Monte Carlo Investigation of the Ratios of Short-lived Radioactive Isotopes in the Interstellar Medium

We study first the regime where δ_c ≫ τ_eq, τ₁, τ₂. In this case, represented by the example in the top left panel of Figure 1, the average abundance ratio is

$$\begin{eqnarray}\mu =\left\{\begin{array}{ll}\displaystyle \frac{{\tau }_{\mathrm{eq}}}{{\delta }_{c}}\,P & \mathrm{if}\ {\tau }_{\mathrm{eq}}\gt 0,\\ \displaystyle \frac{| {\tau }_{\mathrm{eq}}| }{{\delta }_{c}}\,P\,{e}^{{\delta }_{c}/| {\tau }_{\mathrm{eq}}| } & \mathrm{if}\ \ {\tau }_{\mathrm{eq}}\lt 0.\end{array}\right.\end{eqnarray} \tag{ 6 }$$

Given that the ratio τ_eq/δ_c is low, we expect an average value much lower than the production ratio P when τ_eq > 0. For a case where τ_eq < 0, we have an exponential term of δ_c /∣τ_eq∣, which will instead yield an average value much higher than P. In addition, the ratio will vary between the production ratio P and 0 (or P and $P\exp ({\delta }_{c}/| {\tau }_{\mathrm{eq}}| )$) for the case of positive (negative) τ_eq.

The intuitive understanding of this regime is that the time between enrichment events is longer than it takes for both radioactive isotopes and their ratio to decay, which prevents any memory buildup and results in a very large relative uncertainty.

In this regime, δ_c ≪ τ_eq, τ₁, τ₂. This case, represented in the top right panel of Figure 1, has an average equilibrium value of

$$\begin{eqnarray}&&\mu =P\,\displaystyle \frac{{\tau }_{1}}{{\tau }_{2}}.\end{eqnarray} \tag{ 7 }$$

The evolution of the ratio of radioactive isotopes is marked by relatively frequent events, and the time between them is shorter than the mean life of any of the isotopes. This means that the abundance of both isotopes retains the memory of the previous events, and the ratio drifts from the production ratio P to oscillate around the equilibrium average with a low relative uncertainty, behaving in a similar fashion to the case of large τ/δ_c studied in Lugaro et al. (2018).

In this regime, δ_c ≪ τ_eq and δ_c ≫ τ₁, τ₂. This case, represented in the bottom left panel of Figure 1, has an average equilibrium value of

$$\begin{eqnarray}&&\mu =P.\end{eqnarray} \tag{ 8 }$$

Although the value for the average in this case can be recovered from the formula of Regime 2 by using τ₁ ≈ τ₂, we set this case apart because it represents the specific situation when the equivalent mean life is much longer than δ_c , while the individual mean lives of each isotope are not. This regime only arises when the difference between the mean lives is small enough to make τ_eq orders of magnitude longer than them (see Equation (3)). Given the short mean life of the individual SLR, it is likely that each SLR carries information from the last event alone (see Paper I, Figure 9 and the related discussion). At the same time, the variation on the value of the ratio is relatively small because the equivalent mean life is too long for the ratio to change significantly before the next enriching event.

In this regime, δ_c ≫ τ_eq, τ₁, but δ_c ≪ τ₂. The average value in this case, shown in the bottom right panel of Figure 1, is

$$\begin{eqnarray}\mu =\left\{\begin{array}{ll}\displaystyle \frac{{\tau }_{\mathrm{eq}}}{{\tau }_{2}}\,P & \mathrm{if}\ \ {\tau }_{\mathrm{eq}}\gt 0,\\ \displaystyle \frac{| {\tau }_{\mathrm{eq}}| {\tau }_{1}}{{\delta }_{c}^{2}}{e}^{{\delta }_{c}/| {\tau }_{\mathrm{eq}}| }\,P & \mathrm{if}\ \ {\tau }_{\mathrm{eq}}\lt 0.\end{array}\right.\end{eqnarray} \tag{ 9 }$$

Although the evolution resembles that of the first regime when τ_eq > 0, the maximum value attained by the ratio of the radioactive isotopes in the equilibrium becomes much lower than P. The reason is that although the evolution of M₁ does not retain the memory of the previous events, the evolution of M₂ does. We note that in this regime, ${\tau }_{\mathrm{eq}}\approx \min ({\tau }_{1},{\tau }_{2})$.

