Thermal Drilling in Planetary Ices: An Analytic Solution with Application to Planetary Protection Problems of Radioisotope Power Sources

Abstract

Thermal drilling has been applied to studies of glaciers on Earth and proposed for study of the martian ice caps and the crust of Europa. Additionally, inadvertent thermal drilling by radioisotope sources released from the breakup of a space vehicle is of astrobiological concern in that this process may form a downward-propagating “warm little pond” that could convey terrestrial biota to a habitable environment. A simple analytic solution to the asymptotic slow-speed case of thermal drilling is noted and used to show that the high thermal conductivity of the low-temperature ice on Europa and Titan makes thermal drilling qualitatively more difficult than at Mars. It is shown that an isolated General Purpose Heat Source (GPHS) “brick” can drill effectively on Earth or Mars, whereas on Titan or Europa with ice at 100 K, the source would stall and become stuck in the ice with a surface temperature of <200 K. Key Words: Planetary protection—Planetary environments—Ice—Titan. Astrobiology 12, 799–802.

1. Introduction

T he interiors of icy satellites such as Titan, Enceladus, and Europa are of interest in hosting aqueous liquid environments that are, in principle, “habitable.” This makes them of great astrobiological interest and has prompted consideration of means of accessing their interiors by drilling. The martian polar cap is also of interest (perhaps more geophysical than astrobiological) and has similarly been considered as a drilling target.

The challenges of implementing a mechanical drill on a space vehicle and operating it in an unfamiliar environment have directed interest into thermal drilling, wherein a heat source is used to melt a hole in the ice. This drilling approach, which is usually attributed to Karl Philberth (1962), avoids many mechanical complications and allows an instrumented drilling head to move down through the ice without a long heavy drill “string.” Thermal drilling has been proposed for drilling in rock on Earth and the Moon (e.g., Rowley and Neudecker, 1985), but this would likely require temperatures and powers too high for foreseeable implementation; most of the work on this topic is only found in Los Alamos National Laboratory internal reports from the early 1970s.

In ice, however, melting is easier. Ulamec et al. (2007) gave a lucid and comprehensive overview of thermal drilling in ice, and planetary applications are also discussed, for example, by Elliott and Carsey and by Kömle et al. (2002, 2004). For Europa, a “cryobot” has been proposed (e.g., Zimmerman, 2001), wherein a small vehicle with a large radioisotope heat source might bore into the ice to reach the subsurface ocean.

Because these same locations are of astrobiological interest, another thermal drilling consideration arises, namely, that of planetary protection (e.g., NRC, 2000). While it is usually somewhat straightforward (albeit expensive) to assure that a vehicle or drill system will not introduce terrestrial biota into a habitable planetary environment if it works correctly, a space mission may require demonstration that this will not happen even should an “off-nominal” event occur, such as loss of control before hypersonic entry into the atmosphere. Such an event may lead to breakup of the vehicle; the radioisotope heat sources, however, are designed with formidable thermal and mechanical robustness such that they can survive entry (in this connection note that the Radioisotope Thermoelectric Generator from the Apollo 13 lunar module, which of course returned to Earth instead of reaching the Moon, lies intact on the seafloor).

2. Physical Problem

Conceptually, thermal drilling is rather trivial. To drill at a certain speed U, an object with cross-sectional area A must simply supply the heat required to melt ice at a volume rate UA, that is, UAρL where ρ is the density of ice and L the effective latent heat of melting. This in turn is the conventional latent heat for the phase change (which on an airless body where liquid is not thermodynamically stable is the heat of sublimation rather than the latent heat of melting) plus the sensible heat needed to warm the ice from the ambient temperature T _o to the melting temperature T _m. Because of this consideration, a thermal drill is usually considered to be a slender object, such that it has enough volume to contain the required instrumentation but minimizes the cross section.

The above paradigm is approximately true for high melting power where the speed U is rather fast (where fast is defined by whether the thermal skin depth that corresponds to the time for the vehicle to move its own length is small compared with the vehicle length). This may be true for terrestrial polar drilling supported by large electrical or hot-water heat supplies on the surface, but it is not true for the more power-constrained scenarios expected in space exploration.

