The Timing of Potential Last Nucleosynthetic Injections into the Protosolar Molecular Cloud Inferred from ⁴¹Ca–²⁶Al Systematics of Bulk CAIs

Short-lived radionuclides (SLRs; half-lives of ∼0.1–100 Ma), present in the early solar system, are now extinct because they decayed below the detection level (Dauphas & Chaussidon 2011). They provide important constraints on the chronology of the earliest solar system events including the formation of CAIs, chondrules, and accretion of early planetesimals as well as their chemical differentiation and thermal evolution. Extinct nuclides also trace their nucleosynthetic sources and place constraints on the birth environment of the solar system. Many SLRs are believed to be products of the long-term chemical galactic evolution, while those with half-lives <1 Ma require late addition to the protosolar molecular cloud. Such an addition could be caused by either an injection from young stellar sources or in situ production by irradiation of local materials in the protosolar molecular cloud or the solar nebula itself. For example, an explosion of a nearby star, which triggered the collapse of the protosolar molecular cloud and the formation of the solar system, might have injected SLRs into it. Another possible source is the wind from a Wolf–Rayet star (Dwarkadas et al. 2017). Thus, constraining the origin of SLRs may elucidate how the forming solar system was influenced by the ambient astrophysical environment (Adams 2010). Among the SLRs, ⁴¹Ca, with an extremely short half-life of 0.0994 Ma (Jörg et al. 2012), is perhaps the most important nuclide to constrain the timescale of molecular cloud collapse and the eventual formation of a young stellar object like our Sun.

Previously, the abundance of ⁴¹Ca was inferred from the detection of radiogenic ⁴¹K excesses (δ⁴¹K*) in the oldest solar system materials, Ca–Al-rich inclusions (CAIs) in carbonaceous chondrites (Goswami 1998; Srinivasan et al. 1994, 1996). CAIs are the first solids formed in the solar nebula. The absolute U-Pb age of CAIs defines the age of our solar system (Amelin et al. 2010). Because of their early formation, CAIs provide insights into the sources of the SLRs in the solar system. Given their high Ca and very low K concentrations, CAIs are also perfect for studying the origin of ⁴¹Ca. The abundance of ⁴¹Ca expressed as the (⁴¹Ca/⁴⁰Ca)_I isotope ratio of 1.41 × 10⁻⁸ (Srinivasan et al. 1996) has historically been inferred from ⁴¹Ca–⁴¹K isochrons by in situ measurements using secondary ion mass spectrometry (SIMS or ion microprobe). Recently, Liu (2017) reported a much lower (⁴¹Ca/⁴⁰Ca)_I ratio of 4.6 × 10⁻⁹ than that of previous studies (Goswami 1998; Srinivasan et al. 1994, 1996), resulting in a reevaluation of the significance of ⁴¹Ca in the earliest solar system models, as well as the origins of ⁴¹Ca.

Here, we report a new (⁴¹Ca/⁴⁰Ca)_I ratio for the solar system inferred from a bulk CAI isochron established by multicollector inductively coupled plasma mass spectrometry (MC-ICP-MS). The key advantage of MC-ICP-MS is its ability to eliminate the severe matrix interferences characteristic of the ion microprobe technique. As a result, MC-ICP-MS allows, for the first time, a precise ⁴¹Ca–⁴¹K chronometry of bulk CAIs with uncertainties of the ⁴¹K/³⁹K isotope ratios as small as 0.030‰.

We studied four CAI samples (three chips of SJ101 and the whole CH007 inclusion), one matrix sample (Mx), and one chondrule sample (CH001) from the Allende CV3 chondrite, described in detail in the Appendix. The major and trace elements Appendix were measured in small aliquots by a quadrupole ICP-MS; the remainders were passed through ion-exchange columns for K separation, and isotope analyses were performed using the Nu Sapphire (#SP001) MC-ICP-MS equipped with a collision cell. Potassium isotope compositions are expressed relative to our laboratory standard, Merck Suprapur: δ⁴¹K (‰) =[(⁴¹K/³⁹K)_sample/(⁴¹K/³⁹K)_standard –1] ×10³. Uncertainties are reported at the two-sigma level, and an overall sample reproducibility of 0.030‰ is applied to all samples; additional uncertainties were applied to the samples with high Ca/K ratios as discussed in Appendix A.2.3.