In Paper I we analyzed a single SLR and found that three different regimes can be applied depending on the relation between τ and γ. Here we report a brief description of them and and how they connect with the regimes in this work. For the sake of clarity, the three regimes from Paper I are marked in Roman numerals, while Arabic numerals refer to the four regimes considered here.

Regime I refers to τ/γ > 2 and is similar to Regimes 2 or 3 in that statistics can be calculated because the spread is not much larger than the median value. Regime I is associated with the calculation of the isolation time, T_iso, because in this case, the ISM will contain an equilibrium value from where there can be an isolation period before the ESS abundances. In the present work, Regime 2 is that associated with the calculation of T_iso.

Regime III is a case that covers the region of τ/γ < 0.3. In this regime, there is a high probability that the ISM abundance that decayed into the ESS abundance originated from a single event. Therefore this regime is associated with the calculation of the time since the last event, T_LE. Regime 3 of this work is related to Regime III of Paper I in that both most likely carry abundances from only the last event before the formation of the solar system. The difference is that while Regime III allows us to calculate T_LE, Regime 3 also allows us to narrowly determine the production ratio of the last event.

Regime II falls between two well-defined cases described above. This regime has 0.3 < τ/γ < 2, which does not allow for meaningful statistics nor for a clean definition of a last event to which the ISM abundance can be solely or mostly attributed. This regime does not correspond to any of the regimes in this work, and it may be similar to the region between Regime 2 and Regimes 1 and 4.

We also investigated the possibility of calculating the uncertainties using an analytical approach instead of the full Monte Carlo simulations. The aim is to provide a better understanding of the regimes and their uncertainties, as well as give an alternative to calculating approximate numbers without the need of a simulation. To do this, we use the expression for the average given by

$$\begin{eqnarray}&&\mu \approx P\,\displaystyle \frac{{\tau }_{\mathrm{eq}}}{\langle \delta \rangle }\displaystyle \frac{1-\langle {e}^{-\delta /{\tau }_{2}}\rangle }{1-\langle {e}^{-\delta /{\tau }_{1}}\rangle }\left(1-\langle {e}^{-\delta /{\tau }_{\mathrm{eq}}}\rangle \right),\end{eqnarray} \tag{ 10 }$$

derived in the Appendix, and for the relative standard deviation, we use

$$\begin{eqnarray}&&\displaystyle \frac{\sigma }{\mu }\approx F\sqrt{\displaystyle \frac{\langle \delta \rangle }{2{\tau }_{\mathrm{eq}}}\displaystyle \frac{1-\langle {e}^{-2\delta /{\tau }_{\mathrm{eq}}}\rangle }{{\left(1-\langle {e}^{-\delta /{\tau }_{\mathrm{eq}}}\rangle \right)}^{2}}-1},\end{eqnarray} \tag{ 11 }$$

where F is a correction factor applied to Equation (A14) and is defined by

$$\begin{eqnarray}&&F=K\left[1+{\mathrm{log}}_{10}{\left(\displaystyle \frac{\min ({\tau }_{1},{\tau }_{2})}{\langle \delta \rangle }\right)}^{2}\right],\end{eqnarray} \tag{ 12 }$$

where K = 1 unless $\min ({\tau }_{1},{\tau }_{2})$ is larger than the span of the DTD, in which case, K = 0.5. In cases where $\min ({\tau }_{1},{\tau }_{2})\lt \langle \delta \rangle$, then F = K.

