Free Access
Volume 510, February 2010
Article Number A55
Number of page(s) 22
Section Planets and planetary systems
Published online 09 February 2010

Online Material

Appendix A: Validation of the model and parameters

Here we estimate the uncertainty of the conversion of the line area  $\int T_{\rm mB}{\rm d}v$ into the corresponding production rate Q, introduced by the simplifying assumptions of our model and by the uncertainties of the adopted model parameters (Sect. 4.1). We performed the calculations using the values calculated or assumed in Sects. 4.1 and 4.2, i.e., $\langle Q_0 \rangle$ = 2.70 $\times $ 1025 molec s-1, T = 80 K, and $v_{\rm exp} = 0.8$ km s-1. Whenever the water production rate was needed, we used 1.2 $\times $ 1028 molec s-1, which we consider as a mean-diurnal value during our campaign. It is based on the result by Schleicher (2006), who estimated the production of water from his optical observations of OH, but we scaled his result to r = 1 AU (through a power law with an exponent of -4, which we consider as a ``textbook'' value; cf. Sect. 4.3). This is consistent with the results from Villanueva et al. (2006) and Dello Russo et al. (2007), based on direct observations of H2O in IR, which are scattered around the adopted value when normalized in the same way (marginally or not important for the latter). Note that the IR-based production rates are likely to be snapshot values whereas the OH-based result is presumably a mean-diurnal production rate (cf. Sects. 5.1 and 6).

First, working with the model itself, we investigated how much flux is contributed to our observations by the molecules at different nucleocentric distances $\rho$. Such dependence is entirely controlled by the beam pattern: in our model the total number of molecules in a thin shell centered at the nucleus is independent of the shell radius (if the thickness is fixed), and also the light emission properties of the molecules are identical throughout the coma. We also calculated cumulative flux contributions from the nucleocentric distances below a certain radius. Both profiles are presented in Fig. A.1. In the final step, with the aid of these profiles, we investigated the effects introduced individually by each of the assumptions, along with the input parameters, which we present in the following sections.

A.1 Was the coma optically thin?

Let us consider an optical depth $\tau = 0.25$as small enough to validate treatment of the coma as optically thin. Then, assuming for a moment the coma to be completely stationary, this criterion is satisfied at $\rho \geq 100$ km for the J(3-2) transition and at $\rho \geq 150$ km for the J(4-3) one. However, Fig. A.1 shows that even in the extreme case (May 10, 2006), when the size of the beam was smallest (see Table 1), and the optical depth largest, only 13% of the molecules were within the optically thick regime.

In fact the cometary coma is not stationary but rather expands rapidly. Therefore, molecules along a column have different velocity components along the line of sight, and due to different Doppler shifts, the emission from the molecules are slightly misaligned in frequency. For this reason, the optical depth along a column of an expanding coma is smaller than for a static coma with the same number density - hence the net impact on our derivations is $\ll$13%.

A.2 Were LTE conditions present throughout the coma?

We convolved the model flux contributions (Fig. A.1) with the nucleocentric profiles of the non-LTE to LTE occupancy ratios of the J = 3 and J = 4 rotational levels. We found that the determined production rates are overestimated by 6.5-13.5% (depending mostly on the beam size). We adopted the non-LTE distribution of the rotational levels at 1 AU from the Sun from Bockelée-Morvan et al. (2004a), but we scaled the nucleocentric distance to keep the water density profile consistent with SW3-C, which was an order of magnitude less productive than assumed in their model. Although the adopted distribution was calculated for the (constant) kinetic temperature of T = 50 K, this inconsistence is probably negligible.

The analysis shows, that determination of the production rates assuming the Boltzmann distribution of the energy levels was justifiable in our case; by no means, however, can this fact be identified with the LTE conditions present throughout the observed coma, which we do not address here.

\end{figure} Figure A.1:

Flux in the model coma as observed by our model beam. The left vertical scale concerns the relative flux contribution  ${\rm d}F\!/\!{\rm d}x$, generated in a shell of a unit thickness and the nucleocentric radius x. The right vertical scale concerns the cumulated relative flux F, produced in a sphere with the nucleocentric radius x. The x quantity is a relative length, measured in units of the beam radius at half-power.

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A.3 Was the photodissociation process negligible?

A convolution of the model flux contributions (Fig. A.1) with an exponential decay of HCN gives 2.5-4.5% underestimation of the determined production rates (depending on the beam size). We used for this test the quiet-Sun HCN photodissociation rate of 1.26 $\times $ 10-5 s-1 at r = 1 AU (Huebner et al. 1992). The error introduced by this assumption is very small because the comet approached the Earth so closely that we observed the inner coma only (half-power radius of the beam was about 1000 km). The photodissociation process, however, played an important role at much larger nucleocentric distances, since the characteristic photodissociation scale-length for HCN was about 60 000 km.