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Three SJ101 pieces (A, B, and C) show enrichment in ⁴¹K compared to the CH007 CAI (δ⁴¹K = −0.294‰). SJ101-A has a δ⁴¹K value of −0.036‰, while SJ101-B and -C yield more positive values of +0.226‰ and +0.236‰, respectively (Table 1). The K isotopic compositions of CAIs, plotted versus ⁴⁰Ca/³⁹K ratios in Figure 1, yield a ⁴¹Ca–⁴¹K isochron with the (⁴¹Ca/⁴⁰Ca)_I ratio of (2.00 ± 0.52) × 10⁻⁸. By analogy with the initial (²⁶Al/²⁷Al)_I ratio of the bulk CAI isochron, we interpret this value as the initial ratio for the solar system (more discussion of isochron fitting in Appendix A.3–A.4). We also note that the goodness of the fits for the isochron, as indicated by the χ² values in Figure 1, is not perfect. This is further discussed in Appendix A.4 and Table 2. The uncertainties shown in Figure 1 are maximum uncertainties as they are expanded using the square root of the χ² values. The matrix (Mx), chondrule (CH001), and bulk sample (Allende-1) all plot significantly above the CAI isochron, implying K isotope heterogeneity in the Allende meteorite. Such heterogeneity in Allende has been recorded by Jiang et al. (2021).

The previously reported δ⁴¹K values of three CAIs (Jiang et al. 2021), also measured by MC-ICP-MS, are indistinguishable from the bulk Allende value. Given the lack of petrographic description of these CAIs and unreasonably high concentrations of FeO, Na₂O, and K₂O in them, suggestive of contamination, we prefer not to add these data to our isochron.

A two-component mixing model between the matrix and CAIs can, in principle, produce the observed linear δ⁴¹K–⁴⁰Ca/³⁹K correlation (Figure 1), as well as the correlations for the refractory elements like Al and Ti (Figures 2(a)–(b)). Such a mixing model should also produce a linear array among all CAI samples and the matrix in a δ⁴¹K versus 1/K diagram, but that is clearly not the case (Figure 2(c)). Mixing between two components also fails to reproduce the variations in the diagrams involving volatile elements (Mn, Tl, and Pb) (Figures 2(d)–(f)). Thus, the linear CAI array in the isochron diagram cannot be a result of a simple two-component mixing of CAIs and matrix. Moreover, a detailed and thorough investigation of SJ101 (Petaev & Jacobsen 2009) shows only minuscule patches of matrix attached to its exterior and the lack of such material inside. This effectively rules out matrix contamination of the SJ101 interior fragments used in this study. The images of CH007 (Appendix) do show some matrix loosely attached to the inclusion. It is unclear how much matrix is left after fragmentation and sonication of this sample, but its contribution to the sample can be accessed from the concentrations of highly volatile elements (e.g., Pb and Tl), which are enriched in matrix by factors of ∼28 and ∼27, respectively. Assuming all Pb and Tl in CH007 are from matrix (Table 3), the maximum contribution from matrix is only 3.7 wt%. Because the matrix-free samples, SJ101, also contain Pb and Tl at ∼0.5 × CH007 levels, the matrix contribution to CH007 must be <2 wt%. The effect of such a small addition of matrix on either the K isotopic composition or K/Ca ratio of CH007 CAI is less than the analytical uncertainty of the δ⁴¹K measurements. We conclude (1) that the linear CAI array in Figure 1 is a fossil isochron, not a mixing line, and (2) that a very small matrix component, if left on the CAIs, should have no effect on the measured δ⁴¹K values or the interpretation of the data.

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Both the commonly observed aqueous alteration in CV chondritic meteorites and natural K isotope fractionation could have affected or disturbed the isotope systematics of the ⁴¹Ca−⁴¹K system by changing somewhat both the δ⁴¹K and K/Ca ratios of the samples studied. Both processes would lead to nonlinear effects and are not likely to produce the observed isochron.

