Thermal Interaction of Mobile Ground Systems with Boulders on the Lunar Surface ^†

Abstract

The paper at hand evaluates the necessity of depicting topographic features like boulders on the lunar environment in thermal analyses for a size of up to 6.5 m in diameter. The question regarding the thermal influence becomes relevant when analysing a technical system within the lunar environment. This influence on the thermal behaviour of a test object is investigated in sensitivity studies. It is shown that the local surroundings can significantly alter a system’s net heat flux and can lead to overheating or critically cooling down instead of theoretically surviving when not considering local topographic features. Especially for small and lightweight systems ≤20 kg, like micro rovers, the effect of the surrounding on the system’s temperature becomes critical due to the low thermal capacity. Thus, it is a substantial aspect to be accounted for during the design phase as well as in mission operation.

1. Introduction

Over the past few years, national space agencies have increased their efforts in the field of lunar accessibility, exploration, and habitation. In particular, efforts are concentrating on autonomous operations and thermal survivability of lightweight rover systems around 20 kg. Due to their low mass, such ground systems are particularly vulnerable to variations in heat flux from their surroundings. Where heavier systems have sufficient thermal capacity to dampen variations in external heat flux to a certain extent, lightweight systems experience a comparatively rapid change in temperature due to their low thermal capacity. This leads to the assumption that the local surroundings, boulders, and small craters, which a rover system passes by or through, may result in unacceptably high heat fluxes and thus component temperatures.

This aspect is further elicited by the datasets on lunar surface topography. While topography data at the lunar poles have a resolution of approx. 5 m per pixel [1], the currently most accurate and publicly available dataset SLDEM2015 on topography in equatorial regions has an effective horizontal resolution of approx. 60 m per pixel [2]. This dataset was generated from data of the LOLA (Lunar Orbiter Laser Altimeter) instrument onboard the LRO (Lunar Reconnaissance Orbiter) and SELENE (Selenological & Engineering Explorer) TC (Terrain Camera) data and covers the region of ±60° north and south [2]. Thus, topographic features with sizes around and below this resolution are not depicted as distinct topographic features, but rather as topographic elevations of the respective pixel, which makes it difficult to account for in path planning.

This paper aims at evaluating the effect of local topographic features of the lunar surface on the thermal behaviour of a mobile ground system for three different illumination cases and boulder sizes with a diameter of 0.3–6.5 m. The necessity of taking into account local topographic features for the system design and operational pathfinding is discussed. This work is part of the SAMLER-KI project, which aims to provide and demonstrate technological solutions for a lunar micro rover platform [3].

2. Methodology

To evaluate the thermal effect of local topographic features on a mobile ground system, thermal simulations are conducted in a lunar environment setting, containing a distinct topographic feature and a test object substituting the mobile ground system.

2.1. Submodels

The simulation setup consists of three separate submodels: the circular lunar surface section model, a test object model that is placed on this surface, and a generic lunar boulder model located on the surface in the vicinity of the boulder. Each submodel and its respective properties are described in the following sections.

2.1.1. Lunar Surface

The lunar surface mesh is extracted from the SLDEM2015 dataset mentioned in Section 1 as a point cloud. For the analyses in this work, a circular area of approx. 300 m in diameter at 12.004° latitude and 305.0° longitude is used due to the flat terrain and the absence of topographic features. The modelling is conducted according to the two-layer model proposed by [4] and previously used for similar lunar surface models by e.g., [5]. It consists of a 2 cm upper layer of lunar dust and a denser soil layer below, down to a depth of 2 m. The subsurface is discretised by 10 nodes in depth, placed in a logarithmic fashion, with respective thermal conductances. The thermal properties used for the lunar surface are taken from pertinent literature and are similar to the ones used by e.g., [5]. An overview of all relevant parameters, their values, and the respective sources is given in

Table 1. Overview of properties of the thermal model of the lunar surface.

Parameter	Symbol	Value/Formula	Unit	Source
Density	${\rho }_{reg}$	$1.92·{\displaystyle \frac{z+12.2}{z+18}}$	${\displaystyle \frac{\mathrm{g}}{{\mathrm{cm}}^3}}$	[7]
Thermal Conductivity Dust	${\lambda }_{dust}$	$9.22·{10}^{-4}·(1+1.48·{\left({\displaystyle \frac{T}{350K}}\right)}^3)$	${\displaystyle \frac{\mathrm{W}}{\mathrm{mK}}}$	[8]
Thermal Conductivity Soil	${\lambda }_{soil}$	$9.3·{10}^{-3}·(1+0.073·{\left({\displaystyle \frac{T}{350K}}\right)}^3)$	${\displaystyle \frac{\mathrm{W}}{\mathrm{mK}}}$	[9]
Specific Heat Capacity	cp	$-34·T^{0.5}+8·T-0.2·T^{1.5}$	${\displaystyle \frac{\mathrm{J}}{\mathrm{kgK}}}$	[10]
Solar Absorptivity	$\alpha$	0.93	-	[11]
IR Emissivity	$ϵ$	0.93	-	[6]
Lunar Internal Heat	$Q_{int}$	$0.016$	${\displaystyle \frac{\mathrm{W}}{{\mathrm{m}}^2}}$	[12]

