CHARGED-PARTICLE AND NEUTRON-CAPTURE PROCESSES IN THE HIGH-ENTROPY WIND OF CORE-COLLAPSE SUPERNOVAE

If the pressure per unit volume by relativistic particles (photons, electrons, positrons, etc.) is in excess of the total pressure due to non-relativistic particles, the entropy is dominated by radiation. This occurs at high temperatures and moderate to small matter densities. As high temperatures also ensure a nuclear statistical equilibrium composition, it is possible to choose an arbitrary, but sufficiently high, initial temperature, e.g., T₉ ≃ 9 (T₉ = T/10⁹ K). If the thermal energy k_BT is larger than the energy of the rest mass of electrons, we also have to consider the contributions by electrons and positrons. The radiation dominated entropy is given by S_γ = (4/3)aT³/ρ, where S_γ is the entropy per unit mass and a is the radiation constant. Making use of relations for ultrarelativistic fermions, the electron and positron contributions are given by $S_{e^{+},e^{-}}=({7}/{4})S_{\gamma }=({7}/{3}) a{T^{3}}/{\rho }$. The total entropy per unit mass is therefore

$$\begin{equation} S=S_{\gamma }+S_{e^{+},e^{-}}= \frac{11}{3} a\frac{T^{3}}{\rho }, \end{equation} \tag{ 1 }$$

which is a factor (11/4) larger than S_γ. After the temperature has decreased to values where the corresponding energies become comparable or smaller to the rest mass of electrons, the contributions of electrons and positrons can be neglected. The electrons become non-relativistic and the positrons cease to exist. To consider both extreme situations (pure radiation or radiation plus ultra-relativistic electrons and positrons), Witti et al. (1994) introduced a very close approximation for the entropy

$$\begin{equation} S=S_{\gamma } \left[1+\frac{7}{4}f(T_{9})\right] \end{equation} \tag{ 2 }$$

with

$$\begin{equation} f(T_{9})=\frac{T^{2}_{9}}{T^{2}_{9}+5.3}. \end{equation} \tag{ 3 }$$

The fit function f(T₉) varies between 0 and 1. If we write the entropy S in units of k_B per baryon and the density ρ in units of 10⁵ g cm⁻³ ($\rho _{_{5}}$), the entropy S can be expressed as

$$\begin{equation} S=1.21\frac{T^{3}_{9}}{\rho _{_{5}}}\left[1+\frac{7}{4}f(T_{9})\right]. \end{equation} \tag{ 4 }$$

Since the expansion of the hot bubble proceeds adiabatically (TV^1/3 = const), the time evolution of the temperature is given by

$$\begin{equation} T_{9}(t)=T_{9}(t=0)\left(\frac{R_{0}}{{R_{0}}+V_{\rm exp}\, t} \right), \end{equation} \tag{ 5 }$$

where R₀ is the initial radius of an expanding sphere and V_exp is the expansion velocity of that sphere. From Equation (4) one can deduce the matter density ρ

$$\begin{equation} \rho _{_{5}}(t)=1.21\frac{T^{3}_{9}}{S}\left[1+\frac{7}{4}f(T_{9})\right]. \end{equation} \tag{ 6 }$$

Making use of a typical initial value R₀ = 130 km and a sufficiently high initial temperature T₉ = 9, which ensures nuclear statistical equilibrium to consist only of neutrons and protons, we have performed calculations for a large grid of expansion velocities (1875, 3750, 7500, 15,000, and 30,000 km s⁻¹, for a given S) and a large grid of entropies (5 ⩽ S ⩽ 415). It should be noted that for the lowest entropies of this grid (S < 15) the above treatment is probably not valid, as the radiation dominance is not fulfilled. However, for the purpose of an r-process survey this parameterization seems still useful.

Figure 1 shows the time evolution of T₉(t) and $\rho _{_{5}}(t)$, both normalized to 1, for a selected expansion velocity of V_exp = 7500 km s⁻¹. These normalized quantities are identical for different entropies as long as V_exp is kept constant.

Zoom In Zoom Out Reset image size

The electron abundance Y_e = ∑_iZ_iY_i (with proton number Z_i and abundance Y_i = X_i/A_i (mass fraction over mass number)) is equal to the averaged electron or proton to nucleon ratio <Z/A > and thus is a measure of the proton to neutron ratio in the HEW. This is of key importance for the r-process after the freeze-out of the charged-particle reactions. A neutron-rich wind means that Y_e < 0.5. For an initial T₉ = 9 only free protons and neutrons exist, leading to 1 = X_n + X_p = Y_n + Y_p = Y_n + Y_e. Therefore, a given Y_e defines the initial proton and neutron abundances Y_p = Y_e and Y_n = 1 − Y_e. The initial value of Y_e depends ultimately on all weak interactions (including neutrino interactions) with the available free nucleons in the HEW. While in the early phase of core-collapse supernovae a Y_e larger than 0.5 is expected (see, e.g., Fröhlich et al. 2006a, 2006b; Pruet et al. 2006), the late phases are expected to experience Y_e < 0.5 and enable the onset of an r-process. In order to test such conditions we have chosen a large grid of Y_e (from 0.40 up to 0.499) for our calculations.

2.2.1. The Charged-particle Network

2.2.2. The r-process Network

When during the expansion of the hot bubble the temperature drops below T₉ ≈6, the α-particles start to react with each other to form heavier nuclei via ¹²C, either following the triple-α reaction

$$\begin{equation} 3\alpha \rightarrow {^{12}{\mbox{C}}}, \end{equation} \tag{ 10 }$$

or, if neutrons are available, the reaction

$$\begin{equation} \alpha +\alpha +n\rightarrow {^{9}{\mbox{Be}}}, \end{equation} \tag{ 11 }$$

followed by ⁹Be(α, n)¹²C.

