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Astron. Astrophys. 349, 649-659 (1999)

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3. Results and analysis

The techniques used to analyse these data varied, depending on the observed appearance of the comet. Hence we describe separately the results and analysis for the undetected, unresolved and active comets.

3.1. Undetected comets

[FORMULA] 40P/Vaisala 1 & 26P/Grigg-Skjellerup

For each comet the corresponding images were shifted and coadded using their known rates of motion in an attempt to reveal the objects by increasing the signal to noise ratio. The objects remained undetected. Limiting magnitudes were determined by adding artificial stars at certain positions to the individual frames. The frames were then blinked to simulate the motion of the comet between frames. The magnitude of the artificial stars were gradually decreased until they were no longer detectable, by visual inspection, against the background noise. This iterative process eventually determined limiting magnitudes down to an accuracy of 0.1 magnitudes. As the images of 40P/Vaisala 1 & 26P/Grigg-Skjellerup were obtained with sidereal tracking a small correction must be applied to the limiting magnitude due to trailing of the comets during the exposures (see Fig. 3 of Tancredi et al. 1994). The corrected limiting magnitudes are given in Table 3. Taking these limiting magnitudes as the upper limit to the comet brightness, upper limits to their nuclear radii were derived using the following expression (Russell 1916):

[EQUATION]

where Rh[AU] and [FORMULA][AU] are the heliocentric and geocentric distances respectively, C[[FORMULA]] is the geometrical cross-section of the nucleus, [FORMULA] and [FORMULA] are the apparent R magnitudes of the sun and comet respectively. [FORMULA] is the geometric albedo in the R filter, which throughout this paper is assumed to be 4% (see Fig. 6 of Fitzsimmons et al. 1994, and references therein). The denominator expresses the phase angle dependence of the brightness where [FORMULA] and [FORMULA] are the phase angle and phase coefficient respectively. Throughout this paper a value of 0.04 for [FORMULA] is taken from previous studies (Luu & Jewitt 1992).


[TABLE]

Table 3. R magnitude and radii estimates for the inactive comets. Lower limits for [FORMULA], the steady state coma magnitude, are derived using Eq. (2).


3.2. Unresolved comets

[FORMULA] 69P/Taylor, 81P/Wild 2, 43P/Wolf-Harrington, 120P/Mueller 1, 86P/Wild 3 & 79P/du Toit-Hartley

All images in this section were taken through the R filter. Nearby stars were removed from all exposures of these six comets using the IRAF/DAOPHOT package. The average point spread function (PSF) of the background stars was calculated and fitted to the contaminant stars which were then subsequently removed using the substar task. The analysis procedure was similar for all comets in this section. Therefore for reasons of clarity we describe just the analysis of 69P/Taylor in detail.

All exposures of the comet were shifted and coadded. The average airmass was used in the calculation to convert instrumental magnitudes to apparent magnitudes. To determine whether the comet had a resolved coma the PSF for the combined frame was subtracted from the image of the comet. This process completely removed the comet from the frame and it was therefore regarded as a point source. The absence of any residual flux after PSF subtraction shows that any trailing of the comet is negligible. Indeed this was the case for all the sidereally tracked unresolved comets. The apparent magnitude was measured and substituted into Eq. (1) to calculate the nuclear radius. Both values are listed in Table 3.

To quantify any coma contribution to the magnitude the following expression is applied (Jewitt & Danielson 1984):

[EQUATION]

where [FORMULA] is the integrated magnitude of the assumed steady state coma within a circular aperture of radius r[arcsec] and [FORMULA](r)[mag./[FORMULA]] is the measured brightness at r. To calculate an upper limit for [FORMULA](r) the following approach is used. For faint objects the signal to noise ratio (S/N) is given by the usual formula:

[EQUATION]

where n is the number of pixels within the aperture, I is the total measured intensity [ADU's/sec] of the object within the aperture, t is the exposure time and [FORMULA] is the standard deviation of the background noise. By assuming that S/N should be at least 2 for the comet to be detectable 1, then a minimum value for the brightness/[FORMULA] can be obtained and substituted for [FORMULA](r) in Eq. (2). r is taken to be the point where the PSF for the combined frames becomes negligible i.e. at [FORMULA] 2% of its peak intensity. Applying this to 69P/Taylor yields [FORMULA]. The measured total magnitude was [FORMULA]. From this it can be said that the nucleus significantly dominates the observed flux.

