Cliff collapse on Comet 67P/Churyumov–Gerasimenko – II. Imhotep and Hathor

Cliff collapses on Comet 67P/Churyumov–Gerasimenko expose relatively pristine nucleus matter and offer rare opportunities to characterise ice–rich comet material. Here, Microwave Instrument for Rosetta Orbiter (MIRO) observations of two collapsed or crumbling cliffs in the Imhotep and Hathor regions have been assembled. The empirical diurnal antenna temperature curves are analysed with thermophysical and radiative transfer models in order to place constraints on the physical properties and degrees of stratification in the near–surface material. The Imhotep site consists of an exposed dust/water–ice mixture with thermal inertia 100–160 J m − 2 K − 1 s − 1 / 2 , having sublimating CO 2 ice located 11 ± 4 cm below the surface. Its estimated age is consistent with an outburst observed on 2014 April 27–30. The Hathor site has a 0 . 8 ± 0 . 2 cm dust mantle, a thermal inertia of 40 ± 20 J m − 2 K − 1 s − 1 / 2 , no CO 2 ice to within 1 . 0 m depth, and a mantle bulk density of 340 ± 80 kg m − 3 that is higher than the theoretically expected 180 ± 10 kg m − 3 , suggesting that compression has taken place.


INTRODUCTION
One of the biggest surprises during the Rosetta mission (Glassmeier et al. 2007;Taylor et al. 2017) to Comet 67P/Churyumov-Gerasimenko (hereafter, 67P) was the discovery of pervasive, concentric layering of each nucleus lobe (Massironi et al. 2015;Penasa et al. 2017).Due to differential erosion rates, these layers are manifested on the surface as intricate staircase systems of cliffs and plateaus (Sierks et al. 2015;Thomas et al. 2015) with spectrophotometrically peculiar properties (Ferrari et al. 2018;Tognon et al. 2019;Davidsson et al. 2022b).The cliffs are major sources of activity and producers of coma jet-like features (Vincent et al. 2016b(Vincent et al. , 2017)).The frequently occurring talus cones and boulders underneath cliffs (Pajola et al. 2015) show that mass wasting, ranging from small-scale crumbling to large-scale collapse of 100 m-sized wall segments (Pajola et al. 2017a), is an important part of comet nucleus evolution.
The collapse of cliff walls suddenly exposes deep and relatively pristine nucleus material that has been largely shielded from solar heating by the significantly more processed dust mantle material that is dark, ice-poor, and dominated by refractory silicates and organics (Fornasier et al. 2015;Capaccioni et al. 2015;Quirico et al. 2016;Mennella et al. 2020).Collapses provide rare opportunities to study the icy component of comet material, and to characterise physical properties that are more representative of the nucleus bulk than the surface.Developing a better understanding of comet nucleus interiors is crucially important in order to understand their formation and evolution, and by extension, constraining the properties of the Solar nebula in which they formed.
★ E-mail: bjorn.davidsson@jpl.nasa.govDavidsson (2024, hereafter, Paper I) analysed observations by the microwave instrument Rosetta/MIRO (Gulkis et al. 2007) of the Aswan cliff before and after its collapse.By forward-modelling the thermophysics of specific nucleus models, and feeding the resulting temperature profiles to a radiative transfer solver that generates synthetic antenna temperature curves, it is possible to constrain the composition, stratification, thermal inertia, diffusivity, extinction coefficients, and single-scattering albedos of the collapse sites if and when those curves are matching the MIRO observations.The current paper extends this type of analysis to two other sites, located in Imhotep and in Hathor (for nucleus regional names and definitions, see Thomas et al. 2018).
In Paper I it was found that the pre-collapse Aswan site dust mantle had a thickness of at least ℎ m ≥ 3 cm and a thermal inertia of Γ ≈ 30 J m −2 K −1 s −1/2 (hereafter MKS).The post-collapse material had an estimated dust/water-ice mass ratio of  = 0.9 ± 0.5, a diffusivity of D = 0.1 m 2 s −1 , a thermal inertia of Γ = 25 ± 15 MKS, and the observations were consistent with the independently estimated carbon dioxide molar abundance CO 2 /H 2 O = 0.32 relative to water (Davidsson et al. 2022a), and a nucleus bulk density of  bulk = 535 kg m −3 (Jorda et al. 2016;Preusker et al. 2017;Pätzold et al. 2019).A thin (ℎ m ≤ 3 mm) dust mantle developed ∼ 7 months after the collapse, having a thermal inertia in the Γ = 10-45 MKS range.The sublimation front depth of the CO 2 supervolatile (sv) was ℎ sv = 0.4 ± 0.2 cm after 5 months, ℎ sv = 2.0 ± 0.3 cm after 7 months, and ℎ sv = 20 ± 6 cm after 11 months.In the current paper, attempts are made to place similar constraints on the material at the Imhotep and Hathor collapse sites.
The Imhotep collapse site is seen in Rosetta/OSIRIS (Keller et al. 2007) images shown in Fig. 1 (all OSIRIS images are available at the NASA Planetary Data System, PDS 1 ).The context image (left panel) is dominated by the large lobe, centrally showing the large smooth-terrain plain of Imhotep.At its border, where the landscape topography becomes increasingly complex, lays the cliff wall in question.This collapse site, easily recognisable by its higher albedo (right panel) was already present when Rosetta arrived (in fact, this image was acquired less than two weeks after arrival).Figure 2 shows another view, at somewhat better resolution.The bright scar appears rather featureless at ∼ 0.76 m px −1 resolution, in contrast to the rougher and darker terrains surrounding it.
For RGB colour-composite images of the Imhotep collapse region, see Fig. 7B in Auger et al. (2015) and Fig. 8d in Pommerol et al. (2015).Such images reveal that the bright regions are spectrally bluer than their surroundings.Cross-comparison between OSIRIS spectrophotometry and Rosetta/VIRTIS (Coradini et al. 1998) spectroscopy show that bluish terrain displays the characteristic ∼ 3 m absorption feature of water ice (Barucci et al. 2016).It is therefore evident that water ice is abundant at the collapse site, while it is virtually absent in the surrounding dust mantle material.
Spectrophotometrically bluish terrain is also found elsewhere.One particular region that attracted attention early in the Rosetta mission was the part of Hathor dubbed the 'alcove' by Thomas et al. (2015).An RGB colour-composite image of the alcove is seen in Fig. 5 (left) of that paper.The region is also shown here in Fig. 3. Thomas et al. (2015) describe ≤ 10 m bright spots on the cliff wall, and additionally mention 2-5 m spots in the underlying talus that are 20 per cent brighter and bluer than the surroundings.It therefore seems like this cliff wall was actively experiencing small-scale crumbling in 2014 August, that exposed ice-rich material on the wall and deposited ice-rich debris on the valley floor underneath.However, it is clear that the majority of the surface area is covered by dark and refractory 1 https://pds-smallbodies.astro.umd.edu/data_sb/missions/rosetta/index_OSIRIS.shtmlFigure 2.This image shows the Imhotep collapse site, at somewhat higher resolution than in Fig. 1.To brighten the overall image and enhance visibility, the radiance factors of a few unusually bright pixels have been re-set to lower values.They appear as small grey patches within whitish terrain and are artificial.Image MTP7/n20140905t080224552id4ff71.img (Sierks & the OSIRIS Team 2020b) that was acquired on 2014 September 5 with ∼ 0.76 m px −1 resolution when Rosetta was 42.8 km from the comet.material.This could mean that it once resembled the more recently collapsed Aswan and Imhotep cliffs, but now has aged to the point that the dust mantle largely has recovered its pre-collapse appearance.Alternatively, the alcove represent another type of wall on 67P that The alcove is located on the small lobe.A small fraction of the large lobe is visible in the foreground in the lower third of the panel.Note the morphological differences between these parts of the small and large lobes, being dominated by consolidated and smooth terrains, respectively.Right: This is a zoom on the red square from the left panel.Spectrophotometry reveals unusually bright and bluish material at and beneath the Hathor alcove, though much of its surface is covered by dark dust.Both panels show image MTP7/n20140912t195732364id4ff22.img (Sierks & the OSIRIS Team 2020b) that was acquired on 2014 September 12 with ∼ 0.53 m px −1 resolution when Rosetta was 29.9 km from the comet.
is not prone to large-scale collapse, but merely fragments one 10 mblock at a time.However, it is perhaps relevant that the Hathor alcove lies at a peculiar boundary to the region Anuket.The boundary is very sharp and gives the impression of a brittle material from which a big chunk has been broken off along a distinct crack, thus revealing the alcove that is stratigraphically interior to Anuket (El-Maarry et al. 2015).However, there is no way of knowing if that was a one-time big event, or a long series of much smaller events.

