Production of water from a dynamically new comet

ABSTRACT

In our paper, we presented the results of numerical simulations dealing with the production of water from the nucleus of a hypothetical dynamically new comet. The simulations were performed using empirical equation based on the results of laboratory experiments performed in vacuum using ice agglomerates. We presented also the results obtained using simple, popular approach. The model nucleus is a prolate ellipsoid of rough surface. The results were compared with the observed production of water from the nucleus of comet C/2012 K1 PanSTARRS. Our simulations indicate, that the observed production of water can be reproduced without assuming the presence of highly volatile ices when the dust may slide down the slopes and the thermal conductivity of the nucleus is very low.

solid state: volatile, comets: general

1 INTRODUCTION

The nuclei of dynamically new comets may differ from old ones in different respects including the thickness of the dust mantle. When a comet approaches the Sun for the first time, then the dust mantle may be thin, or absent (Bernardinelli et al. 2021).

In this paper, we considered one of the possible models describing production of water from the hypothetical dynamically new comet without a dust mantle. The dust released due to the sublimation of ice is assumed to be either blown out to space, or slide down the inclined surfaces leaving them uncovered. The illumination of the inclined surfaces depends on their orientations. However, in our model the average temperature of a larger area is approximated by the temperature of a smooth surface characterized by the effective emissivity taking into account roughness of the surface. The surface of the nucleus is approximated by 240 triangular sectors. The illumination of each sector depends only on its position on the nucleus.

We calculated the surface temperature and the rate of production of water separately for each sector, based on the energy balance including both the heat of sublimation and the energy transport beneath the surface. The mass flux of subliming water was calculated using the experimentally tested equation taking into account the porosity of material and the subsurface temperature gradient. Models assuming instantaneous equilibrium between the absorbed energy and the energy consumed for the sublimation overestimate both the temperature and the flux of vapour. Additional source of errors is the oversimplification of the equation for the rate of sublimation. We described this problem in the Section 2.2.

The mathematical model applied in this work is a simplified version of the model used in our previous papers, most recently in Kossacki & Czechowski (2018) and Kossacki & Czechowski (2019). In the present investigations we assumed, that the content of ice in the material is small and that ice is hidden within dust agglomerates. In this case, the individual particles of ice may be not in contact one with the other and they do not sinter. Hence, the sintering related formulas included in the original model are not used.

2 MODEL

2.1 Basic features of the used model

The basic features of the model are:

• The model nucleus is a prolate ellipsoid. On the surface are present small-scale slopes and the non-volatile dust does not accumulate on the surface, but slides down the slopes, or is ejected to space.

• The material composing the nucleus of a comet is likely the mixture of porous agglomerates containing silicates, ices, and organic species (Markkanen et al. 2018; Güttler et al. 2019; Fulle et al. 2020). In this work, we consider the model nucleus composed of agglomerates containing non-volatile dust (silicates and other species) and the hexagonal water ice. More volatile ices are present only in the interior of the nucleus, deep beneath the surface. This is only one of the possibilities, but we intended to investigate how the sublimation of water ice affects activity of a dynamically new comet.

• The simulations presented in this work were performed using example experimentally verified equation (see equation 7) for the flux of vapour subliming from porous materials, which can be considered as potential cometary analogues (Kossacki et al. 2022, 2023). This formula depends on the characteristic radius of pores and the total porosity of the material, but not on the internal structure of grains composing material. They can be either solid ice grains with porous cores, or complex agglomerates of dust (refractory material) and ice.

• The thermal conductivity of material λ is either constant, or temperature dependent. In the latter case λ = λ_solid + λ_pores, where λ_solid denotes the conductivity of porous ice–dust agglomerates and λ_pores is the conductivity resulting from the sublimation and condensation of vapour within pores. The mass fraction of ice is assumed to be much smaller than of dust and the whole ice is within agglomerates.

