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

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5. General discussion

5.1. Equilibrium temperature

For the photometric size determination of the nuclear diameters and for the normalisation in the light curves the same model was used, which Altenhoff et al. (1994) had successfully applied to asteroids. This method assumes a radio emissivity of unity and the comet in temperature equilibrium with insolation. For many comets this equilibrium may not be reached, because e.g. evaporating water ice keeps the surface near 195 K (see e.g. the comet model of Fanale & Salvail 1984).

Considering the size of the observed halos and the evaporation time scale most dust particles must be refractory grains and at least the bigger ones should adjust to the equilibrium temperature; this assumption is possibly supported by the observed heliocentric distance dependence of the halo, which will be discussed below. Even if the nuclear temperature would not vary, the results of the light curves are hardly affected.

De Pater at al. (1985, 1998) have discussed the effective nuclear brightness temperature with frequency as function of surface material, emissivity and depth structure, offering a brightness temperature range from 195 K to 280 K at d [FORMULA] 1 a.u.; for their interpretation they adopt 195 K, the sublimation point of water ice, independent of the heliocentric distance.

The very low optical albedo for cometary nuclei (e.g. Halley, Hale-Bopp) of about 3 % indicates that the nucleus is covered by dark material rather than by pure ice. Since the surface depth involved in absorption and emission is of the same order as the wavelength, it seems probable that for mm wavelength (300 GHz) the surface temperature is close to the expected equilibrium temperature. This temperature is supported by far IR observations of the "bare" nucleus of Comet IRAS-Araki-Alcock by Hanner et al. (1985) and of the nucleus of Comet Halley from spacecraft, as reviewed by Yeomans (1991), showing a temperature at least 100 K warmer than expected from a sublimating, icy nucleus. (A more extended observing run on PdBI, sampling a wider heliocentric distance range, could have tested these temperature assumptions for Comet Hale-Bopp, too.) For the discussion here we assume that the effective brightness temperature equals the equilibrium temperature to scale the cometary halo intensities and to calculate the nuclear sizes.

5.2. Model of the halo

SEDs of power-law shape have been found previously for Comet Hyakutake by Jewitt & Matthews (1997) and Altenhoff et al. (1996) and for comet Hale-Bopp by de Pater et al. (1998), Bieging et al. (1997), Jewitt & Matthews (1999); power-law SEDs are well known from observations of interstellar dust, where they are thought to originate from a power-law size distribution, [FORMULA], of dust grains with radii a and [FORMULA] (Mathis et al. 1977; Krügel & Siebenmorgen 1994). In the particle halo model described below we follow this prescription of a power-law distribution of grain radii, characterized by [FORMULA] and a normalization constant [FORMULA]. We first estimate the particle mass contained in the total halo by model fitting the observed SED. The radial distribution of the halo emission, its Gaussian width and the excess above a Gaussian at larger radii, will be modelled in a second step.

In estimating the mass of the particle halo from the observed SED, we need to adopt a wavelength dependent absorption coefficient for the different particle sizes which contribute to the mm/submm emission. Rather than assuming specific optical constants for the cometary grains (Walmsley 1985, Jewitt & Matthews 1997), we adopt the less specific but physically plausible concept that the grain absorption efficiency [FORMULA], defined as the ratio of the absorption cross section [FORMULA] to the geometric cross section, is unity for grains larger than [FORMULA] and varies as [FORMULA] for [FORMULA]. For a homogenous, non-porous, spherical dielectric particle [FORMULA] and [FORMULA] (see e.g. Krishna-Swamy 1986). The cometary particles may have properties departing from these idealizations, and we have therefore kept [FORMULA] and [FORMULA] as free parameters. In maintaining this simplified concept for [FORMULA] we effectively assume that any resonances occuring in the mm/submm wavelength range are smoothed out to a large degree. Calculations of [FORMULA] were made using Mie theory for optical constants of various core/mantle grains. The program (E. Krügel, private communication) gave results in agreement with our simplified concept, while significant departures occured only in some cases at [FORMULA] where they are without relevance to our model.

