Fig. 9. Comparison of the CO production rates with measurements . The modeled nucleus contained initially amorphous or crystalline H2O ice
■8 -6 -4 -2 0 2 4 6 8 10 Heliocentric distance - perihelion distance — O 'jl (AU)
Fig. 10. Comparison of H2O and CO production rates with measurements . The modeled nucleus contained initially 40% amorphous H2O ice, 5% CO trapped in amorphous water ice, 5% CO condensed independently, and 50% dust (by mass). The two orbits shown differ by the assumed orientation of the pole axis
■8 -6 -4 -2 0 2 4 6 8 10 Heliocentric distance - perihelion distance — O 'jl (AU)
Fig. 10. Comparison of H2O and CO production rates with measurements . The modeled nucleus contained initially 40% amorphous H2O ice, 5% CO trapped in amorphous water ice, 5% CO condensed independently, and 50% dust (by mass). The two orbits shown differ by the assumed orientation of the pole axis and observations, even for well-studied comets such as Hale-Bopp. This is not really surprising in view of the model uncertainties. In addition, the comet was not on its first orbit into the inner solar system, and some processing probably has already happened.
We may then ask what we would expect in the other case of a purely crystalline surface layer. As outlined above, the sublimation fronts may have already moved deep into the interior of the nucleus. In the extreme case, the production rates of highly volatiles, such as CO, may not vary significantly along the orbit anymore. Modeling the comet coming into the solar system for the first time with crystalline water ice results in much lower CO gas production rates (Fig. 9), because the CO sublimation front has already moved several meters into the nucleus . This is clearly not observed. Hale-Bopp's nucleus, therefore, seems not to be in a highly differentiated state. Again this is not surprising, because the orbital heat wave will penetrate to several tenths to hundreds of meters for a long-period comet such as Hale-Bopp. As a result, many revolutions are needed to reach an evolved and highly differentiated state. Possibly, this state is never reached completely, if effects such as efficient surface erosion during perihelion passage are taken into account.
When observing the evolution of gas activity, it is interesting to study the region of the onset of activity of the volatiles, because the onset depends on their volatility and in addition on the heat conduction into the nucleus interior. Unfortunately, the onset occurs at large rh for most ices except H2O (Fig. 4). However, HCN sublimation is expected to start around 7-8 AU heliocentric distance. As HCN and and its daughter product CN show strong emission bands, measurements over a wide range of rh were possible for comet Hale-Bopp (Fig. 11), and the region of activity onset could be probed. A comparison of the observations to models solving the surface energy balance on the nucleus has been made for various heat conductivity parameters (Fig. 11). The observations are consistent with sublimation of HCN close to the surface or a very low heat conductivity of the nucleus . Again, no sign of significant differentiation of the nucleus is found from observational data.
Hale-Bopp's activity evolution seems to be different from models treating the nucleus in terms of an old, evolved and differentiated body, but also from models treating it as a newcomer made of amorphous ice. More complex models to simulate the observations have been made in the meantime (see  for a recent overview), but it is still difficult to understand the evolution of the production rates observed. Unfortunately, most sublimation models do not include ices of intermediate volatility, such as HCN, for a comparison to observations.
The release of CO from extended coma sources (see Sect. 5) may complicate the discussion further, because it can lead to a significant increase of the CO production rate with decreasing heliocentric distance resulting from
coma processes. This process is not included in the nucleus outgassing models and may explain part of the discrepancy between models and observations for molecules with coma sources. However, the significance of extended coma sources is reduced for most observed volatiles, showing again that it is important to incorporate several volatile species into the sublimation models.
The interior of short-period comets might show a higher state of differentiation than comet Hale-Bopp, because they remain closer to the Sun and therefore at higher temperatures. Unfortunately, in short-period comets, only daughter products could be monitored over a wide range of rh so far. They provide a less stringent constraint on the differentiation processes taking place, because we add uncertainty by modeling their parent production rates with chemical models (see Sect. 5). In addition, only for few comets, the heliocentric distance range covered by observations of gas emissions is extending beyond rh « 2.5 AU, and we therefore only have data in the water-dominated regime.
The variation of production rate with heliocentric distance is often expressed by a power law, rk, by fitting the slope of log(Q) over log(rh). The resulting exponents, k, can vary a lot from comet to comet. For example, values of k from —0.8 to —10.1 for Q(CN) have been determined , but the most extreme values are usually found for comets observed over only a small range in rh. This already illustrates a major problem when studying a comet's activity evolution. Temporal variations (rotation, outbursts, etc.) require good coverage in rh to disentangle these relatively short-term effects from the long-term orbital evolution. Extrapolation of often only poorly known model parameters, such as scale lengths (see Sect. 5), to large rh can additionally introduce false distance dependencies on the production rates derived. Additional complications are found for species released by extended coma sources, as already mentioned. In view of the large number of uncertainties, one needs to be cautious when interpreting observations based only on few data points, in particular when comparing results derived with different models and parameters. Clearly, a larger statistical sample is needed.
