4 Discussion

   
4.1 Nuclear radii measurements

The effective nuclear radius measurements derived here range from $0.9 \textnormal{ km} \leq {r}_{\rm N} \leq 4.6 \textnormal{ km}$, and the upper limits derived span the range $0.5 \textnormal{ km} \leq {r}_{\rm N} \leq 7.5 \textnormal{ km}$(for an assumed albedo of 0.04). These values are typical for Jupiter-family comets. The quoted errors for these values, as listed in Tables 2 and 3, are due to the photometry alone.

Figure 4: Nuclear radius as a function of the assumed albedo for the unresolved comets of Sect. 3.2. The filled circles highlight the position of each of the unresolved comets on their respective radius/albedo curves for our adopted albedos. The region between the two vertical dashed lines represents the optimal range of assumed albedos, as the bulk of measured albedos reside within this region. The radius/albedo curves have been separated into two separate graphs for reasons of clarity. References: [a] A'Hearn et al. (1989), [b] Millis et al. (1988), [c] Campins et al. (1987), [d] Soderblom et al. (2002), [e] Keller et al. (1987), [f] Lamy et al. (2002), [g] Fernández et al. (2000), [h] Bus et al. (1989), [i] Cruikshank & Brown (1983).

Following the discussion in Paper II, we have illustrated in Fig. 4 how the derived radius values vary as a function of the assumed albedo, for the unresolved comets. Filled circles highlight the position of each of the unresolved comets on their respective radius/albedo curves for the adopted albedos. Note that comets 2P/Encke and 49P/Arend-Rigaux have previously measured albedos of 0.05 (Fernández et al. 2000) and 0.03 (Millis et al. 1988) respectively.

Listed in Cols. 3 and 4 of Tables 2 and 3 are effective nuclear radii upper limits for the undetected and active comets respectively (for assumed albedos of 0.04 and 0.02). Upper limits based on the minimum measured albedo of 0.02 (A'Hearn et al. 1989) may be regarded as firm. Even if one applies a maximum axis ratio of 2.6 (Meech et al. 1993) and a minimum albedo of 0.02 to the undetected comets, their semi-major axes are all constrained to be below 8.7 km. The uncertainty in the assumed phase coefficient of $\pm$ $\textnormal{0.005}$ mag/degree (as introduced in Sect. 3.1), will lead to an additional uncertainty of no greater than 0.2 km for the derived nuclear radii values and upper limits.

As with Papers I and II it is always appropriate to compare radius measurements and/or upper limits of those comets for which previous estimates exist. Such a comparison may allow additional constraints to be placed on nuclear axial ratios. With additional assumptions, this can lead to an estimate of the fractional active surface area present during previous apparitions. Table 5 lists those comets with previous nuclear radius measurements and upper limits derived from photometry, and lower limits derived from the amount of active area required to produce the measured OH production rates listed in A'Hearn et al. (1995).

Each of the 3$\sigma$ upper limits obtained from this work are consistent with the previous measurements, with the exception of 137P/Shoemaker-Levy 2. Licandro et al. (2000) observed this comet at a heliocentric distance of 4.24 AU (post-perihelion). The comet appeared stellar on this occasion and had a mean absolute V band magnitude of 14.5, which corresponds to a mean radius of 4.2 km for an assumed albedo of 0.04. A variation of 0.4 magnitudes was observed, which corresponds to a projected axial ratio of 1.5. Our 3$\sigma$ upper limit of 3.4 km (also assuming $A_{\rm R} = 0.04$) is still significantly lower than this previous estimate even if one considers the lightcurve indicated by Licandro et al. If we assume a typical (V-R) colour index of $0.50 \pm 0.10$ for the nucleus (Mueller 1992; Jewitt & Meech 1988; Boehnhardt et al. 1999; Licandro et al. 2000), then the absolute R band magnitude 3$\sigma$ lower limit of 14.42 derived here increases $\Delta m$ to $0.61 \pm 0.10$ mag, which in turn implies a new axial ratio lower limit of $1.8 \pm 0.2$. Considering that this comet had an observable coma during these observations, it is inevitable that this lower limit will increase further.

