Local activity of comets: an indicator of non-uniform composition

ABSTRACT

In this paper, we investigate the conditions for the local ejection of a metamorphosed surface layer of material from a comet. We have calculated the strengthening of the material of the nucleus of comet 67P/Churyumov–Gerasimenko and the increase of gas pressure in pores. We are interested in the boundary between the regions Anhur and Bes, where an active pit has been identified. We show that material rich in ice can be exposed as a result of the local ejection of the strengthened layer of material several metres thick. This process does not require locally increased concentrations of very volatile components, such as carbon monoxide. A sufficient condition is the heterogeneity of material granulation.

solid state: volatile, comets: general

1 INTRODUCTION

The large number of images obtained during the two-year observation of the nucleus of the comet 67P/Churyumov–Gerasimenko (hereafter 67P/CG) using the Rosetta probe has enabled the quantitative study of the role of topography on local activity. In this respect, the Anhur region in the southern part of the nucleus is particularly interesting, where the most intense explosion identified during the Rosetta mission has its source. In addition, a large active pit was identified at the border between the Anhur and Bes regions (Fornasier et al. 2017). It should be noted that the regions of comet 67P/CG were defined only on the basis of their morphology (El-Maary et al. 2015, 2016; Thomas et al. 2015). Maps based on thermal properties or material composition might be different. The location of the considered pit is shown in Fig. 1. In Fornasier et al. (2017), it is marked in fig. 4 (white arrows) and in fig. 12. According to the SHAP7 model based on pre-perihelion images (Preusker et al. 2017), this pit was formed before the Rosetta mission. An interesting question is why it was formed in this place and why it remains active while the surrounding area is not. The presence of exposed ice requires either for the dust mantle to be ejected, or for both dust and the underlying ice material to be ejected.

Figure 1.

Active pit at the border between the regions Anhur and Bes (see Fornasier et al. 2017). The surface is described by the SHAP7 model (Preusker et al. 2017).

We investigate the possibility that ice–dust material is ejected in different locations in the active pit and in the surrounding area. The mechanism considered is the formation of shallow gas pockets followed by ejection of the overlying material, as described by Kossacki & Czechowski (2018). The shape of the nucleus is described by the high-resolution shape model SPG SHAP7, which is derived from 1531 images using the German Aerospace Center (DLR - Deutsches Zentrum fuer Luft- und Raumfahrt) stereo-photogrammetric (SPG) technique (Preusker et al. 2017). The surface is approximated by 44 million triangular faces, hereafter called sectors. The version cg-dlr-spg-shap7-v1_4M used in this work has 3 999 962 sectors. The alternative stereo-photoclinometry (SPC) SHAP5 model (Jorda et al. 2016) is derived from more than 5500 images, but has a lower spatial resolution.

2 MODEL

The model was initially formulated to simulate evolution of granular ice in a vacuum (Kossacki et al. 1994) and it was subsequently extended. The most recent version is described in Kossacki & Czechowski (2018). The basic features of the present model are the following.

• The nucleus is covered by a layer of non-volatile agglomerates of fine particles.

• The underlying material is a mixture of ice grains with non-volatile cores, which are agglomerates of silicate particles glued with organic material. This structure is consistent with the distribution of halogen-bearing species in the coma of comet 67P/CG. Measurements suggest a deficit of halogens in the gas phase and their presence on dust grains. This could indicate that hydrogen halides freeze on silicate grains and are subsequently covered by ice (Dhooghe et al. 2017). For comparison, Kossacki & Czechowski (2018) consider a mixture of two types of agglomerates, while Markkanen et al. (2018) discuss a matrix of ice grains with embedded particles composed of silicate grains and organic grains.

• The strength of the cores of grains is the same as pure ice.

• The water ice is crystalline and amorphous, depending on the depth. Directly below the dust mantle is a layer of grains composed of crystalline water ice and non-volatile material, while in the centre of the nucleus, the water ice is in the low-density amorphous phase.

