A&A 510, A55 (2010)

The HCN molecule as a tracer of the nucleus rotation of comet 73P-C/Schwassmann-Wachmann 3

M. Drahus¹ - M. Küppers^1,2 - C. Jarchow¹ - L. Paganini¹ - P. Hartogh¹ - G. L. Villanueva³

1 - Max-Planck-Institut für Sonnensystemforschung, 37191 Katlenburg-Lindau, Germany
2 - European Space Astronomy Centre, ESA, 28691 Villanueva de la Cañada, Madrid, Spain
3 - NASA Goddard Space Flight Center, MD-20771 Greenbelt, USA

Received 19 October 2007 / Accepted 30 March 2009

Abstract
Context. The causes of cometary break-ups are still uncertain. One suggested mechanism is splitting due to fast rotation of the nucleus. This can be tested by measuring rotation periods of cometary fragments.
Aims. The exceptionally close approach of the split comet 73P/Schwassmann-Wachmann 3 to the Earth in May 2006 made it an ideal target to investigate the rotation of its fragments. We used the HCN light curve for this purpose, because it is particularly sensitive to the rotation of the nucleus and at the same time it allows us to study the physics of cometary activity.
Methods. Comet 73P/Schwassmann-Wachmann 3 was observed between May 1 and 22, 2006, with the Submillimeter Telescope on Mt. Graham, Arizona, USA. Emission from HCN and CS were clearly detected. In this work we focus exclusively on the observations of the HCN molecule in fragment C, obtained during five nights between May 10 and 22, 2006, which provide the best S/N and the best temporal coverage.
Results. The light curve of comet 73P-C/Schwassmann-Wachmann 3 in HCN shows strong non-random variations, most probably stimulated by the nucleus rotation. The variability has an amplitude of about a factor of 2 on a time scale of hours. Among several plausible solutions for periodicity, we found strong indications for a rotation period of between 3.0 and 3.4 h, consistent with the determination from the Hubble Space Telescope. At 1 AU from the Sun the mean-diurnal HCN production rate was 2.7  10²⁵ molec s^-1 (with an uncertainty of about 20%) and the coma was expanding with a velocity of 0.8  0.1 km s^-1. The line position was evolving with a phase angle that is visible in the night-averaged spectra.
Conclusions. Evolution of the line position is consistent with the solar-stimulated activity. The mean-diurnal HCN production rate should be considered as very high, and it requires an unusually large fraction of the nucleus area to be active, whereas the coma expansion velocity was typical. The proposed rotation period, being the shortest ever determined for a cometary nucleus, cautiously suggests the disruption of the parent body due to a large centrifugal force, though it cannot be considered as a proof of this scenario. On the other hand, the observed stability of 73P-C against the rotational disruption suggests a bulk tensile strength of at least 14-45 Pa. The rotation period was surprisingly stable, indicating that no more than about 0.2% of the total outgassing was effectively accelerating or decelerating the nucleus spin. This is consistent with the large active fraction of the nucleus.

Key words: comets: individual: 73P-C/Schwassmann-Wachmann 3 - submillimeter: planetary systems

1 Introduction

More than 40 comets are known to have split, producing several hundred fragments (Boehnhardt 2004). However, except for rare cases of cometary disruption by tidal forces due to close approaches to the Sun or to Jupiter, the causes of cometary splitting are still unknown. One candidate mechanism is rotational break-up, when the centrifugal force surpasses the nucleus self gravity and tensile strength (cf. e.g. Davidsson 2001,1999).

The hypothesis of rotational splitting is testable: a rotation period of a main fragment, which is close to the disruption threshold shortly after the break-up, would be suggestive of rotational splitting. On the other hand, a long rotation period after break-up does not necessarily exclude rotational splitting, because cometary nuclei may change their spin state very rapidly, especially those highly active and of a small size (cf. e.g. Samarasinha et al. 1986; Gutiérrez et al. 2002; Jewitt 1999).

