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where G = 6.6 x 10-11 Nkg~2 m2 is the gravitational constant and pd and a are the density and radius of the dust grain. The gas drag force is a complicated function of the grain parameters (shape, roughness) and of the ratio of the grain size, a, compared to the mean free path in the gas, Amfp. In the case where the grain size, a ^ Amfp, it is reasonable to consider the momentum of impacting gas molecules as being added one at a time, giving the classical drag force expression

Fd = Cd naa2pmHN1AV2 (14)

in which Cd is the (dimensionless) drag coefficient, p is the molecular weight of the sublimating gas (p = 18 for water), mH = 1.67 x 10~27 kg is the mass of the hydrogen atom, N1 [m~3] is the concentration of the gas at the nucleus surface and AV is the velocity of the gas relative to the grain. We calculate N1 from the thermal equilibrium equation for sublimating ice. The velocity difference AV is roughly the bulk speed of the gas as it leaves the nucleus, which data, physics and models show is of order, Vs, the sound speed in the gas at the temperature of the sublimating surface. Balancing gravitational force on a spherical grain with the gas drag then gives lim^NiV2

Gpn Pd^n for the critical size above which a grain cannot be accelerated to the escape speed from the nucleus and so which must fall back to the surface. We have ignored numerical constants in this expression and, given our state of ignorance, set Cd = 1. Noting that dm

we can rewrite this expression as

GpnPdVn dm

where dm/dt is obtained by solution of (3). Substitution gives us an immediate estimate of ac. Consider a water ice nucleus 1 AU from the Sun and with radius rn = 5 km. The sublimation rate per unit area is dm/dt - 10~4 kg m 2 s 1 (Fig. 10). If we take V — 500 R^i/2 [ms_1] as a first-order approximation to the gas speed at heliocentric distance Rau [AU] and further take pn = pd = 1000kgm~3, we obtain ac — 0.1m. Decimeter-sized bodies can be launched by gas drag against the gravitational attraction to the nucleus. This critical size decreases dramatically with increasing heliocentric distance owing to the rapid decline in the specific sublimation rate as the nucleus temperature drops. Beyond RaU — 5 or 6 AU, we find ac < 0.1 |m, and the particles that can escape the gravity of the nucleus are those that are too small to efficiently scatter optical photons (with wavelengths A — 0.5 | m), rendering them unob-servable. The magnitude of ac is plotted in Fig. 24 as a function of nucleus size and heliocentric distance.

There are many weaknesses in this simple calculation and many papers have been written to refine it since Whipple's (1950) classic exposition. Still, the essential point is that very large particles, if they exist in the nucleus, a c

Heliocentric Distance, R (AUJ

Fig. 24. Solution to (17) computed for dark (albedo 0.04) sublimating water ice nuclei as a function of heliocentric distance for nucleus radii from 1 to 100 km (as marked). The semimajor axes of the orbits of Earth, Jupiter, and Saturn are marked for reference. Particles larger than the wavelength of visible light, A ~ 0.5 |J,m, can be ejected all the way out to Jupiter's orbit but not much beyond

Heliocentric Distance, R (AUJ

Fig. 24. Solution to (17) computed for dark (albedo 0.04) sublimating water ice nuclei as a function of heliocentric distance for nucleus radii from 1 to 100 km (as marked). The semimajor axes of the orbits of Earth, Jupiter, and Saturn are marked for reference. Particles larger than the wavelength of visible light, A ~ 0.5 |J,m, can be ejected all the way out to Jupiter's orbit but not much beyond cannot be easily launched by gas drag into interplanetary space and will remain on the surface where they will impede the heating of surface ice and so diminish the sublimation gas flux. This is the rubble mantle (Fig. 1.25).

It is interesting to consider some consequences of this simple model. First, how thick must such a mantle be? The physical condition for the mantle to seriously impede the heating of ice is that the mantle thickness must rival or exceed the diurnal thermal skin depth. The latter is a measure of the depth to which heat can be carried from the surface by conduction, and is given by LD ~ (KProt )1/2, where k is the thermal diffusivity and Prot is the rotation period of the nucleus. With k = 10-7 m2 s-1 (appropriate for the porous dielectric materials likely to comprise the mantle matter) and Prot = 10 h (typical of the well-observed cometary nuclei; see Table 3), the skin depth is only Ld ~ 0.06m (6 cm!) and the mantle need not be very thick to impede the gas production.