This same procedure was repeated for all exposures of 81P/Wild 2 and 43P/Wolf-Harrington. Unfortunately in the case of 86P/Wild3 and 120P/Mueller 1 background stars were trailed therefore a slightly different method is used to compare the PSF of the background stars with the cometary profiles. Instead, the scaled profiles of the round ends of fairly bright stellar trails were measured and compared directly with those of the comets. The cometary profiles of 86P/Wild3 and 120P/Mueller 1 were identical within measurement errors to the scaled stellar profiles considering the faintness of each comet and the rms scatter of the sky background in each of the exposures. 86P/Wild3 and 120P/Mueller 1 were therefore considered as point sources. The apparent magnitude of 120P/Mueller 1 was 20.3 [FORMULA] 0.2 which corresponds to a nuclear radius of 1.5 [FORMULA] 0.2 km. Also [FORMULA], indicating that the nucleus dominates the observed flux and therefore the nuclear radius estimate for this comet is close to the real value. The measured apparent magnitudes, nuclear radii estimates and [FORMULA] for 43P/Wolf-Harrington, 81P/Wild 2 and 86P/Wild 3 are listed in Table 3. A comparison of [FORMULA] to the measured apparent magnitudes for these three comets shows that the dominant flux source remains uncertain.

Of the two images of 79P/du Toit-Hartley obtained, the comet was only partly visible on the 2nd frame due to a pointing error in the telescope. Although identification was possible, by its known rate of motion, only the first frame could be used for the photometry. Its apparent magnitude was 23.3 [FORMULA] 0.4 corresponding to a nucleus radius of 1.4 [FORMULA] 0.3 km. [FORMULA] was calculated to be [FORMULA] 21.55. Comparison of this value with the measured apparent magnitude again shows that the dominant flux source is uncertain.

3.3. Active comets

In this section the results and analysis for each comet shall be discussed separately and in more detail than the unresolved comets of the previous section. Again the analysis procedures used are similar for each comet. Therefore only the case of 119P/Parker-Hartley will be discussed fully.

3.3.1. 119P/Parker-Hartley

One image in each of the B, V, and R filters was taken of 119P/Parker-Hartley, each with an exposure time of 600s. An extensive dust coma was present. In the R frame the dust tail extended [FORMULA] 20 arcsec, illustrating the degree of activity possible for short period comets at 3.42 AU. For all exposures of the active comets discussed in this section nearby stars and cosmic rays were removed. B, V and R magnitudes were measured through apertures of radius 8.6 arcsec and are listed in Table 4.


[TABLE]

Table 4. B,V & R magnitudes for the active comets.


To measure the relative dust production rate the quantity [FORMULA] is used, first defined by A'Hearn et al. (1984). [FORMULA] is an aperture independent quantity which is roughly proportional to the dust production rate of a comet assuming equal size distributions of particles in the coma. [FORMULA] [cm] can be determined from the observations using:

[EQUATION]

where A is the geometric albedo of the cometary dust grains, the filling factor f is the total cross section of the grains in the field of view, [FORMULA] [cm] is the radius of the field of view, [FORMULA][AU] and [FORMULA][cm] are the heliocentric and geocentric distances respectively. [FORMULA]/[FORMULA] is the ratio of the cometary and solar flux at the observed wavelength. A value for [FORMULA] of [FORMULA] cm was used for all comets discussed in this section and determines the radius of the photometric aperture used in deriving an [FORMULA] value. By measuring the R band magnitude within this radius the right hand side of Eq. (4) can be evaluated to yield an [FORMULA] value for the comet. The R magnitude measured through an aperture of radius 6 arcsec was [FORMULA] which corresponds to an [FORMULA] value of [FORMULA] cm (also listed in Table 5).


[TABLE]

Table 5. [FORMULA] values for the active comets. The following [FORMULA] radius values are equivalent at the heliocentric distances of each comet listed in Table 1.
Notes:
*) Calculated by applying a least squares fit to the data points within the [FORMULA] radii of the cometary profiles of Fig. 1, [FORMULA]) Weighted least squares fit


It should be noted that the quantity [FORMULA] is not strictly applicable to a non steady-state coma, therefore whether or not the coma brightness can be represented by a steady-state coma model must be determined. Hence the surface brightness profile was calculated by measuring the flux from the comet within a series of circular annuli 1-4 pixels wide extending from 2-42 pixels. The resulting plot of surface brightness [mag./arcsec2] vs. log(r)[arcsec] is dependent on the state of the coma (Jewitt 1991). Fig. 1a shows the surface brightness profile plot of 119P/Parker-Hartley. The two solid diagonal lines, with gradients of -1 and -1.5 represent steady state coma models. The steeper gradient takes into account the effects of radiation pressure (Jewitt & Meech 1987). The vertical line is the [FORMULA] radius. Applying a weighted least squares fit to the data points within the [FORMULA] radius gives a value of [FORMULA] for the slope of the profile. Therefore a steady state coma model can be applied to this comet and the above [FORMULA] value is feasible. Table 5 lists the profile gradients of all the active comets.