. T represents the temperature in [K] and z the depth in [cm]. According to a formula by reference [6], the infra-red emissivity depends on the surface temperature and varies between 0.93 and 0.98 for the temperature regime expected in this study. Since the influence of emissivity variation with temperature is not within the scope of this study, a constant value of 0.93 is used as a simplification that is conservative for the hot case and irrelevant for the steady-state cold case. The absorptivity does not change with the angle of incidence.

The lunar surface model was verified, sensitivity studies have been conducted for all relevant parameters, and the surface temperatures throughout one lunation have been compared to previously existing models and bolometric temperature measurements. The resulting values show a large agreement with the models and temperature measurements compared against.

A comprehensive description of the lunar surface model used for this work, as well as the whole verification procedure and its results, can be found in reference [13].

2.1.2. Lunar Boulder

The lunar boulder model is a hemispherical solid with a diameter of 8 m, placed on the surface with its flat side down. This shape approximates a generic boulder on the lunar surface. It is discretised as 7 elements in circumference and 4 elements in vertical direction, as shown in Figure 1. The boulder has a diameter of 5 m and thus a height of 2.5 m. This is larger than lunar rock size abundance distributions, for example, around landing areas provided by reference [14], although still small enough to not distinctly be identified by the SLDEM2015 dataset. Results for steady-state conditions can be transferred to other geometries and sizes by considering the same view factor between the boulder and the test object.

The thermo-optical surface properties of the boulder model are chosen to correspond to those typical for lunar regolith (cp. Section 2.1.1). For the emissivity, a value of $ϵ=0.95$, and for the absorptivity, a value of $\alpha =0.93$ are chosen. The material parameters density, specific heat, and thermal conductivity depend on the composition of the lunar regolith as well as on its temperature. These dependencies are neglected here, as the former two are not relevant for steady-state conditions, and an approximation of the latter does not introduce significant errors for the purpose of relative comparisons. The density is set to 1500 ${\displaystyle \frac{\mathrm{kg}}{{\mathrm{m}}^3}}$ as a constant approximation [15]. The values of specific heat and thermal conductivity, $cp=500$ ${\displaystyle \frac{\mathrm{J}}{\mathrm{kgK}}}$ and $\lambda =1.8·{10}^{-3}$ ${\displaystyle \frac{\mathrm{W}}{\mathrm{mK}}}$, respectively, are in good agreement with the temperature dependent values used by reference [5].

2.1.3. Test Object

The test object is a sphere with a diameter of 0.5 m, discretised as $4\times 6$ elements. Its surface has an emissivity of $ϵ=0.62$ and an absorptivity of $\alpha =0.39$, depicting the specifications of e.g., aluminium-coated polyimide second surface mirror multi-layer insulation (MLI) blankets from Sheldahl [16]. Such coatings with $\alpha /ϵ\approx 0.5$ exhibit a relatively low sensitivity to solar illumination compared to other conventional MLI coatings. This has the potential to significantly influence the absolute temperature of a test object. For the purpose of relative comparisons regarding the influence of position and illumination, the absolute temperature of the test object is of less importance, though. Hence, the aspect of different surface coatings is not the subject of this paper and will be examined in the future.

In order to quickly obtain steady-state conditions, the heat capacity is set to a low value of $0.01$ ${\displaystyle \frac{\mathrm{J}}{\mathrm{KgK}}}$, which approximates the behaviour of MLI. The thermal conductivity is kept realistically low at a value of $0.01$ ${\displaystyle \frac{\mathrm{W}}{\mathrm{mK}}}$. Such a value sufficiently mitigates the heat flow between the individual elements of the test object to obtain a clear picture of the temperature distribution across the test object without being disturbed too much by conductive heat flows. Figure 2 visualises this temperature distribution across the test object for the case of being illuminated from the side without a boulder in its vicinity.

2.2. Simulation Cases

All simulations are conducted with the software ESATAN-TMS. The submodel of the test object is placed in the centre of the lunar surface submodel. The boulder submodel is placed on the lunar surface as well, in the vicinity of the test object. Depending on the simulation case, its distance towards the test object or its size is changed throughout the simulation. This is done under three different lighting conditions, respectively, in order to examine the influence of shadowing and reflection of sunlight under a low elevation angle.

