Probing the Production of Actinides under Different r-process Conditions

We perform our nucleosynthesis calculations using four different libraries of nuclear reactions to run our nuclear network code Winnet (Winteler et al. 2012). The first set is the JINA Reaclib default version (from 2017 October 20), with added rates for neutron-induced fission from Panov et al. (2010), β-delayed fission from Panov et al. (2005), and spontaneous fission as described in Petermann et al. (2012) henceforth referred to as FRDM. Note that this JINA Reaclib set includes theoretical rates from the FRDM set of Rauscher & Thielemann (2000) on the neutron-rich side and the theoretical neutron capture and (γ,n) rates from Panov et al. (2010) for (n,γ) target nuclei with Z > 83. In addition to this reaction library, we also perform calculations based on the Extended Thomas-Fermi with Strutinsky Integral mass model including shell quenching corrections, ETFSI-Q (Aboussir et al. 1995). In this set, the neutron capture rates and their reverse reactions are taken from the ETFSI-Q rate set given in Rauscher & Thielemann (2000), supplemented by neutron capture rates on Z > 83 nuclei from Panov et al. (2010), spontaneous fission rates from Panov et al. (2013), and the rates for the other fission modes from the same sources as above, but based on the corresponding fission barriers (Mamdouh et al. 2001). A third reaction library consists of rates based on the Duflo–Zuker mass model (Duflo & Zuker 1995) with 10 parameters (labeled as DZ10 in the following), newly calculated for this work using the SMARAGD Hauser-Feshbach code as described in Cyburt et al. (2010); Rauscher (2011). A fourth set is a variation of the first library, but with theoretical β-decay half-lives replaced with the predictions of Marketin et al. (2016). This library will be referred to as D3C*(FRDM) throughout this article. Table 1 provides an overview of the reaction rates in the different libraries that are relevant to this study.

Nuclear reactions release energy that can increase the temperature. We include this effect in our nucleosynthesis calculations following the description of Freiburghaus et al. (1999).

In order to test actinide production in the r-process in different conditions, we employ several models of suggested r-process sites. We are using simulations of dynamical ejecta in a binary compact merger (two neutron stars with 1.0 M_⊙ each; in the following called R1010) and a neutron star—black hole merger (1.4 M_⊙ and 5.0 M_⊙; henceforth referred to as R1450) from Korobkin et al. (2012) and Rosswog et al. (2013). We also include a neutron star merger model from Bovard et al. (2017), with both neutron star masses of 1.25 M_⊙ and the SFHO equation of state (Steiner et al. 2013). As discussed in Bovard et al. (2017), the dynamical ejecta in their models cover a wider range of Y_e and entropies than the Rosswog et al. (2013) models, although the bulk of the ejecta also contains rather low electron fractions and entropies. Moreover, we test other possible sites of the r-process: accretion disks in mergers have been shown to host conditions favorable for a strong r-process (Surman et al. 2008; Fernández & Metzger 2013; Perego et al. 2014; Just et al. 2015; Wu et al. 2016; Lippuner et al. 2017; Siegel & Metzger 2018). Therefore, we include two different disk scenarios (S-def and S-s6) which were first described in Fernández & Metzger (2013). The data we are using are from the improved simulations as described in Wu et al. (2016), Lippuner et al. (2017). Furthermore, magnetohydrodynamically driven (MHD) supernovae with fast expanding jets possibly represent an additional r-process site, where heavy elements could be produced under different conditions than in compact binary mergers. Here we are using the model of Winteler et al. (2012). Other MHD SN simulations have been performed by, e.g., Nishimura et al. (2015, 2017) and Mösta et al. (2018). As a summary, all hydrodynamical models are listed in Table 2, together with the nomenclature that will be used throughout the text.

The hydrodynamic trajectories provide data only up to the end of the simulation, which is typically of the order of 0.01 s for explosive models (and a few seconds for the disk models). We therefore extrapolate using a parameterized expansion according to Korobkin et al. (2012) and Eichler et al. (2015). If not indicated otherwise, the results shown represent abundances 1 Gyr after the r-process event.

