ON THE PROVENANCE OF PLUTO'S NITROGEN (N₂)

Using observations of current KBO populations (Schlichting et al. 2012), typical KBO impact velocities onto Pluto of ∼1–2 km s⁻¹ (e.g., Zahnle et al. 2003; Bierhaus & Dones 2015), and Pluto's cross section, one of us, Stern et al. (2015a), estimated ∼14,000 comets 1 km in diameter or larger would impact Pluto over 4 billion years (comparable to an earlier estimate by Durda & Stern 2000). The rates derived in Zahnle et al. (2003) yield a similar order of magnitude estimate of ∼11,200 impactors larger than 1 km in diameter over 4 billion years. Both Bierhaus & Dones (2015) and Greenstreet et al. (2015) conducted detailed analysis of KBO sub-populations and their estimates are 1840–5600 impactors >1 km in diameter (nominal to maximum case), and 1400–8440, respectively. These estimates are based on what is known about the present-day impactor population and are taken to be valid for ∼the past 4 billion years (starting after any Late Heavy Bombardment period and Charon's formation). Greenstreet et al. (2015) estimate a factor of a few is necessary to account for collisional and dynamical erosion of the Kuiper Belt over the past 4 billion years (e.g., Stern & Colwell 1997; Weissman & Levison 1997; Farinella et al. 2000). The impact flux model given by Bierhaus & Dones (2015) assumes the decay factor was outweighed by other uncertainties, and thus it is not considered in the model. Taking into account the maximum estimate of their model, it is possible that the uncertainties in the impactor flux could be up to an order of magnitude.

Adopting the Bierhaus & Dones (2015) estimate of impactor flux at Pluto (their Table 8, nominal case), a cumulative rate of a given size impactor and larger per year can be written as

$$\begin{eqnarray}&&\mathrm{rate}(\gt d)=4.6\times {10}^{-7}{d}^{-1.837},\end{eqnarray} \tag{ 1 }$$

where impactor diameter (d) is in km. The corresponding characteristic timescale between impacts of a certain size and greater, τ_impact, is $1/$rate(>d). To estimate the number of impactors as a function of size, we differentiate 1-rate(>d) to obtain

$$\begin{eqnarray}&&{spdf}(d)=8.4\times {10}^{-7}{d}^{-2.837}.\end{eqnarray} \tag{ 2 }$$

Integrating this function between 1 m impactors and a hypothetical largest impact of ∼60 km, we find that over 4 billion years there have been ∼6 × 10⁸ total impactors on Pluto, the majority of which would be in the smallest size range. Equation (2) further allows us to predict 1.2 × 10⁵ impactors between 100 m and 1 km in diameter over 4 billion years, with only ∼1810 between 1 and 10 km, 26 between 10 and 60 km, and 1 impactor that is 60 km or larger. Because we do not know the actual size of the largest impactor, for the following calculations we will assume the largest impactor in the past 4 billion years was 100 km as a reasonable upper limit. Multiplying Equation (2) by the volume of a spherical impactor $(4/3)\pi {(d\ast 1000/2)}^{3}$ and an estimate of overall comet density (1000 kg m⁻³), and integrating from 1 m to 100 km impactors, yields a total mass of 3 × 10²⁰ g delivered to Pluto. An independent calculation of total mass delivered by comets to Pluto by L. Dones (2015, private communication) found 3.7 × 10²⁰ g, in good agreement with the estimate presented here.

Stern et al. (2015a) estimate an N₂ mass of 5 × 10^10–11 g in a 1 km comet with a 50:50 ratio of volatiles to refractories, an overall density of ∼1000 kg m⁻³, and typical cometary N₂ volatile fractions of 0.002 to 0.0002 (Crovisier 1994; Rubin et al. 2015). However, given that N₂ is quite volatile, this specific molecule may be depleted in comets. In this paper we will consider the total nitrogen mass (i.e., N in all forms) per impactor by simply multiplying the volume of a spherical impactor $(4/3)\pi {(d\ast 1000/2)}^{3}$, by the overall density (1000 kg m⁻³), and the highest N fraction estimates (0.02), for an upper limit (Figure 1(a)). We then estimate the mass of impactor-delivered N to Pluto per year (Figure 1(b)), by multiplying Equation (2) by the N mass per impactor:

$$\begin{eqnarray}&&{M}_{{\rm{N}}\_\mathrm{delivered}\_\mathrm{per}\_\mathrm{year}}(d)=8.8\times {10}^{6}{d}^{0.163},\end{eqnarray} \tag{ 3 }$$

where M and d are in g and km, respectively. Integrating Equation (3) from 1 m up to an assumed largest impactor of 100 km (and multiplying by 4 billion years), would result in 6 × 10¹⁸ g N delivered to Pluto over the past 4 billion years.