This factor F was derived from the Monte Carlo experiments and corrects some of the approximations made in the derivation of A14 in the Appendix. With this correction factor, Equation (11) becomes an accurate estimate of the results of the Monte Carlo experiment.

If the full distribution of δ is unknown, a further approximation to Equation (11) can be used instead, rendering

$$\begin{eqnarray}&&\displaystyle \frac{\sigma }{\mu }\approx F\sqrt{\tfrac{1}{6}\tfrac{\langle \delta {\rangle }^{3}+3{\sigma }_{\delta }^{2}\langle \delta \rangle }{{\tau }_{\mathrm{eq}}^{2}\langle \delta \rangle -{\tau }_{\mathrm{eq}}{\sigma }_{\delta }^{2}}},\end{eqnarray} \tag{ 13 }$$

with the advantage that only 〈δ〉 and σ_δ (the standard deviation of the delta distribution) have to be known. This formula is much easier to calculate because no sampling of the δ distribution is needed.

The validity of Equations (11) and (13) can be tested by comparing them to the 68.2% (1σ) confidence interval calculated from the Monte Carlo experiments. This comparison is presented in Figure 3. In the worst case, with the small-box DTD, the relative difference between the analytical approximations, and the results from the numerical experiments are just above 25%. These are valid for calculations related to Regime 2. For Regime 3, as seen in Table 1, the average instead remains very close to P and introduces asymmetry in the distribution. In this case, the theoretical σ is an average of the lower and upper 1σ threshold. When this σ is such that μ + σ > P, it is better to calculate a lower limit for the distribution with P − 2σ because in this case, the distribution piles up at P, making any value between P and μ functionally equiprobable.

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In this case, the abundances of both SLR nuclei retain significant memory from past events. The average of their ratio, according to Equation (10), is the same as the constant case for the same regime, given by Equation (7). When the uncertainties are compared, however, there is a significant difference between the constant and the stochastic case. As a first-order approximation, and taking σ_δ ≈ 〈δ〉 (see Table 2 of Paper I), we can write Equation (13) as

$$\begin{eqnarray}&&\displaystyle \frac{2\sigma }{\mu }\approx 2F\displaystyle \frac{\langle \delta \rangle }{{\tau }_{\mathrm{eq}}},\end{eqnarray} \tag{ 14 }$$

which, when 〈δ〉 is substituted by δ_c and Equation (12) is divided by Equation (5), reveals that the stochastic case has a larger uncertainty relative to the constant case by a factor of 2F. This factor can be shown to be in the range 2F ∈ [2.5, 35] when we consider τ_eq/〈δ〉 ∈ [3, 10⁴] by using Equation (12) with K = 1 and take $\min ({\tau }_{1},{\tau }_{2})={\tau }_{\mathrm{eq}}$. Therefore the time-stochastic nature of enrichment events can increase the uncertainty by more than an order of magnitude in this regime. The uncertainty on the ratio of two SLR in Regime 2 is still relatively low. For example, for the large box, with τ₁ = 10 Myr, τ₂ = 15 Myr, and γ = 1 Myr, Table 1 has a relative uncertainty of 12%. For a similar example with τ = 10 Myr and γ = 1 Myr in Table 3 of Paper I, the relative uncertainty is 45% for the large box. Even if we take the case of τ = 31.6 Myr and γ = 1 Myr, we still have a relative uncertainty of 25% for the SLR/stable isotopic ratio.