The recent study of secondary aqueous alteration of Allende CAIs (e.g., Krot et al. 2021) shows that during the alteration Si, Na, Cl, K, Fe, S, and Ni could be added to, while Ca, Mg, and some Al could be removed from the host inclusions to a various degree. Such an alteration typically affects only melilite and anorthite; it is highly localized and never extends to a whole inclusion. Our detailed study of SJ101 (Petaev & Jacobsen 2009) detected only a few grains of melilite partially replaced by grossular, monticellite, and rare wollastonite. Minute euhedral grains of andradite were observed in voids. The lack of Na-rich phases among alteration products implies that no Na and, most likely, no K were added. Additionally, Sr, a very mobile element during aqueous alteration, shows coherent concentrations with other refractory elements across all SJ101 chips, implying that aqueous alteration could only have a minor effect, if any, in these samples (Appendix A.1). However, both CH001 and CH007 (Appendix A.1) provide clear mineralogical evidence of metasomatic reactions of their peripheral portions with an aqueous fluid containing Na, Cl, K, Si, Fe, Mg, Mn, Ca. Among them, K, Ca, Na, and Cl are of particular interest because the first two elements directly affect the δ⁴¹K–⁴⁰Ca/³⁹K correlation while Na and Cl are excellent tracers of the extent of alteration. Si, Na, and Cl have been added to and Ca removed from both CH001 and CH007, resulting in substitution of anorthite (CH001) and melilite (CH007) by sodalite and nepheline. The fate of K is different—its concentration is markedly increased in the altered areas of CH001 where some nepheline grains contain up to ∼1.5 wt% K₂O, but the K distribution in the K_Kα X-ray map of CH007 is homogenous throughout the whole sample, regardless of the Na and Cl distributions. Also, the EDS point analyses of alteration phases in CH007 do not detect K. The bulk Na/K ratio in CH007 (∼26) is markedly higher than that of other samples (ranges from 15 to 18), suggesting that the aqueous alteration did not add much K to CH007 and therefore did not disturb the ⁴¹Ca−⁴¹K systematics of CH007.

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It has been argued that the correlated Mg and Si compositions of CAIs were caused by partial loss of these elements during thermal processing accompanied by kinetic isotope fractionation. At the high temperatures of the solar nebula, the near-equilibrium isotopic fractionation of K, if it has occurred, is very small and likely negligible (more discussion in Appendix A.5). The lack of Si and Mg isotope fractionation in SJ101 (Jacobsen et al. 2008b; Petaev & Jacobsen 2009) provides clear evidence that neither SJ101 as a whole nor its precursors have experienced nonequilibrium processing involving kinetic isotope fractionation of these elements. The melilite mantles in Type B1 CAIs like CH007 are typically interpreted as evidence of evaporative loss of Si and Mg during the earlier stages of CAI crystallization before the mantles seal off the interiors. Such a loss is expected to induce kinetic isotope fractionation, which has not been found in two CAIs studied in detail by Bullock et al. (2013). Even if such a kinetic fractionation would have occurred, it should make the K isotope composition of CH007 heavier than that of SJ101. However, CH007 is lighter than SJ101 so this effectively rules out the derivation of the K isotopic composition of CH007 from that of SJ101. Theoretically, the heavier K isotopic composition of SJ101 can be explained by a nonequilibrium evaporative loss of K from CH007, but the depletion of SJ101 in refractory Ca, Al, and Ti along with the enrichment in moderately volatile Mg and Si, relative to CH007, rules out such a formation scenario. Therefore, the overall positive correlation between δ⁴¹K and ⁴⁰Ca/³⁹K ratios is unlikely to be a result of a mass-dependent K isotope fractionation.

Because neither two-component mixing nor chemical or thermal processing of the CAIs studied can account for the observed linear relationship between the δ⁴¹K and ⁴⁰Ca/³⁹K ratios, we interpret this linear correlation as an extinct isochron caused by ⁴¹Ca decay. The small-scale scatter of data points along the isochron could result from an initial ⁴¹Ca/⁴⁰Ca heterogeneity (perhaps not surprising for a nuclide (⁴¹Ca) with such a short half-life) or minor effects of chemical or thermal processing.

