Cometary masses derived from non-gravitational forces (original) (raw)

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1Departamento de Astronomía, Facultad de Ciencias, Iguá 4225, Montevideo 11400, Uruguay

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1Departamento de Astronomía, Facultad de Ciencias, Iguá 4225, Montevideo 11400, Uruguay

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Received:

03 November 2008

Accepted:

03 November 2008

Published:

21 January 2009

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Abstract

We compute masses and densities for 10 periodic comets with known sizes: 1P/Halley, 2P/Encke, 6P/d'Arrest, 9P/Tempel 1, 10P/Tempel 2, 19P/Borrelly, 22P/Kopff, 46P/Wirtanen, 67P/Churyumov–Gerasimenko and 81P/Wild 2. The method follows the one developed by Rickman and colleagues, which is based on the gas production curve and on the change in the orbital period due to the non-gravitational force. The gas production curve is inferred from the visual light curve. We found that the computed masses cover more than three orders of magnitude: ≃(0.3–400) × 1012 kg. The computed densities are in all cases very low (≲0.8 g cm−3), with an average value of 0.4 g cm−3, in agreement with previous results and models of the cometary nucleus depicting it as a very porous object. The computed comet densities turn out to be the lowest among the different populations of Solar system minor bodies, in particular as compared to those of near-Earth asteroids (NEAs). We conclude that the model applied in this paper, in spite of its simplicity (as compared to more sophisticated thermophysical models applied to very few comets), is useful for a statistical approach to the mean density of the cometary nuclei. However, we cannot assess from this simple model if there is a real dispersion among the bulk densities of comets that could tell us about differences in physical structure (porosity) and/or chemical composition.

1 INTRODUCTION

Comets have a great cosmogonical interest since they are thought to be the relics of planet formation. They are ice-rich bodies formed beyond the snowline, i.e. the distance from the Sun at which water ice could condense. Comet material is presumably the most pristine so far known in the Solar system that may keep the record of the grain accretion process in the protoplanetary disc. Micrometre-sized grains will tend to collect after gentle collisions into loose aggregates. Since comets are very small bodies, their self-gravity was never large enough to compact the material, so they are expected to conserve their primordial porous structure. Therefore, the comet nucleus is regarded as a very fragile and low-density structure. Donn (1963) estimated that the mixture of ices and meteoric matter would form an aggregate with a density of a few tenths g cm−3. Later Weissman (1986) proposed a nuclear structure consisting of aggregates of various sizes that he called the ‘primordial rubble-pile’ model.

The comet bulk density is a key parameter to learn about the geochemistry and the degree of compactness of the comet material. Masses and sizes of comets are necessary to derive their bulk densities. The determination of these two physical parameters is not trivial. The main difficulty to determine the comet size is that comets are usually shrouded by a coma of dust and gas. Nevertheless, great efforts have been made to observe Jupiter family comets near aphelion where they are inactive or have only residual activity, so estimates of comet sizes have been possible for a large sample of Jupiter family comets (e.g. Fernández et al. 1999; Tancredi et al. 2000, 2006). It has also been possible to obtain very accurate sizes of a few comets from close-up images taken from spacecraft during flyby encounters. These are the cases of 1P/Halley (e.g. Keller et al. 1987), 19P/Borrelly (e.g. Soderblom et al. 2002), 81P/Wild 2 (e.g. Brownlee et al. 2004; Duxbury, Newburn & Brownlee 2004; Howington-Kraus et al. 2005) and 9P/Tempel 1 (e.g. A'Hearn et al. 2005).

The determination of comet masses has been an even more difficult task. In 1805 Laplace could set an upper limit of 1/5000 of the Earth mass for the mass of comet Lexell by noting that the Earth's motion did not suffer any perceptible perturbation after a close encounter with the comet in 1770. From then new attempts to derive comet masses led to even smaller masses showing that comets were indeed very small bodies. Vorontsov-Velyaminov (1946) derived minimum initial masses of 1018 g for comet 109P/Swift–Tuttle and of 1016 g for 55P/Tempel–Tuttle, based on the mass estimates for the associated Perseids and Leonids meteor streams, respectively, and assuming that the masses of the meteor streams were only a fraction of the masses still remaining in the respective nuclei. Asphaug & Benz (1996) modelled the tidal disruption of comet Shoemaker–Levy 9, and the formation of a long chain of clumps of fragments, during its near-miss with Jupiter in 1992. The authors assumed a strengthless body. By comparing the model results with observations, they obtained as a best-fitting solution a diameter ∼1.5 km and a density of ∼0.6 g cm−3 for SW9.

All the results mentioned before can only be applied to particular cases and are very uncertain. Since we do not know about binary comets or comet satellites, and no spacecraft has measured the gravitational field of a comet (except for the Deep Impact experiment with comet 9P/Tempel 1, as commented in Section 6), the only general effect that can give us a direct estimate of the mass of the comet nucleus is the non-gravitational acceleration as shall be analysed in the next section. A method for computing cometary masses based on the non-gravitational effect was introduced by Rickman (1986, 1989), Rickman et al. (1987), and a somewhat similar approach was presented by Sagdeev, El'yasberg & Moroz (1987).

In this paper we plan to re-evaluate the suitability of the non-gravitational force as a tool to estimate cometary masses following a procedure somewhat similar to that developed by Rickman and colleagues. We will also analyse how the uncertainties involved in the different physical parameters related to the problem affect the final results. Finally, we will take advantage of the progress made in the determination of cometary sizes during the last decade to compute bulk densities from the derived non-gravitational masses.

Section 2 presents a brief description of the non-gravitational forces. The method used, the model parameters, and the data of the selected comets for our study are summarized in Section 3. In Section 4 we discuss the problems in the cometary photometry and the empirical correlation between visual magnitudes and gas production rates. In Section 5 we present and discuss the visual light curves obtained for our sample of comets. The results are presented and discussed in Section 6, with a further analysis in Section 7. Finally, a summary with our main conclusions is made in Section 8.

2 NON-GRAVITATIONAL FORCES

The main non-gravitational effect that can be detected in a periodic comet is a change in its orbital period, with respect to that derived from purely gravitational theory. Whipple (1950) proposed an icy conglomerate model for the cometary nuclei, and showed that the momentum transferred to the nucleus by the outgassing could cause the observed non-gravitational effect, an idea that was already advanced more than a century before by Bessel (1836). Because of the thermal inertia and the nucleus's rotation, Whipple showed that the direction of the net outgassing will be deviated a certain angle (known as the lag angle) with respect to the subsolar point. Today we know that some geometrical factors like the shape, the topography, the spin orientation or the location of the active areas on the nucleus surface, are also crucial in the determination of the lag angle, as suggested by thermophysical models (Davidsson & Gutiérrez 2004, 2005, 2006; Davidsson, Gutiérrez & Rickman 2007). We emphasized that in this paper the term ‘non-gravitational forces’ means the momentum transferred to the nucleus by the outgassing, since other non-gravitational forces acting on minor bodies of the Solar system are negligible in the case of comets.

The change Δ_P_ in the orbital period P due to the non-gravitational force can be expressed in terms of the radial _J_r and transverse _J_t components of the non-gravitational acceleration J by means of the Lagrangian planetary equations under the Gaussian form, which leads to

1

where n is the mean motion, a the semimajor axis, e the eccentricity, f the true anomaly and r the heliocentric distance. The direction of the non-gravitational acceleration (which is in the opposite direction to that of the net outgassing) can be described by the angle η with respect to the antisolar direction (i.e. the lag angle), and an azimuthal angle φ in the plane perpendicular to the antisolar direction. The components of the non-gravitational acceleration can be expressed in terms of the angles η and φ as follows:

2

Equation (1) shows that Δ_P_ depends on the radial and transverse components _J_r and _J_t, but not on the normal component _J_n of the non-gravitational acceleration (i.e. the component perpendicular to the orbital plane).

We find some confusion in the literature as regards to what we understand by Δ_P_. Some authors refer to a delay or advance in the time of the perihelion passage, instead of the change in the orbital period. These quantities are not equivalent since the change in the time of the perihelion passage is determined not only by the change in the period Δ_P_, but also by the change in the angular orbital parameters.

Finally, according to Whipple (1950), we can relate the non-gravitational acceleration J and the comet's mass M by means of the conservation of momentum, i.e.

3

where Q is the gas production rate, u is the effective outflow velocity and m is the average molecular mass.

3 THE METHOD

In this section we introduce the assumptions we made in order to simplify the problem of the mass determination.

By substituting _J_r and _J_t of equation (1) by the expressions given by equations (2) and (3), and then solving for the mass, we obtain the following expression:

4

In order to solve the integrals in equation (4) we should know the outflow velocity and the angles as functions of time or the heliocentric distance. Our first approach will be to consider an average value 〈_u_〉 for all escaping molecules during an orbital revolution of period P. We also consider average values for the angles 〈cos(η)〉 and 〈sin(η) cos(φ)〉. We assume these values as constants for all the comets of our sample. We also assume that water, as the principal volatile component in the cometary composition, dominates the gas production in the inner Solar system (say _r_≲ 3 au), so we take m as the water molecular mass. Therefore, we can express equation (4) as follows:

5

where _I_r and _I_t are given by

The computation of the effective outflow velocity 〈_u_〉 relies on sophisticated thermophysical modelling of the exchange of linear momentum between the sublimating gas and the nucleus surface. The value of 〈_u_〉 depends not only on nucleus temperature, but also on shape and location of active areas, therefore is highly uncertain. As an educated guess, we consider 〈_u_〉=ζ_v_th, where _v_th is the mean thermal velocity (in an equilibrium gas), and ζ is a dimensionless quantity that contains any difference between the physically meaningful parameter 〈_u_〉 and the arbitrary chosen normalization quantity v_th. The mean thermal velocity is given by v_th= (8_kT/π_m)1/2, where k is Boltzmann's constant, T is the temperature of the sublimating gases (which is about 200 K for the range of heliocentric distances of interest ∼0.5–3 au), and m, as before, the water molecular mass. For ζ≃ 0.5–0.6 we obtain the value 〈_u_〉≃ 0.25 km s−1 (Wallis & Macpherson 1981; Rickman 1989). Some other works point to a somewhat higher value of 〈_u_〉 (around 0.3 km s−1) (Peale 1989; Davidsson & Gutiérrez 2004). Taking into account the range of published values, we finally adopted the mean value of (0.27 ± 0.1) km s−1 for the effective outflow velocity.