The main goal of our study is to investigate and explain trends of ²³²Th (actinide) production for different conditions and theoretical nuclear physics models. For this purpose, we summarize here how ²³²Th is built up by decaying r-process nuclei on the example of a trajectory from R1010 and one from Wmhd. In Figure 1 the abundances of ²³²Th, ²³⁶U, and ²⁴⁴Pu are shown as a function of time. Like for any other (quasi-)stable isotope produced in the r-process, there is a direct β-decay feeding channel from more neutron-rich nuclei with mass numbers A ≳ 232 (including the possibility of β-delayed neutron emission), responsible for the initial strong buildup around 100 s (see Figure 1). An additional production channel only sets in much later, when ²³⁶U begins to α-decay with a half-life of 23.4 Myr. Before it decays, ²³⁶U itself is fed by the α-decay of ²⁴⁰Pu, which in turn is added to by the decay chain ${}^{248}\mathrm{Cm}{(\alpha \gamma )}^{244}\mathrm{Pu}{(\alpha \gamma )}^{240}{\rm{U}}{({\beta }^{-}\bar{\nu })}^{240}\mathrm{Np}{({\beta }^{-}\bar{\nu })}^{240}\mathrm{Pu}$. Note that all of these isotopes are also directly produced by the β-decay channel. The ²³²Th abundance reaches its maximum value only after ²⁴⁴Pu has decayed, with a half-life of 80.0 Myr. Figure 1 furthermore reveals that a difference in ²³²Th abundances can have different origins. While the Wmhd case produces less actinides in general (resulting in a lower ²³²Th production after 100 s compared to the R1010 case), the largest difference is in the heavier isotopes ^240,244Pu. Thus, the initial difference further increases when the heavier actinides α-decay around the 10¹⁵ s mark.

Zoom In Zoom Out Reset image size

The plurality of production channels effectively means that nuclei with a wide mass number range (232 ≲ A ≲ 250) are responsible for the eventual production of ²³²Th. This mass range coincides with relatively long-lived nuclei on the r-process path along the N = 162 isotone. In particular, ²⁴²Hg and ²⁴¹Au are strongly produced in our calculations, acting as important precursor nuclei for the final ²³²Th abundance. The dependence of our results on the β-decay half-lives of these two nuclei is discussed in Section 4.3.

