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

The early solar system contained a short-lived radionuclide, 26 Al (its half-life time t 1 / 2 = 0 . 7 Myr). The decay energy 26 Al is thought to have controlled the thermal evolution of planetesimals and, possibly, the water contents of planets. Many hypotheses have been proposed for the origin of 26 Al in the solar system. One of the possible hypotheses is the ‘disk injection scenario’; when the protoplanetary disk of the solar system had already formed, a nearby ( < 1 pc) supernova injected radioactive material directly into the disk. Such a 26 Al injection hypothesis has been tested so far with limited setups for disk structure and supernova distance, and treated disk disruption and 26 Al injection separately. Here, we revisit this problem to investigate whether there are self-consistent conditions under which the surviving disk radius can receive enough 26 Al which can account for the abundance in the early solar system. We also consider a range of disk mass and structure, 26 Al yields from supernova, and a large dust mass fraction η d . We find that 26 Al yields of supernova are required as ≳ 2 . 1 × 10 − 3 M ⊙ ( η d / 0 . 2) − 1 , challenging to achieve with known possible 26 Al ejection and dust mass fraction ranges. Furthermore, we find that even if the above conditions are met, the supernova flow changes the disk temperature, which may not be consistent with the solar-system record. Our results place a strong constraint on the disk injection scenario. Rather, we suggest that the fresh 26 Al of the early solar system must have been synthesized/injected in other ways.


INTRODUCTION
Short-lived radionuclides (SLRs) with half-lives of tens of Myr or less drive the thermal evolution of planetesimals (∼km-sized bodies in planetary systems) and, consequently, influence the formation of planetary systems (e.g., Desch et al. 2022).Among SLRs, 26 Al (its halflife 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, 26 Mg) provided direct evidence and abundance estimate of 26 Al 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 accretion and resulting 26 Al content of each body, the radioactive decay heat led to various physicochemical evolution 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 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 environ-ment of the solar system.The discovery of undecayed 26 Al (hereafter referred as 'fresh 26 Al') in the protosolar disk at the time of CAIs formation (Lee et al. 1976) led to three origin theories: inheritance, irradiation, and injection.The inheritance hypothesis suggests 26 Al 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: molecular clouds forming the solar system take over 1 Myr, during which 26 Al would decay, conflicting with meteorite data.The irradiation hypothesis proposes that 26 Al is synthesized by irradiation of accelerated particles, such as from solar flares, during solar system formation.This model explains 10 Be and also some 26 Al 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.1 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 10 Be; 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. 2007Ouellette et al. , 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, efficient capture of SN ejecta from a d ≈ 0.3 pc explosion would explain the estimated 26 Al amount from the meteorite.Hydrodynamical simulations by Ouellette et al. (2007) (and also Close & Pittard (2017), Portegies Zwart et al. (2018), and Portegies Zwart (2019)) show that the disk resists total destruction by SN impacts, but the contribution of 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 not injected into the disk, large grains (≳ 1 µm) are efficiently injected with ∼ 100% efficiency.If sufficient 26 Al is contained in large dust grains condensed from supernova ejecta, the protosolar disk would receive enough 26 Al to account for the meteorite ratio.Thus the injection of 26 Al 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 26 Al, accepting a certain amount of disk disruption.The previous analytical studies have treated the structure in one-zone model (Chevalier 2000;Ouellette et al. 2005), and previous numerical studies covered only a limited number of disk models (Ouellette et al. 2007(Ouellette et al. , 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 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 and that the explosions remain 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 ⊙ supernova model, at least as frequently cited in previous discussions of early solar system composition (Ouellette et al. 2005(Ouellette et al. , 2007)), inappropriate as a typical value.In addition, over the past decade, there has been much discussion of the indeterminacy of the 26 Al yield based on a more modern understanding of stellar evolution and supernova explosions (e.g., Woosley & Heger 2007;Tur et al. 2010;Brinkman et al. 2019Brinkman et al. , 2021Brinkman et al. , 2023)).Observations of dust mass abundance ratios, also important in this study, are now available for supernovae 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 26 Al from a SN, is injected into an already formed protosolar disk (Figure 1) and investigate the conditions under which the surviving disk radius can capture a sufficient amount of 26 Al for planet formation while allowing some disruption of the disk.We consider a diversity of disk mass and its structure, 26 Al yields of SN, and large-dust mass fraction of 26 Al.We found that for all supernova 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 26 Al injection into an already formed protosolar disk cannot be reproduced in almost all supernova explosion models.
The paper is organized as follows: Section 2 details the system setup and 26 Al injection conditions.Section 3 examines conditions for disk disruption by supernova flows.Section 4 addresses disk structure and mass, determining conditions for sufficient 26 Al capture.Section 5 explores supernova impacts on disk temperature and planet formation.Section 6 discusses limitations of our study.Finally, Section 7 summarizes our findings.

