Self-consistent Conditions for ²⁶Al Injection into a Protosolar Disk from a Nearby Supernova

Short-lived radionuclides (SLRs) with half-lives of tens of Myr or less drive the thermal evolution of planetesimals (∼kilometer-sized bodies in planetary systems) and, consequently, influence the formation of planetary systems (e.g., Desch et al. 2022). Among SLRs, ²⁶Al (its half-life time t_1/2 ≈ 0.72 Myr; Auer et al. 2009) is thought to be the dominant heat source in the early solar system (Urey 1955). Meteorite analysis (measurements of the daughter nuclide, ²⁶Mg) have provided direct evidence and abundance estimates of ²⁶Al at the time of the formation of calcium–aluminum-rich inclusions (CAIs), the oldest material condensed in the early solar system (Gray 1974; Lee et al. 1976). Depending on the timing of the accretion and the resulting ²⁶Al content of each body, the radioactive decay heat led to various physicochemical evolutions of planetesimals, including core–mantle differentiation of achondrite parent bodies (e.g., Lichtenberg et al. 2019b) and aqueous chemistry to form secondary minerals (e.g., Kurokawa et al. 2022) and to drive chemical evolution of organics (Li et al. 2022). The radioactive decay heating of planetesimals and subsequent water loss has been proposed to have controlled the water budgets of planets and their diversity in the solar and extrasolar systems (Ciesla et al. 2015; Lichtenberg et al. 2019a).

Moreover, SLRs provide information on the astrophysical conditions for star formation and the birth environment of the solar system. The discovery of undecayed ²⁶Al (hereafter referred as “fresh ²⁶Al”) in the protosolar disk at the time of the formation of CAIs (Lee et al. 1976) led to three origin theories: inheritance, irradiation, and injection. The inheritance hypothesis suggests ²⁶Al came from the parent giant molecular cloud, enriched by star formation in spiral arms (e.g., Fujimoto et al. 2018). However, this theory faces a timing issue: the molecular clouds forming the solar system take over 1 Myr, during which ²⁶Al would decay, conflicting with meteorite data. The irradiation hypothesis proposes that ²⁶Al is synthesized by the irradiation of accelerated particles, such as from solar flares, during solar system formation. This model explains ¹⁰Be and also some ²⁶Al in CAIs (Sossi et al. 2017; Jacquet 2019), but it is unclear whether the appropriate mechanism for producing accelerated particles can be explained from astronomical sources. ⁴ Thus, the long-standing and most reliable approach is the injection hypothesis: if nearby core-collapse supernova (hereafter we refer as “supernova (SN)”) ejecta are injected into the protosolar system, it is known that the relative fraction of injected SLRs is roughly consistent with the abundances found in meteorites (excluding ¹⁰Be; e.g., Meyer & Clayton 2000).

In the injection hypothesis, two major injection timing models have been considered. One is that the SN ejecta enters the molecular cloud core, leading to its collapse from the SN shock wave (e.g., Cameron & Truran 1977; Boss & Foster 1998; Boss et al. 2008), and the other is that the SN ejecta is injected directly into the already formed protosolar disk (e.g., Ouellette et al. 2007, 2010; Fukai & Arakawa 2021). Also mentioned is the injection into filaments, which assumes both cases (Arzoumanian et al. 2023).

For the case of direct injection into the protosolar disk, several previous studies have investigated the pressure conditions under which the disk can be injected without disruption. Chevalier (2000) analytically estimated the disruption of the protosolar disk by SN shocks and found that, under typical conditions, the disk could be partially stripped but not completely destroyed. Ouellette et al. (2005) also estimated that for a disk with mass M_disk ≈ 0.01M_⊙ and radius R ≈ 30 au, the efficient capture of SN ejecta from a d ≈ 0.3 pc explosion would explain the estimated ²⁶Al amount from the meteorite. Hydrodynamical simulations by Ouellette et al. (2007; and also Close & Pittard 2017; Portegies Zwart et al. 2018; Portegies Zwart 2019) show that the disk resists total destruction by SN impacts, but the contribution of the gas-phase ejecta for injection is minimal, less than 1%. However, Ouellette et al. (2010) showed that while small dust grains (≲0.1 μm) follow the gas and are not injected into the disk, large grains (≳1 μm) are efficiently injected with ∼100% efficiency. If sufficient ²⁶Al is contained in large dust grains condensed from SN ejecta, the protosolar disk would receive enough ²⁶Al to account for the meteorite ratio. Thus, the injection of ²⁶Al into the disk would depend on the ability to inject SN large dust grains.