Here we want to study the impact of astrophysical and nuclear factors on the actinide production in r-process calculations. The models described in Section 2 all produce the heaviest r-process nuclei up to the actinides. However, the conditions in the ejecta differ considerably (i.e., electron fraction, entropy, and ejecta velocity), which leaves an imprint on the compositions. Roederer et al. (2009) tested their sample of stellar abundance ratios against a site-independent nucleosynthesis model consisting of several components with different neutron densities and employing the ETFSI-Q mass model. In Figures 2 and 3 we revisit their approach with real hydrodynamical r-process models for our four sets of nuclear physics input. The four panels belonging to each reaction rate library show four different chronometer pairs: Th/Eu (top left), Th/Hf (bottom left), Pb/Th (top right), and U/Th (bottom right). The dots represent stellar abundance data from Roederer et al. (2009), complemented by the newer sample of Mashonkina et al. (2014), as well as the recent survey of stars in the Sagittarius dwarf galaxy by Hansen et al. (2018) and the recently discovered actinide-boost star J09544277+5246414 (Holmbeck et al. 2018). In addition, we also include the Reticulum II star with known Th yield, DES J033523-540407 (Ji & Frebel 2018). In accordance with Roederer et al. (2009), stars with ${\mathrm{log}}_{\epsilon }\left(\mathrm{La}/\mathrm{Eu}\right)\lt 0.25$ are marked separately, as they are assumed to have no significant contribution from the s-process to the heavy element yields. Actinide-boost stars are marked in green. The calculated abundance ratios are shown in Figures 2 and 3 as horizontal lines for all hydrodynamical models (see Table 2) for a time t = 10 Gyr after the event.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Figures 2 and 3 reveal the dependence of the chronometer pairs on the nuclear physics, as well as the hydrodynamical conditions of the r-process. Scenarios with a very low Y_e generally contain the highest Th/Eu and Th/Hf abundance ratios (i.e., R1010 and R1450), while the more moderate electron fractions prevalent in the merger disks and MHD SN result in lower ratios. An exception from this general trend can be observed in the Th/Hf ratio with the ETFSI-Q mass model, where the Bs125 model produces the lowest abundance ratio. Across the nuclear mass models, FRDM exhibits the highest Th yields, visible here from the highest Th/Eu and Th/Hf and the lowest Pb/Th ratios of all sets considered. The ETFSI-Q model, on the other hand, has the lowest Th/Eu ratio for the merger scenarios. The DZ10 model shows the largest sensitivity on the hydrodynamical scenarios, with theoretical Th/Eu and Th/Hf ratios spanning one and a half orders of magnitude (compared to less than one order of magnitude for both FRDM and ETFSI-Q). The Pb/Th and U/Th abundance ratios are quite insensitive to the hydrodynamical models, suggesting that these chronometer pairs are more reliable than Th/Eu for the determination of stellar ages. With regards to the nuclear physics, Pb/Th and U/Th are also less sensitive than Th/Eu and Th/Hf. A comparison of the D3C* set with FRDM illustrates the impact of the β-decay rates. As already mentioned, the D3C* half-lives are systematically shorter for heavy nuclei, which results in lower actinide abundances (see also Eichler et al. 2015; Wu et al. 2016; Holmbeck et al. 2019b) and thus significantly lower Th/Eu and Th/Hf ratios. However, Pb/Th and U/Th increase for all hydrodynamical models compared to the FRDM calculations. Since the thorium-to-lanthanide abundance ratios are highest for R1010, R1450, and Bs125 (i.e., models of NSM dynamical ejecta), one possibility is that actinide-boost stars have inherited an r-process composition reflecting a larger fraction of dynamical ejecta and a smaller fraction of disk ejecta, compared to stars with a thorium-to-lanthanide abundance ratio closer to the solar value.

Typically, r-process conditions are described by three defining quantities: electron fraction Y_e, entropy S, and the expansion velocity, which describes how fast the ejecta expand and cool. In order to study the dependence of actinide production on the hydrodynamic properties of the r-process environment, we want to cover a wide range of different conditions that are actually present in models of possible r-process sites. To that end, we pick representative trajectories from our hydrodynamical models, i.e., one trajectory with conditions that are characteristic for each model. Their properties (electron fraction Y_e, entropy S, and expansion velocity v) are summarized in Table 3, along with the calculated Th/Eu abundance ratios for each trajectory.

Table 3.  Representative Trajectories for Each Model Employed

Name	log (Th/Eu)	Ye (T = 8 GK)	S	v	References
			[kB/baryon]	(cm s−1)
R1010-rep	0.619	0.044	0.012	3.06 × 109	Rosswog et al. (2013)
R1450-rep	0.766	0.016	1.798	1.65 × 109	Rosswog et al. (2013)
Bs125-rep	0.337	0.225	37.367	7.87 × 109	Bovard et al. (2017)
Wmhd-rep	−0.085	0.175	6.001	2.02 × 109	Winteler et al. (2012)
FMdef-rep	0.857	0.168	12.593	1.70 × 108	Fernández & Metzger (2013)
FMs6-rep	0.052	0.175	12.696	6.02 × 108	Fernández & Metzger (2013)

Note. The second column lists the Th/Eu abundance ratio calculated with the original trajectory and evaluated at t = 1 Gy after the event. The electron fraction Y_e, entropy S, and expansion velocity v are read from the trajectory where the temperature drops below 8 GK for the last time.