• The heat and vapour transport are calculated in one dimension, in the direction normal to the local surface, while the surface illumination is calculated using a 3D model.

2.2 Equations

In this section, we described the set of equations solved by the programme used in this work.

Evolution of the temperature at the surface was described by the equation

$$\begin{eqnarray} \varrho c_\mathrm{ s} (T_\mathrm{ s}) \frac{\partial T_s}{\partial t} = F_{\mathrm{ rad}} - \, H\, F_{\mathrm{ H2O}}- \lambda \nabla T|_\mathrm{ s}, \end{eqnarray}$$

(1)

where F_rad is the total flux of energy absorbed and emitted from the surface in the given location, c_s is the specific heat, ϱ is the average density of material, F_H2O is the flux of vapour, H is the latent heat of sublimation, and λ is the thermal conductivity. The average density is ϱ = ϱ_iv_i + ϱ_sdv_d, where ϱ_i is the density of compact ice, ϱ_sd is the density of material composing dust (when compressed to the non-porous state), v_i denotes the volume fraction of ice, and v_d denotes the volume fraction of dust. The total porosity of material is ψ = 1 − v_i − v_d and ϱc_s = c_sdϱ_sdv_d + c_siϱ_iv_i.

In the case of an object of known shape and topography the flux of radiation is

$$\begin{eqnarray} F_{\mathrm{ rad}} = \frac{\mathrm{ Sc}}{R_\mathrm{ h}^2}(1 - A) \, \text{max}(\cos {\alpha _{\mathrm{ Sun}}}, 0) + F_{\mathrm{ sc}} + F_{\mathrm{ IR}} - \epsilon \sigma T_\mathrm{ s}^4, \end{eqnarray}$$

(2)

where F_sc is the absorbed fraction of the flux of solar radiation scattered from the surrounding terrain toward the given point, F_IR is the absorbed flux of infrared radiation from the surrounding surface, σ is the Stefan–Boltzmann constant, A is the albedo, ϵ is the emissivity, Sc is the flux of solar radiation at 1 AU, R_h is the heliocentric distance of comet, α_Sun is the zenith angle of the Sun, and T_s is the surface temperature. Unfortunately, the shapes of dynamically new comets are unknown and it is impossible to calculate the terms F_sc and F_IR. Thus, we assume that F_sc = F_IR = 0, but the effective emissivity ϵ is reduced due to the roughness of the surface. Davidsson et al. (2013) have found, that the effective emissivity of the nucleus of comet 9P/Tempel 1 fitted using the data acquired by the Deep Impact space probe is lower than 0.5 for most of the area. In our work, the effective emissivity is the model parameter.

The specific heat of the non-volatile component was calculated using equation

$$\begin{eqnarray} c_{\mathrm{ sd}} = -34 T^{0.5} + 8 T -0.2 T^{1.5} \end{eqnarray}$$

(3)

(Winter & Saari 1969), while for the specific heat of H₂O ice we used the formula

$$\begin{eqnarray} c_{\mathrm{ si}} &=& 7.73 T\,\, \left(1 - \mathrm{ e}^{-1.263 \times 10^{-3} T^2} \right) \\ &&\times \,\left(1 {+} \mathrm{ e}^{-3\, T^{\frac{1}{2}}} 8.47 \times 10^{-3}\, T^{6} {+} 2.0825 \times 10^{-7}\, T^4 \mathrm{ e}^{-4.97 \times 10^{-2}\, \mathrm{ T}} \right)\\ \end{eqnarray}$$

(4)

(Shulman 2004).