• Ice mantled grains undergo sintering as a result of micro- and macro-scale diffusion of vapour. The latter was not considered in Kossacki & Czechowski (2018). It is related to the macroscopic migration of vapour in the pores along the temperature gradient. As the surface temperature increases because of the absorption of solar light, a temperature gradient is formed and the molecules sublimating beneath the dust mantle can migrate towards the cold interior of the nucleus. At the depths at which the flux of vapour decreases, the volume fraction of the ice increases, and the radii of bonds between the grains grow. An opposite effect is also possible.

• The model includes the temperature dependence of specific heat and thermal conductivity.

• The heat and vapour transport are calculated using the one-dimensional model, in the direction normal to the local surface, while the surface illumination is calculated using a three-dimensional (3D) model that takes into account the shadows caused by the relief. The absorption of infrared radiation emitted by the surrounding elevated surface is not considered.

2.1 Equations

The solved set of equations is the same as in Kossacki & Czechowski (2018), except for their equations (9)–(14) that are related to the thermal conductivity and the rate of sintering of grains. Also, the dust mantle is assumed to have constant thermal conductivity, which is a parameter of the model, so their equations (9) and (10) are not used. The ice–dust material is a matrix of ice grains with cores instead of the mixture of two types of agglomerates. Thus, their equations (11)–(13) are replaced with

$$\begin{eqnarray*} \lambda = \lambda _{\rm g} h (v_{\rm i} + v_{\rm c})^{2/3} + r_{\rm p} v_{\rm i} v_{\rm c} \left(\frac{32 \mu }{9 \pi R T} \right)^{1/2} H \frac{\rm d}{{\rm d}T} p_{\rm sat} \end{eqnarray*}$$

(1)

(Kömle & Steiner 1992; Kossacki et al. 2015) and

$$\begin{eqnarray*} \lambda _{\rm g} = \lambda _{\rm i} \left[ 1 + \frac{ 3 f (\lambda _{\rm c} - \lambda _{\rm i})}{2 \lambda _{\rm i} + \lambda _{\rm c} - f (\lambda _{\rm c}-\lambda _{\rm i})} \right] . \end{eqnarray*}$$

(2)

Here, λ_g denotes the thermal conductivity of a non-uniform grain (see Haruyama et al. 1993), h is the Hertz factor, which is a measure of the size of bonds between adjacent grains, v_i is the volume fraction of ice, v_c is the volume fraction of porous cores of grains, r_p is the characteristic radius of pores (assumed to be equal to the radius r_g of a grain), μ is the molar mass of water, R is the universal gas constant, H is the latent heat of sublimation and T is the temperature.

The first term on the right-hand side of equation (1) describes the transport of heat through the solid phase, while the second term results from the sublimation, or condensation of vapour migrating through the connected pores along the temperature gradient.

Another modification is related to the description of the changes of the Hertz factor. The original equation (14) describes the micro-scale sintering due to the migration of vapour molecules from the surfaces of grains on to the surface of the bonding neck. In this paper, we also consider changes in the sizes of bonds resulting from the macro-scale migration of vapour. The equation describing both effects is

$$\begin{eqnarray*} \frac{{\rm d}h}{{\rm d}t} &=& \frac{2 h^{0.5}}{{r_{\rm g}}} \frac{\Omega ^2 \gamma \, p_{\rm sat}(T) S}{{(2\pi \mu R T )}^{{1}/{2}} R T} \left[\frac{2}{r_{\rm g}} + \frac{2 (r_{\rm g} - r_{\rm n})}{{r_{\rm n}}^2} - \frac{1}{r_{\rm n}} \right]\nonumber\\ &&+ \frac{2 h^{0.5} \psi r_{\rm g}}{3 (1 - \psi) \tau ^2 \varrho } \left(\frac{32 \mu }{9 \pi R T} \right)^{0.5} \left(\frac{{\rm d}^2p}{{\rm d}z^2} \right). \end{eqnarray*}$$

(3)

The parameter r_n is the radius of the contact area between the grains, S is the area of a grain from which molecules migrate on to the bond, γ is the surface energy and Ω is the molar volume (Ω²γ = 4 × 10⁻¹¹ m⁴ mole⁻² J).

