Impacts of Dust Grains Accelerated by Supernovae on the Moon

There is evidence that nearby supernovae have resulted in the ⁶⁰Fe and other radionuclides detected in deep-ocean samples (Knie et al. 1999, 2004; Feige et al. 2012; Wallner et al. 2016), the lunar surface (Fimiani et al. 2016), cosmic rays (Ruderman 1974; Kachelrieß et al. 2015, 2018), and microfossils (Ludwig et al. 2016). Supernovae ejecta can accelerate dust to subrelativistic speeds ≲0.1c (Spitzer 1949; Bingham & Tsytovich 1999; Weiler 2003; Hoang et al. 2015). Studying dust accelerated by supernovae could elucidate the history of nearby supernovae (Knie et al. 2004; Thomas et al. 2016; Wallner et al. 2016; Melott et al. 2017) and constrain theoretical models of supernovae (Wesson et al. 2015; Bevan et al. 2017; De Looze et al. 2017; Kirchschlager et al. 2019). Searches on the lunar surface for signatures from interstellar objects have been suggested (Lingam & Loeb 2019; Siraj & Loeb 2020a). Dust grains accelerated by supernovae could appear as meteors in the Earth's atmosphere (Siraj & Loeb 2020b) and as chemical enrichments in subsurface layers on the Moon (Crawford 2017). Given that NASA's Artemis program plans to establish a sustainable base on the Moon by 2024,¹ it is now particularly timely to explore the detection signatures of interstellar dust on its regolith and rocks. In this Letter, we explore the flux of dust grains accelerated by supernovae at the lunar surface and the expected rate of the resulting impact tracks in lunar materials.

Our discussion is structured as follows. In Section 2, we consider the survival of dust grains accelerated by supernovae as they travel to the Moon. In Section 3, we investigate the effect of radiation pressure from supernovae on the acceleration of dust grains. In Section 4, we explore the lunar impact rate of such dust grains. In Section 5, we estimate the depth to which these dust grains penetrate lunar materials. In Section 6, we compute the expected track densities in lunar rocks. Finally, in Section 7 we summarize key predictions and implications of our model.

Coulomb explosions from charge accumulation hinder the distance that supernova dust grains can travel in the interstellar medium (ISM). The surface potential ϕ_max above which the grain will be disrupted by Coulomb explosions, assuming a typical tensile strength of ∼10¹⁰ dyn cm⁻², is (Hoang et al. 2015)

$$\begin{eqnarray}&&e{\phi }_{\max }\simeq 0.1\,\mathrm{keV}\,\left(\displaystyle \frac{r}{0.01\,\mu {\rm{m}}}\right),\end{eqnarray} \tag{ 1 }$$

where e is the electron charge. There is also a maximum surface potential U_max,H due to the fact that a large potential can halt electrons from overcoming the surface potential and therefore escaping the grain (Hoang & Loeb 2017),

$$\begin{eqnarray}&&{{eU}}_{\max ,{\rm{H}}}=2{m}_{e}{v}^{2}\simeq 0.1\,\mathrm{keV}\,{\left(\displaystyle \frac{v}{0.01c}\right)}^{2},\end{eqnarray} \tag{ 2 }$$

where m_e is the electron mass. Combining Equations (1) and (2)

$$\begin{eqnarray}&&\left(\displaystyle \frac{r}{0.01\,\mu {\rm{m}}}\right)\gtrsim {\left(\displaystyle \frac{v}{0.01c}\right)}^{2}\end{eqnarray} \tag{ 3 }$$

yields a minimum grain radius of r ∼ 0.01 μm for survival at a speed of v ∼ 0.01c and implies that grains above this size will not undergo Coulomb explosions while traveling through the ISM.

Significant slow-down for dust grains traveling through the ISM occurs at distance where the total momentum transferred by particles in the ISM to the object is comparable to the initial momentum of the object. However, at the speed of ∼0.01c, the stopping distance of a proton in silicate material is comparable to the size of the grain for a radius r ∼ 0.4 μm (Figure 2 of Hoang et al. 2017), so the stopping distance through the ISM for r ≲ 0.4 μm and v ∼ 0.01c is

$$\begin{eqnarray}&&{d}_{\mathrm{ISM}}\sim 250\,\mathrm{pc}\,\left(\displaystyle \frac{\rho }{3\,{\rm{g}}\,{\mathrm{cm}}^{-3}}\right){\left(\displaystyle \frac{{n}_{p}}{0.1{\mathrm{cm}}^{-3}}\right)}^{-1},\end{eqnarray} \tag{ 4 }$$

given the local proton density of the ISM is n_p ∼ 0.1 cm⁻³ (Frisch et al. 2011). At faster speeds, d_ISM ≳ 250 pc, but this contribution does not significantly enhance the flux of dust grains from supernovae, so it is not studied here.