Comet 73P/Schwassmann-Wachmann 3 (hereafter SW3) was discovered in 1930 by Arnold Schwassmann and Arno Arthur Wachmann (Hamburg Observatory, Germany), around the perigee of its orbit at only 0.06 AU from the Earth. It is a short-period Jupiter Family Comet with an orbital period of 5.36 years and a perihelion distance of 0.94 AU. The comet broke up into at least four pieces in 1995 (Boehnhardt & Kaufl 1995; Scotti et al. 1996), and the splitting was accompanied by an outburst (Hale et al. 1995; Morris et al. 1995; Crovisier et al. 1995). Three of the components were re-detected during their next perihelion passage in 2000/2001 (Kadota et al. 2000). While the geometry of the apparition of SW3 in 2000/2001 was unfavorable, the comet approached the Earth to less than 0.1 AU in May 2006, displaying over 60 fragments (Weaver 2006). Of all of them, fragment C is the largest, about 0.5 km in radius (Toth et al. 2006; Nolan et al. 2006; Toth et al. 2005).

The exceptionally close approach of comet SW3 was a good opportunity to search for variability in the emission of parent molecules, stimulated by the nucleus rotation. Using a typical (sub-) millimeter facility, the telescope beam radius was of the order of 1000 km at the comet, ensuring that most of the observed emission originated from the inner coma, so that most of the material left the nucleus within less than an hour before the observation. Therefore periodicities with a period longer than one hour might have been detected.

We undertook such study using the HCN molecule, because it is believed to sublimate directly from the nucleus, and it offers the brightest emission lines for ground-based (sub-) millimeter observations. This makes it ideal for investigating rotation of a cometary nucleus. According to Huebner et al. (1992), 97% of the molecules photodissociate into H and CN, with a lifetime of 22 h at 1 AU from the (quiet) Sun. This corresponds to a scale-length of 60 000 km, assuming a typical gas expansion velocity at this heliocentric distance. Furthermore, CN is one of the brightest radicals observed in cometary spectra in the visible range, thus the HCN-CN connection is the best studied parent-daughter relation in cometary comae. However, it is still not clear if HCN is the only important parent of CN (see e.g. Fray et al. 2005; Bockelée-Morvan & Crovisier 1985). The production rate of HCN is of the order of 0.1% relative to water, and this ratio is remarkably constant over large range of heliocentric distances and very similar for many comets (cf. Biver et al. 2002b; Bockelée-Morvan et al. 2004a).

In this article we show selected results from our observations of comet SW3 at (sub-) millimeter wavelengths in May 2006. In Sect. 2 we present our data and outline the reduction steps. Section 3 shows an evolution of the line position with phase angle. In Sect. 4 we determine and discuss the production rates of HCN, and in Sect. 5 we use them to investigate the rotation of the nucleus. Finally, Sect. 6 summarizes this work, and presents our conclusions for future studies. Sections 4 and 5 are supplemented with Appendices A-C, which contain a detailed justification of our work. Appendix D contains the complete data set which we used for this study.

2 Observations and data reduction

We monitored comet SW3 at (sub-) millimeter wavelengths between May 1 and 22, 2006, searching for different molecules in the two brightest fragments (designated as C and B). However, in this article we focus only on the results of our observations of the HCN emission from fragment C (hereafter SW3-C).

We used the 10.0-m Heinrich-Hertz Submillimeter Telescope (Baars et al. 1999; Baars & Martin 1996) at Mt. Graham International Observatory in Arizona (hereafter SMT). Depending on atmospheric transmission we either used the 1.3-mm JT single-sideband SIS receiver or the 345-GHz double-sideband SIS receiver, to observe HCN through the J(3-2) or J(4-3) rotational transitions. Both receivers are so-called dual channel receivers, allowing us to observe the source object in horizontal and vertical polarization simultaneously.

Spectral analysis was performed using the facility Acousto-Optical Spectrometers AOS-A and AOS-B with a total bandwidth of 1 GHz and 2048 nominal channels each; however, the real resolution of the spectrometers is 934 kHz for AOS-A and 913 kHz for AOS-B, so the linear CCD elements oversample the spectra by about a factor of 2. These two spectrometers were always connected to the two channels of the receivers, allowing us to improve the S/N by averaging intermediate results from the horizontal and vertical polarizations. Co-adding the original spectra from different polarizations was problematic due to the different arrangement and slightly different width of the spectral channels in each spectrometer.

Table 1:   Journal of the observations.

For all derivations we approximate the beam as a symmetric Gaussian, whose FWHM is given (in radians) by:

where c is the speed of light, is observed frequency, D is dish diameter, equal to 10 m for the SMT, and b is a dimensionless factor, intrinsic to every dish, which is equal to 1.25 for the SMT. The half-power angular radius of the beam was therefore  at the lower frequency and  at the higher one.

