5 Ensemble properties of Jupiter-family comets

   
5.1 Cumulative luminosity function

Considering Papers I and II together with this work, the complete range of nuclear radii estimates for the unresolved comets is 0.92 km  $< r_{\rm N} < 4.69$ km, while the range of 3$\sigma$ upper limits for the active and undetected comets is 0.5 km  $< r_{\rm N} < 12.7$ km. An albedo of 0.04 was assumed throughout, with the exception of comets for which the albedo has been previously measured. Appendix A brings together each nuclear radius result from Papers I, II, and III into one table. This table also states whether the comet was active, unresolved, or undetected, and whether the comet was on its inbound or outbound leg of its orbit.

In deriving the absolute magnitude of the active comets the following expression is used:

$$R_{(1,1,0)} = m_{\rm R} - 2.5n\log_{10}(R_{\rm h}) - 5\log_{10}(\Delta) - \beta\alpha$$ (3)

where $R_{\rm h}$ [AU] and $\Delta$ [AU] are the heliocentric and geocentric distances respectively, and $m_{\rm R}$ is the measured apparent magnitude. As normal, $\beta\alpha$ accounts for the effects of phase darkening, where $\alpha$ [deg.] is the phase angle, and $\beta$[magnitudes/degree] is the linear phase coefficient (assumed to be $0.035 \pm 0.005$). A linear phase term is valid when one considers the small phase angles of the active comets observed during this study.

For active comets, the constant n is generally assumed to have a value of 4. Essentially, this term accounts for variations of the total scattering cross-section with heliocentric distance, which is caused by changes in the dust production levels with heliocentric distance. For the unresolved and undetected comets, we assume n = 2, as expected for inert bodies.

Figure 5a plots the cumulative number of comets with magnitudes brighter than R_(1,1,0) versus R_(1,1,0)for the active and unresolved comets combined, as-well as for the unresolved comets only. Considering the curve where the data for the active comets and unresolved comets are combined, it is clear that the slope of this curve is best represented by a broken power law for absolute magnitudes brighter than 15.5. Therefore the Cumulative Luminosity Function (CLF) of the absolute magnitudes of our entire sample of 33 detected comets can be described by:

$$\begin{tiny} Log[\Sigma (< R_{(1,1,0)})] \propto \left\{\beg... ... {12.5 < R_{(1,1,0)} < 15.5} \end{array} \right. \end{tiny}$$ (4)

where the gradients were found by applying a least squares fit to the data points within the given magnitude ranges (see Fig. 5b).

Figure 5: (a) The number of comets < R(1,1,0) versus absolute R band magnitude R(1,1,0) for the active and unresolved comets combined (triangles), as-well as the number of comets < R(1,1,0) versus R(1,1,0) for the unresolved comets only (circles). (b) Same as (a), but with a logarithmic y axis. A slope of $0.32 \pm 0.02$ for the unresolved comets represents these data well, within the range 14.5 < R(1,1,0) < 16.5. These plots are derived from the amalgamation of results from Papers I and II, and this work (see Appendix A).

More important is the CLF of the unresolved comets. Fernández et al. (1999) have addressed this issue, but they used data from many different sources, with some dating back as far as 1950. The vast majority of their `best estimates' for the nuclear magnitudes derive from inconsistent reduction methods and a significant portion rely upon less-than-optimal coma subtraction techniques. In addition, one cannot be sure if the derived magnitudes have been transformed onto a common photometric system. This makes inter-comparison of data between individual comets extremely difficult. Nevertheless, they have used these data in an attempt to constrain the CLF of the Jupiter-family comets. They plot Log $[\Sigma (< V_{(1,1,0)})]$versus absolute visual nuclear magnitude V_(1,1,0) for comets with perihelion distances <2 AU. Using these data, Fernández et al. find that the CLF follows a linear relation within the small magnitude range of 15.25 < V_(1,1,0) < 16, with a gradient of 0.54.

As shown in Fig. 5b, a least-squares fit to our data for unresolved nuclei implies a slope for the CLF of $0.32 \pm 0.02$, within the large absolute magnitude range of 14.5 < R_(1,1,0) < 16.5. This gradient is significantly different from Fernández et al. (1999), but as our data is derived using homogeneous reduction methods and is entirely CCD based, we believe that this slope represents the most realistic estimate of the size distribution of Jupiter-family comets to date.