In contrast to the case of 137P/Shoemaker-Levy 2, the nuclear radii measurements for comets 45P/Honda-Mrkos-Pajdusakova and 97P/Metcalf-Brewington are somewhat larger than the previous measurements. Considering 45P/Honda-Mrkos-Pajdusakova first, Lamy et al. (1999) presents a mean radius value of 0.34 km based on a mean apparent R band magnitude of 19.34 measured with the Hubble Space Telescope (HST). The apparent R band magnitude ranged from approximately 19.30 to 19.43 throughout the two nights of observation. The corresponding absolute R magnitude range is $\sim$19.98 to 20.25 for an assumed phase coefficient of 0.035 mag/degree (in order to be consistent with the measurements presented in this paper). The axial ratio for this comet must therefore be at least 1.3. Table 3 lists an apparent R band magnitude of $23.27 \pm 0.86$, which corresponds to an absolute R magnitude value of $16.48 \pm 0.86$. The absolute R magnitude 3$\sigma$ lower limit is therefore 19.06, and after subtraction from the Lamy et al. value of 20.25, provides a new lower limit to $\Delta m$ of 1.19, which in turn implies a new axial ratio lower limit of $\sim$3.0. Some aspects of this result should be considered before it can be considered reliable. If this comet possessed an unresolved dust coma, then the nuclear absolute R magnitude 3$\sigma$ lower limit would be greater than 19.06. This would then lower the $\Delta m$ value to yield a smaller axial ratio lower limit than the value presented here. However, it is reiterated that this comet was observed at a heliocentric distance of 5.14 AU where the sublimation of surface volatiles, particularly H₂O, is usually low. Also, Lamy et al. (1999) measured an extremely small $Af\rho$ value of 3.7 cm at a heliocentric distance of just 0.96-0.97 AU. Furthermore, A'Hearn et al. (1995) measured a similar $Af\rho$ value of 3.5 cm when the comet was only 1.15 AU from the sun (see Table 6). Hence it is highly probable that the dust production levels were negligible during our observations when the comet was over 5 AU from the sun. This value of $\geq$3.0, if confirmed, sets a new limit to the amount of elongation that can be expected for cometary nuclei. The largest previously-measured projected axial ratio was $\geq$2.6 (Meech et al. 1993).

For 97P/Metcalf-Brewington, the radius value of $2.18 \pm 0.41$ km derived here for an assumed albedo of 0.04 is significantly larger than the value of 1.4 km presented in Licandro et al. (2000) (also for an assumed albedo of 0.04). The comet appeared stellar and was at a post-perihelion heliocentric distance of 3.67 AU. A mean absolute V band magnitude of 16.9 was measured, with a peak-to peak amplitude of 0.7. A $\Delta m$ value of 0.7 implies an above average projected axial ratio of $\geq$1.91. Based on this result, the possible range of effective radius values is therefore 1.18-1.63 km. Our value of $2.18 \pm 0.41$ km is actually consistent with this range at the 2$\sigma$ level.

49P/Arend-Rigaux was observed at an outbound heliocentric distance of 3.34 AU. These observations were performed just over five months after the December 1998 observations that were obtained using the 4.2 m WHT (Paper II). It is apparent that the transition region from an outgassing state to one of complete inactivity may occur somewhere within the 2.1-3.3 AU zone. Of-course, heliocentric lightcurves of comets tend to be asymmetric, hence this finding only applies to the outbound leg of its orbit. The radius measurement derived here of $4.60 \pm 0.11$, for an assumed albedo of 0.03 (Millis et al. 1988), represents the bare nucleus value, and is completely consistent with previous values that span the range 3.8-6.8 km (see Paper II and references therein).