METHODOLOGY
The overall approach, and the numerical models being employed here, are very similar to those of Davidsson et al. (2022c) and Paper I. Therefore, the information provided here is very brief in the interest of saving space, and the reader is strongly encouraged to consult the previously mentioned publications for further details.
The first step is to identify the two regions on the 67P shape model.This is necessary both for being able to calculate illumination conditions that are specific to the regions, and to facilitate searches in the MIRO database for suitable observations.The nucleus shape model SHAP5 version 1.5 (Jorda et al. 2016), is here degraded from 3.1 • 10 6 to 5 • 10 4 facets.The regions are located through visual inspection and cross-comparison of images and the shape model.
The Imhotep site is represented by 7 facets, with a largest mutual separation of 66 m (providing a measure of the size of the region).The number of 'terrain facets' (having the geometric capability of shadowing and self-heating the Imhotep site) is 1,404 and the number of 'surrounding facets' (being capable of shadowing the terrain facets) are 10,629, which means that 12,033 facets were included in the illumination condition calculations.The Hathor site is represented by 20 facets, with a largest mutual separation of 105 m.The number of terrain facets is 2,509.The number of surrounding facets is 14,004, so that a total of 16,513 facets were included in the illumination condition calculation.
The model by Davidsson & Rickman (2014) is used to calculate illumination conditions for Imhotep and Hathor.These calculations take the nucleus shape, topography, spin axis orientation, and the rotational phase (including corrections due to the time-dependent rotational period of the comet) into account during orbital motion, when calculating the direct solar flux onto a given Imhotep or Hathor facet.The equilibrium temperature is calculated for illuminated terrain facets, and parts of their thermal reradiation illuminates the facets at the Imhotep and Hathor sites according to the mutual view factors.The surrounding facets are used to calculate which terrain facets are being in shadow (if so their self-heating contributions are switched off).
The illumination conditions at each of the 7 Imhotep facets and 20 Hathor facets are evaluated every 20 min from the 2012 May 23 aphelion to the end of 2015 January (no MIRO data beyond that date was included in this study).Because illumination conditions change slowly with time, and because these calculations are very time-consuming, the actual conditions were evaluated every 12 th nucleus rotation and copied to the following 11 ones.At a given nucleus rotational phase, the area-weighted mean fluxes were calculated for the 7 Imhotep and 20 Hathor facets, respectively.These mean illumination conditions were fed to the thermophysical model.
The thermophysical model employed here is called the 'Numerical Icy Minor Body evolUtion Simulator' or nimbus.It is fully described by Davidsson (2021), but also see Davidsson et al. (2022c) and Paper I for usage specific to MIRO data analysis.nimbus is here used to study porous mixtures consisting of refractories, hexagonal (crystalline) water ice, and CO 2 .Other capabilities of nimbus, including amorphous ice and release of trapped CO and CO 2 during crystallisation into cubic water ice, the cubic-hexagonal transition and associated CO and CO 2 release, segregation of CO 2 : CO mixtures, and the sublimation of CO ice, are not utilised here.nimbus solves a coupled system of differential equations describing the conservation of energy and masses of ices and vapours.These equations describe the solid-state and radiative transport of heat, the energy consumption during H 2 O and CO 2 sublimation, the transport of energy (advection) and vapour mass during gas diffusion within the porous medium (along temperature and vapour partial pressure gradients), and energy release during recondensation of H 2 O and CO 2 vapour.Nominally, both heat transport and gas diffusion are calculated in two spatial dimensions (radially and latitudinally) for a spherical body with illumination conditions specific for each latitude during nucleus rotation and orbital motion.However, here the variant nimbusd is employed, which sacrifices latitudinal mass and energy transport (which are not important for the relatively short time scales considered here) in order to enable erosion of the upper surface in response to outgassing.In any case, each latitude is here fed with the same Imhotep or Hathor illumination conditions.The ices are considered finite resources, which means that water ice withdraws gradually and leaves behind a dust mantle that only consists of dust (unless vapours recondenses temporarily during night time).The CO 2 withdraws as well, resulting in a stratified near-surface region that contains a layer between the dust mantle and the CO 2 sublimation front that only contains refractories and water ice.The porosity of the nucleus evolves as a result of ice sublimation and vapour recondensation, which affects both the effective heat conductivity, heat capacity, and diffusivity of the medium.In case outgassing is so strong that the entire dust mantle erodes away, the CO 2 activity will start to erode the dust and water ice mixture as well.All expressions for heat conductivity of compacted substances, specific heat capacities, latent heats, and vapour saturation pressures are temperature-dependent and species-specific, and are based on expressions established in laboratory measurements.In short, nimbus is a state-of-the-art thermophysics code that includes all processes that typically are considered important for comet activity in the thermophysics literature.
The final outcome of a nimbus simulation of largest importance here is the nucleus temperature as function of depth and time.These temperature profiles are passed to a radiative transport code called themis, described fully by Davidsson et al. (2022c), but also see Paper I. For a given temperature profile, themis calculates the radiance of emitted radiation as function of emergence angle.The code is evaluated for the specific wavelengths at which MIRO is observing -the millimetre (MM) channel at wavelength  = 1.594 mm and the submillimetre (SMM) channel at  = 0.533 mm.The code uses wavelength-specific extinction coefficients ( MM and  SMM ) and single-scattering albedos ( MM = 0 in this type of problems, but  SMM > 0 is employed when required).The radiances are recalculated to antenna temperatures, that can be directly compared with MIRO measurements.The final output of themis are diurnal MM and SMM antenna temperature curves, evaluated for the relevant emergence angle at each rotational phase according to the MIRO observational geometry.
Ideally, MIRO should have observed the Imhotep and Hathor collapse sites continuously throughout an entire nucleus rotation.However, such observations did not take place.Instead, the instrument observed the regions briefly at various times, separated by several days.Because the illumination conditions change very little with time during sufficiently short time intervals, and because the nucleus activity is highly regular (as evident from the 'clockwork repeatability of jets from one rotation to the next'; Vincent et al. 2016a), MIRO observations acquired during a few weeks are therefore binned and time-shifted to a common 'master period' in order to build em-pirical diurnal antenna temperature curves.Typically ±2.5 K error bars are assigned to the bins.More detailed accounts of how the MIRO antenna temperature curves were constructed are given for the individual locations in sections 3.1 and 3.2.
The MIRO observational archive (PDS website2 ; Hofstadter et al. 2018a,b) contains information about the beam centre interceptions with the nucleus surface (radial, latitudinal, and longitudinal coordinates) of each observation.Because these are known for the Imhotep and Hathor collapse site from the identification of relevant facets on the shape model, the archive can readily be searched for the relevant observations.
The quality of a given synthetic diurnal antenna temperature curve, with respect to the empirical one, is determined by calculating  2 residuals and the incomplete gamma function value  (see Press et al. 2002;Davidsson et al. 2022c, and Paper I).Typically, the modelling is performed in stages, starting with refractories-only models, then moving on to dust+H 2 O and finally dust+H 2 O + CO 2 variants if no fits can be found for the more simple media.The parameter values of nimbus (e. g., effective thermal inertia, diffusivity, composition, and degree of stratification) and themis (extinction coefficients and single-scattering albedos) are gradually estimated by generating a number of test cases, check how the synthetic antenna temperature curves changes in response to those parameter settings, and trying to tweak them until the model fits the data.

The Imhotep collapse site: 2014 Dec and 2015 Jan
The Imhotep collapse site is ∼ 70 m across.The Full-Width Half-Maximum (FWHM) diameters of the MM and SMM beams are 207 m and 65 m at a nucleus distance of 30 km, respectively (Schloerb et al. 2015).In order for the collapse site to dominate the radiation sensed by the beams, Rosetta should ideally be closer than 30 km to the comet.During the mission this was predominantly the case during three periods: 1) 2014 September through 2015 January; 2) the first half of 2016 March; 3) 2016 May through September.These periods were searched for MIRO observations of the Imhotep collapse site.Unfortunately, this region was rarely observed.The highest observational frequency was in 2014 December with 21 data points, and in 2015 January with 53 data points.The data were averaged within 2.4 min-wide bins (roughly corresponding to 1 • nucleus rotation).The bins were time-shifted to a common master period starting 2014 December 22, at 22 : 26 : 48 UTC (corresponding to daynumber3  n = 356.7).
2014 December and 2015 January contributed 8 and 4 bins, respectively.The viewing geometries during each set of observations were scrutinised individually, in order to weed out those where the beams sampled too much adjacent terrain, sometimes with drastically different illumination conditions (e. g., the collapse site being in shadow, but the beams including foreground or background fully illuminated terrain).Among the 2014 December bins, five had to be rejected, while three were selected.Figure 4 (left panel) shows an example of the viewing conditions for a selected bin.In this case, the collapse site is clearly visible, it lies entirely in darkness, and fills most of the MIRO beams.Among the 2015 January bins, three had to be rejected, while only one was selected.Figure 4 (right panel) shows an example of the viewing conditions of a rejected bin.Here, the collapse site is viewed so obliquely that it only fills a small fraction of the beams.
The final Imhotep temperature curve thus consisted of merely four bins, sampling early and late morning, as well as late night.The first observations used for the curve were acquired on 2014 December 8, and the last on 2015 January 15, thus spanning a period of 38 days.Note, that the two late-night bins near  n = 357.23 were acquired at almost the same rotational phase but one week apart in real time.The bins are shown in Fig, 5.The MM bins (left panel) differ by 4.3 K, i.e., their ±2.5 K error bars overlap.The SMM bins (right panel) differ by merely 0.4 K.The reason for the differences is partially due to the dependence of emitted radiance on emergence angle (being  = 14 • and  = 24 • for the two bins), and partially because of a small level of temperature dispersion within the beams (this is particularly the case for the larger MM beam).This exemplifies that the nearsurface temperatures are similar from one nucleus revolution to the next, as assumed when the time-shift were made.A further example concerns the late-morning bin, observed on 2015 January 6, or about 29 nucleus revolutions after the initiation of the master period.It is the only one acquired in full daylight, and sensitive to the applied solar flux at the time (the other points where acquired when only nucleus self-heating contributed flux).The master-period illumination flux at this rotational phase was 74.3 J m −2 s −1 , whereas the actual level of illumination at the time of observation was 70.3 J m −2 s −1 .This is a difference of 5 per cent in flux, corresponding to a difference of 2.6 K in terms of equilibrium temperature.This illustrates that usage of master period fluxes at the time of observation introduces small errors.

