Astron. Astrophys. 348, 1020-1034 (1999)

3. Comet Hale-Bopp

Against the sequence of observations we start with Comet Hale-Bopp, because the data reduction technique can be better shown with the more significant signals and the more complete set of data. The first radio observations of Comet Hale-Bopp, on 1997 February 1 by Kreysa et al. (1997) showed that the radio emission was stronger than expected. From the first observation the signal at Pico Veleta was strong enough to measure the position by scans. Fig. 1b shows such a cut through the comet, an average of 32 subscans, which is well represented by a Gaussian fit. For comparison Fig. 1a shows a scan through the point source BL Lac. The obvious beam broadening can be used to determine the halo size, if there is no significant fine structure around the comet. Figs. 1c and 1d show maps around BL Lac and the comet (an average of four maps on consecutive days to reduce the noise). The average cometary emission can be described by a strong compact Gaussian source (halo), superimposed possibly on a weak, very extended structure. The emission by the nucleus, detected by Wink & Bockelée-Morvan (1997) with the PdB interferometer, would hardly be visible as a weak point source on this scale.

Fig. 1. a  Scan through BL Lac on Mar. 24, 97 at 250 GHz. Gauss fit: 11.39". b  Same scan through Comet Hale-Bopp on March 24, 97. Gaussfit: 17.19". c  Map of Bl Lac (point source) on Mar. 19, 97. d  Average of four bolometer maps of Comet Hale-Bopp between March 13 and 16.

3.1. Halo

The halo size at 250 GHz was derived either from Gaussian fits to the Azimuth and Elevations scans or from a two dimensional Gaussian fit in the bolometer maps with the standard evaluation program NIC, giving the major and minor axis of the ellipse at half power. The results are compiled in Table 2. N is the number of observations; the large number of point source observations indicates that on several days both BL Lac and 3C345 were measured. The deconvolution of the comet was done day by day, thus allowing to derive an error limit for the result. BL Lac was mapped only once, giving a high value for the , obviously caused by anomalous refraction. The default value of , derived for ON-THE-FLY bolometer maps during this period, is 11.0". The deconvolved source size = 11.5" can be ascribed to the halo. This halo size will be used to calculate the integrated flux densities from the observed flux densities per beam. At a mean geocentric distance = 1.326 a.u. the deconvolved Gaussian halo size corresponds to a linear diameter of 11080 km.

Table 2. Observed Gaussian half power widths .

3.2. Near nuclear activity

The bolometer maps allow the investigation of the near nuclear activity, seen in the optical domaine. But there are two limitations: a.) Sources with moving centres like a comet cannot be reduced by the automated "standard" reduction program NIC, but need special treatment as the comet position needs to be recalculated ideally for each integration point in the map. b.) The time scale of the near nuclear activity may be rather short compared to the observing time needed for a single map.

The first problem can be solved interactively with the evaluation program MOPSI. Three bolometer maps, evaluated this way, are shown in Fig. 2. The first impression is that the extended emission feature is not fully symmetric to the halo, seemingly more extended to the south west. This is also true for the other 7 bolometer maps. The fine structure seen in low contours of the extended component seems to be noise, as is suggested by the reduced noise in the averaged map in Fig. 1d. The extended component will be discussed below.

Fig. 2a-c. Bolometer maps of Comet Hale-Bopp at 250 GHz. The lowest contours correspond to 6% of the peak (about 20 mJy/beam), the rms noise is about 15 mJy. Maps are centered on the optimized position of the comet (by pointing). a  Mar. 13 at UT 09:31, PA(sun) = 164. deg, PA(dust tail) = 313. deg, b  Mar. 16 at UT 09:53, PA(sun) = 169.5 deg, PA(dust tail) = 317.5 deg, c  Mar. 24 at UT 08:29, PA(sun) = 180. deg, PA(dust tail) = 329.5 deg. Position angles (PA) of dust tail derived from Kammerer (1997).

A search for correlation of emission features in the bolometer maps (Fig. 2a-c) with directions to the sun or to the dust tail were negative. Detection of the nuclear jet would not be expected within .

