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Astron. Astrophys. 319, 995-1006 (1997)

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6. Results and discussion

Comet 29P/Schwassmann-Wachmann 1 is of particular interest for a better understanding of physico-chemical processes and composition inside a comet nucleus. Compared to other short period comets its activity is a priori not dominated by sublimation of water ice. Furthermore, due to small CO photodissociation rates at a heliocentric distance of 6 AU the photochemistry is straightforward.

The results presented and discussed in this work are restricted to a small set of input parameters. The response of the model depends obviously on the choice of these parameters. We have chosen to take special attention to the tilt of the nucleus spin axis, the amorphous-crystalline phase transition and the surface erosion. We have considered the cases of an obliquity [FORMULA] and [FORMULA], an initial composition with and without amorphous ice, as well as the two cases where no surface erosion is taking place or a parametrised erosion rate.

In this work we used a constant value [FORMULA] for the parametrisation of the hertz factor (Eq. 13). Furthermore we adopted a true anomaly of the vernal equinox of [FORMULA] to obtain a maximal seasonal difference (Eq. 33).

Fig. 2a-d shows the CO production rates in molecules [FORMULA] with respect to time for a nucleus having its spin axis in its orbital plane ([FORMULA]). In the case of an initial amorphous composition the CO production rate is about [FORMULA]. If no erosion is taken into account (a) [FORMULA] decreases regularly to about [FORMULA] after 6 revolutions. The penetrating heat wave induces crystallisation within the first metre below the surface. During strong crystallisation [FORMULA] increases by about a factor 3. Small outbursts are hidden by the continuous activity due to sublimation of condensed CO. If no erosion occurs the amorphous ice front becomes rapidly too deep to be heated above a critical temperature which would induce crystallisation. The activity is different if the surface layer is eroded (b) continuously as a function of the CO outflow ([FORMULA], see Eq. 56). The choice of the correlation function determines if an outburst regime is maintained or not. If no amorphous ice is initially in the ice-dust matrix (case c and d) the CO gas production is about one order of magnitude smaller than in the previous case. This behaviour is explained by the low thermal conductivity of amorphous water ice. Since the conductivity of crystalline ice is several orders of magnitude larger, more heat is conducted to deeper layers so that local heating is prevented. Due to the low CO production rate the dust emission and surface erosion is also weak so that the case without erosion (c) is similar to the case with a parametrised surface erosion (d). We note that [FORMULA] is strongly correlated with the apsides in all 4 cases (a-d). The heat diffusion into the nucleus requires a certain time before the heat wave reaches the sublimation front (a,b). This delay is seen with respect to the heliocentric distance. In the case (c) and (d) the CO sublimation front is still close to the surface after 6 revolutions as the CO mass loss is small. Such a behaviour has also been noted for the observed lightcurve of P/SW1 (Cabot et al. 1996).

[FIGURE] Fig. 2. CO gas production rates in molecules [FORMULA] (full line) with respect to time for an obliquity of [FORMULA] and heliocentric distance [FORMULA] in AU (dashed line) is plotted for each configuration: a initially amorphous ice, no surface erosion; b initially amorphous ice, with surface erosion (see Eq. 56); c initially crystalline ice, no surface erosion; d initially crystalline ice, with surface erosion.

The surface is approximately uniformly eroded. After 6 revolutions the volume corresponding to 0.2 m depth or 2 entire layers have been lost at each latitude (b). The local activity averaged over one revolution is thus more or less comparable for the different surface elements. From this volume loss we can evaluate the average dust production of [FORMULA]. For a better fit of the observation more surface material has to be eroded.

Fig. 3e-h shows the CO production rates with respect to time for a nucleus having its spin axis perpendicular to its orbital plane ([FORMULA]). In none of the studied cases crystallisation occurred as the heating periods are too short to obtain sufficiently high temperatures below surface. The CO production rates are again larger for the cases of amorphous ice: no erosion (e), with erosion (f). If no erosion occurs then the production rate is slowly decreasing. In the other case where the CO drags out ice-dust grains the CO production rate is larger because of the continuous surface erosion. [FORMULA] is smaller if only crystalline water ice is considered: no erosion (g), with erosion (h). The reason is the same as the one given above. In these cases appear a correlation between [FORMULA] and heliocentric distance.

[FIGURE] Fig. 3. CO gas production rates (full line) with respect to time for an obliquity of [FORMULA] and heliocentric distance [FORMULA] in AU (dashed line) is plotted for each configuration: e initially amorphous ice, no surface erosion; f initially amorphous ice, with surface erosion (see Eq. 56); g initially crystalline ice, no surface erosion; h initially crystalline ice, with surface erosion.

