Fictitious neutron sinks to trace radiative s -process nucleosynthesis

Context. Asymptotic giant branch (AGB) stars are strong producers of s -process elements, which are synthesized by successive slow neutron captures on elements heavier than iron. The nucleosynthesis calculation involves solving large nuclear networks with hundreds of nuclei, which in a stellar evolution code can greatly extend the computational time. However, the s -process is often measured using a handful of elements located on the neutron magic shells and grouped into tracers called ls , hs , and vhs . Aims. We propose a ﬁctitious network that approximates the production of ls , hs , and vhs species at a minimal computational expense. The network is speciﬁcally designed for the radiative s -process in AGB stars. It is an alternative to methods using large networks that can be used as a fast exploratory tool to trace the production of s -elements. Methods. The ﬁctitious network was constructed by assembling species with Z ≥ 18 into seven ﬁctitious particles whose abundances and reaction rates model the e ﬀ ective properties of the corresponding groups. The e ﬀ ective reaction rates were tabulated as a function of neutron density and number of neutrons captured per initial heavy seed ( N capt ) using single-zone nucleosynthesis calculations. The accuracy of our network was tested by comparing the abundances obtained with the ﬁctitious and large networks during the radiative burning of 13 C during the interpulse period of a 2 M (cid:12) , [Fe / H] = − 2 star. Results. The ﬁctitious network reliably reproduces the abundances of ls , hs , and vhs species during the radiative s -process. The accuracy of the method increases with the strength of the nucleosynthesis as measured by N capt , but diminishes when the nuclear distribution is di ﬀ erent from the initial distribution. This network is well suited to follow the s -process nucleosynthesis in low-mass AGB stars where neutrons are mainly produced below the envelope by the 13 C( α, n ) reaction.


Introduction
The slow neutron-capture process (s-process) is thought to be responsible for about half of the abundances of trans-iron elements in the Galaxy (Arlandini et al. 1999). It occurs in stellar interior, where preexisting seed nuclei capture neutrons, forming heavier isotopes until they become unstable and decay through the β − channel. The process is defined as slow in the sense that most β − decays occur faster than neutron captures, which forces the nucleosynthetic path to follow the valley of β-stability in the table of nuclides. It is associated with low neutron densities ranging from ∼10 7 to 10 12 cm −3 . With higher neutron densities, branching points open up and slightly modify the path taken by the s-process. This affects the production of specific isotopes such as 96 Mo or 142 Nd (e.g., Bisterzo et al. 2015). The s-process nucleosynthesis stops at 209 Bi, after which the chain of reactions 209 Bi(n, γ) 210 Bi(β + ) 210 Po(γ, α) 206 Pb closes the cycle. The s-process has two main components (Kappeler et al. 1989). The weak component includes elements with atomic number 70 ≤ A ≤ 90 and takes place in the late stages of core helium burning of massive stars. There, neutrons are provided by the reaction 22 Ne(α, n) 25 Mg (Langer et al. 1989). The main component synthesizes nuclei with 90 ≤ A ≤ 204 and occurs in lowand intermediate-mass stars (between 1 and 8 M ) during their late evolutionary stages, which is called asymptotic giant branch (AGB ;Herwig 2005;Goriely & Siess 2005;Cristallo et al. 2011;Karakas & Lattanzio 2014). In AGB stars, neutrons are produced by the reaction 13 C(α, n) 16 O (Straniero et al. 1995;Neyskens et al. 2015), although 22 Ne(α, n) 25 Mg can also be activated in the thermal pulse of massive AGB stars with mass M 2 M (Lugaro et al. 2012).
In an AGB star, the production of the neutron source starts during the third dredge-up (3DU) when protons from the H-rich convective envelope mix with the underlying 12 C radiative layer. There, proton captures on 12 C lead to the formation of the 13 C pocket, which provides the source of neutrons during the interpulse phase. The neutron densities attained through the 13 C(α, n) reaction are relatively low ( 10 8 cm −3 ), yet the main component elements can be produced due to the long neutron exposure. The mass of the 13 C pocket, hence the efficiency of the s-process, depends on the mixing mechanism by which protons are transported below the envelope. Various processes have been invoked; among them, convective overshooting (Goriely & Mowlavi 2000), rotational mixing (Herwig 2005;Cristallo et al. 2011), gravitational waves (Denissenkov & Tout 2003), or mixing induced by magnetic fields (Nucci & Busso 2014;Trippella et al. 2016;Vescovi et al. 2020).
The s-process elements are often grouped into three categories depending on their mass: light (ls), heavy (hs), and very heavy (vhs) species. They are bounded by nuclei with magic neutron numbers (N = 50, 82, 126), which are characterized by small neutron-capture cross sections. They thus act as bottlenecks in the s-process and present enhanced abundances. The relative abundances of the peak elements are often used to measure the efficiency of the s-process (e.g., Cristallo et al. 2011).
It can become computationally expensive to follow the detailed s-process nucleosynthesis during the AGB phase. The traditional approach has therefore been to use a small nuclear network of 10-50 species to accurately follow the energetics of the star and then post-process the stellar models using an extended network of ≥400 species (e.g., Lugaro et al. 2012;Gil-Pons et al. 2021;Herwig et al. 2003;Battino et al. 2019;Campbell & Lattanzio 2008).
To account for the neutron captures on the heavy species that are not present in the energy network, Jorissen & Arnould (1989) introduced a fictitious particle called the heavy neutron sink. This particle models the effective properties of all species that are removed from the extended network. With the production of heavier nuclei with a large neutron cross section, the effective neutron capture changes as well. This effect was modeled in Jorissen & Arnould (1989) by performing nucleosynthesis calculations on a single zone using physical conditions representative of the s-process and calculating the effective cross section of the heavy sink as a function of the neutron exposition. The heavy neutron sink approach was subsequently used in numerous stellar evolution codes with satisfactory effects on the neutron density (Forestini et al. 1992;Herwig et al. 2003;Karakas & Lattanzio 2003;Cruz et al. 2013;Gil-Pons et al. 2021).