BASIC PICTURE
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 supernova only after the protosolar disk is formed.(2) All of the 26 Al in the protosolar disk is supplied by a nearby supernova.(3) For simplicity, the protosolar disk is in face-on contact with the supernova ejecta.
As background for each assumption, (1) at least one supernova, which would directly inject the SLRs into the early solar system, is expected to occur in the Sun's birth cluster during the duration of CAI formation (e.g., Arakawa & Kokubo 2023).And (2) since the timescale for disk formation is sufficiently long compared to the lifetime of 26 Al , one would expect that all of the 26 Al originally contained in the disk, has decayed and can be neglected.Finally, (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 detail).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 26 Al injected into the protosolar disk from 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 M SN,tot is as follows where d is the distance from the protosolar disk to the supernova, 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 26 Al injection via gas and small dust grains are negligible, we can write the injected 26 Al mass as, where M SN ( 26 Al) tot is the ejected mass of 26 Al from the supernova, with a adopted value of about ∼ 1.3 × 10 −4 M ⊙ per event (e.g., Timmes et al. 1995;Limongi & Chieffi 2006;Sukhbold et al. 2016;Limongi & Chieffi 2018, and see Table 2).There are still many uncertainties regarding the nucleosynthesis of 26 Al in supernova, and we listed the theoretical 26 Al yields of several different supernova models in Table 2. Since the typical progenitor mass of recently observed supernovae 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 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 supernova to its injection into the disk is , so that the radioactive decay of 26 Al during the travel is negligible.
Next, from meteorite measurements, we know shortlived nuclide/stable isotope abundance ratios at the time of CAI formation as N r /N s = 26 Al/ 27 Al ≈ 5.23 × 10 −5 (e.g., Jacobsen et al. 2008), which are extrapolated to the time of the formation of CAIs.Using this ratio and the mass fraction of 27 Al in the solar abundance (Asplund et al. 2009), the total amount of undecayed 26 Al in the protosolar disk M disk ( 26 Al) 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 CAIs 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 ( 26 Al) = M inje ( 26 Al), we obtain the following conditions for the distance of the supernova: where the disk radius R only determines the solid angle at which 26 Al can be received from SN, once 100 au is adopted.Similarly, the disk mass M disk here normalizes the total amount of 26 Al required, and we once adopted 0.017M ⊙ .
The injectable distance obtained from Eq. ( 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, the injection Standard-mass progenitor models supported by recent observations.

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 wether the ram pressure of the dense SN ejecta (n ∼ 10 3 cm −3 (d/1 pc) −3 ) 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 supernova at distances ≳ O(0.1) pc 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 completeness of the discussion.We consider that the disk is disrupted if the ram pressure of the supernova 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 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 value 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 will be redefined in Section 3.2.
Next, we calculate the ram pressure that the disk receives from the freely expanding supernova ejecta.For simplicity, we use the thin-shell approximation for the ejecta (e.g., Laumbach & Probstein 1969;Koo & Mc-Kee 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 SN. ∆d is the thickness of the shell, which is ∆d/d = 1/12 in the case of the thin shell for an ideal gas (Laumbach & Probstein 1969).
Here, the stability criterion P grav > P SN for the disk region outside the radius r is given by, The disk disruption distance obtained from Eq. ( 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 = 10 15 cm ≈ 66 au and a uniform density distribution Σ disk = M disk /(πR 2 ) ≈ 6.4 g cm −2 , and also used a uniform density for supernova (∆d → d), so its gravitational force is P grav ≈ 10 −3 dyn cm −2 and the stable criteria becomes d ≳ 0.25 pc.