Download table as:  ASCIITypeset image

Using these representative trajectories, we now vary the initial Y_e and repeat the nucleosynthesis calculations, tracking the actinide production as well as second peak and rare-earth peak elements, similar to the procedure described in Holmbeck et al. (2019b). The Th, U, Pb, Hf, and Eu abundances in dependence of initial Y_e are shown in Figure 4. For R1010-rep the Th, U, and Pb yields have a distinct non-linear dependence on Y_e, with the global maximum around Y_e ≈ 0.16 and a minimum at Y_e ≈ 0.12 (comparable to the results of Holmbeck et al. 2019b), while Eu exhibits the exact opposite trend. The NS-BH trajectory R1450-rep shows the same trend (since it has almost the same initial conditions, see Table 3). For Fmdef-rep, FMs6-rep, and Wmhd-rep the trend is shifted toward lower Y_e values, with the strongest Th abundance peaking below Y_e = 0.15 and a minimum around Y_e = 0.1. Only Bs125-rep shows a different trend, with an increasing Pb abundance toward lower Y_e, at the expense of all other four elements shown here. The peculiar composition at low Y_e in this trajectory is the result of an r-process with very few seed nuclei and a fast expansion, leading to an extreme shift of the r-process peaks due to late neutron captures while the composition is decaying toward stability. This means that under these conditions the third-peak composition is effectively dominated by lead nuclei.

Zoom In Zoom Out Reset image size

In order to verify that the observed trend with Y_e is not an artifact introduced by our method of artificially varying the initial Y_e, we show the calculated Th/Eu ratios after 1 Gyr for all trajectories in our six hydrodynamical models in dependence of initial Y_e and entropy S in Figure 5. Again, a clear maximum at low entropies and around Y_e = 0.15 can be identified, confirming the results in Figure 4. The results also show that going toward higher Y_e, it is possible to maintain a large Th/Eu ratio as long as the tracer particle is ejected with a higher initial entropy.

Zoom In Zoom Out Reset image size

In the following, we discuss the main features of Figure 4: the (global) maximum in actinide production between 0.12 < Y_e < 0.17, followed by a minimum at lower Y_e, and the position of the maximum, which can be found at different Y_e values for the different trajectories.

In our calculations, the thorium abundance does not linearly depend on the initial neutron-richness (see Figure 4). Instead, most of the trajectories we have investigated produce the highest actinide yield around Y_e = 0.16, followed by a minimum (around Y_e = 0.12), and roughly constant abundances for even lower initial Y_e. Here we discuss the origin of these particular trends and why the exact locations of the maximum and minimum Th abundance slightly vary for the individual trajectories. Holmbeck et al. (2019b) have already demonstrated the dependence of the actinide and Eu abundances on the fission cycles. This effect is also apparent in our calculations and can be traced, for instance, by following the abundances of ²⁴¹Au and ²⁴²Hg (i.e., the most abundant nuclei with relatively long half-lives at the N = 162 neutron number and precursor nuclei for ²³²Th) and comparing their abundance evolution to the average proton number $\langle Z\rangle$ of the composition as an indicator of fission cycles. The top and middle panel of Figure 6 clearly show such a relation for the three runs in R1010 where the Th abundance reaches extreme values. The abundances of ²⁴¹Au and ²⁴²Hg start to increase only after 0.1–0.2 s, when the reaction flow reaches the actinide region for the first time. Due to their relatively long half-lives, material starts to pile up in these two isotopes for the next 0.1–0.2 s. During this time the actinide abundances increase, while the abundances of seed nuclei with lower mass numbers gradually decline, leading to a slowdown in the production of new actinide nuclei. If the r-process freeze-out occurs around that time when the actinide abundance is highest (i.e., our case Y_e = 0.16 in Figure 6), the resulting ²³²Th abundance will be very high. If, however, the r-process continues beyond that point, neutron captures carry away material from the A = 240 region and into a part of the nuclear chart where fission occurs, thus destroying actinides and producing fission fragments around the second peak and/or rare-earth peak. This phase coincides with $\langle Z\rangle$ and the ²⁴¹Au and ²⁴²Hg abundances decreasing in Figure 6. If the r-process freezes out just after this first fission cycle has been completed (i.e., our case Y_e = 0.12), the composition decaying to stability will have lower actinide abundances and higher lanthanide abundances than the previously discussed case. In conditions with even higher neutron densities, the fission fragments of the first fission cycle continue capturing neutrons, which eventually leads to an increase in actinide abundances yet again (see our case Y_e = 0.09 in Figure 6).