Here, we raise the question of whether large dust grains can inject a sufficient amount of ²⁶Al, accepting a certain amount of disk disruption. The previous analytical studies have treated the structure in a one-zone model (Chevalier 2000; Ouellette et al. 2005), and previous numerical studies have covered only a limited number of disk models (Ouellette et al. 2007, 2010) and investigated the pressure conditions without disruption. However, to consider this question, a wide range of disk masses and structures should be considered. Also, all of the previous studies focused only on the pressure condition and did not consider the effect of an SN on the disk temperature.

Furthermore, recent advancements in observational astronomy have greatly enhanced our understanding of SN explosions in the last decade. Notably, the recent direct detection of several progenitors suggests that the majority of massive stars above ∼20M_⊙ may collapse quietly to black holes, with the explosions remaining undetected (e.g., Smartt 2015; Adams et al. 2017). Recent numerical calculations have also reported results leading to implosion at progenitor masses above ≳20M_⊙ (e.g., Ugliano et al. 2012; Sukhbold et al. 2016), making the 25M_⊙ SN model, at least as frequently cited in previous discussions of early solar system composition (Ouellette et al. 2005, 2007), inappropriate as a typical value. In addition, over the past decade, there has been much discussion of the indeterminacy of the ²⁶Al yield based on a more modern understanding of stellar evolution and SN explosions (e.g., Woosley & Heger 2007; Tur et al. 2010; Brinkman et al. 2019, 2021, 2023). Observations of dust mass abundance ratios, also important in this study, are now available for SNe with younger timescales of interest in this study (∼40 yr; e.g., SN 1987A and SN 1980K; Matsuura et al. 2011; Zsíros et al. 2023).

To summarize, we assume that ²⁶Al from an SN is injected into an already formed protosolar disk (Figure 1) and we investigate the conditions under which the surviving disk radius can capture a sufficient amount of ²⁶Al for planet formation, while allowing some disruption of the disk. We consider a diversity of disk masses and structures, ²⁶Al yields of SN, and large dust mass fractions of ²⁶Al. We find that for all SN models with progenitor masses of 11–40M_⊙, the disk radius is disrupted to less than 30 au if a sufficient amount of injection is achieved. Thus, we conclude that even if partial disk disruption is allowed, the hypothesis of ²⁶Al injection into an already formed protosolar disk cannot be reproduced in almost all SN explosion models.

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The paper is organized as follows: Section 2 details the system setup and ²⁶Al injection conditions. Section 3 examines the conditions for disk disruption by SN flows. Section 4 addresses disk structure and mass, determining the conditions for sufficient ²⁶Al capture. Section 5 explores SN impacts on disk temperature and planet formation. Section 6 discusses the limitations of our study. Finally, Section 7 summarizes our findings.

In this section, some definitions are summarized below to clarify our model. Figure 1 shows a schematic picture of this study, and Table 1 shows the definitions of the variables in our study. The three major assumptions in this study are: (1) SLRs are injected from a nearby SN only after the protosolar disk is formed; (2) all of the ²⁶Al in the protosolar disk is supplied by a nearby SN; and (3) for simplicity, the protosolar disk is in face-on contact with the SN ejecta.