The position of the comet was continually calculated from the orbital elements provided by the JPL HORIZONS system for the epoch of our observations. The pointing was calibrated on Jupiter, Saturn, and most frequently on the protoplanetary nebula CRL2688 that was close to the comet. From the stability of these observations we concluded that the pointing error was at worst of the order of half the beam radius at half-power (  or  depending on the observed frequency) for the observations accepted for analysis (see further in this section). Such precision is an order of magnitude worse than that guaranteed by the ephemeris but it still ensured that the flux loss was below 10% (see Fig. 1).

The spectra of the comet were taken in a position-switching mode, using a   offset in azimuth where the coma contamination is negligible. We used a 15 sec integration time for the ON and also a 15 s integration time for the OFF position, and each spectrum was composed of 8 or 10 ON-OFF pairs, thus giving us a fundamental time resolution of 4 or 5 min (hereafter exposure).

The moments of observations were not corrected for the travel time of light as the correction was negligible (see the geocentric distances in Table 1). They are approximately mid-points of the exposures, but not exactly: we added halves of the exposures to the initialization moments of the spectra, but neglected the time taken by the initial calibrations, switching time, time losses when the telescope was blown off the position, etc.

The absolute temperature scale was calibrated using the chopper wheel method (see e.g. Ulich & Haas 1976; Kutner & Ulich 1981), which automatically corrects for absorption in the Earth's atmosphere, resistive antenna losses, and rearward spillover and scattering terminated at ambient temperature. Furthermore, we corrected the temperature scale for the main beam efficiency, which we found to be equal to 0.60  0.04 (at both monitored frequencies) using our observations of continuum in Venus, Mars, Jupiter, and Saturn.

Because a typical edge tapering of 12 dB reduces the maximum power level of the first sidelobe to less than 0.3% of the power level of the main beam, the sidelobes do not contribute significantly to the convolution result of the the antenna pattern with planetary discs or coma brightness distribution. Furthermore, as long as the main beam pattern is similar to the assumed Gaussian profile (which is the case for e.g. an edge-tapered Airy profile), the influence of a small error in the assumed beam width on cometary observations will be largely compensated for by the main beam efficiency factor (affected similarly if calculated using the same assumed beam).

The frequency scale was converted to the radial-velocity scale (negative velocities indicate blueshift), and corrected for the radial velocity of the comet with respect to the telescope. We used 265.88643392 GHz as the central rest frequency of the J(3-2) transition, and 354.50548084 GHz for the J(4-3) one. These frequencies are the optocenters of the complete hyperfine structures as listed in the Cologne Database for Molecular Spectroscopy (Müller et al. 2005).

For each single spectrum the line area was determined from a velocity interval between -3 and +3 km s^-1, and two symmetric intervals from -10 to -3 km s^-1 and from +3 to +10 km s^-1 were used to estimate the background level. The background was subtracted upon fitting a linear baseline with a  iterative rejection of ``bad'' spectral channels. Additionally, information about noise in the background was used to estimate a statistical error of the line area, assuming that both the line and the background have identical noise. After this procedure was applied separately to the spectra from AOS-A and AOS-B, the resulting line areas were averaged (with the same statistical weights), and the effective errors were calculated. Figure 2 shows examples of such individual spectra for both the J(3-2) and the J(4-3) transitions. Since the effect of oversampling is not clearly visible, possibly due to an additional source of noise which partly removed the correlation between neighboring channels, for simplicity we analyzed the spectra as their channels were completely independent.

To ensure consistency of the absolute calibration and stability of the instrumentation, we periodically (about each 90 min) monitored standard calibration sources on the sky, namely CRL2688, W51D, DR21, and W3(OH). Within one night and for a fixed instrumental setup we found intensity drops of up to 50%, which we believe to be caused directly or indirectly (via pointing calibration) by anomalous refraction (Altenhoff et al. 1987). To ensure a consistent and properly calibrated time series of cometary HCN intensity, we restricted our observations in the following way:

(i)

we rejected the nights suffering from an insufficient monitoring of the calibration sources;

(ii)

we rejected the nights during which we encountered instabilities of the calibration sources larger than 20%;

(iii)

post-factum we restricted the time series to the measurements with the production rate errors below 1.2  10²⁵ molec s^-1 (calculation of the production rates is further described in Sect. 4.1) and hence rejected 7 observations; additionally, we rejected the last 3 observations from May 10, 2006, whose errors are significantly higher than those of the earlier data from that date.