A.4 Were the molecules isotropically ejected from the nucleus?

Comet SW3-C was sublimating preferentially in (roughly) the Sunward direction, which is perfectly clear from the evolution of the line position with phase angle, visible in the night-averaged spectra (Sect. 3). Isotropic outgassing is additionally refuted by the single-peak shape of the night-averaged spectra, as isotropic models, such as ours, predict an obvious central dip when the coma is larger than the beam. Anisotropic, pro-solar outgassing is understandable, since it is the Sun that controls the temperature distribution over the nucleus. Therefore, there is no question of whether or not the molecules were isotropically ejected from the nucleus, but rather of how important the violation of this assumption is, while determining the production rate using our model.

Let us first consider separately the impact of an anisotropic outgassing on the derived production rates at two extreme geometries: a sublimation restricted to the plane perpendicular to the line of sight, and restricted to the plane parallel to the line of sight, where both planes cross the nucleus. For simplicity we assume here that outgassing is constant in time. In the former situation, there is no influence in the calculated production rates, i.e. the total flux observed by the beam leads to an unambiguous retrieval of this quantity regardless of the angular profile of the local sublimation rates (that may not be constant over the nucleus due to e.g. its dependence on the insolation angle). However, the latter scenario permits either an excess or a deficit of the observed flux with respect to the isotropic sublimation (at the same global rate), depending on a preferred direction of outgassing. An excess of the flux appears when the molecules are being preferentially ejected along the line of sight, thus many of them (i.e. more than for an isotropic sublimation) accumulate in the beam for a long time. On the other hand, if the molecules are being preferentially ejected in a direction perpendicular to the line of sight, many of them leave the beam rapidly, resulting in a deficit of the observed flux. In consequence, the calculated production rate is respectively over- and underestimated, if a model assuming an isotropic sublimation is used for the retrieval. Considering a comet whose nucleus is a uniformly volatile plain sphere whose outgassing is entirely controlled by the zenith angle of the Sun, it is intuitively clear that the largest underestimation of the production rate should be expected when the comet is at the phase angle $\phi = 90\hbox{$^\circ$ }$, whereas the highest overestimation should occur at $\phi = 0\hbox{$^\circ$ }$ and  $180\hbox{$^\circ$ }$.

\end{figure} Figure A.2:

Dependence of the relative model flux F on phase angle $\phi $ for different exponents $\gamma $ (see text).

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This agrees with our quantitative analysis which is presented in Fig. A.2. It was obtained in the framework of our model, generalized however to account for the anisotropy of the outgassing. As the activation function we used  $\cos^\gamma z$, where z is local zenith angle, and $\gamma = 1$ (though we also show the results for $\gamma = 0.5$ and $\gamma = 2$ for comparison). If $\gamma = 0$ is used, the model reduces itself to the standard isotropic version. For simplicity we assumed that the molecules are ejected in the direction strictly perpendicular to the nucleus surface. Although this formally implies no collisions and T = 0 K (in a thermodynamical sense), such ``artifacts'' of the model are not very important while investigating the influence of geometric projections. The obtained result is very general, e.g. independent of the beam size and pattern, as long as the other assumptions of our model are satisfied (e.g. negligible photodissociation).

Unfortunately, it is very difficult to model an anisotropic outgassing in a realistic, time-dependent way (cf. Appendix A.5), thus the isotropic model of Haser (1957) is the choice of almost every author (see e.g. A'Hearn et al. 1995; Bockelée-Morvan et al. 2004a). We followed this approach - even though it is not very realistic - for the obvious reason of simplicity, and also to keep the basic consistence with most of the other published results (note that the photodissociation process, naturally tackled by the Haser model, is negligible in our case - see Appendix A.3). For our observations of comet SW3-C, which was around $\phi = 90\hbox{$^\circ$ }$ in that time (see Table 1), this may result in about a 20% systematic underestimation of the production rates (see Fig. A.2). In spite of this, the self-consistence of the values should be practically unaffected, thanks to the fairly constant phase angle, which varied only  ${\pm}15\hbox{$^\circ$ }$ around the minimum of this dependence (see Table 1).

A.5 Was the production rate constant in time?

We analyze the heliocentric evolution (Sect. 4) and the short-term variability (Sect. 5) of the HCN production, although we had determined the rates under the assumption that they are constant. This assumption, which greatly simplifies our model making it time-independent, is however well satisfied in both cases, considering each process as a sequence of isolated states, such as the production is constant within each particular state, although it varies from state to state.

The characteristic time scale for this problem, calculated in the framework of our model, is only about 45 min, and after this time most of the molecules (73-87% depending on the beam size) would have left the beam (encircled by its radius at half-power). This is a clear advantage of the exceptionally close approach of SW3 to the Earth (see Table 1), making our beam unusually small at the comet distance. Provided that the production rate evolved on a time scale significantly longer than 45 min, and bearing in mind that the individual exposures (4-5 min) never exceeded this characteristic time scale, the obtained single spectra indeed can be considered as snapshots of a steady-state coma.