Both Equations (10) and (11) depend on the square of the matter density, and Equation (11) depends in addition on Y_e. They represent bottle-necks for the subsequent nucleosynthesis toward heavier nuclei and are the reason for the well-known difference between a normal and an α-rich freeze-out of charged-particle reactions. Further reactions of possible importance like ³He + ⁴He → ⁷Be + γ, ⁷Be + ⁴He → ¹¹C + γ, ⁴He + ²H → ⁶Li + γ, ⁶Li + ⁴He → ⁹Be + ¹H, and ⁹Be + ⁴He → ¹²C + γ are already included. In addition, test calculations with Fynbo's triple-alpha reaction rates (Fynbo et al. 2005) have been performed for two entropies S = 100 and 250 k_B/baryon, although in several recent publications the data analysis with respect to the "missing" 2⁺ rotational-band member of the ¹²C Hoyle state and its astrophysical consequences have been questioned (see, e.g., Freer et al. 2009). We also have tested the NACRE triple-α rate for the above two entropies, and no significant differences in the seed contributions were observed at the charged-particle freeze-out. For high matter densities, i.e., low entropies (S ∝ T³/ρ), the three-body reactions of Equations (10) and (11) will remain effective, leading to a total recombination of α-particles and free neutrons. The final abundances at charged-particle freeze-out close to T₉ ≈3 are dominated by iron group elements. For a moderate neutron excess or 0.4 < Y_e < 0.5, the free neutrons will be consumed completely and no subsequent rapid neutron capture (r-process) will be possible. This is referred to as a normal freeze-out of charged-particle reactions (see, e.g., Woosley et al. 1994; Freiburghaus et al. 1999, and references therein).

In the case of lower matter densities, i.e., higher entropies, the reactions of Equations (10) and (11) cease to be effective for further recombination of α-particles toward heavier nuclei. The degree to which such a production of heavier nuclei is prohibited is a gradual function of S and Y_e, ranging from α-particle mass fractions of a few percent to close to 100 X_max = 200Y_e% (e.g., for Y_e = 0.45 the maximum amount of the α-particles is 90%) and accordingly small amounts of heavy "seed" nuclei (see Figure 3).

Zoom In Zoom Out Reset image size

In an α-rich freeze-out the full nuclear statistical equilibrium (NSE) among all nuclei breaks down into quasi-equilibrium (QSE) subgroups (see, e.g., Hix & Thielemann 1999). For a strong α-rich freeze-out and moderately neutron-rich Y_e, the dominant abundances of heavy (seed) nuclei during the charged-particle freeze-out can be shifted to mass numbers between 80 and 110, thus overcoming the shell closure at N = 50. The ratio of neutrons to heavy seed nuclei at this point is a function of Y_e (measuring the neutron excess) and entropy (determining the ratio of heavy nuclei to α-particles and indirectly also the amount of neutrons consumed in these heavy nuclei). This can be seen from Figure 3 and will be further elaborated in the following subsections. The neutron to seed ratio is a measure of the amount of neutron captures on seed nuclei and the resulting average mass number after an r-process, and indicates therefore the strength of the r-process. The fact that the seed abundances are dominated by nuclei with mass numbers between 80 and 110, is one reason that an r-process in the HEW can be faster than in classical calculations, which typically starts below N = 50.

We define the expansion time τ_exp between our arbitrarily chosen initial temperature T₉ = 9 and its decrease to T₉ = 3 as the timescale for the charged-particle reactions and as the time when the freeze-out of such reactions occurs within the expanding and cooling wind. According to Equation (5) one gets

$$\begin{equation} \tau _{\rm exp}=\frac{2 R_{0}}{V_{\rm exp}}, \end{equation} \tag{ 12 }$$

where R₀ is the initial radius of the hot bubble and V_exp is its expansion velocity during the successful supernova explosion. For instance, an expansion velocity of 7500 km s⁻¹ corresponds to an expansion timescale of 35 ms.

After an α-rich freeze-out, matter which surpassed the bottle-neck above A = 4 is predominantly accumulated in nuclei of the Fe-group or beyond. At sufficiently high entropies these nuclei are shifted to mass numbers in the range 80 ⩽ A ⩽ 110. If free neutrons are still available, neutron captures on those seed nuclei can proceed after the freeze-out of the charged-particle reactions, thus leading to a subsequent r-process. For a given Y_e, (V_exp, S)-combinations can be chosen which result in a fixed neutron to seed ratio (Y_n/Y_seed) and a unique distribution of seed nuclei at freeze-out (independent of the expansion timescale for such choices). This is shown in Figure 4 for a selected Y_e-value with a constant neutron to seed ratio and for four different expansion velocities. This means that for an r-process with a given strength for a specific Y_e, i.e., with a given initial neutron to seed ratio, both the initial temperature (T₉ ≈ 3) and the seed distribution are always the same. However, in order to get the same r-process ejecta, i.e., in some sense a universal r-process, similar expansion velocities (timescales) are required. The reason for this is that the r-process path for a given strength also depends on the temperature profile which is related to the expansion timescale. Similar to other authors (Arcones et al. 2007; Wanajo 2007), we use the terms "hot" or "cold" r-process depending on the average temperature experienced when the neutron captures occur (which enables photodisintegrations to be effective or not). For instance, a small expansion velocity (V_exp < 3000 km s⁻¹) leads to a "hot" r-process which is completely governed by an (n, γ) − (γ, n) equilibrium. On the other hand, higher expansion velocities lead to a "hybrid" r-process, which is first equilibrium dominated at high temperatures (T₉ ⩾ 1), then followed by a non-equilibrium phase at lower temperatures (T₉ < 1), i.e., the "cold" r-process where neutron captures and β-decays compete. In Figure 5, we show the dependence of the seed composition for the same wind characteristics (i.e., the same strength and timescale), but for different electron abundances Y_e. In contrast to the quite robust seed distribution of Figure 4, here we observe for a constant V_exp a significant variation with Y_e, with the major differences at the highest seed masses above A ≈ 90. Since the mass fraction of α-particles increases with decreasing Y_e, the amount of the ejected heavy material (X_heavy = 1 − X_α) is different for the same r-process strength, provided that the expelled matter contains roughly the same amount of mass per entropy interval.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

There is a natural upper limit of the r-process-relevant entropies, i.e., the last entropy which still yields a significant amount of seed nuclei. We label those maximum entropies as S_final. Beyond S_final the recombination of the α-particles (see Equations (10) and (11)) fails completely and the yields at such entropies only consist of α-particles, free neutrons which decay to protons, and traces of ²H,³H, ³He, ¹²C and ¹⁶O. This is similar to the big bang nucleosynthesis where the dominant protons and the less abundant neutrons recombine to roughly 75% protons, 24%⁴He, and traces of ⁶Li and ⁷Be. Table 3 shows an example of the yields of such a boundary entropy beyond S_final. In the limiting case of disappearing protons, the abundance and mass fraction of α-particles and the neutrons are given by Equations (7)–(9), and are shown for some values of Y_e in Table 2.