The case of comet 43P/Wolf-Harrington is interesting in the sense that each of the previous measurements are similar to the value derived here of $3.43 \pm 0.22$ km. The previous measurements are $3.3 \pm 0.7$ km (Paper II), and $\leq$2.9 km (Licandro et al. 2000). Similar effective radii measurements and upper limits implies either the nucleus may be approximately spherical, or that the measurements were performed at similar points in the rotational lightcurve (excluding the Licandro et al. measurement). The only clue towards an elongated nucleus would be the fact that the Licandro et al. measurement was performed when the comet was highly active. The 2.9 km upper limit will undoubtedly be a significant overestimation of the effective radius, therefore the actual effective radius would not be consistent (at the 3$\sigma$ level) with our value.

   

Table 5: Comparison of nuclear radii measurements with previous values.

Comet	$r_{\rm N}$ [km]	Previous values$^{\diamond}$[km]	Ref.
2P/Encke	$4.43 \pm 0.32$	0.25-5	1
		$\geq$0.24	2
19P/Borrelly (Night 4)	$1.90 \pm 0.32$	4.4$\times$1.8	3
		$\geq$0.73	2
43P/Wolf-Harrington	$3.43 \pm 0.22$	$3.3 \pm 0.7$, $\leq$2.9	4,5
		$\geq$0.42	2
44P/Reinmuth 2	$\leq$3.0$^{\star}$	$\leq$1.4$^{\star}$, $\sim$1.63	6,7
45P/H-M-P	$1.34 \pm 0.55$	0.34	8
		$\geq$0.11	2
46P/Wirtanen	$\leq$2.6$^{\star}$	$0.60 \pm 0.02$, $0.73 \pm 0.23$	9,10
		$\geq$0.39	2
47P/Ashbrook-Jackson	$\leq$7.5$^{\star}$	3.0	5
49P/Arend-Rigaux	$4.60 \pm 0.11$	3.8-6.8	11
64P/Swift-Gehrels	$\leq$1.9$^{\star}$	1.5	5
67P/C-G	$\leq$2.9$^{\star}$	$2.4 \pm 0.1$ $^{\dagger}$	12
69P/Taylor	$\leq$4.6$^{\star}$	$3.6 \pm 0.7$	4
		$\geq$0.25	2
97P/Metcalf-Brewington	$2.18 \pm 0.41$	1.4	5
		$\geq$0.75	2
103P/Hartley 2	$\leq$6.7$^{\star}$	$\leq$5.0, 0.56, $\leq$5.9$^{\star}$	5,13,14
		$\geq$0.62	2
137P/Shoemaker-Levy 2	$\leq$3.4$^{\star}$	4.2	5

r_N is the nuclear radius, $\diamond$ The lower limits are based on the OH production rates from A'Hearn et al. (1995) (see text), $\star$ 3$\sigma$ upper limits, $\dagger$ Based on a mean apparent R band magnitude of $22.04 \pm 0.09$ (as presented in the respective reference) and an assumed albedo and phase coefficient of 0.04 and 0.035 respectively. References: [1] Fernández et al. (2000) and references therein, [2] A'Hearn et al. (1995), [3] Lamy et al. (1998b), [4] Paper I, [5] Licandro et al. (2000), [6] Paper II, [7] Lamy et al. (2000), [8] Lamy et al. (1999), [9] Lamy et al. (1998), [10] Möhlmann (1999), [11] See Paper II and references therein, [12] Mueller (1992), [13] Jorda et al. (2000), [14] Paper II.

Several of the targets from this particular observing run are possible or established targets of current and future spacecraft missions. Such targets include 2P/Encke, 19P/Borrelly, and 46P/Wirtanen. Comet 2P/Encke was one of the possible three or more targets of the recently failed Comet Nucleus Tour (CONTOUR). This comet has been studied extensively in the past with a range of nuclear radius values being published (see Fernández et al. 2000 and references therein). The effective radius value derived here of $4.43 \pm 0.32$ km falls within the total range of previous values (i.e. 0.25-5.00 km). As with 49P/Arend-Rigaux, this comet is one of the few to have its nuclear albedo measured. Using simultaneous visible and infrared observations, Fernández et al. (2000) measured an albedo of $0.05 \pm 0.02$ and an effective radius of $2.4 \pm 0.3$ km. If we consider the observed rotational lightcurve from Fernández et al. (2000) then we can argue that our photometry is consistent with their measurements. In other words, our observations may have taken place near the peak of the rotational lightcurve.