Here a significant and, in fact, dominant term must be considered, namely, the heat that leaks conductively sideways as the vehicle moves down. This heat leak, which becomes progressively more significant the slower the drill rate, in effect reduces the efficiency of the system; but if the heat loss is sufficient, melting will no longer occur at all, and the vehicle will stall and be frozen in place in the ice. A rather forbidding analytic expression for this loss was proposed by Aamot (1967), which involves a double integral and two types of Bessel functions (the solution is originally from the text of Carslaw and Jaeger, 1959). This integral typically requires numerical solution (and, in fact, full finite-element thermal models can be usefully applied to this problem—see, e.g., Chumachenko and Nazirov, 2009). Ulamec et al. (2007) supplied an analytic approximation, although this is unfortunately rather restricted in the parameter range over which it applies [Shreve, (1962) also presented a solution to the problem, but for ice almost at its melting point, which is not the case in planetary applications considered here].

However, for planetary protection considerations, at least, it is possible to draw on a different approximation, namely, that the heat source is a point or sphere, since radioisotope heat sources are not slender. Here, Carslaw and Jaeger (1959) gave the following expression for the temperature due to a point source moving at a speed U in an infinite conductive medium: \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*}(T - T_{\rm o} ) = ( q / 4 \pi kR ) { \rm exp} ( - U [ R - x ] / 2 \kappa ) \tag{1}\end{align*}\end{document}

Where T is the temperature at a distance R from the source, x is the coordinate along the line of motion, q is the heater power, k is thermal conductivity, and κ is the thermal diffusivity (=k/ρc_p ) with ρ the density and c_p the specific heat capacity.

In the limiting case where the source no longer moves, the surface temperature will tend toward the first term in parentheses with R equal to the source radius. Clearly, if the surface temperature is below the melting temperature, then melting does not occur, and the object stalls. In other words, the criterion for melting is simply \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6} \begin{document} \begin{align*}T_{\rm o} + ( q / 4 \pi kR ) > T_{\rm m} \tag{2}\end{align*}\end{document}

3. Results

We consider three objects [for a brief overview of radioisotope power sources, see NRC (2009) or Bennett (2006)]. The first is a bare General Purpose Heat Source (GPHS) unit, often informally termed a “brick.” This is a graphite block about 10×10×6 cm that delivers about 245 W of heat when loaded with fresh plutonium dioxide. The graphite housing ensures tolerance of entry temperatures. For an approximation to spherical geometry, a diameter of 8 cm will be used.

The second object is the Advanced Radioisotope Thermoelectric Generator (ASRG—see Lockheed Martin, 2010). This is a unit presently under development to support future planetary missions, including a Discovery mission concept presently undergoing a Phase A study, the Titan Mare Explorer (TiME). The ASRG contains two GPHS blocks inside an insulated housing that also holds a mechanical Stirling-cycle generator to efficiently provide ∼120 W of electrical power from the ∼500 W of thermal power. The unit is roughly 0.8×0.4×0.4 m, so a dimension of 0.5 m will be used.

The third object is the Multi-Mission Radioisotope Thermoelectric Generator (MMRTG). This also develops ∼120 W of electrical power but uses less efficient thermoelectric converters and therefore requires 8 GPHS units developing 2 kW of thermal power to produce this output. This unit is 0.7 m long and has fins that span ∼0.6 m. Taking into consideration that the fins might be damaged in a breakup scenario, I have adopted 0.5 m as the dimension, which is the same as for the ASRG. Since power/radius is the relevant parameter in the temperature expression, it can be seen that the MMRTG and bare GPHS are rather close, with the MMRTG being (just) the worst case for Mars and Titan: an intact ASRG is the most benign.

Since water is the biologically relevant material, I consider water ice to be the medium. A key point not normally significant in terrestrial thermal drilling work is that the thermal conductivity of ice is strongly temperature-dependent. Specifically, k is ∼651/T, where k is in W/m/K and T is in K [e.g., Petrenko and Whitworth (1999) and Klinger (1980) gave 567/T, and a slightly more elaborate expression was given by Slack (1980)—all are within ∼12% of each other]. Thus, near the melting point, the conductivity of ice is ∼2 W/m/K, and for Mars ice temperatures of, say, 200–250 K, k is ∼2.6–3.2 W/m/K. However, in the outer Solar System, near-surface temperatures are much colder, Titan has a surface temperature of about 94 K, and a design value of ∼100 K may be considered for Europa in general (its poles may be colder). Enceladus' temperatures are similar, except for the famous “hot spots” of ∼150 K or more.