Table 4 finally shows the results of simulations for two expansion velocities and a selection of Y_e-values. The second and the sixth columns contain the initial entropies (S_initial) with a strength equal to unity, so that from the respective seed composition a subsequent r-process can occur. The third and seventh columns contain the highest entropy (S_final), for which seed nuclei can be formed. Beyond S_final, no Fe-group seed nuclei are produced, and hence under such conditions an r-process is no longer possible. Obviously, Y_e-values larger than 0.49 will exhibit a depletion of the third peak elements and the actinides Th and U since the maximum Y_n/Y_seed ratios are smaller than 100.

Table 4. The α-rich Freeze-out Conditions for an r-process at Two Expansion Velocities V_exp = 7500 and 15000 km s⁻¹ which Correspond to Expansion Timescales of 35 and 15 ms, Respectively

Ye	7500 km s−1				15,000 km s−1
	Sinitial	Sfinal	ΔS	$\left(\frac{Y_{n}}{Y_{\rm seed}}\right)_{\rm final}$	Sinitial	Sfinal	ΔS	$\left(\frac{Y_{n}}{Y_{\rm seed}}\right)_{\rm final}$
0.499	206	253	47	17	156	193	37	17
0.498	185	258	73	37	143	194	51	32
0.496	162	263	101	60	128	198	70	50
0.494	153	267	114	72	121	202	81	62
0.492	147	272	125	84	117	205	88	68
0.490	142	277	135	96	113	209	96	78
0.486	136	283	147	109	109	216	107	92
0.482	131	295	164	132	105	223	118	107
0.478	126	303	177	148	101	229	128	120
0.474	122	311	189	167	98	236	138	139
0.470	118	319	201	188	95	242	147	155
0.466	114	327	213	212	92	248	156	174
0.462	111	334	223	233	89	254	165	195
0.458	107	341	234	258	87	260	173	219
0.454	104	348	244	286	84	265	181	239
0.450	100	354	254	311	81	270	189	261
0.400	53	416	363	755	43	320	277	655

Notes. For a given Y_e, S_initial is the entropy which initiates an r-process (Y_n/Y_seed ≈ 1), S_final is the last entropy which is able to build heavy seed nuclei (beyond it no r-process can take place), ΔS = S_final − S_initial, and $\big(\frac{Y_{n}}{Y_{\rm seed}}\big)_{\rm final}$ represents the maximum Y_n/Y_seed ratio which can be attained when S = S_final. For further details, see discussion in the text.

Download table as:  ASCIITypeset image

From Table 4, one can also see that for a given expansion speed, S_final changes and increases as Y_e decreases. This is because the bottle-neck reaction of Equation (11) additionally depends on Y_e, and—provided neutrons are available—it can proceed at densities smaller than those needed by the triple-α reaction since an uncharged particle is involved.

Here we present a first survey of detailed network calculations for a variety of entropies, followed until the total consumption of neutrons. According to the entries in Table 4, entropies of 175, 195, 236, 270, and 280 k_B/baryon result in neutron to seed ratios of about 23, 34, 66, 107, and 121. The first two components lead to abundance patterns with maxima between A = 100–130. The third component populates the rare earth elements (hereafter, REE) in the mass range around A = 162. The fourth component forms the abundances of the A = 195 peak, and the fifth component proceeds to even heavier isotopes, also populating the actinides. These results are displayed in Figures 6–10, making use of five different mass models/formulae. Fission is here implemented in a simplified way, assuming complete and symmetric fission for nuclei with A > 260. With a superposition of different entropy yields under the assumption that the ejected neutrino wind elements per equidistant entropy interval have equal volumes, final isotopic r-process abundances are obtained, which are displayed in Figures 24–28. Deficiencies to the N_r,☉ will be discussed in more detail later in Section 5.2.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

While the r-process proceeds, the neutron abundance Y_n decreases and the amount of the newly formed fresh r-process material increases as function of time. Since neutron captures destroy the seed nuclei, we replace the seed abundance Y_seed by the r-process material abundance Y_r. With this, we define the neutron freeze-out as the instant when

$$\begin{equation} \frac{Y_{n}(t)}{Y_{r}(t)}<1, \end{equation} \tag{ 13 }$$

and we label the corresponding time as t_freeze. After t_freeze, the remaining neutrons and those produced by β-delayed neutron emission will not change the final abundances dramatically. Figures 6–10 show for four mass models and one mass formula the freeze-out times t_freeze for five selected entropies, i.e., S = 175, 195, 236, 270, and 280, and the final abundance curves after β-decay back to stability. Table 5 shows a comparison between two specifically selected mass models, i.e., FRDM and ETFSI-Q, for the same astrophysical conditions (V_exp, Y_e, and S). Obviously, the nuclear-physics input seems to have a significant impact on the r-process abundances. For instance, with the FRDM mass model a longer process duration (by about a factor 1.5) is needed to produce the second (A ≃ 130) and third (A ≃ 195) r-process peaks than for ETFSI-Q. Similar differences in the timescales are also observed for ETFSI-1, DUFLO–ZUKER, and HFB-17. These differences are mainly due to the strong N = 82 shell closure of the FRDM mass model which in addition leads to an underproduction of the REE mass region.

The weaker N = 126 shell closure of FRDM causes an underproduction of Pb and an earlier onset of fission cycling than the other mass models. Hereafter, when speaking about freeze-out without any further specifications, the neutron freeze-out (Y_n/Y_r < 1) is meant.

