3 Results and analysis

   
3.1 Undetected comets

Out of the original 25 targeted comets, 10 were undetected. These comets are listed in Table 2. Note that comet 118P/Shoemaker-Levy 4 was not found at the first attempt on night 3, but was subsequently detected on night 6. Considering the photometric data first, all available exposures of each comet were first aligned with respect to the background stars, then shifted according to their known rates of motion, and then finally median combined to increase the S/N. As this proved unsuccessful, limiting magnitudes were then determined by introducing artificial stars to the median combined frames at each comet's expected position. For each comet, the artificial stars were constructed using the average PSF of several bright, well isolated, background stars. To calculate the average background PSF, the individual frames for a given comet were shifted and coadded with the background stars aligned. This therefore provides an optimal representation of the brightness profile of a point source comet had it been visible on the shifted, median combined frame. The magnitudes of these artificial stars were gradually increased until they were no longer detectable, by visual inspection, against the background noise. An accuracy of $\pm$0.1 mag was attainable for the limiting magnitudes.

Comets 64P/Swift-Gehrels, 111P/Helin-Roman-Crockett, 113P/Spitaler, and 118P/Shoemaker-Levy 4 (Night 3), were imaged during non-photometric conditions. Artificial stars were added to the median combined non-photometric frames to obtain instrumental R band limiting magnitudes. These instrumental limiting magnitudes were converted to real limiting magnitudes via the use of calibration fields taken on subsequent photometric nights. As no trailing occurred, no correction to the limiting magnitudes was necessary. The limiting magnitudes were then used to derive upper limits to their effective nuclear radii by substituting them for $m_{\rm R}$ in the following equation (Russell 1916):

$$A_{\rm R} r^2_{\rm N} = \frac{2.25 \times 10^{22} R_{\rm h}^{... ...a^{2} 10^{0.4(m_{\odot} - m_{\rm R})}}{10^{-0.4 \alpha \beta}}$$ (1)

where $A_{\rm R}$ is the geometric albedo in the R filter, $r_{\rm N}$[m] is the effective radius of the nucleus, $R_{\rm h}$[AU] and $\Delta$[AU] are the heliocentric and geocentric distances respectively, $\alpha$ and $\beta$ are the phase angle and phase coefficient respectively. $m_{\odot}$ and $m_{\rm R}$ are the apparent R magnitudes of the sun and comet respectively. Table 2 lists the 3$\sigma$ limiting magnitudes and corresponding effective nuclear radii 3$\sigma$ upper limits. As normal, empirically derived values for the albedo of 0.04 (see Fig. 4) and phase coefficient of $0.035 \pm 0.005$ (Sekanina 1976; Meech & Jewitt 1987; Jewitt & Meech 1987; Jewitt & Meech 1988; Jewitt & Luu 1989) are assumed, and additional uncertainties introduced through these assumptions will be discussed later.

   

Table 2: Limiting R band magnitudes, nuclear radius upper limits (based on assumed albedos of 0.04 and 0.02), and $Af\rho$ upper limits for the undetected comets. Note that comet 118P/Shoemaker-Levy 4 was subsequently detected three nights later under improved weather conditions.

Comet	$m_{\rm R}$$^{\ast}$	$r_{\rm N}$$^{\star}$ [km]		$Af\rho$$^{\star}$ [cm]
		$A_{\rm R}$ = 0.04	$A_{\rm R}$ = 0.02
30P/Reinmuth 1	21.8	$\leq$3.8	$\leq$5.3	$\leq$13.6
44P/Reinmuth 2	21.2	$\leq$3.0	$\leq$4.2	$\leq$11.4
46P/Wirtanen	21.7	$\leq$2.6	$\leq$3.7	$\leq$10.0
64P/Swift-Gehrels	20.7	$\leq$1.9	$\leq$2.6	$\leq$4.9
67P/Churyumov-Gerasimenko	22.1	$\leq$2.9	$\leq$4.1	$\leq$17.1
75P/Kohoutek $^{\diamond}$	22.6	$\leq$1.5	$\leq$2.2	$\leq$3.6
83P/Russell 1 $^{\diamond}$	23.3	$\leq$0.5	$\leq$0.6	$\leq$0.7
111P/Helin-Roman-Crockett	22.3	$\leq$1.5	$\leq$2.2	$\leq$4.9
113P/Spitaler	21.6	$\leq$2.0	$\leq$2.8	$\leq$9.3
118P/Shoemaker-Levy 4 (Night 3)	21.4	$\leq$2.6	$\leq$3.6	$\leq$12.8
P/1993 X1 (Kushida-Muramatsu)	22.0	$\leq$2.0	$\leq$2.8	$\leq$5.6

$\ast$ 3$\sigma$R band limiting magnitudes, $\star$ 3$\sigma$ upper limits, $A_{\rm R}$ is the assumed nuclear albedo, $r_{\rm N}$ is the nuclear radius, $\diamond$ The effective radii upper limits for comets 75P/Kohoutek and 83P/Russell 1 should be treated with caution until further astrometric observations have been performed (see Sect. 3.1).