Comet 19P/Borrelly was observed previously with the HST in November 1994 (Lamy et al. 1998). By assuming an albedo of 0.04 and that the rotational axis is pointing in the direction determined by Sekanina (1979), the semi-axis were shown to be $4.40 \pm 0.30$ km and $1.80 \pm 0.15$ km. A rotational period of $25.0 \pm 0.5$ hours was also found. Recently, this comet was the flyby target of the Deep Space 1 mission, the encounter occurring on September 22, 2001. Imaging by the spacecraft resolved the nucleus and revealed an average geometric albedo of $0.030 \pm 0.005$ (Soderblom et al. 2002). We observed this comet with the JKT on two separate nights. The observations were separated in time by 48 hours, and on both occasions an effective radius of $\sim$ $1.91 \textnormal{ km}$ was found. The comet was therefore observed on both nights near the same point in the rotational lightcurve (i.e. at the brightness minimum in this case). This acts as independent support for the above rotation period at the 2$\sigma$ confidence level. It is also apparent that the rotational state has remained virtually unchanged since the HST observations, despite prolonged outgassing during this period.

Comet 46P/Wirtanen is the target of the upcoming ROSETTA comet orbiter mission, scheduled for launch in January 2003. Previous observations show this cometary nucleus to be an extremely small object with a radius of $\sim$0.6-0.7 km (Lamy et al. 1998; Möhlmann 1999). An effective radius upper limit of $\leq$2.6 km was found, which is consistent with the previous work.

   
4.2 Af$\rho$ measurements

Brightness profile analysis revealed coma activity for only 4 of the 25 targeted comets. The $Af\rho$ values obtained for these active comets range from 5.6 cm  $\leq Af\rho \leq 37.8$ cm, whereas the $Af\rho$ upper limits obtained for the undetected and unresolved comets range from 0.7 cm  $\leq Af\rho \leq 32.0$ cm. The complete range of $Af\rho$ values and upper limits observed on this occasion is rather small relative to that observed in Papers I and II. Poor seeing may explain why a large fraction of the detected comets appeared stellar, but from the observed range of $Af\rho$ values (which are a measure of the actual dust production levels) it would appear that the reduced levels of distant activity are real and not an artifact of instrumental or atmospheric limitations.

   

Table 6: Comparison of $Af\rho$ values with previous values listed in A'Hearn et al. (1995) [1], Paper I [2], Paper II [3].

Comet	$Af\rho$ [cm]	$R_{\rm h}$ [AU]	Previous $Af\rho$	$R_{\rm h}$ [AU]	Ref.
			value [cm]
2P/Encke	$\leq 25.9$$^{\star}$	3.93I	28.84	0.86P	1
19P/Borrelly	$\leq 20.9$$^{\star}$ $^{\diamond}$	5.36I	645.7	1.38P	1
43P/W-H	$\leq 32.0$$^{\star}$	4.46O	134.9	1.82I	1
			$\leq 28.4$$^{\star}$	4.87I	2
44P/Reinmuth 2	$\leq 11.4$$^{\star}$	4.26I	$\leq 2.3$$^{\star}$	4.73I	3
45P/H-M-P	$\leq 4.2$$^{\star}$	5.14I	3.5	1.15I	1
46P/Wirtanen	$\leq 10.0$$^{\star}$	5.02O	112.2	1.12O	1
49P/Arend-Rigaux	$\leq 23.8$$^{\star}$	3.34O	107.2	1.56O	1
			$\leq 17.1$$^{\star}$	2.11O	3
67P/C-G	$\leq 17.1$$^{\star}$	5.72A	208.9	1.38O	1
69P/Taylor	$18.39 \pm 1.69$	4.03O	$\leq 13.6$$^{\star}$	4.89I	2
97P/M-B	$\leq 9.8$$^{\star}$	4.76I	275.4	1.61O	1
103P/Hartley 2	$37.83 \pm 2.75$	4.57O	245.5	1.04O	1
			$49.3 \pm 4.8$	3.63O	3

I - Inbound (Pre-perihelion), P - At perihelion, O - Outbound (Post-perihelion), A - At aphelion, $\star$ 3$\sigma$ upper limits, $\diamond$ Based on night 4 observations.