   
5.2 Size distribution and comparison to other populations

Now that we have an estimate of the CLF slope for the Jupiter-family comets, we can derive the size and mass distributions of the Jupiter-family population. The CLF for Jupiter-family comets is of the form:

$${\rm Log}[\Sigma (< R_{(1,1,0)})] \propto \alpha R_{(1,1,0)}$$ (5)

where $\alpha$ is the slope of the CLF. This implies $\Sigma (> r_{\rm N}) \propto 10^{\alpha R_{(1,1,0)}}$, and from Eq. (1) we have $r_{\rm N} \propto 10^{-0.2R_{(1,1,0)}}$. If we approximate the size distribution of Jupiter-family comets by a power law, such that the number of nuclei with radii > $r_{\rm N}$ is:

$$\Sigma(> r)\propto \begin{array}{ll} r^{1-q}, & q>1, \end{array}$$ (6)

then $10^{\alpha} \equiv 10^{0.2(q-1)}$ i.e. $\alpha=0.2(q-1)$. Following on from Sect. 5.1, we therefore find:

$$\Sigma(> r)\propto \begin{array}{ll} r^{-1.6 \pm 0.1}. \end{array}$$ (7)

It is important to compare the observed CLF of the Jupiter-family comets with that of the Edgeworth-Kuiper belt, as this is the most probable source region of the majority of the Jupiter-family population (Fernández 1980). Many investigations into the size distribution of Trans-Neptunian Objects (TNOs) have been performed. Typical values for the slope of the CLF are $0.52 \pm 0.02$ (Chiang & Brown 1999), $0.58 \pm 0.05$ (Jewitt et al. 1998), 0.60 _-0.10^+0.12 (Trujillo et al. 2001), 0.69 (Gladman et al. 2001), and 0.76 _-0.11^+0.10 (Gladman et al. 1998). These values are much larger than our measurement of $0.32 \pm 0.02$ for Jupiter-family comets, but are for significantly larger objects. The TNO size distributions are measured only for objects with supposed diameters $\geq$100 km, while that of the cometary nuclei is for objects $\leq$10 km. As of yet, there is no independent confirmation of the detection of cometary-nucleus sized bodies in the Kuiper belt, and so direct comparison of their size distributions is uncertain at best. Intriguingly however, Kenyon & Windhorst (2001) have shown that $\alpha\leq 0.48$for radii less than $\sim$1 km by using Olbers's paradox, thereby proving the size distribution to be shallower for smaller TNOs.

It is unlikely that the size distribution of the ejected TNOs would be preserved upon entering the inner solar system, due to the various processes acting upon the nucleus that would inevitably change their physical characteristics. Such processes include tidal disruption by the giant planets (Sekanina 1997), fragmentation due to intense solar heating (Delahodde et al. 2000; Filippenko & Chornock 2000), and nuclear sublimation. Unfortunately, such processes would increase the slope of the CLF, and not decrease it as required by our data. Therefore how can we account for a CLF slope for the Jupiter-family comets of $0.32 \pm 0.02$ presented here? One explanation for this effect could be the observational bias towards the discovery of larger Jupiter-family comets, or at least those with a significant active surface area. If our measured value is truly intrinsic to the population however, then perhaps Solar heating leads to complete disintegration of small cometary nuclei or at least their rapid diminution to below observational detection limits, which would result in a further decrease in the slope of the CLF.

If indeed there is a progressive decrease in the slope of the CLF as comets evolve from the Edgeworth-Kuiper belt to the realm of the Jupiter-family comets, then a precise determination of the CLF of the Centaur population may prove valuable. Unfortunately, the discovery rate of Centaurs is relatively slow at present, but wide field CCD surveys are being conducted, with preliminary values for the slope of the Centaur CLF of $\sim$0.6 (Sheppard et al. 2000) and 0.54 (Larsen et al. 2001).

Finally, it is interesting to note that our derived value for $\alpha$ of $0.32 \pm 0.02$ is similar to that for main-belt asteroids according to Jedicke & Metcalf 1998, who found $\alpha \sim 0.3$ but with large variations. Also, recent estimates for the CLF slope parameter of Near Earth Objects are 0.35 (Rabinowitz 2000; Bottke et al. 2001), and 0.39 (Stuart 2001). Hence these two collisionally dominated populations display size distributions significantly different from the theoretically expected value of $\alpha=0.5$ (Dohnanyi 1969; Williams & Wetherill 1994).