Table 2. Regimes and Values of the Ratios from the Monte Carlo (MC) Experiments Applied to the Specific Cases of Ratios of Two SLRs (Column SRL₁/SRL₂) and of the SLRs and Their Corresponding Stable or Long-lived Reference Isotopes (Columns SRL₁/stable and SRL₂/stable, See the Main Text for the List of Reference Isotopes)

	τ1	τ2	τeq	γ	SLR1/stable		SLR2/stable		SLR1/SLR2
					Regime	MC Values	Regime	MC Values	Regime	MC Values
247Cm/129I	22.5	22.6	5085	1	I	${22.37}_{-3.22}^{+3.45}$	I	${22.47}_{-3.23}^{+3.46}$	2	${1.00}_{-0.00}^{+0.00}$
				3.16	I	${7.01}_{-1.77}^{+1.98}$	I	${7.04}_{-1.77}^{+1.99}$	2	${1.00}_{-0.00}^{+0.00}$
				10	II	<3.29	II	<3.30	3	${1.00}_{-0.00}^{+0.00}$
				31.6	II	<1.29	II	<1.29	3	>0.99
				100	III	<0.54	III	<0.54	3	>0.96
				316	III	<0.08	III	<0.08	3	>0.89
107Pd/182Hf	9.4	12.8	35.4	1	I	${9.28}_{-2.05}^{+2.27}$	I	${12.68}_{-2.41}^{+2.63}$	2	${0.73}_{-0.04}^{+0.03}$
				3.16	I	${2.86}_{-1.10}^{+1.32}$	I	${3.94}_{-1.31}^{+1.53}$	2	${0.73}_{-0.08}^{+0.06}$
				10	II	<1.62	II	<2.06	3	${0.70}_{-0.19}^{+0.12}$
				31.6	III	<0.69	III	<0.86	−	${0.50}_{-0.32}^{+0.30}$
53Mn/97Tc	5.4	5.94	59.4	1	I	${5.29}_{-1.53}^{+1.75}$	I	${5.83}_{-1.61}^{+1.83}$	2	${0.91}_{-0.02}^{+0.02}$
				3.16	II	<2.63	II	<2.84	3	${0.90}_{-0.05}^{+0.03}$
				10	II	<1.04	II	<1.12	3	${0.86}_{-0.15}^{+0.08}$
				31.6	III	<0.41	III	<0.45	−	${0.68}_{-0.31}^{+0.22}$
97Tc/98Tc	5.94	6.1	226	1	I	${5.83}_{-1.61}^{+1.83}$	I	${5.99}_{-1.63}^{+1.85}$	2	${0.97}_{-0.01}^{+0.00}$
				3.16	II	<2.84	II	<2.91	3	${0.97}_{-0.02}^{+0.01}$
				10	II	<1.12	II	<1.14	3	${0.96}_{-0.05}^{+0.02}$
				31.6	III	<0.45	III	<0.47	3	${0.90}_{-0.13}^{+0.07}$
				100	III	<0.05	III	<0.06	−	${0.73}_{-0.29}^{+0.19}$

Note. Production ratios are always 1. Also indicated are τ₁, τ₂, τ_eq, and the adopted γ, all in Myr. The values of γ are selected such that it is possible to remain within Regimes 2 or 3, for which cases we can model the uncertainties. The Roman numerals correspond to the regimes of Paper I (SRL_1,2/stable), while the Arabic numerals correspond to the regimes described in this work. The dashes correspond to cases that do not fit neatly in any of the regimes.

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Table 3. Timescales Derived by Decaying the Reported ISM Ratios to the ESS Ratios in Column 2 for a Subset of Ratios and γ Values Considered in Table 2 to Represent Possible Realistic Values in the Galaxy for the Corresponding Production Event

	ESS Ratio	τeq	γ	SLR1/stable		SLR2/stable		SLR1/SLR2
				ISM Ratio	Time	ISM Ratio	Time	ISM Ratio	Back-decayed Ratio
247Cm/129I	2.28 × 10−3	5085	316	9.63 × 10−2	171	1.15 × 10−1	153	1.22 × 10−2 a	2.35 × 10−3
107Pd/182Hf	4.25	35.4	1	3.56 × 10−4	${16}_{-3}^{+2}$	5.20 × 10−4	${21}_{-3}^{+2}$	2.41	7.17
			3.16		${16}_{-6}^{+4}$		${21}_{-6}^{+4}$
			31.6	1.20 × 10−3	27	1.28 × 10−3	32	3.28	9.78
53Mn/97Tc	>1.70 × 105	59.4	1	1.58 × 10−4	${17}_{-2}^{+2}$	3.84 × 10−5	>7	1.65 × 106	>2.26 × 105
			31.6	9.23 × 10−4	26	2.04 × 10−4	>17	1.82 × 106	<2.63 × 105