The thermal conductivity was either constant, or calculated using the equation

$$\begin{eqnarray} \lambda &=& \lambda _{\mathrm{ solid}} + \lambda _{\mathrm{ pores}}, \qquad \text{where}\\ \lambda _{\mathrm{ pores}} &=& r_{\mathrm{ p}} \psi {\left(\frac{32 \mu }{9 \pi R T} \right) }^{1/2} H \frac{\mathrm{ d}\, \mathrm{ p}_{\mathrm{ sat}}}{\mathrm{ d}T}. \end{eqnarray}$$

(5)

The terms on right-hand side of equation (5) denote: the heat transport within solid grains and the heat transport due to the sublimation and condensation of vapour in pores, respectively. The latter term was described in Kömle & Steiner (1992). The symbol μ denotes the molar mass of water, R is the universal gas constant and p_sat(T) is the pressure of the phase equilibrium at the temperature T.

The evolution of the volume fraction of ice was calculated using the approximate equation

$$\begin{eqnarray} \frac{\partial v_\mathrm{ i}}{\partial t} &= & \frac{r_\mathrm{ p} \psi }{\varrho \tau ^2} \frac{\mathrm{ d}}{\mathrm{ d}z} \left(\frac{32 \mu }{9 \pi R T} \right)^{0.5} \left(\frac{\mathrm{ d}^2p_{\mathrm{ sat}}}{\mathrm{ d}z^2} \right)\\ &=& \frac{ r_\mathrm{ p} \psi }{\tau ^2 (v_\mathrm{ i} \varrho _\mathrm{ i} + v_\mathrm{ d} \varrho _{\mathrm{ sd}})} \left(\frac{32 \mu }{9 \pi R T} \right)^{0.5} \frac{a b}{T^2} \mathrm{ e}^{-\mathrm{ b T}^{-1}} \\ &&\times \,\left[ \frac{\mathrm{ d}^2T}{\mathrm{ d}z^2} + {\left(\frac{\mathrm{ d}T}{\mathrm{ d}z} \right) }^2 \left(\frac{b}{T^2} - \frac{2.5}{T} \right) \right] \end{eqnarray}$$

(6)

(Kossacki, Czechowski & Skóra et al. 2020). The parameter |$\tau = \sqrt{2}$| (Carman 1956) is the tortuosity of the pores.

The flux of subliming water F_H2O was calculated using equation

$$\begin{eqnarray} F_{\mathrm{ H2O}} &= &\alpha _{\mathrm{ subl}}(T_\mathrm{ s}) {\left(\frac{\mu }{2 \pi R_\mathrm{ g} T_\mathrm{ s}} \right)}^{0.5} p_{\mathrm{ sat}}(T_\mathrm{ s}) \\ &&\times \,\left[1 + 127.86\, \frac{r_\mathrm{ p}}{p_{\mathrm{ sat}}(T_{\mathrm{ sub}})} \left| \nabla p_{\mathrm{ sat}}\right|\right] \\ &&\times \,\left[ 1.73 - 5.16 \psi + 3.96 \psi ^2 \right] \end{eqnarray}$$

(7)

(Kossacki et al. 2023, equation 1). This equation was originally derived for the rate of recession of the surface (equation 15 in Kossacki et al. 2022). The numerical coefficients in the last term in square brackets were described in Kossacki et al. (2020). Equation (7) is based on laboratory investigations of evolution of samples made of different kind porous ice–dust agglomerates in a vacuum chamber, described in our previous works. The symbol α_subl denotes the temperature-dependent sublimation coefficient (Kossacki et al. 1999, 2017), ∇p_sat is the pressure gradient beneath the surface. For comparison, the simplest equation for the flux of subliming vapour

$$\begin{eqnarray} F_{\mathrm{ H2O}} = {\left(\frac{\mu }{2 \pi R_\mathrm{ g} T_\mathrm{ s}} \right)}^{0.5} p_{\mathrm{ sat}}(T_\mathrm{ s}), \end{eqnarray}$$

(8)

which is still used due to its simplicity (e.g. Bernardinelli et al. 2021) is reasonably accurate only in the case of smooth compact ice at the temperature below 200 K (Kossacki et al. 1999; Gundlach, Skorov & Blum 2011).