In total, 4 different test setups are simulated and analysed, which are described in the following paragraphs. Graphics of these test setups are depicted in Figure 3. The brief overview of each test setup is as follows:

• Case 1: Sun in the zenith with the boulder and the test object next to each other. The distance in between is varied from 150 m to 0.3 m, with a constant boulder diameter of 5 m.
• Case 2: Boulder behind the test object with respect to the sun illumination, with an elevation angle of 3°. The distance between the boulder and the test object is varied from 150 m to 0.3 m, with a constant boulder size of 5 m in diameter.
• Case 3: Boulder in front of the test object with respect to the sun illumination, with an elevation angle of 3°. The distance between the boulder and the test object is varied from 150 m to 0.3 m, with a constant boulder size of 5 m in diameter.
• Case 4: Boulder behind the test object with respect to the sun illumination, with an elevation angle of 3°. The distance between the boulder and the test object is set to a constant value of 2 m. The boulder size is varying in diameter between 0.5 m and 6.5 m.

In general, the analysis results shown in this work all refer to steady-state conditions, when a thermal equilibrium has been reached. The heat flux values, which are shown in the plots in the subsequent chapters, are heat fluxes onto or between the geometries and averaged over the faces of the respective geometries. This way, differences in the heat flows between objects and respective changes in temperature become apparent quickly.

3. Analysis Results

3.1. Distance Between Boulder and Test Object

Case 1: The influence of the proximity of an environmental feature to a test object is evaluated by solar illumination directly from above onto the test object as well as onto the boulder, such that shadowing effects stay minimal. Figure 4 depicts the effective heat fluxes towards the test object, as simulated by Monte Carlo Ray Tracing (MCRT). The horizontal lines in the plots serve as comparison values of a case not affected by a boulder. Up to a distance of approx. 10 m, no significant heat flux is present. Below, it increases to a value of 4 W with decreasing distance. The higher view factor between the boulder and the test object comes at the cost of a lower view factor towards space, resulting in less heat loss to space. The heat flux received from the lunar surface decreases by approx. 10.5 W, which is 2.6 times the amount received additionally from the boulder at the same time. This can be attributed to the colder boulder elements that are facing the test object, which in turn is attributed to the boulder curvature receiving solar flux under a higher angle of incidence. At the same time, the heat flux radiated towards space decreases by 8.5 W, resulting in an increase in net heat flux of 2 W.

While the heat flux from the boulder towards the test object increases steadily below 10 m distance, the changes in heat flux from the surface and towards space increase sharply at a distance of less than 2 m. In conjunction with the boulder size, this already indicates that the interaction effects between the boulder and test object are visible for view field angles >±${15}^{\circ }$ and become significant at an occupied view field angle of roughly $\pm {50}^{\circ }$. The decrease in heat flux below 0.5 m boulder diameter is attributed to a lower temperature of boulder elements, as they see the test object instead of the hot lunar surface. The deviations of some temperature points from a smooth curve, as it would have been expected with decreasing distance, are caused by the discretisation of the boulder. Since the temperature of an element is constant over its face in the Finite Difference Method used for these analyses, large temperature differences between elements may occur, and the change in view factor to a specific element may thus have a large influence on the heat flux. Such deviations can be seen throughout the analyses. Minor deviations are caused by the quasi-random nature of the MCRT, especially in scenes with such a large range in element sizes.

Case 2: Considering the results from Case 1, it seems useful to have a look at the illumination hot case for a test object. This hot case can occur during morning and evening hours, respectively, when the test object is illuminated from one side by the sun and receives reflected and infra-red heat flux from the illuminated hot boulder on its other side.

The top elements of the test object are illuminated less, decreasing their temperature and thus decreasing the overall loss to deep space significantly. Since the test object is illuminated directly from the side, its lower elements are warmer than the lunar surface that is illuminated under a shallow angle, which can be seen e.g., in Figure 2. It needs to be considered, though, that the temperature of the test object is contingent on its thermo-optical surface properties and will thus vary for different coatings. The heat flux results on this case are shown in Figure 5. It can be observed that there is a net heat flux towards the surface instead of originating from it. Within a distance of about 10 m, the boulder again exhibits a net heat flux onto the test object, about one order of magnitude larger than during noon, as shown in Case 1. This extra heating power leads to a rise in temperature on the side facing the sun. This in turn affects an increase in heat loss towards the lunar surface as well as towards deep space.