The list of model parameters in Table 8 further includes the temperature [FORMULA] of the particles and their minimum and maximum radii. We assumed that the particles are weakly reflective (optical albedo 0.03) and are in equilibrium with the solar radiation field. The range of particle radii was divided into logarithmic intervals and the emission spectrum from each interval was integrated numerically. The model parameters [FORMULA], and [FORMULA] were varied until the observed SED was reproduced. The resulting mass of the particle halo, [FORMULA], is relatively insensitive to the values of the model parameters. The allowed ranges are listed in Table 8, from which we conclude that, within a factor 2, the model-derived mass [FORMULA] g for Comet Hale-Bopp.


Table 8. Particle halo model parameters

The mass of the particle halo for Comet Hyakutake was derived in the same way. Since the spectral indices of the SED are the same for both comets ([FORMULA] = 2.8), their model particle size distributions must also be identical. Using the equilibrium temperature of 270 K for the comet at the position and time of our observations, even though the smallest halo particles may not quite behave as grey bodies, this model yields, within a factor 2, [FORMULA] g.

5.3. Limitation of the flux density determination

The "integrated" cometary flux densities were derived from ON-OFF measurements, corrected for the apparent size of the halo at the observation. This size information was derived from Gaussian fits to the halo emission. The contribution of the extended structure, not included in the Gaussian fit, is significant. A first estimate was done by measuring planimetrically the difference, giving a contribution of roughly 30 %) to the total flux density. But this difference is dependent on both the integration limits and the accuracy of the long restored scans and is therefore not very precise. Fortunately the first observations with SCUBA by Matthews et al. (1997) allow an independent estimate. They report a total flux density of 2.18 Jy at 353 GHz within the synthetic aperture of 60". Scaled with spectral index [FORMULA] = 2.8, the total flux at 250 GHz is 830 mJy, about 40 % higher than given above for Gaussian fits. This value should be a good estimate for the contribution of the very extended structure at 250 GHz, and it may be a valid estimate for the other frequencies, too.

5.4. Dust production rate

The dust production rate Q of the comet can be estimated if we assume that the dust particles form a spherically symmetric halo around the nucleus and drift away from it at a constant velocity [FORMULA]. The average dust particle therefore resides a time [FORMULA] inside the observed halo radius [FORMULA], and must be replaced after one residence time [FORMULA]. The dust production rate is then simply the mass of the particle halo [FORMULA] as derived above, divided by [FORMULA].

While the observed [FORMULA] emission profile in Fig. 3 clearly supports the spherical constant velocity outflow model, the value of [FORMULA] is observationally not well constrained. Hydrodynamical models of particle entrainment in the gas stream from the nucleus usually give particle velocities dependent on their radius roughly like [FORMULA] (Crifo 1987). The micron-sized (and smaller) particles attain the maximum velocity, the expansion velocity of the gas observed to be [FORMULA] km s-1 by Biver et al. (1997). Lisse et al. (1999) report a particle expansion velocity of 0.4 km s-1, projected on the plane of the sky. Their results, derived from direct mid-IR images of the motions of features in the coma near perigee, probably refers to smaller particles than those that dominate the mm/submm emission. The particles dominating the 250 GHz signal are however much larger and should therefore have lower velocities. For a particle radius of [FORMULA]m, probably the smallest size contributing efficiently, the model gives [FORMULA] m s-1. The largest contributing particle would have a radius of [FORMULA] cm and a velocity of 20 m s-1, just large enough to escape from the gravity of the nucleus. An estimate of the particle expansion velocity obtained by taking the geometric mean of the latter two values gives [FORMULA] m s-1. The average particle residence time in the halo would then be [FORMULA] hours, and the dust production rate [FORMULA] g s-1.