The aim of this work is to improve the neutron sink approach introduced in Jorissen & Arnould (1989) by considering seven additional fictitious particles to trace the abundances of the ls, hs, and vhs species. In this way, we can approximately follow the efficiency of the s-process using a small nuclear network at a minimal expense in computational complexity. We also improve the method used in Jorissen & Arnould (1989) by tabulating the effective capture cross sections as a function of neutron density and neutrons captured per initial heavy seed instead of neutron exposure. This provides a more physical interpolation of the cross sections to variable physical conditions that may impact the branching points of the s-process.
The paper is organized as follows. Section 2 is devoted to the description of the formalism of the fictitious network. In Sect. 3, we perform the one-zone simulations we used to tabulate the effective cross sections of the fictitious particles. Section 4 analyzes the radiative s-process using our fictitious network using the stellar evolution code STAREVOL. Finally, conclusions are drawn in Sect. 5.

Defining the fictitious particles
Our goal is to reduce an extended network including 1160 species and 2125 reactions to a minimum network capable of mimicking the s-process nucleosynthesis. This network is referred to as the large network, and its details can be found in Goriely & Siess (2018) and Choplin et al. (2021). The reduction was made by grouping real nuclei with Z i > Z Cl into fictitious particles whose physical properties mirror the properties of the entire set. The nuclei with Z i ≤ Z Cl and their reactions are part of our default AGB network that is used to determine the energetics and follow the abundances of the light elements. It is composed of 54 elements up to 37 Cl and a fictitious neutron sink coupled through 168 reactions as described in Siess & Arnould (2008) and is referred to as the energy network. Each fictitious particle k is defined by a couple of mass numbers (A k,min ; A k,max ) such that it contains all nuclei i with atomic mass number A k,min ≤ A i ≤ A k,max . Because the groups are delineated along isobars, unstable species and the products of their decay remain in the same fictitious group. This is crucial to our reduced network, as it implies that the abundances of fictitious particles are not affected by β − decays and are effectively stable.
Three of the fictitious particles that trace the s-process elements (ls, hs, and vhs) were constructed such that they contain the stable species with the same magic neutron numbers (N = 50, 82 and 126, respectively) and all of their isobars.
Owing to their small neutron-capture cross sections, they act as bottlenecks for the s-process, and their abundance is greatly enhanced compared to other elements. As an example, the ls fictitious particle is composed of stable nuclei with N = 50, namely 86 Kr, 87 Rb, 88 Sr, 89 Y, and 90 Zr, plus all the other nuclei with mass number between A ls,min = 86 and A ls,max = 90. This adds the nuclei 86 Sr, 87 Sr, 89 Sr, 90 Sr 86 Rb, and 90 Y to the ls group. For vhs, we also included all elements involved in the Pb recycling process, so that this fictitious particle effectively plays the role of the end point of the s-process.
The nuclei in between the ls and hs form the lse particle and are bounded by 91 ≤ A i ≤ 136, while those between hs and vhs form the hse particle and have 143 ≤ A i ≤ 205. The nuclei with 56 ≤ A i ≤ 85 form the fictitious particle fe. At the start of the s-process, this group is essentially dominated by the abundant 56 Fe. Finally, the particle light connects the energy and fictitious parts of our network and is an exception to the isobar criterion as it contains nuclei with Z i > Z Cl and A i ≤ 55. The specific condition on Z i allowed us to keep all chlorine isotopes in the energy network.
Our large network also contains reactions that bring back nuclei from the light group to the energy network: 37 Ar(n, p) 37 Cl, 37 Ar(e − , ν e ) 37 Cl, 37 Ar(n, α) 34 S, 39 Ar(n, α) 36 S, and 40 K(n, α) 37 Cl. We have verified in our simulations that neglecting these reactions has very limited effects on the abundance of the light particle and almost none on the abundances of the heavier fictitious groups. When these reactions are neglected in the fictitious network, only neutron captures affect the abundances of the fictitious particles. The definition of the fictitious particles is summarized in Table 1.
Before defining the physical properties of the fictitious particles, we introduce several notations. The subscript i is reserved for real nuclei, while the subscript k relates to fictitious particles. When the atomic mass number and charge (A i , Z i ) need to be specified, the notation i (A,Z) is used. Numbers are assigned to fictitious particles by order of increasing mass number of nuclei they contain (e.g., light = 1, fe = 2 and so on). The set of nuclei i that are grouped together to form the fictitious particle k is called E k .
We distinguish two kinds of reactions between fictitious particles. First, the internal reactions in which both the reagent and the product belong to the same set E k . By construction, it follows that all β − decays are interior reactions. Most of the neutron captures in the large network are also internal reactions, and while they modify the distribution of nuclei, the total number of particles in E k is conserved. They are denoted by k → k. The second kind of reactions includes the neutron captures that transfer nuclei from group E k to E k+1 . They are called border reactions and are referred to by k → k + 1. By definition of the fictitious particles, only nuclei that are on the isobars A max,k are involved in border reactions. These nuclei, which are also the heaviest species in E k , are referred to as border nuclei.