As we said before, in general η and φ will vary with the orbital position. Unfortunately, we cannot know the values of these angles, unless we can study the comet nucleus in situ. Therefore, we depend on educated guesses and averages to sort this problem out. For most reasonable combinations of tentative values for the rotational period, the thermal inertia of the outermost layers of the nucleus and the heliocentric distance, Rickman (1986) and Rickman et al. (1987) obtained, by means of thermal models, η≤ 30°. In more recent works with thermophysical models Davidsson & Gutiérrez (2004, 2005, 2006) also found small values for η. These results confirm our guess that the net non-gravitational force may be deviated from the radial direction just by a small angle.

So we have only positive values for the parameter 〈cos(η)〉, while the parameter 〈sin(η) cos(φ)〉 has not such a constraint. This leads to an indetermination in the sign of the transverse non-gravitational contribution, as defined in equation (5). To sort this problem out, we will use the following approach:

To explain this assumption, we refer to equation (5). There we can see that, for a comet with a symmetric curve Q(t) with respect to perihelion (this implies a symmetric light curve, as assumed in this paper), the first integral will vanish, since a symmetric Q will also imply a symmetric J_r, so the change Δ_P will depend only on the transverse component J_t. In this case it is easy to see that Δ_P and J_t must have the same sign, and also Δ_P and 〈sin(η) cos(φ)〉.

Yet, as pointed out by Rickman (1986), most comet light curves are moderately or highly asymmetric (see e.g. the light curves in Section 5), so the integral of the term containing _J_r will not vanish, and in some cases may become dominant. When one of the terms defined in equation (5) clearly dominates the other, there is no ambiguity in the result: (i) when _I_r≫_I_t, the uncertainty in the sign of the transverse term contributes to the overall uncertainty in the mass as any other source of error and (ii) when _I_r≪_I_t, there is no ambiguity as explained before. The problem of the indetermination in the sign of the transverse term arises when both terms have similar absolute values, in which case we cannot easily discern if _I_t adds or subtracts to _I_r. Another problem related to the case where _I_r≈_I_t arises because the errors may make the difference _I_r−_I_t either positive or negative, leading for one of the signs to an unrealistic value M < 0.

If we further assume a uniform random distribution in the range [0, 2π] for the angle φ, we finally get the nominal values for our model parameters:

Now we turn our attention to the remaining parameters or quantities involved in the computation of the nucleus mass: the gas production curve, the relevant orbital parameters (semimajor axis and eccentricity) and the non-gravitational effect Δ_P_. Unlike the parameters discussed before, these parameters are specific for each comet.

As we can see from equation (5), the computation of the cometary mass requires to know from observations the shape of the curve Q(t). There are only a few cases for which we have good measurements of gas production rates at the different orbital positions, so in general we have to rely on light curves, and hence, assume some formula to approximately convert total visual heliocentric magnitudes _m_h (i.e. the apparent magnitudes m corrected for the geocentric distance Δ: _m_h= _m_− 5 log Δ) to water production rates. In this regard we use the following empirical law, introduced by Festou (1986):

6

We adopt for the coefficients _a_1 and _a_2 the values found by Jorda, Crovisier & Green (2008) (see Section 4.2 for a discussion of the empirical correlation between total visual magnitudes and gas production rates). In Table 1 we summarize the values adopted for the model parameters m, 〈_u_〉, 〈|cos(η)|〉, 〈|sin(η) cos(φ)|〉, _a_1 and _a_2.

Table 1

Summary of model parameters.

Table 1

Summary of model parameters.

The orbital elements of the selected comets are shown in Table 2. For the orbital parameters a and e we take the values corresponding to the epoch closer to the perihelion passage. The symbols represent: τ the time of the perihelion passage, q the perihelion distance (au), e the eccentricity, P the orbital period (yr), ω the perihelion argument (°), Ω the longitude of the ascending node (°) and i the inclination (°). The orbital elements and the epoch are referred to the 2000.0 equinox. The elements were obtained from the catalogue of cometary orbits of Marsden & Williams (2008). For those comets for which the light curves were based on photometric data corresponding to more than one apparition, an average of the values for each epoch was made. The mean motion n and the orbital period P were computed from the value used for the parameter a.

Table 2

Orbital parameters of the studied comets.

Table 2

Orbital parameters of the studied comets.

The non-gravitational effect Δ_P_ was computed from the non-gravitational orbital solution for the comet. In this case the equation of motion has three non-gravitational parameters _A_1, _A_2, A_3 for the radial, transverse and normal components of the non-gravitational acceleration. The change Δ_P will be independent of _A_3 and, for the symmetric case, also of _A_1, so it will only depend on _A_2 through the equation (Marsden, Sekanina & Yeomans 1973; Festou, Rickman & Kamél 1990)

7

The computed values of _A_2 are shown in Table 3. Although the model used by Marsden et al. (1973) (known as the standard symmetric model) was questioned in a physical sense to be not too realistic (since it assumes a symmetric outgassing curve with respect to perihelion), it has achieved in practice good fits to the astrometric observations of a large number of comets.

Table 3

Non-gravitational effect in the studied comets.

Table 3

Non-gravitational effect in the studied comets.

The parameter _A_2 represents the transverse component of the non-gravitational acceleration at 1 au from the Sun, and g(r) is an empirical function which describes the variation of the water snow sublimation rate with respect to the heliocentric distance

8

where α= 0.1113, _m_= 2.15, _n_= 5.093, _k_= 4.6142 and _r_0= 2.808 au (Marsden et al. 1973).

It may be argued that there may be a contradiction between the derivation of Δ_P_ from the symmetric model and the use of a more realistic asymmetric model. The explanation of this seemingly contradictory procedure is that A_1, A_2 arise from the best-fitting orbital solution from Marsden & Williams (2008) which is based on the symmetric model developed by Marsden et al. (1973). Therefore, for consistency Δ_P has to be obtained from the A_2 value for the symmetric model. We expect that Δ_P values derived from asymmetric models should lead to a reasonable agreement with the values derived from equation (7), as far as we get good-quality orbit solutions for the comet's astrometric positions based either on a symmetric or an asymmetric model. In order to check this assumption we also derived Δ_P from Yeomans & Chodas's (1989) asymmetric model. In this regard the JPL Small-Body Database Browser (http://ssd.jpl.nasa.gov/sbdb.cgi) provides sets (_A_1, _A_1, _A_1, DT) for a few Jupiter family comets derived from the latter model. DT represents the time offset of the non-gravitational force maximum from the time of the perihelion passage. From the JPL web site we chose comets 6P, 10P, 46P, 67P and 81P. The results (shown in Table 4) indicate that the values derived from both models are consistent, as expected.

Table 4

Non-gravitational effect according to two different models.

Table 4

Non-gravitational effect according to two different models.

4 COMETARY PHOTOMETRY

In this section we refer to the photometric data base and the criteria applied to construct the cometary light curves _m_h(t). In this regard we analyse some problems in the determination of cometary magnitudes, and discuss how we can relate these magnitudes to the gas production curve.

4.1 Determination of cometary total magnitudes

We understand as ‘total’ magnitudes the ones that comprise the light coming from the nuclear region and the extended coma. By contrast, the term ‘nuclear’ magnitude refers to the ones comprising only the central condensation. It should be noted that in most cases the latter magnitudes do not correspond to the bare nucleus, since it is very difficult to avoid contamination by coma light. Nevertheless, efforts are being made to determine true magnitudes of comet nuclei by observing them far from the Sun or by a coma subtraction method (e.g. Tancredi et al. 2006).

The precise determination of cometary total magnitudes presents serious problems since comets do not appear as point sources like the stars but as nebulous sources. The measured magnitude will depend on the instrument employed (e.g. Bobrovnikoff 1941; Morris 1973). A telescope of small aperture will bring within the field of view the central condensation of the comet plus the extended coma, while a telescope of large aperture will only bring the central condensation. In the latter case the comet will thus look fainter than in the former case. This aperture effect varies both from comet to comet and among different telescope types. The average aperture effect also depends upon the degree of condensation of the comet. Therefore magnitudes measured by different observers with different instruments can differ by several units of magnitude. As an example, Fig. 1 shows the apparent total magnitudes for the 2005 apparition of comet 9P/Tempel 1, measured by a visual method (i.e. an estimate made with a small telescope or binoculars by eye) as a function of the instrumental aperture. All the magnitude data used in this paper come from the International Comet Quarterly (ICQ) (courtesy of D. Green), unless we specify another source. Following the ICQ's recommendations, to minimize the aperture effects the magnitudes should be obtained using the smallest objective aperture and magnification needed to easily see the comet. In Fig. 1 we can see a general trend to reporting brighter magnitudes with smaller apertures, and vice versa. We also note that even for a given aperture, there is a noticeable dispersion in the magnitudes reported by different observers.

Figure 1

Apparent visual total magnitudes (circles) as a function of time from the perihelion passage for the 2005 apparition of 9P/Tempel 1 (data from the ICQ archive). The size of the circles indicates the size of the aperture used: the largest circles corresponds to apertures of 40 ≤ diameter < 45 cm, while the smallest circles corresponds to apertures of diameter <10 cm.

We are restricted to the spectral region where the human eye is more sensitive, i.e. all the magnitudes considered in this paper are visual magnitudes. So in the following, by ‘visual magnitudes’ we refer to the cometary magnitudes estimated by a visual method, while by ‘CCD magnitudes’ we refer to estimates based on a CCD detector with a V broad-band filter or on an unfiltered CCD detector. According to the ICQ, it has been found by some observers that V and unfiltered magnitudes of comets do not differ by more than several tenths of a magnitude. Mikuz & Dintinjana (2001) have compared also CCD V and CCD unfiltered observations finding no meaningful differences among them.