For 1997 March 15 and 24 (near observations of Fig. 2b-c) Aguirre (1997) presented IR pictures of multiple expanding dust shells. From these pictures and the rotation period of 11.47 h, derived by Lecacheux et al. (1997), one can derive an arc spacing of about and a projected expansion rate of 1.6"/h for the observing epoch. Such structure would be detectable with the given resolution of and a rather short observing time below 1 hour per map. But there is no indication of a shell structure in the radio maps. The missing information on the exact observing time of the IR-maps prevents a more accurate analysis of the correlation of IR and radio data. The short rotation period of the nucleus and the high expansion rate of the arcs do not allow to average maps of different days to search for weak structure near the nucleus.

The observations with the 30m telescope at 250 GHz with its high angular resolution permitted to map the global radial brightness profile of comet Hale-Bopp with an accuracy unprecedented for radio observations of any comet. The mean observed brightness profile of the comet was derived from the bolometer maps in Fig. 2. Pairs of orthogonal cuts through the maps have been extracted and averaged; the result is shown in Fig. 3.

Fig. 3a and b. Cross section through the radio halo of Comet Hale-Bopp at 250 GHz plotted in linear a and logarithmic b scales. The heavy dots represent the observed mean radial brightness distribution derived from the bolometer maps (Fig. 2) obtained at the 30m telescope. The curves show, from the inside outward, the response to a point source (continuous line), the gaussian fit to the data (dotted), and the model density distribution convolved with the telescope beam.

The heavy dots represent the derived brightness distribution. The linear representation (a) shows that the inner part of the particle halo is well represented by a Gaussian fit with the half power width , which is the convolution of the half power beam width of and the equivalent gaussian half power width of the halo . At radii larger than , however, the observed emission is clearly stronger than the Gaussian approximation. This figure also shows that the mean emission of the halo extends to , corresponding to a nuclear distance of km. We have therefore tried to model the radial particle distribution with a profile which falls off at larger radii more slowly than a Gaussian. Such a profile naturally arises if the dust particles lost from the nucleus expand at a constant velocity with a constant size distribution. The resulting brightness distribution was then convolved with a Gaussian beam of . Since the beam switching observations set an instrumental baseline at scan offsets of , the model was set to zero at these offsets. This model is shown in the figures as the outer, heavy line; it is a good approximation to the observed mean brightness distribution. The logarithmic representation (b) emphasizes the intermediate range of radii and shows that the surface brightness variation is a power law of the radius, strongly supporting the model used. The given error limits of the observed surface brightness distribution reflect the uncertainty in the zero level determination; it corresponds to  K.

Naturally, time variation of the dust production rate, and in particular the presence of jets, and possibly instrumental effects like sidelobes and error pattern, may cause departures from a smooth density profile. The instrumental effects seem to be small, because the narrowest error pattern of the 30m telescope near 250 GHz, observed by Garcia-Burillo et al. (1993) has a Gaussian size of 170", much bigger than the particle halo ; this is supported by pointing scans (e.g. Fig. 1a), which show no indication of significant sidelobes or error pattern. The underestimate of the total flux density by the Gaussian approximation will be discussed later.

3.3. Light curve

The main purpose of the ON-OFF observations was the derivation of the "light curve", i.e., the observed radio signal as function of time. In the simplest case when the constitution of the comet's nucleus and particle halo do not change with time and the cometary radiation is in equilibrium with insolation, the observed flux density only depends on the helio- and geocentric distance, d and respectively; it is proportional to and to , modified by the partial resolution. This is the same model as used e.g. by Altenhoff et al. (1994) to determine asteroidal sizes from flux densities near 250 GHz. Since the respective distances are known, this prediction for the light curve can be tested with observations.

