Cometary Nuclei

The nucleus is the fundamental component of any comet because it contains most of the mass. Unfortunately, it is also the hardest to study, because most of the cross-section is carried by dust and gas ejected from the nucleus and not by the nucleus itself. As a result, physical studies of comets have, until recent times, been biassed toward the study of gas and dust released from the nucleus by its sublimation. These are the subjects of Heike Rauer's [128] lectures in this Saas Fee workshop. A great deal of important information about comets has been gleaned, for example, from the study of molecular fragments from dissociated parent molecules. In this section, though, I want to focus on what we have learned about the nucleus itself.

The first well-established detections of nuclei were achieved from the ground in 1984, quickly followed by close-up images of the nucleus of 1P/Halley in 1986 (Fig. 14). Before that time, direct observation of the nucleus was held by many to be impossible because of contamination of the nuclear signal by scattering from nearby dust and gas. A common misperception is that the nucleus is invisible from the ground because it is shielded from view by near-nucleus dust. This is almost never the case for a very simple reason: dust is ejected from the nucleus by the drag forces exerted on it by sublimated ice. If the coma were to become optically thick, the source of heat driving the sublimation would be shut down, reducing the dust opacity. Feedback, then, stabilizes the optical depth along a line of sight from the nucleus to the Sun, to be smaller than unity. Transient exceptions to this feedback control can

Fig. 14. Nucleus of lP/Halley imaged from the ESA Giotto spacecraft. This classic image was the first to show a nucleus at high spatial resolution. While various surface features can be discerned, it is obvious that important structure lurks beneath the resolution of the data. Dust jets are seen to emanate primarily from the sun-facing side of the nucleus. Courtesy Giotto camera PI H. U. Keller and ESA

Fig. 14. Nucleus of lP/Halley imaged from the ESA Giotto spacecraft. This classic image was the first to show a nucleus at high spatial resolution. While various surface features can be discerned, it is obvious that important structure lurks beneath the resolution of the data. Dust jets are seen to emanate primarily from the sun-facing side of the nucleus. Courtesy Giotto camera PI H. U. Keller and ESA

be imagined and might occur, but few or none of the observed properties of comets require large broadband optical depths to be understood. Cometary comae are, to a good level of approximation, optically thin. Since 1986, the nuclei of comets Borrelly (Fig. 15), Wild 2 (Fig. 16), and Tempel 1 (Fig. 17) have been imaged by spacecraft.

Nucleus size

Cometary nuclei subtend minuscule angular diameters (milliarcseconds) and are unresolved in optical ground-based data. No occultation of a field star by a nucleus has ever been observed: most nucleus sizes must be inferred by indirect means. The size of the cometary nucleus can be inferred from the "classical" technique first used by Dave Allen [3] in which simultaneous optical (scattered) and infrared (thermally emitted) flux densities are compared. This method is so important to the study of small bodies that it is worth describing in more detail: in essence it is very simple. Photons from the Sun strike a body and are either reflected or absorbed. The fraction reflected is called the "Bond albedo," A. Photons not reflected are absorbed, raising the temperature of the body and producing thermally emitted photons at longer wavelength. The fraction of the incident photons that is absorbed is (1 - A).

Fig. 15. Nucleus of P/Borrelly imaged from NASA's Deep Space 1 spacecraft. The effective radius is ~2.2 km and surface albedo ~0.03. Note the lobed structure of the nucleus (perhaps caused by a composite structure consisting of two major bodies in contact) and the smooth "pond" material above the waist. Courtesy NASA

Fig. 16. Nucleus of P/Wild 2 imaged from the NASA Stardust spacecraft. The effective radius is ~2.1km and surface albedo ~0.03. Note the remarkably smooth shape of the nucleus, which resembles that of a rotational figure of equilibrium. Courtesy Don Brownlee and NASA

The optical flux density scattered from a body is proportional to the product Cep, where Ce is the cross-section, while the thermally emitted flux density is proportional to Ce(1— p). Here, p is the "geometric albedo," which is related to the Bond albedo by A = pq, where q is a measure of the angular dependence of the scattering function called the "phase function." Provided q is known, measurements at optical and thermal wavelengths permit us to solve for the two unknowns Ce and p (cf. Fig. 18). The measurements should be simultaneous because small bodies are usually not spherical, causing Ce to vary with time.

