%\baselineskip 14pt \baselineskip 22pt \hoffset 25 true mm \hsize 130 true mm \font\bfS=cmbx10 scaled\magstep1 \def\lta{~\raise.4ex\hbox{$<$}\llap{\lower.6ex\hbox{$\sim$}}~} \def\gta{~\raise.4ex\hbox{$>$}\llap{\lower.6ex\hbox{$\sim$}}~} \rm \topskip 2cm \centerline{\bfS PHYSICAL AND DYNAMICAL EVOLUTION OF } \centerline{\bfS JUPITER FAMILY COMETS:} \medskip \centerline{\bfS SIMULATIONS BASED ON THE OBSERVED SAMPLE} \vskip 2 true cm \centerline {\bf G. Tancredi} \bigskip \centerline {Astronomical Observatory, Box 515, {\sl S}-75120, Uppsala, Sweden} \bigskip \centerline {Depto. Astronom\'{\i}a, Fac. Ciencias, Trist\'an Narvaja 1674, Casilla Correo 10773, Montevideo 11200, Uruguay} \centerline {E-mail : gonzalo@fisica.edu.uy \hskip 5mm Tel : (+598 2) 41 80 05 \hskip 5 mm Fax : (+598 2) 40 99 73} \vskip 3cm \noindent Address for proofs: Depto. Astronom\'{\i}a Fac. Ciencias Trist\'an Narvaja 1674 Casilla Correo 10773 Montevideo 11200 Uruguay \vfill\eject %\vskip 2 true cm \noindent{\sl ABSTRACT.}\quad We describe a method to simulate the combined physical and dynamical evolution of Jupiter-family comets. We integrate all the observed Jupiter family comets and 3 variant orbits until they leave the family. Regarding the physical modelling, the parameters are the nuclear mass (or radius) and the fraction of the surface area covered by dust mantles formed during previous orbital evolution. This picture is motivated by recent findings concerning the coupled orbital and physical evolution of cometary nuclei. We present results concerning the lifetime of comets, both in the active and dormant phase, and snapshot distributions of the number of comets vs. perihelion distance. \vfill\eject \noindent{\bf 1. Introduction} \medskip\noindent The origin and evolution of the Jupiter family (JF) has wide-ranging implications. The comets evolve dynamically by frequent close encounters with Jupiter that ultimately return the objects to the Jupiter-Saturn region or eject them from the solar system on hyperbolic orbits. Since the lifetime for definitive loss of the objects is estimated to be $\sim 10^6\,$yr, much shorter than the age of the solar system, either we assume that the Jupiter family is a temporary phenomenon or the canonical picture that the Jupiter family is a steady-state structure maintained close to its present form by a more or less steady influx of comets from a rich and enduring source. The current interest in capture scenarios stems largely from the view that the Jupiter family is a probe of hitherto unobserved cometary reservoirs in the outer solar system (Bailey 1990a). Revealing the size and structure of these reservoirs has important cosmogonical implications. Comets are objects with a complex evolutionary process where physics and dynamics are closely interrelated. As a consequence of a possibly complicated aging process, involving periods of dormancy followed by reactivation in response to orbital perturbations, the steady state picture of the JF population can not be modelled simply by considering independent estimates of physical and dynamical lifetimes -- we should rather try to simulate the combined process. This is the purpose of our investigation, and as a spin-off we expect also to find the orbital distribution of extinct and dormant comets. The latter has important implications for estimating the cometary contribution to the population of near-Earth asteroids (Weissman {\it et al.} 1989). This paper is a continuation of a previous study (Rickman {\it et al.} 1992, hereafter Paper I), where the problem was analysed using Monte Carlo simulations based on jump probabilities between different perihelion distance bins as a stochastic dynamical model. The present approach mainly differs in the modelling of the orbital evolution, as explained in Section~2. The physical modelling described in Section~3 is largely maintained unchanged. The results are presented in Section~4 and discussed in Section~5, where we also preview further developments of the modelling procedure. \vfill\eject %\vskip 2 true cm \noindent{\bf 2. Dynamical modelling} \medskip\noindent The dynamics of JF comets is basically chaotic due to frequent close encounters with Jupiter (Froeschl\'e 1990). Thus, although regular patterns may occur during limited or even extended periods of time, the long-term evolution of most orbital parameters is ultimately unpredictable. Several authors (including Paper I) tried to describe the dynamical evolution by stochastic modelling, {\sl i.e.}, the orbital perturbation experienced over a certain time interval was picked at random according to a prescribed probability distribution. The increase in computing power in the last few years has made a more realistic approach possible, since nowadays we can accurately compute the orbital evolution of hundreds of objects moving in cometary-type orbits for several $10^4$ years in a few hours of CPU time. We thus decided to create a database of the dynamical evolution of ``comets'' (objects in cometary-type orbits) by numerically integrating the orbits of observed JF comets and variants of these orbits. As it was recently proposed (Valsecchi 1992), we will consider a comet to formally belong to the JF if the orbital period $P<20$ yrs, the perihelion distance $q<5$ AU and the Tisserand parameter $2 < T \lta 3$ (in addition excluding P/Encke whose motion is decoupled from Jupiter). From the Catalogue of Cometary Orbits, 7th Edition (Marsden and Williams 1992) we extract all the comets that fulfil the previous criteria, {\sl i.e.