Notes. Time and τ_eq are in Myr. The ISM SLR_1,2/stable ratios in Roman are calculated using the steady-state formula from Côté et al. (2019a) and K = 2.3. These are cases within Regime I and can provide T_iso (also in roman). The ISM SLR_1,2/stable ratios in italics are calculated instead using the last-event formula, i.e., Equations (3) (with K = 2.3) and 4 (with K = 1.2) of Côté et al. (2021) and the selected value of γ = δ. These are cases within Regime III and can provide T_LE (also in italics). The ISM SLR₁/SLR₂ values are calculated as = P(τ₁/τ₂) (Equation (7)) for roman values and as = P (Equation (8)) for italic values. The production ratios used in all the formulas are reported in the text in each subsection. The "back-decayed" ratios are calculated by decaying the ESS ratio back by the average of T_iso, or T_LE, from both SLR_1,2/stable ratios, except for the case of ⁵³Mn/⁹⁷Tc, where only the times derived from ⁵³Mn were used. Differences between the values in the last two columns highlight the problems discussed in the text.

^a We calculated this possible r-process production using the average ²⁴⁷Cm/²³²Th ratio from Goriely & Janka (2016) and assuming the solar ratio ¹²⁷I/²³²Th of 31 from Asplund et al. (2009). This is to avoid using ²³⁵U, which decays much faster than ²³²Th (with a mean life of roughly 1 Gyr instead of 20 Gyr) and would complicate the assumption that the produced ¹²⁷I/²³⁵U was solar.

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As discussed in the constant δ_c scenario, this regime shows a low variation around the average P while retaining no memory of previous events. The difference between the constant and stochastic case is similar to that in Regime 2 (because Equation (13) only depends on 〈δ〉 and τ_eq), that is, a factor of 2F. The factor F = K is a constant here (when $\min ({\tau }_{1},{\tau }_{2})\lt \langle \delta \rangle$), equal either to 0.5 or to 1, which means that the relation between the uncertainties in the constant and stochastic cases is a factor of two at most. Additionally, the stochastic case results in a nonsymmetric distribution around the median. The reason is that the ratio is always bounded between 0 and the production factor P (when τ₂ > τ₁): when the enriching events are more frequent than average, the ratio will remain at P, while when the enriching events are less frequent than average, the ratio decays away from this average. In any case, the characteristic of Regime 3 is that the average ratio always remains very close to P.

These two isotopes are made by the rapid neutron-capture (r) process, and typical estimates for the time interval at which r-process nucleosynthetic events that are believed to enrich a parcel of gas in the Galaxy range between 200 and 500 Myr (Hotokezaka et al. 2015; Tsujimoto et al. 2017; Bartos & Marka 2019; Côté et al. 2021). Therefore the case of ²⁴⁷Cm/¹²⁹I is the best example of Regime 3 because τ_eq = 5085 Myr (Table 2) is much longer than γ, while each τ (≃22.5 and 22.6 Myr, respectively) is much shorter than γ. The ratios to the long-lived or stable references isotopes, ²⁴⁷Cm/²³⁵U and ¹²⁹I/¹²⁷I, allow us to derive a T_LE, for example, for the specific γ value of 316 Myr considered in Table 3, and derive typical production ratios of 1.35 for ¹²⁹I/¹²⁷I and 0.3 for ²⁴⁷Cm/²³⁵U. While our T_LE values are not perfectly compatible with each other, the more detailed analysis shown by Côté et al. (2021) demonstrates that there is compatibility for T_LE in the range between 100–200 Myr, depending on the exact choice of the K parameter (Côté et al. 2019a), γ, and the production ratios. The short mean lives of ²⁴⁷Cm and ¹²⁹I ensure that there is no memory from previous events, while the long τ_eq of ²⁴⁷Cm/¹²⁹I instead ensures that this ratio did not change significantly during T_LE and has a high probability to be within the 10% of the production ratio. Therefore the production ratio of the last r-process event that polluted the ESS material can be accurately determined directly from the ESS ratio. If we assume that the last event produced a ²⁴⁷Cm/²³²Th ratio similar to the average predicted by Goriely & Janka (2016) and assume a solar ratio for ¹²⁷I/²³²Th, then we find an inconsistency between the numbers in the last two columns of Table 3. The back-decayed value is more than five times lower than the assumed production ratio, which indicates a weaker production of the actinides from this last event with respect to the production ratios that we are using here. The number in the last column therefore represents a unique constraint on the nature of the astrophysical sites of the r process in the Galaxy at the time of the formation of the Sun and needs to be compared directly to different possible astrophysical and nuclear models (Côté et al. 2021).

