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Fig. 19. Color-color plane showing the Damocloids, KBOs, and Centaurs. While the KBOs, and Centaurs show a wide range of surface colors (and, presumably, compositions) the Damocloid surfaces are entirely lacking in ultrared matter (spectral gradient >25%/1000Â, corresponding to the upper right in this color-color diagram). From [79]

Fig. 19. Color-color plane showing the Damocloids, KBOs, and Centaurs. While the KBOs, and Centaurs show a wide range of surface colors (and, presumably, compositions) the Damocloid surfaces are entirely lacking in ultrared matter (spectral gradient >25%/1000Â, corresponding to the upper right in this color-color diagram). From [79]

a lower limit to the nucleus axis ratio, because in general the rotational axis will not be aligned perpendicular to the line of sight. Repeated measurements under different geometries are needed to remove these effects of projection.

Figure 20 shows a range vs. period plot for those comets thought to be well-measured. Added to the plot are curves computed for two models. First, I show curves for prolate bodies in rotation about a minor axis, computed under the assumption that gravity at the tip of the spheroid exactly equals the centripetal acceleration there [68]. Second I assume that the nuclei are figures of rotational equilibrium and plot curves taken from Chandrasekhar's (in)famous book [17] in which the shapes of strengthless bodies are computed as a function of their density and angular momentum. The figure shows that the nuclei do not need to be very dense (in general, the critical densities are <1000 kg m-3) in order to be stable against centripetal effects, regardless of which model is used. It is not known whether the nuclei behave at all like

Fig. 20. rotation period vs. axis ratio (derived from lightcurve range) for cometary nuclei. Prolate spheroid curves (dashed lines) were computed as described in the text. The equilibrium spheroids (solid lines) were computed by Pedro Lacerda. The densities of the models are given in the figure and can be interpreted as limits to the nucleus densities under the assumption that the nuclei are strengthless

Fig. 20. rotation period vs. axis ratio (derived from lightcurve range) for cometary nuclei. Prolate spheroid curves (dashed lines) were computed as described in the text. The equilibrium spheroids (solid lines) were computed by Pedro Lacerda. The densities of the models are given in the figure and can be interpreted as limits to the nucleus densities under the assumption that the nuclei are strengthless strengthless bodies, but the consensus view (influenced very strongly by the split comets, see [6]) is that this is likely to be a good approximation.

Nucleus Density

There are no good measurements of the densities of cometary nuclei, but there are many strong opinions held by planetary scientists about what those densities are! Perhaps because of preconceived ideas about the way in which comets formed, most planetary scientists believe that the nuclei are less dense than water. This might be true, but we do not know.

Several indirect methods have been invoked to measure the densities of the cometary nuclei.

- The range vs. period plot was first used to argue that the densities must be low [68]. A prolate ellipsoid nucleus model was used to estimate the density from the period and the lightcurve range. There is no particular reason to assume that the nuclei are well described by prolate ellipsoids and, as can be seen from Fig. 20, an alternate assumption gives substantially lower densities for a given period, range pair.

- D/Shoemaker-Levy 9 was disrupted while passing close to Jupiter (Fig. 21). Measurements of the spreading rate of the "string of pearls" comet after disruption, when interpreted as the product of tidal stresses acting on an aggregate body of negligible tensile strength, give a relatively robust estimate of the density p = 600 kg m~3 [6].

- Asymmetrical outgassing exerts a "rocket" acceleration on the nucleus of magnitude an = fiwn— (12)

where Mn is the nucleus mass, V is the bulk speed of the material launched from the nucleus by sublimation and fr is a dimensionless constant. The value of fr depends on the angular distribution of the momentum flux in material launched from the nucleus. For a nucleus that ejects matter in a perfectly collimated beam fr = 1 while for isotropic ejection fr = 0. Consider a 1km radius comet (mass x 1012 kg) ejecting mass at 103kgs-1 in a collimated beam (fr = 1) while at 1 AU from the Sun. The rocket acceleration is an ~3 x 10~7 ms_1, or about 10~5 times the Solar gravity at this distance. Although small, the long action time allows the rocket acceleration to produce measurable deviations from Kep-lerian motion. To use (12) to determine nucleus density, the acceleration an must first be measured from astrometry of the comet. Spectroscopy gives V from the Doppler shift of lines resonantly scattered from escaping gas and dMn / dt can be estimated from the strengths of molecular

Fig. 21. Multiple components of the nucleus of D/Shoemaker-Levy 9 imaged from the Hubble Space Telescope. Each component sports a stubby tail, created by radiation pressure sweeping of emitted dust. Photometry shows that the emission was largely impulsive and occurred at the moment of break-up of the nucleus as it passed Jupiter (minimum distance 93,500 km or about 1.31 Rj)

emission lines. Then, given a value of fr, this equation gives the nucleus mass. Coupled with an estimate of the nucleus volume, the density can be determined.

This method has been used to estimate the densities of 81P/Wild 2 (p < 600 to 800 kgm-3; [30]), 67P/Churyumov-Gerasimenko (p < 600 kg m-3; [29]), 19P/Borrelly (100 < p < 300kgm-3; [28]). The low densities are interesting and in accord with the value obtained for D/Shoemaker-Levy 9 by a different method but, given the large amount of modeling needed to estimate fr, I suspect that this method can give almost any density the user wants.

