e with H + 7Be 8B and f with 8B 8Be + /e + n, which produce the 8Be needed by the final reaction through a different mechanism. Astrophysicists refer to these three pathways as the pp1, pp2 and pp3 chains, respectively.
When asked to generate reaction chains from hydrogen to helium, the ASTRA system finds all of these reaction chains including the CNO cycle. Yet, ASTRA formulates another reaction chain which involves an electron capture, as in pp2:


H + H D + n
D + H 3He
3He + 6Li 9B
e + 9B 9Be + n
H + 9Be 4He + 6Li.


which has the same net effect. We did not see any record of this chain in the literature we examined. The program also finds 44 other processes of helium synthesis that differ in their last link of the chains. Many of these would be disregarded by astrophysicists for their small cross sections, including the chain:

H + H D + n
D + 4He 6Li
H + 6Li 7Li + n
H + 7Li 4He + 4He ,


As D is believed to be quickly destroyed by the reaction D + H 3He. Both Cujec & Fowler (1980) and Harris, Fowler, Caughlan, and Zimmerman (1983) argue that the reactions involving D are unlikely due to their low abundance. However, Clayton (1983, pp. 371-2) notes that the density of deuterium in the interstellar medium and the sun remains unknown, and suggests that the substance might be more common that usually believed.

4.4 Generation of Carbon and Oxygen
The origin and the relative abundance of carbon and oxygen has been one of the main concerns of astrophysics. The standard account (e.g., Fowler, 1986, pp. 5-6) relies on the process of helium-burning, in which helium nuclei react to form carbon and oxygen in the following steps:


4He + 4He 8Be
4He + 8Be 12C
4He + 12C 16O .


However, there were theoretical problems with this account; the first reaction is endothermic and the lifetime of 8Be is very short (2x 10-16s). Later calculations showed that 8Be resonances were sufficiently stable to allow the reaction with an alpha particle to produce carbon as in the second reaction. ASTRA does not formulate the reaction 4He + 4He 8Be because it is slightly endothermic, but the system finds 20 other reactions that produce 8Be, such as

D + 6Li 8Be
3He + 7Li 8Be + D
n + 7Be 8Be .


Once 8Be is available, 4He + 8Be 12C can take place exothermically, so ASTRA formulates this reaction. The system produces 24 additional chains that differ in their final steps to 12C. These include:

n + 8Be 9Be
4He + 9Be 12C + n ,


which relies on one of the neutron capture reactions that we discussed earlier. An alternative and even more plausible pathway produced by ASTRA involves a proton capture:

H + 8Be 9Be + n
4He + 9Be 12C + n .
Briefly, if 8Be captures a neutron or proton before it decays, then it transforms into its stable isotope 9Be. This in turn produces carbon by reacting with 4He, where the emitted neutron from the latter reaction can combine with another 8Be. Once 12C is formed, in whatever manner, it can react with 4He exothermically to produce oxygen


4He + 12C 16O .

In summary, ASTRA finds a number of reaction chains to carbon and oxygen that do not appear in astrophysics literature, all of which are theoretically possible, but the final judgement about their scientific value requires further evaluation, as we discuss next.

5. Discussion of Results
We have carefully compared ASTRA. s outputs, at both the reaction and pathway level to those available in astrophysics texts (Clayton, 1983; Audouze & Vauclair, 1980; Kippenhahn & Weigert, 1994; Fowler et al., 1967, 1975, 1983; Cujec & Fowler, 1980). We have examined the results of the system only on exothermic reactions, but ASTRA can formulate reactions in any energy band.

Although ASTRA calculates the Q-values of all reactions that it formulates, the current version does not take into account the reaction rates which are used by astrophysicists in determining the more likely reactions and reaction chains. Due to this limitation, the current version cannot decide which reactions must be dominant in a given burning phase in the star. We are considering to implement this capability in the program's future versions, in such a way that, given a stellar model (e.g., a model of the sun), when the reaction rates are given, it should eliminate some of the low-rate reactions before constructing the reaction chains.

It would be impossible for astrophysicists even to formulate all theoretically possible reactions for an exhaustive research, without a computational aid like ASTRA. For example, Fowler, et al. 1967; 1970 and 1983 cite 88 reactions for elements from H to 16O in their research, while ASTRA uses 276 such reactions. In the 88 reactions, the same authors cite 33 H-capture, 17 n-capture and 8 D-fusion reactions, while ASTRA formulates 46 H-captures, 59 n-captures and 75 D-fusions for the same range of elements.

The ASTRA program can handle a very large volume of data for constructing reaction chains, and although the hydrogen and helium burning processes have been dealt with extensively in the current literature, there may still be room for research. We understand that there is even more room for research on the synthesis of the heavier elements. Therefore, a complete analysis on the reactions and pathways can only be carried out with the aid of a computational tool such as our program.

Despite its current limitations the program still has the potential of being useful in several subfields of nuclear astrophysics. Although we tested ASTRA on the exothermic reactions of the light elements from H to 16O, the system can be used in exploring the reactions of heavier elements from oxygen to iron and further, which take place in stellar and interstellar processes.

Our research is continuing in three strands: on the one, we are in the process of evaluating the current results of the system, while on the other, we plan to add to it, the ability to use the rates of the reactions to distinguish more likely mechanisms, and finally, we are improving its interface to make the system a more useful research aid for astrophysicists.

6. Conclusions
In this paper we described ASTRA, a computational tool which formulates nuclear reactions and pathways for researchers in astrophysics. Although we have been generally satisfied with ASTRA's performance to date, there clearly exists a number of directions in which we can extend our work. We are planning to present the program's predictions to domain experts to further evaluate the behavior of the current system in terms of the novelty and plausibility of its results. Our work is continuing to improve ASTRA, and to make it a more useful research tool for astrophysicists.

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