   
5.3 Absolute magnitudes and orbital parameters

Considering the activity levels of the Jupiter-family population as a whole, this survey clearly illustrates the diverse levels of activity present beyond 3 AU from the Sun, and that for several comets the levels of activity are substantial. From Papers I, II, and III, the measured $Af\rho$ values for the active comets range from $5.6 \pm 0.7$ cm to $299 \pm 11$ cm.

Correlations between the activity levels of the comets in our sample and their various orbital parameters were investigated. The only correlation we found for our sample was between intrinsic brightness and perihelion distance. Figure 6 plots the absolute R band magnitude R_(1,1,0) versus perihelion distance for every comet observed throughout the survey, and also includes most of the 3$\sigma$ upper limits obtained for the undetected comets. For the active comets in Fig. 6, the plotted magnitudes are the total magnitudes, i.e. nucleus plus coma. Several of the comets in this sample were targeted on separate observing runs, therefore upper limits for undetected comets that were previously or subsequently detected, have been discarded. Also, for comets that were observed to be stellar in appearance on several occasions, the mean absolute magnitude is taken.

Figure 6: This figure is a plot of the absolute R band magnitude R(1,1,0) versus perihelion distance for the comets observed in this survey, and also includes most of the 3$\sigma$ upper limits obtained for the undetected comets. The active comets are represented by filled circles, whereas the unresolved and undetected comets are represented by open circles and stars respectively. For the active comets, there appears to be a distinct correlation between absolute total R band magnitude and the comet's perihelion distance. A similar correlation between intrinsic brightness and perihelion distance is not seen for the unresolved and undetected comets, despite a wide range of perihelion distances (see Fig. 7).

In Fig. 6, the active comets are represented by filled circles, whereas the unresolved and undetected comets are represented by open circles and stars respectively. For the active comets only, there appears to be a distinct correlation between absolute R band magnitude and the comets perihelion distance, i.e. the intrinsic brightness increases with perihelion distance. Performing a least squares fit to these data points yields a slope of $-0.89 \pm 0.40$. Accurate knowledge of the parameter n for each individual comet may reduce the scatter in the data points and hence the associated uncertainty, however it is believed that the upward trend seen in Fig. 6 for the active comets is a genuine feature.

This effect can be interpreted either as a discovery bias towards brighter comets, or in terms of mantle formation, specifically the "Rubble'' mantle hypothesis. Recent arrivals to the inner Solar system should have relatively large fractional sublimating areas. As these new Jupiter-family comets are perturbed inwards, prolonged sublimation produces a rubble mantle which reduces the amount of free sublimating area and hence the brightness of the cometary coma at all heliocentric distances. The rubble mantle would continue to spread across the nuclear surface as the comet spends a larger fraction of its orbital period at progressively smaller heliocentric distances. This would imply the existence of a correlation between the amount of active area and/or composition, with perihelion distance. Such a correlation between the amount of active area and perihelion distance has been seen previously by (A'Hearn et al. 1995), albeit extremely weakly.

Figure 7: This figure plots the "best estimates'' for the visual nuclear magnitude V(1,1,0) versus perihelion distance. These data are from Fernández et al. (1999).

A similar correlation between intrinsic brightness and perihelion distance is not seen for the unresolved and undetected comets, despite a wide range of perihelion distances. This finding is compared with the data presented in Fernández et al. (1999). Figure 7 plots their "best estimates'' for the visual nuclear magnitudes versus perihelion distance. An apparent upward trend is seen and if one performs a least squares fit to the data points, then a linear relation with gradient $-0.81 \pm 0.12$ describes the data well. This value is remarkably similar to the slope seen for the active comets of Fig. 6. Hence, the illusion of an upward trend in Fig. 7 may be created if the comets with large perihelion distances are actually outgassing. Indeed, if one removes comets with perihelion distances beyond 3 AU (which is only 16% of the comets in their sample), then one is left with a random scatter of data points, spread over a magnitude range similar to that of the unresolved and undetected comets of Fig. 6. Therefore, based on this argument and the data presented in Fig. 6, we conclude that there is, as of yet, no correlation between absolute nuclear magnitudes and perihelion distance.