For faint comets, detectors (e.g. CCDs) tend to record only the nuclear condensation while losing the broad (faint) coma that fades into the sky background. Therefore, the measured ‘total’ magnitudes will be fainter than they actually are. The question of how CCD magnitudes – whether filtered or unfiltered – relate to visual magnitudes is still open. Towards the limit of visual observations (near magnitudes 12–15), it has been noted that CCD total magnitudes are typically 1–3 mag fainter than visual estimates (Green 1996). As an example, Fig. 2(a) shows the apparent visual magnitudes as compared to the apparent CCD magnitudes, for the 2005 apparition of 9P/Tempel 1. The figure shows that the CCD magnitudes, as a group, are systematically fainter than the visual magnitudes. The CCD magnitudes are also more scattered, while the visual magnitudes depict a more compact group.

Figure 2

Apparent visual total (asterisks) and CCD (plus signs) magnitudes as a function of time (days from the perihelion passage), for the 2005 apparition of 9P/Tempel 1 (data from the ICQ archive). (b): The same as the previous figure, except that the CCD magnitudes were corrected for the sky fading effect according to the Krésak & Kresáková (1989) formula.

To take into account the sky fading effect, Krésak & Kresáková (1989) suggest the following empirical correction for the measured apparent magnitudes of faint comets with _m_≥ 9:

9

Fig. 2(b) shows that the corrected CCD magnitudes are much less dispersed (resulting in a more flat curve). Yet, in spite of a good agreement with respect to the visual magnitudes group, as we go backwards in time (≳50 d before perihelion), we see an increasing departure of the CCD group from the visual group: the CCD light-curve pre-perihelion branch seems to be significantly different from the visual pre-perihelion branch, with a less steep slope. We conclude that the corrected CCD light curve does not agree well with respect to the visual light curve.

Since we could not find an adequate correction to CCD magnitudes for the sky fading effect, we use only the visual observations in this paper. We also found that for the comets of our sample the visual observations are much more numerous than the CCD observations, during the most active period of the comet (see Fig. 2a for instance).

But the visual observations also show a considerable dispersion, as we saw before (the observations are made with different instrumentation, methodology and/or reference stars, and probably in different local conditions), so we have to adopt some criteria to homogenize the visual magnitude data.

As the first step, we discarded the observations made when the comet was at low altitude without an atmospheric extinction correction applied, or when the magnitude estimate is not very accurate, or when it was made under poor weather conditions according to the information provided by the observer.

Whereas there are not obvious effects that may lead to an increase in the perceived brightness, other than observation during an outburst, there are several effects (like large apertures or an excess of magnification as we mentioned before, and others like moonlight, twilight, haze, cirrus clouds, dirty optics, etc.) that will decrease the brightness and hence overestimate the observed visual magnitudes. As noted by Ferrín (2005), the top observations follow a smooth trend, defining a sharp boundary, while the lower part of the visual observations is more diffuse, poorly defined (see the visual light curve of Tempel 1 in Fig. 2a). Following Rickman et al. (1987), we take a third-, fourth- or sixth-order polynomial fit to the upper envelope of the ensemble of photometric observations as the light curve _m_h(t). In practice this polynomial is obtained as a least-squares fit to a set of selected data points. This set is defined by the following procedure: we divide the time domain (defined by the observations) in N bins of a fixed size. Then we select, for each one of these time bins, the three brightest observations (namely the ones with the lowest values of the heliocentric magnitude), which do not depart more than 3σ from the mean value of the observed magnitudes within the bin. We neglect those time bins with very few observations. Therefore, we allow for a certain range of magnitudes to define our upper envelope, not just the brightest one within each time bin, which gives us more confidence that such a range should cover all potential sources of errors in the magnitude measurements. We used a time bin of 3 d, except for comet 46P/Wirtanen for which we used a time bin of 5 d due to the relatively small number of observations involved.

The definition of the light curve, as a third-, fourth- or sixth-order polynomial, implies that it varies smoothly with time, so comets suspected of having outbursts or another ‘anomalous’ photometric behaviour are excluded of our sample. The polynomial fit does not have any symmetry restriction either, so asymmetric fits (with respect to perihelion) are allowed, and actually are usually the case. The only inconvenience we find with this light-curve definition is that we cannot extrapolate the magnitude beyond the time interval defined by the observations, which is equivalent to neglect the gas production outside such an interval. Nevertheless, outside the observational time interval the gas production usually drops several orders of magnitude with respect to the maximum, so we do not expect to introduce a significant bias in the computed mass due to this constraint (in Section 6 we discuss the uncertainty in the computed cometary masses due to the assumptions we make as regards to their light curves).

We also neglect short-term variations in the visual magnitudes (like the amplitude of the rotational light curve), and variations due to the phase angle, since these sources of intrinsic brightness scattering would be included in the fitted light curve by considering sets of the three brightest magnitudes in every time bin, as discussed before.

4.2 Visual magnitudes and gas production rates

As we pointed out earlier, at heliocentric distances _r_≲ 3 au the cometary activity is governed by the sublimation of water ice. The gas production rate of water ice is thus an adequate quantitative indicator of cometary activity. Direct observations of the H2O production rate are difficult and sparse because they involved sophisticated and oversubscribed instrumentation. Indirect observations (e.g. observation of water-derived products like the OH radical from its radio lines at 18 cm and from narrow-band photometry in the near ultraviolet) are more accessible, but also limited. The huge data base of visual magnitudes gives another approach to this problem (Crovisier 2005). In this regard many authors have studied the empirical relation between total visual magnitudes (corrected for geocentric distance) and gas production rates (Festou 1986; Roettger et al. 1990; Jorda, Crovisier & Green 1992; de Almeida, Singh & Huebner 1997; Jorda, Crovisier & Green 2008). From a statistical analysis of 37 comets, based upon _m_h values from the ICQ and from the Nançay data base, Jorda et al. (2008) derived the linear relation shown in equation (6) where and _a_2= 30.675. This correlation is based on a larger data base than that of the former work of Jorda et al. (1992), and it would be more reliable (L. Jorda, private communication). This relation (or similar ones) has been used to predict gas production rates of comets in the absence of direct measurements (e.g. Festou et al. 1990; Jorda & Rickman 1995; de Almeida et al. 1997). It has also been explored a similar relationship for the CO molecule by Biver (2001), for r > 3 au.

Festou (1986) argued that the relationship (equation 6) seems to hold very well if a set of conditions is fulfilled: (1) a steady state is established in the coma (which would imply that the light curve varies smoothly with time), (2) the visible part of the comet spectrum is dominated by C2 emissions and (3) the ratio of C2 to OH does not vary from comet to comet and with the heliocentric distance. This latter condition could suggest a similar composition for all comets. In this regard A'Hearn et al. (1995) found that there is little variation of relative abundances with heliocentric distance, and there is also little variation from one apparition to the next for most short-period comets.

In Fig. 3 we show the empirical water production rate Q(t) as a function of time that we obtained for comets 9P/Tempel 1, 19P/Borrelly, 67P/C–G and 81P/Wild 2. Each curve Q(t) was obtained from the respective visual light curve (shown in Section 5), where the heliocentric visual magnitudes (_m_h) were converted to water production rates through equation (6), using Jorda et al. (2008) calibration's coefficients. Fig. 3 also shows an individual calibration for each comet, obtained from a least-squares linear fit between the observed and the estimated magnitudes _m_h at the respective observing times (we did not applied any filter or weight to the compiled observational data). We cannot tell at this point if these individual calibrations make a better fit than the Jorda et al. (2008) calibration, due to the scatter (and relatively scarcity) of the observational measurements. We also computed the water production rates predicted by the Jorda et al. (1992) calibration's coefficients, and found that the residuals between predicted and observed water production rates were about a factor of 2 larger than the residuals obtained with the Jorda et al. (2008) constants. So we chose the Jorda et al. (2008) calibration as our nominal one, for all comets of our sample, and take the differences in the computed masses (with respect to the computed masses using the individual calibrations), as an estimate of the uncertainty due to the calibration's coefficients (see Section 6.1 for more details on the error's estimate).

Figure 3

Empirical water production rate as a function of time Q(t) (solid line), superposed to observational data (small circles), for the 2005 apparition of 9P/Tempel 1 (top, left-hand panel), for the 1987, 1994 and 2001 apparitions of 19P/Borrelly (top, right-hand panel), for the 1982 and 1996 apparitions of 67P/C–G (bottom, left-hand panel) and for the 1990, 1997 and 2003 apparitions of 81P/Wild 2 (bottom, right-hand panel). The empirical function Q(t) was obtained from the visual light curve (shown in Section 5) via equation (6), using Jorda et al. (2008) calibration's coefficients. The dotted line shows an individual calibration for each comet, obtained from a least-squares linear fit between the observed and the estimated _m_h from the light curve at the same observing times. The data correspond to observed water production rates extracted from Davidsson et al. (2007) for comet Tempel 1 (courtesy of P. J. Gutiérrez), from Davidsson & Gutiérrez (2004) for comet Borrelly, from Davidsson & Gutiérrez (2005) for comet C–G and from Davidsson & Gutiérrez (2006) for comet Wild 2.

We consider, as Crovisier (2005), that the empirical relation (equation 6), even if it is physically poorly understood, it is a very useful tool. May be the present lack of a physical interpretation of this empirical relationship reflects our poor knowledge of how the cometary activity takes place. By assuming a very simplified physical model for the comet nucleus that consists of a spherical nucleus with a smooth surface and that most of the solar radiation is evenly spread across the surface and spent in the sublimation of ices, we then have _Q_∝_r_−2 (_r_≲ 3 au), so log _Q_∝−2 log r. Since _m_h=H_+ 2.5_n log r, where H is the absolute total magnitude and n is the photometric index that gives the comet's brightness variation with r, we can relate the previous two relations obtaining

10

where C is independent of r and only depends on H. Equation (10) allows us to relate the index n with the coefficient _a_1 of the empirical calibration (equation 6). It is interesting to stress a few consequences of this very simple approach: if the comet brightens exactly like a bare solid body (i.e. the solid nucleus without gaseous activity), we have _n_= 2, and then _a_1=−0.4. But comets usually show activity indices n > 2 which indicates that when approaching the Sun they brighten much more than expected for a bare solid body, because in this case most of the comet light comes from sunlight scattered by dust particles in the coma and fluorescent emission from gaseous molecules, which greatly increases their photometric cross-section. So we expect n > 2 and then _a_1 > −0.4. This is consistent with the value _a_1=−0.2453 from Jorda et al. (2008), for which we derive _n_∼ 3. This latter value is closer to the average of the observed slopes of a large number of comets light curves (_n_= 4). As it was pointed out by Biver (2001), the ultimate goal would be to find the scientific justification of this photometric index. This should be related to physical parameters like the amount of dust relative to gas, the size distribution of the dust particles, the relative abundance of C2, etc., as a function of the heliocentric distance. Of course all these imply a much more sophisticated model for the comet nucleus that is beyond the scope of this paper.