Obviously the heliocentric evolution of the production rate happens on a time scale much longer than 45 min (Sect. 4.2). However, is the same argumentation valid to justify our analysis of the short-term variation (Sect. 5.3)?

First, a rotation phase profile of the production rate which one observes may be noticeably different from the outgassing pattern intrinsic to the nucleus (cf. Biver et al. 2007). Generally, it is delayed in phase, its amplitude is decreased and shape deformed, which is caused by the non-zero beam size and non-zero exposure, that imply collecting the flux from molecules released at different moments of time - thus different rotation phases or even cycles. Most importantly, however, the periodicity of the observed phase profile remains unchanged thus can be directly linked to the rotation period of the nucleus. The faster the molecules leave the beam, and the slower the nucleus rotates, and the shorter the individual exposures are - the closer the observed phase profile is to the intrinsic one. However, to determine only the rotation period correctly, it is already sufficient to satisfy the three aforementioned conditions to be constant in time at any values.

In such a case, a physical interpretation of the phase profile (e.g. determining an instantaneous total production rate, locating and comparing sources of activity) may be done upon full non-steady-state modeling. Consider for example a nucleus which does not sublimate at all at some rotation phase, but the line is still visible due to the emission from the molecules released earlier. In such a case one will determine a substantial production rate if a steady-state model, such as ours, is used for the retrieval. But even if a non-steady-state modeling is performed, or if we deal with a truly intrinsic phase profile, such an interpretation may still be problematic. This is because the phase profile is entirely controlled by superposition of instantaneous ejections from the active regions at a given rotation phase, so we can barely resolve the individual sources at similar nucleocentric longitudes as well as those over which the insolation does not change noticeably over the rotation cycle (though in both cases they can be isolated by modeling the spectral line shape and/or by imaging). Therefore, when investigating a phase profile of the production rate, one should rather refer to the effective active sources.

We are convinced that the discussed effects of flux processing do not affect our observations significantly, and hence the observed phase profile is close to the effective intrinsic activity pattern of the nucleus (regardless of the slightly varying exposure and the noticeably varying beam size). This is because both the characteristic time scale for a significant loss of the molecules from the beam and the individual exposures were shorter than the time scale of any clear feature stimulated by the nucleus rotation (see Fig. 8). Interestingly, this justification applies to all the eight phase profiles, because those that are highly structured are found at low frequencies, and the complexity decreases fairly linearly with increasing frequency.

Note also a comparison of properties of the phase profiles for parent coma vs. daughters and dust presented in Sect. 5.1.

A.6 Were the expansion velocity and the temperature of the molecules constant in space and time? Were they equal to the adopted values?

Let us restrict this analysis to the nucleocentric distances $\rho$ = 200-5000 km, from where about 80% of the model flux contribution comes. For a comet at r = 1 AU with the production rate of water as estimated for SW3-C, in this range the nucleocentric profile of the outflow velocity should be practically constant at about 0.9 km s-1 (see e.g. Bockelée-Morvan & Crovisier 1987). This is very close to the value of 0.8 $\pm$ 0.1 km s-1, which we derived from the line widths in the night-averaged spectra (see Sect. 4.1), and which is consistent with the determinations for other comets around the same heliocentric distance (cf. e.g. Biver et al. 2002a,1999).

However, we are conscious that this result may be affected by the several simplifications of our model. Particularly, if most of the outgassing goes permanently in one direction, the corresponding spectral line may be very narrow regardless of the outflow velocity, and thus may result in significant underestimation of the velocity if a model assuming an isotropic sublimation (such as ours) is used to fit the line width. For a rotating nucleus with discrete active source(s), the situation may be improved by the flux processing (see Appendix A.5), which simulates isotropization of the coma. This is because such a narrow line would be drifting in frequency as the gas radial velocity changes over the rotation cycle, and hence the processed line would be broader. Our night-averaged spectra are definitely subjected to this effect due to their long effective exposures, however, it is not likely to significantly isotropize the coma as we did not find any hint of the line position variability due to the nucleus rotation (see Sect. 5.1). Moreover, since the observations were obtained in fairly ``frozen'' geometry (see Table 1), even if we used a campaign-averaged spectrum, the Sunward anisotropy (cf. Sect. 3) would remain barely reduced by the processing. As we concluded in Appendix A.4, it is very difficult to model an anisotropic outgassing in a realistic, time-dependent way, therefore we determined the expansion velocity in a standard manner, which is easy to implement and which ensures basic consistence with most of the other published results.