Finally, we obtained 124 individual measurements of the line area  (Table D.1 in Appendix D), collected over 5 days (Table 1), which we used for further analysis. The errors on the line area include only the statistical component, since the other error sources, discussed in this section, are of a different nature (systematic and/or maximum) and are often negligibly small. Visual inspection of Table D.1 shows that occasionally a difference between neighboring measurements exceeds very significantly an acceptable range indicated by the errors. Although we cannot explain this effect, we assume it has a negligible influence on our analysis due to its rareness and apparently stochastic nature. Additionally, in Fig. 3 we present the weighted-mean spectra, obtained upon averaging the selected observations separately for each night, where the weights were inversely proportional to the square of the background noise.

3 Line position in the night-averaged spectra

Using the night-averaged spectra (illustrated in Fig. 3) we calculated positions of the lines (on a common velocity scale), defined as points splitting line area into halves. An error of the position was estimated using the Monte Carlo approach: on each spectrum we superimposed 1000 noise realizations, and in each individual simulation we determined the line position. The scatter of the obtained values is a very good measure of the investigated uncertainty, therefore, as the error we adopted the standard deviation calculated with respect to the original line position. While simulating noise we assumed a Gaussian noise distribution, with a variance determined from the background noise (see Sect. 2) - thus identical for all the channels within one spectrum. Although as the references for the simulations we ideally should have used noiseless representations of the real data, in this specific case Gaussian fits featured noticeably shifted centers, thus we used the investigated spectra themselves. We followed these procedures separately for the night-averaged spectra from AOS-A and AOS-B, hence preserving the original arrangement of their channels. Finally we determined the mean positions from both spectrometers, which are presented versus phase angle  in Fig. 4.

It is clearly visible that the lines became redder with the increasing phase angle . This effect is expected for outgassing stimulated by the Sun, whose position controls the temperature, and hence the sublimation rate, over the nucleus. Moreover, the line position equal to nearly zero at indicates that at that moment the distribution of the outgassing rate was symmetric with respect to some direction perpendicular to the line of sight (reaching a maximum in this direction), which might have been exactly Sunward. The scatter of the line positions about the global trend well corresponds to their errors estimated from the background noise (which would be even slightly underestimated if the spectral channels were partly correlated - see Sect. 2). Clearly, the observed trend contains a lot more information about the activity of this comet, however, in order to retrieve them we would have to model the complete line profiles in the individual (i.e. not night-averaged) spectra, which is impossible due to the too large noise. Note that this effect should be accompanied by an evolution of the line area  with phase angle , as we model it in Appendix A.4, but we did not investigate our data in this respect due to its small magnitude and because it partly overlaps with the heliocentric trend (see Sect. 4.2).

4 Production rate of the HCN molecule

4.1 Model and parameters

In order to retrieve a production rate Q (that is the global outgassing rate of a given molecule) from a measurement of the line area  , a model involving kinematics, chemistry, and excitation of the molecules, along with light propagation properties, must be used. For the purpose of our derivation we assume that:

(i)

the coma is optically thin at the observed frequencies;

(ii)

LTE conditions are present throughout the coma;

(iii)

the volume density of the molecules is inversely proportional to the square of the nucleocentric distance, i.e. the photodissociation process is neglected;

(iv)

the molecules are isotropically ejected from the nucleus;

(v)

the production rate Q is constant in time;

(vi)

the expansion velocity and the temperature T of the molecules are constant in space and time; under LTE, the excitation temperature is equal to the kinetic temperature, therefore we simply refer to the gas temperature.

Under these assumptions the production rate Q depends linearly on the line area  :

where k and h are the Boltzmann and the Planck constants respectively,  is geocentric distance, and I(T) is the integrated line intensity (at temperature T) as defined in the JPL spectral line catalog (Pickett et al. 1998). In this equation the same approximation of the beam is used as given by Eq. (1). Note that in order to calculate Q in molecules per second, must be provided in the commonly used unit of K m s^-1.

Unfortunately, we could not observe several HCN lines simultaneously. This would have enabled us to derive the gas temperature T by solving Eq. (2) simultaneously for the production rate and the temperature itself. Instead, we adopted T = 80 K.