The observed intensities at 250 GHz are plotted in Fig. 4 as a function of Julian date. Dots stand for values of Pico Veleta, triangles for data of the Heinrich-Hertz-Telescope. Both sets are normalized to the heliocentric distance of d = 0.925 a.u. The observed flux densities per beam from both telescopes have been integrated to (total) flux densities, normalized to = 1.315 a.u. Aside from a small calibration scale error between the two telescopes the average normalized flux density of the comet at 250 GHz is = 590 mJy. This average is the constant flux density of our model; the predicted flux densities per beam for both telescopes are calculated from this flux density, considering the changing geocentric distance and the partial resolution. The predicted values are shown as solid lines in Fig. 4. They seem to be a reasonable fit to the observations, consistent with this simple cometary model. The scatter relative to the predicted curves during the main period of coordinated observations may be random noise, rather than an indication of variability. This is supported by the fact that the small deviations of both sets of observations are not correlated with each other. Our observing accuracy of 10 % is clearly an upper limit to the variability of the cometary radio emission. This raises some questions about the relation of the rather constant thermal emission and the possibly variable molecular production rates. Additionally, there is no evidence of any transient icy grain halo (IGH) event, as described e.g. by Hobbs et al. (1975) in our or any other published data of this comet. The possibly systematic deviations in the light curve at both ends of the observing time interval will be discussed later.

Fig. 4. The observed "light" curves of Comet Hale-Bopp at 250 GHz as function of Julian date. (Julian date = 2450000 + day.) Dots are data of Pico Veleta, triangles data of the Heinrich-Hertz-Telescope. Intensities are normalized to heliocentric distance 0.925 a.u. The plotted curves give the predicted flux densities per beam for the given geocentric distance.

3.4. The nucleus

The observations on Plateau de Bure resulted in the first interferometric detection of continuum emission of any comet by Wink & Bockelée-Morvan (1997). It is dominated by its nuclear emission. Fig. 5 shows the cleaned maps for four different days of simultaneous observations near 90 and 218 GHz. The circumstances of these observations (date, time, frequency, geo- and heliocentric cometary distance) are listed in Table 3 and also the intensity of a point source, fitted to the data. Assuming that the nuclear brightness temperature is near the equilibrium temperature with solar insolation, the diameter of the nucleus can be calculated from the flux density of the point source, using the Planck formula. The derived values for equilibrium temperature and nuclear diameter are included in Table 3. The interpretation of this diameter depends on the structure of the halo. If e.g. the halo is spherical, the point source fit would fully represent the nucleus, but if the halo can be represented by a distribution, it has a broad variety of spatial frequencies, and the observed diameter is an upper limit to the real nucleus. Considering only the data in the higher frequency band because of the higher angular resolution the average nuclear diameter from the point source fits becomes formally = 57.1 km.

Fig. 5a-d. Cleaned maps of Comet Hale-Bopp, obtained with the PdBI, near 90 GHz (left) and near 220 GHz (right). Arrows indicate directions to the sun and of proper motion. Contour level spacings are 1 mJy for 90 GHz and 5 mJy for 220 GHz. Crosses indicate ephemerides from Yeoman's solution # 55 for the epochs: a  Mar. 9 (left) UT 05:00, (right) UT 08:00, b  Mar. 11 (left) UT 14:00, (right) UT 06:00, c  Mar. 13 (left) UT 15:00, (right) UT 08:00, d  Mar. 16 (left) UT 16:00, (right) UT 16:00. The corresponding ephemerides are listed in Table 4. Offset positions are in arcsec.

Table 3. Comet Hale-Bopp with Plateau de Bure Interferometer

The visibility plots for the observations of March 13, 1997 at 90 and 218 GHz are shown in Fig. 6 to analyse the source structure. To obtain these points, the uv-data were phase shifted on the nucleus, and vector averages of amplitudes were performed in circles of 300 wavelengths width. A series of models were calculated with the density structure found in connection with the extended structure in Fig. 3 and with various nuclear diameters; the visibility of these models was calculated and compared with the visibility plots for all 4 pairs of observations. The model with the nuclear diameter of 44.2 km gave the best fit to all visibility plots; the solid line in Fig. 6 represents this model. The visibilities at 218 GHz at the longer uv-spacings, which are not sampled at 90 GHz, are scaled to the 90 GHz plot; they fit perfectly into this visibility plot, demonstrating the consistency of the model. Thus we can take the extended structure in the bolometer maps and the partial resolution of the interferometer observations as strong indications for the brightness distribution of the halo. The average diameter of the nucleus, derived from all the visibility plots, is accurate to at least 5 %, with a value

Fig. 6. Correlated amplitudes as function of effective baselines for March 13, 1997 at 218 GHz (triangles) and at 90.7 GHz (circles). Small filled triangles are data of 218 GHz, transformed to 90.7 GHz under assumption of thermal emission. The curves indicate a model with an exponential density distribution as in Fig. 3 together with a thermal point source of 15 mJy at 218 GHz. Above the figure the angular resolution corresponding to the effective baseline is given in arcsec.

