Cosmic-ray bath in a past supernova gives birth to Earth-like planets

Abstract

A key question in astronomy is how ubiquitous Earth-like rocky planets are. The formation of terrestrial planets in our Solar System was strongly influenced by the radioactive decay heat of short-lived radionuclides (SLRs), particularly ²⁶Al (aluminum-26), likely delivered from nearby supernovae. However, current models struggle to reproduce the abundance of SLRs inferred from meteorite analysis without destroying the protosolar disk. We propose the “immersion” mechanism, where cosmic-ray nucleosynthesis in a supernova shockwave reproduces estimated SLR abundances at a supernova distance (~1 parsec), preserving the disk. We estimate that solar mass stars in star clusters typically experience at least one such supernova within 1 parsec, supporting the feasibility of this scenario. This suggests that Solar System–like SLR abundances and terrestrial planet formation are more common than previously thought.

A nearby supernova’s trapped cosmic rays supply crucial ²⁶Al, turning young disks into cradles for rocky, water-poor worlds.

INTRODUCTION

The prevalence of Earth-like planets is one of our fundamental questions to the universe. Earth has a small but certain amount of liquid water, which allows the atmosphere-ocean-crust interactions and characterizes its habitable environment. It has been proposed that the desiccation of planetesimals is crucial for forming water-depleted rocky planets (a bulk mass fraction lower than 1%) such as Earth (1, 2). Parent bodies of differentiated meteorites in the Solar System are known to have experienced substantial heating due to radioactive decay of a short-lived radionuclide (SLR) ²⁶Al (3), and lost originally accreted water and other volatiles (2, 4). Meteorite analysis has found that SLRs with half-lives shorter than 5 million years (Myr) (¹⁰Be, ²⁶Al, ³⁶Cl, ⁴¹Ca, ⁵³Mn, and ⁶⁰Fe) existed globally in the early Solar System (5). In contrast, SLR-depleted systems, if they exist, may only form ocean planets whose bulk water content is a few tens of percent (6). Therefore, understanding the origin of SLRs in the Solar System is crucial to answering the abovementioned question of the prevalence of Earth-like planets in other stellar systems.

Origin of SLRs

The excess of SLRs in the early Solar System provides critical insight into the formation of the Solar System (see Supplementary Text for more details). The abundance of SLRs in the “initial solar system” is derived from meteorite analysis (see in Table 1) (5), inferring their levels at the formation of Ca-rich, Al-rich inclusions (CAIs), the first solids in the Solar System (7). Given their short half-lives, the estimated SLR abundances are too high to have been inherited solely from the parent molecular cloud before the onset of Solar System formation (8–10). Moreover, inheritance from the molecular cloud alone cannot account for the coexistence of ²⁶Al-rich and ²⁶Al-poor CAIs as such a process would not introduce spatial heterogeneity on the scale of the protosolar disk (11). Hence, our Solar System must have undergone either an in situ production or an external injection of SLRs shortly before the formation of the CAIs.

Table 1. SLR abundances in the early Solar System and model predictions.

The evaluated values are expressed as the ratio of the number density of each SLR to its corresponding SI ($N^{\text{SLR}}/N^{\text{SI}}$). The nominal values and half-lives of the listed SLRs were obtained from ref. (5). The last column presents the predicted values from our immersion model.

SLR	Half-life (Myr)	Ratio	Nominal values	Our model
10Be	1.387	10Be/9Be	7.1 × 10−4	2.6 × 10−3
26Al	0.717	26Al/27Al	5.2 × 10−5	2.2 × 10−5
36Cl	0.301	36Cl/35Cl	2.0 × 10−5	4.5 × 10−6
41Ca	0.099	41Ca/40Ca	4.2 × 10−9	6.3 × 10−9
53Mn	3.98	53Mn/55Mn	7.8 × 10−6	2.1 × 10−5
60Fe	2.62	60Fe/56Fe	0.9 × 10−8	1.0 × 10−8

A nearby supernova explosion has long been believed to be a strong candidate for the source of SLRs (12, 13). However, the supernova injection scenario faces an unresolved problem in that existing supernova models could not reproduce both the relative and absolute abundances of SLRs without disrupting the protosolar disk. For instance, these models predict that, if a supernova provided ²⁶Al and ⁴¹Ca to the Solar System, it would also supply 100 times more ⁵³Mn than its estimated nominal abundance [see Fig. 1A and refs. (14, 15)]. Moreover, regarding absolute abundances, ref. (16) demonstrated that supernova explosions within 0.3 pc can disrupt the protosolar disk and that a supernova injection event capable of supplying a sufficient SLR amount would likely prevent the Solar System formation altogether.