5 THE STUDIED COMETS AND THEIR LIGHT CURVES

In this section we present the criteria applied to select the comets for this study, and the light curves we obtained for the selected comets.

We will limit this study to the short-period comets (i.e. P < 200 yr) since these comets have been observed in more than one apparition, so for them it is possible to obtain a good measurement of the non-gravitational effect Δ_P_.1 To select the comets, the following set of conditions was adopted: (1) a measured non-gravitational change Δ_P_, (2) an adequate photometric coverage during the most active period around the perihelion passage, (3) a good estimate of nucleus size (in order to compute a bulk mass density) and preferentially (but not exclusive) (4) a measured water production rate (in order to check the coefficients for the empirical calibration of equation 6). The comets selected were 1P/Halley, 2P/Encke, 6P/d'Arrest, 9P/Tempel 1, 10P/Tempel 2, 19P/Borrelly, 22P/Kopff, 45P/Honda–Mkros–Pajdusakova, 46P/Wirtanen, 67P/Churyumov–Gerasimenko and 81P/Wild 2.

The data base of apparent total magnitudes used in this paper is from the ICQ archive for observations after 1990, and from the Comet Light Curve Catalogue/Atlas Part I (Kamél 1991a) for observations prior to 1990.

For some comets we use observations involved in more than one apparition to make a composite light curve, to compensate for the relative low number of observations in a single apparition. This is possible since no significant variation of the orbital elements is noted on two or more consecutive observed apparitions of the chosen comets, and also by assuming that the shape of the light curve remains almost unchanged during consecutive perihelion passages of the comet.

Figs 4–14 show the light curves obtained as plots of the total visual heliocentric magnitude _m_h as a function of the time relative to the perihelion passage t (the observations were previously filtered according to the criteria explained in Section 4). The horizontal line indicates the cut-off magnitude _m_hC (i.e. the polynomial fit is considered a good approximation to the light curve only within the interval where _m_h≤_m_hC). For the conversion from apparent to heliocentric magnitudes we compute the comets' distances with the Mercury orbital integrator (Chambers 1999), and use some routines from Press et al. (1992) for function interpolation, and other routines from the novas package of the US Naval Observatory, written by G. H. Kaplan, for the conversion of time from Gregorian to Julian Date.

Figure 4

Heliocentric total visual magnitudes as a function of time (days from the perihelion passage), for comet 1P/Halley. The polynomial fit _m_h(t) to the upper envelope of the broad distribution of photometric measurements is shown. The horizontal line indicate the cut-off magnitude _m_hC, as defined in Section 5. The value of _m_hC is also shown. The magnitude data for this comet and the rest of the sample were extracted from Kamél (1991a) (for observations before 1990) and from the ICQ archive (for observations from 1990 and forward).

Figure 5

Same as Fig. 4, but for comet 2P/Encke.

Figure 6

Same as Fig. 4, but for comet 6P/d'Arrest.

Figure 7

Same as Fig. 4, but for comet 9P/Tempel 1.

Figure 8

Same as Fig. 4, but for comet 10P/Tempel 2.

Figure 9

Same as Fig. 4, but for comet 19P/Borrelly.

Figure 10

Same as Fig. 4, but for comet 22P/Kopff.

Figure 11

Same as Fig. 4, but for comet 45P/H–M–P.

Figure 12

Same as Fig. 4, but for comet 46P/Wirtanen.

Figure 13

Same as Fig. 4, but for comet 67P/C–G.

Figure 14

Same as Fig. 4, but for comet 81P/Wild 2.

As we discussed in Section 3, most comet light curves are asymmetric with respect to the perihelion. As a first approach, the perihelion asymmetries of the comet light curves could be mainly related to structural changes of the nucleus that occur during the approach to the Sun, like the expulsion of an insulating dust mantle that would leave exposed areas of fresh ice. This effect, combined with the thermal inertia, would increase the gaseous activity after the perihelion passage. Most of the studied comets effectively show a post-perihelion brightness excess. These are the cases of comets 1P/Halley, 6P/d'Arrest, 10P/Tempel 2, 19P/Borrelly, 45/H–M–P and 67P/C–G. On the other hand, comets 2P/Encke, 22P/Kopff and 81P/Wild 2 show a pre-perihelion brightness excess. Hence the gaseous activity could have its maximum before perihelion. This could be taken as another evidence of the complexity of the cometary activity, suggesting that other causes, like the concentration of the gaseous sublimation in a few discrete active areas, and the orientation of the spin axis, which may, or may not, favour the illumination of the active areas, may also play a role in the asymmetric behaviour. Comets 9P/Tempel 1 and 46P/Wirtanen show the most symmetric curve of the sample. Comet Tempel 1 shows a slightly pre-perihelion excess of activity, but in this case the near- and post-perihelion coverage is very poor, due to unfavourable geometrical conditions.

The maxima of the comet light curves seem to occur typically around or about a few days before or after perihelion, although greater departures are common (see e.g. the light curves of comets d'Arrest and C–G), as it was already noted by Festou (1986).

We note that comet Halley at its maximum brightness is several magnitudes brighter than the rest of the comets of the sample at their maxima. This should be correlated with a much greater gaseous activity for Halley as compared to the other comets. For instance, Halley shows a gas production rate of about 1031 s−1 near perihelion, while the rest of the studied comets are approximately in the range [0.1– 2.0]× 1028 s−1 near perihelion (A'Hearn et al. 1995; Crovisier et al. 2002). This result may be partly due to the different comet sizes, and partly due to the different source regions: while Halley is the prototype of the population of the Halley-Type (HT) comets – for which the source region would be the Oort cloud (Fernández 2005, chapter 7), the rest of the studied comets belongs to the Jupiter Family (JF) population – for which the main source region would be the trans-Neptunian belt (Fernández 1980; Duncan, Quinn & Tremaine 1988). Since JF comets may have spent up to hundreds or thousands of revolutions on short-period orbits near the Sun, they may be more physically evolved than HT comets, so 1P/Halley may still be a comet with a fresher and more active surface, leading to its observed substantially higher activity.

Because of their short orbital periods (P < 20 yr), most of the discovered JF comets have been observed in several apparitions, and in some cases through all the orbit up to their aphelia, allowing to gather a very valuable wealth of physical data. Furthermore, several JF comets have been or will be the targets of space missions, which will help to greatly increase our knowledge about their physical nature. This is the case of four out of the 11 comets studied in this paper. In the following we make a brief summary for each one of the light curves obtained.

5.1 1P/Halley

Comet 1P/Halley was the target of several international space missions during its last apparition in 1986, which could take images of its nucleus of irregular shape, and confirmed, in general terms, Whipple's model. From the ground the comet was bright enough to be visible to the naked eye, though far below the spectacular previous apparition of 1910. This comet presents the best photometric coverage in one single apparition, as compared to the rest of the studied comets, resulting in a good-quality light curve. The plot in Fig. 4 contains 1221 observations corresponding to the 1986 apparition, and shows the fitted light curve _m_h(t). The light curve obtained is similar to others found in the literature (e.g. Ferrín 2005).

5.2 2P/Encke

For this comet we combined the observations of four consecutive apparitions: 1990 (186 observations), 1994 (202 observations), 1997 (29 observations), 2000 (24 observations). The observations of 1997, in spite of their low number, were important to define better the post-perihelion branch, for which only the 1994 apparition has post-perihelion observations. Fig. 5 shows a good consistency between the different apparitions, confirming our assumption that the light curve has not changed significantly from one apparition to the next, during the last decade. Yet we observe a secular decrease of the non-gravitational effect during this period. This was already noted by Kamél (1991b) who studied the light-curve evolution of this comet from 1832 to 1987. Kamél also found a secular decrease of the perihelion asymmetry (which he related to a shift in the time of maximum brightness from three weeks before perihelion to a few days after perihelion in the latest apparitions). He concluded that the shape of the light curve has not significantly changed during short intervals of time (like the last apparitions he studied), in particular the perihelion asymmetry has remained, showing a faster brightness decrease after the maximum and a slower increase before it. Our composite light curve, based on most recent observational data, is in good agreement with the results from Kamél.

5.3 6P/d'Arrest

The light-curve data set for this comet contains 424 observations from the 1995 apparition (the last apparition in 2002 was unfavourable since the comet was in conjunction with the Sun close to the time of the perihelion passage). The light curve is shown in Fig. 6. This comet shows an extraordinary strong asymmetry with respect to perihelion. The brightness rises rapidly as the comet passes perihelion, reaching the maximum around ∼40 d after perihelion, with a slow decline after the maximum. The light curve obtained is similar to others found in the literature (e.g. Szutowicz & Rickman 2006).

5.4 9P/Tempel 1

This comet has been the most recent target of a successful space mission, the Deep Impact, which encountered the comet about 1 d before perihelion on 2005 July 4. Therefore a great amount of data has been collected for this comet, including a good photometric coverage. Unfortunately, like the former apparition of 1994, there are very few post-perihelion observations due to geometrical circumstances. The light-curve data set for this comet contains 447 observations from the 2005 apparition. The light curve is shown in Fig. 7. We also studied the light curve corresponding to the 1994 apparition (with a data set of 603 observations), and found an average brightness decrease of about 1–2 mag from 1994 to 2005. The perihelion asymmetry seems to remain, with a slightly pre-perihelion predominance, but this feature is quite uncertain since there are not enough post-perihelion observations to define a good post-perihelion light curve. Ferrín (2005, 2007) has published a light curve for this comet, but only based on the 1994 data, which is similar to the 1994 light curve we obtained.