2.2 Parameters

Our model comet has orbital elements found for comet 67P/CG. Following Keller et al. (2015), these are the semimajor axis a_a = 3.463 au, orbital period P = 6.44 yr and eccentricity e = 0.64102. The initial temperature is 40 K, constant through the whole nucleus. The assumed properties of the dust mantle and of the underlying material are described below.

2.2.1 Dust mantle

The dust mantle consists of agglomerates of small compact particles. It is characterized by five parameters: the density of grains ϱ_gd, the average density ϱ_d, the specific heat c_d, the thermal conductivity of grains λ_bd and the average thermal conductivity λ_d.

The density of solid particles composing the agglomerates is assumed to be ϱ_gd = 3500 kg m⁻³, which is likely the highest estimate (Herique et al. 2016). The density of agglomerates depends on their internal structure. If the particles constitute 35 per cent of the agglomerate volume (i.e. v_d,int = 0.35 and micro-porosity ψ_int = 0.65), then the corresponding density of the agglomerate is 1225 kg m⁻³. Similarly, when ψ_int = 0.59 (Fulle et al. 2017), then the density of the dust agglomerate is 1435 kg m⁻³. Finally, if the packing density of the agglomerates is v_ad = 0.35, then the bulk density of the dust mantle is ϱ_d = 430 kg m⁻³. The corresponding total porosity of the dust is ψ_d = 0.878. This value matches well the porosity of 0.87 indicated by the spectrophotometric analysis performed by Fornasier et al. (2015).

The thermal conductivity of compact olivine is 5 W m⁻¹ K⁻¹ (Opeil, Consolmagno & Britt 2010). The values derived for stony meteorites, except meteorite Abee, range from 0.5 W m⁻¹ K⁻¹ for the CM sample to 1.9 W m⁻¹ K⁻¹ for H sample (Opeil et al. 2010). The values of thermal inertia derived from measurements performed during the Rosetta mission range from 10 to 160 m⁻² s^−0.5. According to early measurements, I_th = 10–50 J K⁻¹ m⁻² s^−0.5 (Gulkis et al. 2015). Later measurements performed using VIRTIS and MIRO indicate that I_th = 10–160 J K⁻¹ m⁻² s^−0.5 and I_th < 80 J K⁻¹ m⁻² s^−0.5, respectively (Marshall et al. 2018). The thermal inertia of other comets can be different, but the value considered in our work matches the average thermal inertia 30 ± 20 J K⁻¹ m⁻² s^−0.5 found by Groussin et al. (2019).

In this paper, λ_bd = 2.7 W m⁻¹ K⁻¹, which is in the middle of the smallest and largest possible values, the specific heat is c_d = 550 J kg⁻¹ and the thermal conductivity of the dust layer λ_d = 3–10.6 × 10⁻³ W m⁻¹ K⁻¹. The considered values of λ_d correspond to the thermal inertia 27 and 50 J K⁻¹ m⁻² s^−0.5. When it is not stated otherwise, λ_d = 3 × 10⁻³ W m⁻¹ K⁻¹.

In our numerical model, we used a thickness of the dust mantle equal to the size of the numerical cell −1.0 cm.

2.2.2 Ice–dust material

The material underlying the dust mantle is composed of ice grains with porous cores. The volume fractions of ice and the volume fraction of cores are v_i = 0.109 and v_c = 0.35, respectively. The latter corresponds to the volume fraction of the compacted dust (silicate-organic material) v_d = 0.123. These values correspond to the dust–to–ice mass ratio 4.3, which is in the range 4 ± 2 derived for the nucleus of comet 67P/CG (Rotundi et al. 2015). The average density 531.4 kg m⁻³ is in the range 533 ± 6 kg m⁻³ derived by Pätzold et al. (2016), while it is smaller than the value 537.8 ± 0.7 kg m⁻³ found by Preusker et al. (2017).