Fig. 1. Normalized ratios (calculated/inferred nominal) for ¹⁰Be, ²⁶Al, ³⁶Cl, ⁴¹Ca, ⁵³Mn, and ⁶⁰Fe supplied by our immersion model, with the gray band indicating agreement with meteoritic constraints.

(A) Our immersion model (red circles; d = 1 pc, $t_{\text{delay}}$ = 0.45 Myr, and $13 M_{\odot }$ progenitor for optimal parameters) is directly compared to two other previous cases: only injection cases (orange diamonds; d = 0.3 pc, $t_{\text{delay}}$ = 0.9 Myr, and 40 $M_{\odot }$ progenitor for optimal parameters) taken from ref. (14) and the case of supernova injection + flare synthesis taken from ref. (29) (see Materials and Methods for more details). (B) Decomposition of the immersion result into CR-synthesis (orange dashed squares) and direct supernova injection (purple dashed diamonds) components, illustrating each individual contribution to the total SLR inventory. N.A., not applicable.

To solve the discrepancy in relative abundance, an alternative combined scenario has been proposed in which ⁵³Mn and ⁶⁰Fe are injected from supernovae, whereas ²⁶Al and ⁴¹Ca are synthesized by different processes (15), such as energetic particle irradiation from protosolar flares (17). This flare synthesis also could account for the presence of ¹⁰Be, which is absent in stellar nucleosynthesis and must originate from spallation reactions (18, 19). However, even when adding in the contribution from the flare synthesis to the supernova injection (see Materials and Methods), the resulting SLR abundances differ by more than an order of magnitude from the nominal Solar System values (see Fig. 1). It should also be noted that this flare synthesis has serious drawbacks in explaining the global distribution of SLRs in the Solar System (5). Because the flare synthesis process works only in a minimal area of the protosolar disk, it would require extensive mixing on a scale not yet understood.

Immersion model

We propose a unified scenario, the immersion mechanism, that explains the origin of all SLRs to be consistent with the nominal abundances inferred from meteorites (Fig. 1). In this scenario, when certain SLRs—specifically ⁵³Mn and ⁶⁰Fe—are injected into the protosolar disk from a nearby supernova, the disk is naturally immersed in accelerated particles confined within the shockwave of the supernova. This process can, in principle, drive in situ synthesis of ¹⁰Be, ²⁶Al, ³⁶Cl, and ⁴¹Ca via nonthermal nucleosynthesis, a phenomenon we refer to as the immersion mechanism (see Fig. 2).

Fig. 2. Schematic picture of the system assumed in this study.

(A) A supernova explosion occurs at a distance d from the protosolar disk, and (B) the expanding supernova shockwave contacts the protosolar disk. At this time, a huge number of accelerated particles are trapped in the shockwave region. With the contact, (Ca) SLRs synthesized inside the supernova are directly injected into the disk, and (Cb) particles trapped inside the shockwave irradiate the disk, causing nucleosynthesis within the disk.

Our immersion mechanism assumes that a supernova explosion occurs in close proximity to the Sun ($d\lesssim 10$ pc) during the lifetime of the protosolar disk (see Birth environment for the estimated event rate). In this model, the distance (𝑑) plays a critical role in determining the supplied abundance of SLRs. When a supernova explodes, it generates a collisionless shock mediated by plasma instabilities, where charged particles, primarily protons, undergo diffusive shock acceleration to reach high energies (20–22). Theoretical studies on particle escape from supernova shockwaves indicate that most accelerated particles, including subrelativistic particles with energies of ≲ 1 GeV, remain confined within the shocked region. Hereafter, these particles are referred to as “trapped cosmic rays (trapped CRs)” (23, 24). As the supernova interacts with the protosolar disk, the heliosphere is compressed by the pressure of the supernova to a scale of ≲1 astronomical unit (au)—smaller than Earth’s orbital radius (25–27). This compression allows cosmic rays to impinge on the protosolar disk without hindrance from the magnetosphere. The disk could become largely exposed to trapped CRs, initiating nonthermal nucleosynthesis throughout its extent while simultaneously incorporating nuclides originating from the supernova, as in the injection model (16, 28). This nucleosynthesis process operates on gas and small grains in the disk. Given the stopping depth of ≲1 GeV protons in rock (29), the trapped CRs can penetrate grains as large as 3 to 4 cm in radius. Thus, although the current model assumes that the supernova event happened before the formation of CAIs [typically smaller than this threshold (30)] as its baseline, this assumption can be relaxed.To evaluate the predicted SLR abundances from our immersion mechanism, we formulated the ratio of the SLR abundance to the stable isotope (SI) abundance at the time of CAI formation, $N^{\text{SLR}}/N^{\text{SI}}$, under the assumption that preexisting SLRs in the protosolar disk are negligible, as