Mass and molar fraction
The molar fraction Y k of the fictitious particle k is given by the sum of the molar fractions Y i of the nuclei composing E k , Defining the mass fraction of a fictitious particle X k as the sum of the mass fractions of nuclei in E k implies that its mass number A k writes . This strongly complicates the numerical implementation of the fictitious network because A k is not constant, but depends on the distribution of elements in E k , which is not available in the fictitious network. An additional complication arises when neighboring shells, containing fictitious particles with different values of A k , are mixed. Our attempts to keep track of A k at all time have hurt the precision of the solutions. To avoid these problems, we instead assigned every fictitious particle a constant mass (2) This specific value of A fict is explained at the end of the section and has to do with the last element of our energy network, which stops at 37 Cl. To keep the fictitious-particle mass constant during neutron captures, we introduced the inert neutronñ, which has the same mass as a neutron. The neutrons reacting with fictitious particles are not captured, but are rather tagged and become free neutrons that are unable to react a second time. Formally, the neutron-capture reaction now writes k(n,ñ)k , where k = k stands for internal reactions and k = k + 1 stands for border reactions. Only the number of reacting particles is relevant for nucleosynthesis. Inert neutrons therefore do not affect the nucleosynthesis. Consequently, the mass fraction of a fictitious particle k does not represent the true mass fraction of real nuclei belonging to E k , as all nuclei i ∈ E k have mass numbers A i > A fict , so The difference in the mass fraction between the fictitious and real particles corresponds to the mass fraction of the inert neutrons such that As the nucleosynthesis equations are independent of the fictitious particle mass, the value of A fict is a matter of choice. We chose A fict = 36, so that the 36 Cl(β − )light reaction that connects the energy part of the network to the fictitious part respects the mass conservation and keeps the usual form of a β − decay. Imposing the mass conservation on the reactions light(n,ñ)fe, fe(n,ñ)ls, and so on implies that every fictitious particles has the same value of A k = A fict .

Defining the fictitious reaction rates
In Sect. 2.1, we made the distinction between internal neutroncapture reactions of the type k(n,ñ)k that do not change the abundance of the fictitious particle k and border neutron-capture reactions symbolized by k(n,ñ)k + 1 that connect adjacent groups of fictitious particles E k and E k+1 . We first describe how the reaction rate corresponding to the border reactions was calculated.
In the framework of the s-process, the general time evolution equation of the abundance of an element i (A,Z) is where σv n,i (A,Z) and λ i,(A,Z) are the neutron capture and β − decay rates of the element i (A,Z) , respectively, and n n is the neutron density. The first and third terms on the right-hand side describe the creation and destruction of the particle i (A,Z) through neutron captures, while the second and fourth terms account for β − decays. For stable particles, λ = 0. From the definition of Y k (Eq. (1)), it follows that the time evolution equation of the abundance of the fictitious particle is equal to dY k /dt = i∈E k dY i (A,Z) /dt. Performing the sum, most of the terms in Eq. (5) cancel out: i) elements on the same isobar are in the same group E k , thus all of β − decay terms vanish. ii) All the neutron-capture terms also cancel, except those corresponding to border reactions. The evolution equation for Y k finally writes The first term corresponds to the production of lightest nuclei in E k by neutron captures on the border particles of E k−1 . Analogously, the second term is associated with neutron captures on the border particles in E k . We now define now the evolution reaction rate of the particle k as and introduce it into Eq. (6), yielding which has the familiar form of a neutron capture for a stable particle. The effective reaction rate depends on the current distribution Y i , which is not accessible in the fictitious network. The tabulation of the effective reaction rates is detailed in Sect. 3. The definition of the evolution reaction (Eq. (7)) is central to our work. The numerator of this equation involves only reactions with the border nuclei of group E k . Because of the definition of the fictitious groups (Table 1), the border nuclei are located on the isobar A = A max,k . For example, for the fictitious particle fe, the summation in Eq. (7) is done only over 85 Kr and 85 Rb. The neutron-capture rates on the other nuclei of the group (with A min,k ≤ A i∈E k < A max,k ) do not enter the calculation of σv evo,k . The numerator also traces the neutron-capture path taken during the s-process. At high neutron densities, more neutron-rich nuclei are produced. Their molar abundance Y i is higher and their weight in Eq. (7) increases. We thus expect σv evo,k to be lower when the most neutron-rich nuclei with the lowest cross section are more abundant.
Finally, we note that the light particle is an exception to the evolution equation as there is no fictitious particle preceding it.
The term corresponding to the production of light (first term on the right-hand side of Eq. (8)) is replaced by all the actual reactions from the energy network that leak into the light group. In our network, this involves the β − decay of 36 Cl and neutron and proton captures on 36 Cl and 37 Cl.
We now look at the internal reactions. As previously stated, they affect the neutron abundance, but leave the fictitious particles unchanged. This was the original goal of the introduction of neutron sinks in Jorissen & Arnould (1989). The time evolution equation for the abundance of neutrons, Y n , writes where R + n and R − n correspond to the production and destruction of neutrons by reactions in the energy network. The last term includes the neutron captures by species in the fictitious groups. The decomposition of the second sum into internal and border reactions yields Similarly to the evolution reaction rate (Eq. (8)), we define the internal neutron-capture rate as Combining Eqs. (7), (10) and (11), we can rewrite the evolution equation for neutrons as The reaction corresponding to internal neutron captures by fictitious particles takes the form of k(n,ñ)k, so that the production rate of inert neutrons is equal to the capture rate of neutrons on fictitious particles, To summarize, to mimic the full s-process, we introduced seven fictitious particles representing the abundances of seven groups of nuclei and an inert neutron. These eight particles are connected via seven internal reactions and six border reactions.