The model by Bockelée-Morvan & Crovisier (1987) also suggests that the kinetic temperature of the parent coma has a nearly constant nucleocentric profile at about 15 K in our case. The very low model temperature is a direct result of the cooling process induced by the adiabatic expansion of gas, with only very weak heating by highly energetic photodissociation products, whose number was simply too low for comet SW3-C to efficiently drive the process. However, e.g. Lis et al. (1997) determined a (rotational) temperature equal to 73 K for comet Hyakutake at very similar geo- and heliocentric distances, and using a similar observing facility (beam radius 400-800 km depending on the transition frequency), whereas the same model suggests T < 40 K already accounting for the 10 times higher productivity of that comet. This suggests that the temperature yield of this model should be considered as a robust lower limit for the mean gas temperature within our beam.

Table A.1:   List of published HCN rotational temperatures  $T_{\rm rot}$ based on IR spectroscopic observations of SW3-C.

Additional constraints on the HCN temperature come from IR observations of SW3-C (Table A.1). As expected, the temperature was rising while the heliocentric distance and the size of the sampled coma were decreasing. Unfortunately, the helio- and nucleocentric dependencies cannot be decoupled, nor are any of these measurements compatible with our beam size. However, since the temperatures by Dello Russo et al. (2007) were determined at very similar heliocentric distances, our best guess is that they are at the high end of what is reasonable to expect for our observations of SW3-C. We realize that they were sensitive only to the high-temperature region close to the nucleus, whereas our beam sampled the more distant colder gas too; however, the warmer gas, heated by collisions with the photodissociation products further from the nucleus, should also contribute to our beam if the process was efficient (see the characteristic nucleocentric scalelengths for the adiabatic cooling and the photolytic heating in e.g. Bockelée-Morvan & Crovisier 1987). Temperature determinations from other molecules and for fragment B (also reported by other authors), although educative, unfortunately do not remove the ambiguity of this discussion.

\end{figure} Figure A.3:

Dependence of the line area $\int T_{\rm mB}{\rm d}v$ on the coma temperature T (assumed to be constant) for both considered transitions (black lines). It was derived with our model for the HCN production rate Q = 2.7 $\times $ 1025 molec s-1, the geocentric distance $\Delta = 0.1$ AU, and the beam properties as of the SMT (see Sect. 2). The line ratio is also displayed (gray line).

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The observed J(4-3)-to-J(3-2) line-area ratio (cf. Fig. 3) is also suggestive about the temperature, although it cannot be used to measure it directly. One problem is that we did not observe both lines simultaneously, hence their ratio is influenced by the nucleus rotation if different rotation phases were sampled on different nights (cf. Sect. 5), and, to a smaller extent, by the slight change in observing geometry (see Table 1). Another problem is that the beam size was different for each of the observed transitions (cf. Table 1), hence the line ratio is influenced by the nucleocentric temperature profile (see before), and also because of a possible drop of the main-beam efficiency at the higher frequency (though undetected - see Sect. 2). Nevertheless, using the night-averaged spectra from May 10 and 11, 2006 (see Fig. 3), we computed the ratio of 1.34 $\pm$ 0.05, which - in LTE conditions - suggests a gas temperature of only 30 $\pm$ 2 K (see Fig. A.3). However, taking the aforementioned effects into account it is clear that this estimation should be considered as another lower limit on the temperature.

Taking all the arguments into account we eventually adopted T = 80 K, which may be, however, slightly overestimated.

Nevertheless, for both monitored transitions we checked with our model how the line area  $\int T_{\rm mB}{\rm d}v$ depends on the temperature T, and the result is presented in Fig. A.3. It shows that if for example we took the temperature as low as 45 K, the production rate obtained from the J(3-2) transition would be 38% lower, and 14% lower from the J(4-3) one. Therefore, if the temperature were lower than adopted, then the production rates we report in this work would be slightly overestimated, and the heliocentric dependence would be overly steep (Sect. 4.2). On the other hand, the influence of the uncertainty on the temperature is too weak to affect our periodicity analysis (Sect. 5), which is fairly stable against much larger changes of the heliocentric reduction (see Appendix C.5).

The facts that the temperature probably varies noticeably with the nucleocentric distance (cf. the profiles in Bockelée-Morvan & Crovisier 1987, for higher production rates), and with direction (cf. analysis of this effect for the production rate in Appendix A.4) are not very important here, since its constant value is a model concept only, aimed at optimum retrieval of the production rates. Therefore it should be considered as an effective temperature for a given beam rather than a precisely defined thermodynamic property of the coma. The expansion velocity is less sensitive to these effects, which are practically negligible compared to the problems discussed in the second paragraph of this section.

The heliocentric evolution of the coma expansion velocity and of the temperature must be practically negligible on the time scale of our campaign, which is due to the excellent stability of the observing geometry (see Table 1). Similarly, we ensured that the uncertainty about the temperature is negligible for the determination of the expansion velocity.