Still, accepting for the moment that the densities are <1000kgm-3 and that the strengths are small, it is interesting to speculate about the possible internal structures of the nuclei. Most probably, the nuclei are porous dirt-ice mixtures with a broken internal structure consisting of blocks each much smaller than the aggregate size (middle panel of Fig. 22).

Fig. 22. Schematic of possible internal structure of the cometary nucleus. On the left, a differentiated nucleus in which the material properties (strength, composition) vary radially as a result of past heating, concentrated at the core. This model seems unlikely, given the high-volatile contents and low tensile strengths of comets. However, some of the larger nuclei could have experienced non-negligible internal heating from radioactive decays (enough to mobilize interior volatiles). In the middle, a multi-component (sometimes called "rubble pile") nucleus in which sub-elements in the body are loosely bound by gravity. This is probably closest to the real structure inside cometary nuclei. On the right, a monolithic nucleus with structural integrity over its whole diameter. Very small comets (like asteroids of <100m scale), could be like this. The red skin on each object symbolizes the non-volatile mantle

Fig. 22. Schematic of possible internal structure of the cometary nucleus. On the left, a differentiated nucleus in which the material properties (strength, composition) vary radially as a result of past heating, concentrated at the core. This model seems unlikely, given the high-volatile contents and low tensile strengths of comets. However, some of the larger nuclei could have experienced non-negligible internal heating from radioactive decays (enough to mobilize interior volatiles). In the middle, a multi-component (sometimes called "rubble pile") nucleus in which sub-elements in the body are loosely bound by gravity. This is probably closest to the real structure inside cometary nuclei. On the right, a monolithic nucleus with structural integrity over its whole diameter. Very small comets (like asteroids of <100m scale), could be like this. The red skin on each object symbolizes the non-volatile mantle

3.1 Mantles

Observations show that the surfaces of cometary nuclei are largely nonvolatile, consisting of refractory matter generally described as a "mantle" (crust might be a better word, and certainly less confusing given the strati-graphic relationship between the Earth's mantle and crust). Evidence for the existence of mantles includes

- Images from the ground and from space show that the mass loss from comets occurs from only a fraction of the total surface, suggesting that surface volatiles are not widely distributed (note: this says nothing about the distribution of volatiles inside the nucleus). Specifically, the mass loss occurs in jets and the total rate of production of water is less than would be expected if the whole nucleus were covered in water ice. The derived fractional "active areas" range from ~0.01 to ~10% [1].

- Spectral maps of comet 9P/Tempel 1 obtained from the NASA Deep Impact spacecraft show evidence for water only in a few locations occupying about 0.5% of the total surface [140].

- Temperatures of some nuclei are higher than can be sustained by a sublimating ice surface. Examples include 1P/Halley (peak temperature >360K [41]), C/1996 B2 (Hyakutake) (320K [98]) and 9P/Tempel 1 (peak temperatures ~330K [140]).

The physical properties of the mantles remain poorly determined. This is a more serious problem for cometary science than it at first sounds, because almost everything we know about the comets is either controlled or at least strongly modulated by the mantles. Likewise, the physics behind mantle formation and destruction is not well known.

Figure 23 compares the colors of objects within each of several small-body populations. Color is parametrized by the normalized reflectivity gradient, S' [%/1000 A], essentially the slope of the spectrum of the object after division by the spectrum of the Sun. Several features in Fig. 23 deserve comment.

(a) The nuclei of comets, both dead and alive, show a spread in color that matches that observed in the Trojans but which is distinct from the KBO color distribution. A few blue nuclei are known. We will argue below (Sect. 3.1) that these are most likely surfaces covered by rubble mantles.

(b) The Trojans (which are often but incorrectly described as consisting of very red D-type asteroids) in fact show a wide range of surface colors, down to neutral (S' = 0), and they are much less red than the majority of KBOs.

(c) Very red material is found only on the surfaces of the KBOs and the Centaurs. Specifically, if we define ultrared matter as having S' > 25%/1000A [71], then the figure shows that ultrared matter is absent in the inner Solar system populations including the Jovian Trojans, the nuclei of active and inactive Jupiter family comets and the Damocloids (not shown here, but see Fig. 19; [79]). As the progression of objects from top to bottom in Fig. 23

Fig. 23. Histogram showing the normalized reflectivity gradients measured in various small-body populations. Negative (positive) spectral gradients indicate blue (red) reflection spectra, relative to the Sun, which by definition has a spectral reflectivity gradient of zero. Material with S' > 25%/1000Â is defined as ultrared matter. Figure from [79]

Fig. 23. Histogram showing the normalized reflectivity gradients measured in various small-body populations. Negative (positive) spectral gradients indicate blue (red) reflection spectra, relative to the Sun, which by definition has a spectral reflectivity gradient of zero. Material with S' > 25%/1000Â is defined as ultrared matter. Figure from [79]

represents (except for the Trojans) a dynamical progression from the Kuiper belt source inward, a plausible conclusion is that the ultrared matter cannot survive in the inner Solar system. One guess is that the ultrared objects are coated in organic matter that has been irradiated by long-term exposure to cosmic rays and other particles, creating an "irradiation mantle" (Sect. 3.1).

Rubble Mantles

A rubble mantle consists of refractory, particulate debris that is left behind on the surface of the nucleus by the sublimating gases. Particles bigger than a certain critical size, ac, are too heavy to be launched against the gravitational attraction to the nucleus and remain behind. Assuming a spherical nucleus of radius rn and density pn, the surface gravitational force is just

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