Number of neutrons captured per heavy seed
Equations (7) and (11) show that the effective reaction rates depend on the distribution of the real nuclei. This distribution is not accessible in the fictitious network. The values of σv ncap,k and σv evo,k are thus tabulated as a function of two variables: the number of neutrons captured per initial heavy seed N capt and the current neutron density n n . The number of neutrons captured per heavy seed at a time t in the large network is defined as Using Eqs. (1), (3) and (4), the definition of N capt in the fictitious network becomes The effective reaction rates σv evo,k and σv ncap,k were tabulated based on a one-zone nucleosynthesis calculation for a given initial composition, temperature, and mass density. The initial shell composition in molar fractions was the same as in Jorissen & Arnould (1989) and was given by Y4 He ≈ 0.1247, Y12 C = 0.0275, and Y13 C = 0.0123. Neutrons are produced by the 13 C(α, n) 16 O reaction. Hydrogen was assumed to have been completely used to produce the 13 C pocket, so that its abundance, as well as the abundances of Li, Be, and B, which are not involved in the s-process nucleosynthesis, were set to zero. Nuclei with A ≥ 14 had a solar distribution given by the Asplund et al. (2009) mixture. The burning of 13 C produces enough neutrons for the total conversion of the initial 56 Fe into Pb. The temperature of the simulation (expressed in units of T 6 = 10 6 K) controls the neutron density as the reaction 13 C(α, n) 16 O strongly depends on it. The mass density was kept constant and equal to 500 g cm −3 .
In the resulting simulations, the neutron density shows little variations. It is maximum at the beginning, and n n decreases due to the depletion of 13 C and the production of species with higher neutron-capture cross sections. With temperatures ranging from T 6 = 60 to T 6 = 260 by steps of ∆T 6 = 5, we probed neutron densities between n n = 10 6 cm −3 and n n = 10 15 cm −3 . The simulations were pursued until the value of N capt = 200 was reached, at which time all the initial iron has been converted into lead. For each simulation, we extracted the values of σv evo,k and σv ncap,k as a function of N capt and log 10 n n and interpolated the data to construct a table of the effective reaction rates, containing 64 points carefully spaced in N capt and 32 in log 10 n n . The dependence of the reaction rates on temperature was not considered, since neutron-capture rates depend only weakly on it.
The large nuclear network used in this paper is described in detail in Choplin et al. (2021). Elements between Pu and Cf have been added since. It contains 1160 species up to Cf and 2125 reactions (n-, p-, α-captures, α-decays, electron captures, β − decays, and photodisintegrations). Nuclei with a decay halftime longer than 1 s are accounted for, allowing this network to describe neutron-capture processes associated with neutron densities up to 10 17 cm −3 . If not stated otherwise, all computational results presented in this section were performed under the above conditions. Figure 1 shows the evolution of N capt as a function of neutron exposure φ for simulations where 13 C burns at different temperatures (the initial neutron densities are shown for each T 6 ). The neutron exposure is another common variable that is used to measure the progress of the s-process nucleosynthesis and is defined by where v th = 3k b T m n is the thermal velocity of neutrons at temperature T and m n the neutron mass.
In the large network, the value of N capt changes through three types of reactions: a) neutron captures on heavy nuclei, b) production of nuclei with mass number A = 56 from neutron captures on lighter nuclei, and c) ejection of α particles during the Pb recycling chain of reactions. The first two reactions increase N capt , while the third reaction decreases it.
The evolution of N capt is primarily driven by neutron captures on 56 Fe and its s-process progeny due to its dominant abundance. As is observed in Fig. 1, for higher neutron densities, greater neutron exposures are needed to attain the same value of N capt . The curves move from left to right with increasing temperature. The value in parentheses represents the typical neutron densities in cm −3 associated to the burning temperature. The T 6 = 100, No Light case has the same composition as T 6 = 100, but the abundance of nuclei with 13 < A i < 56 is set to zero. The NP case accounts for enhanced abundances of neutron poisons (see text). The T 6 = 100 -Fict case uses the fictitious network. This is explained by the fact that for increasing n n , the path of the s-process is shifted away from the valley of β − stability, toward more neutron-rich isotopes that have generally smaller neutroncapture cross sections. Therefore, a greater neutron exposure is needed to capture the same number of neutrons.
The effect of light species on N capt can be assessed by comparing the solid and dashed curves for T 6 = 100. In the simulation without light species (elements with 13 < A < 56), N capt approaches the maximum value of N capt = 152, corresponding to the conversion of all the initial heavy seeds into 208 Pb. Thereafter, the Pb recycling reactions are in equilibrium and freeze the abundances. On the other hand, when light species are included (solid black line), the limit on N capt is lifted as it can be further increased by the production of A = 56 nuclei from lighter species. These reactions affect N capt only in the very advanced stages of the s-process, when N capt 130.
To investigate the impact of a change in the species involved in the energy network (Z ≤ 17), we performed a simulation referred to as NP, in which the mass fraction of the light neutron poisons 14 N, 16 O, and 20 Ne are multiplied by 10 with respect to their solar values and 22 Ne, 23 Na, and 25 Mg are multiplied by 100, keeping all other nuclei with A > 13 at their solar abundance. With higher abundances of neutron absorbers, n n decreases, so that a slightly higher temperature is required to reach the same neutron density as in the T 6 = 100 simulation. In this way, the differences between the two simulations result solely from the variation of the initial composition. As shown in Fig. 1, the effect of introducing the light neutron poisons is negligible. The NP simulation (dotted red line) closely follows the T 6 = 100 simulation (solid black line). The NP simulation ends earlier (N capt = 178), however, because a larger fraction of the neutrons is captured before 13 C is exhausted. We performed additional tests in which we varied the temperature, and we found in all cases that the addition of light neutron poisons has a very small effect: For the same neutron density, we have T 6 =80 (~10 6 ) T 6 =100 (~10 8 ) T 6 =120 (~10 10 ) T 6 =150 (~10 12 ) T 6 =200 ( ~10 14  the same run of N capt as a function of exposure. In other words, for a given (n n , N capt ), the reaction rates are almost not affected by the enhanced abundance of neutron poisons. The comparison between the large (solid black line) and fictitious (dotted black line) networks in Fig. 1 at T 6 = 100 shows good agreement, except at high exposures (φ 1.5 mb −1 or N capt 150), where N capt is overestimated in the fictitious network. The reason for this slight discrepancy is due to the emission of α-particles in the Pb-recycling reactions in the large network, which limits the increase in the mass fraction of the heaviest species. In this high N capt region, the fictitious network will overestimate N capt and potentially affect the interpolation of σv (N capt , n n ). Our simulations (Sect. 4) show, however, that this has a negligible impact on the final abundances of the fictitious particles.