In case of a chemical equilibrium between neutron captures and reverse photodisintegrations, or a so-called (n, γ) − (γ, n) equilibrium between neighboring nuclei (Z, A) and (Z, A + 1) within an isotopic chain, the abundance ratio of these two nuclei is given by the nuclear Saha-equation

$$\begin{eqnarray} \frac{Y(Z,A+1)}{Y(Z,A)} &=& \frac{n_{n}}{2}\frac{G(Z,A+1)}{G(Z,A)} \left(\frac{A+1}{A}\right)^{3/2}\left(\frac{2\pi \hbar ^{2}}{m_{u}k_{B}T} \right)^{3/2} \nonumber\\ && \times\mbox{exp}\left(\frac{-S_{n}(Z,A+1)}{k_{B}T}\right). \end{eqnarray} \tag{ 14 }$$

The neutron separation energy S_n, the nuclear partition function G, and the mass number A represent the nuclear-physics input, whereas the temperature T and the neutron number density n_n represent the conditions of the astrophysical environment. In the following, we want to analyze our r-process calculations, testing if in high-entropy expansions an (n, γ) − (γ, n) equilibrium is achieved at all, how long it prevails, and at which time it breaks down. In addition, we will monitor the effect of abundance changes after the freeze-out from such an equilibrium via final captures of remaining free neutrons and those from β-delayed neutron emission. If we express the product k_BT in units of T₉ Equation (14) can be rewritten as

$$\begin{eqnarray} \frac{Y(Z,A+1)}{Y(Z,A)}&=& 8.4\times 10^{-35}\, n_{n}\, \frac{G(Z,A+1)}{G(Z,A)} \, \left(\frac{A+1}{A}\right)^{3/2} \nonumber\\ &&\times T_{9}^{-3/2}\mbox{exp}\left(-\frac{11.604\, S_{n}(Z,A+1)}{T_{9}}\right). \nonumber\\ \end{eqnarray} \tag{ 15 }$$

From Equation (15), we then obtain

$$\begin{eqnarray} && \hspace{-10pt}S_{n}^{(n,\gamma -\gamma,n)}(Z,A+1) = -\frac{T_{9}}{11.604} \nonumber \\ && \times\ln \bigg[1.2\times 10^{34}\, \frac{Y(Z,A+1)}{Y(Z,A)} \,\frac{G(Z,A)}{G(Z,A+1)}\nonumber\\ && \times\bigg(\frac{A+1}{A}\bigg)^{-3/2}\,\frac{T_{9}^{3/2}}{n_{n}}\bigg]. \end{eqnarray} \tag{ 16 }$$

S_n(Z, A + 1) can also be calculated via

$$\begin{equation} S_{n}^{\rm real}(Z,A+1)=m_{\rm ex}^{Z,A+1}-(m_{\rm ex}^{Z,A}+m_{\rm ex}^{n}), \end{equation} \tag{ 17 }$$

where m^{Z,A + 1}_ex, m^Z,A_ex, and mⁿ_ex are the mass excesses of the nuclei (Z, A + 1), (Z, A) and the neutron, respectively. We define the freeze-out with regard to the (n, γ) − (γ, n) equilibrium as the instant, when the ratio of S^{(n,γ−γ,n)}_n and S^real_n of the nuclei (Z, A + 1) next to those lying in the r-process path (Z, A) which represent the abundance maxima in each isotopic chain computed via the relations given in Equations (16) and (17) diverge from unity, and we label the corresponding time as t_chem.

Figures 11–13 show the validity and duration of the (n, γ) − (γ, n) equilibrium as a function of time for the entropies which synthesize the second, the REE pygmy and the third r-process abundance peaks. Obviously, the second peak is formed under a full chemical equilibrium. The ratios of the real neutron separation energies and those predicted by the Saha-equation are about unity for the whole Z range (blue color). The break-out from the chemical equilibrium happens 60 ms later, i.e., clearly after the neutron freeze-out. Therefore, for the A ≃ 130 r-process peak one has t_freeze < t_chem . The broad REE pygmy peak with its abundance maximum around A ≃ 162 is formed at the limit of a chemical equilibrium, where

$$\begin{equation*} t_{\rm freeze}\approx t_{\rm chem}\,. \end{equation*}$$

However, the A ≃ 195 peak is no longer built under chemical equilibrium conditions. This means

$$\begin{equation*} t_{\rm freeze}>t_{\rm chem}\,. \end{equation*}$$

Therefore, the morphology of the rapid neutron-capture process in the neutron-rich HEW can be depicted as a combination of three types.

• 1.  
  A "hot" r-process, which proceeds and ends already at relatively high temperatures (T₉ ≈ 0.8), maintaining a full chemical equilibrium until the A ≃ 130 peak is formed.
• 2.  
  A "hybrid" r-process, which proceeds at high temperatures and ends at low temperatures (T₉ ≈ 0.5), maintaining a partial chemical equilibrium until the REE pygmy peak is formed.
• 3.  
  A "cold" r-process, which proceeds at high temperatures and ends at even lower temperatures (T₉ ≈ 0.35) with no chemical equilibrium until the A ≃ 195 peak is formed (see also Arcones et al. 2007; Wanajo 2007).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Thus, for the latter two r-process types the neutron-capture rates will play a significant role for the mass region beyond A ≃ 140.

The dynamical freeze-out occurs when the neutron depletion becomes inefficient, and thus the timescale τ_n is larger than the hydrodynamical timescale τ_hyd, i.e.,

$$\begin{equation} \left|\frac{Y_n}{\dot{Y}_n}\right|> \left|\frac{\rho (t)}{\dot{\rho }(t)}\right|. \end{equation} \tag{ 18 }$$

We label the corresponding time with t_dyn. Then, Δt = t_dyn − t_freeze is the time interval when the last free neutrons are captured and β-delayed neutrons are recaptured during the decay back to stability. Depending on the capability of the radioactive progenitor nuclei to emit β-delayed neutrons after t_freeze, the freeze-out times t_dyn and t_freeze may be identical or can differ from each other. Beyond t_dyn the neutron number density n_n drops to values ${\le }10^{18} \,\mbox{\mbox{\it n} cm}^{-3}$, thus representing the signature for the end of the r-process.