Fig. 4a and b shows the CO molar concentration after 6 revolutions (only free CO, no trapped CO) within the nucleus. Due to the CO mass loss the sublimation front penetrates into the interior. We have found that the stronger the sublimation at a given region within the nucleus, the stronger the recondensation in the deeper neighbouring layers.

[FIGURE] Fig. 4a and b. CO molar concentration within the first few metres below surface after 6 revolutions. Top: obliquity [FORMULA] (a), bottom: obliquity [FORMULA] (b). [FORMULA], [FORMULA] and [FORMULA] molar concentration [FORMULA].

Fig. 5 shows the molar concentration of amorphous ice after 6 revolutions for case (a). The crystallisation takes places in the polar regions where the highest surface temperatures are obtained.

[FIGURE] Fig. 5. Molar concentration of amorphous ice within the first few metres below surface after 6 revolutions. Plotted only for configuration (a). [FORMULA], [FORMULA] and [FORMULA] molar concentration [FORMULA].

The model results are rather different for the diverse chosen configurations. The results of configuration (b) with an obliquity of [FORMULA], an initial amorphous ice composition and a parametrised grains emission fits the best the observations. The CO gas production rate is initially about [FORMULA], sudden peaks during a short period of time and the correlation with the apsides correspond to the observed features. None of them is found if only crystalline ice is considered. Nevertheless a comparable CO gas production rate could be obtained for crystalline ice for an initially extremely low hertz factor which may increase close to surface due to grain sintering. But for this case we would need another cause to explain the sudden release of CO during an outburst.

Fig. 6 shows the surface temperature at a pole and at the equator for the configuration (a). In this case the simulated temperatures at both poles vary from 155 K and 150 K during the period of maximal insolation to about 40 K at the antipode. No direct measurement of the nucleus surface temperature exists. However, the coma expansion velocity can be derived from the Doppler shift of a radio line in the spectroscopic signal. Following Crovisier et al. (1995) the coma expansion velocity on the day side of P/SW1 is [FORMULA] and [FORMULA] on the night side. The energy balance of a CO dominated coma without any photochemical processes is


where [FORMULA] is the enthalpy (without rotational energy) of the gas at surface and [FORMULA] is the translational energy. [FORMULA] represents energy sinks like the energy transfer to solid particles. Since collisions are not so efficient in producing rotational cooling as for translational cooling the adiabatic cooling following the coma expansion is more important than the rotational cooling. Therefore the energy balance can be written without the terms for rotational energy. As the grains are only weakly coupled to the gas coma we adopt [FORMULA]. The nucleus surface temperature [FORMULA] corresponds to the effusion temperature if the gas is completely thermalised. After computation of Eq. (60) we obtain temperatures [FORMULA] K on the day side and [FORMULA] K the night side respectively. These temperatures are significantly higher than those suggested (100 K and 40 K) from an analogue [FORMULA] coma by Crovisier et al. (1995). The high temperature on the day side is consistent with the black body temperature at 6 AU. The day side temperature corresponds quite well to our model result for a inclined spin axis, but so does not the night temperature. A better fit would be obtained for an intermediate obliquity.

[FIGURE] Fig. 6. Maximal surface temperatures for a nucleus with an obliquity [FORMULA] (a), plotted for the pole directed to the Sun at perihelion (full line) and the equator (dashed line).

The results of the presented simulation are an important clue for the presence of amorphous ice in comets. Amorphous ice is observed in molecular clouds (see Léger et al. 1979) and suspected to exist in comets. It has been proposed by Haruyama et al. (1993) and Prialnik & Podolak (1995) that a nucleus containing amorphous water ice and minerals could have completely crystallised as a result of radiogenic heating during its stay in the Oort cloud in case of a very low thermal conductivity. Nevertheless the result could be different if one assumes the presence of volatiles which reduce any local temperature increase by the latent heat of sublimation and a larger effective thermal conductivity.

A final remark concerns the nucleus spin period. The chosen period of [FORMULA] days has the advantage to allow a larger numerical integration step [FORMULA] but could be too large compared to the real rotation period which is unknown so far. A recent estimation by Meech et al. (1993) comes up with periods of 14 and 32 hours depending on the rotational state. However, our results would not be very different as in the case of maximal tilt of the nucleus spin axis ([FORMULA]) the period plays only a minor role. For [FORMULA] the crystallisation of amorphous ice would even be less likely. The maximal surface temperatures would be lower as the heat wave through heat conduction propagates roughly proportional to [FORMULA]. However using a much lower hertz factor crystallisation could occur.

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© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998