The presence of CO (taken as an example of a material more volatile than H₂O) can increase the density by a few per cent. If the volume fraction v_CO is 0.005, then the density becomes 537.7 kg m⁻³, which is in the range found by Preusker et al. (2017). It should be noted that the concentration of components can be very non-uniform. Thus, we consider two different values of v_CO = 0.01 and 0.005.

Another parameter is the initial value of the Hertz factor h₀. If the tensile strength σ_T is proportional to the product of h and (1 − ψ), as considered in Skorov & Blum (2012) and Kossacki et al. (2015), then h₀ can be derived from the initial tensile strength. Unfortunately, there is poor knowledge of the strength of cometary material. Local strength was determined for the first touch-down place and the final resting place of the probe Philae. The derived values of the compressive strength are 1 kPa (Biele et al. 2015) and about 2 MPa (Spohn et al. 2015), respectively. Groussin et al. (2019) have provided a useful review of estimates and laboratory measurements dealing with the strength of cometary materials. The former value characterizes granular material depleted in volatile components (i.e. the so-called dust mantle), while the latter is for consolidated material. The sintering process can lead to the formation of a cohesive layer like that found in the resting place of Philae (Kossacki et al. 2015). The tensile strength can be an order of magnitude smaller than the compressive strength, σ_T0 > 0.1 kPa. The initial tensile strengths σ_T0 = 1 and 0.1 kPa correspond to h₀ = 0.0001 and 0.00001, respectively.

The range of radii of ice particles derived for ice-rich material on comet 9P/Tempel 1 is 5–25 μm (Sunshine et al. 2006), while the distribution of radii of particles in patches of ice-rich material on the surface of comet 67P/CG has two peaks at 25 and 1000 μm (Raponi et al. 2016). The values of r_g considered in our work are 12.5, 25 and 250 μm. The characteristic radius of pores in dust r_dm, the thickness of the dust mantle Δ_d and h₀ are free model parameters.

3 RESULTS

Simulations of the thermal and structural evolution of material were performed for 40 sectors. They are located both in the active pit and in the surrounding area. The calculated dependence of the gas pressure on time and depth has been used to select locations where the layer can be ejected. It is assumed that the ejection of ice–dust material is possible when the pressure of gas in pores exceeds the tensile strength of the layer of metamorphosed material forming beneath the dust mantle. In the extreme case, the strengthened subsurface layer is thick enough that it remains not lifted when the gas pressure exceeds the compressive strength σ_c of the underlying pristine material. In such a case, a gas pocket may form.

3.1 Gas pressure

The first of our calculations were performed assuming r_dm = 200 μm, v_CO = 0.01, r_g = 12.5 μm and h₀ = 0.001. The latter corresponds to σ_T0 = 1 kPa and the initial compressive strength σ_C0 ∼ 10 kPa. We define two threshold pressures: p_crit,1 = 1 kPa and p_crit,2 = 10 kPa. The considered locations are shown in Fig. 2. The condition p > p_crit,1 is met in all considered locations. Moreover, in all these places the condition p > p_crit,2 could be also met. However, this is possible only if the strengthened layer is thick and heavy enough that it is neither lifted nor broken when p > p_crit,2.

Figure 2.

Locations considered in this paper.