We acknowledge that in this discussion we consider the rotational temperature to be the same as the kinetic temperature - hence we directly compare the observational measurements of the former with the model predictions for the latter. However, this identity is true only under LTE, which was not necessarily well satisfied in our case (see Appendix A.2).


Considering the determined production rates as isotropic, it is clear that all the other model assumptions do not affect them significantly. Note that the stability of our instrument was about 20% (see Sect. 2), and the statistical error of an individual measurement is also 20% on average. Furthermore, the estimated deficit of our production rates resulting from the assumptions discussed in Appendices A.1 and A.3 cancels to some extent the excess resulting from the assumption discussed in Appendix A.2 and from the adopted coma temperature discussed in Appendix A.6. Thus, the overall systematic error introduced by the deviation of our model from reality should be lower than 20%, and the self-consistence of our production rates much better, justifying the use of this simple approach.

Appendix B: Validation of the heliocentric correction

The heliocentric evolution of the production rate is superimposed onto the short-term variability due to the rotation of the cometary nucleus. The determined heliocentric dependance may be different from the real one if the rotation phase of the nucleus is sampled irregularly. To evaluate the uncertainty introduced by this effect on our mean-diurnal HCN production rate at r = 1 AU, and on its heliocentric evolution, we assumed a simple sinusoidal rotation phase profile, coupled with a power-law dependance on heliocentric distance. Then the production rate Q at the time tand the heliocentric distance r is given by:

Q = \left(1 + \frac{A-1}{A+1}\sin(2\pi(t-t_0)f)\right)~Q_0(r/r_0)^n,
\end{displaymath} (B.1)

where Q0 is the mean-diurnal production rate at r = r0 = 1 AU, n is an exponent of the power-law heliocentric dependence, A is the ratio between maximum and minimum production rate during one rotation cycle, f is the rotation frequency, and t0 is the moment of the zero rotation phase.

We used A = 2, as typically found in our analysis (see Sect. 5.3), and n = -8.2 as obtained from our data (see Sect. 4.2). For the frequency f between 0.02 and 0.8 h-1, and the unknown zero-phase moment t0 ranging between 0 and 1/f, we calculated the model production rates according to Eq. (B.1), using the moments of time t of our observations. Then we determined Q0* and n* from a power-law fit to such artificially created data, as we did for our HCN measurements (cf. Sect. 4.2).

Generally, the reconstructed quantities Q0* and n* are different from the input ones, Q0 and n, because the heliocentric reduction cannot take into account the rotation term, that is unknown at that point. We found that the measured Q0* tends to be slightly lower than the real Q0. Its mean value  $\langle Q_0^* \rangle$, calculated from the complete range of the zero-phase moments t0, is between 1.3 and 2.9% lower than Q0, depending on the frequency f. The standard deviation of Q0* around  $\langle Q_0^* \rangle$, measuring how strongly the determined Q0* depends on the zero-phase moment t0 at a given frequency f, is found between 0.1 and 17.3%, with a median of only 2.9%. Generally, the error introduced on Q0* by the influence of the nucleus rotation is smaller than those from the other error sources, which are discussed in Appendix A.

\end{figure} Figure B.1:

Standard deviation among the reconstructed power-law exponents, $\sigma _{n^*}$, as a function of the rotation frequency f. The simulated production rates were calculated from Eq. (B.1). Positions of the eight best solutions (see Table 2) are indicated by the arrows: red for those from the system and blue for the others, though solutions B and C are unresolved.

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The mean value of the reconstructed power-law exponent, $\langle n^* \rangle$, is always equal to the real n, regardless of the frequency f. This is because the variations average out when varying t0 over a full rotation period. The standard deviation  $\sigma _{n^*}$, as a function of frequency f, is shown in Fig. B.1. It shows that it is possible to obtain from our data n* which is very different from the real n, especially for lower frequencies f, but the discrepancy is very sensitive to the actual f. Moreover, we investigated the probability of obtaining $n^* \ge -4$ (which we consider as a ``textbook'' exponent; cf. Sect. 4.3), strongly differing from the real n = -8.2, and found it to be equal to 2.2%. The same analysis, limited to the rotation periods in the range 3.0-3.4 h (as the most likely rotation period of SW3-C - see Sect. 5.4), yielded the probability of 3.6%. On the other hand, if we would have assumed n = -4, then the probabilities of deriving $n^* \le -8.2$ are 2.1% and 3.8% for the overall and restricted period ranges respectively.

Interestingly, the discussed functions of the rotation frequency f, such as the standard deviations of the reconstructed Q* and n*, as well as  $\langle Q_0^* \rangle/Q_0$, do not depend on the actual input values of Q0 and n. But since the distribution of n* around the assumed n is slightly asymmetric, they cannot be interchanged while calculating probabilities, which is illustrated by our results shown in the previous paragraph.