5.5 10P/Tempel 2

The light-curve data set for this comet contains 153 observations from the 1983 apparition and 143 observations from the 1988 apparition. The composite light curve is shown in Fig. 8. There are few pre-perihelion observations in 1983, but those from 1988 are enough to properly define the pre-perihelion branch of the light curve. We found a good consistency between the observations of both apparitions, which implies that the light curve remains practically the same. Rickman et al. (1991a) have studied the light curve of this comet during 13 consecutive apparitions from 1899 up to 1988. They find that after perihelion the 1983 and 1988 apparitions yield quite different light curves. The disagreement between their result and ours may be due to a different filtering criterion applied to the observations (since the original data base –Kamél 1991a– is the same). In correspondence with a negligible variation of the light curve between both apparitions, we do not find any significant change in the orbital parameters nor in the non-gravitational effect (see Tables 2 and 3, respectively). Anyway, both works agree in the remarkable asymmetry showed by this comet light curve, which is very similar to the d'Arrest case: a rapid brightness increase before the maximum, followed by a slow decrease after it.

We also studied the 1999 light curve, based on 239 observations (the 1994 and 2005 apparitions were too poor in number of observations). We found a significant brightness decrease of 1–2 mag from 1988 to 1999, and a shift in the time of the maximum brightness from around 14 d after perihelion to a few days after perihelion. Notwithstanding this remarkable change in the light curve, the asymmetry remains in shape: faster brightness increase before the maximum, and a slower brightness decrease after it (although the pre-perihelion increase is less steep than that of the 1988 light curve). This brightness decrease could be related to a small but noticeable increase (of about 7 per cent) of the perihelion distance from 1988 to 1999 (see Table 2). We also noted a certain decrease (of about 20 per cent) of the non-gravitational change Δ_P_ (see Table 3). This comet shows the smallest non-gravitational effect of the sample (closely followed by Tempel 1). Rickman et al. (1991a) suggested a possible correlation between the lower values of the non-gravitational effect on Tempel 2 and the apparent drop in the gas production as inferred from the light curves.

We conclude that there is a remarkable difference of the 1983 and 1988 composite light curve with respect to the 1999 light curve. Since the quality of the former is much better, we adopt it as our nominal light curve.

5.6 19P/Borrelly

This comet has been the target of the Deep Space 1 mission, which encountered the comet on 2001 September 22, about 8 d after perihelion. The light-curve data set for this comet contains 360 observations from the 1987 apparition, 853 observations from the 1994 apparition and 242 observations from the 2001 apparition. The composite light curve is shown in Fig. 9. We found no meaningful changes in the light curve from one apparition to the next, during the studied period. We discarded a small group of bright measurements of the 2001 apparition bunched together around 20 d before perihelion as a sort of spike, since they depart from the general trend of the bulk of the observations. This spike could be due to a little outburst produced around 30–20 d before perihelion. Borrelly shows the same asymmetrical behaviour as, for example, d'Arrest or Tempel 2: a faster brightness increase before the maximum followed by a slower decrease after it. The composite light curve obtained in this paper resembles very much the one from Ferrín (2005).

5.7 22P/Kopff

The light-curve data set for this comet contains 349 observations from the 1983 apparition and 586 observations from the 1996 apparition. The composite light curve is shown in Fig. 10. This comet presents a curious light curve since, even though it shows a faster brightness increase before the maximum, like those comets with a post-perihelion excess (e.g. C–G, d'Arrest, Tempel 2 or Borrelly), it shows however a pre-perihelion excess. We note however that the brightness maximum is quite uncertain, since the comet seems to hold a similar activity level for several weeks, so we cannot exclude the possibility that the real maximum could be a few days after perihelion. Anyway, it is clear from Fig. 10 that the comet shows more activity before than after perihelion.

5.8 45P/Honda–Mkros–Pajdusakova

The light-curve data set for this comet contains 117 observations from the 1990 apparition, 108 observations from the 1995 apparition, and 40 observations from the 2001 apparition. The composite light curve is shown in Fig. 11. This comet shows a post-perihelion excess but not so strong as C–G, d'Arrest, Tempel 2 or Borrelly. We can see an increasing scatter of the observations as the comet moves away from perihelion (the post-perihelion observations after about 60 d were not considered for this reason).

5.9 46P/Wirtanen

This comet was the former target of the Rosetta mission, which will finally be encountering comet C–G in 2014. The light-curve data set for this comet contains 105 observations from the 1991 apparition. The light curve is shown in Fig. 12. We also studied the light curve corresponding to the 1997 apparition (with a data set of 104 observations). We note that the 1997 observations seems to be somewhat brighter (about several tenths of magnitude) than those from 1991, so we decided not to combine both apparitions into one single composite light curve, since it is not clear that the light curve has not changed from one to the other. Since the 1997 observations are much more scattered than those from 1991 as we go far from the perihelion, which results in a more poorly defined light curve, we decided to take the 1991 light curve as the nominal one. As inferred from this light curve, besides a post-perihelion brightness excess, this comet also shows a rapid decrease of the gaseous activity for increasing heliocentric distances. This is in agreement with Jorda & Rickman (1995), who also studied the 1991 light curve of this comet.

5.10 67P/Churyumov–Gerasimenko

This comet will be encountered by the Rosetta spacecraft in 2014, which will orbit it during its journey through the inner Solar system, and land a probe on its surface. Hence a great wealth of physical data for this particular comet is expected in the next years.

The light-curve data set for this comet contains 260 observations from the 1982 apparition, and 127 observations from the 1996 apparition. The composite light curve is shown in Fig. 13. This comet shows a strong post-perihelion asymmetry. We also note that this comet is the faintest one of the sample (followed closely by Tempel 1), since its maximum brightness is somewhat below 10 mag, while most JF comets show maxima at magnitudes between 7–9, as we pointed out before. The post-perihelion excess and the maximum brightness obtained is in agreement with other works (e.g. Ferrín 2005).

5.11 81P/Wild 2

This comet was visited by the Stardust spacecraft on 2004 January 2, which obtained high resolution images of the nucleus, showing remarkable differences with previous images of Halley and Borrelly, like the evidence of impact craters (Brownlee et al. 2004).

The light-curve data set for this comet contains 78 observations from the 1990 apparition, 626 observations from the 1997 apparition and 72 observations from the 2003 apparition. The composite light curve is shown in Fig. 14. This comet shows the strongest pre-perihelion asymmetry of the sample. The bulk of the observations which defines the light curve around the maximum brightness corresponds to the 1997 apparition, while the 2003 observations contributes to define the pre-perihelion branch, and the 1990 observations help to define the outermost part of the post-perihelion branch. This is the only comet of the sample which has a visual photometric coverage up to heliocentric distances of about 3 au. Our light curve is in good agreement with that studied by Sekanina (2003). Ferrín (2005) also derived a similar light curve for this comet, except that ours shows a steeper post-perihelion slope, which implies a more rapid fading of the activity.

6 THE RESULTS

For the computation of the non-gravitational change Δ_P_ and the comet mass (from equations 5 and 7, respectively) we use some routines from Press et al. (1992) for numerical integration of functions and for root finding, and write our own routine to solve the Kepler equation.

We first introduce the main sources of uncertainty in the computation of the cometary masses, and then the results obtained for the comets of the sample. Our intention was not to proceed to a rigorous computation of errors, since that would be meaningless under the general assumptions we made and the statistical scope of this work, but instead to evaluate in an empirical way the effect of the main sources of uncertainty in the computed masses.

6.1 Sources of uncertainty

The most important sources of uncertainty are the comet light curve, the model parameters (particularly the effective outgassing velocity, which may be is the most uncertain of the parameters assumed), and the computed non-gravitational change Δ_P_.

6.1.1 The light curve

In order to estimate the uncertainty due to the light curve, we proceeded as follows: as we explained in Section 4.1, the light curve is defined by a least-squared fitted polynomial to a set of selected observations. This set of observations was obtained by dividing the observational time interval in bins of fixed size, and selecting, in each bin, the three brightest observations with the condition that they do not significantly depart from the bulk of observations. By varying the bin size (from 2 to 6 d, with 1-d step), we picked different sets of brightest magnitudes leading to different light curves (one per bin size), and hence different cut-off magnitudes (_m_hC). For each one of these light curves (with its corresponding cut-off magnitude) we computed the mass (leaving the rest of the parameters involved fixed with their nominal values). In this way we obtained a set of different mass estimates for each comet of the sample. Hence, for a given comet, we take the mean value of the relative differences in the computed masses, with respect to the nominal value, as an empirical estimate of the uncertainty in the comet mass due to the light-curve uncertainty.

Comet d'Arrest was found to be the comet with the smaller uncertainty due to its light curve [(Δ_M_/M)light curve= 3 per cent], followed by, in increasing order of uncertainty: Halley and C–G (6 per cent), Tempel 1 and Tempel 2 (8 per cent), Kopff (9 per cent), Encke (11 per cent), Borrelly (13 per cent), Wild 2 (28 per cent), H–M–P (43 per cent) and Wirtanen (44 per cent).

We note that these values do not necessarily reflect the quality of the light curve in all cases, since, for instance, Tempel 1 has a relatively poorly defined post-perihelion branch of the light curve, though we got a small uncertainty in the computed mass. Nevertheless, in most cases, we find a good correlation between the light-curve quality and the estimated uncertainty in the mass, as in the case of Wirtanen. This comet shows the most poorly defined light curve of the sample, due to a relatively low number of observations, which reflects in the largest estimated uncertainty in the mass.

6.1.2 The model parameters

From the discussion of Section 3, we assumed an error of ±0.1 km s−1 in 〈_u_〉. We also assumed 0.05 ≤〈|sin(η) cos(φ)|〉≤ 0.15 and 0.9 ≤〈cos(η)〉≤ 1.

Although a thorough study of the correlation between water production rates and visual magnitudes is beyond the scope of this paper, we perform individual calibrations for some comets of our sample (as explained in Section 4.2), just to evaluate, in an empirical way, how the uncertainty related to the constants {_a_1, _a_2} propagates to the computed masses. We found that the relative difference in the mass computed using the individual calibration, with respect to the nominal mass (i.e. the mass computed using the Jorda et al. (2008) calibration) varied between 13 per cent (for Wild 2) and 46 per cent (for Tempel 1). The average of these relative differences was about 30 per cent. We also found that the relative difference in the mass computed using the Jorda et al. (1992) calibration, with respect to the nominal mass, was about 30 per cent for the comets of the sample, except for comet Halley for which the difference was about 20 per cent. Hence, we estimate the uncertainty in the computed masses due to the calibration constants {_a_1, a_2} to be about (Δ_M/M)calibration≅ 30 per cent, for all the comets of the sample.