Refractories only
The brightness of the Imhotep collapse site, as well as its spectrally blue colour (Auger et al. 2015;Pommerol et al. 2015), excludes that the surface is covered by a dust mantle.Yet, it is interesting to compare its MIRO antenna temperatures with those predicted by nimbus and themis models for refractory material.The reason is primarily that the empirical antenna temperature curve consists of few bins, and it is important to understand whether this type of models can fit those particular available parts of the curve.If fits are possible, the evidence for presence of sublimating ice that MIRO might bring (in addition to the OSIRIS brightness and colour information), cannot be considered particularly strong.However, if dust-only models do not fit, but models with sublimating ice work better, it could be considered a MIRO-based verification that ice is exposed at the surface, that is independent from that of OSIRIS.
Three nimbus models with different porosities (0.770 ≤  ≤ 0.805) where considered (see Fig. 5 for additional parameter values).Because the Shoshany et al. (2002) formalism was used to calculate the Hertz factor (the reduction of heat conductivity with respect to that of a compact substance due to porosity), this resulted in models having thermal inertia fluctuating around ∼ 20, ∼ 30, and ∼ 50 MKS.This range is relevant based on previous work on the thermal inertia of 67P (e. g.Schloerb et al. 2015;Marshall et al. 2018;Davidsson et al. 2022c, Paper I).First, attempts were made to fit the MM data. MM was adjusted in themis until the model fitted the late-night bins.The result can be seen in Fig. 5 (left panel).For the lowest thermal inertia the model is 2.3 K warmer than the nominal late-morning bin (at 66 per cent of the noon illumination flux of 145 J m −2 s −1 ), and barely consistent with its upper error margin.Increasing the thermal inertia only makes the model warmer (by ≤ 8.7 K).However, all models are 4.3-6.9K too cold at the early morning bin (at 10 per cent of the noon illumination flux), the worst being the model with low thermal inertia.In essence, the modelled morning slope is too steep compared with the data, when  MM is chosen to force a fit late at night.The best such model (R13_001A with ∼ 30 MKS) has  2 MM = 8.9 for  MM = 70 m −1 .This residual can be reduced somewhat if relaxing the requirement that the model curve should pass between the two late-night bins.Model R13_003A with ∼ 20 MKS has  2 MM = 6.5 for  MM = 50 m −1 .
A similar investigation was made at the SMM wavelength (see Fig. 5, right panel), with similar results.When all models were made to fit late at night by adjusting  SMM , they were under-shooting in the early morning (by 2.5-5.7 K, improving as Γ increased), and over-shooting in the late morning (by 1.3-9.3K, deteriorating as Γ increased).Again, the modelled morning slope appears to be steeper than the data and this problem cannot be fixed merely by Γ and  MM adjustments.The best of these models where R13_001A with ∼ 30 MKS, with  2 SMM = 4.9 for  SMM = 100 m −1 .By relaxing the requirement to fit the late-night bins, a somewhat lower  2 SMM = 3.1 for  SMM = 80 m −1 could be obtained for the same thermophysical model.
In terms of simultaneous MM and SMM fits, model R13_003A is the best with  MM = 0.011 for  MM = 50 m −1 and  SMM = 0.017 for  SMM = 100 m −1 , noting that a  > 0.01 success-criterion has been applied by Davidsson et al. (2022c) and in Paper I based on recommendations by Press et al. (2002).Therefore, in a blind test, these MIRO observations would not be sufficient to exclude that a dust mantle is being observed.The fact that exposed ice fills most of the beams is therefore entirely based on OSIRIS observations.

Refractories and H 2 O
The OSIRIS observations suggest that the Imhotep collapse site has surface ice at the time of observation.Therefore, a number of nimbus models considered mixtures of refractories and water ice.First, a dust/water-ice mass ratio  = 1 and diffusivity based on tube lengths and radii { p ,  p } = {100, 10} m and tortuosity  = 1 were applied (see Paper I for their relation to D), based on previous reproductions of the 67P water rate production curve (Davidsson et al. 2022a).It was assumed that the original material had contained carbon dioxide at a molar abundance CO 2 /H 2 O = 0.3 relative to water (Davidsson et al. 2022a, Paper I), and had the same bulk density  bulk = 535 kg m −3 as the nucleus (Preusker et al. 2017).Removal of the CO 2 , without any other structural changes than the resulting increase of porosity , yielded  = 0.73 and  bulk = 390 kg m −3 for the remaining dust/water mixture.The heat conductivity was corrected for porosity according to Hertz factors ℎ following Shoshany et al. (2002), resulting in a thermal inertia reaching ∼ 100 MKS at day.If water ice is removed from the surface layer, a  = 0.94 porosity dust mantle is left behind.A lower limit ℎ ≥ ℎ min = ℎ( = 0.79) was enforced to prevent the dust mantle from having a thermal inertia falling below ∼ 20 MKS.
Such a medium was modelled without erosion during a period of three weeks just prior to the master period, to understand the level of water activity.In this time, the water withdrew merely ∼ 2.5 mm, showing that the water ice was almost dormant.The main effect of adding water is therefore to produce a denser medium with higher volumetric heat capacity, and lower porosity which leads to a higher heat conductivity, thus a higher thermal inertia.Figure 6 (left panel) compares the MM curves for R13_001A without water ice (from Fig. 5) with R13_005A (considering water ice, including at the surface) for the same extinction coefficient  MM = 70 m −1 .Increasing the thermal inertia has the effect of elevating the MM antenna temperature, and more significantly at night than at day.The quasi-fit of the dust model is thereby ruined ( 2 MM increases from 8.9 to 47).Increasing the extinction coefficient has the effect of lowering nighttime temperatures, but elevate daytime temperatures.Figure 6 (left panel) shows that the error bars of the late night bins are barely reached when  MM = 200 m −1 , and additional simulations show that the antenna temperature is not reduced further by stronger extinction.However, because of increased discrepancies in the late morning, this attempt to improve the fit late at night only leads to a somewhat higher  2 MM = 51.An  MM this high may be nonphysical, because it would suggest an opacity so high that the MM channel only probes the top ∼ 5 mm.Furthermore, Schloerb et al. (2015)  Figure 6 (left panel) shows that R13_005A has similar problems at SMM wavelengths, with the antenna temperature  A being too high at all bins for  SMM = 600 m −1 (and  A,SMM would not decrease more if  SMM was further increased).For this model,  2 SMM = 40.It therefore seems that a dust/water mixture performs worse than dust alone, at least when the diffusivity is as low as suggested by { p ,  p } = {100, 10} m and  = 1.This is surprising, considering that the OSIRIS observations suggests that water ice indeed is present at the surface at the collapse site.Therefore, higher diffusivities were considered to increase the degree of sublimation cooling, including one based on { p ,  p } = {10, 1} cm and  = 1.Such large values appear to have been common post-perihelion on the northern hemisphere (Davidsson et al. 2022a), presumably due to the replenishment of airfall debris around perihelion, that have typical sizes in the millimetre-decimetre range (Mottola et al. 2015;Pajola et al. 2016Pajola et al. , 2017b)).On the inbound orbit, the much lower diffusivity used above as default is typical, suggesting some sort of pulverisation and settling of material around aphelion.If such a transition indeed took place, some locations may have persisted at high diffusivities for longer.For example, Davidsson et al. (2022c) found that a location in Hapi appears to have had a three orders of magnitude drop in diffusivity between 2014 October and November.However, airfall debris would not accumulate on a steep cliff side.Still, there might be other mechanisms having similar effects.In Paper I it was found that { p ,  p } = {1, 0.1} cm and  = 1 characterised the Aswan collapse site five months after the wall fell, and I speculated that vigorously sublimating CO 2 (estimated to be located 0.4 ± 0.2 cm below the surface) was responsible for 'puffing up' the surface layer and increase the diffusivity.Here, it is assumed that a similar mechanism might have been active at the Imhotep site.
Increasing the diffusivity to a level determined by { p ,  p } = {10, 1} cm and  = 1 intensifies sublimation only slightly.In three weeks, without erosion, the dust mantle thickness increases from ∼ 0.25 to ∼ 1.3 cm.Adding  e = 2 H 2 O (kg m −2 s −1 ) erosion during the last week before the master period brings the water ice to the surface and maximises the cooling due to its sublimation.Figure 6 (left panel) shows that the MM antenna temperature becomes somewhat lower (R13_005AC), particularly during day.The late night MM antenna temperature approaches the data bins only if  MM ≥ 100 m −1 (the 200 m −1 case is shown here to allow direct comparison with the low-diffusivity case R13_005A).Yet, the model is ∼ 13 K warmer than the late-morning bin, which is typical of all models including water.At SMM wavelengths ( Fig. 6, right panel) there are similar discrepancies, reaching ∼ 12 K.Both displayed R13_005AC models have  2 MM ≈  2 SMM ≈ 30.Lowering to  MM = 50 m −1 and  SMM = 200 m −1 worsen the fits late at night (6-9 K discrepancies) but improve them in the late morning, lowering overall residuals to  2 MM ≈  2 SMM ≈ 18.Therefore, not even the strongest achievable water sublimation cooling is capable of cancelling the effect of increased thermal inertia, introduced by the presence of water itself.However, it indicates that sublimation cooling by the more volatile CO 2 is worth testing.