$$\begin{array}{c}\frac{N^{\text{SLR}}}{N^{\text{SI}}}=\frac{N_{\text{new}}^{\text{SLR}}·\text{exp}(-t_{\text{delay}}/{\mathrm{\tau }}_{\text{SLR}})}{N^{\text{SI}}}\hfill \\ \approx \frac{(N_{\text{inj}}^{\text{SLR}}+N_{\text{syn}}^{\text{SLR}})·\text{exp}(-t_{\text{delay}}/{\mathrm{\tau }}_{\text{SLR}})}{N^{\text{SI}}}\hfill \end{array}$$ (1)

where $N_{\text{inj}}^{\text{SLR}}$ and $N_{\text{syn}}^{\text{SLR}}$ denote the number densities of SLRs injected from the supernova and synthesized via trapped CRs, respectively. The sum of $N_{\text{inj}}^{\text{SLR}}$ and $N_{\text{syn}}^{\text{SLR}}$ is denoted by $N_{\text{new}}^{\text{SLR}}$, which represents the number density of SLR newly supplied to the protosolar disk. The exponential term represents the decay of SLRs with a mean lifetime ${\mathrm{\tau }}_{\text{SLR}}$ over the time interval $t_{\text{delay}}$, which is the duration between the supply of new SLRs and the formation of CAIs. In Eq. 1, we assumed that injection and synthesis occur simultaneously. This assumption is justified because the time difference between these processes is expected to be much less than 1 Myr (see Materials and Methods). The free parameters in this model are the time interval $t_{\text{delay}}$ and the distance 𝑑 between the supernova and the protosolar disk. The optimal model parameters are identified by minimizing the deviation between the predicted abundances of SLRs (see Eq. 1) and the nominal values derived from meteorite analyses (see Table 1).

For the nonthermal synthesis of SLRs in this system, we assumed that the disk is uniformly exposed to trapped CRs without temporal variation over the time interval $\mathrm{\Delta }t$, which corresponds to the duration of the supernova shockwave traveling through the disk. The number density of SLRs, $N_{\text{syn}}^{\text{SLR}}$, synthesized by the bombardment of trapped CRs on a target nucleus labeled 𝑗 (i + j → SLR), can be expressed using the thin target approximation as

$$N_{\text{syn}}^{\text{SLR}}=\mathrm{\Delta }t\sum _{(i,j)}\mathit{[}{\mathrm{\gamma }}_i N_j\underset{E_{\mathit{0}}}{\overset{\mathit{\infty }}{\mathit{\int }}}{\mathrm{\sigma }}_{ij}\mathit{\left(}E\mathit{\right)}\frac{d{F}_{\mathit{\text{CR}}}}{dE}dE\mathit{]}$$ (2)

where N_j is the number density of the target nuclei 𝑗, γ_i is the relative abundance of trapped CRs 𝑖 relative to protons, and σ_ij (E) and E₀ represent the energy-dependent cross section and the threshold energy of the reaction, respectively. These quantities are calculated using the TALYS code (31, 32) (see Materials and Methods). The number flux of accelerated particles in the supernova shock region, $F_{\text{CR}}$, is assumed to follow a standard power-law momentum distribution, $d{F}_{\text{CR}}/\mathit{dE}\propto p{\left(E\right)}^{-s}$, where $s\approx 2.1$ is the spectral index of CRs inferred from observations (33). The normalization of this flux is derived from a model that reproduces observed Galactic CR results (34), where 10% of the supernova kinetic energy density, $U_{\text{sh}}$, at the shock position is converted into CR energy density, $U_{\text{CR}}$. For the injection component, we adopted typical assumptions of the supernova injection model (16, 28). In this model, the supernova ejecta spread spherically, and only the fraction of SLRs intercepted by the protosolar disk is injected, depending on the distance d. We bracket uncertainties in CR acceleration efficiency, spectral index, and progenitor mass (see Materials and Methods for more detail).