Table 1. Summary of Some Important Variables

Name	[Unit]	Description	Typical Value
	[M⊙]	Total ejecta mass of a nearby SN explosion	10
MSN	[M⊙]	Mass of the SN ejecta in contact with the protosolar disk (referred to as the “contact mass”), assuming an isotropic explosion from a distance of d
	[M⊙]	Ejected masses of 26Al from SN	1.3 × 10−4
d	[pc]	Distance from the protosolar disk to the SN	0.1
ηd	[ − ]	Mass fraction of large dust (a > 1μ m) in SN ejecta	0.20
Mdisk	[M⊙]	Mass of the protosolar disk	∼0.017
R	[au]	Radius of the protosolar disk	∼100
Σdisk	[g cm−2]	Surface density of the protosolar disk	∼1.7
q	[ − ]	Radius dependence of the disk surface density (Equation (8))	1 − 3/2
tdelay	[Myr]	Time delay from injection to formation of CAIs	⋯

Note. The “typical value” is the value adopted in this study to estimate the value. And “[−]” denotes dimensionless variables. Those marked with “()” depend on other variables—see the text.

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As background for each assumption: (1) at least one SN, which would directly inject the SLRs into the early solar system, is expected to occur in the Sun’s birth cluster during the formation of the CAIs (e.g., Arakawa & Kokubo 2023); (2) since the timescale for disk formation is sufficiently long compared to the lifetime of ²⁶Al, one would expect that all of the ²⁶Al originally contained in the disk has decayed and can be neglected; and (3) the assumption of a face-on disk gives upper limits on both the disk disruption and injection efficiencies. In practice, the disk is likely to be inclined at an angle, but the face-on disk is the most injectable assumption (see Section 6.1 for more details). Moreover, as discussed in Wijnen et al. (2017), when the disk contacts the external outflow, the disk can be readjusted to be perpendicular to the flow. Therefore, the inclination angle will not change the overall conclusion.

We first discuss the amount of ²⁶Al injected into the protosolar disk from a nearby SN. To avoid confusion, the mass of the SN ejecta in contact with the protosolar disk is called the “contact mass” M_SN. The relation of this value to the total mass of the SN ejecta is as follows:

where d is the distance from the protosolar disk to the SN and R is the radius of the protosolar disk. The contact mass is composed of gas and dust, and we define the mass fraction of large dust (here defined as grains sufficiently large to be injected into the disk; typically >1 μm; Ouellette et al. 2010) as η_d. Assuming that the ²⁶Al injection via gas and small dust grains is negligible, we can write the injected ²⁶Al mass as

where is the ejected mass of ²⁶Al from the SN, with an adopted value of about ∼1.3 × 10⁻⁴ M_⊙ per event (e.g., Timmes et al. 1995; Limongi & Chieffi 2006, 2018; Sukhbold et al. 2016; and see also Table 2). There are still many uncertainties regarding the nucleosynthesis of ²⁶Al in SNe, and we have listed the theoretical ²⁶Al yields of several different SN models in Table 2. Since the typical progenitor mass of recently observed SNe is M ≈ 8–17M_⊙ (Smartt 2015), the typical value of ∼1.3 × 10⁻⁴ M_⊙ used in this study is considered a sufficiently robust upper limit. Here, we have adopted the large dust mass fraction η_d ∼ 20% as the typical value (e.g., SN 1987A; Matsuura et al. 2011; and also see Section 6.2). Note that the timescale from the ejection of the SN to its injection into the disk is , so that the radioactive decay of ²⁶Al during the travel is negligible.

Next, from meteorite measurements, we know the short-lived nuclide/stable isotope abundance ratios at the time of CAI formation as N_r/N_s=²⁶Al/²⁷Al ≈ 5.23 × 10⁻⁵ (e.g., Jacobsen et al. 2008), which are extrapolated to the time of formation of the CAIs. Using this ratio and the mass fraction of ²⁷Al in the solar abundance X_⊙(²⁷Al) ≈ 5 × 10⁻⁵ (Asplund et al. 2009), the total amount of undecayed ²⁶Al in the protosolar disk M_disk(²⁶Al) can also be written as

where the exponential function in the equation is the correction factor between the actual injected mass and the observed mass due to the time delay t_delay from injection to CAI formation. In this study, we have chosen a value of t_delay = 0. Although this term is not insignificant, it represents the minimum amount that should be introduced to give robust conditions. Here, to satisfy M_disk(²⁶Al) = M_inje(²⁶Al), we obtain the following conditions for the distance of the SN:

where the disk radius R only determines the solid angle at which ²⁶Al can be received from the SN, once 100 au is adopted. Similarly, the disk mass M_disk here normalizes the total amount of ²⁶Al required, and we adopt 0.017M_⊙.