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From these considerations we conclude that the N mass delivered by comets is three-to-four orders of magnitude less than the 7 × 10^21–22 g N₂ lost by atmospheric escape over this same time period (the last ∼4 billion years). We conclude that even given uncertainties in the impactor flux (up to an order of magnitude), it does not appear that comets can deliver enough N to supply Pluto's N₂ atmosphere. The material influx onto Pluto during the first ∼0.5 billion years of its history would contribute additional N, but this influx would need to be 3–4 orders of magnitude higher than that of the last 4 billion years, and remain near Pluto's surface, to supply the atmosphere at current escape rates. Hence, except in the extremely improbable scenario that a recent, relatively large impact is supplying the current atmosphere, these calculations imply that either escape rate models overestimate total N₂ losses over time or endogenic sources must be responsible for replenishing Pluto's N₂.

The surface and subsurface layers of Pluto likely contain a mixture of materials that are less volatile than N₂ (e.g., H₂O, CH₄, and other hydrocarbons). As N₂ sublimates from Pluto's surface, a lag deposit of these more refractory components is likely to build up and will tend to choke off Pluto's N₂ layer. The thickness T_lag of a lag deposit can then be calculated by ${T}_{\mathrm{lag}}={H}_{\mathrm{sub}}{f}_{\mathrm{invol}}({\rho }_{{{\rm{N}}}_{2}}/{\rho }_{\mathrm{invol}})$, where H_sub is the height of the sublimated layer, f_invol is the involatile fraction of the surface (involatile/N₂), which we assume for simplicity to be identical everywhere, and $({\rho }_{{{\rm{N}}}_{2}}/{\rho }_{\mathrm{invol}})$ is the density ratio of the volatile to involatile materials. The distribution of materials of the surface and near-surface layers is not well known, thus we present a range of scenarios for f_invol in Figure 2. It is unknown at this time if this N₂ ice detected on Pluto's surface is a thin veneer, or a deeper layer, or how thick of lag deposits may have built up over time. We take $({\rho }_{{{\rm{N}}}_{2}}/{\rho }_{\mathrm{invol}})$ of ∼one for N₂ mixed with involatile ices, and $\sim 1/3$ for N₂ with a $\mathrm{rocky}/\mathrm{organic}$ component.

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We thus calculate that if ∼0.3–3 km of N₂ have been lost to escape (the thickness of a globally averaged surface layer), this loss could leave a total lag deposit built up over 4 billion years of ∼0.15–1.5 km (Figure 2) for an involatile fraction of 0.5. Although this estimate is crude, it serves for subsequent analysis we will use to explore how lag deposits would affect N₂ resupply against escape. We expect such a lag deposit to lie beneath a thin veneer of N₂ deposited by the most recent seasonal atmospheric deposition cycle (Stern et al. 1988). Pluto's complex seasonal cycles (Young 2013), potential polar caps (Olkin et al. 2015), and varied surface albedo patterns (Buie et al. 2010) suggest the surface ices are heterogeneously distributed. If the N₂ must be supplied by a smaller percentage of Pluto's surface area then sublimation would need to proceed to larger depths to supply the same N₂ mass, hastening the development of lag deposits.

The N₂ mass excavated by impacts reaching below Pluto's lag deposit would be available to resupply the atmosphere. For simple craters on Pluto (less than 4 km in diameter; Moore et al. 2015) the final crater diameter, D_f, is taken as 1.2^*D_tr based on lunar data (Pike 1977). For the final diameter of complex craters we use McKinnon & Schenk (1995; their Equation (1)). Using this, a 5 km impactor makes an ∼32 km final crater (scaling explained below), and excavates to a depth of ∼2 km. This crater excavates ∼3 × 10¹⁷ g of material, which is similar to Pluto's current atmospheric mass (see Table 1 for additional examples).

To calculate the total N₂ excavated by cratering on Pluto, we use Equation (2), which gives the cratering rates per year, multiply it by an estimate of the N₂ excavated by a given size impact (derivation below), sum over all impactor sizes, and multiply by 4 billion years. This involves scaling from the impactor size to the crater size (e.g., Holsapple 1993). Little data exist on the mechanical behavior of N₂ ice, so we follow the example of previous authors in using H₂O ice as an order of magnitude estimate (e.g., Bierhaus & Dones 2015; Stern et al. 2015b). N₂ ice is a weaker material, thus in theory craters would scale differently in the strength regime. This would only affect craters on Pluto smaller than a few hundred meters, which are created by impactors smaller than a few tens of meters. For larger impacts in the gravity regime, the density contrast between the impactor and target plays a role, but the density of N₂ is similar to that of H₂0 ice at Pluto temperatures (Scott 1976). The scaling formula derived for primary craters $(\mathrm{depth}/\mathrm{diameter}=0.2)$ for cometary impactors into water ice in the gravity regime with a 45° impactor (details in Singer et al. 2015) is

$$\begin{eqnarray}&&{D}_{\mathrm{tr}}(d)=0.86{d}^{0.783}{(g/{U}^{2})}^{0.217},\end{eqnarray} \tag{ 4 }$$

where D_tr is the transient crater diameter, g is surface gravity (0.66 m s⁻² on Pluto), U is the impact velocity (we use 2 km s⁻¹ as a typical speed), and all variables are in MKS units. Here we use a paraboloid of revolution as the crater shape, where the volume is $(1/2)\pi {R}_{\mathrm{tr}}^{2}{H}_{\mathrm{tr}}$. R_tr and H_tr are the transient crater radius and depth, respectively, and both can be written in terms of D_tr and thus the impactor diameter (d) through Equation (4).