The effective radii upper limits for comets 44P/Reinmuth 2 and 118P/Shoemaker-Levy 4 are regarded as extremely reliable as both objects were subsequently detected on nights 2 and 6 respectively, and were both within a few arcseconds of their predicted positions. Although 44P/Reinmuth 2 was re-imaged on night 2, the data could not be photometrically calibrated and thus unsuitable for effective radius measurements and was excluded from the analysis.

Astrometric measurements of comet 113P/Spitaler were performed within 1.5 years after these observations were performed and it was found that any change in the orbital elements was negligible. Its orbit was therefore accurate during the observations and the derived effective radii upper limit can be regarded as firm. For comets $\textnormal{30P/Reinmuth 1}$, 46P/Wirtanen, 67P/Churyumov-Gerasimenko, and 111P/Helin-Roman-Crockett, astrometric observations have been performed within 3 years prior to June 1999. It is highly unlikely that non gravitational forces would alter the predicted positions of these objects sufficiently enough to remove them from our field of view when one considers the size of the field of view that was available to us (i.e. 5.58 $\hbox{$^\prime$ }\times 5.58$$^{\prime }$) and the short timescales involved. Hence, the derived effective radii upper limits for these comets may also be regarded as firm. The most recent astrometric observations of comet P/1993 X1 (Kushida-Muramatsu) were obtained in June 1995. The effect of non gravitational forces on its orbital motion cannot yet be ascertained and only future observations can confirm the predicted position used for these observations. However, like comets 30P, 46P, 67P, and 111P, it is unlikely that non-gravitational forces acting upon the nucleus between June 1995 and June 1999 will remove the object from our large field of view.

Only comets 64P/Swift-Gehrels, 75P/Kohoutek and 83P/Russell 1 show reason for concern. It had been at least 8 years since they were last observed astrometrically. The probability of nuclear splitting or spontaneous disintegration occurring is therefore much higher. The derived effective radii upper limits for these comets should be treated with caution until additional astrometric observations are obtained. It is also noted that recent nuclear splitting has not been reported for any of the undetected comets of this section.

$Af\rho$ upper limits were also calculated based on the derived limiting magnitudes and are listed in Table 2 along with the effective radii upper limits. For each comet, $\rho$ is taken to be the point where the background PSF is $\sim$2% of its peak intensity. For example, in the case of 30P/Reinmuth 2, the background PSF of the median combined frame drops to $\sim$2% of its peak intensity at $\sim$3.3 $^{\prime\prime}$ from the central peak. 3.3 $^{\prime\prime}$ is equivalent to $\sim$12 240 km at the comet, hence $\rho$ is taken to be this distance in the $Af\rho$ calculation.

   
3.2 Unresolved comets

Figures 1a-1c show typical R band CCD images of some of the brighter unresolved comets that were observed. Where appropriate, stars and cosmic rays were removed from the vicinity of the comets using various tasks within the IRAF software package. In order to search for signs of activity, the scaled brightness profiles of background stars were compared with those of the comets, and in every case the profiles were indistinguishable, i.e. the objects appeared as unresolvable point sources. Aperture photometry was performed on these objects to yield R filter apparent magnitudes and corresponding effective radii for assumed red geometric albedos of 0.02 and 0.04. These values are listed in Table 3 along with upper limits for $Af\rho$.

To ascertain the significance of any possible coma contribution to the R band apparent magnitudes and hence the nuclear radius measurements, the following expression is applied (Jewitt & Danielson 1984):

$$m_{\rm coma} = \Sigma (r) - 2.5 \log (2 \pi r^{2})$$ (2)

where $m_{\rm coma}$ is the integrated magnitude of the assumed steady state coma within a circular aperture of radius r[arcsec] and $\Sigma$(r)[mag/ $\rm arcsec^{2}$] is the measured brightness at r. Here r is taken to be the radius of the photometric aperture used in the determination of the comets real apparent magnitude. If we assume a minimum S/N (per arcsec²) for detection of coma at r, we can calculate an lower limit to $\Sigma (r)$ and therefore a lower limit to $m_{\rm coma}$. The derived $m_{\rm coma}$ values are also listed in Table 3.