Our (⁴¹Ca/⁴⁰Ca)_I ratio for the solar system of 2.00 × 10⁻⁸ is about four times higher than the previous value of 4.6 × 10⁻⁹ inferred from SIMS measurements (Liu 2017). The latter study was able to analyze samples with an extremely wide range of Ca/K ratios (10³ to 10⁶) on a microscale. However, because the precision of the SIMS measurements was low (uncertainties exceeding 70‰), isobaric interferences with other species could result in a false correlation in a fossil isochron diagram (such an example for the ⁶⁰Fe–⁶⁰Ni system is discussed by Trappitsch et al. 2018). In addition, the resultant ⁴¹K/³⁹K–⁴⁰Ca/³⁹K correlation and its slope heavily rely on very few points with extremely low K. Such analytical challenges are likely the main cause of the large variations in (⁴¹Ca/⁴⁰Ca)_I ratios (from 1.4 × 10⁻⁸ to 4.2 × 10⁻⁹) among published mineral isochrons (Srinivasan et al. 1996; Sahijpal et al. 2000; Ito et al. 2006; Liu et al. 2012; Liu 2017). Moreover, the mineral isochrons themselves could also reflect resetting by the multiple events in many CAIs. In contrast, the MC-ICP-MS technique removes such isobaric interferences by analyzing chemically separated high-purity K solutions, with high precision.

Previously, the (⁴¹Ca/⁴⁰Ca)_I ratio for the solar system was established by SIMS analyses of a few CAIs (Table 5). The discrepancy in (⁴¹Ca/⁴⁰Ca)_I ratios between the SIMS and our MC-ICP-MS studies likely results from the differences between the bulk and mineral isochrons, which date the initial closure of the ⁴¹Ca−⁴¹K system of a whole CAI and subsequent disturbances, respectively. The mineral isochrons for the Type B1 E65 and Type A NWA 3118 1Nb CAIs are different, with both having lower ⁴¹Ca/⁴⁰Ca ratios (Liu 2017; Liu et al. 2012) than our bulk Type B CAI isochron. The existence of two distinct ⁴¹Ca reservoirs for Type A and Type B CAIs is inconsistent with the δ²⁶Mg isotope data (MacPherson et al. 2017) that reveal no temporal evolutionary sequence among CAIs of different types. This means that the younger ages with lower (⁴¹Ca/⁴⁰Ca)_I ratios inferred from mineral isochrons, which date the last mineral reequilibration events, probably reflect nebular processing of CAIs. Specifically, the mineral ⁴¹Ca–⁴¹K isochrons of Liu et al. (2012) and Liu (2017) are ∼0.4 Ma younger than our bulk CAI isochron; this time difference likely corresponds to an interval of CAI processing after the ⁴¹Ca–⁴¹K closure age determined by the bulk CAI isochron.

A similar CAI processing interval of 0.2 Ma was used by MacPherson et al. (2017) to explain the variations of (²⁶Al/²⁷Al)_I ratios among 25 CAI mineral isochrons relative to the bulk CAI isochron. A combination of ⁴¹Ca–⁴¹K and ²⁶Al–²⁶Mg extinct systems extends the CAI processing interval (Figure 3) to ∼0.26 Ma when considering the large uncertainty of the E65 (²⁶Al/²⁷Al)_I ratio.

The SJ101 mineral ²⁶Al–²⁶Mg isochron (MacPherson et al. 2017) yields an age and an (²⁶Al/²⁷Al)_I ratio indistinguishable from the bulk ²⁶Al–²⁶Mg Allende CAI isochron (Jacobsen et al. 2008a), implying that SJ101 experienced very little or no mineral disturbances after its initial formation. This suggests that the inferred (⁴¹Ca/⁴⁰Ca)_I and (²⁶Al/²⁷Al)_I ratios of SJ101 currently provide the best estimate of the initial abundances of ⁴¹Ca and ²⁶Al in the solar system.