Summarizing, a small possibility exists that the steep instantaneous slope of the heliocentric dependance, derived in Sect. 4.2, is an artifact. However, this is still unlikely to significantly affect the periodicity analysis, as in Appendix C.5 we show that the presence of the solutions between 3.0 and 3.4 h is stable against such large changes in the assumed heliocentric evolution of the production rate.

Appendix C: Validation of the periodicity analysis

The analysis of periodicity presented in Sect. 5.3 is clearly insensitive to the systematic uncertainties of the production rates introduced by the simplifications of our model, which we extensively discuss in Appendix A. There are, however, several other problems which might have potentially affected the analysis. We discuss them in this section, and present direct tests for the robustness of our results.

C.1 Seasonal effects?

Following the statistical approach of Drahus & Waniak (2006), who adopted the routine of Michaowski (1988), we investigated the extent to which the orbital motions of the comet and Earth might have affected the observed periodic variability of the HCN production rate driven by the nucleus rotation (cf. Sect. 5.3). The results are summarized in Table C.1. They show that the changes in the observing geometry might have produced shifts of the rotation phase, but they are relatively small compared to the scatter of the individual data points. Furthermore, since such phase shifts increase roughly linearly with time, they are well compensated for by a (very small) adjustment of the rotation period, which is then called synodic (in contrast to the ``real'' one, so-called sidereal). Small phase shifts additionally suggest that the amplitude and shape of the phase profile were also stable during our observations.

Table C.1:   Maximum possible phase shiftsa.

C.2 Angular acceleration of the nucleus?

Small yet active cometary nuclei are subjected to rapid changes of their rotation periods (see Sect. 5.2). The effect is produced by torques which appear as a result of the outgassing in non-radial directions. Even though the torques may be small, and may largely cancel out, the effective residual torque is still expected to efficiently drive the process, and consequently to cause a measurable change in the rotation period on a time scale of a single apparition.

Several authors, e.g. Samarasinha et al. (1986), Jewitt (1999), and recently Drahus & Waniak (2006), presented a very similar simple description of this effect. Using the notation of the last authors, an instantaneous rate of change in the rotation frequency  ${\rm d}f/{\rm d}t$, can be described as:

{\rm d}f/{\rm d}t = \frac{15}{16\pi^2}\frac{v_{\rm subl}~Q_{\rm tot}}{R^4~ \varrho}S,
\end{displaymath} (C.1)

where $v_{\rm subl}$ is the gas sublimation velocity, $Q_{\rm tot}$ is the production rate of all gaseous species, R is the effective radius of the nucleus, $\varrho$ is the nucleus bulk density, and S is a dimensionless scaling factor which is the fraction of the total outgassing $Q_{\rm tot}$ which effectively accelerates or decelerates the rotation.

Using this equation we estimated the magnitude of angular acceleration for SW3-C during our campaign. The sublimation of gas was characterized by $Q_{\rm tot} = 1.2$ $\times $ 1028 molec s-1 (the adopted water production rate; see the beginning of Appendix A), and $v_{\rm subl} = 0.8$ km s-1 (the expansion velocity of the HCN coma; see Sect. 4.1), and the nucleus by R = 0.5 km (Toth et al. 2006; Nolan et al. 2006; Toth et al. 2005), and $\varrho = 600$ kg m-3 (as found for comet Tempel 1; cf. A'Hearn et al. 2005). The least known parameter is S, which we assumed to be also the same as for Tempel 1. Since the nucleus of the latter was recently discovered to be slowly spinning up (Belton & Drahus 2007), the rate of change in the frequency, measured in the Deep Impact photometry at the level of 7 $\times $ 10-8 h-2, suggests S = 1.5%, when linked with the other measured properties through Eq. (C.1). Finally, as the estimation for SW3-C we obtained ${\rm d}f/{\rm d}t = 1.4$ $\times $ 10-4 h-2. Although it is a factor of 2000 stronger than measured in comet Tempel 1, it is not a surprise that comet SW3-C should be changing its spin much faster than the six-times-larger nucleus of comet Tempel 1 (cf. A'Hearn et al. 2005), bearing in mind the biquadratic dependence on the radius, and that the comets had comparable production rates during both campaigns (cf. Biver et al. 2007).

Integration of ${\rm d}f/{\rm d}t$ over the duration time of our campaign yields the expected change of the rotation frequency equal to 0.041 h-1, regardless of the frequency itself. Double integration gives the expected change of the rotation phase, which is equal to 1.51 when referred to the middle moment of the campaign. The latter shows that the classical approach (see Sect. 5.2) is irrelevant in such a case because the phase shift of the first and last data point is more than one full rotation cycle.