Supplying our model with this temperature we calculated the HCN expansion velocity  using the line widths determined from the night-averaged spectra (Fig. 3). We found   0.1 km s^-1, which is a mean value from the results obtained individually for each single night, and its error is the standard deviation among the constituent measurements.

A detailed justification of the assumptions of our model and of the input parameters is presented in Appendix A.

4.2 Individual production rates

Using the simple model (Sect. 4.1) we converted each individual measurement of the line area  (see Table D.1) into the corresponding HCN production rate Q, and analyzed how they change with the heliocentric distance r of the comet (see Fig. 5). An unweighted power-law fit to this dependence yields the power index of -8.2  1.4 and the mean production rate   =   10²⁵ molec s^-1 at r = 1 AU (indicated by index ``zero''), corresponding to 1.22  0.04 kg s^-1. Since heliocentric evolution of the HCN production rate typically follows power laws with indices between 2 and 5 (see e.g. Biver et al. 2002a,1999), the obtained dependence should be considered as unusually steep. The likely reason for this difference is the very limited range of heliocentric distances covered by our observations, thus the obtained evolution shows only an instantaneous behavior of this comet.

Nevertheless, the ultimate goal of obtaining this dependence was to remove the instantaneous heliocentric trend from the production rate in order to search for its cyclic changes that might have been driven by the nucleus rotation. As further shown in Sect. 5, this effect may be responsible for the large scatter of the measurements about the fit. Thus, the presented snapshot of the heliocentric dependence concerns, strictly speaking, the mean diurnal production rate, provided that the rotation phase was sampled fairly uniformly. This is discussed in more detail in Appendix B.

This also means that the derived errors of the fit parameters - resulting purely from the scatter of the measurements - are artificially boosted by the short-term variability of the production rate about its mean-diurnal value. On the other hand, they do not account (at least not directly) for the simplifications of our model and the uncertainties of its input parameters (Sect. 4.1), the instrumental effects and the individual errors resulting from the noise statistics (Sect. 2), and sampling of the nucleus rotation phases (Appendix B).

Using the determined trend we reduced all the individual measurements to r = 1 AU, and denoted them as Q₀. Their errors are the statistical uncertainties on the line area  (see Sect. 2) converted with Eq. (2) and reduced with the same trend. Hence they neither account for the several effects discussed in Sect. 2, nor for the simplifications of our model (Sect. 4.1) or the uncertainty on the heliocentric trend derived in this section. The reason is that all these effects are of a different nature (systematic and/or maximum), whereas it is the relative uncertainty between the individual Q₀, not the absolute one, which matters for the periodicity analysis.

4.3 Discussion

The determined expansion velocity is typical for a comet at this heliocentric distance (cf. e.g. Biver et al. 2002a,1999), and is consistent with theoretical models (see e.g. Bockelée-Morvan & Crovisier 1987).

The mean diurnal HCN production rate at r = 1 AU is generally consistent with other corresponding determinations, published by Milam et al. (2006), Villanueva et al. (2006), and Dello Russo et al. (2007), though discrepancies up to a factor of 3 exist. They can be explained by different heliocentric distances and observing geometries of the comet (the latter being further discussed in Appendix A.4), along with modulation induced by the nucleus rotation (further discussed in Sect. 5). Inconsistencies between the different techniques also cannot be excluded (cf. e.g. Magee-Sauer et al. 1999; Fray et al. 2005).

Furthermore, we compared our result with the corresponding values for eight other comets. Whenever necessary we normalized their production rates using a power-law heliocentric dependence with an exponent of -4. This is between the values of -4.5 and -3.4, obtained from the long-term pre- and post-perihelion evolution of the HCN production rate in comet Hale-Bopp (Biver et al. 2002a), though the slope was noticeably flatter for comet Hyakutake (Biver et al. 1999). The correction was significant only for comet Tempel 1 (observed at r = 1.62 AU), and moderate for comet Borrelly (observed at r = 1.36 AU). The comparison is presented in Fig. 6, where the production rates Q₀ are plotted vs. the nuclei radii R, along with the best fit of a square function (calculated in logarithmic space); that is because the total nucleus area is the main parameter controlling the global sublimation. The dependence shows that the HCN production rate in SW3-C was unusually high. Since the relative abundance of this molecule was found to be typical (Villanueva et al. 2006), only an exceptionally high nucleus volatility - that is a large active fraction - can explain this. The conclusion is on one hand not surprising, bearing in mind the former break-up which excavated much fresh and highly volatile material; on the other hand, the comet has been continually losing its excessive activity since then (Boehnhardt 2004), and even though, it was still unusually active a decade later. The dependence also shows that at r = 1 AU a typical nucleus with the radius of R = 1 km produces HCN at the rate of 1.5^+8.7_-1.3  10²⁵ molec s^-1 (corresponding to 0.7^+3.9_-0.6 kg s^-1). Covering nearly 4 orders of the nucleus area, the analysis is fairly insensitive to the errors on individual data points, introduced by observing noise, particular models, normalizations, etc. Therefore, the scatter of the points is most likely dominated by diversity among the nucleus volatilities, with some contamination from differences in composition and shape.