The ⁴¹Ca in the solar system could have been produced by in situ irradiation in the early solar nebula or, alternatively, by stellar nucleosynthetic injections from outside the solar nebula. Irradiation by energetic protons and alpha particles could result in local ⁴¹Ca heterogeneity in the solar nebula (Lee et al. 1998), as is the case for ¹⁰Be. Irradiation calculations depend on various parameters, including the magnitude of X-ray emission from young stars, quantified as fluence, and the initial compositions of the target nuclides (i.e., ⁴⁰Ca(p,pn)⁴¹Ca). The fluence, as a function of flux and time, could be estimated from the astrophysical observations of T Tauri stars, pre-main-sequence stars that may be similar to our young Sun. The resultant (⁴¹Ca/⁴⁰Ca)_I value is highly sensitive to the magnitude of fluence; for instance, increasing fluence by a factor of 10 would increase the (⁴¹Ca/⁴⁰Ca)_I ratio by an order of magnitude (Leya et al. 2003). The target chemical composition is typically assumed to be either the solar composition or the nominal CAI composition. The (⁴¹Ca/⁴⁰Ca)_I ratio produced by irradiation is tied to the (¹⁰Be/⁹Be)_I ratio from the same sample, but because the latter is not available for SJ101, we could only rely on theoretical calculations. Leya et al. (2003) used the average (¹⁰Be/⁹Be)_I for CAIs to calculate (⁴¹Ca/⁴⁰Ca)_I of 2.6 × 10⁻⁸ for the solar composition and (⁴¹Ca/⁴⁰Ca)_I of 3.3 × 10⁻⁷ for the present-day CAI compositions. While the irradiation conditions inferred for ¹⁰Be can potentially account for the observed ⁴¹Ca, ²⁶Al will be underproduced by a factor of ∼2–5 (Dauphas & Chaussidon 2011). The irradiated ²⁶Al is produced by the reaction ²⁷Al(p,pn)²⁶Al, so the solar ²⁷Al abundance will result in an (²⁶Al/²⁷Al)_I ratio of ∼10⁻⁶, much less than the measured value (Jacobsen et al. 2008a). In addition, the similar δ²⁶Mg values for Earth, CAIs, and chondrules hint at a homogeneous distribution of ²⁶Al, which is inconsistent with the expected local heterogeneity developing during local irradiation. Also, ²⁶Al and ¹⁰Be abundances show no correlation in hibonite-bearing inclusions and CAIs (Liu et al. 2009), making the production of ²⁶Al by irradiation less plausible (Dauphas & Chaussidon 2011). The higher (⁴¹Ca/⁴⁰Ca)_I and (²⁶Al/²⁷Al)_I ratios inferred from the bulk ⁴¹Ca–⁴¹K and ²⁶Al–²⁶Mg isochrons likely point to a similar stellar origin of ⁴¹Ca and ²⁶Al.

Pollution of the protosolar molecular cloud by freshly produced nuclides injected by supernova shock waves from supernova explosions, AGB, or Wolf–Rayet star winds is the main working hypothesis for explaining the higher abundances of the shortest-lived extinct nuclides in the solar system relative to the galactic background (Table 6). For example, the (²⁶Al/²⁷Al)_I ratio of (5.23 ± 0.13) × 10⁻⁵ for the solar system is more than three orders of magnitude higher than that of the protosolar molecular cloud (1.1 × 10⁻⁸; Table 7). The new solar system (⁴¹Ca/⁴⁰Ca)_I of (2.00 ± 0.52) × 10⁻⁸ is also about three orders of magnitude higher than that of the molecular cloud (4 × 10⁻¹¹; Table 7). Clearly, the inferred concentrations of both ⁴¹Ca and ²⁶Al in CAIs require late injection(s) from outside the protosolar molecular cloud.

The constraints on the timing of such injections are dependent on the initial solar system abundances and nucleosynthetic sources of injections. As discussed above, the initial (⁴¹Ca/⁴⁰Ca)_I and (²⁶Al/²⁷Al)_I ratios of the solar system are defined by the bulk CAI (not mineral) isochrons. Then, by tracing the closed-system evolution of both isotope systems back in time and comparing the calculated values with those modeled for TP-AGB stars, Type II supernovae (Wasserburg et al. 2006), and Wolf–Rayet stars (Young 2014), one can estimate the time of injection for each nucleosynthetic source (Figure 3(b)). This time ranges from 0.62 Ma for a 60 solar-mass (M_⊙) Wolf–Rayet star to 0.87–1.06 Ma for a low-mass (1.5 M_⊙–2 M_⊙) AGB star or ∼1.47 Ma for a 15 M_⊙ Type II supernova.