Additionally, thermal sublimation in both the ISM and the solar radiation field and Coulomb explosions in the solar wind do not limit further the sizes and speeds of grains considered here (Hoang et al. 2015). We note that interstellar dust particles of similar sizes considered here that travel at typical speeds are excluded from the inner solar system due to heliospheric and radiation pressure effects (see Sterken et al. 2019 for a review on interstellar dust in the solar system).

Radiation pressure from a supernova accelerates dust grains to subrelativistic speeds (Spitzer 1949; Bingham & Tsytovich 1999; Weiler 2003; Hoang et al. 2015). Assuming a typical bolometric luminosity of ∼10⁸ L_⊙, Equation (12) in Hoang et al. (2015) yields

$$\begin{eqnarray}&&v\sim 0.01c\,{\left(\displaystyle \frac{d}{5\times {10}^{18}\mathrm{cm}}\right)}^{-1/2}{\left(\displaystyle \frac{r}{0.01\mu {\rm{m}}}\right)}^{-1/2},\end{eqnarray} \tag{ 5 }$$

where r is the grain radius, d is the initial distance, and v is the grain speed. Since dust grains are sublimated within a distance of d ∼ 10¹⁶ cm, Equations (3) and (5) jointly constrain the range of sizes that this method is applicable to grains of size ∼0.01–0.07 μm, corresponding to speeds of ∼0.01–0.03c.

Adopting the canonical dust mass fraction in the ISM of ∼0.01, ∼2 × 10⁻⁴ M_⊙ of preexisting dust around each supernova exists within a distance of 5 × 10¹⁸ cm, interior to which r ∼ 0.01 μm dust grains can be accelerated to ∼0.01c. The fraction of ISM dust with size r ≳ 0.01 μm is ∼0.5 (Mathis et al. 1977), so ∼10⁻⁴ M_⊙ of dust is accelerated per supernova, corresponding to ∼10⁴⁴ grains. Additionally, the size distribution of ISM dust grains follows a power law with exponent −3.5 (Mathis et al. 1977).

Dust grains produced in supernovae travel at the ejecta speed of ∼0.01c because the drag timescale due to gas (Sarangi & Cherchneff 2015) is short relative to the acceleration timescale by radiation pressure (Hoang et al. 2017). However, the abundance and size distribution of supernova ejecta dust are model dependent, unlike preexisting dust grains in the ISM. The ejecta model for typical Type II-P supernovae, presented in Figure 4 of Sarangi & Cherchneff (2015), peaks for grain radii of r ∼ 0.02 μm, a size bin at which ∼5 × 10⁻³ M_⊙ of dust grains is produced per supernova, which also corresponds to ∼10⁴⁴ grains. Further modeling of the ejecta of core-collapse supernovae in general will reveal the true abundance of subrelativistic dust grains produced relative to preexisting ISM dust that is accelerated by the radiation pressure of the supernova. Additionally, ejecta-formed dust likely undergo different dynamics relative to accelerated ISM dust grains (Fry et al. 2018; Fields et al. 2019).

In addition, massive stars such as luminous blue variables can reach stellar wind mass-loss rates of ∼10⁻⁵ M_⊙ yr⁻¹ (Andrews et al. 2011), which could lead to ∼10⁻⁴ M_⊙ of preexisting dust that would be accelerated to ≳0.01c due to radiation pressure from the supernova. However, luminous blue variables are a small fraction of all supernovae, so we do not consider them here.

The local timescale between core-collapse supernovae is estimated to be τ_SN ∼ 2 Myr within a distance d_SN ∼ 100 pc (Knie et al. 2004; Thomas et al. 2016; Wallner et al. 2016; Melott et al. 2017), implying a timescale of τ_SN ∼ 0.1 Myr within a distance d_SN ∼ 250 pc.

SNe II are an order of magnitude more common than SNe Ib/c, and so we focus our discussion on them (Guetta & Della Valle 2007).

The impact area density on the lunar surface due to one supernova at ∼250 pc of r ∼ 0.01 μm dust grains traveling at ∼0.01c accelerated by core-collapse supernovae is therefore

$$\begin{eqnarray}&&{\sigma }_{\mathrm{SN}}\sim {10}^{3}\,{\mathrm{cm}}^{-2}\,{\left(\displaystyle \frac{{d}_{\mathrm{SN}}}{250\mathrm{pc}}\right)}^{-2}\left(\displaystyle \frac{{\tau }_{\mathrm{SN}}}{0.1\,\mathrm{Myr}}\right).\end{eqnarray} \tag{ 6 }$$

Lunar surface material is composed primarily of silicates (Keller & McKay 1997; Prettyman et al. 2006; Melosh 2007; Greenhagen et al. 2010), and so we adopt the properties of quartz for our impact depth penetration analysis. Since the impacts occur at subrelativistic speeds, we consider the dust grains as collections of constituent nuclei (Si, as a fiducial example). The stopping power dE/dx as a function of Si nucleus speed for impacts in quartz is adopted from Figure 2 of Hoang et al. (2017), allowing the penetration depth l of a single ion to be computed.