If T_LE for the last r-process event is longer than 100 Myr, as discussed in the previous section, the presence of these two SLRs in the ESS should primarily be attributed to the slow neutron-capture (s) process in asymptotic giant branch (AGB) stars, which are a much more frequent event due to the low mass of their progenitors because their r-process contribution would have decayed for a time of about 10 times their mean lives (Lugaro et al. 2014). Experimental results on the SLRs ¹⁰⁷Pd (τ = 9.8 Myr) and ¹⁸²Hf (τ = 12.8 Myr) are reported with respect to the stable reference isotopes ¹⁰⁸Pd and ¹⁸⁰Hf, respectively. The ISM ratios reported in Table 3 are calculated using production ratios of 0.14, 0.15, and 3.28 for ¹⁰⁷Pd/¹⁰⁸Pd, ¹⁸²Hf/¹⁸⁰Hf, and ¹⁰⁷Pd/¹⁸²Hf, respectively, derived from the 3 M_⊙ model of Lugaro et al. (2014). For the short γ values considered in Table 3 (1 and 3.16 Myr), the SLR_1,2/Stable ratios belong to Regime I and the SLR₁/SLR₂ ratio belongs to Regime 2. Therefore we can calculate T_iso from all the ratios. As shown in Table 2, the ratios relative to the stable reference isotopes have larger uncertainties (40% or 85% depending on γ, and assuming no uncertainty on the stable isotope abundance) compared to the ratio of the two SLRs (less than 20%). However, when the actual ISM ratios are considered, the uncertainties on the evaluation of T_iso become comparable because they are relative uncertainties and the ratio of the two SLRs and the equivalent mean life have a much higher absolute value than the other two ratios. While the T_iso values derived from the SLR_1,2/stable ratios are consistent with each other, the value calculated from SLR₁/SLR₂ would need to be much shorter. In the last column of Table 3 we report the back-decayed ratio as the ISM ratio that is required to obtain a self-consistent solution.

The discrepancy between the ISM and back-decayed values may be due to problems with the stellar production of these isotopes: a main caveat to consider here is that while the ¹⁰⁷Pd/¹⁰⁸Pd ratio produced by the s process is relatively constant because it only depends on the inverse of the ratio of the neutron-capture cross sections of the two isotopes, both the ¹⁸²Hf/¹⁸⁰Hf and ¹⁰⁷Pd/¹⁸²Hf production ratios can vary significantly between different AGB star sources. The ¹⁸²Hf/¹⁸⁰Hf ratio is particularly sensitive to the stellar mass (Lugaro et al. 2014) due to the probability of activating the ¹⁸¹Hf branching point, which increases with the neutron density produced by the ²²Ne(α, n)²⁵Mg neutron source reaction, which in turn increases with temperature and therefore stellar mass. The ¹⁰⁷Pd/¹⁸²Hf involves two isotopes belonging to the mass region before (¹⁰⁷Pd) and after (¹⁸²Hf) the magic neutron number of 82 at Ba, La, and Ce. This means that this ratio will also be affected by the total number of neutrons released by the main neutron source ¹³C(α, n)¹⁶O in AGB stars, which has a strong metallicity dependence (see, e.g., Gallino et al. 1998; Cseh et al. 2018). This means that a proper analysis of these s-process isotopes can only be carried out in the framework of full GCE models, where the stellar yields are varied with mass and metallicity. This work is submitted (Trueman et al. 2021, submitted), and the uncertainties calculated here will be included in this complete analysis.