Our choice of tabulating the reaction rates as functions of (N capt , n n ) contrasts with the seminal work of Jorissen & Arnould (1989), where the authors considered only one fictitious particle (called heavy particle) to account for the neutrons captured by all particles with A i ≥ 56 and tabulated the effective neutroncapture reaction rate as a function of exposure, yielding where Y heavy (0) = A i ≥56 Y i (0). As in our case, the function σv heavy (φ) was extracted from one-zone nucleosynthesis calculations.
We illustrate in Fig. 2 our choice of interpolating variables on σv heavy , but the conclusion that N capt is better suited than the exposition holds for all other effective reaction rates. Figure 2 shows σv heavy as a function of N capt and φ. The burning temperature of 13 C determines the neutron density during the nucleosynthesis and is indicated in parentheses. At the start (φ = 0 and N capt = 0), σv heavy is mainly determined by the initial 56 Fe, as it is the dominant nucleus. For increasing neutron densities, the s-process path encounters more n-rich nuclei that have smaller cross sections. Since σv heavy is a weighted function of the individual capture rates, it is reduced when n n is larger (right panel of Fig. 2). For N capt 150, σv heavy enters an asymptotic regime where all the heavy nuclei have been transformed into Pb and where light species (A ≤ 55) start being depleted. This slightly affects the capture rate.
We also note that with increasing neutron density, σv heavy (φ) is shifted toward greater neutron exposures. This is a consequence of the global decrease in the effective capture rate with increasing n n . As neutrons are less efficiently captured, a higher exposure is required to reach the same value of N capt (Fig. 1).
When the variable N capt is used (right panel of Fig. 2), the shift is no longer present and the curves present an homothetic behavior. Consequently, if the neutron density changes, it is much easier to interpolate the rate as we can keep the same value of N capt . If the capture rates were tabulated as a function of φ, we would also need to adjust the exposure to account for the change in the neutron density. From now on, we only use N capt to describe the evolution of the reaction rates.

Effective reaction rates
The goal of this subsection is to explain the dependences of the reaction rates on N capt and n n . The behavior of σv evo,k and σv ncap,k is explained in terms of the distribution of nuclei to which they are connected via Eqs. (7) and (11). To discuss the dependence on N capt , we performed a one-zone nucleosynthesis simulation at T 6 = 80 using a large network. This temperature was chosen so that the burning of 13 C yields neutron densities lower than 2.5 × 10 6 cm −3 and the s-process follows the valley of stability closely. The resulting abundances and effective reaction rates of the fictitious particles are presented in Fig. 3. We start the discussion on σv evo,k by recalling that this reaction rate is driven mainly by the abundances of the border nuclei in E k (Eq. (7)). If at a given time the distribution inside the fictitious particle is dominated by its lightest species, the evolution reaction rate will be low. Conversely, if the proportion of "border" nuclei increases with neutron captures, σv evo,k will increase. We illustrate the discussion using lse as an example, for which we show in Fig. 4 several nuclear compositions for 88 ≤ A ≤ 137 taken at different values of N capt from our onezone simulations. All the conclusions made for this particle can be extended to the other fictitious particles.
We identify three successive regimes that govern σv evo,k : the initial, intermediate, and asymptotic regimes. The initial regime affects particles with k ≥ ls, during which the behavior of the reaction rates depends heavily on the initial nuclear distribution. During the intermediate regime, the abundances and reaction rates are primarily driven by neutron captures on fe. Finally, during the asymptotic regime, the effects of neutron captures on nuclei lighter than 56 Fe become apparent.
The initial regime is characterized by a decrease in σv evo,k from its starting value until it reaches a global minimum. The middle panel of Fig. 3 shows that the minimum appears at higher N capt for heavier fictitious particles (ls = 5.6, lse = 13, hs = 22, and hse = 30). The lse and hse particles additionally show an initial spike at N capt = 0.8 and 1.75. To understand the behavior of σv evo , we focused on the distribution of lse, whose evolution is presented in Fig. 4. Between N capt = 0 and 0.8, the number of lse border nuclei (with A = 136) increases by a factor ≈4 and A40, page 6 of 13 light × 5 T 6 =80 (~10 6 ) T 6 =100 (~10 8 ) T 6 =120 (~10 10 ) T 6 =150 (~10 12 ) T 6 =200 (~10 14 ) yields the early rise in σv evo,lse . After this initial adjustment, the lightest nuclei of lse start to be populated as a result of neutron captures on the abundant fe seeds. The relative abundance of lse border nuclei decreases until N capt = 11 and σv evo,lse attains its minimum. At the end of the initial regime, the abundances of the various fictitious particles has increased by 500. The shape of the distribution, and by extension the magnitude of σv evo,k during this phase, strongly depends on the initial composition. The intermediate regime describes the region of moderate values of N capt where σv evo,k increases steadily. It starts when σv evo is minimum and ends when it reaches a maximum (Fig. 3). Again, heavier fictitious particles reach the end of the intermediate regime at higher N capt , ranging between 56 for fe and 160 for hse. The increase in σv evo,k occurs in two phases, and we explain this behavior again using lse as a proxy. First, as shown in Fig. 4, between N capt = 11 and 40, neutron irradiation produces a rearrangement of the initial distribution, which flattens. The abundance of the border nuclei significantly increases and σv evo,k rises. Then, around N capt = 40, neutron captures on 56 Fe can no longer supply fresh nuclei to the fictitious particles. At this turning point, the abundance of the lightest elements (with A = 91) is maximum, and it subsequently decreases for higher N capt , while nuclei with A = 136 continue to be produced. At N capt = 126, σv evo,lse reaches its peak value. During this intermediate phase, the shape of the elemental distribution is primarily determined by neutron captures on the abundant 56 Fe and is almost independent of the initial mass fractions of the other nuclei.