The timescale for such a mantle to form is

where fM is the fraction of the solid mass that cannot be ejected by gas drag because it is contained in bodies with a > ac. For a power-law distribution in which the number of solid particles with sizes in the range a to a + da is given by n(a)da = ra-qda (r and q are constants), the fraction fM is easily calculated from

Fig. 25. Schematic cross-section in a cometary nucleus showing the formation of a rubble mantle. At initial time T0, the nucleus consists of a mix of "rocks" (dark) and ices (light). The nucleus is heated from above by sunlight, leading to the sublimation of the ices. Gas drag forces expel smaller rocks into the coma while larger solid particles are left behind. Movement of the sublimation surface into the nucleus exposes more rocks, including large ones that eventually clog the surface, creating a thermally insulating, non-volatile rubble mantle. Any mantle thicker than the diurnal skin depth (~5 cm) can inhibit sublimation. The interval from TO to T3 is a function of nucleus size and the pattern of insolation on the nucleus, but can be shorter than the orbit period for comets in the inner Solar system

JO Tl T2 T3

Fig. 25. Schematic cross-section in a cometary nucleus showing the formation of a rubble mantle. At initial time T0, the nucleus consists of a mix of "rocks" (dark) and ices (light). The nucleus is heated from above by sunlight, leading to the sublimation of the ices. Gas drag forces expel smaller rocks into the coma while larger solid particles are left behind. Movement of the sublimation surface into the nucleus exposes more rocks, including large ones that eventually clog the surface, creating a thermally insulating, non-volatile rubble mantle. Any mantle thicker than the diurnal skin depth (~5 cm) can inhibit sublimation. The interval from TO to T3 is a function of nucleus size and the pattern of insolation on the nucleus, but can be shorter than the orbit period for comets in the inner Solar system

where a- and a+ are the minimum and maximum sizes in the dust size distribution. This integral takes a particularly simple form when q = 4, and this happens to be not too different from the size distribution measured in the coma of lP/Halley by the dust detectors of the Giotto spacecraft, at least for sizes near 100 |m [87]. Then

provided a+ > ac, and fM =0 otherwise. The size of the largest "particle" in the cometary nucleus is unknown, but studies of bolides show that comets eject bodies of decimeter and larger sizes when near the sun. We take a+ = 0.1m and, based on observations of tiny dust particles in lP/Halley, set a- = 10-8 m.

Combining (16, 17, 18, 19) and using (3) to calculate dm/dt, we obtain an estimate of the mantling time, tm, and the results are plotted in Fig. 26. Two volatiles have been used to estimate the timescales, water and carbon monoxide; the main difference being that the latent heats of sublimation of these materials are in the ratio of about 10:1. I further show curves computed

1 10

Heliocentric Distance, R [AU]

Fig. 26. Timescale for mantle formation from a simple model (18) as described in the text. Curves are shown for two volatiles (CO and H2O) and two nucleus radii

(5 and 50 km), with assumed density of 1000 kg m-3. From [71]

1 10

Heliocentric Distance, R [AU]

Fig. 26. Timescale for mantle formation from a simple model (18) as described in the text. Curves are shown for two volatiles (CO and H2O) and two nucleus radii

(5 and 50 km), with assumed density of 1000 kg m-3. From [71]

for to values of the nucleus radius (at constant assumed density 1000 kg m-3)

to indicate the effect of size. Several features of Fig. 26 are worthy of note.

- The mantling timescales for water are less than 1 year for heliocentric distances < 3 AU, for nuclei of both 5 and 50 km radius. This very short timescale means that rubble mantles can potentially grow within a single orbit. A patch of ice exposed to the Solar insolation would, in this model, seal itself against continued sublimation on a timescale of a year. If true, we should think of the mantle as a dynamic structure that can adapt to changes in the insolation.

- Mantling of the water nuclei slows with increasing heliocentric distance. At distances Rau > 6, the mantling time exceeds the —0.5 My dynamical lifetime of the Jupiter family comets [96]. Rubble mantles should not form at larger distances if formed only by the sublimation of water ice.

- Cometary activity powered by CO sublimation extends to much lower temperatures and larger heliocentric distances than for water. Indeed, CO is so volatile that it sublimates strongly across the entire planetary region of the Solar system. The mantling time because of CO is therefore very short even out to the orbits of the KBOs. One conclusion is that CO should not be found on the surfaces of the KBOs (unless held there by gravity on the largest objects). Another is that the past presence of CO in the Kuiper Belt would have led to rapid and complete encrustation of these bodies by rubble mantles.

- Figure 26 shows that the mantling times rise at the smallest heliocentric distances. This is most obvious for the CO, 5 km radius model, which rises toward infinity at about 2.5 AU. The physical reason for this is that when sublimation is very strong, the gas drag forces are able to eject even the largest solid bodies in the distribution (i.e., ac > a+), and no mantles can form.

This simple model illustrates many of the key features of the rubble mantle. It needs to be only centimeters thick to protect nucleus ice from the heat of the Sun. It can form very quickly. A mantle formed at large heliocentric distance can be unstable to ejection at smaller distances. Mantles on large nuclei are more stable than on small nuclei. Depending on the size distributions in the refractory particles, very tiny nuclei might be unable to retain rubble mantles at all. Considerations like these have induced some researchers to consider models that couple mantle development with orbital evolution, particularly with the drop in perihelion distance that has occurred to most observed comets. The results are very interesting, and parallel to the qualitative ones presented here [130].