5 Rotation period of the nucleus

5.1 Basic concepts

An active cometary nucleus may manifest its rotation by stimulating a periodic variation of the global outgassing rate, induced by diurnal changes of the insolation over the active region(s). Also a uniformly volatile nucleus may produce such variability provided that its shape is irregular - hence the effective sublimation area varies as the insolation changes during the rotation cycle. In principle, for both scenarios the observed rotation phase profile of the production rate may be noticeably different from the one which is intrinsic to the nucleus; they should be, however, quite similar in our case (see Appendix A.5).

Rotation periods have been determined for many cometary nuclei, both active and inactive, using several different techniques (see e.g. Jorda & Gutiérrez 2000; Samarasinha et al. 2004). However, periodic variability of the parent molecule production rate, stimulated by the nucleus rotation, has been detected only very recently, by Biver et al. (2007) for HCN in comet 9P/Tempel 1 (the Deep Impact target). Although the rotation period they found is consistent with the determination from optical photometry of the bare nucleus (Lamy et al. 2007) - thus providing a double-peak phasing of the light curve - the phase profile for HCN features only one peak. Moreover, this profile is well correlated with periodic oscillations of the line position, which were also detected by Biver et al. (2007). This demonstrates that the phase profiles for the HCN production rate and for the optical light curve are indeed controlled by different underlying mechanisms.

A clear advantage of using a time series of parent molecule production rate to investigate the rotation of active comets is high probability of detecting a phase profile with a large amplitude A (which we define as a ratio of maximum to minimum of the phase profile). This will be the case if the sublimation is indeed restricted to a limited number of isolated sources, and the observations closely reflect an intrinsic activity pattern of the nucleus, satisfying the conditions discussed in Appendix A.5. This contrasts with low amplitudes of the phase profiles for daughter species and for dust. The former one is damped by the significant excess velocities (in random directions) gained during their photodissociative creation, and also due to the exponential nature and long time scale of this process, both effects making the spacetime distribution of daughter species more uniform. The latter is damped because of low and simultaneously size-dependent velocities of dust grains; it was also postulated that dust ejection may be driven more by sublimation of the most volatile ices (CO, CO₂) than by water ice (Feaga et al. 2007), which is weakly sensitive to the diurnal changes of the insolation conditions. Moreover, if a large field of view or long exposures are used, then the rotation-driven variability of any coma component (i.e., daughter species, dust, or parent molecules) is additionally damped (cf. Appendix A.5). An example of these effects are observations of comet Tempel 1: compare the HCN phase profile (A = 2.3) reported by Biver et al. (2007) - although it is expected to differ noticeably from the intrinsic one (thus they call it apparent) - with the CN and dust phase profiles (A = 1.5 and A < 1.5 respectively) inferred by Manfroid et al. (2007).

Another approach is to investigate a (diurnal) variability of the outgassing direction instead of (or along with) variations of the production rates. Such an effect stimulates the variability of the line position and shape over the rotation cycle. Requiring spectral resolutions of at least one million (for realistic gas outflow velocities in comets), this approach is feasible only for radio techniques. The rotation-driven variability of the line shape and/or position was first identified for comet Hale-Bopp (Boissier et al. 2007; Henry et al. 2002), and later found for Tempel 1 (Biver et al. 2007).