Examined closely, the Allen size method is more complicated. The scattered radiation is both anisotropic and wavelength dependent, introducing two extra parameters. The phase function q is not in general known and has only been measured for a few bodies that can be observed over a very wide range of phase angles. Real surface materials will have (wavelength dependent) thermal emissivities <1, introducing another parameter. Most seriously of all, heat absorbed on the day-side of a rotating body can be carried by rotation onto the night-side, meaning that the emitted flux density depends on the heat-retaining capacity of the surface layers (traditionally characterized by the "thermal inertia parameter" I = kpcp, where k is the thermal conductivity, p is the bulk density and cp is the specific heat capacity, or by the "thermal diffusivity" defined by k = k/pcp. The magnitude of this interaction

Fig. 17. Nucleus of P/Tempel 1 imaged from the NASA Deep Impact spacecraft. The effective radius is ~3.1km and surface albedo —0.05. Note the craters, the left-right gash across the nucleus, and the two regions of smooth terrain apparently occupying lowland positions. Courtesy Mike A'Hearn and NASA

Fig. 17. Nucleus of P/Tempel 1 imaged from the NASA Deep Impact spacecraft. The effective radius is ~3.1km and surface albedo —0.05. Note the craters, the left-right gash across the nucleus, and the two regions of smooth terrain apparently occupying lowland positions. Courtesy Mike A'Hearn and NASA

between the rotation and the thermal emission introduces more parameters, for the thermal constants of the surface, and for the magnitude and orientation of the rotation vector relative to the line of sight. What looked like a conceptually simple method is in fact horribly complicated: the number of unknown parameters in the model generally exceeds the number of observational constraints.

What saves the Allen method is the empirical finding that assumed values for a great many of the unknown parameters can nevertheless give object cross-sections and albedos of useful accuracy. The "Standard Thermal Model" (STM) has arisen as a way to bundle many assumptions in such a way that they are not too visible to the user and so not too frightening! In STM, the thermal emission is assumed to emanate from a spherical body in which the surface temperature is set by instantaneous equilibrium with sunlight and where the effects of rotation are unimportant. This could mean that the surface heat retention is very small, so that heat is lost before rotation carries it away from the day-side, or it could mean that the rotation vector points exactly at the Sun, so that rotation does not change the surface heating pattern. Even with these and other assumptions for the emissivity (generally —0.9) and the angular dependence of the scattering, STM must include a fudge factor called n, the "beaming parameter," that is supposed to represent

Radius [km]

Fig. 18. Example of the thermal-optical method of determining the size and albedo of an object. The optical data place a constraint on the product pRr2, where pR is the geometric albedo and r is the effective radius. The thermal data place, through a model of the surface temperature distribution, a constraint on (1 - pR)r2. The two curves labeled x = 2 and x = n refer to the STM and ILM surface temperature approximations. The dots mark plausible solutions for these two models. Both yield low geometric albedos for this object. From [74]

Radius [km]

Fig. 18. Example of the thermal-optical method of determining the size and albedo of an object. The optical data place a constraint on the product pRr2, where pR is the geometric albedo and r is the effective radius. The thermal data place, through a model of the surface temperature distribution, a constraint on (1 - pR)r2. The two curves labeled x = 2 and x = n refer to the STM and ILM surface temperature approximations. The dots mark plausible solutions for these two models. Both yield low geometric albedos for this object. From [74]

the angular dependence of the emission from the surface caused by surface roughness and topographic effects. The value of n in STM is often taken to be n = 0.756 ( [89]) but in fact it is very uncertain and recent work suggests that n = 1 may apply. For our purposes, the point is that the interpretation of thermal emission data in terms of object size and albedo depends on poorly specified parameters such as n.

A counterpart to the STM is the "Isothermal-Latitude Model" (ILM) which is best thought of as applying to a spherical body with the Sun in its equator and a rotation period so short that the temperature is independent of azimuth and a function only of latitude. The ILM model has lower mean surface temperatures than the STM and so requires a larger Ce (and smaller p) to generate a given thermal emission signal.

The strength of the Allen method is that it is widely applicable and seems mostly to give diameters accurate to ^5 or 10% when appropriately "tuned" by the selection of the uncertain parameters. It has been used to measure the cross-sections and albedos of about a dozen comets, as listed in Table 3.

Table 3. Well-Measured Cometary Nuclei

Object

' e

P6

P c

b/a'

lP/Halley

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