RESULTS

Our immersion model offers a consistent explanation for the observed SLR abundances in the early Solar System. Figure 1 shows that our immersion model successfully reproduces all SLR abundances in the early Solar System to within one order of magnitude of their nominal values. This level of agreement falls well within the combined uncertainties, which stem from nuclear reaction cross sections and the estimated nominal SLR abundances. Each factor contributes to an overall uncertainty of approximately one order of magnitude. By contrast, each of the previously proposed models contains at least one SLR whose predicted abundance deviates from its nominal value by more than an order of magnitude. This discrepancy persists even when summing up the contributions of the flare and supernova injection. We also derived the optimal values of $d=1 \text{pc}$ and $t_{\text{delay}}=0.45 \text{Myr}$, where d is the distance from the supernova to the protosolar disk and $t_{\text{delay}}$ is the time delay of CAI formation for the new SLR supply. Meteorite analyses constrain $t_{\text{delay}}$ to lie in the range of 0.2 to 0.7 Myr (see Materials and Methods). Notably, the ⁴¹Ca/⁴⁰Ca ratio indicates that $t_{\text{delay}}$ should exceed 0.1 Myr, suggesting that immersion before CAI formation is desirable.

Beyond matching the SLR abundances, the immersion scenario addresses several limitations noted in previous scenarios. First, compared to the inheritance model from the molecular cloud, this mechanism can provide sufficient amounts of SLR while accounting for the nuclear decay that occurs during the time delay to CAI formation. Second, compared to the direct injection model by supernova explosions, this mechanism can reproduce both the relative and absolute amounts of SLR by a supernova at the distance that does not destroy the Solar System $(d>0.3 \text{pc})$ (16). Third, compared to the flare synthesis model, this mechanism does not require unexplained large-scale mixing in the disk and can distribute SLR throughout the protosolar disk. Even the inner part of the protosolar disk can be exposed to trapped CRs because the ram pressure of the supernova at 1 pc can compress the heliosphere to a scale of $\lesssim 0.1 \text{au}$ (26).

The possible existence of a reservoir shielded from trapped CRs may help explain lower ²⁶Al/²⁷Al ratios reported for a small fraction of CAIs (35). In our model, we assumed that the protosolar disk was optically thin to the CRs. However, the inner region of the disk could have been optically thick; on the basis of the surface density profile of the minimum-mass solar nebula (36) and the cross section of H₂ gas to $\sim 1$ GeV protons (37), the region within $\sim 10 \text{au}$ is likely to have been opaque to CRs.

DISCUSSION

Birth environment

We find that typical young star-cluster environments readily permit the enrichment conditions required by our immersion model. Figure 3 shows that at least one supernova event occurs within 1 pc of the early Solar System at high probability, if the Sun is formed within a star cluster (38, 39). The early Solar System should have been in a young star cluster, and a large fraction of young star clusters are below the solid curve. Most observed clusters are located above the solid curve simply because they are old. They should have been below the solid curve when they were young (40, 41). This situation is further consistent with recent studies of star-forming regions, which depict environments characterized by dynamic interactions and frequent supernova explosions (42, 43). By contrast, the traditional supernova injection model requires a much closer explosion within 0.3 pc (14, 16), which is statistically less probable. Only a small fraction of young star clusters are below the dashed curve.

Fig. 3. Diagram of half-mass radii and total masses of several types of star clusters: Open clusters, old globular clusters, young massive clusters, and young open clusters.