Unfortunately, investigating periodicity of the line positions measured in our individual spectra of SW3-C we found only random noise. Although an evolution with phase angle is evident in the night-averaged spectra (see Sect. 3), there is no hint of the rotation-driven variability also there, though it might have manifested itself by producing an excess scatter in the trend (if different rotation phases of the nucleus were sampled on different nights). A possible explanation for this non-detection is simply an insufficient S/N of the individual spectra, causing too large an uncertainty on the line position. Alternatively, a specific combination of the spin axis orientation and the nucleus activity pattern might have kept the line position and shape fairly constant over the rotation cycle. Therefore, we limit our further discussion to the periodic variability of the production rate, which is independent of the line position variability.

5.2 Methods

Performing the periodicity analysis we generally followed the classical approach of Drahus & Waniak (2006), where a constant frequency f of the periodicity in the input time series is being determined. Although they also introduced a dynamical approach, where an evolution of the frequency with time  is simultaneously searched for, we did not apply it to our data.

It has been well established, both in theory (e.g. Samarasinha et al. 1986; Gutiérrez et al. 2002; Jewitt 1999) and through observations (e.g. Drahus & Waniak 2006; Belton & Drahus 2007, and references therein), that the cometary nuclei, especially those that are small and active, can substantially change their rotation periods on a time scale of a single perihelion passage. Using the simple model of Drahus & Waniak (2006) we estimated that the expected change of the rotation frequency of SW3-C during our campaign, regardless of the frequency itself, is about 0.04 h^-1 (Appendix C.2). In the rotation phase profile for the middle frequency this corresponds to the shift of the first and last data point of more than a full rotation cycle, making the classical periodicity analysis completely irrelevant. At the same time their dynamical approach, naturally accounting for this effect, would be ideal.

Nevertheless, we limited our investigation to the classical analysis only - although it seems incorrect at the moment - due to the insufficient quality of our data for the dynamical approach. Drahus & Waniak (2006) have successfully tested the routine using simulations with a S/N of about 3, i.e. comparable with their best observing results (where S/N is defined as half of the difference between maximum and minimum of the phase profile over the standard deviation of the points about their smooth representation), and using 128 synthetic data points, i.e. the same number as observed. On the other hand, the additional parameter may lead to the failure of this approach when the S/N is worse and/or the number of input data points is smaller (Waniak & Drahus, in preparation). In our case the number of measurements is very similar, but the S/N, being below 2 for half of the considered period solutions (further discussed in Sect. 5.3, particularly in Fig. 8), does not guarantee sufficient reliability of the dynamical investigation. Nevertheless, we show in Appendix C.2 that - in contrast to our expectations - the analysis done within the framework of a constant periodicity approximation is probably justifiable in our case.

We performed the periodicity analysis using two different methods: the Phase Dispersion Minimization, originally proposed by Stellingwerf (1978), and generalized by Drahus & Waniak (2006) in order to take statistical weights of individual data points into account (hereafter WPDM, where ``W'' indicates weighting), and the  minimization using a sum of harmonics fitted to the data, that also accounts for the statistical weights of the individual measurements (hereafter HF, which stands for harmonics fit). Both methods return a variance ratio  which measures the quality of a phase profile at given frequency f. Generally, the lower the value of  is, the better the profile it indicates. The working parameters of WPDM are the number of bins and the number of covers (denoted as  and  respectively) and the only parameter of HF is the number of frequencies used for fitting (denoted as ).

Table 2:   Possible solutions for periodicity and their properties.

The error on the frequency f of a given periodicity solution was estimated using the Monte Carlo approach, in a similar way as the error on line position (see Sect. 3). As the reference for simulations we used a smooth counterpart of the original phase profile, which we calculated using the HF method. (Note that we used HF for two different reasons: as an independent periodicity-search routine, and in order to obtain a noiseless representation of a given phase profile.) The smooth profile was then disturbed with a large number of noise realizations, which we produced assuming a Gaussian noise distribution with a variance given by the square of the data point error. The simulated noisy phase profiles were analyzed with the same method and setting which originally had returned the solution whose error was under study. As the investigated error we adopted a standard deviation of the values from the simulations, calculated with respect to the actual frequency of the investigated solution. Note that such an approach naturally converts the uncertainties of the data points in the input time series into a standard error of periodicity (Drahus & Waniak 2006).

Moreover, we also retrieved the probabilities that any phase profile - characterized by a certain value of  - does not result entirely from noise. Such confidence levels were derived also using the Monte Carlo method. Again, we simulated a number of noise realizations, taking however a constant reference curve this time. Then, they were analyzed using both methods with given settings, yet only global minima of  were retrieved. This made it possible to constrain individual statistics for each method and setting, which provided the confidence levels for .