The asymptotic regime starts after σv evo has passed its maximum. The reaction rates undergo an initial decline due to the consumption of the border nuclei. They then remain roughly constant as abundances reach an equilibrium (Fig. 3).
We now analyze the behavior of σv ncap,k (bottom panel of Fig. 3). In the initial regime, the behaviour of the internal neutron-capture rates depends on the initial composition, similarly to σv evo,k . We see large variations (in particular in σv ncap,hse ) until the distribution of nuclei has readjusted. However, for such low N capt , the overabundance of fe with respect to the other fictitious particles implies that the latter have a negligible effects on the neutron density. In the intermediate and asymptotic regimes, the internal reaction rates remain roughly constant and weakly depend on N capt . This indicates a composition that is close to the nuclear equilibrium and that is illustrated for lse distributions at N capt = 40, 126, and 160.
The dependence of the reaction rates on the neutron density was assessed by performing one-zone simulations at temperatures varying between 80 ≤ T 6 ≤ 200, thus probing neutron densities in the range 10 6 cm −3 −10 14 cm −3 . The resulting σv evo and σv ncap are presented in Figs. 5 and 6, respectively. All reaction rates generally decrease with increasing n n . The reason is that the s-process path includes more nuclei with smaller cross sections.
A major branching point of the radiative s-process is at 85 Kr. It activates at n n ≈ 10 8 cm −3 and greatly affects σv evo,ls , as can be seen by comparing the T 6 = 80 and 100 simulations in the middle left panel of Fig. 5. For n n ≥ 10 8 cm −3 , the s-process path is diverted from 85 Kr→ 85 Rb→ 86 Sr towards 85 Kr→ 86 Kr→ 87 Rb. The last two nuclei are on the magic N = 50 isotone. They have small cross sections and cause a bottleneck for neutron captures. Their increased abundance causes the decrease in the evolution rate of ls following Eq. (7). For n n ≤ 10 10 cm −3 , the evolution reaction rates of the other fictitious particles conveniently show a weak dependence on n n .
For T 6 = 80 and 100, σv ncap varies only slightly with n n for all fictitious particles, with the exception of ls (Fig. 6). This is because they are associated with modest neutron densities (10 6 cm −3 < n n < 10 8 cm −3 ), so the path of the s-process essentially follows the valley of β − stability. The internal capture rate of ls is highest for T 6 = 80 due to the absence of branching at 85 Kr, as explained previously. For higher neutron densities, the presence of n-rich nuclei is more noticeable, especially for fe, lse, and hse particles, as seen by the much lower rates for T 6 = 150 and 200. The internal reaction rates for fe, lse, and hse are most relevant for the accurate determination of n n because their cross section is highest (Eq. (12)). Particles lse and hse present an exceptionally flat behavior for N capt 30 because a nuclear equilibrium is reached in this regime.

Results using STAREVOL
Using the STAREVOL evolution code, we calculated the evolution of a 2 M star with [Fe/H] = −2 using the large network up to the end of the first 3DU to test the fictitious network during the radiative s-process. The convective-overshooting model described in Goriely & Siess (2018) was used to allow protons to spread past the convective envelope with an exponentially decreasing diffusion coefficient. This creates the 13 C pocket necessary for the subsequent radiative s-process. For this case, we used diffusion parameters f over = 0.1, p over = 0.5, and D min = 10 6 . Starting from this model, the interpulse nucleosynthesis was simulated using both the large and fictitious networks. The model was adapted to the fictitious network by converting mass fractions using Eq. (3). We first analyze the shell at mass coordinate M r = 0.599 M , where the neutron density is maximum, reaching 5.17 × 10 7 cm −3 . The corresponding evolution of neutron density, abundances, and effective reaction rates in both networks as functions of N capt are presented in Fig. 7. As discussed in Sect. 3.1, the Pb recycling reactions were not accounted for in the fictitious network, resulting in a noticeable increase in N capt in the region N capt 160. This effect can be observed in the uppermost panel of Fig. 7, where the final N capt reaches 205 in the large network and 210 in the fictitious network. The uppermost panel also agrees remarkably well for the neutron density profile in both networks; the differences do not exceed 3% (when the comparison is done at a given time t, as opposed to a comparison at a similar N capt ). The abundances of fictitious particles in both networks also agree well. Slight dif-ferences that do not exceed 0.3 dex are observed and mainly affect ls.
The third row of Fig. 7 shows the values of σv ncap,k as a function of N capt . The effective neutron-capture rates are well reproduced for all fictitious particles. The largest differences are observed for hse and for ls in the region N capt ≥ 150, but they never exceed 5%.
The evolution reaction rates (final row of Fig. 7) are also well reproduced. The largest differences arise during the intermediate phase, in which the fictitious reaction rates are slightly underestimated. They have a limited impact on the abundances of the fictitious particles, however.
The reaction rate σv evo,ls shows the greatest deviations and can be underestimated by as much as ≈20%. This is explained by the differences in the evolution of n n during the nucleosynthesis shown here and the one used to tabulate σv evo,ls and the activation of the branching point at 85 Kr occurring at n n = 10 8 cm −3 . During the radiative s-process, the temperature rises gradually, and the neutron density increases over time as the 13 C(α, n) reaction is more efficiently activated. The neutron densities initially favor the decay of 85 Kr, and 85 Rb is produced at the expense of 86 Kr, which is more difficult to destroy due to its low cross section. In the one-zone simulations, the neutron density is approximately constant, and a larger amount of 86 Kr is produced. As 86 Kr is not a border nucleus of ls, its higher abundance in one-zone simulations than during the radiative s-process is responsible for the lower value of σv evo,ls in the fictitious network.