Refractories, H 2 O, and CO 2
The first model that included CO 2 (R13_005B) had the default parameters  = 1, molar CO 2 /H 2 O = 0.3,  bulk = 535 kg m −3 , { p ,  p } = {100, 10} m and  = 1.Such a medium was first propagated during a period of three weeks from an initially homogeneous state, assuming that no erosion took place.At the time of the master period, the CO 2 sublimation front had withdrawn to a depth of 10.9 cm, illustrating its higher sublimation rate with respect to water.Interestingly, no dust mantle formed in this time period.The reason is the cooling effect of CO 2 , that quenched the water production rate substantially with respect to models R13_001A and R13_001AC.
The CO 2 -free top layer, consisting of refractories and water ice, has the same ∼ 100 MKS thermal inertia as the models in section 3.1.2.Yet, because of the stronger cooling of CO 2 sublimation compared to that of H 2 O, the temperature in the near-surface layer is reduced.Thereby, the antenna temperatures are more easy to reproduce, than for media where the only volatile is water ice.That can be seen in the left panel of Fig. 7, where the model has  2 MM = 8.9 and  MM = 0.003 when forced to pass between the two late-night bins by setting  MM = 70 m −1 .The model works even more convincingly at SMM, seen in the right panel of Fig. 7. Here, the residuals are characterised by  2 SMM = 3.2 and  SMM = 0.072 when  SMM = 130 m −1 .A model (R13_005C) with  e = 1.5(H 2 O +  CO 2 ) erosion initiated a week before the master period was tested to see if a more shallow CO 2 front (at 6.9 cm) was compatible with the data.In this case, the model curve was too cold, with  2 MM = 31 and  2 = 18 for  MM = 50 m −1 and  SMM = 80 m −1 respectively.Another model (R13_005D) tested the effect of increasing the thermal inertia in the dust/water layer to Γ = 160 ± 10 MKS by lowering the CO 2 abundance to CO 2 /H 2 O = 0.15 while maintaining  bulk = 535 kg m −3 .With  e = 3.0( H 2 O +  CO 2 ) erosion, this model had the CO 2 at a depth of 14.4 cm during the master period.This combination of higher thermal inertia and weaker CO 2 sublimation cooling had the interesting effect of reducing the morning curve slope, thereby bringing it closer to that of the data (see Fig. 7).However, such a model starts to lose contact with the late-night bins, yielding  2 MM = 12,  2 SMM = 13 for  MM = 50 m −1 and  SMM = 200 m −1 , respectively.
If the residuals for the various Imhotep models are considered (see Table 1 for a summary) it appears that CO 2 -rich media perform nearly as well as the dust-only media.The cooling provided by CO 2 sublimation compensates for the higher thermal inertia introduced by adding H 2 O and CO 2 ices.It is also evident that the less volatile water ice is not capable of such compensation on its own.Unfortunately, the placement of the few available bins prevents us from clearly distinguishing between the dust-only and dust/H 2 O/CO 2 cases.However, the additional observations by OSIRIS favours the exposure of ices at the surface at the Imhotep collapse site.Based on the available simulations, a tentative depth of the CO 2 sublimation front is 11 ± 4 cm.

The Hathor collapse site: 2014 Nov and Dec
The Hathor collapse site constitutes a larger target than that at Imhotep, measuring ∼ 105 m across.The highest concentration of high-resolution observations took place in 2014 November and December.First, the data acquired during the two months were considered separately.Whereas the 2014 November data set covers the night and forenoon parts of the diurnal curve, the 2014 December data set covers late afternoon, night, and morning.Because the antenna temperatures are similar at the portions of the curve where there is overlap, the two sets were merged to obtain a more complete diurnal curve.The combined dataset consisted of 171 individual 1-second continuum observations, acquired between November 8 and December 29 during a 50.6 day period.The observations took place when Rosetta was 20.6-30.4km from the comet.A master period starting on  n = 337.0452was selected (2014 December 3, 01:05:05 UTC) and usage of 2.4 min-wide binning intervals resulted in 12 data bins.Viewing and illumination conditions were scrutinised individually and all bins represented unobscured views of Hathor.For bins #1-2 Hathor was fully illuminated, while for bins #3-8 the region was in full darkness, considering the FWHM extensions of both the MM and SMM beams.The FWHM extension of the smaller SMM beam contained only dark terrain for bins #9-10, and only faintly illuminated terrain for bins #11-12.Because the SMM footprint did not include drastically different illumination conditions at any point, all 12 bins were selected for the SMM analysis.However, the larger MM footprint sampled some faintly illuminated terrain when most of the Hathor collapse site was in darkness (bins #9-10), and it sampled some shadowed terrain when most of the collapse site was faintly illuminated (bins #11-12).Therefore, bins #9-12 were rejected in the MM analysis.
Figure 8 shows the MM and SMM data in the left and right panels, respectively.To illustrate the consistency in antenna temperatures measured on different occasions but at similar nucleus rotational phases, it is noted that: 1) the three bins at  n = 337.31-337.35 were acquired on November 10, December 18 and 29, yet agree to within 1.4 K (MM) and 2.2 K (SMM); 2) the three bins at  n = 337.42-337.45 were acquired on November 8 and 29, and December 26, yet agree to within 0.5 K (MM) and 2.6 K (SMM).The four rejected MM bins at  n = 337.51-337.52 are shown for completeness (they agree to within 2.3 K).Interestingly, the daytime MM antenna temperatures are not significantly elevated above the nighttime data, i. e., the MM curve seems to have a very small amplitude (4.4 K) compared to the SMM curve (15.1 K).
It is interesting to note that the Hathor MM antenna temperatures (∼ 180 K) are much hotter than those of .This suggests that the physical conditions that are governing each site ought to be rather different.

Refractories only
Though several bright and bluish ≤ 10 m spots are seen at the Hathor alcove (Thomas et al. 2015), suggesting the presence of exposed water ice, most of the area has a dark dust coverage.The first step is therefore to test whether the MIRO antenna temperature curves can be reproduced by assuming an ice-free medium consisting of a porous assembly of refractory grains.Nominally, the nimbus simu- lations considered the standard temperature-dependent expressions for specific heat capacity and heat conductivity for compacted material used by Davidsson (2021).Assuming a compacted density  1 = 3250 kg m −3 for the dust monomer grains, and applying the Shoshany et al. (2002) heat conductivity correction ℎ due to porosity, the medium was assigned a porosity  = 0.79 in order to achieve a baseline thermal inertia of Γ ≈ 30 MKS.The resulting bulk density of the porous dust mantle was  bulk = 682 kg m −3 .
This model (R14_001A) was propagated from the 2012 May 23 aphelion of Comet 67P, up to and including the alcove master period, using the illumination conditions specific to this location in Hathor.This model had Γ = 33 ± 3 MKS during the master period (with the dispersion in thermal inertia caused by diurnal temperature variations).Other models considered 2 or 4 times lower or higher thermal inertia by multiplying or dividing ℎ by factors 4 or 16.These models used the R14_001A conditions ∼ 28 days prior to the master period as initial conditions before being propagated just beyond that period.Using themis to calculate the corresponding MM antenna temperature, the extinction coefficient  MM was adjusted until the models roughly reproduced the nighttime data (assuming a singlescattering albedo  MM = 0, as expected for rocky material at these wavelengths; Gary & Keihm 1978).
The results for a few models can be seen in Fig. 8 (left panel).Model R14_004A with Γ = 63 ± 4 MKS roughly fits the nighttime data for  MM = 50 m −1 , but the daytime temperatures are far too high.Note that the 'double-peaked' antenna temperature curve is due to a shadowing episode during daytime, when the large lobe temporarily is blocking the Sun.By reducing the thermal inertia, the near-surface layer with substantial temperature variations (i.e., the thermal skin depth) is shrinking.If this is done while keeping  MM fixed, the layer probed by MIRO becomes increasingly isothermal, and the MM curve amplitude is thus reduced as Γ is falling.However, the curve then approaches an average level that is too warm compared to the nighttime data.It can be brought down by increasing the extinction coefficient to  MM = 60 m −1 for R14_001A (with Γ = 33 ± 3 MKS) and to  MM = 110 m −1 for R14_003A (with Γ = 9 ± 1 MKS).But then the measured radiance again primarily comes from within the skin depth.As the left panel of Fig. 8 shows, it is difficult to obtain a net amplitude reduction just by adjusting the thermal inertia.
Adjusting the assumed bulk density  bulk = 682 kg m −3 does not help.As explained by Schloerb et al. (2015), the same antenna temperature curve is obtained (at a given thermal inertia, and for homogeneous media) as long as the / ratio is fixed.To verify this, model R14_001A with  = 682 kg m −3 and  MM = 50 m −1 was compared to a model R14_005A with  = 169 kg m −3 and  MM = 10 m −1 (both having ∼ 30 MKS).Having comparable / ratios, their MM antenna temperature curves were indeed very similar.If a bulk density  (thus, heat capacity ) is randomly chosen, and a fit to measured data cannot be obtained for any , there is no point in trying other bulk density values because the achievable antenna temperature curves will be the same (except that a given curve will be obtained for a somewhat different -value than previously).For this reason, a refractories-only model is not capable of reproducing the low-amplitude MM data.
At SMM wavelengths, the modelled antenna temperature is typically too high throughout the rotation period when  SMM = 0, as illustrated for model R14_003A in the right panel of Fig. 8.In order to lower the nighttime modelled values towards the data, it is beneficial to increase the extinction coefficient.This is illustrated in the figure by the difference in applying  SMM = 100 and 160 m −1 .However, a large  SMM also enhances the curve amplitude, making the daytime temperatures increasingly too high.The only way to achieve a match is to increase  SMM to the point that the model curve has the correct amplitude (though at too high absolute temperatures), and then introduce scattering in the material by considering  SMM > 0 (this pulls down the curve rather evenly at all rotational phases).Among the various models considered (50 combinations of Γ,  SMM , and  SMM ), the best fit ( SMM = 0.11) was obtained for Γ = 9±1 MKS,  SMM = 160 m −1 , and  SMM = 0.2.Acceptable solutions ( SMM > 0.01) included the entire 120 ≤  SMM ≤ 180 m −1 range for Γ = 9 ± 1 MKS and  SMM = 0.2, as well as marginal fits ( SMM ≈ 0.04) for Γ = 16 ± 2 MKS,  SMM = 110 ± 10 m −1 , and  SMM = 0.2.Despite these SMM solutions, refractory-only models are here considered failed.The most important reason is that there are no MM solutions.But additionally, the high SMM modelled antenna temperatures (here dealt with by introducing significant scattering), could be an artificial consequence of having neglected sublimation cooling.To test whether cooling by subliming ices might produce more compelling simultaneous MM and SMM solutions, the following section therefore deals with mixtures of refractories and water ice.