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 proto-solar 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 Eq.( 10).Also, when q = 3/2, R ≈ 55 au is obtained.Then, the stability criterion of Eq.( 7) can be rewritten as Here, a simple comparison of the 26 Al injection distance (Eq.( 4)) and the disk disruption distance (Eq.( 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 26 Al 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 with assuming the distance from the supernova to the disk: By rewriting Eq. ( 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 Eq. ( 4) with the disk radius as a function of distance, we also obtain the radius r req required to receive a sufficient amount of 26 Al for a given distance d from SN (call this the 'required radius for 26 Al'): Then, there are three conditions that must hold for the different radii given by Eqs.( 10), (12), and (13) as 7 follows, (i) The meaning are following: (i) the disk radius that gives the necessary solid angle for sufficient 26 Al injection must be smaller than the radius R estimated from the disk mass in Eq.( 10), (ii) the disk radius that survives the disruption by the supernova flow should remain at least to the Neptune formation radius R min = 30 au, and (iii) the surviving disk radius r surv is more than the required disk radius r req that can intercept a sufficient amount of 26 Al.

Achievement Conditions Assuming Supernova Model
First, given the 26 Al yield in the current supernova model, we investigate whether the disk injection scenario can be achieved.In this case, solving the condition (iii) (i.e., r req < r surv ) gives the achievable distance from the supernova to the disk.For q ̸ = 1, the condition (iii) is satisfied in the following: Figure 3 shows the achievable distance conditions by substituting the 26 Al yield of Table 2 into Eq.( 14).Also shown in the figure is the distance conditions for the disk radius to remain above 30 au, obtained from Eq. ( 11).It is shown that for all current supernova 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 supernova explosion models cannot reproduce the hypothesis of direct 26 Al injection into an already formed protosolar disk.

More General Achievement Conditions
Now, more generally, let us seek what conditions are required for sufficient 26 Al injection to be achieved while allowing partial disk disruption.We can obtain the minimum 26 Al yield required to satisfy all the conditions when the equations relating the distance and 26 Al yield obtained from each of the conditions (i) through (iii) are coupled.Figure 4 shows the solution conditions obtained from the coupled equations.
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, the condition (iii) is satisfied in the following: For there to exist a distance d from the supernova 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 Eq. ( 18) as a requirement for the mass of 26 Al contained in the large dust ejected from 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 Eq. ( 11) with the disk radius as a function of distance, we obtain the surviving radius r surv as r surv ≈ 30 au d 0.21 pc The required radius r req for 26 Al is the same with Eq.( 13).In the q = 1 case, only the condition (iii) to Al can be injected into the surviving disk while allowing partial disk disruption in the case that the 26 Al yield for each supernova model is adopted (with the large dust mass fraction η d = 20%).Error bars correspond to cases where the large dust mass fraction is from 5% to 100%.For all supernova models with progenitor masses of 11 − 40M⊙, we show that the disk radius can be disrupted to less than 30 au if a sufficient amount of injection is achieved.the supernova is given as The result is a continuous solution with q ̸ = 1.
In summary, the condition (i), (ii), and (iii) are satisfied and the disk injection scenario is realized as As can be seen from Table 2, there is no supernova 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. 2023and Brinkman et al. 2019, 2021).The uncertainty of the nuclear reaction rate also has the potential to increase the 26 Al yield, but it seems to be difficult to do so as far as it is currently suggested by Woosley & Heger (2007) and Tur et al. (2010).

TEMPERATURE CONDITIONS FOR DISK
In this section, we discuss the impact on planet formation if a supernova explosion satisfies the conditions we found in Section 4.
We consider the case where the shock from a supernova is thermalized when it contacts the disk.We assume that the kinetic energy of the supernova is thermalized throughout the disk.Here, the energy flux produced in the shock-heated region can be written approx-Supernova Energy Flux: ℱ SN shock-heated layer Schematic picture of a supernova flow being thermalized at the top of the disk, resulting in heating of the disk.
imately 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 thermal radiation from the disk (cooling).The following equations are satisfied: where black-body radiation is assumed as the cooling term for the disk.From Eqs.( 22), ( 23) and ( 24), the thermal equilibrium temperature of the disk is given as follows Finally, we estimate the disk temperature by considering a supernova explosion as a suitable source of 26 Al.When substituting Eq.(4) into Eq.(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 eyecatching 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.

Effect of inclination angle
As in section 2, this study assumes that the flow from a supernova hits the face-on disk.However, in reality, the disk can be inclined at an angle.
In terms of injection efficiency, the inclination reduces the contact solid angle, so that Eq (2) is rewritten as where θ is the angle between the disk axis and the direction from the solar system to the supernova.Therefore, the resulting distance required for injection is On the other hand, in terms of the disk disruption efficiency, the supernova flow that the disk receives per unit area is smaller by cos θ as well as the injection case.That is, the stability criterion yields P grav > P SN • cos θ, and Eq. ( 11) can be rewritten as The conditions required for injection in Eq. ( 28) become more strict with cos 1/2 θ, whereas the conditions for avoiding disk disruption in Eq. ( 29) are relaxed with cos 1/3 θ.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.