The gravity acceleration on comets is low, and on the nucleus of comet 67P/CG it is 2 × 10⁻⁴ m s⁻² (Groussin et al. 2015). Thus, a subsurface strengthened layer has to be very large and thick to avoid lifting by gas released beneath it. If we consider a disc of strengthened material with a 50-m radius and a 10-m thickness, then for a density of 531 kg m⁻³ and g = 2 × 10⁻⁴ m s⁻², the gravity force exerted on the material underneath (e.g. gas cavity, pristine material) is F_grav ∼ 8340 N. If the pressure is enhanced under a small fraction of the disc (i.e. 1 per cent), then the condition F_gas > F_grav can be met when p ∼ 0.1 kPa. When the edges of the strengthened layer are under elevated surrounding terrain, then the critical gas pressure is higher. In any case, the tensile strength of the strengthened material is important.

3.2 Tensile strength

Fig. 3 plots the profiles of σ_T versus depth, calculated when the initial tensile strength σ_T0 = 1 kPa. The curves are drawn for selected locations for the times when the gas pressure in given locations satisfies conditions: p < 0.1 kPa (upper panel), p > p_crit,1 = 1 kPa (middle panel) and p > p_crit,2 = 10 kPa (lower panel). Profiles shown in the middle panel are taken 113 h (nine rotation periods) after the profiles presented in the upper panel. During this period, the thickness of the strengthened layer increased by at least 0.5 m. In one location, a secondary strengthened layer formed above the crystallization front of the amorphous water ice. The thickness of the main strengthened layer is ∼1 m when the gas pressure approaches 0.1 kPa, 0.8–1.8 m when p > p_crit,1 and 0.8–3.3 m when p > p_crit,2. A strengthened layer about 1 m thick will probably be broken or lifted. Thus, in locations marked with white triangles, the presence of uncovered ice-rich material should be expected. Increased emission of more volatile components (such as carbon monoxide) is also likely.

Figure 3.

Tensile strength versus depth, when the CO pressure p beneath the strengthened layer is p < 0.1 kPa (upper panel), p > p_crit,1 = 1 kPa (middle panel) and p > p_crit,2 = 10 kPa (lower panel). The initial tensile strength is 1 kPa. The remaining parameters of the model parameters are given in the legends.

It is interesting that when σ_T0 = 1 kPa the ejection of material is possible in all considered locations, while the uncovered ice was found only in the pit. This difference can be explained by the non-uniform concentration of highly volatile CO, but other options should also be considered, such as the non-uniform granulation or porosity of the material.

Fig. 4 is similar to the upper and lower panels of Fig. 3. The profiles shown in Fig. 4 are calculated for σ_T0 = 0.1 kPa instead of σ_T0 = 1 kPa. It can be seen that the strengthening of the loosely bonded mixture of grains is slower than with moderate initial strength. The thickness of the strengthened layer at the moment when the CO gas pressure reaches the threshold value p_crit,1 is ∼1 m for σ_T0 = 0.1 kPa and 1.1–1.9 m for σ_T0 = 1 kPa. The next threshold p_crit,2 is exceeded when the strengthened layer is 1.5–2.3 m and 2.9–3.3 m thick, respectively. Hence, the decrease of σ_T0 by an order of magnitude results in a decrease of the thickness of the strengthened layer by only about 50 per cent.

Figure 4.

The same as in the upper and lower panels of Fig. 3, but the initial tensile strength is equal to 0.1 kPa instead of 1 kPa. These values correspond to the initial Hertz factors 0.001 and 00001.

One more problem is the possible role of the thermal conductivity of the dust mantle. Fig. 5 is similar to the lower panel of Fig. 3. The profiles drawn in Fig. 5 have been calculated assuming that the dust mantle has high thermal conductivity λ_d = 10.6 × 10⁻³ W m⁻¹ K⁻¹, which corresponds to the thermal inertia I_th = 50 J K⁻¹ m⁻² s^−0.5. A comparison of the profiles shows that the layer of σ_T > 50 kPa is about 3 m thick. This thickness is independent of the thermal conductivity of dust λ_d. However, the time when the condition p > p_crit,2 is satisfied depends on λ_d. It could differ by more than 50 d.

Figure 5.