Refractories and H 2 O
Two different types of media consisting of refractories and H 2 O were considered.Both types assume that the material once had dust/waterice mass ratio  = 1, molar abundances CO 2 /H 2 O = 0.32, CO/H 2 O = 0.16, and bulk density  bulk = 535 kg m −3 (for motivations, see Davidsson et al. 2022a).The first type of models (R14_006A-R14_008A) then assumes that the CO 2 and CO ices Here, the effect of lowering the thermal inertia is shown, while adjusting  MM until the nighttime data are reproduced.The slight reduction of the curve amplitude associated with thermal inertia reduction is not sufficient to reproduce the daytime temperatures even for the very low Γ = 9 ± 1 MKS.
have been removed from the top region without any other structural changes to the material.That produced a dust/water-ice medium with  = 0.76 and  bulk = 337 kg m −3 , used for the simulations.In case water ice additionally is removed from the top layer, the dust mantle would have  = 0.95 and  bulk = 169 kg m −3 .A Hertz factor ceiling of ℎ ≥ 2.3 • 10 −3 was introduced to force a thermal inertia reference value Γ ≥ 30 MKS for the dust mantle.The dust/water-ice mixture would have Γ ≈ 45 MKS.Because water ice is not visible on most of the Hathor alcove surface, one should expect that such a dust mantle indeed has been formed at this location at that time.
The second type of medium (models R14_009A-R14_011A) assumes that the removal of CO 2 and CO has led to a compaction of the remaining dust/water-ice mixture.Observational reasons and theoretical mechanisms for such a compaction were primarily discussed by Davidsson et al. (2022c), but also see Davidsson et al. (2022a).Here, such a hypothetical compaction is mainly introduced to test the influence of the dust mantle heat capacity on the solution.As explained in section 3.2.1,homogeneous refractories media essentially only have two free parameters: the thermal inertia Γ and the ratio / (i.e.,  and  cannot be disentangled and determined individually).However, when the medium is not homogeneous but layered, and having a significant density discontinuity near the surface, two solutions with different  and  are not necessarily identical even when having the same / ratio and thermal inertia Γ.To investigate this effect, the denser media had  = 1 and  bulk = 535 kg m −3 for the dust/water-ice mixture, resulting in  = 0.63.In case a dust mantle forms, it would have a bulk density  bulk = 268 kg m −3 and porosity  = 0.91.In order to achieve a nominal Γ ≈ 30 MKS thermal inertia for the dust mantle, a Hertz factor ceiling of ℎ ≥ 1.4•10 −3 was used.
The first priority, once water ice has been introduced, is to understand the level of sublimation activity and the associated cooling.All models started off with the temperature profile of model R14_001A obtained 28 days prior to the master period and were propagated up to and including that period.No mantle erosion was applied, therefore the dust mantle growth directly shows the speed by which water ice withdraws below the surface.Three different values for the diffusivity were applied, based on  = 1 and { p ,  p } = {100, 10} m, {1, 0.1} mm, or {10, 1} mm.The low baseline value reproduced the inbound water production rate according to Davidsson et al. (2022a).
The intermediate and high values corresponds to diffusivities being ×10 and ×100 times higher than the baseline, respectively.A higher diffusivity facilitates escape of vapour from the sublimation front, resulting in a higher net sublimation rate and a higher front withdrawal speed (because we here have weak sublimation, whereas the net sublimation rate is insensitive to diffusivity at strong sublimation, see Davidsson et al. 2021).During the 28 days, the mantles grew to 0.8, 1.9, and 3.2 cm, respectively, for the lowerdensity models R14_006A-R14_008A.The higher-density models ( R14_009A-R14_011A) had 0.4, 0.8, and 1.3 cm thick mantles formed in the same period.Here, the withdrawal is slower because the initial concentration of water ice is higher (268 versus 169 kg m −3 ), thus requiring more time for ice removal.
These mantle formation rates are relatively low, suggesting that water-sublimation cooling may not be particularly efficient under the considered conditions.This suspicion is confirmed by Fig. 9 showing the MM antenna temperature curves for the higher-density models for the three different diffusivities when initial conditions have been adjusted to yield the same mantle thickness of 1.9 cm at the master period.If water sublimation is important, the antenna temperature should be reduced with increasing diffusivity because of a more efficient cooling.However, the curves are very similar, i. e., the diffusivity value has negligible effect on the antenna temperature.The antenna temperature is even somewhat elevated at the highest diffusivity, contrary to expectations.This is because a larger tube radius  p also increases the radiative heat transport, here resulting in a slightly warmer interior.A similar investigation was made for the lower-density models when the mantle thickness was reduced to 0.8 cm (thus the degree of sublimation quenching caused by the mantle is even weaker).Yet, there was no difference between the low and intermediate diffusivity values, compared to a case where water sublimation was switched off completely by artificially forcing the water saturation pressure to zero.From a MIRO observational point of view, the water ice can be considered completely dormant at the Hathor alcove.That is to say, while the long-term effect of water sublimation is clearly seen in terms of dust mantle growth, the energy consumption during a given nucleus revolution is too small to have a measurable effect.
Next, the influence of the assumed bulk density (and resulting heat capacity) is investigated.Figure 10 shows lower-density Γ ≈ 30 MKS model R14_007A that developed a 1.9 cm mantle after 28 days for the intermediate diffusivity value.At  MM = 20 m −1 the model reproduces the nighttime MM data, though the curve amplitude is too high, resulting in a daytime temperature excess.This model has / = 8.4 kg m −2 (here omitting  for simplicity because it changes rather slowly with temperature and is similar for all models).The figure also shows higher-density model R14_010C for the same dust mantle thickness and extinction coefficient (resulting in / = 13.4 kg m −2 ).Here, increasing the mantle density has the effect of lowering the antenna temperature by 11 K on average, while slightly decreasing the amplitude.Both models have similar physical surface temperatures (differing by 4 K on average).The lower-density model has a skin depth of  = Γ/ √  ≈ 2.5 cm (for the nucleus ro-tational angular velocity  = 2/, where  is the rotational period), which means that the region with strong diurnal temperature variations extends well below the dust mantle.The higher-density model has  ≈ 1.5 cm, so that most diurnal temperature variations are confined to the mantle.Because the thermal inertia Γ = √  is fixed, the higher-density model R14_010C has a lower heat conductivity than lower-density model R14_007A, which is why the former becomes colder at depth.For a homogeneous medium, the R14_007A solution would have been restored by setting  MM = 32 m −1 for model R14_010C (recovering the / = 8.4 kg m −2 value of model R14_007A).However, Fig. 10 shows that the antenna temperature of such a model only recovers partially at day, and remains cold at night.Evidently, the presence of the density discontinuity at shallow depth (caused by the presence of inert water ice) distorts the temperature profiles to the point that the simple /-rule for homogeneous media breaks down.
For the Hathor alcove (that evidently has dust on its surface), the solutions therefore depend on dust mantle thickness, thermal inertia, and mantle bulk density, because it is strongly different from the bulk density of the near-surface dust/water-ice mixture.Note that this density-dependence does not apply to the Imhotep dust/waterice investigation in section 3.1.2because in that case the dust/waterice mixture extended up to the surface (i.e., strong near-surface density gradients are not expected).Single-layer media (exemplified by Imhotep) do not allow for unique density solutions (merely identification of the appropriate / ratio), while two-layer media (exemplified by Hathor) allow for constraints to be placed on the density.
In order to proceed, the MM antenna temperature curves are now studied as function of dust mantle thickness.This is first done for the nominal ∼ 30 MKS mantle thermal inertia, and for the two different mantle density values.That basic comparison in made in Fig. 11.The left panel shows the low-density cases.Here, the nighttime data can be fitted for  MM = 20 m −1 .However, the curve amplitude is high and the models are too warm at day (as previously pointed out, in fact, model R14_007A appears in Fig. 10 as well).The nighttime temperature decreases with growing mantle thickness, and for mantle thicknesses ℎ > ∼ 3 cm it is too cold even for an extinction coefficient as low as  MM = 20 m −1 .The MIRO MM data cannot be reproduced for a mantle density as low as  m ≈ 170 kg m −3 , at least not when Γ ≈ 30 MKS.
The right panel of Fig. 11 shows the corresponding plots for the higher mantle density of  m ≈ 340 kg m −3 .In these cases, the curve amplitudes are significantly smaller.Again, the antenna temperature drops with increased mantle thickness.A number of  MM -values were considered for each mantle thickness, and the best overall fit was obtained for model R14_010A, having  MM = 0.08 for  MM = 20 m −1 .Figure 11 shows all models at that extinction coefficient to ease a direct comparison.Acceptable ( MM ≥ 0.01) solutions were only obtained for R14_010A for the interval  MM = 20 ± 10 m −1 .
Once one MM solution had been found, attention turned to the SMM data.Among the six Γ ≈ 30 MKS models studied here, the best one was indeed R14_010A for mantle thickness ℎ m = 0.8 cm, with  SMM ≤ 0.26 for  SMM = 160 ± 90 m −1 , though it was necessary to apply a small level of scattering by setting  SMM = 0.10.Model R14_011A with ℎ m = 1.3 cm also had  SMM ≤ 0.12 solutions for  SMM = 50 ± 30 m −1 and  SMM = 0.However, a mantle as thick as ℎ m = 1.3 cm is here excluded for two reasons: 1) there are no corresponding MM solutions; 2) the SMM extinction coefficient is uncharacteristically low.Considering those solutions spurious, the conclusion is that at least one simultaneous MM and SMM solution exist for mantles with Γ ≈ 30 MKS, ℎ m = 0.8 ± 0.2 cm,  m ≈  The next step is to test other mantle thermal inertia values than the nominal Γ ≈ 30 MKS.The main questions to be answered are: 1) can solutions be extended to the lower-density mantle scenario by modifying the thermal inertia?; 2) can solutions be extended to a wider range of mantle thicknesses in the higher-density mantle scenario by changing the thermal inertia?First, four models were run with  m = 169 kg m −3 for mantle thicknesses ℎ m = 0.8 and 1.9 cm, considering Γ ≈ 15 MKS or Γ ≈ 50 MKS for the mantle.At MM, it was found that increasing the thermal inertia makes the situation somewhat worse, by enhancing the problematic amplitude (Fig. 11) slightly.Lowering the thermal inertia reduces the amplitude, but the change is so small, that sufficient amplitude reductions for reasonable Γ ranges is not deemed possible.For that reason, the SMM curves were not considered.
Next, the higher-density case was tested for mantle thicknesses ℎ m = 0.4, 0.8, and 1.3 cm, for a lower thermal inertia of 15 MKS.This destroyed the fit for ℎ m = 0.8 cm seen at Γ ≈ 30 MKS (by significantly lower the antenna temperature), and it did not introduce new fits at other mantle thicknesses.Therefore, attention turned to higher thermal inertia values, first testing Γ = 50 MKS at ℎ m = 0.4 and 0.8 cm.The changes to the MM curve were very marginal for the thinner mantle.However, the quality of the MM fit improved slightly for ℎ m = 0.8 cm and Γ = 50 MKS compared to the lower thermal inertia.The SMM fits were not as good but still acceptable, though the single-scattering albedo had to be increased to  SMM = 0.15.For this reasons, even higher mantle thermal inertia values of Γ = 80 and 100 MKS were tested for ℎ m = 0.8 cm.These models were still statistically consistent with the data, but reaching amplitudes as low as for R14_010A with Γ ≈ 30 MKS required  MM ≤ 10 m −1 .Such transparent media (significant contribution to the MIRO signal from 0.1 m depth) are probably nonphysical, particularly considering that this is supposed to be a relatively high-density medium.The best SMM solutions had  SMM ≤ 0.02, thus merely representing marginal fits.Therefore, the thermal inertia range is tentatively restricted to Γ = 40 ± 20 MKS.
Note, that increasing Γ leads to higher daytime antenna temperatures, while keeping nighttime values virtually fixed (on the Γ = 30-100 MKS range).Therefore, the low nighttime temperature associated with thicker mantles (i.e., R14_011A in Fig. 11) are not expected to change much with mantle thermal inertia, thus the ℎ m = 1.3 cm case was not considered.In conclusion, the most likely dust mantle thickness at the Hathor alcove is therefore ℎ m = 0.8 ± 0.2 cm.