Observational constraints on large dust masses of supernovae
Dust formation in supernova ejecta, identifiable by infrared (IR) excess, has been confirmed in SN 1987A (Matsuura et al. 2011;Wesson et al. 2015).Considering that SN flows take decades to reach protosolar disks, SN 1987A, about 40 years after the explosion, serves as an excellent case to study dust formation in supernovae relevant to our research.Matsuura et al. (2011) observed far-infrared and submillimeter emission from SN 1987A.The shapes of the spectral energy distributions (SEDs) were consistent with continuous dust emission, indicating the presence of 0.4 − 0.7M ⊙ of dust.This was further supported by extrapolating the synchrotron flux measured at shorter wavelengths up to 8014 days after the explosion to farinfrared wavelengths, which showed an order of magnitude lower emission than observed, indicating dominant dust emission.Given the total metal mass of SN 1987A of ∼ 3M ⊙ , a dust fraction η d ∼ 20% seems reasonable.
Further research by Wesson et al. (2015) over 24 years, starting 615 days after the explosion, found 0.6 − 0.8M ⊙ of dust, consisting of grains larger than ≳ 2µm, exists in SN 1987A, suggesting that most of the small dust was formed early and grew through accretion and aggregation.These results underscore the importance of considering supernovae at appropriate post-explosion timescales for understanding dust formation rates relevant to solar system formation.
More recently, mid-infrared imaging of SN 1980K over 40 years post-explosion by Zsíros et al. (2023) reported its dust mass M d ≈ 0.02M ⊙ .However, the SED analysis indicates that a much greater amount of dust (∼ 0.24 − 0.58M ⊙ ), suggesting a dust mass fraction of η d ≲ 0.20.
To summarize, for a supernova within the timescale of our study, a dust mass fraction η d of up to 0.20 seems realistic.This assumes spherical symmetry and nonextreme conditions, but higher η d values are possible in clumpy structures.To understand further details, comprehensive studies are needed, including simulations of multidimensional supernova explosions with dust forma-tion and observations of dust emission in supernovae several decades after the explosion.

Effect of disk temperature on planet formation
Heating of a protoplanetary disk due to a nearby supernova (Section 5) may cause sublimation and/or melting of materials in the disk and, consequently, influence planet formation.Moreover, such heating may not be consistent with the early solar-system record, as we explain below.
Organic matter on dust grains in the protosolar disk can be irreversibly lost due to the heating.While amorphous carbon component survives at temperatures up to ∼ 1000 K, organic materials start to be removed due to pyrolysis and evaporation at ≃ 250-400 K (Gail & Trieloff 2017).Loss of organic mantles significantly reduces the stickiness of dust grains and thus may limit rocky planet(esimal) formation (Homma et al. 2019).Although the origins of organic matter in carbonaceous chondrites and in returned samples of primitive asteroids in the solar system are poorly understood, some organic matter is considered to trace back to the interstellar medium (Glavin et al. 2018).Thus, the presence of these pristine materials may put a limit on the heating of the early solar system due to supernova exposure.
Sublimation of highly volatile ices (here defined as materials whose sublimation temperature is lower than water ice) may impose a much lower limit on the temperature experienced during the supernova heating.For instance, major volatile molecules, H 2 O and CO sublimates at ≃ 150 K and ≃ 20 K, respectively (Okuzumi et al. 2016).Sublimation is a reversible process, and thus, re-condensation after the cooling of the disk may recover its original state.However, sublimation and recondensation may turn pristine amorphous ice into crystalline ice; this may not be consistent with the property of cometary ice, which is widely thought to be amorphous (e.g., Rubin et al. 2020;Prialnik & Jewitt 2022).Moreover, isotopic compositions of highly volatile elements in a comet 67P/Churyumov-Gerasimenko measured with the Rosetta spacecraft suggest that cometary ice has never sublimated and been mixed with the inner solar-system materials (Marty et al. 2017;Rubin et al. 2020).Thus, depending on the re-condensation processes and the efficiency of mixing in the protosolar disk, the heating due to a nearby supernova may be constrained or ruled out.
Lastly, we note that heating to a much higher temperature (≳ 1300 K) leads to sublimation and/or melting of silicates.Such high-temperature heating events formed silicate grains, including CAIs and chondrules from their precursor aggregates in our solar system (e.g., Krot et al. 11 2009), which likely changed their stickiness and aerodynamic properties and influenced planet formation; for instance, efficient accretion of chondrules onto large planetesimals is proposed to have assisted forming planetary embryos (Johansen et al. 2015).
To summarize, the supernova heating may cause irreversible changes in protoplanetary-disk materials and thus influence planet formation processes.Heating of the protosolar disk associated with implantation of 26 Al from a supernova (Section 5) would have caused sublimation of highly volatile ices and, possibly, processing of organic matter, which may contradict the solar system record.Thus, we suggest that future studies need to perform a more detailed analysis of the heating and cooling processes of disks during supernova exposure and physicochemical impacts on the disk materials.