The evolution of the abundances of the fictitious particlesȲ k averaged over the radiative shells where N capt > 3 is presented in Fig. 8. In these layers, the s-process has taken place, and the resulting nucleosynthesis is ingested in the thermal pulse. By performing the average, we assessed which elements are effectively produced during the interpulse and are brought to the surface of the star. In this low-mass star model, the nucleosynthesis occurring in the thermal pulse is unlikely to change the distribution of the fictitious particles, but this may be different in highermass stars (see Appendix A). The averaged abundance contains the nucleosynthesis products of layers exposed to a wide range of neutron densities.
At the start of the radiative s-process, the fictitious particles ls, lse, hs and hse are successively produced (Fig. 8). The slight decrease that follows the maximum ofȲ k results from the layers where the neutron density was the highest and the s-process was sufficiently advanced to start the destruction of the fictitious particles. This destruction was counterbalanced by the production of fictitious particles in neighboring layers, however, where the neutron density was lower. In this simulation, fe is only consumed and vhs is produced. There is a very good agreement in Y k for all fictitious particles at all times. The largest differences are for ls and lse and do not exceed <0.1 dex.
Pulse after pulse, the composition of the layers below the convective envelope is progressively enriched in heavy species and can show substantial differences with respect to the solarscaled composition that was used to compute σv evo,k and σv ncap,k . To test how robust our estimation of the fictitious reaction rates is, we performed new calculations with the fictitious and the large networks using the conditions at the beginning of the seventh interpulse of our 2 M , [Fe/H]= −2 stellar model. At this time, the abundance of the heavy nuclei is increased by a factor of ≈ 50 compared to the initial solar value, but for some s-process elements, this enhancement can reach almost 3 dex, such as for 208 Pb. On the other hand, the abundance of 56 Fe is almost not affected because it was replenished by the previous 3DU episode. In this test simulation, the composition of the fictitious abundances was adjusted to this new structure using Eq. (3), but the fictitious reaction rates were kept unchanged and were tabulated using the initial solar-scaled composition. Figure 9 shows the evolution of the abundances and reaction rates of the fictitious particles in the most irradiated shell as a function of N capt , which is a proxy for time. The enrichment in heavy species can be assessed by comparing the upper panel of Fig. 9 with the middle panel of Fig. 7. Small differences are observed for hs and hse at the beginning of the interpulse, but they disappear as soon as N capt 30, that is, when nucleosyn-thesis enters the intermediate regime.
For the other elements, the agreement is excellent. Relatively large variations are present initially for particles more massive than fe concerning σv evo (middle panel of Fig. 9), but again, they are significantly reduced when the intermediate regime is reached (after σv evo has passed its global minimum). For higher N capt , the differences are comparable to those presented in Fig. 7. The same conclusions hold for σv ncap shown in the bottom panel: The reaction rates are somewhat discrepant for low N capt , but become very comparable in the intermediate regime. We also computed the averaged abundance of the fictitious particles over the shells that have reached N capt > 3 A40, page 9 of 13 at the end of the interpulse. The resulting abundances between the large and fictitious network do not differ by more than 0.1 dex for the early pulse case. In conclusion, our tests show that the fictitious network is weakly dependent on the composition in heavy elements of the intershell as long as the nucleosynthesis enters the intermediate regime, which is characterized by N capt ≥ 30 and corresponds to a neutron exposure φ 0.5 mb −1 (Fig. 1). For completion, we present in the appendix the performance of the fictitious network under convective s-process nucleosynthesis and highlight some limitations of this model.

Discussion
We have presented a fictitious network that contains an inert neutron and seven fictitious particles, each representing the abundances of groups of nuclei involved in the s-process nucleosynthesis. Their effective reaction rates were tabulated as functions of N capt and n n using one-zone nucleosynthesis simulations. In contrast to Jorissen & Arnould (1989), we used N capt instead of the exposure φ. This resulted in an easier and more consistent interpolation of the rates when the neutron density varied strongly.
The network was tested on a 2 M , [Fe/H] = −2 star for the radiative s-process. We found that the production of fictitious particles and the neutron density are correctly reproduced. For a typical radiative s-process that reaches N capt 30 (φ 0.5 mb −1 ), we found that the reaction rates enter the so-called intermediate regime in which the final abundances are primarily determined by the initial 56 Fe abundance and weakly depend on the initial distribution of heavy nuclei.
In Appendix A, the convective s-process is tested on the 19th pulse of a 3 M , [Fe/H] = −2.5 star. In this case, the fictitious network performance is lower. While the final abundances of the fe, ls, hse, and vhs are fairly well reproduced, the abundances of lse and hs showed some disagreements. This is ascribed to two factors. First, the composition irradiated during the thermal pulse can differ significantly from the initial composition as a result of past s-process episodes, and the tabulated reaction rates are no longer correct. Second, the convective s-process does not lead to sufficient neutron captures, so that the intermediate regime cannot be reached, where abundances are mainly determined by the initial 56 Fe. We tried to approximate the correct reaction rates based on the composition of the material entering the thermal pulse. This led to a significant improvement for the lse particle, but hs remained poorly determined. To conclude, our fictitious network is able to trace the production of s-process elements in radiative layers fairly well, but shows limitations when dealing with the convective s-process. It is suited for low-mass AGB stars and represents a substantial improvement compared to the previous formalism developed by Jorissen & Arnould (1989).