The same as in the lower panel of Fig. 3, but for λ_d = 10.6 × 10⁻³ W m⁻¹ K⁻¹. The number of profiles is reduced for better readability.

3.3 Granulation of material and concentration of CO ice

To constrain the influence of the granulation of material, the simulations were repeated assuming that r_g = 25 μm instead of r_g = 12.5 μm, while the values of the other parameters were the same. In addition, the simulations were performed for one location, assuming very coarse-grained material (r_g = 250 μm and r_d = 500 μm). Fig. 6 shows the distribution of the gas pressure p beneath the strengthened layer, when r_g = 25 μm. The white triangles indicate places where p > p_crit,1, while the black triangles are for the remaining locations. The increase of the grain size has two consequences: an increase of the characteristic radius of pores, in our model r_p = r_g, and a decrease of the rate of sintering due to the micro-scale diffusion of molecules. A comparison of Figs 2 and 6 indicates that the condition p > p_crit,1 is met in all locations when r_g = 12.5 μm, but only in ∼50 per cent of the locations when r_g = 25 μm. The maximum value of gas pressure calculated for location 1 is p > 10 kPa (r_g = 12.5 μm, r_d = 200 μm), p ∼ 0.85 kPa (r_g = 25 μm, r_d = 200 μm), and p < 0.01 kPa (r_g  = 250 μm, r_d = 500 μm). Our findings indicate that the granulation of material is critical to the possibility of the strengthened subsurface material being ejected.

Figure 6.

Gas pressure beneath the strengthened subsurface layer in selected locations, when the grain radius r_g = 25 μm. White triangles are for locations where the condition p > p_crit,1 is met, while black triangles are for the remaining locations.

Another problem is the possible influence of the CO ice concentration on the metamorphism of the material and the rise of gas pressure in the pores. We considered two values of the volume fraction of the CO ice: 0.01 and 0.005. The remaining parameters were r_dm = 200 μm, r_g = 12.5 μm and h₀ = 0.001. The comparison was performed for location 5 in the centre of the pit. Fig. 7 plots the tensile strength versus depth, when the condition p > p_crit,2 is met. The depletion of material in the CO ice reduces the rate of growth of the gas pressure. Thus, the threshold values are exceeded when the strengthened layer is thicker than in the case of material rich in CO. If v_CO is doubled, then the strengthened layer becomes thinner. The difference is ∼30 per cent when p > p_crit,1, and only ∼20 per cent when p > p_crit,2.

Figure 7.

Tensile strength versus depth, when the volume fraction of the CO ice is 0.01 and 0.005. The profiles are drawn for location 5 in the centre of the pit, for p > p_crit,2.

The results presented in this section indicate that if the concentration of CO ice in the pristine material is in the considered range 0.005–0.01 by volume, it has only a moderate influence on the possibility of strengthened material being ejected. Another considered parameter, the initial strength of the mixture, of grains is more important, but it remains far less significant than the size of the ice grains.

4 SUMMARY AND CONCLUSIONS

We aim to determine the conditions sufficient for local ejection of a layer of strengthened material forming beneath the surface of a comet. Numerical simulations were performed for the nucleus of comet 67P/CG. The investigated region is the border between regions Anhur and Bes, where an active pit was identified (Fornasier et al. 2017). The evolution of temperature, density and strength of material, as well as the gas pressure in the pores between the grains, were calculated. It is found that the material rich in ice can be exposed as a result of local ejection of a strengthened layer of material up to several metres thick. This process does not require a high concentration of volatile components, such as carbon monoxide or carbon dioxide. The sufficient condition is the non-uniformity of granulation of material. Hence, the presence of areas of exposed ice can only indicate possible enhancement of the material in volatile components.

ACKNOWLEDGEMENTS

This work was supported by Poland’s National Science Centre (Narodowe Centrum Nauki; decision no. 2018/31/B/ST10/00169). The authors thank the reviewer for valuable comments, which helped to improve the paper.