A half-mass radius encloses half the mass of a star cluster. Data were obtained from refs. (48, 58). The translucent blue region and the left-tilted arrows indicate the dynamical evolution of open clusters over time, as suggested by recent N-body+SPH simulations (40, 41). The open blue circle, open blue square, and arrowhead correspond approximately to $t=0$ Myr, $t\sim 20$ Myr, and $t\gtrsim 20$ Myr, where $t$ denotes the age of the cluster since its formation. A star experiences at least one supernova event within 1 pc (0.3 pc) when it is in a star cluster below the solid (dashed) curve. Here, we modeled each star cluster as follows. Its stellar distribution follows the Plummer’s distribution (59). We adopt a Kroupa initial mass function (60). Massive stars with $8 \text{to} 20 M_{\odot }$ undergo supernova explosions (57).

Even if all the stars in a cluster formed during a few Myr, the clusters with total stellar masses ≳ 500 𝑀_⊙ (solar mass) provide conditions under which a Sun-like star inevitably experiences a nearby supernova within its disk lifetime (39). Moreover, observations reveal that star formation in most clusters extends over 10 Myr or more (44), suggesting that the mass threshold is lower. Although a full, coupled model of disk survival, cluster dynamics, and supernova immersion is beyond our present scope, the joint evidence from cluster masses and age spreads offers statistical support for our scenario.

Once a massive star appears, its ultraviolet (UV) radiation clears residual molecular gas around the Solar System (42), facilitating the direct injection of ejecta into the disk and efficient particle acceleration. Moreover, at distances of ≳0.3 pc, such UV flux remains too weak to photoevaporate the protosolar disk (45). Thus, its UV clearing would prepare the stage without destroying the target, leaving the immersion of the disk in ejecta physically plausible.

More than 50% of stars are born in massive star-forming regions comparable to or more massive than the Orion Nebula Cluster (ONC) (46, 47). Moreover, of stars still remain in bound clusters even after 30 Myr (48), corresponding to a timescale longer than the lifetime of massive stars that undergo supernovae. Thus, we conclude that at least 10%, possibly 50%, of Sun-like stars are likely to host protoplanetary disks with SLR abundances similar to those of the protosolar disk.

Universality of our Solar System

Our results suggest that Earth-like, water-poor rocky planets may be more prevalent in the Galaxy than previously thought, given that ²⁶Al abundance plays a key role in regulating planetary water budgets (1, 2). Because a measurable fraction of stars form in clusters, Solar System–like SLR abundances are likely to be common rather than exceptional. This challenges previous interpretations that classified the Solar System as an outlier with a particularly high ²⁶Al abundance (6). Given our estimate that $~10\%$ of stellar systems in the Galaxy likely acquired Solar System–like SLR abundances with the immersion mechanism, we predict that upcoming exoplanet surveys targeting habitable zones around several tens of nearby solar-type stars, as proposed with the Habitable World Observatory (49), will lead to the detection of a few Earth-like rocky planets.

MATERIALS AND METHODS

This study aims to reproduce the observed abundance ratios of SLRs at the time of CAI formation. By treating the protosolar disk as a one-zone model, we investigated the optimal parameters that minimizes the deviation between the model-predicted values, as defined by Eq. 1, and the nominal $N^{\text{SLR}}/N^{\text{SI}}$ ratios derived from meteorite analyses (Table 1). As outlined in the main text, the predicted values in our immersion model (Eq. 1) represent the sum of contributions from supernova injection and nonthermal nucleosynthesis. In the following sections, we address three key assumptions of our analytical model: (i) the timing of supernova injection and nonthermal nucleosynthesis, (ii) the reaction processes and cross sections incorporated in the nonthermal nucleosynthesis term, and (iii) the modeling of the supernova injection term.

The timing of injection and immersion

This section explains the rationale behind the assumption that supernova injection and nonthermal synthesis occur simultaneously in our model. This assumption is reflected in Eq. 1, where the two terms are added and assigned the same delay time. In practice, a time difference, $\mathrm{\Delta }t$, exists between the moment the supernova ejecta make contact with the protosolar disk and the time the supernova shock region completely traverses the disk. At the point of contact (𝑑 ∼ $R_{\text{sh}}$), the scale of the shocked region ($\mathrm{\Delta }{R}_{\text{sh}}$ ∼ 0.1 pc) is considerably larger than the scale of the protosolar disk (∼100 au). Here, hydrodynamic simulations of supernova remnants show that the width of the shocked shell is of the shock radius $\mathrm{\Delta }{R}_{\text{sh}}\approx R_{\text{sh}}/10$ (50, 51). This configuration yields a timescale of $\mathrm{\Delta }t\approx \mathrm{\Delta }{R}_{\text{sh}}/v_{\text{sh}}\approx 43 \text{years} (d/1 \text{pc}){(E_{\text{exp}}/{10}^{51} \text{erg})}^{-1/2}$, which is considerably shorter than the CAI formation timescale (∼10⁶ years). Thus, the assumption in Eq. 1 is considered valid.