The given picture of rubble mantle development is highly simplistic, however. For example, the role of centripetal acceleration has been ignored. An elongated nucleus in rotation about its short axis will experience net reduction in gravity toward the tips that could render the rubble mantle unstable, producing bald spots. The existence of even a small tensile strength would overwhelm the significance of the tiny nuclear gravity and could give the mantle properties quite different from those inferred above. Lastly, and most importantly, what we have presented is no more than a hideous cartoon compared to the complex surface structures imaged by spacecraft on the nuclei of comets (Figs. 14, 15, 16 and 17). Making a deeper connection between the properties of the mantle and the surface morphology will require mechanical and other data from a surface lander. Perhaps ESA's Rosetta will do the job?

Irradiation Mantles

An entirely different type of mantle has long been postulated for the surfaces of cometary nuclei. This mantle is formed by the long-term bombardment of ices on the nucleus surface by energetic particles from the Sun, the Solar wind and galactic cosmic rays and is generally known as the "irradiation mantle" (see Fig. 27). Ironically, there is no specific evidence for irradiation mantles on the nuclei of comets. Instead, if they exist anywhere, they are most likely to be found on the exposed surfaces of the Kuiper belt objects. The reason for this is simple: rubble mantle formation timescales are much shorter than the timescales for radiation damage, given the known fluxes of energetic particles.

Energetic particles dissipate their energy in a complicated cascade of interactions that results in breaking the covalent bonds that hold common molecules together. New bonds can form, producing molecules that were not present in the initial mix. Hydrogen liberated from parent molecules in this way is small enough and sufficiently volatile to be able to escape, leaving behind C, N, and O to form complex molecules with whatever hydrogen remains.

cosmic cosmic

TO TI T2 T3

Fig. 27. Schematic cross-section in a cometary nucleus showing the formation of an irradiation mantle. At initial time T0, the nucleus consists of a mix of "rocks" (dark) and ices (light). Cosmic rays bombard the surface layers, breaking bonds in the ice molecules, allowing the formation of radicals, the preferential escape of hydrogen and the formation of a carbon-rich, low albedo "irradiation mantle." The thickness of the layer is of order 1m (for bulk density 1000 kg m-3). The interval from T0 to T3 is uncertain, but probably —100 My for complete processing. A thinner surface layer (affected only by low energy particles) could form on a shorter timescale

TO TI T2 T3

Fig. 27. Schematic cross-section in a cometary nucleus showing the formation of an irradiation mantle. At initial time T0, the nucleus consists of a mix of "rocks" (dark) and ices (light). Cosmic rays bombard the surface layers, breaking bonds in the ice molecules, allowing the formation of radicals, the preferential escape of hydrogen and the formation of a carbon-rich, low albedo "irradiation mantle." The thickness of the layer is of order 1m (for bulk density 1000 kg m-3). The interval from T0 to T3 is uncertain, but probably —100 My for complete processing. A thinner surface layer (affected only by low energy particles) could form on a shorter timescale

Experiments show that the result is a chemically complex mixture of organics, both aliphatic (carbon chain molecules) and aromatic (carbon ring molecules), in some cases polymerized to a very high molecular weight (| > 100s). High molecular weight corresponds to low volatility and the resulting irradiation mantle is stable against sublimation relative to the common ices. The mantle is also of low albedo, a reflection (pun intended) of the high carbon content. In fact, the molecular and chemical nature of this type of material is poorly defined. Related complex organic materials called "Tholins" are sometimes used as analogs, but these are produced by spark discharge in low pressure gases, and they may not be an appropriate analog for the mantle material. "Kero-gens," high molecular weight hydrocarbons found in terrestrial oil shales, may be a good analog, although these are not produced by irradiation.

The depth to which material can be damaged by energetic particles is a function of the particle energy. In the planetary region, the largest fluxes are for low energy particles in the Solar wind (energy —1 to 10keV), and these particles have very small penetration depths in ice. Much more energetic particles (MeV to GeV and beyond) are found in the cosmic rays but at relatively low fluxes. Damage occurs fastest at the surface but, given billions of years should extend to column densities —1000 kg m-2 (1 m in ice of density 1000 kg m-3). Calculations of the timescale for delivery of 100 eV per oxygen atom are shown in Fig. 28, for heliocentric distances of 40 AU, 85 AU, and "to AU" (corresponding to the local interstellar medium). This energy dose is chosen because it corresponds to heavy damage to the exposed material. Major differences exist between these locations both because the flux of low energy particles from the Solar wind declines with the inverse square of the distance and because the magnetic interaction of the wind with the interstellar medium

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