Theoretical predictions and astrophysical observations demonstrate the plausibility of the triggered formation of the solar system, known as the rapid collapse of the protosolar molecular cloud (Hartmann et al. 2012; Dale 2015). If the majority of both ⁴¹Ca and ²⁶Al is due to a late injection, then the well-determined initial abundance of both nuclides places an upper limit on the time span between the collapse of the protosolar molecular cloud and the formation of the solar system. Among the three possible sources mentioned above, a supernova injection is currently inconsistent with the low abundance of ⁶⁰Fe in solar system materials (Trappitsch et al. 2018). Thus, the event that might have triggered the collapse of the parental molecular cloud would have taken place between 0.6 and 1.1 Ma before CAI formation, implying a very short timescale of formation for our Sun.

We are very thankful to the late Dr. Ursula B. Marvin for donating the CH007 CAI and the CH001 chondrule as well as two Allende specimens used for the preparation of the Allende whole-rock powder and matrix samples. We are also grateful to Dr. Natasha V. Krestina for preparing the Allende matrix sample and Dr. C. A. Parendo for help in the laboratory. We thank Dr. E. Young, Dr. M.-C. Liu, Dr. L. Zeng, and Dr. J. Mitrovica for help with some aspects of this paper. Y. K. thanks the Harvard Origins of Life Initiative, NASA (grant number 80NSSC20K0346 to S.B.J.), and the DOE-NNSA (grant number DE-NA0003904 to S.B.J.) for supporting her graduate research. This work was primarily supported by the NASA Emerging Worlds Program grant number 80NSSC20K0346 to S.B.J.

A.1. Sample Descriptions

A.2. K Isotope Analytical Methods and Data

A.2.1. Acid Dissolution and Chemical Purification of Potassium (K)

Aliquots (10–40 mg) of the matrix, as well as chondrule and CAI powders, were dissolved in multiple steps using mixtures of concentrated HF, HCl, and HNO₃. The first step requires adding a mixture of 0.5 mL HF, 2 mL HNO₃, and 1 mL HCl acids, which was then heated to 160°C on a hotplate for a day. For the second dissolution step, the resulting solutions were dried down, and a mixture of 2 mL HNO₃, 6 mL HCl, and 1 mL purified MQ H₂O was added to Teflon EasyPrep vessels for high-temperature and high-pressure dissolution applications. The samples were then repeatedly heated to 210°C in a CEM MARS 6 microwave digestion system. After complete digestion, the aliquots were redissolved in a dilute 0.5N HNO₃ solution for K ion-exchange column chemistry.

To separate K, the bulk solutions were loaded into 13 mL fused silica chromatography columns filled with Bio-Rad AG50W-X8 cation-exchange resin (100–200 mesh) using our established procedure (Ku & Jacobsen 2020). Here, all samples were passed through the columns three times in order to purify the K cuts (>99% yield). The K solutions were then dried down and redissolved in dilute HNO₃ (∼2%) for MC-ICP-MS analysis.

A.2.2. Acquisition of K Isotope Ratios by Mass Spectrometry

Potassium (K) isotopes were measured at Harvard University using a multicollector ICP-MS Nu Sapphire (serial #SP001) that is equipped with a hexapole collision cell to eliminate mass interference from Ar-bearing ions such as ArH⁺, Ar⁺, and ArO⁺. The samples and standard were matched for ³⁹K intensity to within 1%. The measured K isotope values are expressed relative to our lab standard, a batch of Merck KgaA high-purity KNO₃ Merk Suprapur:

$$\begin{eqnarray*}&&{\delta }^{41}{\rm{K}}=\left[\displaystyle \frac{{\left(\tfrac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{sample}}}{{\left(\tfrac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{suprapur}}}-1\right]\,\times \,{10}^{3}.\end{eqnarray*}$$

For each analytical run, we monitored the matrix effect arising from concentration mismatch among solutions by analyzing an extra standard solution with up to 15% concentration mismatch. The data reported here are obtained from analytical runs where the δ⁴¹K values from the "mismatched" standard are less than 0.05 (Ku & Jacobsen 2020).