Refractories, H 2 O, and CO 2
Thus far, it has been found that refractories-only models are unable to fit the Hathor alcove data, but that dust/water-ice mixtures underneath a thin dust mantle are capable of providing simultaneous MM and SMM fits under the right conditions.For this reason there is no strong incentive to introduce CO 2 ice into the study, if the goal is to prove its existence.Even if the presence of CO 2 would enable additional simultaneous MM and SMM solutions for other combinations of mantle thicknesses, densities, and thermal inertia values, that would not constitute compelling proof that CO 2 necessarily exists near the surface.Clearly, it is possible to do without CO 2 , as shown in section 3.2.2.
However, such fits involving CO 2 (if they are found) could potentially call into question the currently established constraints on mantle density, thermal inertia, and thickness.It is therefore important to understand how presence of CO 2 affects the antenna temperature curves.For example, the main problem with the lower-density mantles as well as Γ > 50 MKS thermal inertia, was the high amplitudes of the model MM antenna temperature curves.If presence of CO 2 would bring down the daytime antenna temperatures without disturbing the nighttime values, that could potentially justify lower-density and/or higher-thermal-inertia mantle scenarios.Another reason for considering CO 2 is to understand at what point it would destroy the solutions already established in section 3.2.2.Finding the smallest CO 2 front depth for which the current solutions are still acceptable, places constraints on how shallow CO 2 possibly can be at the Hathor alcove.
For this reason, a number of models were run where carbon dioxide ice was introduced up to a given depth.Above the CO 2 front, the parameters of the previously successful R14_010A higher-density model were applied, except that the diffusivity was reduced an order of magnitude, by using { p ,  p } = {100, 10} m.Below the CO 2 front, CO 2 /H 2 O = 0.32 was applied, resulting in  bulk = 744 kg m −3 and  = 0.49 at such depths.Figure 12 shows some of these models.For example, if the CO 2 front is located ∼ 0.4 m below the surface (model R14_010O), there are drastic antenna temperature drops both at MM (the mean curve is ∼ 17 K below the mean data) and SMM.If the diffusivity is larger than currently assumed, the discrepancies would grow further.Even when the CO 2 front is moved down to ∼ 0.8 m below the surface (model R14_010Q), the MM solution is disqualified, although the SMM solution now is compatible with the data.As it turns out, the CO 2 front must be located at ≥ 0.99 m depth to allow for  MM ≈ 0.01 solutions (compared to  MM = 0.08 for the CO 2 -free variant).
Importantly, Fig. 12 shows that the effect of CO 2 sublimation cooling is to lower the entire antenna temperature curve, such that the effect on its amplitude is negligible.As a result, CO 2 would not be capable of introducing new solutions at low mantle densities or for high mantle thermal inertia values, as discussed previously.Not only does it mean that the constraints placed of mantle density, thermal inertia, and thickness in section 3.2.2 are confirmed, it also means that CO 2 cannot be allowed to ruin the only solution that has been obtained.That means that the ≥ 1.0 m lower limit on the CO 2 front depth is a necessary criterion.