In spite of this expectation, an argument suggesting a fairly constant periodicity may also be raised. As we noted in Sect. 5.4, the rotation period of 3.2 $\pm$ 0.2 h, obtained by Toth et al. (priv. comm.) for April 10, 2006, is similar to our three shortest-period solutions (i.e. F, G, and H - see Table 2). If both their result and any of these three solutions are indeed correct at their epochs, then the corresponding change of the rotation period between these two campaigns does not exceed 0.2 h. This corresponds to ${\rm d}f/{\rm d}t \approx \pm 2.2$ $\times $ 10-5 h-2, and consequently yields the scaling factor of approximately $S \approx \pm 0.23$%; which is a factor of 6.5 lower than for comet Tempel 1, and hence the net torque in SW3-C was much less efficient. This may result from the exceptionally large active fraction of the nucleus of SW3-C, which was about a factor of 20 more volatile than the one of comet Tempel 1 (see Sect. 4.3 and especially Fig. 6). Jewitt (1999) shows that the scaling factor (which he calls a dimensionless moment arm and denotes as kT) is inversely proportional to the square-root of number of active sources randomly distributed in azimuth. If therefore we interpret the difference in nucleus volatility as a difference in number of active sources, we will obtain the scaling factor ratio of both comets of approximately 4.5, which is comparable to what we derived before taking the simplicity of our argumentation into account.

This result should be now confronted with the duration of our campaign. The change of the rotation frequency is implied to be equal to  ${\pm}0.0063$ h-1, and the change of the rotation phase equal to 0.23 when referred to the middle moment of the campaign. To satisfy the latter we need to assume that if the rotation period were not constant, the frequencies of the considered solutions (determined in the classical way) would correspond to the middle time of the campaign - which is, however, very likely, since the time arrangement of our data is rather symmetric, and the quality does not change too much with time. Normally, such phase shifts should visibly decrease the quality of the phase profiles. But our observations cluster at the beginning and at the end of the campaign, therefore the phase shifts are fairly constant within each group (although formally proportional to the square of time), and they are well compensated for by a (very small) adjustment of the rotation period. The residual phase shifts are then insufficient to affect the obtained periodicities, and introduce only some small and ``safe'' excess scatter.

Note that the low efficiency of the accelerating torque, and consequently, the small changes of the rotation parameters during our campaign, are still the upper limits. That is because any other rotation period, within the uncertainty range of the result by Toth et al. (priv. comm.), would be even closer to one of the three shortest-period solutions from our list - thus being in support of it and suggesting slower changes of the rotation period. At the same time the increased difference with respect to the other solutions would make them unlikely. That is because faster changes of the rotation period would be required, which would consequently make their reality (as detected in the classical way) questionable, though we acknowledge that in such case the scaling factor would be closer to its theoretically-predicted value. An indication towards one of our three shortest-period solutions would also fix the sign of the period changes: currently we find the spin-up and the spin-down scenarios equally possible as the rotation period by Toth et al. (priv. comm.) is almost exactly between the extreme solutions H and F.

We conclude that our analysis, being done within the framework of a constant rotation period, is fully justifiable for the shortest-period solutions (i.e. F, G, and H). All the other solutions may be considered as realistic only under the assumption that the period found by Toth et al. (priv. comm.) is incorrect, and the rotation was sufficiently stable during our campaign. Otherwise, they must be regarded as artifacts, as the presence of any of them would require a very rapid evolution of the rotation period, making detection of such a solution strongly problematic, and actually requiring disruption of SW3-C, due to an overly fast rotation, shortly before the HST run, which was not observed.

C.3 Changes of the spin axis orientation?

Clearly, torques induced by outgassing may also excite the nucleus rotation and reorient the total angular momentum vector (cf. e.g. Gutiérrez et al. 2002; Samarasinha et al. 2004), producing changes of the spin axis orientation.

However, the possibility that the considered periodicities (Table 2) are affected or even completely generated by complex rotation instead of simple is very unlikely. In such a case the periods of rotation and precession must have been very close to a resonance, probably yielding a much longer apparent periodicity in our analysis. Only the 1:1 resonance would enable detection of the true period, although not distinguished as complex with our methods. But it is difficult to imagine a reason for such a (or even any) resonance, and even harder a comet rotating with a period much shorter than 3 h - as stability of the nucleus is then problematic (cf. e.g. Davidsson 2001,1999). Moreover, if the obliquity of the instantaneous spin axis to the total angular momentum vector was not large or the period of precession was long compared to the duration of our campaign, and if the time scale for substantial reorientation of the total angular momentum vector was long, then the spin axis orientation could not change much during our campaign. Consequently, the true periodicity could be correctly detected with our methods, as the discussed effects would only increase scatter in the phase profiles.