For long γ values, such as 31.6 Myr considered in Table 3, the ¹⁰⁷Pd/¹⁰⁸Pd and ¹⁸²Hf/¹⁸⁰Hf ratios would likely mostly reflect their production in one event only (Regime III). In this case, we derive an T_LE. Because ¹⁰⁷Pd/¹⁸²Hf is between Regimes 1 and 3, this isotopic ratio changes more significantly during the time interval T_LE than in the case of the r-process isotopes discussed in the previous section. In Table 3 we report the production value predicted by decaying the ESS ratio back by T_LE. As in the case of the r-process isotopes, in this regime, this number can be used to determine the stellar yields of the last AGB star to have contributed to the s-process elements present in the ESS (Trueman et al. 2021, submitted).

These two SLRs are next to each other in mass and are both p-only isotopes, i.e., they are nuclei of higher mass than Fe that can only be produced by charged-particle reactions or the disintegration (γ) process. While the origin of p-only isotopes is currently not well established especially for those in the light-mass region, and the main sites may be both core-collapse and Type Ia supernova, recent work has shown that the main site of production of the SLRs considered here is probably Chandrasekhar-mass Type Ia supernova (see, e.g., Travaglio et al. 2014; Lugaro et al. 2016; Travaglio et al. 2018). Because their mean lives are remarkably similar (τ = 5.94 and 6.1 Myr, respectively, for ⁹⁷Tc and ⁹⁸Tc), their τ_eq = 226 Myr and as shown in Table 2, the theoretical uncertainties related to their ratio are very low for values γ up to 31.6 Myr.

The full GCE of these isotopes was investigated by Travaglio et al. (2014). Expanding on that work, in combination with the present results, could provide us with a strong opportunity to investigate both the origin of these p-nuclei and the environment of the birth of the Sun. There are many scenarios that might be investigated. If the γ value of the origin site of the Type Ia supernova was around 1 Myr, then we could derive a T_iso from all the different ratios and check for self-consistency. If the γ value of the origin site was above 30 Myr, we would instead be in a similar case as the r-process isotopes discussed above, and the ⁹⁷Tc/⁹⁸Tc would give us directly the production ratio in the original site, to be checked against nucleosynthesis predictions. For γ values in between, the ⁹⁷Tc/⁹⁸Tc ratio would still provide us with the opportunity to calculate T_iso. Unfortunately, we only have upper limits for the ESS ratio of these two nuclei, relative to their experimental reference isotope ⁹⁸Ru, which means that an ESS value for their ratio cannot be given and a detailed analysis needs to be postponed until these data become available.

From a chemical evolution perspective, the origin of Mn (and therefore ⁵³Mn) is still unclear (Seitenzahl et al. 2013; Cescutti & Kobayashi 2017; Eitner et al. 2020; Kobayashi et al. 2020; Lach et al. 2020). Nevertheless, the ⁵³Mn/⁹⁷Tc ratio can be assumed to be synchronous as there are indications that the main site of origin of ⁵³Mn is the same as that of ⁹⁷Tc ⁸ (see, e.g., Lugaro et al. 2016). Table 2 shows that the uncertainty for the ratio of the two SLRs is below 30% for most cases (and as low as 5% when γ = 1 Myr), while for each one of the individual isotopes it is larger than 60%. Similar to the ⁹⁷Tc/⁹⁸Tc ratio discussed above, the ⁵³Mn/⁹⁷Tc ratio can also provide the opportunity to investigate T_iso for γ values up to 2 Myr because even if τ_eq = 59.4 Myr, the shorter mean lives of each SLR do not allow to built a memory, making this a case of Regime 3, which cannot be treated here. The ISM values reported in Table 3 were calculated with a production ratio of 2.39 × 10⁻² for ⁹⁷Tc/⁹⁸Ru, 0.108 for ⁵³Mn/⁵⁵Mn, and 1.82 × 10⁶ for ⁵³Mn/⁹⁷Tc (Travaglio et al. 2011; Lugaro et al. 2016).