[FIGURE] Fig. 1a-g. Azimuthally averaged surface brightness profiles of the active comets. The two solid diagonal lines on each graph, with gradients of -1 and -1.5, represent steady state coma models. The steeper gradient takes into account the effects of radiation pressure. The vertical dashed lines are the [FORMULA] radii of the respective comets.

Unlike the comets of the previous section the quantity [FORMULA] of Eq. (2) can be evaluated for 119P/Parker-Hartley. As the surface brightness profile of 119P/Parker-Hartley can be represented by a steady state coma model Eq. (2) can be applied to 119P/Parker-Hartley to give [FORMULA] within the [FORMULA] radius. The total apparent R magnitude within the [FORMULA] radius was [FORMULA] and any nucleus contribution to the calculated [FORMULA] value is negligible.

To derive an upper limit to the nuclear radius of 119P/Parker-Hartley, it is not desirable to use a large aperture due to the overwhelming contribution from the dust coma. The apparent magnitude was hence measured through a circular aperture of radius 3 arcsec. This radius was chosen to minimise the coma contribution and yet include all flux from the nucleus, as here the intensity of the PSF becomes negligible. An R magnitude value of [FORMULA] was found corresponding to an upper limit to the nuclear radius of [FORMULA] km. As a way of refining this upper limit the PSF for the frame was scaled and subtracted from 119P/Parker-Hartley. The residual flux was measured and subtracted from the total flux from the comet to calculate the magnitude of the scaled PSF. The PSF magnitude for 119P/Parker-Hartley was [FORMULA] corresponding to an upper limit of [FORMULA] km. Although the error in this new value is significantly larger the upper limit itself has been reduced by [FORMULA] 46%. Both the preliminary and refined upper limits to the nuclear radius are listed in Table 6.


[TABLE]

Table 6. Preliminary and refined nuclear radii upper limit estimates for the active comets. See text.


3.3.2. 32P/Comas-Solá

As in the case of 119P/Parker-Hartley only one image in each of the B, V, and R filters was taken, each with exposure times of 600s. In the R frame the dust tail extended [FORMULA] 10 arcsec. B, V, and R magnitudes for 32P/Comas-Solá were measured through apertures of radius 6.6 arcsec and are listed in Table 4. The R magnitude within a 6.6 arcsec radius aperture was [FORMULA] which corresponds to an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated by measuring the flux from the comet through a series of circular annuli 1-4 pixels wide extending from 2-40 pixels. The surface brightness profile for 32P/Comas-Solá is shown in Fig. 1b. A steady state coma is assumed for this comet and [FORMULA] within the [FORMULA] radius. Therefore the dust coma dominates the observed flux within the [FORMULA] radius. The R magnitude, measured through a 9 pixel radius aperture, was [FORMULA]. Corresponding to an upper limit to the nuclear radius of [FORMULA] km. The scaled PSF magnitude, as described for 119P/Parker-Hartley, was [FORMULA] reducing the upper limit to [FORMULA] km.

3.3.3. 74P/Smirnova-Chernykh

4 images were taken of 74P/Smirnova-Chernykh, 2 with the R filter and 1 in each of the B, and V filters. Each had an exposure time of 300s. The R frames were shifted and coadded to give a combined image with an effective exposure time of 600s. 74P/Smirnova-Chernykh was by far the most active comet considered here. Even at a heliocentric distance of 4.61 AU a dust coma extending [FORMULA] 8.3 arcsec was observed in the R frame. The B, V and R magnitudes for 74P/Smirnova-Chernykh were measured through an aperture of radius 8.3 arcsec and are listed in Table 4. The R magnitude within an aperture of 4.3 arcsec radius was [FORMULA] which gives an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated by measuring the flux from the comet through a series of circular annuli 1-2 pixels wide extending from 2-30 pixels (Fig. 1c). This plot shows that a steady state coma model is viable for this comet. Therefore using Eq. (2) we find a modelled coma magnitude [FORMULA] of [FORMULA] within the [FORMULA] radius. Comparison with the observed magnitude shows that the dust coma dominates the [FORMULA] measurement. The R magnitude measured through a 9 pixel radius aperture was [FORMULA] giving an upper limit of [FORMULA] km for the nuclear radius. The R magnitude of the scaled PSF was [FORMULA] reducing the upper limit of 74P/Smirnova-Chernykh to [FORMULA] km.