Using the scaling in Equation (4), half the transient crater volume as the approximate amount of excavated material, N₂ ice density of 1000 kg m⁻³ (Scott 1976), and for an upper limit assuming an 100% N₂ ice layer, we arrive at the mass in grams excavated per impact size (Figure 3(a)) of

$$\begin{eqnarray}&&{M}_{{{{\rm{N}}}_{2}}_{\mathrm{excavated}}}(d)=7.0\times {10}^{15}{d}^{2.353},\end{eqnarray} \tag{ 5 }$$

where again d is in km. Multiplying Equation (5) by Equation (2), we derive an equation for the N₂ mass (g) excavated per year (Figure 3(b)) as

$$\begin{eqnarray}&&{M}_{{{\rm{N}}}_{2}\_\mathrm{excavated}\_\mathrm{per}\_\mathrm{year}}(d)=5.9\times {10}^{9}{d}^{-0.484}.\end{eqnarray} \tag{ 6 }$$

The negative exponent on impactor diameter in Equation (6) reflects the fact that both the cratering efficiency and the number of impacts decreases as impactor size increases. Integrating from 0 to 100 km for impactor diameters results in a total of 5 × 10²⁰ g N₂ excavated over 4 billion years. The low primary impact speeds on Pluto mean only a small fraction of the ejecta, less than 1% of the impactor mass, will have velocities higher than Pluto's escape speed, and not much material is vaporized (Bierhaus & Dones 2013).

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Even with uncertainties in the impactor flux, and the extremely conservative assumption that all excavated material is pure N₂, excavation from cratering does not seem to be a likely source for supplying all of Pluto's atmospheric mass loss over time. In fact, it apparently falls short by an order of magnitude or more.¹

Cratering can also contribute new N₂ to atmospheric resupply by exposing buried N₂ next to crater walls/floors and as the thermal wave from solar insolation drives subsurface N₂ sublimation. We use 2 times the annual thermal skin depth as a rough upper limit length scale over which this sublimation is likely to occur. The thermal skin depth, or L_T, is given by $\sqrt{(\kappa \tau )/(\pi \rho {c}_{p})}$, where κ is thermal conductivity, τ is the thermal variation timescale (conservatively we adopt the annual solar insolation on Pluto of 248 years), ρ is material density, and c_p is heat capacity. For a pure slab of N₂ ice, we estimate that L_T, or one e-folding depth of the annual wave would be ∼20 m.

The N₂ mass sublimated from the ice near a crater cavity is estimated by taking the volume of a paraboloidal shell with an inner radius of D_f/2 and an outer radius of D_f/2 + 2L_T, where 2L_T is 0.04 km in this case (note the use of final, rather than transient, diameter here). Multiplying the volume of the paraboloidal shell by Equation (2) gives

$$\begin{eqnarray}{M}_{{{\rm{N}}}_{2}\_\mathrm{sublimated}\_\mathrm{per}\_\mathrm{year}}(d) & = & 2.0\times {10}^{10}\\ & & \times {(0.04+d)}^{2.353}{d}^{-2.837}\\ & & -2.0\times {10}^{10}{d}^{-0.484}\end{eqnarray} \tag{ 7 }$$

(d in km) and integrating over all 1 m to 100 km diameter impactors results in 8 × 10²¹ g N₂ lost through crater wall and floor sublimation over 4 billion years, assuming pure N₂ under the cavity of each crater. In this simple upper limit calculation, the sublimation contribution is ∼one order of magnitude higher than that excavated by cratering due to the fact that the amount excavated by small craters (of which there are many) is less than the amount sublimated. This simple estimate does not attempt to take into account the actual composition of subsurface materials, any lag deposits (discussed below), the actual final shape of large versus small craters, or any effects of the varying insolation that would be received by a crater cavity at different latitudes on Pluto (there are no crater shadows in this model). These additional effects would work to reduce the amount of N₂ sublimated. It is possible that impact heating or other effects could enhance sublimation temporarily at the impact site, but we consider this is to be a second order effect because of the low impact velocities found at Pluto.

If a lag deposit is present, smaller craters would not expose fresh N₂ surfaces. To calculate the effect of a lag deposit, we assume that the excavation depth plus 40 m (2L_T for pure N₂) must be greater than the lag deposit thickness. For the lower limit lag deposit of 0.15 km, craters smaller than ∼1.3 km in final diameter will not punch through. Integrating over impactors from 125 m to 100 km, the material sublimated is considerably reduced, down to 4 × 10¹⁹ g N₂ sublimated over 4 billion years (for a pure N₂ surface). For a 1.5 km lag deposit, again integrating from 3.5 to 100 km in impactor diameter, the mass sublimated is reduced to 5 × 10¹⁷ g N₂ over 4 billion years.

All of these various estimates of exposed N₂ based on the presence of a lag deposit fall short by two-to-five orders of magnitude of the amount necessary to resupply atmospheric escape.