Figure 1: R band CCD images of selected unresolved comets. For each frame, the comets heliocentric distance and the direction of the sun are given. The size of each image is indicated by the arcsec scale along the x-axis. The total field of view is 5.63$^{\prime }$ squared.

   
3.3 Active comets

Figures 2a-2d show median combined R band CCD images of the active comets. The observations were performed at airmasses $\leq$2, and as with the comets of the previous section, nearby stars and cosmic rays were removed before any photometry was performed. For comets such as 103P/Hartley 2 and 137P/Shoemaker-Levy 2, cometary activity was easily recognized from their coma morphology. For 47P/Ashbrook-Jackson and 69P/Taylor, a profile analysis was performed whereby the observed brightness profile of the comet was directly compared to that of nearby field stars.

Figure 2: R band CCD images of the active comets before the removal of nearby stars. Note that only 47P/Ashbrook-Jackson and 103P/Hartley 2 were observed under photometric conditions. For each frame, the comets heliocentric distance and the direction of the sun are given. The size of each image is indicated by the arcsec scale along the y-axis. The total field of view is 5.63$^{\prime }$ squared.

Figure 3: Azimuthally averaged surface brightness profiles of the active comets. The two diagonal lines on each graph, with gradients of -1.5 and -1, represent steady state coma models with and without the effects of radiation pressure respectively. The vertical dashed lines are the $Af\rho$ radii (i.e. the radii of photometric apertures used in the derived $Af\rho$ values).

For the active comets in this section, the value for $\rho$ used in the evaluation of the $Af\rho$ quantity is related to the aperture radius used for the total R band magnitude measurement. In other words, a photometric radius of 6.6 $^{\prime\prime}$ is equivalent to $\sim$15 500 km at the comet, therefore $\rho$ is taken to be this distance. Azimuthally averaged surface brightness profiles were determined for each of the four active comets and are shown in Fig. 3. In each case the cometary profiles are comparable to steady state coma models, within the experimental errors. Therefore the derived $Af\rho$ values listed in Table 3 are reliable.

(V-R) and (R-I) colour indices were measured and are listed in Table 4. The total R band magnitude was used to determine an upper limit to the effective nuclear radius. Also listed in Table 3, and labeled with the PSF superscript, are the 3$\sigma$ effective radius upper limits as determined using the scaled PSF method outlined in Sect. 3.3 of Paper I. This PSF method simply utilizes the stellar background PSF to reduce the effective radius upper limits of diffuse/active comets. Essentially, the stellar background PSF is scaled so that its peak brightness matches that of the comet, therefore the magnitude of the scaled stellar PSF is taken to be the maximum possible brightness of the nucleus at the time of the observations. Again, we assume typical values for the albedo of 0.02 and 0.04 to place upper limits on the nucleus sizes.

   

Table 3: R band magnitudes, nuclear radius measurements and upper limits (based on assumed albedos of 0.04 and 0.02), and $Af\rho$ values and upper limits for the detected comets. Note that the albedo values used for comets 49P/Arend-Rigaux and 2P/Encke are different as they have been previously measured.