6.1.3 The non-gravitational effect

For some comets we found in the literature an estimate of the absolute error Δ(Δ_P_) (e.g. Borrelly by Davidsson & Gutiérrez 2004, Tempel 1 by Davidsson et al. 2007, C–G by Davidsson & Gutiérrez 2005 and Wild 2 by Davidsson & Gutiérrez 2006). For the rest of the comets we have to adopt some criteria in order to estimate the uncertainty of this parameter. For those comets which have showed some variation in the non-gravitational effect Δ_P_ during the apparitions considered in this paper (e.g. Encke, Kopff, H–M–P, Wirtanen), we estimate the error as the standard deviation with respect to the mean value (although for these comets intrinsic variations of the change Δ_P_ cannot be ruled out, we neglect them by considering a composite light curve). For comets Halley and d'Arrest, in the absence of a better criterion, we assume an error given by the average of the normalized (i.e. relative) errors estimated for the former comets. For comet Tempel 2, which shows a very small Δ_P_ like Tempel 1, we assumed the same error than for this latter comet. The estimated uncertainties range from ∼4 per cent for comets Kopff and H–M–P, up to ∼58 per cent for comet Wild 2. The intermediate values of the estimated uncertainty are: ∼6 per cent (comets Halley, d'Arrest and Borrelly), ∼13 per cent (comet Wirtanen), ∼20 per cent (comet C–G), ∼27 per cent (comets Tempel 1 and Tempel 2), and ∼50 per cent (comet Encke).

6.1.4 Estimate of the overall error in the mass and density

Although for some parameters involved in the computation of the mass (like the non-gravitational effect Δ_P_, and the effective outflow velocity 〈u_〉) their effect in the mass can be easily evaluated separately, since the mass is proportional to them, we found more convenient to evaluate the overall uncertainty in the computed mass in an empirical way. In this regard we considered, for each comet of the sample, N points of a parametric space defined by the parameters 〈_u_〉, 〈cos(η)〉, 〈|sin(η) cos(φ)|〉, Δ_P and F, where F is a factor that accounts for the uncertainty due to the light curve and the calibration constants {a_1, a_2}, defined as F_= 1 + (Δ_M/M)light curve+ (Δ_M/M)calibration (a value of F_= 1 implies a negligible uncertainty due to the light curve and the calibration constants). We chose N_= 1000. We denote by F L the computed values of F for the comets of the sample. We obtained values ranging between F L_= 1.3 for comet d'Arrest and F L_= 1.7 for comets H–M–P and Wirtanen. Each point of the parametric space was obtained by generating a random value with an uniform distribution for each parameter, with the constrains: 0.17 ≤〈_u_〉≤ 0.37, 0.9 ≤〈cos(η)〉≤ 1, 0.05 ≤〈|sin(η) cos(φ)|〉≤ 0.15, Δ_P_min≤Δ_P_≤Δ_P_max and 1 ≤_F_≤_F L, where Δ_P_min=Δ_P_nom−Δ(Δ_P) and Δ_P_max=Δ_P_nom+Δ(Δ_P), according to the estimated absolute error Δ(Δ_P) (computed from the relative errors presented in Section 6.1.3), and to the nominal value Δ_P_nom for the non-gravitational effect given in Table 3. Then, each one of these N points represents a set of parameter values from which a mass is computed, by means of equation (5), where the factor F multiplies the right-hand member of this equation. By computing the standard deviation of these N mass estimates with respect to the nominal value we evaluate the overall uncertainty or error in the mass (Δ_M) of a given comet.

Finally, by propagating the estimated errors in the mass (Δ_M_) and in the effective nucleus radius (Δ_R_) (given in [Sections 6.2.1–6.2.11](#ss6-2-1 ss6-2-11)), we obtain the error in the mass density

11

where R represents the effective nuclear radius.2

6.2 Computed masses and densities

We present the computed masses from the nominal values for the model parameters (summarized in Table 1), with a rough estimate of the mass uncertainty due to the main sources of uncertainty as considered before. We also show the derived bulk mass density with an error estimate as derived from the size and the mass uncertainties, respectively. The results are compared with those from other works, and summarized in Table 5.

Table 5

Masses and densities.

Table 5

Masses and densities.

6.2.1 1P/Halley

• Mass. We derive a mass of [3.2 ± 1.2]× 1014 kg for this comet.
• Size and density. This comet has a determination of its nucleus size measured on close-up images from fly-by missions, which is the best procedure to determine unambiguously the size, albedo, or shape of a comet nucleus. Keller et al. (1987) determined an effective radius of 5.2 ± 1.0 km. According to this value for the comet size we derive a bulk density of 0.5 ± 0.3 g cm−3 for this comet.
• Other works. Rickman (1989) estimated _M_∼[1.3–3.1]× 1014 kg. Later, Skorov & Rickman (1999) revised the Rickman's (1989) estimate and obtained ρ=[0.5–1.2] g cm−3, with a preference for the lower value. Sagdeev et al. (1987) derived _M_= 3 × 1014 kg, and a bulk density of 0.6+0.9−0.6 g cm−3, also based on a non-gravitational model. Peale (1989) estimated ρ=[0.03–4.9] g cm−3, with a preference for ρ = 0.7 g cm−3. These results are consistent with ours.

6.2.2 2P/Encke

• Mass. We derive a mass of [9.2 ± 5.8]× 1013 kg for this comet.
• Size and density. Encke has the shortest orbital period among the known comets. It is also a NEC (i.e. a near-Earth comet). As a consequence of its short periodicity, proximity to Earth and orbital stability, this comet has been extensively studied from ground-based observations, and hence has several size determinations, also with different methods. Luu & Jewitt (1990) estimate an effective nucleus radius _R_Neff of 3.28 ± 0.06 km from near-aphelion and time-series CCD photometry. Lowry & Weissman (2007) determine _R_Neff= 3.95 ± 0.06 km by using the same technique. From radar and IR study Harmon & Nolan (2005) estimate _R_Neff= 2.42+0.86−0.06 km. Tancredi et al. (2006) derive _R_Neff= 1.95+1.96−0.67 km from the estimated absolute nuclear magnitude. Since there are for this comet several different estimates of the nucleus size, for which some of them yield unphysical values of ρ when the corresponding errors are considered, we preferred to assume a sort of averaged radius, with a certain error estimate. After analysing the different estimates, we adopted _R_Neff∼ 3 ± 1 km. The corresponding bulk density would be then ρ= 0.8 ± 0.8 g cm−3.
• Other works. Rickman et al. (1987) estimated M_∼[2.4–3.2]× 1013 kg, which is about half the value we obtained. We find that the cause of this discrepancy could be that Rickman et al. considered a non-gravitational Δ_P that was a factor of 2 larger than ours.

6.2.3 6P/d'Arrest

• Mass. We derive a mass of [2.8 ± 0.8]× 1012 kg for this comet.
• Size and density. Tancredi et al. (2006) estimate _R_Neff= 1.7 ± 0.1 km. According to this value for the comet size we derive a bulk density of 0.15 ± 0.05 g cm−3 for this comet.
• Other works. Szutowicz & Rickman (2006) estimate M_= 7.0 × 1012 kg, from non-gravitational modelling. This value is larger than ours by a factor of 2.7. The light curve obtained and the Δ_P considered by these authors are similar to ours, so we looked for differences in the values assumed for the model parameters. Szutowicz & Rickman determine an empirical value for the lag angle η by fitting an asymmetric model to astrometric observations performed between 1851 and 2001. They obtained η∼ 14°, which agrees very well with our assumption. On the other hand, they assume a mean effective outflow velocity 〈_u_〉= 0.6 km s−1, which is larger than the value we assume by a factor of 2.2. This would imply a very collimated flux with a very high momentum transfer efficiency. Since the mass is directly proportional to this parameter, this explains the difference between both mass estimates. Szutowicz and Rickman also used different calibration coefficients: they assumed the value given by Jorda et al. (1992) for the slope coefficient (_a_1=−0.240), and determine _a_2= 30.5 by fitting the observations to measured gas production rates. We conclude that the different values assumed for the mean effective outflow velocity, as well as for the calibration coefficients, account for the discrepancy between both results. In principle, Szutowicz & Rickman's adopted value of 〈_u_〉 seems to be extremely high.

6.2.4 9P/Tempel 1

• Mass. We derive a mass of [2.3 ± 1.6]× 1013 kg for this comet.
• Size and density. A'Hearn et al. (2005) estimate _R_Neff= 3.0 ± 0.1 km from close-up images taken by the Deep Impact fly-by spacecraft. According to this value for the comet size we derive a bulk density of 0.2 ± 0.1 g cm−3 for this comet.
• Other works. Richardson & Melosh (2006) obtain _M_= 5.0 × 1013 kg by studying the expansion of the base of the conical ejecta plume created by the impact of a 370 kg projectile delivered by the Deep Impact spacecraft on the comet nucleus. This is the most direct measurement obtained to date (and independent of the non-gravitational effect, as we would like to emphasize). They obtain a bulk density ρ= 0.4 ± 0.3 g cm−3. Davidsson et al. (2007) obtain _M_= (5.8 ± 1.6) × 1013 kg, and ρ= 0.45 ± 0.25 g cm−3, by utilizing a sophisticated thermophysical model of the non-gravitational forces (which in addition relies on the measured gas production rates instead of the light curve). Our results are consistent with both independent works, which gives support to the usefulness of the modelling method used here.

6.2.5 10P/Tempel 2

• Mass. We derive a mass of [3.5 ± 1.5]× 1014 kg for this comet.
• Size and density. Jewitt & Luu (1989) obtain _R_Neff= 5.3+0.2−0.7 km from time-series CCD photometry. Tancredi et al. (2006) estimate _R_Neff= 4.02+1.49−0.96 km from the estimated absolute nuclear magnitude. From the different estimates we adopted an averaged value _R_Neff∼ 4.8 ± 0.7 km. According to this value for the comet size, we derive a bulk density of 0.7 ± 0.4 g cm−3.
• Other works. Rickman et al. (1991a) obtain _M_= (1.6 ± 0.5) × 1014 kg, from non-gravitational force modelling. This is ∼50 per cent of the value derived in this paper. The main cause of this discrepancy could be that Rickman et al. computed the mass considering only the radial non-gravitational force component, and hence neglecting the contribution from the transverse component (according to our results, the transverse component contributes to ∼50 per cent of the mass estimate). From the rotational light curve Jewitt & Luu (1989) derived parameters a/b and _P_rot (the axial ratio and the rotational period, respectively), and obtained a minimum bulk density ρmin by assuming a negligible tensile strength of the nucleus material. They calculated ρmin= 0.3 g cm−3, which is consistent with our result.