During the r-process the abundance flow from each isotopic chain to the next is governed by β-decays. We can define a total abundance in each isotopic chain Y(Z) = ∑_AY(Z, A), and each Y(Z, A) can be expressed as Y(Z, A) = Y(Z)P(Z, A), where P(Z, A) represents the population coefficient of the nucleus (Z, A). In case of a steady flow of β-decays between isotopic chains, in addition to an (n, γ) − (γ, n) equilibrium, we also have

$$\begin{eqnarray} \sum _{A}Y(Z,A)\lambda _{\beta }(Z,A) &=& Y(Z)\sum _{A}P(Z,A)\lambda _{\beta }(Z,A) \nonumber\\ &=& Y(Z)\lambda ^{\rm eff}_{\beta }(Z)={\rm const}. \end{eqnarray} \tag{ 19 }$$

In other words, each Z-chain can be treated as an effective nucleus with a β-decay rate λ^eff. Under such conditions the assumption of an abundance Y(Z_min) at a minimum Z-value would be sufficient to predict the whole set of abundances as a function of A, provided that an (n, γ) − (γ, n) equilibrium and a steady flow has been reached. Already in the past, various groups have discussed the possibility of either a global r-process steady-flow equilibrium or local (n, γ) − (γ, n) and β-flow equilibria in between neutron shell closures (see, e.g., Burbidge et al. 1957; Seeger et al. 1965; Hillebrandt et al. 1976; Cameron et al. 1983a, 1983b), assuming different astrophysical sites and different theoretical nuclear-physics input. First experimental evidence for a combination of an (n, γ) − (γ, n) equilibrium and a local steady flow at N = 50 and N = 82 came from the work of Kratz et al. (1986), Kratz (1988), and Kratz et al. (1993) in their site-independent approach. This result that the β-flow is not global was shortly afterward confirmed by Meyer (1993) for the specific site of a high-entropy hot bubble. In the present paper, we extend the above steady-flow studies and show under which entropy and temperature conditions which type of equilibrium is reached and when it breaks down.

In Figures 14–16, we show the quantity Y(Z)λ^eff_β(Z) (color-coded) as a function of time for entropies of S = 195, 236, and 270 which produce the second peak, the REE pygmy peak and the third r-process peak regions, respectively. Those entropies were selected because they indicate best how the β-flow proceeds through both shell closures (N = 82 and 126) and the REE region between them. As a function of time, the following conclusions can be drawn.

• 1.  
  Before approaching the freeze-out, a β-flow equilibrium is established for isotopic chains with Z < 43, i.e., where the abundance distribution does not populate the N = 82 shell closure.
• 2.  
  After reaching the freeze-out, another late β-flow proceeds through the most populated Z-chains: (1) Z = 46–49 for the A ≃ 130 mass region, (2) Z = 50–63 for the REE pygmy peak region, and (3) Z = 64–70 for the A ≃ 195 mass region.

Thus, we confirm the earlier conclusions (see, e.g., Kratz et al. 1993; Meyer 1993; Thielemann et al. 1994; Pfeiffer et al. 2001b), that in the r-process a global β-flow equilibrium does not occur, but local equilibria are established in between neutron shell closures where short β-decay half-lives are encountered. These equilibria are essential for a successful reproduction of the N_r,☉ distribution (see Section 5).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

¹³⁰Te, ¹⁶²Dy, and ¹⁹⁵Pt represent the maxima of the A ≃ 130, the REE and the A ≃ 195 r-process abundance peaks. While it is generally accepted that the two sharp main peaks (both with FWHM widths of about 15 m.u.) are due to the retarded r-matter flow at the N = 82 and N = 126 shell closures, the origin of the broad structured REE pygmy peak distribution (with a total width of about 50 m.u.) as either being a signature of nuclear deformation or as representing the result of a late abundance enhancement by fission cycling, still seems to be under debate. In the following, on the basis of Table 6 and Figures 17–28, we will discuss in some detail the formation of the three r-abundance peaks. Table 6 shows the isobaric progenitors of the three peak maxima at A = 130, 162, and 195, respectively, for the three freeze-out types at different times, temperatures, and neutron densities discussed above. Figures 17–19 indicate the effects of the late recapture of β-delayed neutrons on the final r-abundances. Figures 20–23 show the calculated abundance heights and widths as a function of entropy of the above three peak isotopes for the two selected mass models ETFSI-Q and FRDM. And finally, Figures 24–28 present overall fits to the N_r,☉ distribution for the five different mass models chosen in the present study.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In the historical WP approximation of the r-process at the commonly assumed freeze-out around T₉ ≃ 1, the main isobaric progenitor of stable ¹³⁰Te is the N = 82 isotope ¹³⁰Cd, for which in the meantime the most important nuclear-physics properties (T_1/2, P_n, and Q_β) are experimentally known (Kratz et al. 1986; Hannawald et al. 2000; Dillmann et al. 2003). In a fully dynamical r-process model like our present HEW study, however, the situation is more complicated. In the following, we will discuss in detail the formation of ¹³⁰Cd during the freeze-out phases (see Table 6, and Figures 11, 14, and 17). For this purpose, we consider here the (representative) conditions of an entropy of S = 195, for which the highest partial ¹³⁰Te abundance is obtained.

Under these conditions, the initial r-matter flux at a neutron density of n_n ≃ 10²⁷ occurs further away from the β-stability line than in the later freeze-out phases. As in the classical WP approach, the r-process in the A ≃ 130 region still starts to "climb up the N = 82 staircase" (Burbidge et al. 1957), however only up to Z = 45 ¹²⁷Rh. Already at Z = 46 ¹²⁸Pd, the r-process begins to break out from the magic shell, with comparable timescales for the two competing reactions of further neutron capture and β-decay. In the Z = 46 isotopic chain, the r-process does produce some N = 82 ¹²⁸Pd, but proceeds up to N = 84 ¹³⁰Pd (with the highest isotopic abundance) and N = 86 ¹³²Pd. Similarly, also for Z = 47 the r-process partly breaks out from N = 82, still producing ¹²⁹Ag, but again continuing up to N = 84 ¹³¹Ag and N = 86 ¹³³Ag with comparable initial abundances. In this early stage, for the classical Z = 48 ¹³⁰Cd waiting point, neutron capture out of N = 82 still dominates over β-decay (τ_nγ/τ_β ≃ 0.1) thus forming only a small fraction of its total abundance. Here, the highest initial isotopic yield is obtained for N = 86 ¹³⁴Cd. Finally, the classical N = 82, Z = 49 "break-out" isotope ¹³¹In is passed quickly (τ_nγ/τ_β ≃ 0.02), and N = 86 ¹³⁵In is formed with the highest initial isotopic yield.