Nuclear reaction

Here, we provide a detailed account of the nonthermal nuclear reaction processes in our immersion model. All the reaction processes considered in this study and their respective cross sections are presented in Fig. 4. The production of ⁶⁰Fe by nonthermal synthesis was not considered in this study, and the cross section for the production of ¹⁰Be was adopted from ref. (29). The cross sections for the production of other SLRs were calculated using the TALYS code (31).

Fig. 4. Energy-dependent nuclear reaction cross sections used in this study for the synthesis of ²⁶Al, ³⁶Cl, ⁴¹Ca, and ⁵³Mn.

Each panel shows cross sections calculated by the TALYS code (31): (top left) ²⁶Al, (top right) ³⁶Cl, (bottom left) ⁴¹Ca, and (bottom right) ⁵³Mn. For ¹⁰Be, we adopt the spallation processes ¹⁶O(𝑝,𝑋)¹⁰Be and ¹⁶O(α,𝑋)¹⁰Be, whose cross sections are data from ref. (61). No nonthermal production pathway is assumed for ⁶⁰Fe under charged-particle collisions.

The calculations presented in Eq. 2 do not encompass the full range of potential reaction processes. Instead, our analysis focuses on the interactions between protons/alpha particles with stable nuclei, which have relatively high abundances, while excluding reactions with other CRs such as ³He particles. This exclusion is justified through the following estimation. Among the target stable nuclei 𝑗, the fraction 𝑓 converted to SLR by collisions with CR, labeled 𝑖, can be expressed as

$$f=\frac{N_{\text{syn}}^{\text{SLR}}}{N_j}\sim ⟨{\mathrm{\sigma }}_{\mathit{ij}} F_{\text{CR}}⟩·\mathrm{\Delta }t$$ (3)

Using typical values for each parameter, the estimated fraction of SIs depleted or synthesized in this nuclear reaction process is 𝑓 ∼ 10⁻⁵γ_𝑖 (σ_𝑖𝑗 /100 mb)(𝑑/1 pc)⁻², where γ_𝑖 is the relative number abundance of the CR nuclei 𝑖 compared to protons. This estimation suggests that the influence of the injected particles other than protons and alpha particles, as well as reactions involving low-abundance target nuclei and processes with small cross sections, is negligible. This suggests that we can neglect the relative abundance γ_𝑖 of impact particles other than protons and alpha particles, processes with a small cross section σ_𝑖𝑗, and cases where the number density 𝑁_𝑗 of parent nuclei is small by an order of magnitude.

It should be noted that the small value of $f\sim {10}^{-5}$ indicates that the reduction in parent nuclei is sufficiently small, ensuring that any isotopic anomalies induced by our model remain minimal. This finding supports the robustness of our model in predicting the isotopic composition.

Injection model from supernova

This section evaluates the amount of SLRs injected into the protosolar disk from a nearby supernova, reviewing the material presented in ref. (16). Supernova ejecta spread spherically and is intercepted by the protosolar disk, which has a radius R_disk and is located at a distance 𝑑. Hydrodynamical simulations have shown that the disk can resist complete destruction from supernova impacts at distances greater than 𝑑 > 0.3 pc (28, 52). However, the contribution of gas-phase ejecta to SLR injection is minimal, accounting for less than 1% (28). Reference (53) demonstrated that small dust grains (≲0.1 μm) follow the gas flow and are not injected into the disk, whereas larger grains (≳1 μm) are injected with nearly 100% efficiency. On the basis of these findings, we assume that the supernova ejecta consist of gas and dust, define the mass fraction of large dust grains (≳1 μm) as η_𝑑, and disregard SLR injection via gas and small dust.