Appendix A: Convective s-process
In the large network, the calculation of the nucleosynthesis in convective zones is done in the one-zone approximation, in which the mass fraction of all elements, except neutrons, is averaged across the mixed region. The reaction rates are computed locally and are then mass averaged over the extent of the convective region, where the subscript j refers to quantities evaluated at the shell j, and dm j is the mass of this shell. This expression applies to all reactions except neutron captures because the short lifetime of neutrons means that their local abundance must be used. For the averaged neutron-capture rate, the following expression is used: We applied the fictitious network on the 19th thermal pulse of a 3 M , [Fe/H]=-2.5 star. The pulse-driven convective zone covers the mass coordinates between m bot = 0.810002 M and m top = 0.817528 M at its maximum extent. The temperature at the base of the convective zone reached T = 3.8 × 10 8 K, activating the 22 Ne(α, n) 25 Mg reaction and resulting in a peak neutron density of 8.2 × 10 13 cm −3 . The thermal pulse lasted for 39 years. In these conditions, ten neutrons are captured by heavy nuclei in the convective shell by the end of the thermal pulse on average. Starting from the same model computed with the large network just before the thermal pulse, we followed the nucleosynthesis using our formalism, setting the initial composition of the fictitious model using Eq. (3). We followed the abundances Y k of fictitious particles averaged over the maximum extent of the convective zone (i.e., between m bot and m top ) throughout the duration of the pulse. The results are presented in Fig. A.1. The convective zone is present between models 60609 and 61002. The figure shows that the changes inȲ k occur between models 60640 and 60680, where the neutron density exceeds ≈ 10 12 cm −3 . Outside these models, n n is too low and the neutroncapture nucleosynthesis is inefficient.
In the large network, the abundances of ls, lse, hs, and hse have increased by 0.6 − 0.7 dex and by 0.3 dex for vhs. The abundance of fe has dropped by only 0.02 dex. The lse and hs particles are underproduced by the fictitious network by 0.4 and 0.3 dex, respectively. The agreement for ls and vhs is better, as the final abundances are within 0.1 dex of the large network. Only hse is close to the large network value. The discrepancies between the two networks for the convective s-process are more pronounced than in the radiative case.
Two main factors can explain the weaker performance of the fictitious network. The low efficiency of the convective s-process prevents the nucleosynthesis from entering the intermediate regime, where the abundances are mostly determined by neutron captures on 56 Fe and weakly depend on the distribution of the other elements. The convective s-process thus depends on the composition of heavy elements at the start of the neutron irradiation episode, which in the convective shell can differ strongly from the initial distribution used to tabulate the reaction rates due to 3DU episodes, for example.
We present in Fig. A.2 the composition of the matter entering the thermal pulse (Y pulse ). It is made of layers, located below m 3DU , the mass coordinate corresponding to the maximum extent of the convective envelope during the previous 3DU, which belonged to the previous pulse. It carries the products of the last convective s-process, and its composition Y 1 shows depletion in 56 Fe and enrichment in light s-elements. The mass of this layer is ∆m 1 = m 3DU − m bot , and Y 1 is characterized by N capt = 10.15, meaning that it mostly contains light s-elements. The layers between m 3DU and m top contain the envelope composition after the dredge-up and has been processed by H-burning during the interpulse period. Its composition Y 2 has 56 Fe close to the initial value and is slightly enriched in heavier s-elements from the previous 3DU episodes. The mass of this layer is ∆m 2 = m top − m 3DU , and its composition is characterized by N capt = 1. In our model, we have M pulse = ∆m 1 + ∆m 2 and Y pulse = (Y 1 ∆m 1 + Y 2 ∆m 2 )/M pulse . In Fig. A.2, Y 1 is scaled up to match the value of Y pulse at A = 136 to show that it is almost identical to Y conv for most nuclei. In fact, Y 1 provides 92% of nuclei with 60 ≤ A ≤ 142 and 82% of nuclei with A ≥ 143. The distribution near the iron peak is determined by Y 2 , which is very rich in these elements. The mixing of Y 1 and Y 2 has two important consequences. First, the value of N capt in the pulse drops to ∼ 3.17; the exact value depends on ∆m 1 and on the efficiency of the previous convective s-process. In this regime of low N capt , the reaction rates for fe are fairly insensitive to n n (Figs. 5 and 6), even though neutron densities can vary sharply over several orders of magnitude in the pulse. Second, σv ncap, f e (Eq. 11) depends on the most abundant element in fe, that is, 56 Fe. Since the abundance of fe is weakly affected in the pulse, σv ncap, f e will not be strongly impacted by the change in the distribution of the heavy nuclei.
A40, page 12 of 13 Concerning σv evo, f e , the numerator of Eq. (7) is proportional to the abundances of nuclei with A = 85 and the denominator is mainly determined by the abundance of 56 Fe. The abundance of A = 85 nuclei in the pulse cannot be known beforehand, but it will always be greater than its initial value, since the products of the last s-process are mixed in the convective shell. It is thus expected that the tabulated σv evo, f e will be underevaluated and the fe particle will not be consumed as efficiently in the fictitious network as it is in the large one.
Concerning the reaction rates of the heavier fictitious particles (for nuclei with A ≥ 86), given the similarity between the pulse (Y pulse ) and irradiated (Y 1 ) distributions, we expect that the effective reaction rates that were calculated in the radiative case will remain applicable. With this insight, we modified the calculation of the fictitious rate to improve our treatment of the convective s-process. In practice, for k > f e, the reaction rate for the fictitious particle k in the convection zone was calculated taking the average rate over the pulse overlap region 1 only, σv k = j∈1 σv k, j (N ncap, j , n n, j )Y k, j dm j j∈1 Y k, j dm j , (A.3) and for the rates involving fe, we kept the standard procedure. The results of the corrected fictitious network (Fictitious + Approx) are also displayed in Fig. A.1. The agreement for lse and vhs particles is considerably improved and remains the same for ls and hse. Unfortunately, hs is now overproduced, but the difference with the large network is on the same order as before.
To summarize, the dilution of irradiated material in the pulse modifies the distribution of chemical elements that is expected from a pure radiative s-process. This alteration does not impact the rate of σv ncap, f e because this rate mainly depends on the fe abundance, which is weakly affected. For the heavier fictitious particles, the agreement is acceptable and improved if the reaction rates are averaged over the region that is rich in heavy nuclei between m bot and m 3DU .