When focusing in the following on A = 130, we recognize that several isotopes lying beyond N = 82 with high initial abundances act as β-decay and β-delayed 1n to 3n progenitors of neutron magic ¹³⁰Cd. As mentioned above, the most important isobaric r-process nuclide now is N = 84 ¹³⁰Pd, which contains about 60% of the total abundances of the early progenitors of ¹³⁰Cd. However, due to its high β-delayed neutron branching ratios (P_1n ≃ 73%; P_2n ≃ 18%), and the total P_an ≃ 24% of the daughter ¹³⁰Ag, only about 6% of the initial ¹³⁰Pd abundance lead to ¹³⁰Cd. The next important initial β-xn progenitors of ¹³⁰Cd are ¹³²Pd and ¹³³Ag, both with about 20% progenitor abundance. With these different decay branches, during the early neutron and chemical freeze-out phases (see Table 6), only a part of the total ¹³⁰Cd abundance is produced. Another sizeable fraction comes from neutron captures of the remaining "free" neutrons. Here again, the most important contribution comes from the initial ¹³⁰Pd abundance via its strong β-delayed 1n decay branch to ¹²⁹Ag. When considering the different neutron-capture cross sections for N = 82 ¹²⁹Ag and its β-decay daughter N = 81 ¹²⁹Cd (about a factor 55 in favor of ¹²⁹Cd) under the relevant time, temperature, and neutron-density conditions at the end of the chemical freeze-out (see Table 6), it is clear that neutron capture predominantly occurs after ¹²⁹Ag β-decay to ¹²⁹Cd into N = 82 ¹³⁰Cd. The respective production modes of ¹³⁰Cd are reflected by the abundance ratios of Y(¹³⁰Cd)/Y(¹²⁹Ag) at the beginning and at the end of the chemical equilibrium. In this freeze-out phase, within about 240 ms, this ratio changes by about a factor of 20 from 0.14 to 2.7. These early abundances are further modulated during β-decay back to stability by the capture of the remaining "free" neutrons and the β-delayed neutron recapture. The resulting abundance shifts in the A ≃ 130 peak region from the late recapture of previously emitted β-delayed neutrons are shown in Figure 17. We see that significant effects occur in the rising wing of the peak. At the top, the abundance of ¹²⁹Xe is reduced by a factor of 2.5, whereas the ¹³⁰Te abundance increases by about a factor 2.2.

Early debates (see, e.g., Burbidge et al. 1957; Cameron 1957) were related to the possible origin of the REE pygmy peak, either due to mass-asymmetric fission cycling from the trans-actinide region or from the nuclear shapes of the REE progenitor isotopes. In their site-independent approach Kratz et al. (1993) had already recognized obvious deficiencies of the FRDM mass model in the shape-transition regions before and behind the N = 82 and N = 126 shell closures, which resulted in an inadequate REE pattern. More detailed network calculations were then performed by Surman et al. (1997), again using the FRDM mass model and a simplified set of "adapted" β-decay half-lives. They concluded that the REE peak is due to a subtle interplay of nuclear deformation and β-decay, not occurring in the steady-flow phase of the r-process. At about the same time, the conclusion of the above authors were in principle confirmed by r-abundance calculations of Kratz et al. (1998), who compared the REE distributions resulting from (1) the spherical mass model HFB/SkP of Dobaczewski et al. (1984), and (2) the deformed mass model ETFSI-Q of Pearson et al. (1996).

In the present paper, we use five selected (deformed) mass models with different microscopic sophistication to reproduce the overall N_r,☉ distribution. From Figures 24–28 one can clearly see the different success of our dynamical network calculations in reproducing the overall shape of the REE pygmy peak. However, with the exception of FRDM, for the other four cases we can consistently confirm and strengthen the earlier conclusions of Surman et al. (1997) and Kratz et al. (1998), that (at least for the HEW scenario) fission cycling seems to be unimportant. And, when considering the FRDM masses, it is highly questionable if its nuclear deficiencies around the magic neutron shells would be "repaired" more or less artificially by assuming strong fission cycling at extremely high entropies (e.g., S > 450 k_B/baryon).

From Table 6 one can also see that the REE pygmy abundance peak is formed with the "correct" shape and relative height in a continuous r-matter flow during freeze-out. It is interesting to note that in the early phase (begin of the chemical equilibrium at about 220 ms) the build-up of the whole REE region is still dominated by neutron captures, forming Z = 54 ¹⁶²Xe and Z = 56 ¹⁶²Ba with the highest A = 162 isotopic abundances. Within the following about 100 ms, when approaching the end of the chemical equilibrium, the matter flow has continued to the maximum A = 162 abundance for Z = 58 ¹⁶²Ce. Now, up to this mass region β-decay clearly dominates over further neutron capture. Only in the late phase of the dynamical freeze-out (after about 480 ms), the continued matter flow has formed Z = 59 ¹⁶²Pr and Z = 60 ¹⁶²Nd with their maximum yields. Under these low-temperature conditions (T₉ ≃ 0.3), the density of "free" neutrons has already dropped to n_n ≃ 10¹⁷ so that by now the whole REE region between Z = 54 and Z = 62 is dominated by β-decays. Finally, as is indicated in Figure 18, during even later times of the decay back to stability, an additional gradual abundance shift to A = 162 occurs, which originates from the recapture of previously emitted β-delayed neutrons. Thus, the final shape of the REE pygmy peak is established only at very late non-equilibrium.

The formation of the A ≃ 195 N_r,☉ peak, which is related to the N = 126 shell closure, is similar to that of the A ≃ 130 (N = 82) peak but not exactly the same. There are several differences with respect to both astrophysical as well as nuclear-physics parameters. Starting with a similar initial neutron density of n_n ≃ 10²⁷, the higher entropy of S = 270 (compared to S = 195 for the A ≃ 130 peak) leads to a higher neutron to seed abundance ratio of Y_n/Y_seed ≃ 105 (compared to Y_n/Y_seed = 36 for S = 195). This results in an initial r-process path further away from β-stability than in the A ≃ 130 region, involving extremely neutron-rich progenitor isotopes with on the average higher β-delayed neutron branching ratios. In consequence, the description of the formation of the A ≃ 195 abundance peak and its modulation during the decay back to β-stability is more complicated. Apart from the "bottle-neck" behavior of the N = 126 shell closure, we have to face a detailed interplay between captures of "free" neutrons, β-decays, emission of β-delayed neutrons and their recapture. For some snapshots of the freeze-out during the first 750 ms, see Table 6 and Figures 13, 16, and 19. And at the end, we will argue why the top of the A ≃ 195 N_r,☉ peak occurs with ¹⁹⁵Pt at an odd mass number, and not—as one would expect—at an even mass number.