The injectable distance obtained from Equation (4) is the same as the distance obtained by matching the parameter η_d with previous studies: Ouellette et al. (2005) assumed a disk radius of R = 30 au, an injection rate of 100%, and chose an SN that can eject more ²⁶Al than a typical SN (the 40M_⊙ progenitor model for Woosley & Weaver 1995, assuming ∼3.6 × 10⁻⁴ M_⊙ yields), deriving the distance d ≈ 0.3 pc. Also, Ouellette et al. (2007) suggested that ²⁶Al is insufficient at d = 0.1 pc for the 4% injection rate suggested from hydrodynamical simulations. These estimates are consistent with our result obtained from Equation (4), while we note that gas and dust are not distinguished in these estimates, and thus the correspondence to η_d values in our model has no physical meaning regarding the actual η_d value.

3.1. Basic Picture of Disk Disruption

Next, we discuss the stability of the protosolar disk due to dynamical pressure, as shown in Figure 2. Since it is straightforward to speculate whether the ram pressure of the dense SN ejecta can destroy the protosolar disk, it has been investigated analytically/numerically in many previous studies (e.g., Chevalier 2000; Ouellette et al. 2007; Close & Pittard 2017). Our conclusion that the disk can survive the SN at distances is the same as in previous studies. The pressure stability condition derived below is almost identical to that in Chevalier (2000), but here we briefly review the condition for the completeness of the discussion.

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We consider the disk to be disrupted if the ram pressure of the SN flows P_sn exceeds the gravitational force per unit area, P_grav, which keeps the disk bound to the central pre-main-sequence star. An estimate of the gravitational force per unit area is

where M_⋆ is the mass of the central star and is adopted as M_⋆ = 1M_⊙, r is the distance of the radial direction from the center, and Σ_disk is the surface density at r. Here, typical values of the disk surface density Σ_disk and radius r were adopted once from the minimum-mass solar nebula model. For a more detailed discussion, the values are redefined in Section 3.2.

Next, we calculate the ram pressure that the disk receives from the freely expanding SN ejecta. For simplicity, we use the thin-shell approximation for the ejecta (e.g., Laumbach & Probstein 1969; Koo & McKee 1990). In this scenario, we consider an isotropic ejecta, which forms a hot bubble, and the pressure of this bubble is given by

where E_expl is the explosion energy of the SN. Δd is the thickness of the shell, which is Δd/d = 1/12 in the case of the thin shell for an ideal gas (Laumbach & Probstein 1969).

Here, the stability criterion for the disk region outside the radius r is given by

The disk disruption distance obtained from Equation (7) is the same as the distance obtained from the disk surface density and radius in previous studies: Chevalier (2000) assumed a disk radius of R = 10¹⁵ cm ≈ 66 au and a uniform density distribution Σ_disk = M_disk/(π R²) ≈ 6.4 g cm⁻², and also used a uniform density for the SN (Δd → d), so its gravitational force is P_grav ≈ 10⁻³ dyn cm⁻² and the stable criterion becomes d ≳ 0.25 pc.

3.2. Formulation with Disk Structure Assumption

Here, based on the minimum-mass solar nebula model (e.g., Hayashi 1981; Armitage 2010), we adopt the following disk model for a broad range of disk masses and structures:

where q is the radius dependence of the disk surface density, 1 < q ≤ 3/2. For instance, q = 3/2 in the minimum-mass solar nebula model (Hayashi 1981), while q = 1 in a viscous accretion disk model with a constant turbulent alpha parameter (Armitage 2010). In this case, the radius R of the protosolar disk in Section 2 is then defined as

and we can rewrite the disk radius R as

For q = 1, R ≈ 81 au is obtained from Equation (10). Also, when q = 3/2, R ≈ 55 au is obtained.