3.3.4. P/1993 K2 (Helin-Lawrence)

2 images in each of the B, V, and R filters was taken of Helin-Lawrence. All images with a common filter type were shifted and coadded to produce single B, V, and R images of Helin-Lawrence. The dust tail of Helin-Lawrence extended out to 14 arcsec in the R image. B, V, and R magnitudes measured through apertures of radius 6 arcsec are listed in Table 4. The R magnitude measured through an aperture of 4 arcsec was [FORMULA] giving an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated by measuring the flux from Helin-Lawrence through a series of circular annuli 1-2 pixels wide extending from 2-30 pixels (Fig. 1d). A steady state coma is assumed and [FORMULA]. The R magnitude measured through a 9 pixel radius was [FORMULA] corresponding to an upper limit of [FORMULA] km for the nucleus radius. The R magnitude of the scaled PSF was [FORMULA] reducing the upper limit to [FORMULA] km.

3.3.5. 9P/Tempel 1

As in the case of 74P/Smirnova-Chernykh 4 images were taken of 9P/Tempel 1, 2 with the R filter and 1 in each of the B and V filters. The R frames were shifted and coadded. In each of the B, V, and R frames 9P/Tempel 1 was almost stellar in appearance. Only PSF subtraction revealed any sign of a faint coma. B, V, and R magnitudes measured through an aperture of radius 6.3 arcsec are listed in Table 4. The R magnitude measured through an aperture of radius 5.4 arcsec was [FORMULA] giving an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated using the same procedure used for 74P/Smirnova-Chernykh and Helin-Lawrence (Fig. 1e). [FORMULA] within the [FORMULA] radius. The R magnitude measured within a 9 pixel radius aperture was [FORMULA] giving an upper limit of [FORMULA] km for the nucleus radius. The R magnitude of the scaled PSF was [FORMULA] reducing the upper limit to [FORMULA] km.

3.3.6. 89P/Russell 2

Only V and R filter images were taken of 89P/Russell 2, each with exposure times of 900s. The 2 R filter images were shifted and coadded to give a single image with an effective exposure time of 1800s. Similar to the case of 9P/Tempel 1, PSF subtraction revealed a faint coma around the comet and R and V magnitudes measured through circular apertures of radius 7.3 arcsec are listed in Table 4. The R magnitude measured through an aperture of radius 6.6 arcsec was [FORMULA] giving an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated by measuring the flux within a series of circular annuli 1-3 pixels wide extending from 2-30 pixels (see Fig. 1f). A steady state coma is assumed for 89P/Russell 2 and [FORMULA] within the [FORMULA] radius. Although 89P/Russell 2 is nearly a point source the large uncertainty in [FORMULA] suggests the dust coma could dominate the observed flux. The R magnitude measured within a 9 pixel radius aperture was [FORMULA] giving an upper limit of [FORMULA] km for the nucleus radius. The R magnitude of the scaled PSF was [FORMULA] reducing the upper limit to [FORMULA] km.

3.3.7. 87P/Bus

Similar to 89P/Russell 2 only R and V images were taken of 87P/Bus, but the exposure times here were 300s. Each image with a common filter type was shifted and coadded to give single R and V images. Even in the coadded R image the comet was extremely faint. R and V magnitudes measured through apertures of radius 6.9 arcsec are listed in Table 4. The R magnitude measured through an aperture of 6.5 arcsec was [FORMULA] giving an [FORMULA] value of [FORMULA] cm. The surface brightness profile was calculated using the same procedure used for 89P/Bus (Fig. 1g). A steady state coma is assumed and [FORMULA] within the [FORMULA] radius. Therefore the dust coma dominates the observed flux. The R magnitude measured within a 9 pixel radius aperture was [FORMULA] giving an upper limit of [FORMULA] km for the nucleus radius. The R magnitude of the scaled PSF was [FORMULA] reducing the upper limit to [FORMULA] km.

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© European Southern Observatory (ESO) 1999

Online publication: September 2, 1999
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