In principal similarity to the formation of the early A ≃ 130 N_r,☉ peak, the initial r-matter flux enters the A ≃ 195 peak region at the "lighter" N = 126 waiting points, producing small fractions of Z = 62 ¹⁸⁸Sm to Z = 66 ¹⁹²Dy. Already from Z = 67 on upward, the early r-process breaks through the closed shell. In this phase, in the whole mass region neutron capture strongly dominates over β-decay.

At the end of the chemical equilibrium freeze-out phase after about 420 ms (see Table 6), the abundances of N = 126 ¹⁹¹Tb to ¹⁹⁶Yb have increased between 1 up to 7 orders of magnitude, all approaching their maximum values during the first 750 ms of the freeze-out. For all "lighter" N = 126 isotopes up to Z = 68 ¹⁹⁴Er, now β-decay dominates over neutron capture. Under these conditions, ¹⁹⁴Er has the highest N = 126 isotopic abundance. For Z = 69 ¹⁹⁵Tm and Z = 70 ¹⁹⁶Yb neutron capture and β-decay timescales become comparable; hence, the break-out from N = 126 now has shifted to higher Z values. At the time when the dynamical freeze-out occurs (after about 750 ms; see Table 6), the abundances of the lighter N = 126 isotopes up to Z = 67 (Ho) have already decreased by more than an order of magnitude due to their β-decays, whereas under these late conditions finally also the abundances of the "heaviest" N = 126 isotopes from Z = 71 (Lu) to Z = 74 (W) have reached their maximum values. Interestingly, the abundance of Z = 68 ¹⁹⁴Er has only decreased by about a factor of 4, and the abundance of ¹⁹⁵Tm has remained constant. In this late dynamical freeze-out phase, for all N = 126 nuclei β-decay dominates over neutron capture.

As mentioned above, at the end of this section we want to come back to the question why the top of the A ≃ 195 N_r,☉ peak occurs at an odd mass number and not at the—as expected—even mass A = 194 of the most abundant classical N = 126 waiting-point isotope ¹⁹⁴Er. At the end of the chemical equilibrium and in the subsequent dynamical freeze-out phase, a subtle interplay of several β-decay, β1n- to β3n-decay, and neutron-capture channels occurs, which leads to kind of reaction cycles. Recapture of β-delayed neutrons does not yet play a significant role. Under these conditions, the maximum abundance in the peak is still ¹⁹⁴Er closely followed by ¹⁹⁵Tm. When neglecting β-delayed neutron recapture, obviously this signature is held during the whole decay back to stability, resulting in a final abundance ratio of Y(¹⁹⁴Pt)/Y(¹⁹⁵Pt) ≃ 1.3. However, when allowing the recapture of previously emitted β-delayed neutrons, which occurs relatively late during the β-back decay, a significant upward abundance shift with Z occurs in the whole rising wing up to the top of the peak. Beyond A = 195, for a few masses the reverse effect is observed. This is shown in Figure 19. In our calculations, the final abundance ratio now changes to Y(¹⁹⁴Pt)/Y(¹⁹⁵Pt) ≃ 0.5. Hence, we again conclude that the recapture of β-delayed neutrons has to be included in detailed r-process calculations.

In our attempt to not only understand the behavior of individual entropy components, but rather to find an explanation of the solar system isotopic r-abundance residuals N_r,☉ ≃ N_☉ − N_s,☉, we need to consider a recipe for the weights in such entropy superpositions, similar to the superposition of neutron number densities in classical r-process calculations (Kratz et al. 1993, 2007; Cowan et al. 1999; Pfeiffer et al. 2001a). Hereafter, we will discuss two simple approaches, assuming the ejection of neutrino-wind elements per equidistant entropy interval with equal masses (dM(S)/dS = const) or with equal volume (dV(S)/dS = const), where the entropy spans from S = 5 to S_final. This is a pure assumption not resulting from dynamic explosion calculations; but we will show that this simple weighting leads to excellent fits to the solar r-process abundances between the rising wing of the A ≃ 130 peak and the Pb-Bi peak.

5.1.1. Ejecta per Equidistant Entropy with Equal Volumes

If the ejected neutrino-wind elements are equal-sized (hereafter, assumption 1), one obtains

$$\begin{equation*} \frac{M(S)}{\rho (S)}={\rm const}. \end{equation*}$$

Since S ∼ 1/ρ, one obtains for an entropy S

$$\begin{equation} M(S)\, S=M(S_{\rm ref})\, S_{\rm ref}={\rm const}, \end{equation} \tag{ 20 }$$

where S_ref represents an arbitrary reference entropy which we set equal to 5 k_B/baryon, the smallest entropy in our network calculations. From Equation (20), one infers that the entropy weight is

$$\begin{equation} w(S)=\frac{M(S)}{M(S_{\rm ref})}=\frac{S_{\rm ref}}{S}. \end{equation} \tag{ 21 }$$

Hence, by integrating over different entropies, the accumulated abundance Y_sum(Z, A) of a given nucleus (Z, A) is

$$\begin{equation} Y_{\rm sum}(Z,A)=\sum _{S=5}^{S_{\rm final}} \frac{S_{\rm ref}}{S} Y_{S}(Z,A). \end{equation} \tag{ 22 }$$

Therefore, the ejected mass fraction of heavy material per entropy is given by

$$\begin{equation*} X_{\rm heavy}(S)=w(S)(1-X_{\alpha }). \end{equation*}$$

Figures 20 and 21 show the calculated abundances of ¹³⁰Te, ¹⁶²Dy, and ¹⁹⁵Pt as a function of entropy. The agreement of the peak heights of our simulations with the corresponding solar-system r-abundances is excellent for the mass models ETFSI-Q and FRDM. However, the latter mass model underproduces the REE mass region because of its very strong N = 82 shell closure.

5.1.2. Ejecta per Equidistant Entropy with Equal Masses

Here we show (as a different example) superpositions with equal amounts of ejected mass per entropy interval (hereafter, assumption 2). When the ejected masses per entropy are equal, the entropy weights will be the same, i.e., w(S) = 1. The accumulated abundance of a given nucleus then is

$$\begin{equation} Y_{\rm sum}(Z,A)=\sum _{S=5}^{S_{\rm final}} Y_{S}(Z,A). \end{equation} \tag{ 23 }$$

Figures 22 and 23 show the results for ETFSI-Q and FRDM. Both mass models show similar (but slightly worse) fits to the solar r-abundances and thus indicate that within the expected uncertainties both assumptions 1 and 2 yield good agreement. This might indicate what one would expect from realistic scenarios. For the attempt to reproduce the solar r-abundances, in the following we use the assumption 1.