Biver et al. (1997) deduced a gas production rate near perihelion of [FORMULA] g s-1 (including both water and CO), which, combined with our estimates of the dust rate, implies a ratio of dust/gas production of [FORMULA].

For Comet Hyakutake the dust production rate, Q, is also derived from an estimate of the average residence time [FORMULA] of a dust particle inside the [FORMULA] halo (equivalent radius 1000 km). Assuming as for Hale-Bopp an expansion velocity of 60 m s-1 for the most massive dust particles we obtain [FORMULA] hours and a dust production rate Q of about [FORMULA] g s-1.

5.5. Particle size distribution

We derive a best fit index [FORMULA], not too different from its presumed interstellar value of 3.5. This latter value is believed to characterize a size distribution which derives from collisional fragmentation (Hellyer 1970). Does this mechanism also work in comets? We can show, based on our particle halo model, that the mean free path of the smaller particles is only of the order of 1 km, within 50 km from the nucleus. Collisions among dust particles should therefore be frequent.

5.6. Nature of dust particles

We detect dust out to distances of 100", i.e. [FORMULA] km from the nucleus. If these were ice particles whose lifetime is believed not to exceed one day, the particles must have moved at speeds of [FORMULA] km s-1. This is close to or even exceeds the gas expansion velocity. According to hydrodynamical models (Crifo 1987) such high speeds may only be reached by the smallest particles whose radii are smaller than a few microns. If particles lose some fraction of their kinetic energy in a near nucleus collision zone, their initial velocities must have been even higher, i.e. their radii even smaller, in order to reach the 70000 km evaporation radius in a few days. Yet our best fit size distribution implies the presence of a substantial mass component with sizes 10 µm up to [FORMULA] mm. It is therefore plausible that the particles dominating the mm/submm light are in fact refractory. Our modelling of the radial brightness distribution gives a near-perfect fit assuming that the size distribution of particles is constant throughout the halo. Our modelling is therefore fully compatible with refractory grains whose size distribution does not change when they flow away from the nucleus.

5.7. Precision of mass estimates

In Table 8 we estimated the error of the halo masses as about a factor of 2 for both comets. This error reflects only the uncertainty in the observations and in the fitting procedure. The error resulting from our poor knowledge of the dust properties is more difficult to assess. A useful recent compilation of models of the dust absorption cross section at submm/mm wavelengths is given in Fig. 15 of Menshchikov & Henning (1996). While there appears to be little scatter between the various models at wavelengths up to 30 µm, the spread between models increases toward the submm, and reaches about a factor 30 near [FORMULA] mm. The lowest value 3, [FORMULA](1 mm) = 0.4 cm2 g-1 (per gram of dust), applies to normal interstellar dust (Draine & Lee 1984; Pollack et al. 1994).

In their theoretical study of dust in protostellar cores and circumstellar disks Krügel & Siebenmorgen (1994) came to a similar range of [FORMULA](1 mm). Their analysis shows quantitatively how grain growth coupled with the deposition of ice mantles onto porous refractory cores can increase [FORMULA] by up to two orders of magnitude above its interstellar value. Our grain model, though very rudimentary in comparison, successfully reproduces their [FORMULA] for those grain size distributions which are dominated by mm sized particles. Such particles are sufficiently large and heterogenous that any resonances due to specific grain materials are smeared out. In particular, we reproduce within 30 percent their model labelled [FORMULA] Pic which characterizes the peculiar dust in the circumstellar disk of [FORMULA] Pic. Since some of this dust may well originate from disintegrating comets (Vidal-Madjar et al. 1994), this agreement is especially relevant.