6.2.6 19P/Borrelly

• Mass. We derive a mass of [2.7 ± 2.1]× 1012 kg for this comet.
• Size and density. From Hubble Space Telescope (HST) observations Lamy, Toth & Weaver (1998) obtain semi-axes _a_= 4.4 km and _b_=_c_= 1.8 km by approximating the nucleus to a triaxial ellipsoid, which corresponds to _R_Neff= 2.4 ± 0.2 km. This was later confirmed by direct imaging of the nucleus by Deep Space 1 (Soderblom et al. 2002). According to this value for the comet size we derive a bulk density of 0.05 ± 0.04 g cm−3 for this comet.
• Other works. Davidsson & Gutiérrez (2004) constrain the mass within the range _M_=[8–24]× 1012 kg, and the bulk density in the range ρ=[0.10–0.30] g cm−3, based on a thermophysical modelling of the non-gravitational force. Rickman et al. (1987) derive an upper limit M < 8.4 × 1012 kg, which is consistent with our result. Davidsson (2001) obtains a minimum bulk density ρmin= 0.04 g cm−3 by requiring that the nucleus withstands disruption due to centrifugal forces, assuming a strengthless nucleus material. By comparison with these results, we may conclude that we somewhat underestimated the mass and bulk density of Borrelly. Nevertheless, all the results suggest that we are dealing with an extremely low-density comet. In disagreement with these results, Farnham & Cochran (2002) obtain a mass estimate that is substantially higher: they derive M_= 3.3 × 1013 kg and ρ= 0.49 g cm−3 by following the method introduced by Rickman et al. (1987), but using a different vaporization model for the water production. They observed a strong primary (sunward) jet that was aligned with the nucleus' spin axis (according to their derived pole position). They also assumed a relatively small active area (4 per cent of the nucleus' surface area), close enough to the pole that it would be illuminated almost continuously during the rotation of the nucleus. The combination of these two features (spin orientation and location and size of the primary source of activity) will produce a strong collimation of the jet, making the non-gravitational force much more efficient (e.g. the non-gravitational force would produce the same non-gravitational effect Δ_P acting on a larger nucleus mass, than a less efficient non-gravitational force acting on a smaller nucleus mass). They assumed 〈_u_〉= 0.33 km s−1.

6.2.7 22P/Kopff

• Mass. We derive a mass of [5.3 ± 2.2]× 1012 kg for this comet.
• Size and density. Tancredi et al. (2006) estimate _R_Neff= 1.8 ± 0.2 km from the estimated absolute nuclear magnitude. According to this value for the comet size we derive a bulk density of 0.2 ± 0.1 g cm−3 for this comet.
• Other works. Rickman et al. (1987) derive an upper limit M < 2.9 × 1013 kg, which is consistent with our result.

6.2.8 45P/Honda–Mkros–Pajdusakova

• Mass. We derive a mass of 1.9 × 1011 kg for this comet, with an upper limit of 5.4 × 1011 kg. The lower limit turns out to be negative which is unphysical. Therefore the computed mass of this comet is very uncertain and should be taken with extreme caution. The main reason for this uncertainty is that the radial and transverse terms _I_r, _I_t that intervene in the computation of the mass (cf. equation 5) turn out to be of similar absolute values but different signs (see Table 5), so small errors in _I_r and _I_t can lead to either positive or negative values in (_I_r+_I_t).
• Size and density. Tancredi et al. (2006) estimate _R_Neff= 0.33 ± 0.07 km from the estimated absolute nuclear magnitude. This would be the smallest comet of the sample, and one of the smallest known. According to this value for the comet size we derive a bulk density of 1.2 g cm−3 for this comet. But we must emphasize that the size estimate is very uncertain, according to the referred authors. Because of the large uncertainty (not only in the nuclear size but also in the computed mass) we decided to exclude this comet from the general discussion between masses and densities presented in Section 7.
• Other works. Rickman et al. (1987) derive _M_=[1.5–5.4]× 1011 kg, which is consistent with our result.

6.2.9 46P/Wirtanen

• Mass. We derive a mass of [3.3 ± 2.3]× 1011 kg for this comet.
• Size and density. Tancredi et al. (2006) estimate _R_Neff= 0.58 ± 0.03 km from the estimated absolute nuclear magnitude. According to this value for the comet size we derive a bulk density of 0.4 ± 0.3 g cm−3 for this comet.
• Other works. We have not found another mass estimate for this comet.

6.2.10 67P/Churyumov–Gerasimenko

• Mass. We derive a mass of [1.5 ± 0.6]× 1013 kg for this comet.
• Size and density. From HST observations Lamy et al. (2006) obtain _R_Neff= 1.98 ± 0.02 km. According to this value for the comet size we derive a bulk density of 0.5 ± 0.2 g cm−3 for this comet.
• Other works. Davidsson & Gutiérrez (2005) obtain _M_=[0.35–2.1]× 1013 kg, and the bulk density in the range ρ=[0.10–0.60] g cm−3, based on a sophisticated thermophysical modelling of the non-gravitational force. Our result is consistent with their work.

6.2.11 81P/Wild 2

• Mass. We derive a mass of 8.1 × 1012 kg for this comet, with an upper limit of 2.5 × 1013 kg. The lower limit turns out to be negative which is unphysical. Therefore the computed mass of this comet is very uncertain and should be taken with extreme caution. The main reason for this uncertainty is the same as for comet 45P/H–M–P.
• Size and density. Brownlee et al. (2004) and Duxbury et al. (2004) found that the comet can be approximated by a triaxial ellipsoid with semi-axes {a, b, c}={2.75, 2.00, 1.65}± 0.05 km, from Stardust images. This corresponds to _R_Neff= 2.09+0.05−0.06 km. Howington-Kraus et al. (2005) performed a refined image analysis obtaining {a, b, c}={2.607, 2.002, 1.350}± 0.001 km, which corresponds to _R_Neff= 1.917 km. According to this value for the comet size we derive a bulk density of 0.3 g cm−3 with an upper limit of 0.8 g cm−3.
• Other works. Davidsson & Gutiérrez (2006) determined an upper limit _M_≤ 2.3 × 1013 kg, and ρ≤ 0.60 g cm−3, based on a sophisticated thermophysical modelling of the non-gravitational force. Our result is consistent with their work.

In Table 5 we summarize our results, namely the components _I_r, _I_t, the computed masses and bulk densities, the latter based on the radii found in the literature, also included in the table.

According to the results of Table 5, the range of cometary masses of our sample cover several orders of magnitude, ranging from ∼3–4 × 1014 kg (Halley and Tempel 2) down to ∼2–3 × 1011 kg (H–M–P, Wirtanen), while the densities are almost all of the same order (tenths of g cm−3), and below ∼0.8 g cm−3 with a mean value ≈0.4 g cm−3 (we excluded comet H–M–P due to its larger uncertainty in the size, as compared to the rest of the comets). This result is consistent with the cut-off value ∼0.6 g cm−3 for cometary densities suggested by Lowry & Weissman (2003) and Snodgrass, Lowry & Fitzsimmons (2006). In the case of Borrelly we obtain en extremely low density of 0.05 g cm−3 (very close to the lower limit imposed by the centrifugal forces, according to Davidsson 2001) though the large uncertainty of this estimate admits values more in line with the rest of the comets. Thermophysical modelling provides a higher value for Borrelly's density, though still pointing to a very low-density comet.

6.3 The correlation between the non-gravitational change Δ_P_ and the perihelion asymmetries of the light curves

We can quantitatively evaluate the perihelion asymmetry of the gas production curves by introducing the following definition:

12

where t, f and P are defined as before, and _Q_m is a normalization factor arbitrarily chosen to be the maximum water production rate of the comet. According to this definition, A is measured in days. In Fig. 15 we plot the perihelion asymmetry against the time of the maximum observed brightness as derived from the light curves (we remind the reader that the time is referred to the time of perihelion passage). As it should be expected, we can observe in general a linear relationship between both parameters, which also show very similar values.

Figure 15

Top: Perihelion asymmetry A as function of the time of the brightness maximum T for the comets of the sample. A lineal fit to the computed values is also shown. Bottom: Perihelion asymmetry A as a function of the standard non-gravitational effect Δ_P_′ for the comets of the sample. Those comets with a dominant contribution from the radial force component to the non-gravitational effect are marked with open triangles, and those comets with a dominant transverse force component contribution are marked with filled triangles.

We find that for those comets which show a strong positive perihelion asymmetry (i.e. a notorious post-perihelion excess of gaseous activity), the radial _I_r contribution dominates, with the exception of Borrelly. These are the cases of comets Halley, d'Arrest, Tempel 2 and C–G (actually, in the case of Tempel 2 the radial dominance is very weak). Wirtanen shows the most symmetrical light curve of the sample, with a weak positive perihelion asymmetry. In this case the transverse term _I_t turns out to be the dominant one, as expected. Comets Kopff and Wild 2 show a strong negative perihelion asymmetry (i.e. a notorious pre-perihelion excess of activity) and a dominant transverse contribution. Even though the strong asymmetric light curves cause large radial components in the latter two comets, their transverse contributions are actually large too (a factor of ∼1.4–2 greater than the radial ones). Tempel 1 also shows a negative asymmetry though a very weak one, so the transverse term dominates. Encke shows an intermediate negative perihelion asymmetry, but a slight dominant radial contribution instead.