As K solutions could be contaminated with very small amounts of Ca, we corrected for any possible Ca-related interference on mass 41 (⁴¹K), such as Ca-hydride ⁴¹(⁴⁰CaH⁺), of both sample and standard solutions. This formation of ⁴¹(⁴⁰CaH⁺) is also described in the literature (Moynier et al. 2021). The correction is based on monitoring the mass 40 intensity for every analytical run and is calculated using the difference in intensity (I) ratios of mass 41 and mass 39 (⁴¹I/³⁹I) between the Ca-spiked and unspiked standard solutions:

$$\begin{eqnarray*}&&\displaystyle \frac{{}^{41}\mathrm{CaH}}{{}^{40}\mathrm{Ca}}=\displaystyle \frac{{\left({}^{41}I{/}^{39}I\right)}_{\mathrm{Ca}-\mathrm{spiked}}-{\left({}^{41}I{/}^{39}I\right)}_{\mathrm{std}}}{{\left({}^{40}I{/}^{39}I\right)}_{\mathrm{Ca}-\mathrm{spiked}}-{\left({}^{40}I{/}^{39}I\right)}_{\mathrm{std}}}.\end{eqnarray*}$$

To determine an accurate value of this correction, the Ca concentration in the spiked solution was always selected to be substantially larger than that of the sample solutions. This correction factor is then used to calculate the interference-corrected δ⁴¹K values using the following equation:

$$\begin{eqnarray*}&&\begin{array}{l}{\delta }^{41}{{\rm{K}}}_{\mathrm{corr}-\mathrm{std}}={\delta }^{41}{{\rm{K}}}_{\mathrm{meas}-\mathrm{std}}\\ \quad -\,\left(\displaystyle \frac{{}^{39}{\rm{K}}}{{}^{41}{\rm{K}}}\right)\left(\displaystyle \frac{{}^{40}\mathrm{Ca}}{{}^{39}{\rm{K}}}\right)\,\times \,\left(\displaystyle \frac{{}^{41}\mathrm{CaH}}{{}^{40}\mathrm{Ca}}\right)\,\times \,{10}^{3}.\end{array}\end{eqnarray*}$$

In most of our routine K isotope measurements, this correction is not significant as the measured ⁴⁰Ca/³⁹K is typically <0.001 leading to CaH corrections of <10 ppm. When the correction is significant, it may also contribute to the increase of the uncertainty of the δ⁴¹K value, as discussed in the next section.

A.2.3. Analytical Uncertainties and Precisions

We used the analytical uncertainty of 0.03‰ of the K isotope data of each sample, which was established following the experimental, chemical, and instrumental procedures used in Ku and Jacobsen (Ku & Jacobsen 2020). The within-run uncertainty of 0.03‰ is the calculated standard error (2σ_m) of a K standard solution that was repeatedly measured within an analytical run, and the external reproducibility of 0.027‰ is the calculated variation obtained by measuring a rock sample, BHVO-2, repeatedly on different days. The consistency between the two values implies that no additional variability of the dissolution or column chemistry was present in the data, so we used the former value as the best estimate of the analytical uncertainty for each sample.

The formation of the ⁴¹(⁴⁰CaH⁺) species was monitored in every run and the effect on the δ⁴¹K values from the ⁴⁰CaH⁺ interference has been corrected. The results of the measurements used for these corrections (³⁹K/⁴¹K, ⁴⁰Ca/³⁹K, and ⁴¹CaH/⁴⁰Ca) are given in Table 8. Columns (2) and (3) report the uncorrected values with uncertainty, and columns (10) and (11) list the interference-corrected δ⁴¹K values with their uncertainties expanded based on the uncertainty in the CaH corrections.

Table 8. Source Data for the CaH Corrections and Corrected δ⁴¹K Uncertainties

(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
Sample	δ41Kmeas-std	2σm δ41K (analytical)	39K/41K	40Ca/39K	41CaH/40Ca	41CaH/41K	CaH corr. (‰)	2σ-CaH -corr.	δ41Kcorr-std	2σm δ41K (final)
Mx	−0.368	0.030	12.639	0.00130	0.000838	0.000014	0.014	0.004	−0.382	0.034
CH001	0.106	0.033	12.638	0.03526	0.000762	0.000340	0.340	0.092	−0.234	0.125
CH007	−0.255	0.030	12.639	0.00370	0.000838	0.000039	0.039	0.011	−0.294	0.041
SBJ-101-A	0.022	0.030	12.632	0.00755	0.000612	0.000058	0.058	0.016	−0.036	0.046
SBJ-101-B	0.343	0.030	12.635	0.01218	0.000762	0.000117	0.117	0.032	0.226	0.062
SBJ-101-C	0.569	0.030	12.632	0.03459	0.000762	0.000333	0.333	0.090	0.236	0.120