Implications for exoplanetary systems
Provided that injection into protoplanetary disks is a major source of SLRs in planetary systems, our results indicate that systems like our solar system, whose planet(esimal) formation and evolution are highly influenced by the radioactive decay energy of 26 Al, may be rare.As shown in Section 4, a parameter space that results in injection of 26 Al comparable to the amount found in the solar system without disrupting the disk, is fairly limited.In other words, a typical outcome of supernova exposure is either limited injection of 26 Al , or disruption of the protoplanetary disk, which implies that existing exoplanetary systems are formed without a significant (solar-system-like) amount of 26 Al.It is not fully understood and beyond the scope of this study how the deficit of 26 Al in the protoplanetary disk changes the architecture of resulting planetary systems, but a proposed outcome (Lichtenberg et al. 2019a) is limited loss of water from planetesimals due to insufficient 26 Al heat budget and the dominant formation of water-rich planets.Thus, our study ultimately suggests that Earth-like, water-poor planets may be minor in our universe.

SUMMARY
In this paper, we have assumed that 26 Al from a supernova is injected into an already formed protosolar disk and investigated whether there are conditions under which the surviving disk radius can capture enough amount of 26 Al for planet formation while allowing for some disk distruption.We consider a diversity of disk mass and its structure, 26 Al yields of SN, and largedust mass fraction.In our model, given the disk mass and its structure, the disk radius that can accept 26 Al 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 supernova to the disk.Under each of these being determined selfconsistently, we conclude that, as shown in Eqs. ( 18) and ( 21), an ejecta mass of 26 Al is required as follows: . This value is difficult to reproduce given the diversity of ejected 26 Al mass, and large dust mass fractions from the supernovae, 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-26 Al of the early solar system is injected from a supernova into an already formed protosolar disk is quite challenging.We rather suggest that the fresh-26 Al of the early solar system, should have been synthesized/injected in other ways.

Figure 1 .
Figure 1.Schematic picture of supernova dust containing 26 Al being injected directly into an existing protosolar disk a few tenths of a parsec away.

Figure 2 .
Figure 2. Schematic picture considering whether the disk disruption is caused by ram pressure from a supernova flow.

Figure 3 .
Figure 3. Summary of results for the distance required for 26 Al injection for each supernova model and disk disruption condition.Results are shown assuming disk mass M disk = 0.017M⊙ and surface density structure Σ disk ∝ r −3/2 .The orange dotted line indicates the nearest distance where the disk radius remains undisrupted up to r = 30 au.The points indicate the distance at which a sufficient amount of 26Al can be injected into the surviving disk while allowing partial disk disruption in the case that the 26 Al yield for each supernova model is adopted (with the large dust mass fraction η d = 20%).Error bars correspond to cases where the large dust mass fraction is from 5% to 100%.For all supernova models with progenitor masses of 11 − 40M⊙, we show that the disk radius can be disrupted to less than 30 au if a sufficient amount of injection is achieved.

Figure 4 .
Figure 4. Conditions (i) through (iii) are illustrated on the plane of distance d and 26 Al yield.Each solid line corresponds to a different condition, in the case of the disk mass M disk = 0.017M⊙, surface density structure Σ disk ∝ r −3/2 and large-dust mass fraction η d = 20%.

Table 1 .
Summary of some important variables index Mass of the supernova ejecta in contact with the protosolar disk (referred to as the 'contact mass'), assuming an isotropic explosion from a distance of d.
−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 text.

Table 2 .
Summary of theoretical 26 Al yield of 5 different supernova models.All values in the table are in M⊙