Comet	R band	$r_{\rm N}$ [km]		$m_{\rm coma}$	$Af\rho$	Ap. radius $^{\dagger}$
	magnitude	$A_{\rm R}$ = 0.04	$A_{\rm R}$ = 0.02		[cm]	[arcsec]
UNRESOLVED COMETS
2P/Encke	20.14 $\pm$ 0.16	4.43 $\pm$ 0.32 $^{\diamond}$	4.43 $\pm$ 0.32 $^{\diamond}$	19.06 $\pm$ 0.03	$\leq$25.9$^{\star}$	4.6
14P/Wolf	20.96 $\pm$ 0.11	2.33 $\pm$ 0.12	3.30 $\pm$ 0.17	19.90 $\pm$ 0.05	$\leq$24.3$^{\star}$	2.1
19P/Borrelly (Night 4)	22.63 $\pm$ 0.36	1.90 $\pm$ 0.32	2.69 $\pm$ 0.45	20.76 $\pm$ 0.03	$\leq$20.9$^{\star}$	2.1
19P/Borrelly (Night 6)	22.61 $\pm$ 0.57	1.91 $\pm$ 0.51	2.70 $\pm$ 0.72	18.92 $\pm$ 0.03	$\leq$19.9$^{\star}$	2.3
43P/Wolf-Harrington	20.81 $\pm$ 0.14	3.43 $\pm$ 0.22	4.86 $\pm$ 0.32	20.35 $\pm$ 0.03	$\leq$32.0$^{\star}$	2.3
45P/H-M-P	23.27 $\pm$ 0.86	1.34 $\pm$ 0.55	1.89 $\pm$ 0.76	18.47 $\pm$ 0.03	$\leq$4.2$^{\star}$	2.0
49P/Arend-Rigaux	19.51 $\pm$ 0.05	4.60 $\pm$ 0.11 $^{\ddagger}$	4.60 $\pm$ 0.11 $^{\ddagger}$	21.27 $\pm$ 0.03	$\leq$23.8$^{\star}$	3.3
61P/Shajn-Schaldach	23.27 $\pm$ 0.56	0.92 $\pm$ 0.24	1.31 $\pm$ 0.34	19.47 $\pm$ 0.06	$\leq$9.8$^{\star}$	2.0
97P/Metcalf-Brewington	22.23 $\pm$ 0.40	2.18 $\pm$ 0.41	3.09 $\pm$ 0.57	20.81 $\pm$ 0.06	$\leq$9.8$^{\star}$	3.3
104P/Kowal 2	23.05 $\pm$ 0.93	1.04 $\pm$ 0.46	1.47 $\pm$ 0.65	20.14 $\pm$ 0.03	$\leq$4.6$^{\star}$	4.0
118P/S-L 4 (Night 6)	21.54 $\pm$ 0.20	2.42 $\pm$ 0.22	3.43 $\pm$ 0.31	20.23 $\pm$ 0.03	$\leq$12.0$^{\star}$	4.0
121P/Shoemaker-Holt 2	22.66 $\pm$ 0.76	1.62 $\pm$ 0.57	2.29 $\pm$ 0.82	19.38 $\pm$ 0.03	$\leq$16.7$^{\star}$	2.6
ACTIVE COMETS
47P/Ashbrook-Jackson	19.07 $\pm$ 0.15	$\leq$6.1PSF	$\leq$8.6PSF	-	28.95 $\pm$ 4.10	6.6
69P/Taylor	19.93 $\pm$ 0.10	$\leq$3.4PSF	$\leq$4.8PSF	-	18.39 $\pm$ 1.69	4.6
103P/Hartley 2	20.10 $\pm$ 0.08	$\leq$5.8PSF	$\leq$8.2PSF	-	37.83 $\pm$ 2.75	3.3
137P/Shoemaker-Levy 2	18.71 $\pm$ 0.14	$\leq$3.4PSF	$\leq$4.8PSF	-	5.59 $\pm$ 0.72	9.6

$\star$ 3$\sigma$ upper limits based on R band magnitudes, $\dagger$ Radius of photometric aperture used to determine the R band magnitudes and $Af\rho$ values, $A_{\rm R}$ is the assumed nuclear albedo, $r_{\rm N}$ is the nuclear radius,
$\ddagger$ $A_{\rm R} = 0.03$ (Millis et al. 1988), $\diamond$ $A_{\rm R} = 0.05$ (Fernández et al. 2000), ^PSF These upper limits were determined using the scaled PSF method outlined in Sect. 3.3 of Paper I.

   

Table 4: Additional V and I filter photometry for several unresolved and active comets. Their R band magnitudes are reproduced to allow a direct comparison.

Comet	V	R	I	(V-R)	(R-I)
UNRESOLVED COMETS
14P/Wolf	$20.98 \pm 0.19$	$20.96 \pm 0.11$	$20.71 \pm 0.33$	$0.02 \pm 0.22$	$0.25 \pm 0.35$
19P/Borrelly (Night 4)	$22.88 \pm 0.69$	$22.63 \pm 0.36$	-	$0.25 \pm 0.78$	-
49P/Arend-Rigaux	$19.99 \pm 0.09$	$19.51 \pm 0.05$	$18.97 \pm 0.13$	$0.49 \pm 0.11$	$0.54 \pm 0.14$
ACTIVE COMETS
47P/Ashbrook-Jackson	$19.42 \pm 0.17$	$19.07 \pm 0.15$	$18.88 \pm 0.27$	$0.36 \pm 0.23$	$0.19 \pm 0.31$
103P/Hartley 2	$20.41 \pm 0.09$	$20.10 \pm 0.08$	-	$0.32 \pm 0.12$	-
137P/Shoemaker-Levy 2	$19.24 \pm 0.25$	$18.71 \pm 0.14$	$18.26 \pm 0.30$	$0.53 \pm 0.29$	$0.45 \pm 0.33$

The solar (V-R) and (R-I) colours are 0.36 and 0.28 respectively (Allen 1973; Fernie 1983).