In the past, many attempts were undertaken to reproduce the solar r-process abundances. In the frame of the WP approximation, based on an (n, γ) − (γ, n) equilibrium with static temperatures and neutron number densities, it has been shown that only a superposition of several of neutron number density components with weights given by the realistic heights of the three N_r,☉-peaks (Y(⁸⁰Se) ≃24.3, Y(¹³⁰Te) ≃1.69, Y(¹⁹⁵Pt) ≃0.457; Käppeler et al. 1989) can achieve this goal (Kratz et al. 1993). Freiburghaus et al. (1999) used an entropy-based superposition with an analytical fit function $g(S)=x_1e^{-x_2S}$, where x₁ and x₂ represent two fit parameters. We follow the method described in this work by using our above assumption 1. Figures 24–28 show the results for a selected constant expansion velocity of V_exp = 7500 km s⁻¹ and a specific electron abundance Y_e = 0.45. It is surprising that this simple parameterization of the HEW components provides a good fit to the overall solar r-abundances above A = 110, especially with the mass models ETFSI-1 and ETFSI-Q. The mass formula of DUFLO–ZUKER slightly underproduces the A ≃ 195 and the Pb mass regions, and the new mass model HFB-17 shows deficiencies in the A ≃ 145 and A ≃ 175 mass regions where nuclear phase transitions occur. In these regions, the mass model prescriptions may still have systematic deficiencies. HFB-17 also underproduces the A ≃ 195 and the Pb mass regions. The mass model FRDM has deficiencies in producing the REE, the A ≃ 200 and the Pb mass regions.

The overproduction of the mass region below A < 110 by the HEW is well-known and was discussed by several authors in the past (see, e.g., Woosley et al. 1994; Freiburghaus et al. 1999). It has been referred to as a model deficiency due to the α-rich freeze-out component, where the r-process cannot start because the neutron to seed ratio is still smaller than unity. Figure 29 shows the fit to the solar elemental r-abundances by using the mass model ETFSI-Q. It is quite evident from this figure that the abundance overproduction mentioned above concerns the elements between Sr and Ag. However, in our new understanding of the HEW nucleosynthesis one can avoid this disagreement by considering an additional superposition in terms of Y_e.

Zoom In Zoom Out Reset image size

Since S increases and Y_e decreases with time, it is reasonable to consider the ejecta of a core-collapse supernova explosion as a mixture of different S and Y_e-components. Figure 30 shows the mass fraction of the expected heavy ejecta (A > 4) for a selection of Y_e-values as a function of S. Thereby, one can see that the weight of the ejecta increases as Y_e decreases. Figure 31 shows that the whole mass region from Sr (Z = 38) up to U (Z = 92) can be fitted by using four different Y_e-values (0.498, 0.496, 0.490, and 0.482). Thereby, we have normalized the solar r-abundances so that $Y(\mbox{Nb})_{r,\odot }=Y(\mbox{Nb})_{Y_{e}=0.498}$. We have chosen the element Nb because it can serve as normalization for the elemental and isotopic solar r-abundances since it has only one stable isotope. From Figure 31 one can infer:

• 1.  
  Sr, Y, Zr, and Nb (Z = 38–41) are formed under condition of Y_e = 0.498;
• 2.  
  Mo and Ru (Z = 42 and 44) are formed under conditions of Y_e = 0.496;
• 3.  
  Rh, Pd, and Ag (Z = 45–47) are formed under further decreasing Y_e-conditions of Y_e = 0.490; and
• 4.  
  Cd and the elements beyond (Z ⩾ 48) are formed under more neutron-rich conditions of Y_e = 0.482.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Thus, the well-known overproduction of the mass region from A = 88 to 110 in the HEW in Figures 24–28, is due to the fact that only one Y_e is used. A superposition of several Y_e-values (as, e.g., performed by Meyer et al. (1992) for a selected entropy), however in addition to the superposition of S-values used in our study, could help to resolve this problem.

Based on our assumptions concerning the entropy weights, which provide a good fit to the solar r-abundances beyond the A = 110 mass region, we will estimate the amount of the (neutron-capture) r-process material which is ejected,

$$\begin{equation} {M}_{r}\approx \frac{4}{3}\, \pi \, R_{0}^{3}(t=0)\, \int _{S_{\rm initial}}^{S_{\rm final}} dS (1-4Y_{\alpha })\rho (S,t=0). \end{equation} \tag{ 24 }$$

The range from S_initial to S_final indicates the entropy interval which permits the production of r-process nuclei. Furthermore, our calculations show that the entropy-dependent r-process mass fraction X_r = 1 − 4Y_α can be well approximated by a function $f(S)=x_{1}S^{x_{2}}$. Equation (24) then reads

$$\begin{equation} {M}_{r}\approx \frac{4}{3}\, \pi \, R_{0}^{3}(t=0)x_{1}\int dS\; S^{x_{2}}\rho (S,t=0). \end{equation} \tag{ 25 }$$

Replacing the term of the matter density in Equation (25) by its term in Equation (6), one obtains

$$\begin{equation} {M}_{r}\approx 1.1\times 10^{-3} \;M_{\odot }\, x_{1}\int dS\; \frac{S^{x_{2}}}{S}, \end{equation} \tag{ 26 }$$

in units of solar masses M_☉. The parameters x₁ and x₂ vary with V_exp and Y_e. If we complete the integration in Equation (26), one finally gets

$$\begin{equation} {M}_{r}\approx 1.1\times 10^{-3} \;M_{\odot }\, \frac{x_{1}}{x_{2}}\left(S_{\rm final}^{x_{2}}-S_{\rm initial}^{x_{2}}\right). \end{equation} \tag{ 27 }$$

Table 7 shows the results according to Equation (27) for two expansion velocities. The amount of the estimated mass of the ejecta then depends only on the electron abundance Y_e. For instance, at a given expansion speed a Y_e = 0.45 will produce roughly 10 times more r-process material than 0.49. This is due to the higher entropy needed to start an r-process with the latter Y_e-value.