Zoom In Zoom Out Reset image size

However, we observe an additional effect that comes into play in the case of R1010-rep, where the variations in the individual abundances are strongest (see Figure 4). The bottom panel in Figure 6 shows the neutron separation energies at which nuclei are most abundant in (n,γ)–(γ,n) equilibrium during the r-process. The shaded area highlights the range 1.5 MeV < S_n < 1.7 MeV, where the reaction flow can easily bypass nuclei with mass numbers A = 232–238 in the FRDM, as shown in the following. In hot r-process conditions, nuclei that are located on the reaction path can be identified based purely on their two-neutron separation energy, as well as the neutron density n_n, and the temperature T (see, e.g., Thielemann et al. 2017). The S_2n/2 values predicted by the FRDM mass model in the mass region 220 < A < 260 are shown in Figure 7, together with typical r-process paths favoring nuclei around S_2n/2 = 1.9, 1.7, 1.5, 1.3 MeV (red dots). Every line represents an isotopic chain, with every fifth line drawn in black. Note the saddle point structure for Z = 75–80 at mass numbers A = 230–240. If the freeze-out conditions favor nuclei close to S_2n/2 = 1.7 MeV, a gap opens at these mass numbers and the flow moves from (Z, A) = (77, 231) directly to (Z, A) = (78, 239), bridging all isotopes in between. The resulting low abundances of nuclei with mass numbers A = 232–240 directly impede the production of ²³²Th and its long-lived precursor isotopes, ²³⁶U and ²⁴⁰Pu (see Section 3). If, however, the conditions at freeze-out favor nuclei with higher (i.e., closer to "stability"; top left panel) or lower neutron separation energy (more neutron-rich; bottom right panel), the gap is closed and nuclei with A = 230–240 are produced in larger amounts.

Zoom In Zoom Out Reset image size

The lower panel of Figure 6 shows that the Y_e = 0.09 case freezes out in conditions with S_n < 1.5 MeV, thus enabling the efficient production of precursor nuclei for ²³²Th in the mass range A = 232–238 and enhancing the effect of the fission cycles.

The elemental abundances displayed in Figure 4 show similar trends for most trajectories examined here. However, for a given electron fraction, the obtained abundances for the three elements (and their abundance ratios) are different for each case. Furthermore, the Bs125-rep case differs notably from the other cases. These observations reveal that the initial Y_e does not solely determine the final Th and Eu abundances, and that other factors need to be taken into account. The Bs125-rep trajectory has the highest initial entropy (see Table 3). We therefore test the impact of the initial entropy on the elemental abundances. In order to do this, we have picked from the simulation data of Bovard et al. (2017) eight additional trajectories with starting entropies at S = 5, 10, 20, 30, 40, 50, 60, 70 k_B/baryon, and calculated their Th, U, Pb, Eu, and Hf abundances for different initial Y_e values. Figure 8 shows that for entropies below S = 20 k_b/baryon, the abundance trends roughly follow the trends of the low-entropy cases of Figure 4, with a Th abundance maximum around Y_e = 0.15 coinciding with a local minimum in the Eu abundance. S = 20 k_b/baryon also marks the point where the abundances for all three elements drop at the low-Y_e end, since the high entropy counters the buildup of seed nuclei, resulting in lower r-process abundances for all mass ranges. For trajectories with S > 30 k_b/baryon, the Th/Eu abundance ratio is equal to unity or less for all initial Y_e values. This means that while the Th and Eu abundance curves are following opposing trends for S < 20 k_b/baryon, they switch to a positive correlation for higher starting entropies. Furthermore, at any given initial electron fraction, S < 20 k_b/baryon conditions lead to higher Th/Eu abundance ratios than conditions with higher entropies. Figure 8 also reveals that ejecta with purely high-entropy conditions (S > 30 k_b/baryon) are unlikely to produce large variations in the actinide-to-lanthanide abundance ratios.