We obtain potential self-consistent isolation times, mostly determined by the accurate ESS value of ⁵³Mn/⁵⁵Mn. Consistency between the last two columns of the table, which could inform us on the relative production of nuclei from nuclear statistical equilibrium (such as ⁵³Mn) and nuclei from γ-process in Chandrasekhar-mass Type Ia supernova (such as ⁹⁷Tc), could be found only if the ⁹⁷Tc/⁹⁸Ru ratio in the ESS was 7.3 times lower than the current upper limit.

Similarly to the s-process case described above, for high values of γ (e.g., 31.6 and 100 Myr shown in Table 3), the ⁵³Mn/⁵⁵Mn and ⁹⁷Tc/⁹⁸Ru ratios would record one event only (Regime III) and the derived T_LE are consistent with each other. The value from ⁵³Mn/⁵⁵Mn can then be used to decay the ESS ratio of the ⁵³Mn/⁹⁷Tc back and derive a direct constraint for the last p-process event that polluted the solar material. Overall, a more precise ⁹⁷Tc ESS abundance would allow us to take advantage of the low theoretical uncertainties and give a more accurate prediction of the ISM ratio or the production ratio at the site.

Finally, we consider the case for ⁶⁰Fe/²⁶Al. This ratio is of great interest in the literature because both isotopes are produced by core-collapse supernovae (Limongi & Chieffi 2006), and they can be observed with γ-rays (Wang et al. 2007) as well as in the ESS (Trappitsch et al. 2018). There are strong discrepancies between core-collapse supernova yields and observations as the yields typically produce a ⁶⁰Fe/²⁶Al ratio at least three times higher than the γ-ray observations (e.g., Sukhbold et al. 2016), and orders of magnitude higher than the ESS ratio (see the discussion in Lugaro et al. 2018).

We cannot apply our analysis to interpret the γ-ray ration because it is derived by measuring first the total abundance of ⁶⁰Fe and ²⁶Al separatedly, and then dividing them. In this case, the average abundance ratio is given simply by the ratio of the averages, mixing the ⁶⁰Fe and ²⁶Al productions from several different events, which do not correspond to our synchronous framework.

When the ESS abundance is considered, however, we can apply our methods because the ESS ratio represents the abundance at one time and place in the ISM, generated by a synchronous set of events. In this case, τ₁ = 3.78 Myr (for ⁶⁰Fe) and τ₂ = 1.035 Myr (for ²⁶Al) results in τ_eq = −1.45 Myr. If we consider a γ = 1 Myr for the core-collapse supernova-enriching events, we fall somewhere between Regime 2 and 4, with ⁶⁰Fe and ²⁶Al building memory and almost no memory, respectively, between successive events. As a consequence, when we consider our statistical analysis, the average ISM value given by Equation (10) predicted for the ⁶⁰Fe/²⁶Al ratio is a factor of 3.9 of the production ratio. This is 7% higher than the traditional continuous enrichment steady-state formula P τ₁/τ₂ (i.e., the limit of Equation (2) when δ_c , Δt → 0) used in the literature (see, e.g., Sukhbold et al. 2016) because that gives a factor of 3.65 of the production ratio instead. In conclusion, our analysis does not help to solve the problem that core-collapse supernova yields produce much more ⁶⁰Fe relative to ²⁶Al than is observed in the ESS.