Assuming that the injected SLR mass is uniformly mixed with the disk mass 𝑀_disk, the number density of injected SLRs, $N_{\text{inj}}^{\text{SLR}}$, can be expressed as

$$\frac{N_{\text{inj}}^{\text{SLR}}}{N^{\text{SI}}}\approx \frac{M_{\text{inj}}^{\text{SLR}}}{X^{\text{SI}}{M}_{\text{disk}}}\sim {\mathrm{\eta }}_d\left(\frac{\mathrm{\pi }{R}_{\text{disk}}^2}{4\mathrm{\pi }{d}^2}\right)\frac{M_{\text{SN}}^{\text{SLR}}}{X^{\text{SI}} M_{\text{disk}}}$$ (4)

where $M_{\text{SN}}^{\text{SLR}}$ represents the mass of SLRs ejected from the supernova, with a value adopted from ref. (54), and $M_{\text{disk}}\gg M_{\text{SN}}^{\text{SLR}}$. The mass fraction of large dust grains is taken as η_d = 20%, based on typical values observed in supernova SN 1987A (55, 56). The timescale for SLRs to move from the supernova to the protosolar disk is approximately $t_{\text{SN}}\sim d/v_{\text{SN}}\sim 430 \text{years} (d/1 \text{pc}){(E_{\text{exp}}/{10}^{51} \text{erg})}^{-1/2}$, suggesting that radioactive decay during transit is negligible.

Combined model

Last, we summarize the calculation method used to predict the values of the combined model, developed for comparison with our immersion model. To facilitate this comparison, Fig. 1 presents results from both our model and previous studies, including the combined model, which adds the contribution of the flare synthesis to the supernova injection. The development of the combined model was motivated by the recognition that both accelerated particle irradiation from the protosun flare and stellar nucleosynthesis likely contributed to the origin of SLRs in the early Solar System. Despite this understanding, previous research lacked a quantitative model that integrated these processes. To address this gap, we derived the combined model as follows. The total amount of SLRs, 𝑁^SLR, in the combined model is given as the sum of SLRs supplied by flare synthesis and supernova injection, expressed as $N^{\text{SLR}}=N_{\text{flare}}^{\text{SLR}}+N_{\text{inj}}^{\text{SLR}}$. The flare synthesis contribution is based on ref. (29), assuming a negligible delay time due to a nearly instantaneous synthesis. For the supernova injection term, we use the reference values from ref. (14), denoted as $N_{\text{inj},\text{ref}}^{\text{SLR}}$, and account for the distance 𝑑 using the relation $N_{\text{inj}}^{\text{SLR}}\left(d\right)=N_{\text{inj},\text{ref}}^{\text{SLR}}·{(d/0.3 \text{pc})}^{-2}$. This term already includes a delay time of $t_{\text{delay}}=0.4$ Myr, as specified in ref. (14). To align this model with the nominal values of SLRs in the early Solar System, we determined the distance d by minimizing deviations between the model predicted SLR abundances and their nominal abundances. The optimal distance for the combined model, shown in Fig. 1, is 𝑑 = 0.9 pc.

Model uncertainties

To quantify the robustness of our immersion model predictions, we have explored the sensitivity of the calculated SLR abundance ratios to key input parameters. In particular, we focus on two classes of uncertainty: (i) the properties of the CR population accelerated at the supernova remnant shock and (ii) the choice of supernova yield models. Figure 5 illustrates the results of these tests.

Fig. 5. Uncertainty analysis for cosmic-ray parameters and supernova yields. Normalized ratios (calculated/inferred nominal) for ¹⁰Be, ²⁶Al, ³⁶Cl, ⁴¹Ca, ⁵³Mn, and ⁶⁰Fe supplied by our immersion model, with the gray band indicating agreement with meteoritic constraints.