Therefore, we investigated our phase profiles using this scatter as a tracer of the discussed effects. Taking into account the rms error of our reduced production rates Q0, which is equal to 0.582 $\times $ 1025 molec s-1, we derived that $\Theta$would be equal to 0.448 if the data points were scattered as suggested by the rms. In contrast, the three discussed periodicities are detected at $\Theta$ of only 0.641-0.785 (cf. Fig. 7), corresponding to 20-32% excess of scatter (though noticeably lower than 49% for the unphased data). Thus, it may indeed hint at some small changes of the spin axis orientation or the observing geometry (discussed in Appendix C.1), but it may also result from some instrumental effects (see Sect. 2), and/or intrinsically imperfect repeatability of the sublimation pattern. In any case, the detected periodicity should be practically unaffected.

C.4 Where do the noise-induced solutions preferentially occur?

\end{figure} Figure C.1:

Histograms of the noise-induced solutions. Here N indicates the number of simulations (from the total number of 1000) featuring global minima within a 0.01 h-1 frequency interval centered at f. Positions of the eight best solutions (see Table 2) are indicated by the arrows: red for those from the system and blue for the others, though solutions B and C are unresolved.

Open with DEXTER

Another argument that should be considered in this discussion refers to the statistical properties of the noise-induced solutions (see Sects. 5.2 and 5.3). So far, by analyzing the number N of solutions satisfying the condition $\Theta < \Theta_0$ (which is fairly independent of the frequency f), we showed that all the solutions listed in Table 2 are statistically significant (see the confidence levels in Fig. 7). Nevertheless, another statistic $N(f_0 < f < f_0 + {\rm d}f)$, indicating at which frequencies the noise-induced (global) solutions preferentially occur, provides an additional constraint on the robustness of our results. We present it in Fig. C.1.

As might have been expected, most of the noise-induced solutions can be found at high frequencies. This means that if some of the solutions obtained for our observational data were in fact induced by noise, they would be the shortest-period ones rather than the others, which are practically impossible to generate from noise at f < 0.15 h-1. On the other hand, one should bear in mind that all the solutions we list were found at sufficiently low $\Theta$ to be certain that their presence is due to real short-term variability of the production rate. Therefore, the matter of the debate is only which of the phasings is the true one, but this ambiguity cannot be resolved with such an analysis.

C.5 Analysis of two test data sets

\end{figure} Figure C.2:

Periodograms for our two test data sets (described in the text) calculated with both methods. The horizontal lines indicate the confidence levels of 75% (blue) and 95% (red). Positions of the eight best solutions (see Table 2) are indicated by the arrows: red for those from the system and blue for the others. Solutions B and C are barely resolved in this figure.

Open with DEXTER

We also verified our results by performing the same periodicity analysis as before (cf. Sect. 5.3) for two additional data sets.

Data set A is based on the same AOS observations as used previously (cf. Sect. 2), but a different heliocentric trend was subtracted: we used a much shallower power law than before (cf. Sect. 4.2), with an exponent of -4, which we consider as a ``textbook'' value (cf. Sect. 4.3).

Data set B was constrained from the spectra provided by the facility Forbes Filterbanks (FFB-A and FFB-B), which were working simultaneously with the AOS spectrometers. They have total bandwidths of about 1 GHz with 1024 channels each (a resolution of 1000 kHz). These spectra were reduced in the same way as the AOS observations (cf. Sect. 2), and their own power-law heliocentric trend was subtracted (slightly steeper than the one calculated for the AOS observations - cf. Sect. 4.2). In theory, the meaningfulness of data set B is rather limited, because the observing noise comes mainly from the receiver, thus all the spectrometers should feature the same noise level and even pattern; in practise, however, this is not exactly true, because additional sources of noise also might have been present (cf. Sect. 2).

All the solutions we found before are also detected in these two tests, and most of them are again represented by the deepest minima (see Fig. C.2). However, two additional significant solutions are also present: one at f = 0.16569 h-1 in data set A (although it is weakly visible in the presented periodogram for HF with $N_{\rm f} = 5$, it is already very striking for $N_{\rm f} = 7$, and appears at nearly the same frequency), and one at f = 0.11711 h-1 in data set B. However, they do not appear in the accompanying tests, nor in the original analysis.

Interestingly, the best stability is demonstrated by our shortest-period solutions (i.e. solution F, G, and H - see Table 2), which are prominent in both test cases. The reason for that is the very simple shape of the corresponding phase profiles - the shape which makes them insensitive to small distortions. In contrast, the solutions at lower frequencies, including those newly found, with more complicated phase profiles, are more sensitive to the searching routine and settings, heliocentric reduction, noise realization and spectral sampling in a specific spectrometer, as well as any geometric and spin instabilities (cf. Appendices C.1-C.3).

Appendix D: List of the observations

Table D.1:   Observations of the HCN molecule with the AOS spectrometers (see Sect. 2).

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