I show resulting surface temperatures in Table 1 for terrestrial ice (260 K), a low-temperature Mars case (200 K—coincidentally approximately the surface temperature in Antarctica above Lake Vostok, where thermal drilling has been proposed), and Titan/Europa (100 K). The reader may readily interpolate the results for other situations such as Enceladus. It is emphasized that the thermal conductivities quoted above are for solid polycrystalline ice with no pores; fracturing and porosity may be present, especially near the surface. However, deeper levels where ice will have annealed are of ultimate interest for the planetary protection problem; hence the bulk thermal conductivities above can be used.

Table 1.

Heat Source Temperatures (K) Are Calculated from Eq. 2 ( i.e., Zero Melting Speed from Eq. 1)

	Earth T=260 K k=2.5 Wm⁻¹K⁻¹	Mars T=200 K k=3.2 Wm⁻¹K⁻¹	Titan T=100 K k=6.5 Wm⁻¹K⁻¹
MMRTG	514	374	197
P=2000 W, R=0.25 m			no melting
GPHS	487	395	187
P=250 W, R=0.035 m			no melting
ASRG	323	248	124
P=500 W, R=0.25 m		(marginal melting)	no melting

When the temperature is below the melting point, melt drilling cannot occur.

It can be seen that all the objects considered can cause melting on Earth. On Mars, the MMRTG and GPHS units are well above the melting point too, so planetary protection is a concern. The Mars-ASRG case is marginal—the simple expression yields a temperature below the melting point but is close enough that the approximate approach used may be inadequate for a definitive answer (and even if melting did not occur, it is conceivable that some kind of slow sublimation drilling could occur via vapor diffusion). In contrast, on Titan and Europa, the high thermal conductivity yields surface temperatures far below the melting point, and thermal drilling does not occur.

There is reassuring consistency of the results for Eqs. 1 and 2 when applied to the small-scale experiments of Treffer et al. (2006) with a 4 cm diameter electrically heated sphere. In ice at 220 and 250 K, heated at 60 and 25 W, respectively, the predicted surface temperatures are 312 and 293 K, respectively, above the melting point. And indeed, in these experiments drilling successfully occurred as one would predict. (Of course, these temperatures were not actually reached, in that a significant fraction of the power went into the latent heat of the ice as tunneling proceeded. Note that the specific case of a low-temperature ice at low power density does not appear to have been examined experimentally, since most studies are directed toward showing that thermal drilling is possible rather than exploring situations where it does not occur.)

The geometry assumed in this relationship is of course idealized, and especially in low-pressure conditions, Kaufmann et al. (2009) noted that melting performance is very sensitive to the shape of the hot body and the effectiveness of contact with the ice. It may also be noted that, at temperatures near but below the melting point, some sublimation can occur, and Kaufmann et al. (2009) observed the formation of a sublimation “crater” in one instance of a “stalled” probe. The exponential dependence of vapor pressure on temperature, however, means such phenomena are only likely close to the melting point.

4. Conclusions

An approximate analytic criterion has been developed to determine whether a heat source can drill in ice. This relation has been applied to various radioisotope sources used or planned to be used in planetary exploration in different environments, and it should be noted that the high thermal conductivity of low-temperature ice is a principal factor. Naturally, this problem could benefit from experimental study and analysis with finite-element models, but the expression suggested here gives a quick and useful initial estimate.

The sources considered can all drill on Earth; the stronger sources (GPHS and MMRTG) can drill on Mars, whereas the lower-power-density ASRG may or may not. None of the sources considered can drill in ice at 100 K as expected on Europa or Titan. This has important implications for planetary protection; sterilization measures may not be required if there is no way to introduce biota into subsurface oceans.

Footnotes

Acknowledgments

The author thanks Torrence Johnson of JPL for encouraging treatment of this problem and two anonymous referees for comments.

Author Disclosure Statement

No competing financial interests exist.

Abbreviations

ASRG, Advanced Radioisotope Thermoelectric Generator; GPHS, General Purpose Heat Source; MMRTG, Multi-Mission Radioisotope Thermoelectric Generator.

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