The total rate of production of water from the nucleus was calculated as |$F_{\mathrm{ Tot}} = S_{\mathrm{ act}}\, F_{\mathrm{ H2O}}$|⁠, where S_act is the active surface of the nucleus. The assumed roughness of the surface of the model nucleus implies some increase of the surface area. The area of a slope inclined by 30° (expected to be sufficient to cause sliding of the dust) is about 15 per cent larger than the area of the horizontal surface. If the whole surface of the nucleus is covered by such slopes, then it should be larger than the surface of the smooth ellipsoid of the same axes by the factor 1.15. However, some part of the rough surface can be inactive due to the local shadowing, or the accumulation of dust in lows. Thus, we assume that the active surface is S_act = f_actS, and S is the surface of the smooth ellipsoid.

2.3 Parameters

The presented example calculations were performed assuming that the model comet is orbiting similarly to the dynamically new comet C/2012 K1 PanSTARRS (hereafter called C/2012 K1), with perihelion distance q = 1.0546 AU and eccentricity e = 1.00014 (Nakano 2017). The observed coma structure of C/2012 K1 were used by McKay et al. (2016) to derive a rotation period for the nucleus of 9.2 ± 0.4 h. In the later work the period of rotation was established to be 9.4 h, which we treat as the most probable rotation period of this comet (Betzler et al. 2020).

Initial estimates performed for the shape of C/2012 K1 are: the effective radius R_eff = 8.67 ± 0.08 km and the ratio of axes 1.44 ± 0.03 Betzler et al. (2020). Recent estimate for the effective radius R_eff = 2.352 ± 0.13 km (Paradowski 2020) indicates, that the surface of the nucleus can be about ten times smaller. The model comet considered in this work is a prolate ellipsoid with semi-axes a = b = 1.9 km, and c = 2.74 km. In this case, the area is approximately the same as of the sphere of the radius 2.17 km. Unfortunately, the accurate determination of the size of a nucleus requires observations performed in the period of inactivity, while the considered comet was observed only when it was active. Thus, the nucleus can be even smaller.

The rotation axis Z along one of the short semi-axes of the ellipsoid is assumed to be perpendicular to the orbital plane. In this work, the surface of the model nucleus is approximated by 240 plane triangular sectors.

The nucleus is assumed to be composed of porous dust agglomerates containing some ice. The characteristic radius of pores between agglomerates is: 0.25 mm, 0.5 mm, 2.5 mm, and 5 mm. The mass ratio dust-to-ice is assumed to be within the same range as of the samples for which weobserved complete sliding of sand i.e. 2.3 < f_m < 3.5 (work under review). The volume fractions of ice and dust are: (v_d = 0.28, v_i = 0.26), (v_d = 0.14, v_i = 0.13), and (v_d = 0.07, v_i = 0.065). Formally, v_d and v_i do not depend one from the other, but their ratio determines the mass ratio dust-to-ice f_m. We considered different values of v_d and v_i, which correspond to v_d/v_i = 1.077 and the density of the material composing dust (if compressed to the state of zero porosity) is 2680 kg m⁻³, then f_m = 3.12. The values of the total porosity including the internal porosity of the dust–ice agglomerates, corresponding to the three considered pairs of v_d and v_i were: ψ = 0.46, ψ = 0.73, and ψ = 0.865. For comparison, the porosity of the nucleus of comet 67P/Churyumov–Gerasimenko is within the range 0.75–0.85 (Henrique et al. 2016). In the case of comet 9P/Tempel 1 the density 240–1250 kg m⁻³ (Thomas et al. 2013) converted for the porosity assuming the mass ratio dust-to-ice f_m = 3 and the density of material of dust (when non-porous) 2680 kg m⁻³ gives ψ  = 0.44–0.89.