3.5. Position offsets from ephemerides

In the interferometer maps in Fig. 5 the expected cometary positions, derived with Yeoman's solution 55, are marked by crosses. The observed radio positions deviate systematically. The observed positions and the deviation from the ephemerides are listed in Table 4. These deviations exceed the expected error limits, quoted by Yeomans. A comparison with ephemerides derived from Yeoman's orbital solution 58 (including optical data near perigee) still confirm the systematic deviations. To test the observing technique asteroid Ceres was observed the same way as the comet; it showed no position error. De Pater et al. (1998) found a similar discrepancy between ephemerides and radio positions for their observing epochs. Also the bolometer observations on Pico Veleta showed a similar position offset. Obviously the nuclear and halo positions coincide. Additionally it should be noted that several molecular line observations show their peak at the observed continuum positions. This discrepancy of all measured radio positions to the optical positions is not yet understood.

Table 4. Observed positions of Comet Hale-Bopp

3.6. Spectral energy distribution

The flux densities at 250 GHz, derived from the Gaussian fits and normalized to the epoch of 1997 March 24, were reported above. The corresponding values for the other frequencies were reduced similarly, and all data are compiled in Table 5.

Table 5. Integrated flux densities and photometric diameter, , for Comet Hale-Bopp, normalized to = 1.315 a.u. and d = 0.925 a.u. (24 march 1997).

The comet is clearly detected at all frequencies. The quoted error is either the internal error or the uncertainty of the absolute calibration, whichever is higher. All flux densities include a correction for the halo size and therefore represent total flux densities for a Gaussian-shaped source. This is a good representation for the inner halo, as seen in Fig. 1. At larger radii, weak excess emission above a Gaussian shape, coming from the brightness distribution in the halo, becomes significant as shown before. Its exact amplitude and frequency dependence is not known, however, so it is omitted here.

We also give in Table 5 the photometric diameter, 2R_ph, for each observation. R_ph is the radius of a circular black body at the temperature of the comet which emits the observed flux density. (The temperature is taken as the equilibrium value, discussed in Sect. 5.1 below.) At all frequencies the photometric diameters are both significantly larger than the size of the nucleus and also much smaller than the observed extent of the halo. The observed flux densities are therefore always dominated by emission from the particle halo. We may also conclude that the halo emission is optically thin at all radio frequencies.

The spectral energy distribution (SED) of the particle halo is seen in Fig. 7 to follow a power law with high accuracy over the whole frequency range from 32 to 860 GHz. The slope of the power law, , is 2.8.

Fig. 7. Spectral energy distribution (SED) of Comet Hale-Bopp. The continuous line fits the data (filled triangles) presented in this paper (Table 5). The dashed line shows the observations of the JCMT, scaled to an aperture of 15.3 arcsec (open circles; Jewitt & Matthews 1999). Measurements near 90 GHz (un-filled triangles) are from de Pater et al. (1998). The dotted line is an estimate of the nuclear emission.

For comparison the SED, measured at the JCMT by Jewitt & Mathews (1999), is shown. The results are scaled to an aperture of with a spectral index of 0.13, in good agreement with our data. Since epochs are different and there are no overlapping size determinations, a calculation of the total flux densities was not attempted.

The photometric diameter allows an order of magnitude estimate for the mass in the particle halo. Assuming the thickness of the photometric disk at 250 GHz is one wavelength and the particle density is 1 g cm^-3, the resulting mass of the halo of Comet Hale-Bopp is 5 g. A detailed model for the halo particle emission for Comet Hale-Bopp will be presented in Sect. 5.

© European Southern Observatory (ESO) 1999

Online publication: August 13, 199
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