The grain size distribution which we postulate for Comets Hale-Bopp and Hyakutake is similar to the [FORMULA] Pic dust. We obtain [FORMULA](1 mm)= 75 cm2 g-1, a factor of 2 higher than the [FORMULA] Pic model. This difference is due to slightly different model assumptions on [FORMULA] and [FORMULA]. Most importantly, our grain size distribution index [FORMULA] (Table 8), indicating a larger number of small grains than in the [FORMULA] Pic model ([FORMULA]). Since smaller grains are more efficient (per mass) absorbers, the overall mass absorption cross section is increased in our dust model. In summary, our rudimentary dust model leads us to a mass absorption coefficient at 1 mm wavelength which is at the extreme high end of the range of published [FORMULA](1 mm), at variance with the much lower values for normal interstellar dust (e.g. Pollack et al. 1994), but close to model predictions for dust possibly produced in part by disintegrating comets.

By way of comparison, de Pater et al. (1998) and Jewitt & Matthews (1997, 1999) assumed dust opacities appropriate for interstellar dust [[FORMULA](1 mm) = 0.5 - 0.55 cm2 g- 1], and derived halo dust masses for Hale-Bopp in the range (3 - 5) [FORMULA] g. It is likely, however, that the dust grain population in the cometary halo is quite different from "standard" interstellar dust. The observed spectral index of 2.8 (cf. Fig. 7) suggests that the halo is dominated by large grains, possibly of a "fluffy" or fractal structure. Lisse et al. (1999) find that the IR SED requires a particle size distribution dominated by a mixture of small (1-5 µm) and large ([FORMULA]m) grains, of which the latter would contribute most of the mm and submm flux.

For the Hyakutake halo, Jewitt & Matthews (1997) derive a dust mass of [FORMULA] g from a submm observation within two days from ours. This mass is a factor of 30 higher than our result (Table 8). The difference becomes even larger, a factor of [FORMULA], when we take into account the 4 times smaller volume observed by them. This enormous discrepancy is due entirely to different values assumed for [FORMULA](1mm). Jewitt & Matthews (1997) adopted a value of [FORMULA](1mm) = 0.5 cm2 g-1, very close to that of normal interstellar dust.

These discrepancies in the derived halo dust masses for both comets reflect to some extent our ignorance of cometary dust properties. We argue that the opacity of cometary dust (at one mm wavelength) may be bracketted by the values for normal interstellar dust, [FORMULA], and for the more porous dust, [FORMULA], described here. Since our models (Sect. 5.2) show that the mm/submm continuum of both comets is clearly optically thin, the derived masses scale inversely with the opacity. The adopted opacitiy range from [FORMULA] to [FORMULA] cm2g-1 therefore implies a mass range by a factor 150. The consequent masses of the particle halos and the dust production rates are assembled in Table 9 for the two comets. We propose that the grain opacities derived for interstellar dust probably underestimate the values appropriate to these comets by at least an order of magnitude, and so lead to corresponding overestimates of the particle halo mass. If the opacity used here, being at the high end of the range of publishes [FORMULA](1mm), overestimates the real value of the opacity [FORMULA](1mm), the derived halo masses may be too low and so lead to corresponding overestimates of the dust production rate and the dust/gas ratio.


Table 9. Mass and dust production rates, Q, for Hale-Bopp and Hyakutake. For each quantity two values are given, one for [FORMULA]fluffy dust" ([FORMULA](1mm) = 75 cm2/g) and one for [FORMULA]normal interstellar medium" ([FORMULA] = 0.5 cm2/g). Dust in comets may have properties bracketted by these extremes (see text).

5.8. Discussion of nuclear diameters

In Table 10 the published size determinations are compared with our mean results. The accuracy of the optically-derived sizes is hard to assess, because the values differ even for identical observations. The average value of 48 km may be a good approximation; it agrees well with our estimate derived from the PdB interferometer observations. The quoted lower limits of the VLA observations are derived assuming the equilibrium temperature; if calculated with the sublimation temperature the upper limit on the diameter would be lower by [FORMULA]20 %). The VLA result by Fernandez, as reported by de Pater et al. (1998), is hard to interpret; the full observing details, the separation of halo and nuclear contribution, the uncertainty of the emissivity at this low frequency are not known yet and may need further discussions. The mean nuclear diameter, derived from the PdBI observations, is quite precise and in agreement with most other determinations of the nuclear size of Comet Hale-Bopp. Unfortunately, this comet was out of reach for radar measurements for an additional confirmation.