Festou et al. (1990) and Rickman et al. (1991b) studied the correlation between the perihelion asymmetries of the gas production curves of periodic comets and the non-gravitational perturbations of their orbital periods. In this regard they defined the _standard non-gravitational change_Δ_P_′ as the non-gravitational effect in the orbital period that the comet would experience if its semimajor axis was 3.5 au, which is given by

13

where a is the semimajor axis. Δ_P_′ then represents the non-gravitational effect Δ_P_ corrected for different semi-major axes. Their result was that, in general, the perihelion asymmetries give the dominant contribution to the non-gravitational effect of the orbital period. Actually they found a linear correlation between the perihelion asymmetry and Δ_P_′. This is expected if the _I_r contribution dominates, but not otherwise.

In Fig. 15 we plot the perihelion asymmetry (as defined in equation 12, which is not exactly the same definition given by Festou et al. 1990, but it should be equivalent) as a function of Δ_P_′ for the comets of the sample. According to Festou et al. (1990) and Rickman et al. (1991b), it should be expected an increase of the perihelion asymmetry with the increasing non-gravitational Δ_P_′ (in absolute value). As can be seen in the plot (bottom panel), we did not find such a linear correlation. As shown there, for some comets the transverse component _I_t becomes dominant, which weakens or completely blurs any possible correlation between Δ_P_′ and the radial component _I_r.

7 DENSITIES VERSUS MASSES IN THE SOLAR SYSTEM

We analyse the results obtained for our comet sample placing them into a broader context of different populations of bodies in the Solar system. In Fig. 16 we plot the bulk density as a function of the mass for the different populations of minor bodies in the Solar system as indicated in the figure caption. In the case of the comets we plot the computed masses and densities, indicating the estimated errors for the computed masses and bulk densities as derived in Section 6.2. Fig. 16 shows that the computed comets form a more compact group than the other populations of minor bodies, and also present the lowest densities (ρ≲ 0.8 g cm−3). Our computed comets and the NEAs occupy the left-hand side of the diagram, corresponding to the smaller masses (except one NEA). The main distinction between both populations in this space of parameters is that the NEAs of the sample have larger bulk densities. Only two NEAs show ρ < 1 g cm−3, but still above the upper limit of ∼0.8 g cm−3 derived for the comets. Of course due to the smallness of the samples and the large uncertainties involved in the computation of masses and densities, both populations could actually somewhat overlap. This would be consistent with the idea that some NEAs could be deactivated JF comets (either extinct or dormant). Nevertheless, the clear separation in the density domain shown by comets and NEAs suggests that most NEAs are bona fide asteroids in agreement with Fernández et al. (2002).

Figure 16

Bulk density as a function of the mass for the different populations of minor bodies of the Solar system. For clarity of the figure, we only give error bars for our sample comets. The represented objects are: the 10 comets of our sample (comet 45P/H–M–P is omitted due to its large uncertainty, and for comet 81P/Wild 2 only the nominal values and upper limits are given), 19 objects from the main-belt asteroids (MBAs) population, six objects from the near-Earth asteroids (NEAs) population (Eros, Itokawa, 1999 KW4, 2000 DP107, 2000 UG11, 2002 CE26), one Trojan of Jupiter (Patroclus), one Centaur (Ceto-Phorcys), three trans-Neptunian objects or TNOs (Pluto/Charon, 1999 TC36, 2003 EL61), three terrestrial satellites (Moon, Phobos and Deimos) and 10 Jovian satellites. The mass and density (or size) data for these objects comes from: Yeomans et al. (2000) (433 Eros); Shinsuke et al. (2006) (Itokawa); Merline et al. (2002) (1999 KW4, 2000 DP107, 2000 UG11); Shepard et al. (2006) (2002 CE26); Kovac̆ević & Kuzmanoski (2007) (Ceres); Thomas et al. (2005) (size of Ceres and Pallas); Goffin (2001) (Pallas); Viateau & Rapaport (2001) (Vesta, Phartenope); Michalak (2001) (Hygiea, Eunomia); Lupishko (2006) (Psyche); Marchis et al. (2003) (Kalliope); Marchis et al. (2005b) (Eugenia, Camilla, Elektra, Emma, Huenna); Marchis et al. (2005a) (Sylvia); Marchis et al. (2005c) (Hermione); Veverka et al. (1997) (Mathilde); Descamps et al. (2005) (Antiope); Merline et al. (2002) (Ida, Pulcova); Marchis et al. (2006) (Patroclus); Grundy et al. (2007) (Ceto-Phorcys); Tholen & Buie (1997) (Pluto/Charon); Stansberry et al. (2006) (1999 TC36); Rabinowitz et al. (2006) (2003 EL61); Anderson et al. (2005) (Amalthea) and references in De Pater & Lissauer (2001) (Moon, Phobos, Deimos, Amalthea, Io, Europe, Ganymedes, Calysto, Titan, Titania, Oberon, Triton, Nereida).

We find other objects not catalogued as comets but also with ρ < 1 g cm−3; these are: Patroclus, Amalthea, 1999 TC36, Antiope and Emma (the last two are MBAs). But in these cases the masses are at least about four orders of magnitude greater than the computed comet masses.

We note in Fig. 16 that no bodies with masses _M_≳ 1020 kg have bulk densities ≲1.5 g cm−3. If we assume that comets and other ice-rich bodies of the outer Solar system have similar compositions consisting of dust–ice mixtures, then the different densities would suggest that compaction of the material by self-gravity has taken place, so fluffy objects should not be expected for masses bigger than about 1020 kg. We also have in the figure rocky bodies (NEAs and main-belt asteroids) whose differences in density may be due in this case to the different chemical composition, though different degrees of porosity might also be allowed. This is the case of most main-belt asteroids, TNOs, and, of course, the Moon and the largest Jovian satellites. Therefore, self-gravity would lead to noticeable variations in the primordial physical structure, ranging from loose aggregates to compacted and hardened material, and hence to a wide range of densities. In the case of the studied comets, due to the ranges of masses and sizes involved, we do no expect them to increase their densities by self gravity. Hence comets would be fluffy, low-density objects which have preserved their primordial fragile and porous structure.

8 CONCLUDING REMARKS

From non-gravitational force modelling we derive masses and densities for 10 short-period comets of known sizes: 1P/Halley, 2P/Encke, 6P/d'Arrest, 9P/Tempel 1, 10P/Tempel 2, 19P/Borrelly, 22P/Kopff, 46P/Wirtanen, 67P/Churyumov–Gerasimenko and 81P/Wild 2. For another comet, 45P/H–M–P, it was not possible to compute a reliable mass and density. Our procedure follows the pioneer work of Rickman and colleagues (e.g. Rickman 1986, 1989; Rickman et al. 1987), and confirms the importance of the knowledge of non-gravitational effects on the comet's motion, as a fundamental tool to derive cometary masses. This paper is based on light-curve data, the non-gravitational term _A_2, and different assumptions for some physical parameters. Our main conclusions can be outlined as follows.

• The derived cometary masses cover more than three orders of magnitude, within the range ∼[0.3–400]× 1012 kg. Our best estimate for the bulk densities are below 0.8 g cm−3, which is in agreement with models of the comet nucleus depicting it as a very low-density object. The mean bulk density of our sample is 0.4 g cm−3.
• The results obtained are in general consistent with other works. Particularly, we remark the consistency with results from sophisticated thermophysical non-gravitational force modelling, and from the Deep Impact team using a different method for Tempel 1. In particular, this agreement gives us some reassurance of the usefulness of the non-gravitational method used here for deriving cometary masses, despite all the uncertainties inherent to it.
• The method also relies on the empirical calibration between visual magnitudes and water production rates, which, in spite of its poor physical understanding, it has proved to be a very useful tool. Since there is an ample and comprehensive data base of total magnitudes, the method could be extended to many more periodic comets, for which the non-gravitational change Δ_P_ has been computed. If we also know comet sizes (whose knowledge is also rapidly improving) we can derive the comet's bulk density. Even though the computed bulk density for a given comet has a large uncertainty, by computing them for a large sample of comets we could narrow down the uncertainty of the whole sample to meaningful values.
• In connection with the previous points, we want to stress that our results are meaningful from a statistical sense, by considering an average value of a large sample of comets. Of course, this later statement is true if there are not systematic errors in the method by overestimating or underestimating any of the parameters used in the computation of the masses. Even tough this possibility cannot be ruled out, we have been very careful in choosing the values of the input parameters within physically realistic ranges. We should warn, however, that, given the large errors bars, we cannot tell at this point whether there is a real dispersion in the bulk densities of comets reflecting different degrees of compaction of the material and/or differences in the chemical composition.
• We conclude that a strong perihelion asymmetry of the light curve should not be necessarily related to a negligible transverse force component contribution to the non-gravitational change in the orbital period. Actually, in some cases (like comets Kopff and Wild 2), the transverse component is found to be more important than the radial contribution.
• Limitations of the method. The mass solutions from equation (5) are quite robust to small changes in the input parameters except when the terms _I_r and _I_t turn out to be of similar absolute value but different signs (as in the cases of 45P/H–M–P and 81P/Wild 2).
• Sources of uncertainty. The light-curve uncertainty and the values assumed for the model parameters, particularly for the effective outflow velocity 〈u_〉, would be in general the most important contribution to the uncertainty in the computed masses, although in some cases the non-gravitational change Δ_P could become a major source of uncertainty too. An underestimate or overestimate of, e.g. 〈_u_〉, would lead to a systematic underestimate or overestimate of the computed mass. A more detailed study of this parameter, like its dependence on the heliocentric distance, has to wait for more elaborated results from thermophysical models as well as more observations, particularly in situ.

1

The evaluation of non-gravitational forces is more difficult for long-period comets, since these have not been observed in a second apparition to check for a change in the orbital period. Nevertheless, non-gravitational terms have been fitted to the equations of motion of several LP comets leading to more satisfactory orbital solutions.

2

The effective nuclear radius defines the radius of a sphere whose volume is equal to that of the comet nucleus (usually modelled as a triaxial ellipsoid of semiaxes a, b, c).

We thank Pedro J. Gutiérrez for helpful discussions on this work and for providing data on gas production rates, and the referee for valuable comments and criticisms that helped to improve the presentation of the results. We also thank Daniel Green for providing data on electronic form from the ICQ. AS acknowledges financial support from the Proyecto de Desarrollo de las Ciencias Básicas (PEDECIBA) programme to her MSc thesis performed at Universidad de la República of Uruguay. The present work has been performed as the main part of her thesis work.

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