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The uncertainty in such correction is generally insignificant, except for the two measurements, which have larger-than-usual measured ⁴⁰Ca/³⁹K of 0.035, yielding the total CaH corrections in δ⁴¹K values to be 0.333‰ to 0.340‰. The correction factor

$$\begin{eqnarray*}&&\left(\displaystyle \frac{{}^{39}{\rm{K}}}{{}^{41}{\rm{K}}}\right)\left(\displaystyle \frac{{}^{40}\mathrm{Ca}}{{}^{39}{\rm{K}}}\right)\,\times \,\left(\displaystyle \frac{{}^{41}\mathrm{CaH}}{{}^{40}\mathrm{Ca}}\right)\,\times \,{10}^{3}\end{eqnarray*}$$

is obtained by multiplying columns (4), (5), and (6) and 1000 and is given in permil in column (8). During the measurements reported here we obtained an average CaH/Ca ratio of 6.6 × 10⁻⁴ with an uncertainty of 1.8 × 10⁻⁴ (Figure 8). This gives a multiplier of 0.27 (=1.8/6.6) with column (8) to obtain the 2σ uncertainty of the CaH correction in column (9). Under normal circumstances, this CaH uncertainty would only yield a 3 ppm additional uncertainty to the δ⁴¹K values. However, in the case of large CaH corrections, it resulted in the final 2σ uncertainty of 0.120‰ (SBJ-101-C) and 0.125‰ (CH001). The final δ-values (column 10) are calculated by subtracting column (8) from column (2). The interference-corrected uncertainty is calculated by adding column (9) to column (3). Table 1 reports only these final, interference-corrected values. This is the best estimate that could be done for these samples as all of the material was consumed due to the difficulties of the K isotope measurements.

A.3. Calculation of the Initial ⁴¹Ca/⁴⁰Ca from the ⁴¹Ca−⁴¹K Isochron in Figure 1

The equation for a ⁴¹Ca−⁴¹K fossil isochron is

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{meas}}={\left(\displaystyle \frac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{initial}}+{\left(\displaystyle \frac{{}^{41}\mathrm{Ca}}{{}^{40}\mathrm{Ca}}\right)}_{\mathrm{initial}}{\left(\displaystyle \frac{{}^{40}\mathrm{Ca}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{meas}}.\end{eqnarray*}$$

This equation can be converted to use δ⁴¹K values by using

$$\begin{eqnarray*}&&{\delta }^{41}{{\rm{K}}}_{\mathrm{meas}}=\left[\displaystyle \frac{{\left(\tfrac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{meas}}}{{\left(\tfrac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{std}}}-1\right]\,\times \,{10}^{3}\end{eqnarray*}$$

Thus, in this notation the fossil isochron is

$$\begin{eqnarray*}&&{\delta }^{41}{{\rm{K}}}_{\mathrm{meas}}={\delta }^{41}{{\rm{K}}}_{\mathrm{initial}}+\left[\displaystyle \frac{{10}^{3}}{{\left(\tfrac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{std}}}\right]{\left(\displaystyle \frac{{}^{41}\mathrm{Ca}}{{}^{40}\mathrm{Ca}}\right)}_{\mathrm{initial}}{\left(\displaystyle \frac{{}^{40}\mathrm{Ca}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{meas}}\end{eqnarray*}$$

The value of ⁴¹K/³⁹K = 0.0721677 for the standard (Naumenko et al. 2013). Thus, the slope of the isochron determined in the δ⁴¹K versus ⁴⁰Ca/³⁹K diagram (slope) can be converted to give the initial (⁴¹Ca/⁴⁰Ca)_I ratio:

$$\begin{eqnarray*}&&{\left(\displaystyle \frac{{}^{41}\mathrm{Ca}}{{}^{40}\mathrm{Ca}}\right)}_{\mathrm{initial}}=\mathrm{slope}\,\times \,\left[{10}^{-3}{\left(\displaystyle \frac{{}^{41}{\rm{K}}}{{}^{39}{\rm{K}}}\right)}_{\mathrm{std}}\right].\end{eqnarray*}$$

A.4. Isochron Fitting and the Mean Squared Weighted Deviation

A.5. Possible K Isotope Fractionation Due to Evaporation