Several surveys have been made to study the gas activity evolution with heliocentric distance so far, with different results:
- Surveys of comets made by photometric and spectroscopic measurements in the optical range [2,41,82,167] show similar production rate evolutions for CN, NH, and C3, resulting in constant production rate ratios over distance.
- For the NH2/CN ratio, a decrease with rh has been reported [17,41]. However, this is difficult to understand, because NH does not seem to show this effect. Possibly, heliocentric distance dependencies for NH2 are introduced by uncertainties in the excitation models used for these measurements.
- The C2/CN ratio has been reported to decrease with heliocentric distance in some observations. While the study of five comets  showed a strong dependence of the C2/CN ratio on rh, only little or no variation was found in larger samples [2,41]. However, C2 is likely to be a grand-daughter product from C2H2 and possibly additional parents, such as C2H6 and other organic species, which complicate the determination of production rates. The formation of C2 needs to be clarified further before we can finally conclude to what extent the production rate of C2 evolves different to other species or whether the differences seen at present are an expression of incomplete modeling (see Sect. 5). - Comparing the evolution of daughter products originating from parent ices more volatile than water to the OH production rates shows in general no strong correlation. It seems OH production rates can as well vary more steeply or shallow than the minor volatiles with rh . This is difficult to understand in view of sublimation models. However, the database is sparse and may simply reflect the lack of a statistically significant number of good quality data points.
The large error bars in production rate determinations of short-period comets make the study of their orbital evolution difficult. Nevertheless, we note that all species vary over the orbit. Unfortunately, we do not have data on the CO production rate of a Jupiter family comet, which we could compare to model predictions. However, other volatile parent molecules, such as C2H2 and C2H6, would also be expected to show a different activity evolution compared with less volatile ices, such as HCN (Fig. 4) if the nuclei would be highly differentiated.
To summarize, the observations of gas production rates over heliocentric distance in comets are still insufficient to give a clear and statistically significant picture of the activity evolution of volatile species. We could turn the argument around and conclude that so far no signs for highly evolved and differentiated nuclei can be derived from the observations. However, we will need more observational constraints to further improve our understanding of the sublimation activity of cometary nuclei.
The aim of our discussion was to outline the principle of how production rate observations can be used to constrain nucleus models. Three parameters are important to provide helpful data in future:
• Observe species of different volatility, such H2O and CO, but also ices with intermediate volatility to enlarge the data base for model comparisons.
• Observe over a wide range of heliocentric distances with good coverage to smooth out short-term temporal effects.
• Concentrate at large heliocentric distances, where the onset of activity can be observed.
The latter point is important, because the onset of activity is sensitive to the nucleus heat conduction. Unfortunately, it is usually difficult to observe, because the onset occurs at large rh for the volatiles (Fig. 4). It has been done for H2O and HCN/CN in comet Hale-Bopp but will, however, be extremely difficult for normal comets.
Additional clues may come from comets active at very large distances, e.g., objects such as Chiron and other Centaur objects. Their outgassing is not driven by water, but by highly volatile ices such as CO, and they provide additional clues to the sublimation of these volatiles in addition to the study of normal comets (see ).
Finally, the lander of ESA's Rosetta mission will study the surface layers of comet 67P/Churyumov-Gerasimenko in situ and measure the composition and structure of the top surface layers. Obviously, this will provide data input for a big step forward in refining sublimation models. A major question, also addressed by the ground-based observations outlined above, is of course whether the composition measured in the top layers is pristine or suffered from severe differentiation.
In this section, we discuss the dynamical processes of gas molecules after their sublimation from the nucleus surface. We take the parameters rh = 1 AU and Q = 1030 molecules s as a reference case. The conditions are similar to the values of comet Halley near the encounter of the Giotto spacecraft, which provided us with detailed in situ values of the inner coma. This is still the most comprehensive data set about a comet to date. Halley, therefore, is our reference comet in the following, unless specified explicitly. Obviously, a full and comprehensive description of gas-dust dynamics and its application is beyond the scope of this introductional text. We refer to recent reviews, such as  and , for a detailed overview.
The gas molecules sublimate from the cometary nucleus and accelerate into the coma by (adiabatic) expansion into vacuum. Figure 12 illustrates schematically the principle processes in the coma:
• The main gas acceleration occurs within the first few kilometers above the surface. After a few tens to hundreds of kilometers a mean gas velocity of the order of 1kms_1 is reached. Beyond a few 103 km, the gas accelerates again, because it is heated in the intermediate coma by photolytic processes.
• The gas density decreases quickly as the gas expands (as 1/r2 in case of isotropic expansion and constant velocity).
• The gas coma temperature drops from about 200 K above an active region on the surface to approximately 100 K at a nucleocentric distance of about 102-103 km. In case of pure adiabatic gas expansion, the temperatures are
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