5.3 Results

Both methods were used to investigate the time series of our individual production rates Q₀, normalized to r = 1 AU using the best-fit heliocentric dependence (see Sect. 4.2). We scrutinized the range of frequencies f from 0.02 to 0.8 h^-1 (corresponding to the periods P from 1.25 to 50 h), with a step size of 0.00002 h^-1. As the statistical weights of the individual measurements we used the inverse squares of their errors. The WPDM algorithm was run with ranging from 15 to 35, and , whereas the HF method was used with  ranging from 3 to 7. The most relevant periodograms were found for (corresponding to 5 measurements per bin on average) and . This is because when working with noticeably lower values of these parameters, the methods do not return the whole set of existing solutions, whereas when using much higher values they do not provide any additional solutions and are less stable due to the limited number of input measurements. For all the periodograms we retrieved the confidence levels of detection, simulating each time 1000 realizations of noise. The optimal periodograms are displayed in Fig. 7, along with the ones calculated for and , as well as for and , which we show for comparison.

The figure shows a good consistence between the results from the different methods and parameters. Although a number of solutions are present, only few of them are significant in a statistical sense. We selected eight most likely solutions, which were found in the following way: from the full set of periodograms we identified all the solutions characterized in at least one of them by the  value not higher than 5% above the minimum value in the considered periodogram. Next, we rejected all the solutions at f < 0.07 h^-1 (P > 14.3 h) as they suffer from an insufficient phase coverage with our data. For each of the accepted solutions we retrieved the Monte Carlo error, considering separately the two positions from the two methods with the optimum settings (representing the same solution). As the reference profiles for the simulations we used smooth fits from the HF method with for the solutions at f < 0.1 h^-1 and for those at f > 0.1 h^-1, which provide the most realistic representations of the data. Onto each of them we superimposed 1000 noise realizations. As the final solutions we adopted the combined results from both methods. The positions and properties of the selected solutions are listed in Table 2, and the corresponding phase profiles are displayed in Fig. 8.

Half of these solutions, i.e. A, D, E, and F (see Table 2), form a system of period multiples which starts with solution F. Some further multiples are also visible in the periodograms, although they do not satisfy our aforementioned requirements for a ``good'' solution. The system is so precise that all its constituents are found within a fraction of their errors from the expected frequencies. The reason that our periodograms are dominated by this structure is the very simple shape of the phase profile for its base frequency (solution F), which is nearly sinusoidal and features only one peak (what actually defines this solution as a base of the system). Therefore, it can be easily decomposed into the associated multiples, with slowly increasing complexity - each succeeding solution in the system features one additional peak in its phase profile, and all the peaks are images of the phase profile for the base frequency. Since the complexity of the profiles rises slowly, the system can be traced up to high period multiples (i.e. low frequencies), but of course at some point the profiles become overly complex, and the number of data points too low, thus the system vanishes. On the other hand, we cannot exclude the possibility that some multi-peak solution from this system is actually the correct one, which - just because of an unusually high symmetry - created the whole structure. However, in such a case, multiple, equiproductive, and equidistant (in the nucleocentric longitude) active sources are required, or an irregular nucleus with fairly uniform volatility must be assumed (if a double-peak solution E is considered) - but both scenarios are very specific, thus unprobable in our opinion. At the same time, all the independent solutions, unrelated to the system, are physically very plausible. The requirements of symmetries are not relevant to them, being in fact violated (otherwise the solutions would not be independent, being multiples of simpler periodicities). Therefore, the phase profiles are irregular, often featuring asymmetric multiple peaks, typically characterized by uneven amplitudes and/or unequidistant positions. Although these solutions are bases, they do not produce sharp systems of their multiples, due to rapid growth of the complexity (although in some cases first multiples are weakly visible).

Summarizing, we found a periodic oscillation of the reduced production rate Q₀ with the amplitude factor of 1.6-2.4 and S/N of 1.7-3.7, depending on phasing. The most likely explanation of this effect is a variation of activity over the nucleus rotation cycle. However, our observations alone are insufficient to unambiguously point at the correct rotation period, which is due to the insufficient S/N and large gaps in sampling. Possible problems which might have affected our analysis are discussed in Appendix C.

5.4 Discussion

Table 3:   Other determinations of the rotation period P of SW3-C.

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(B.1)

(C.1)