DISCUSSION
The main final conclusions about the estimated conditions at the Imhotep and Hathor collapse sites may be summarised as follows: These conclusions, along with those for Aswan in Paper I, allow for an assessment of similarities and differences among three specific nucleus locations.First, exposed dust and water-ice mixtures were characterised at Aswan and at Imhotep (both located on the large lobe).Importantly, those are the best available representatives of the deep, less processed, nucleus materials.To the extent that abundances, densities, and optical properties can be constrained by the currently applied method, there are no substantial differences between interior compositions at Aswan and Imhotep.Both are consistent with dust/water-ice mass ratios near unity, CO 2 /H 2 O molar abundances near 30 per cent, absence of SMM scattering, and / SMM = 2.2 ± 0.2 kJ m −2 K −1 .Here, specific heat capacities  1 = 400 J kg −1 K −1 and  4 = 1200 J kg −1 K −1 are applied for refractories and water ice, respectively (valid near  = 150 K according to the laboratory data used by Davidsson 2021), and using the average 800 J kg −1 K −1 for the mixture.The only differences concern thermal inertia (Imhotep having a factor 4-6 higher values than Aswan) and gas diffusivity (Imhotep having two orders of magnitude lower values than Aswan).Both may be considered structural properties (heat conductivity, thus thermal inertia, are highly dependent on the level of grain-to-grain connectivity, while diffusivity is determined by pore-to-pore connectivity).If these differences are evolutionary, it could be related to the discrepancy in CO 2 sublimation front depths at the times of observation: 0.4 ± 0.2 cm for Aswan but 11 ± 4 cm for Imhotep.Note that the Aswan diffusivity appears to have dropped to levels similar to those of Imhotep as its CO 2 front withdrew to 20±6 cm, indicative of a temporal behaviour that perhaps is shared by most freshly exposed surfaces.The inferred range in thermal inertia could easily be accommodated on the rather narrow 0.7 ≤  ≤ 0.8 interval (see Table 8 in Davidsson 2021) due to the strong dependence of heat conductivity on porosity, exemplified by the work of Shoshany et al. (2002).If so, the thermal inertia difference may only constitute a temporary fluctuation in the near-surface porosity during a period of intense sublimation.Alternatively, these are more deeprooted (perhaps primordial) differences, that would suggest some level of large-lobe heterogeneity.However, I would caution against such an interpretation until further evidence emerges in support of intrinsic variability on the large lobe.
Second, dust-mantle materials were characterised at Aswan on the large lobe (old as well as freshly formed variants) and at Hathor on the small lobe.The thermal inertia of Hathor (Γ = 40 ± 20 MKS) is similar to the Γ = 30 MKS estimate for pre-collapse Aswan.Pre-collapse Aswan had / SMM = 0.2 kJ m −2 K −1 and a substantial level of SMM scattering ( SMM = 0.17-0.20).The sudden exposure of the underlying dust/water-ice mixture led to an order-of-magnitude increase of / SMM (to 2.2 ± 0.2 kJ m −2 K −1 ) and removal of SMM scattering, as mentioned above.When the mantle re-formed (though merely being a few millimetres thick) the / SMM value dropped (to 0.4 and later to 0.3 kJ m −2 K −1 ), i. e., did not quiet reach its original level.Notably, SMM scattering was not re-established, perhaps indicating that the mantle needs a certain thickness and/or age to display such properties.The reduction of / SMM as a consequence of mantle formation is consistent with the removal of water ice, nominally cutting both the specific heat capacity and the bulk density in half.However, this nominally comprises a factor ∼ 4 reduction, while the observed drop is a factor ∼ 11.The discrepancy could be due to a higher water ice abundance in the real object than inferred here (i.e., the comet may have  < 0.9), and/or a substantially lower specific heat capacity of cometary dust compared to the forsterite model analogue, and/or the result of dust-matrix compaction following the ice removal.The last mechanism would require that the opacity increases faster than bulk density during dust-matrix compression (i.e., that  SMM grows quicker than ).The Hathor dust mantle had SMM scattering as well ( SMM = 0.1) and a comparably large / SMM = 1.3 ± 0.7 kJ m −2 K −1 .That is a factor ∼ 6 higher than for the Aswan pre-collapse mantle.The MM values are a factor ∼ 5 higher for Hathor (/ MM = 9 ± 5 kJ m −2 K −1 ) than for pre-collapse Aswan (/ MM = 2.0 ± 0.1 kJ m −2 K −1 ), noting that the re-established Aswan mantle first had / MM = 2.3 and later 1.4 kJ m −2 K −1 .This suggests that the dust mantle at Hathor is denser, and/or has higher specific heat capacity, and/or is more transparent to microwave radiation than the one at Aswan.
The Aswan/Hathor differences in dust properties may reflect a level of temporal variability that could be common to many nucleus locations, i. e., they are merely expressions of normal fluctuations.However, the similarity in pre-and post-collapse Aswan dust mantle / SMM and / MM values may suggest that time variability is not that important.If so, there may be systematic physical and/or chemical differences between Aswan and Hathor, causing their mantles to behave dissimilarly.This is an intriguing possibility, considering that Aswan and Hathor are located on different lobes, and remembering that: the small lobe cliffs appear to be stronger than those on the large lobe (El-Maarry et al. 2016); polygonal blocks (also referred to as 'goosebumps' or 'clods', e. g.Davidsson et al. 2016) on the small lobe are twice as large as those on the large lobe (Fornasier et al. 2021); bright icy exposures are six times less com-mon on the small lobe as on the large lobe (Fornasier et al. 2023).It is therefore desirable to analyse MIRO data for additional locations on both lobes, to establish whether or not the differences inferred here for Aswan and Hathor are systematic.
The Imhotep CO 2 front depth estimate enables an attempt to obtain an approximate age of the feature.This age estimate is based on the results for the Aswan collapse site presented in Paper I. Such a comparison assumes that the differences in thermal inertia and diffusivity discussed above do not drastically change the time scale of evolution for freshly exposed interior nucleus material, as long as compositions (using / SMM as a proxy) are similar.The Aswan structure collapsed on 2015 July 10 (based on OSIRIS observations of a major outburst, see Pajola et al. 2017a) and the analysis of MIRO observations in Paper I demonstrated that the CO 2 front was at ℎ sv = 0.4 ± 0.2 cm after 5 months, at ℎ sv = 2.0 ± 0.3 cm after 7 months, and at ℎ sv = 20 ± 6 cm after 11 months.If the Imhotep site has followed the same temporal evolution, the inferred depth of ℎ sv ≈ 11±4 cm in the current work would place the estimated time of collapse 9 ± 1 months prior to the last days of 2014, or roughly in the 2014 March-May time frame.The OSIRIS cameras monitored the brightness of Comet 67P (unresolved at the time) during approach in 2014 March through July, as reported by Tubiana et al. (2015).Interestingly, the comet displayed a larger ∼ 0.6 magnitude outburst starting 2014 April 27-30, and a smaller ∼ 0.2 magnitude outburst starting 2014 June 10-20.The rest of the time, the comet magnitude was stable to within roughly ±0.05 magnitudes.Given the similarity of the estimated age of the Imhotep collapse site with the time of the documented larger outburst, I here propose that the larger outburst was caused by the collapse that exposed the large patch of dust/waterice at Imhotep seen in Figs. 1 and 2. Admittedly, this proposal is speculative and impossible to verify.
According to the results for Hathor, the dust mantle has a thickness of 0.8 ± 0.2 cm.This can be compared to the average dust mantle thickness of the northern hemisphere of Comet 67P, estimated as ∼ 0.6 cm based on jet switch-off during sunset according to Shi et al. (2016), and as ∼ 2 cm based on nimbus model reproduction of the empirical water production rate curve of 67P according to Davidsson et al. (2022a).Also, a location in Hapi had a mantle thickness of ℎ m = 2.3 cm in 2014 October, growing to ℎ m = 21 cm one month later (possibly due to the presence of a pre-existing ice-free layer) according to the analysis of MIRO data presented by Davidsson et al. (2022c), and the pre-collapse dust mantle thickness at Aswan was ℎ m ≥ 3 cm according to Paper I. In view of these numbers, the recovery of the Hathor dust mantle after a presumed collapse in the past is almost or fully complete.Additionally, the necessity of having CO 2 at a depth of ≥ 1.0 m also suggests that the collapse site in Hathor has evolved to a more pronounced level of stratification compared to those in Aswan and Imhotep.For comparison, the average depth of the CO 2 sublimation front is 3.8 m according to nimbus model reproduction of the empirical CO 2 production rate curve of 67P presented by Davidsson et al. (2022a).
Interestingly, the bulk density  m = 340 ± 80 kg m −3 estimated for the mantle, and the assumed  bulk = 535 kg m −3 for the dust/waterice mixture (that reproduces data better than for a  bulk = 340 kg m −3 assumption), are higher than the corresponding values of  m = 170-190 kg m −3 and  bulk = 340-390 kg m −3 expected if the bulk nucleus with a measured  bulk = 535 kg m −3 simply is deprived of its CO and CO 2 .The higher values suggest compression from porosities  = 0.76 to  = 0.63 for the dust/water-ice mixture, and from  = 0.95 to  = 0.91 for the dust mantle.A major outcome of the Rosetta mission, based on measurements by CONSERT and SESAME-PP, is that the near-surface region is compressed com-pared to the deeper interior, particularly in the upper metre (Ciarletti et al. 2015(Ciarletti et al. , 2018;;Brouet et al. 2016;Lethuillier et al. 2016).Prior to Rosetta, Davidsson et al. (2009) also proposed the existence of a near-surface compacted layer based on discrepancies in density estimates from radar (valid for the upper few metres) and from nongravitational force modelling (valid for the bulk nucleus).The current results for Hathor is consistent with that global property.Davidsson et al. (2022c) has proposed that the CO 2 vapour pressure profile, that falls steeply on both sides of the CO 2 front (according to nimbus simulations) is responsible, by forcing movements in the solid dust/water-ice matrix (also see Davidsson et al. 2022a).The inward gradient below the front is hypothesised to cause compression that is considered the ultimate reason of the observed near-surface compaction.The outward pressure gradient above the front (of both H 2 O and CO 2 vapour) has traditionally been considered the cause of ejection of material into the coma (e.g., Fanale & Salvail 1984), so it is obvious that the vapour is capable of forcing these structural changes.
It is perhaps significant that the SMM observations of Hathor seem to require a certain level of scattering at such wavelengths.A similar single-scattering albedo ( SMM ) was obtained for a location in Hapi by Davidsson et al. (2022b), and seemed to be related to a compaction event of the surface that additionally increased the thermal inertia, MM and SMM extinction coefficients, reduced the diffusivity, and removed a previously detectable solid-state greenhouse effect.Scattering is believed to be caused by small-scale dielectric constant variations (Gary & Keihm 1978), caused by millimetre-scale lumpiness.Such lumpiness would form if an initially uniform mixture of m-sized monomer grains is compressed in such a manner that millimetre-sized regions of higher compaction are mixed with lower-density regions of similar size.The emergence of SMM scattering after compression of cometary material, may provide clues on how the small-scale structure of such media are modified when compressed.
The analysis of the Imhotep collapse site resulted in an ambiguity -whereas solutions with dust and water ice could be excluded, the MIRO data was not sufficient to distinguish between (nearly) equally good solutions for refractories-only and refractories mixed with both CO 2 and H 2 O.The reason for this difficulty was the quality of the antenna temperature curves, that lacked data for rotational phases that would have allowed for a disentanglement.Had it not been for the OSIRIS images that exclude dust coverage at the Imhotep site, it would have been impossible to select the dust/H2O/CO 2 solution as the correct one.Based on this difficulty, and similar ones described in Davidsson et al. (2022c) and Paper I, the following operational and observational recommendations are being made for future spacecraft missions carrying a microwave instrument: (i) Close collaborations between the camera and microwave instrument teams are necessary.The camera team will identify regions of particular interest (e. g., cliff collapses) in real time, but it is crucial that such information is passed on to the microwave instrument team and others, to enable urgent follow-up observations with other types of instrumentation.It also requires a certain level of flexibility in spacecraft operations, i. e., to allocate time for target-of-opportunity observations, which is not possible when observations are planed in month-long blocks, weeks in advance, and executed without regard to developing events.
(ii) It is crucially important that a microwave instrument observes a given location during large fractions of the comet day and night, in order to cover as many rotational phase angles as possible.Only with access to a complete thermal diurnal antenna temperature curve is it possible to determine the thermophysical parameters of the location with high accuracy and without ambiguities.It is far better to observe a limited number of carefully selected regions repeatedly, than to continuously scan the nucleus in an attempt to cover as much ground as possible.
(iii) The attempt to analyse microwave data has revealed the importance of access to observations at different wavelength intervals.Sometimes, cases can be found where one physical model of the nucleus reproduces the MM measurements, whereas another physical model of the nucleus reproduces the SMM measurements (e. g., section 3.3 in Paper I). Obviously, none of those physical models are correct, and a third physical model that simultaneously satisfies both the MM and SMM observations needs to be found.Otherwise, false solutions risk to confuse and distort the description of the comet nucleus.
Finally, I echo the conclusion of Paper I, that newly collapsed cliff sites may be the best option for a cryogenic comet nucleus sample return mission that aims at finding super-volatiles like CO 2 in addition to the more readily available water ice.Based on the analysis of Aswan in Paper I, and of Imhotep in the current work, it is found that CO 2 ice manages to stay close (centimetres to decimetres) to within the surface for up to a year after collapse.Extracting ice from such depths requires less technically complex and less costly engineering solutions, than targeting locations elsewhere that may require drilling through several metres of material.It is primarily the long-lived exposures of dusty water ice that reveals the presence of shallow CO 2 .Combined with the rarity of exposed water ice elsewhere on the comet, Paper I and the current work suggest that it is the strong near-surface CO 2 sublimation that prevents the dust mantle from forming, whereas water ice is incapable of self-cleaning.