Table 10. Derived nuclear diameters

For Comet Hyakutake the first size estimate came from radar measurements; the JPL press release from 1996 March 29 reported a size between 1 and 3 km. An improved estimate was given by Harmon et al. (1996). The first reported radio results are given by de Pater et al. (1997) and Fernandez et al. (1997), both assuming the evaporation point of water ice for the brightness temperature. Their limits should be about 20 % higher, if compared to limits based on equilibrium temperatures. Harmon et al. (1997) revised their size estimate, following a re-discussion of the size of Comet IRAS-Araki-Alcock (a secondary calibrator for radar albedo) by Sekanina (1988). This re-discussion leads to bigger sizes of Comet IRAS-Araki-Alcock, which possibly may fit better to the optical data, but which are hard to reconcile with the original radio observations by the VLA and the 100m telescope.

From the radio results of this paper, only the more significant upper limits are included in Table 10. For Comet Hale-Bopp the inner halo contributed to the observed point source; if the same is true for Hyakutake, the radio size of its nucleus might even be below the estimated lower limits. The first estimate of [FORMULA] = 2 km by Harmon et al. (1996) may be the better one. Table 10 suggests also that mm-interferometry is best suited for measuring the sizes of cometary nuclei, even though the smallest upper limit in units of Jy is measured at 8.4 GHz (Fernandez et al. 1997).

5.9. Heliocentric distance dependence

In connection with the light curves of the two comets, we considered systematic deviations from the proposed model. In Fig. 10 the integrated and normalized flux densities at 250 GHz are shown as functions of the heliocentric distance. Dots represent data of Comet Hale-Bopp from Pico Veleta, triangles data of Comet Hyakutake from the Heinrich-Hertz-Telescope; the lines are empirical fits to the observations. They confirm qualitatively for radio data, what is known for the optical regime: the activity (brightness) of a comet depends on its heliocentric distance and may be different for each comet. Why do different comets get active and generate a halo at different heliocentric distances? A quantitative confirmation of this result would be helpful for an improved comet model.

[FIGURE] Fig. 10. The normalized flux density of Comets Hale-Bopp and Hyakutake as function of heliocentric distance

5.10. Comparison of the three best observed radio comets

Table 11 summarizes the physical properties of Comets Halley, Hyakutake and Hale-Bopp (which are the best-observed comets at radio frequencies). The size of Comet Halley was taken from the Giotto measurements; the other two are from this paper. The mass is calculated from the mean geometric size by assuming a density of 1 g cm-3; the rotation periods are from literature. The data on the halo of Comet Halley are from Altenhoff et al. (1989), except for the mass loss rate, which was taken from McDonnell et al. (1987); the other values are from this paper. The photometric diameter is an observational parameter; it seems to correlate linearily with the nuclear size. The SED of Comet Halley is ill defined because of the small frequency basis and the limited accuracy of the heterodyne receiver data. In hindsight we would not be surprised if the spectral index [FORMULA] of Comet Halley were similar to that of the other comets, i.e. showing a similar particle mix, which is also supported by the particle mix seen by Giotto (McDonnell et al. 1987). The halo mass estimate for Comet Halley was derived by a higher size cutoff as for the other comets, thus giving higher masses. The dust production rate of Comet Halley with an upper particle cutoff of 1 g was taken from McDonnell et al. (1987); with an extrapolated cut off of 1 kg the dust production rate would become a factor of 7 higher, equalizing dust and gas production rates for Comet Halley.


Table 11. Comparison of cometary parameters

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© European Southern Observatory (ESO) 1999

Online publication: August 13, 199