The effect on the different heat fluxes becomes significant below a distance of 2 m, which again emphasises the strong interaction at view field angles >$\pm {50}^{\circ }$ occupied by a boulder. On the side facing the boulder, the temperature increases by up to 140 K for the surface properties used here (cp. Section 2), as it can be seen in Figure 6.

Case 3: Due to the missing influence of thermal convection and low conductivity of the upper regolith layer on the lunar surface, the influence of direct illumination and shadowing, respectively, is of considerable importance for the thermal behaviour. The effect of a test object being shadowed by a boulder is investigated. As expected, the solar incidence drops to zero as soon as the test object is shadowed by the boulder, which is accompanied by a rapid decline in temperature as shown in Figure 7. This results in the net heat flux between the test object and the lunar surface changing direction, and the net heat flux towards deep space decreasing by one order of magnitude, which can both be seen in Figure 8.

3.2. Boulder Size

Case 4: A variation in boulder size for a constant boulder distance depicts the influence of the size of a topographic feature on the thermal interaction with a mobile ground system. The test object is therefore placed in front of a boulder, partly shadowing it.

Figure 9 visualises, that the heat flux from a boulder is insignificant up to a radius of twice the test object diameter for a distance of 2 m. This behaviour may be due to the shadowing of a smaller boulder by the object. Nevertheless, the loss towards the lunar surface decreases instantly with increasing diameter, leading to a rise in temperature, which in turn elevates the heat flux to deep space. With increasing heat flux from the boulder towards the test object, the loss to the lunar surface stays on a plateau and even slightly increases again. This behaviour may again be attributed to the rising temperature of the test object and may differ for other coatings.

All heat fluxes reach a plateau around the boulder diameter of 7 m due to the geometric tangent relation between ${\displaystyle \frac{width}{distance}}$ of the object seen and the viewfield angle occupied.

4. Discussion & Conclusions

The results shown in this work refer to steady-state conditions, which differ from the transient behaviour due to the thermal inertia of mass. During sunrise, the temperatures of the whole scene would actually be lower than simulated and higher during sunset.

On the contrary, the shift in solar illumination is slow, with a change in elevation angle of about ${0.5}^{\circ }/\mathrm{h}$. A significant error would only be introduced at the times shortly after sunrise and sunset. Furthermore, the lunar dust covering the whole lunar surface inhibits a low thermal conductivity in the order of $1.2-3.5·{10}^{-2}\mathrm{W}/\mathrm{mK}$ [8], leading to quick responses of the surface temperature. Thus, the steady-state results are expected to be transferable to the actual transient behaviour throughout most of the lunar cycle.

The results shown in Section 3.2 suggest, that the direct and indirect effects of a boulder in the vicinity of an object become relevant for occupied view field angles >$\pm {15}^{\circ }$ and significant for >$\pm {50}^{\circ }$. The actual rate of change in heat fluxes depends on the object’s surface properties and illumination condition, but in general this finding can be used as a guideline for traverse planning to estimate the critical distance for a given boulder size.

The results show, that low solar illumination is potentially more critical for mobile ground systems due to large local differences in solar heat flux. Particularly polar missions are affected by this finding and should thus be tailored accordingly regarding system design as well as mission operation. Especially for small ground systems with a high $ϵ$ value, the impact of the environment may be even more pronounced than described here.

On the contrary, the different solar illumination conditions in a local area on the lunar surface can as well benefit the thermal condition of a system when used deliberately. Additional IR heat flux can enhance temperatures in cold conditions, and the blocking of solar illumination during the day can cool down a system. This requires the knowledge of the temperatures around the system, which can to date not precisely be determined by the system itself or in a suitable resolution from orbit. Further research and development is required to obtain such data in situ and adopt the traverse accordingly.

An aspect not taken into account in this work is the influence of lunar dust deposition on surface features and the ground system itself. For coatings with $\alpha /ϵ\approx 1$, for example black paint, lunar dust will have a minor influence on the behaviour under solar illumination due to the similar $\alpha /ϵ$ ratio of lunar dust. On the contrary, for example, many metals with $\alpha /ϵ\gg 1$ or whitish coatings with $\alpha /ϵ\ll 1$, dust accumulation can change the thermo-optical properties significantly, altering the net heat flux and thus the steady-state surface temperature.

The investigation of dust effects on the thermal behaviour will be addressed in future work, as well as a detailed examination of the thermal influence of different coatings under lunar surface illumination conditions.

The results presented emphasise the necessity of considering local topography for the thermal design of mobile ground systems but do not hold any claim in terms of representative values. Especially during phases of low solar inclination, unexpected heat fluxes may occur. This is of greater consequence when one considers that the environment emits radiation within the infra-red spectral range, whereby surfaces exhibit a different absorption of incident radiation in comparison to solar irradiation.