The sideways shock moves at a speed ≲10⁻² of the dust speed, producing a "track" is in which the depth greatly exceeds the width, which is comparable to the size of the dust grain. Only a fraction of the total energy is shared with the target per penetration depth of a single ion. This energy fraction is found by lowering the sideways speed v_shock per penetration depth x such that the timescale to traverse the dust grain radius r sideways, r/v_shock, equals the timescale to traverse the penetration depth l at the dust speed, l/v. This means that v_shock = (rv/l), implying an energy deposition fraction,  = r/l. Since we deposit the fraction of the dust energy per penetration depth x, the total penetration depth is D ∼ l²/r.

With a grain radius of $r\sim 0.01\,\mu {\rm{m}}$ and a penetration depth per ion of l ∼ 1 μm the total penetration depth is D ∼ 0.1 mm. The penetration depth for a constant size scales as D ∝ v⁴, since dE/dx is constant to order unity over the speeds considered; however, due to the size–speed constraint of Equation (3) and the fact that D ∝ r⁻¹, the penetration depth actually scales as dD/dv ∝ v². The abundance of dust grains, N, scales as the volume (d³) multiplied by the ISM dust grain size distribution (r^−3.5). Since Equation (5) yields rd ∝ v⁻² and Equation (3) gives r ∝ v², in total the abundance scales as dN/dv ∝ v⁻⁵. As a function of depth, the abundance scales as dN/dD ∝ v⁻⁷, which, using Equation (5), yields a total dependence of the dust grain flux at the lunar surface of dN/dD ∝ D^−7/2, as indicated in Figure 1.

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The lunar regolith is steadily overturned by micrometeroid impacts (see Grün et al. 2011 for a review of the lunar dust environment). The approximate depth D to which the lunar regolith is overturned as a function of timescale τ, given in Figure 9 of Costello et al. (2018), is

$$\begin{eqnarray}&&D\sim 10\,\mathrm{cm}\,{\left(\displaystyle \frac{\tau }{0.1\mathrm{Myr}}\right)}^{1/4},\end{eqnarray} \tag{ 7 }$$

which implies that tracks resulting from the penetration of dust grains accelerated by nearby supernovae into the lunar regolith cannot be discovered, given that the overturned depth between supernovae within ∼250 pc is 1000 times larger than the track depth for $r\sim 0.01\,\mu {\rm{m}}$ dust grains at ∼0.01c.

Lunar rocks, on the other hand, are eroded by micrometeroids at a rate of ∼0.1–1 mm Myr⁻¹ (Comstock 1972; Neukum 1973; Fechtig et al. 1977; Nishiizumi et al. 1995; Eugster 2003; Eugster et al. 2006). We conservatively adopt a rate of ∼1 mm Myr⁻¹. They have surface lifetimes of 1–50 Myr (Walker 1980; Heiken et al. 1991). Impact tracks can be discovered in lunar rocks at rates shown in Figure 2.

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Micrometeoroid impacts result in craters with depths comparable to their widths, and cosmic rays develop tracks with widths of ∼1 pm, so subrelativistic dust grain tracks are uniquely identifiable by their characteristic ∼0.01 μm widths and depths that greatly exceed what would be expected from a micrometeoroid impact.

We demonstrated that tracks resulting from r ∼ 0.01–0.07 μm dust grains accelerated by supernovae to speeds of r ∼ 0.01–0.03c can be discovered in the lunar rocks. Studies of lunar rocks could shed light on the history of supernovae within the past ∼1 Myr and within ∼250 pc, with the potential to reveal the timings, luminosities, and directions of recent supernovae. The expected density of tracks is ∼10³ cm⁻² for depths of ∼0.1 mm and widths of ∼0.01 μm and ∼1 cm⁻² for depths of ∼0.7 mm and widths of ∼0.07 μm. However, since lunar rocks may be covered by regolith material with typical grain sizes of ∼100 μm (Heiken et al. 1991), impacts of the subrelativistic grains considered here could catastrophically disrupt such grains instead of forming tracks within a rock, potentially reducing the actual density of tracks observed on lunar rocks.

The six Apollo missions brought to Earth 2200 lunar rocks² that could be searched for these tracks. While cosmic-ray tracks have been discovered in lunar samples (Drozd et al. 1974; Crozaz 1980; Bhandari 1981), these tracks would be differentiable by their widths, which would be at least an order of magnitude larger. Extraterrestrial artifacts, such as microscopic probes akin to Breakthrough Starshot,³ could also form such tracks.

We thank Brian J. Fry and Thiem Hoang for helpful comments. This work was supported in part by a grant from the Breakthrough Prize Foundation.