Zoom In Zoom Out Reset image size

The dependences of the actinide and europium abundances on the mass model, β-decays, and the fission fragment distribution have been discussed in Goriely & Clerbaux (1999), and, more recently on the example of hydrodynamical trajectories from NSM dynamical or disk ejecta, in Wu et al. (2016), Vassh et al. (2019), and Holmbeck et al. (2019b). Holmbeck et al. (2019b) found that in nucleosynthesis calculations using β-decay rates from Marketin et al. (2016) (corresponding to our model D3C*(FRDM)), the variations in Th and Eu abundances with initial Y_e are much smaller, and the resulting Eu abundance is higher, while the Th yield is smaller. They relate these differences to two distinct regions in the nuclear chart where the Marketin et al. (2016) rates are faster than the corresponding FRDM rates: (a) around A = 130, carrying material away from the second peak and filling the rare-earth peak more efficiently, and (b) above A = 190, resulting in a similar effect, where nuclei are guided to the fissioning region more quickly, and therefore less material is stored in α-decaying actinides after the freeze-out. In addition to these effects, we found that the β-decay half-life predictions of the nuclei on the N = 162 isotone also play an important role. As mentioned before, ²⁴²Hg and ²⁴¹Au are especially abundant during the r-process, and a strong connection can be observed between their abundances at the r-process freeze-out and the final abundance of ²³²Th (see Figure 6). The half-lives of these two nuclei are not (yet) known experimentally, and the JINA reaclib β-decay rate (Möller et al. 2003; Mumpower et al. 2016) predicts a half-life of 0.03 s for ²⁴²Hg, while Marketin et al. (2016) predict 0.001 s. For ²⁴¹Au the predictions are 0.006 s (Möller et al. 2003; Mumpower et al. 2016) and 0.001 s (Marketin et al. 2016), respectively. We test the impact of these half-live predictions on our results by exchanging the decay rate of ²⁴²Hg (along with its β-delayed neutron emission probabilities) in our nuclear reaction rate libraries FRDM and D3C*(FRDM) with the rate from the other set. Indeed, for R1010-rep, R1450-rep, FMdef-rep, and FMs6-rep this leads to a change in the final ²³²Th abundance by factors between 1.49 and 2.25 (more Th whenever a Marketin et al. 2016 rate is replaced by an FRDM rate, and vice versa). In a second step, we exchange the rates of both ²⁴²Hg and ²⁴¹Au simultaneously. This increases the effect slightly, with the range of factors changing to 1.63–2.49. The original Wmhd-rep and Bs125-rep trajectories are only weakly affected by these changes, since the reaction paths in the relatively high-Y_e (or high entropy in the case of Bs125) environments run closer to stability at the time of freeze-out, bypassing ²⁴²Hg and ²⁴¹Au. To further quantify the impact of the chosen mass model, Figure 9 shows the relative changes in elemental abundances for several chosen elements beyond the second r-process peak between ETFSI-Q (left panel), DZ10 (middle panel), and D3C*(FRDM) (right panel) with respect to FRDM. The colors and symbols represent the integrated ejecta yields for the different hydrodynamical models, and the values are the relative abundance differences, (Y₁ − Y₂)/(Y₁ + Y₂), where the indices 1 and 2 refer to the different mass models. This enables an assessment of the uncertainties of the individual elements with respect to the nuclear mass model: an elemental yield is sensitive only to the mass model if the points lie far away from 0, but close to each other and sensitive to both the nuclear and hydrodynamical model if the spread of points is large. Although it is difficult to draw conclusions based on Figure 9 alone, the data suggest that the ejecta composition of Wmhd, FMdef, and FMs6 are generally more sensitive to the nuclear mass model (with respect to the elements shown here), while the more neutron-rich scenarios reveal a more robust behavior (i.e., they are often closer to the equality line). Furthermore, Th and U have the almost identical dependencies on both the nuclear and the hydrodynamical models, suggesting once more that U/Th is the most reliable chronometer pair at present.

Zoom In Zoom Out Reset image size