Several sources of uncertainties can be identified in the determination of the adopted nuclear magnitudes and in the computation of the effective nuclear radius. Let us now analyze these in some detail.
We may broadly distinguish between methodological errors due to shortcomings of the observing techniques and intrinsic errors due to unknown properties of the nuclei themselves. The former include all random and systematic errors of photometry, which are often particularly bad for attempted measurements on cometary nuclei. Some of the reasons are that many of the data were taken for non-photometric purposes where magnitudes resulted as a spinoff without much attention to the photometric conditions of the sky, and that the data often refer to objects near the plate limit without appropriate standard sequences, or uncertainty of the CCD extrapolation.
It must be kept in mind that most of the magnitudes in the literature, even if referred to as nuclear, are not truly nuclear magnitudes according to our physical definition. This is the main concern of our investigation, and our most important aim is to minimize these errors, which are due to coma contamination. Nonetheless, the discrepancies remaining between our adopted nuclear magnitudes and the true values have an important component that can be referred to as residual nuclear activity. This component borders between the methodological and intrinsic errors, because:
To the above errors, we should finally add some further intrinsic errors due to the nucleus itself: variations of the photometric nucleus cross-section S due to rotation; and a phase angle effect. Furthermore, the computation of the nuclear effective radius requires precise knowledge of the geometric albedo, which is generally not available.
The error of each individual observed magnitude may be of the order of 0.1 magnitudes or more, except for the case of CCD observations obtained from a few papers where photometry was done carefully; errors as low as a few hundredths of a magnitude can be attained in this case. A few tenths of a magnitude is the usual error of photographic observations. While most CCD magnitudes reported in the MPCs are more indicative values than precise determinations, the great improvement of the recent CCD data with respect to the old photographic data is the huge number of magnitude determinations at larger heliocentric distances. Unfortunately, in many cases the color of the magnitude is not clearly established, or it can only be crudely approximated by a standard color, as occurs, for instance, with Scotti's unfiltered magnitudes.
Cometary nuclei seem to be very irregular and elongated objects. Therefore, the photometric nucleus cross-section may vary with the position of the nucleus with respect to the line of sight. If the spin axis of the nucleus is not aligned with the line of sight, rotation produces fast variations of the photometric cross-section S. Rotational lightcurves have been used to determine the rotation period of cometary nuclei (see e.g. Jewitt & Meech 1987). Assuming that the nucleus is a triaxial ellipsoid with axes a > b > c, that it has a homogeneous surface, and that its spin is relaxed to pure rotation around the fixed axis c, it is easy to see that the maximum amplitude of its rotational lightcurve is
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where
and
are the maximum and minimum brightnesses.
For instance, the lightcurve of a nucleus with a/b=2 fulfilling the
previous conditions will show a maximum amplitude of 0.75 mag. So, for
typical complex shapes, variations up to the
order of one magnitude may be observed due only to the rotation of the
comet (see e.g. Lamy et al. 1998b).
As regards the phase coefficient ,
Scotti (private communication) has
pointed out that it might depart significantly from our adopted value.
Since we have taken a limit for the phase angle of
,
an uncertainty
on
of
would introduce a maximum uncertainty of
in the adopted nuclear magnitude.
The three effects discussed so far imply that, even for a non-active nucleus, a dispersion of the order of at least one magnitude in the plots should be considered as normal.
Some comets present much larger dispersions or a trend in the plots (see Sect. 8.3) that suggests some non-detected activity. The detection of a small coma, especially when the comet is far from the Earth, is not a simple task. Licandro et al. (1999a) compare the comet brightness profile to the profile of field stars, which proves to be very useful, but even with such methods some faint coma may remain undetected. But for most of the data included in the catalog the presence of a coma is reported only when it is easily detected by eye or, in the best case, by comparing the FWHM of the comet with respect to the one of the stars in CCD images.
The only way to determine if the comet is active with the data of the catalog is to compare the nuclear magnitudes obtained at a wide range of heliocentric distances. If the trend of the data becomes horizontal beyond a certain heliocentric distance, and the dispersion of the data is not large (say, less than one magnitude), we should expect that the corresponding average magnitude will correspond to the one of the bare nucleus. Yet, some residual dust coma might remain at some constant level over the whole observed part of the cometary orbit, even if the comet does not present any signs of activity. The main problem with this uncertainty, with respect to the other ones, is that it introduces a systematic effect that produces brighter "nuclear'' magnitudes.
A method was already explained in Sect. 3.2 as developed and applied by James Scotti. We showed that it involves in itself several sources of errors and uncertainties.
The HST team also applied a method of coma subtraction to their observations. A narrow PSF and a very good sampling enables them to obtain a more reliable estimate of the nucleus contribution, even in cases similar to the one sketched in Fig. 1a.
Let us also consider the uncertainties in the determination of
the effective nuclear radius. Let us consider an uncertainty in the
geometric albedo of
and in the absolute
magnitude of
,
leading to a combined uncertainty in the radius of
.
Applying Eq. (1) to
and
and taking the
difference, we obtain
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