Then the stability criterion of Equation (7) can be rewritten as

Here, a simple comparison of the ²⁶Al injection distance (Equation (4)) and the disk disruption distance (Equation (11)) shows that the answer is not clear and is a very complicated problem to discuss. Therefore, we here treat this as a more realistic problem by investigating whether a sufficient ²⁶Al injection can be achieved while allowing some disk disruption. In this section, we describe how to solve this.

To solve this problem, we first define the two radii, assuming the distance from the SN to the disk: by rewriting Equation (11) with the disk radius as a function of distance, we obtain the outermost radius r_surv of the disk that remains unbroken at a given distance d from the SN (we refer to this as the “surviving radius”):

Also, by rewriting Equation (4) with the disk radius as a function of distance, we also obtain the radius r_req required to receive a sufficient amount of ²⁶Al for a given distance d from the SN (we call this the “required radius for ²⁶Al”):

Then, there are three conditions that must hold for the different radii, given by Equations (10), (12), and (13), as follows:

These mean the following: (i) the disk radius that gives the necessary solid angle for sufficient ²⁶Al injection must be smaller than the radius R estimated from the disk mass in Equation (10); (ii) the disk radius that survives the disruption by the SN flow should remain at least the size of the Neptune formation radius ; and (iii) the surviving disk radius r_surv is more than the required disk radius r_req that can intercept a sufficient amount of ²⁶Al.

4.1. Achievement Conditions Assuming SN Model

First, given the ²⁶Al yield in the current SN model, we investigate whether the disk injection scenario can be achieved. In this case, solving condition (iii) (i.e., r_req < r_surv) gives the achievable distance from the SN to the disk. For q ≠ 1, condition (iii) is satisfied in the following:

Figure 3 shows the achievable distance conditions by substituting the ²⁶Al yield of Table 2 into Equation (14). Also shown in the figure is the distance conditions for the disk radius to remain above 30 au, obtained from Equation (11). It is shown that for all current SN models, their injectable conditions lead to mostly destruction with only less than 30 au disk remaining. From this, we conclude that even if partial disk disruption is allowed, almost all SN explosion models cannot reproduce the hypothesis of direct ²⁶Al injection into an already formed protosolar disk.

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4.2. More General Achievement Conditions

Now, more generally, let us seek what conditions are required for sufficient ²⁶Al injection to be achieved while allowing partial disk disruption. We can obtain the minimum ²⁶Al yield required to satisfy all the conditions when the equations relating the distance and the ²⁶Al yield obtained from each of the conditions (i) through (iii) are coupled. Figure 4 shows the solution conditions obtained from the coupled equations.

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The first two conditions (i) and (ii) are satisfied as follows:

We can confirm that there exists a suitable distance d, independent of q (in the range 1 ≦ q ≦ 3/2), such that the above two conditions are satisfied. For q ≠ 1, condition (iii) is satisfied in the following:

For there to exist a distance d from the SN such that all three conditions hold, it is necessary that

Because the disk needs to be sufficiently massive to form the solar system, it is practically difficult to consider a smaller value for the disk mass here. We can read Equation (18) as a requirement for the mass of ²⁶Al contained in the large dust ejected from the SN.

The solution is different for the q = 1 case, but agrees with the extrapolated value. We attach below a solution for q = 1 as an example. By rewriting Equation (11) with the disk radius as a function of distance, we obtain the surviving radius r_surv as

The required radius r_req for ²⁶Al is the same with Equation (13). In the q = 1 case, only the condition (iii) for the SN is given as

The result is a continuous solution with q ≠ 1.

In summary, conditions (i), (ii), and (iii) are satisfied and the disk injection scenario is realized as

As can be seen from Table 2, there is no SN model that can achieve such a value, even with a large estimate of the dust fraction η_d. We can also see that this value is not achievable even within the yield diversity that comes from stellar evolutionary processes (see Figure 5 in Brinkman et al. 2023 and Brinkman et al. 2019, 2021). The uncertainty of the nuclear reaction rate also has the potential to increase the ²⁶Al yield, but it seems to be difficult to do so, as currently suggested by Woosley & Heger (2007) and Tur et al. (2010).