CONCLUSIONS
This paper has analysed observations by Rosetta/MIRO of Comet 67P/Churyumov-Gerasimenko with the aid of the thermophysical model nimbus (Davidsson 2021) and radiative transfer model themis (Davidsson et al. 2022c), with the goal of constraining the physical properties (including thermal inertia, density, dust mantle thickness, CO 2 sublimation front depth, extinction coefficients, and singlescattering albedos) of near-surface comet nucleus material at two cliff collapse sites -the relatively fresh Imhotep location, and the relatively aged Hathor alcove.The main results are summarised as follows: (i) The MIRO observations of the Imhotep collapse site (an exposed dust/water-ice mixture) are consistent with a dust/water-ice mass ratio  = 1, a molar CO 2 abundance relative to water of CO 2 /H 2 O = 0.3, an interior bulk density  bulk = 535 kg m −3 , a thermal inertia Γ = 100-160 MKS (with preference for the lower value), a gas diffusivity parameterised by { p ,  p } = {100, 10} m and  = 1, extinction coefficients  MM = 60 ± 10 m −1 (with preference for the upper range) and  SMM = 170 ± 40 m −1 (with preference for the lower range), and single-scattering albedos  MM =  SMM = 0.The CO 2 ice is located at a depth ℎ sv ≈ 11 ± 4 cm below the surface.
(ii) The estimated age of the Imhotep site (9 ± 1 months at the time of observation) is consistent with an outburst observed by Rosetta/OSIRIS during approach to Comet 67P, here proposed to pinpoint the time of collapse to 2014 April 27-30.

Figure 1 .
Figure 1.Left: Context image of the nucleus of Comet 67P showing the Imhotep collapse site within the red square.It is located on the large lobe, while the small lobe is visible in the top third of the panel.Right: This is a zoom on the red square from the left panel.The Imhotep collapse site clearly stands out as a brighter region.Both panels show image MTP6/n20140816t165914570id4ff22.img (Sierks & the OSIRIS Team 2020a) that was acquired on 2014 August 16 with ∼ 1.8 m px −1 resolution when Rosetta was 103 km from the comet.Axis labels show pixel ID numbers.

Figure 3 .
Figure 3. Left: Context image of the nucleus of Comet 67P showing the Hathor alcove within the red square.The sharp boundary to Anuket runs parallel just within the right side of the square.The alcove is located on the small lobe.A small fraction of the large lobe is visible in the foreground in the lower third of the panel.Note the morphological differences between these parts of the small and large lobes, being dominated by consolidated and smooth terrains, respectively.Right: This is a zoom on the red square from the left panel.Spectrophotometry reveals unusually bright and bluish material at and beneath the Hathor alcove, though much of its surface is covered by dark dust.Both panels show image MTP7/n20140912t195732364id4ff22.img (Sierks & the OSIRIS Team 2020b) that was acquired on 2014 September 12 with ∼ 0.53 m px −1 resolution when Rosetta was 29.9 km from the comet.

Figure 4 .
Figure 4. Examples of Imhotep viewing conditions during acquisition of two different bins.The colour scale shows level of illumination, with grey areas being in darkness.The FWHM of the MIRO beams are shown as red (SMM) and blue (MM) circles.Left: Observations of a selected bin took place on 2014 December 15, at 23 : 47 : 09 UTC, when Rosetta was 20.4 km from the comet.The collapse area is viewed nearly face-on (the emergence angle is  = 13 • ).Right: Observations of a rejected bin took place on 2015 January 20, at 01 : 27 : 37 UTC, when Rosetta was 27.7 km from the comet.Here, the collapse region is viewed edge-on and does not fill the beams.

Figure 12 .
Figure 12.Hathor data compared with a refractories+H 2 O model (providing simultaneous MM and SMM fits) and refractories+H 2 O + CO 2 models.All models have ℎ m = 0.8 cm dust mantles with  m = 337 kg m −3 and Γ ≈ 30 MKS, overlaying a dust+H 2 O interior with  bulk = 535 kg m −3 .Some models also have dust+H 2 O + CO 2 with  bulk = 744 kg m −3 below the front depths as indicated in the legends (unless not applicable or 'n/a').The CO 2 models have diffusivities based on {  p ,  p } = {100, 10} m and  = 1.Left: MM data and models, all having  MM = 20 m −1 and  MM = 0. Right: SMM data and models with  SMM = 150 m −1 and  SMM = 0.1.
Imhotep data compared with refractories+H 2 O models.The legends indicate whether a given model considered water ice (Y/N for yes/no), the approximate thermal inertia Γ, the applied extinction coefficient ( MM or SMM ), and whether the diffusivity (diff) was low (L) with {  p ,  p } = {100, 10} m and  = 1, high (H) with {  p ,  p } = {10, 1} cm and  = 1, or not relevant (n/a for not applicable).Left: MM data and model antenna temperatures.Right: SMM data and model antenna temperatures.Both models have  SMM = 600 m −1 .
Imhotep data compared with refractories+H 2 O + CO 2 models.Both models have  = 1,  bulk = 535 kg m −3 below the CO 2 sublimation front, {  p ,  p } = {10, 1} cm and  = 1.Model R13_005B has CO 2 /H 2 O = 0.30 and model R13_005D has CO 2 /H 2 O = 0.15, resulting in different thermal inertia as indicated by the legends.Furthermore, different erosion rates are applied, which places the CO 2 front at different depths.Left: MM data and simulations.
Right: SMM data and simulations.

Table 1 .
Summary of the nimbus and themis models for the Imhotep case that performed best for the three different types of composition (dust-only, dust+H 2 O, and dust+H 2 O + CO 2 ).Y/N for yes/no on presence of the volatile in question.Note that the thermal inertia values applies to either dust or the dust+H 2 O layer, depending on what is relevant (also for models that include CO 2 ).
Hathor data compared with refractories-only models.Left: Measured and modelled MM antenna temperatures.
, Hathor data compared with refractories+H 2 O models with Γ ≈ 30 MKS: density (thus heat capacity) dependence.Model R14_007A with mantle density  m = 169 kg m −3 and extinction coefficient  MM = 20 m −1 and model R14_010A with  m = 268 kg m −3 and  MM = 32 m −1 have different antenna temperatures.Unlike homogeneous materials, these layered media differ despite identical  m / MM ratios.Note that the legend shows the density of the dust/water-ice mixture (the dust mantle values are half as large).
Hathor data compared with refractories+H 2 O models with Γ ≈ 30 MKS: dust-mantle thickness dependence.Left: At relatively low densities ( m = 169 kg m −3 for the mantle,  bulk = 337 kg m −3 for the dust/water-ice mixture) MM fits cannot be obtained at any mantle thickness.Right: At relatively high densities ( m = 337 kg m −3 for the mantle,  = 535 kg m −3 for the dust/water-ice mixture) a MM fit can be obtained for a 0.85 cm mantle.