(A) Sensitivity to cosmic-ray acceleration efficiency ε_𝑝: The fiducial value (ε_𝑝 = 10%, red circles) is compared to ε_𝑝 = 5% (blue squares) and ε_𝑝 = 20% (purple triangles). (B) Sensitivity to the cosmic-ray spectrum index 𝑠: The fiducial slope (𝑠 = 2.1, red circles) is compared to 𝑠 = 2.0 (blue squares) and 𝑠 = 2.4 (purple triangles). (C) Dependence on the adopted supernova progenitor mass: The fiducial case (13 $M_{\odot }$ model, red circles) is compared to yields from 11 $M_{\odot }$ (blue triangles), 15 $M_{\odot }$ (orange squares), 25 $M_{\odot }$ (violet inverted triangles), and 40 $M_{\odot }$ (cyan diamonds) models, alongside the prior work of ref. (14) with the 40 $M_{\odot }$ case (green diamonds). (D) Sensitivity to the delay time $t_{\text{delay}}$ between the supernova encounter and CAI formation. The optimum model ($t_{\text{delay}}$ = 0.45 Myr, red circles) is compared to $t_{\text{delay}}$ = 0.1 Myr (blue diamonds), 0.2 Myr (orange diamonds), 0.7 Myr (green diamonds), and 0.9 Myr (purple diamonds). Only the short-lived isotopes ($\mathrm{\tau }<1$ Myr; ²⁶Al, ³⁶Cl, and especially ⁴¹Ca) show strong dependence.

The CR acceleration efficiency, ε_𝑝, and the spectral index, 𝑠, govern both the total energy injected into high-energy particles and the shape of their momentum distribution. We vary ε_𝑝 by a factor of 4 around our fiducial value of 10% (testing ε_𝑝 = 5 and 20%; Fig. 5A) and vary 𝑠 around the nominal slope of 2.1 (testing 𝑠 = 2.0 and 2.4; Fig. 5B). Whereas lower acceleration efficiencies (ε_𝑝 = 5%) reduce the CR-synthesis contribution and diminish the ratios of ¹⁰Be, ²⁶Al, ³⁶Cl, and ⁴¹Ca by up to a factor of ∼2, higher efficiencies (ε_𝑝 = 20%) enhance them by similar factors. Likewise, a harder spectrum (𝑠 = 2.0) depletes ∼100-MeV CR particles, which are most efficient at driving disk synthesis, and thus reduces CR-synthesis yields by up to a factor of ∼2, as the effect of lowering ε_𝑝. Conversely, a softer spectrum (𝑠 = 2.4) enhances the low-energy CR flux and increases synthesis-derived isotopes by a similar factor.

Our baseline of the immersion model uses a 13 𝑀_⊙ progenitor; however, nucleosynthesis yields depend on stellar mass. Supernova models alter the injected ⁵³Mn and ⁶⁰Fe yields via direct supernova-disk interaction, but their ability to reproduce meteoritic ratios varies with mass (Fig. 5C). Whereas our fiducial 13 𝑀_⊙ model provides the closest overall match, lighter progenitors (11 to 15 𝑀_⊙) achieve comparably excellent agreement. This is an important result because an initial mass function weighting favors these lower-mass stars. In contrast, the 25 and 40 𝑀_⊙ models were adopted in a previous study (14) because they contain very high amounts of ²⁶Al, but their feasibility has been questioned by recent observations (57), and they may not represent typical supernova progenitors. Consequently, in our Birth environment section, we restrict the supernova mass range to 8 to 20 𝑀_⊙, ensuring both astrophysical realism and robust reproduction of the full SLR inventory.

Together, these sensitivity tests confirm that, despite variations of a factor of 2 from CR parameter choices and order-of-magnitude yield shifts from progenitor mass, our immersion model remains consistent with meteoritic SLR abundances within the adopted uncertainty envelope.

Temporal constraint on the immersion scenario

To evaluate the temporal constraint of the immersion scenario, we varied the delay time $t_{\text{delay}}$ between the supernova encounter and CAI formation from 0.1 to 0.9 Myr (Fig. 5D). ³⁶Cl and, most notably, ⁴¹Ca—whose mean lifetime is only ≈0.99 Myr—respond sensitively to this parameter. The meteoritic ratios remain within the acceptable uncertainty band for 0.2 Myr ≲ t_delay ≲ 0.7 Myr, with an optimum around ≲ t_delay = 0.45 Myr. This time window corresponds to the epoch at which our immersion model can reproduce the standard ⁴¹Ca/⁴⁰Ca ratio. The 0.5 Myr of time window allowed for this model imposes physically reasonable constraints on the immersion process. At the same time, although immersion nucleosynthesis itself can occur after CAI formation, the requirement for $t_{\text{delay}}$ 0.1 Myr suggests that immersion before CAI formation is preferable.

Supplementary Materials

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Supplementary Text

References