The efficiency of the heat transport was characterized either by the thermal conductivity λ (when we assumed, that the thermal conductivity is constant), or by the thermal conductivity resulting from the heat transport within solid phase of the material λ_solid and in the pores. For the calculations with the constant thermal conductivity we assumed, that it was λ = 0.1 mW m⁻¹ K⁻¹ and 0.001 mW m⁻¹ K⁻¹. In the case of the thermal conductivity that is temperature dependent, we considered several values for λ_solid between 10 mW m⁻¹ K⁻¹ and 0.4 mW m⁻¹ K⁻¹. In the case of low values of λ almost whole absorbed energy of the solar light was converted to the heat of sublimation and we obtained the upper estimate for the production of water from the nucleus.

The remaining parameters were: the albedo A (0.05–0.5), the effective emissivity ϵ (0.4–0.9) and the active fraction of the surface 0.2 < f_act < 1.2. The albedo of known comets is close to 0.05, higher values were considered for completeness. If the whole surface is horizontal, then f_act > 1 means hyperactivity of a comet. However, the presence of slopes increases the geometric surface area. When the whole surface is a mixture of slopes of the inclination angle 30° and the whole surface is active, then f_act = 1.15. Thus, f_act ≳ 1.2 should mean hyperactivity of a comet. Sosa & Fernandez (2011) found very large values for the fractions of the active surface area for their samples of long-period comets, i.e. f_act close to 1 and higher. In our work f_act = 1 unless stated otherwise.

2.4 Observational data

Comet C/2012 K1 reached perihelion on 2014 August 27. The maximum water production of this very bright comet was 2 × 10²⁹ molecules per second. The SWAN (Solar Wind Anisotropies) instrument onboard SOHO (Solar and Heliospheric Observatory) observed comet from 23 April to 7 December 2014, and heliocentric distances from 2.2 AU before perihelion to 1.9 AU after. As a result, 138 measurements of water production rates were obtained from observations of the hydrogen Lyman-alpha coma (Combi et al. 2018). Roth et al. (2017) characterized the volatile composition of the comet on 2014 May 22 and 24 using near-infrared spectrograph.They determined production rates for six volatiles among others for H₂O. Observations of OH with the Swift spacecraft were used to measure the water production rate on 2014 May 7 at a heliocentric distance 2.04 AU (McKay et al. 2016). Observations of the forbidden oxygen lines with ARCES echelle spectrometer were used to derive water production rates on 2014 June 4 (McKay et al. 2016). To assess the shape of the gas production curve along the orbit also at distances from the Sun greater than direct measurements allowed, we used photometric data of C/2012 K1. Total visual magnitudes and CCD (charge-coupled device) magnitudes were taken from the Comet Observation Database (COBS ¹) Zakrajsek & Mikuz 2018. The collected magnitudes estimations show a considerable dispersion due to the different instrumentation, methodology, and local conditions of observations. We discarded measurements made in poor-quality local observation conditions as well as those whose total number performed by one observer was less than five. The original total magnitudes were corrected for geocentric distance to derive heliocentric magnitudes, m_h. Next we translated each m_h into water production rate, Q_H2O (molecules s⁻¹) by means of an empirical law given by many authors: |$\text{log} Q_{\mathrm{ H2O}} = A - B\, m_\mathrm{ h}$|⁠. We have established the first coefficient, A = 30.7417 ± 0.0116 by comparing the levels of the brightness estimations and the observed production rates. The value of the second coefficient B = 0.2453 is taken from Jorda, Crovisier & Green (2008). Observed water production rates versus heliocentric distance are shown in Figs 1, 3, and 4 by big symbols with error bars. The water production rates derived from the brightness measurements Q_{|$\mathrm{ w}$|} are marked with small symbols, see Figs 3 and 4. All used values of emissions were recalculated to kilograms per second.

Figure 1.

The production of water versus heliocentric distance, calculated using equation (7) and different sets of parameters. For comparison, we shown the observed production of water from comet C/2012 K1 PanSTARRS.

Figure 2.