In this section, we discuss the impact on planet formation if an SN explosion satisfies the conditions we found in Section 4.

We consider the case where the shock from an SN is thermalized when it contacts the disk. We assume that the kinetic energy of the SN is thermalized throughout the disk. Here, the energy flux produced in the shock-heated region can be written approximately as follows:

Given that the shock-heated layer (Figure 5) is thin, the radiation from the layer is thought to be released as upward and downward radiation fluxes. Additionally, we assume a balance between the downward irradiation flux (heating) and the thermal radiation from the disk (cooling). The following equations are satisfied:

where blackbody radiation is assumed as the cooling term for the disk. From Equations (22), (23), and (24), the thermal equilibrium temperature of the disk is given as follows:

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Finally, we estimate the disk temperature by considering an SN explosion as a suitable source of ²⁶Al. When substituting Equation (4) into Equation (25), we can obtain

The internal energy due to this temperature is well below the gravitational bound energy (e_th < e_grav = GM_⋆/R), so the disk does not thermally evaporate. However, we find that this disk temperature is an eye-catching value for the evolution of the protosolar disk, and its impact on planet formation cannot be ignored. Therefore, we next discuss in Section 6.3 the effect of the temperature we found on planet formation.

6.1. Effect of Inclination Angle

As in Section 2, this study assumes that the flow from an SN hits the face-on disk. However, in reality, the disk can be inclined at an angle.

In terms of the injection efficiency, the inclination reduces the contact solid angle, so that Equation (2) is rewritten as

where θ is the angle between the disk axis and the direction from the solar system to the SN. Therefore, the resulting distance required for injection is

On the other hand, in terms of the disk disruption efficiency, the SN flow that the disk receives per unit area is smaller by , as in the injection case. That is, the stability criterion yields , and Equation (11) can be rewritten as

The conditions required for injection in Equation (28) become more strict with , whereas the conditions for avoiding disk disruption in Equation (29) are relaxed with . A simple comparison reveals that a larger inclination angle makes SN disk injection more difficult. In other words, the assumption of a face-on disk gives the most conservative condition for the injection scenario. This supports the overall conclusion that the disk injection scenario is unlikely to work.

6.2. Observational Constraints on Large Dust Masses of SNe

6.3. Effect of Disk Temperature on Planet Formation

6.4. Implications for Exoplanetary Systems

In this paper, we have assumed that ²⁶Al from an SN is injected into an already formed protosolar disk and investigated whether there are conditions under which the surviving disk radius can capture enough ²⁶Al for planet formation, while allowing for some disk disruption. We consider a diversity of disk masses and its structures, ²⁶Al yields of SN, and large dust mass fractions. In our model, given the disk mass and its structure, the disk radius that can accept ²⁶Al is obtained without any other assumptions. We also obtain the position at which the disk is disrupted by ram pressure, depending on the distance from the SN to the disk. Under each of these being determined self-consistently, we conclude that, as shown in Equations (18) and (21), an ejecta mass of ²⁶Al is required as follows:

This value is difficult to reproduce, given the diversity of the ejected ²⁶Al mass and large dust mass fractions from the SN, which are shown in Figure 3.

Furthermore, we find that even if the above conditions are achieved, the SN shock changes the disk temperature. And this temperature change is not negligible in the context of planetary formation.

Our finding places a strong constraint on the “disk injection scenario”—a scenario in which fresh ²⁶Al of the early solar system is injected from an SN into an already formed protosolar disk is quite challenging. We rather suggest that the fresh ²⁶Al of the early solar system should have been synthesized/injected in other ways.

We thank T. Suzuki, S. Inutsuka, K. Maeda, S. Arakawa, and E. Kokubo for fruitful discussions. This work has been supported by Japan Society for the Promotion of Science (JSPS) KAKENHI grants (18H05437, 20H00174, 20H01904, 20KK0080, 21H04514, 21K13964, 21K13976, 21K13983, 22KJ0528, 22H01290, 22H04571, and 22H05150).