Table 6 compares the present $Af\rho$ values with those listed in Paper I, Paper II, and A'Hearn et al. (1995) derived at different heliocentric distances. Note that the A'Hearn et al. measurements were performed when the comets were at heliocentric distances $\leq$ $1.82 \textnormal{ AU}$, whereas the previous measurements from Papers I and II were performed when the comets were at heliocentric distances $\geq$3.63 AU (with the exception of comet 49P/Arend-Rigaux). For comets 19P, 43P, 46P, 49P, 67P, 97P, and 103P the present $Af\rho$ values or upper limits are very much less than the A'Hearn et al. measurements, which is expected when one considers the difference in heliocentric distance.

None of the comets listed in Table 6 are obvious candidates for continuous sublimation throughout their orbits, but it is strongly suspected that several are inactive during aphelion. Such comets include 19P, 44P, 45P, and 97P. This conclusion is based on the comets displaying either, a) similar $Af\rho$ values at widely different heliocentric distances, indicating an inert body with a constant scattering cross-section (i.e. comet 45P), b) small $Af\rho$ upper limits at large heliocentric distances (i.e. comet 44P), or c) small $Af\rho$ upper limits at large heliocentric distances coupled with a rapid rate of change of dust production with heliocentric distance (i.e. comets 19P and 97P). Each of these four comets are detectable at large heliocentric distances, and given that they are almost certainly inactive during aphelion, they are excellent candidates for photometric studies of nuclear properties. For comets 43P, 46P, 67P, 69P, and 103P, the level of activity during aphelion remains uncertain, although their $Af\rho$ values or upper limits remain small at large heliocentric distances.

   
4.3 Colour indices

Multi-filter photometry was performed on a total of 6 comets (see Table 4). Where possible, both (V-R) and (R-I) colour indices were obtained, but for others such as 19P/Borrelly and 103P/Hartley 2, only V and R band photometry was performed. The photometric errors associated with the bulk of these colour measurements are large, due to either low signal to noise and/or the limited accuracy that can be achieved through the photometric calibration of non-photometric data. Each of the active comets listed in Table 4 have colour indices consistent with the solar values at the 1$\sigma$ confidence level. The solar (V-R) and (R-I) colours, as transformed onto the photometric system used here, are 0.36 and 0.28 respectively (Allen 1973; Fernie 1983). For comet 14P/Wolf, the (R-I) colour index is similar to the solar value, but the (V-R) value is rather small, indicating a blue colour at shorter wavelengths. Such a small (V-R) colour index could be explained by a lack of substantial mantle coverage, which acts to redden the surface.

In the case of comet 49P/Arend-Rigaux, good seeing, combined with relatively high signal to noise resulted in reduced error bars. The (V-R) and (R-I) colours are $0.49 \pm 0.11$ and $0.54 \pm 0.14$ respectively, which are significantly redder than the solar values. Given that this comet was certainly inactive on this occasion, it is clear that these colours represent the actual colours of the nuclear surface. In-phase optical and infrared rotational lightcurves have been observed for this comet (Millis et al. 1988), indicating a uniform surface composition. Hence, the (V-R) and (R-I) colour indices derived here should be unaffected by rotation. These colours are comparable to some TNOs (Green et al. 1997; Barucci et al. 2000), which are among the reddest known objects in the solar system. 49P/Arend-Rigaux has been studied extensively during past apparitions at visible and infrared wavelengths (see Paper II and references therein), and in each case the nucleus or dust coma was red in nature. Luu (1993) presents a nuclear spectrum of 49P/Arend-Rigaux at optical wavelengths and is also consistent with a "Type 2" rubble mantle. We therefore conclude that there has been no significant change in the nature of the mantle coverage since the previous investigations, despite recent periods of low activity (Paper II).