The theoretical production of water from the model comet calculated using equation (7), when the porosity was 0.87 (v_d = 0.07, v_i = 0.065) and ϵ was the model parameter. The profiles are drawn for the heliocentric distance R_h > 2 AU, because closer to the Sun the profiles are indistinguishable.

Figure 3.

The comparison of the theoretical production of water with the observational data. The latter are of two types: observed water production rates (the values with error bars) and the Q_{|$\mathrm{ w}$|} derived from the brightness estimations of comet C/2012 K1 (small circles). The data set covers wide range of the heliocentric distances up to 8.7 AU. However, at the distance larger than 4 AU the activity of comets is driven mostly by the sublimation of ices more volatile than H₂O ice.

Figure 4.

The results of simulations performed using simplified models including popular equation (8) instead of the experimentally tested equation (7) and the thermal conductivity constant and variable. The material parameters are the same as in Fig. 3.

3 RESULTS

In this section, we presented the calculated rate of production of water from the model comet moving in the orbit of C/2012 K1.

In Fig. 1, we presented the results obtained using equation (7), when the assumed porosity was 0.46 (v_d = 0.28, v_i = 0.26), and 0.87 (v_d = 0.07, v_i = 0.065). The material was assumed to be composed only of large agglomerates (the radius of pores r_p = 2.5 mm), or small and large agglomerates (r_p = 0.5 mm). The temperature dependence of the thermal conductivity was taken into account (equation 5). For comparison, we have shown the observed production of water from comet C/2012 K1. We note, that the simulations were performed only for the pre-perihelion branch. Correspondingly, only the pre-perihelion data were presented.

Close to the perihelion the production of water corresponding to the porosity 0.46 (upper panel of Fig. 1) is about three times lower than observed, while at the porosity 0.87 (middle- and lower panel) the calculated production of water can be in agreement with the observations.

Other effect is the heat outflow from the warmed surface to the cold interior of the nucleus. This process depends on the characteristic radius of pores r_p and the thermal conductivity of the agglomerates. Comparison of the profile corresponding to r_p = 2.5 mm and λ_solid = 10 mW m⁻¹ K⁻¹ (middle panel of Fig. 1) with the profile for r_p = 0.5 mm and λ_solid = 10 mW m⁻¹ K⁻¹ (lower panel of Fig. 1) shows very little difference. However, the profiles corresponding to the different values of the thermal conductivity λ_solid are visibly different. The effect is strongest when the radius of pores is small (lower panel of Fig. 1).

In Fig. 2, we presented the results obtained, when the thermal conductivity was constant. The values of ϵ were the model parameters. The remaining parameters were: the porosity 0.87 (v_d = 0.07, v_i = 0.065) and the radius of pores r_p = 5.0 mm.

The influence of the effective emissivity is small, much smaller than the the scatter of the data points. The ratio of the rate of production of water corresponding to ϵ = 0.5 (possible for a very rough surface) to the rate of production of water at ϵ = 0.9 is: 1.19 at R_h = 4 AU and only 1.02 at R_h = 2 AU. Hence, close to the perihelion the poorly constrained effective emissivity of the rough surface can be assumed to be the same as the emissivity of the flat surface. The influence of the characteristic radius of pores r_p is also only minor.

Small influence of ϵ and r_p together with small probably inaccuracy of the albedo A, the relation between the ratio v_d/v_i and probably mass ratio dust-to-ice indicate that there are three important and not well-constrained model parameters: λ_solid, ψ, and f_act.

The thermal conductivity λ_solid has clearly visible influence on the slope of the profile F_H2O(R_h). When λ_solid = 10 mW m⁻¹ K⁻¹, then the production of water exhibits too strong dependency on R_h, while the changes of λ_solid within the range 0.4–2 mW m⁻¹ K⁻¹ have small influence.

The total porosity ψ and the active fraction of the surface f_act affect the rate of production of water independently on the heliocentric distance, see Figs 3 and 4. The observations are well reproduced when ψ = 0.87, but ψ = 0.73 is too low.

The active fraction of the total surface f_act should be large. The best fit corresponds to f_act ∼ 0.7 and 2/3 of the data point lie between the profiles of F_H2O corresponding to f_act = 0.57 and f_act = 0.86.

In Fig. 3, we presented the selected theoretical profiles of the rate of production of water, as well as the observed water production rates and the values of Q_{|$\mathrm{ w}$|} calculated from the observed brightness of the comet. The data set covers wide range of the heliocentric distances up to 8.7 AU. However, at the distance larger than 4 AU the activity of comets is driven mostly by the sublimation of CO ice.

In Fig. 4, we presented the results of simulations performed using simplified model including popular equation (8) for the sublimation flux instead of the experimentally tested equation (7). The thermal conductivity is either variable, or constant with the values close to zero: λ = 0.1 mW m⁻¹ K⁻¹, and λ = 0.001 mW m⁻¹ K⁻¹. The material parameters are the same as in Fig. 3. The active fraction of the nucleus surface is f_act = 0.7, as for the curve best matching the data in Fig. 3. It can be seen, that the rate of production of water calculated using the simplified model is lower than predicted by our base model. The difference is highest at high heliocentric distance, which results in an overestimate of the steepness of the curves by the simplified models. We note, that the models can be further simplified assuming instantaneous equilibrium of the surface temperature i.e. ignoring the term on the left-hand hand side of equation (1). This simplification can result in additional differences, most likely increase of the calculated rate of sublimation.

4 DISCUSSION AND SUMMARY

In this work, we presented numerical simulations of the production of water from a comet in the specific situation, when the dust mantle does not accumulate on the surface of the nucleus. The whole non-volatile dust released during sublimation of ice is either ejected to space, or slides down the slopes of different sizes. This may be the case when the comet approaches the Sun for the first time i. e. is dynamically new. Such comets may have rough surfaces initially free, or almost free of dust. Thus, the release of dust due to the sublimation of ice from slopes of sufficiently high inclinations may be followed by the sliding of dust. The sliding dust is assumed to accumulate beneath the slopes. This process may result in some decrease of the active surface. The significance of this effect should depend on the unknown height of slopes and the travelling distance of the slid material. In this work, we investigated only the period before the first approach of the considered comet to the Sun. Thus, the accumulation of dust was not taken into account. The inclination angle of slopes is assumed to be high enough to cause sliding of dust already when the model comet is at the heliocentric distance 5 AU. We performed laboratory experiments using illumination corresponding to the heliocentric distance 1.2 AU (work under review). The results indicate, that the sublimation of moderately volatile water ice may result in sliding of dust at the inclination as low as 20°. The critical inclination of the slopes yielding occurrence of landslides at higher heliocentric and its dependence on the content of highly volatile ices needs further investigation.

The considered mathematical model is able to reproduce the production of water derived from different observations of comet C/2012 K1 PanSTARRS in the range of the heliocentric distance R_h < 3.5 AU i.e. when the sublimation of water may be responsible for the activity of the comet. The material composing nucleus should be characterized by high porosity (ψ ∼ 0.9), low thermal conductivity of the material, and small characteristic radius of pores r_p ∼ 0.5 mm. The active fraction of the surface should be very large, f_act ∼ 0.7. Moreover, the nucleus of comet C/2012 K1 may be smaller than the model nucleus considered in this work. In such case, the active fraction of the surface need to be even larger, f_act > 0.7. Large active area predicted by the presented simulations is consistent with the assumption, that the surface of comet C/2012 K1 is very rough.

ACKNOWLEDGEMENTS

This work was supported by Poland’s National Science Center (Narodowe Centrum Nauki) [decision no. 2018/31/B/ST10/00169]. We thank anonymous reviewers for their valuable comments.

DATA AVAILABILITY

There are no new data associated with this article.

Footnotes

1

2018, Observations from the COBS International Database, https://www.cobs.si