}, 145 objects. We create four sets of initial orbits: the actual orbits plus three different variants constructed as follows. We divide the orbital elements into two groups: semimajor axis ($a$), eccentricity ($e$) and inclination ($i$) on one side; and argument of perihelion ($\omega$), longitude of the ascending node ($\Omega$) and mean anomaly at the epoch ($M_o$) on the other side. A new set of 145 fictitious objects is created by combining element triplets randomly picked from each group. This procedure was repeated three times, checking that we did not get the same combination of six elements in different sets of 145 objects. By this procedure, the distributions of the above orbital elements are unaltered and even combinations of elements within each group ({\sl e.g.} perihelion and aphelion distance, Tisserand parameter and approximately the longitude of perihelion $\tilde\omega$ ) , but the distributions of elements representing combinations of the two groups ({\sl e.g.}, $e \cos \tilde\omega$, $\sin i \cos \Omega$) may of course change. Considering that Jupiter is the governing body in shaping the Jupiter family and its orbital eccentricity is very small, exchanging the angular orbital elements ($\omega, \Omega$ and $M_o$) of the observed orbits to create a new set of variant orbits represents an attempt to increase the number of simulated objects while keeping the dynamical characteristics of the observed population of comets. The numerical integrations were performed with the 15th order RADAU integrator (Everhart 1985) and a model of the solar system with only the Sun, Jupiter and Saturn was assumed. The integrations started at present and they were performed backward and forward in time. The output time step was 10 yr. The evolution of a comet was followed until it leaves the JF, {\sl i.e.}, the comet enters into an orbit with $q > 5$ AU and/or $P>$ 20 yr for more than 100 yr. There were a small number of objects that, during the $\pm$ 75000 yr spanned by our integrations, are not ejected from the JF either in the past or in the future ( an ``escape'' in the backward integrations would be better termed a capture into the family). The proportion of non-escaping objects varies between the sets of 145 objects; for the ``real'' objects we only get two cases (P/Tempel 2 and P/Pigott), while for the fictitious sets the numbers go from 20 to 25. We can distinguish three types of cases: i) high-inclination objects ($i \gta 35^o$); ii) objects with aphelion distances less than Jupiter's perihelion distance and lines of apsides not aligned with Jupiter; and iii) objects trapped in mean motion resonances with Jupiter. In all cases $a$ remains almost unchanged while $q$ oscillates between fixed values. Since we only found two non-escaping objects in the ``real'' sample and we want to start the simulations of the physico-dynamical evolution with the comets outside the JF, we decided to disregard all the non-escaping objects. Anyway, as comets moving in this kind of stable orbits do exist (and P/Encke should be added to this list), it is a challenge for any evolutionary theory for short-period comets to explain how they could still be active. We will return to this point in the discussion. There were also around 5 to 10 objects in each set that escape only in one direction of time but remain in the family after 75000 yr in the other direction. Also the number of objects of this type was less in the ``real'' sample than in the others. We decided to disregard these too for the same reasons stated above. From the original 4$\times$145 objects, after applying the previous selection criteria, we end up with 467 different dynamical evolutions, and 135 of them correspond to ``real'' comets. Figure 1 shows a few examples of the dynamical evolutions of ``real'' comets (their names are shown in the plots). The upper panel shows an object for which the orbit is almost unchanged during a long period of time, locked in the 3/7 resonance with Jupiter, and suddenly the trapping mechanism breaks and the object is rapidly ejected from the JF. The middle panel represents a comet that is captured from the Jupiter-Saturn region into the JF. It then starts wandering in the $a$--$q$ parametric plane, reaching a minimum perihelion distance ($q_m$) and then finding its way back to the Jupiter-Saturn region. For this case, the complete visit into the JF takes several ten thousand years. The third example corresponds to a very short visit into the JF (a few thousand years). The comet experiences an encounter with Jupiter that dramatically reduces its $q$ by several AU. The object is put into an orbit near the 4/5 commensurability with Jupiter's period. Subsequent encounters are very frequent until a new encounter ejects the object from the family. Due to the commensurability with Jupiter, when we take the $a$, $e$ and $i$ of this kind of object and use the angles of another, this is usually what causes objects to be locked into resonances for the whole span of our integrations. The very long-term trapping mechanisms are not seen for the real JF comets but only for our fictitious objects -- at least the observed sample of real comets tends to be more chaotic on a short time scale than objects randomly created with the same orbital distribution. From the dynamical point of view, we classify the evolutions with regard to the difference in $q$ ($\Delta q$) between two consecutive output time steps. If $|\Delta q| < 0.5 $ AU, we refer to a smooth evolution; if $|\Delta q| > 0.5 $ AU, then we have a jump. %\vskip 13 true cm %\noindent {\bf Fig. 1.--} Dynamical evolutions of comets P/Neujmin 2, P/Shoemaker 2 and P/Kearns-Kwee as computed from the numerical integrations explained in the text. The dashed horizontal lines at 7.37 AU mark the corresponding $a$ for a period of 20 yr. \vfill\eject \noindent{\bf 3. Modelling of physical evolution} \medskip\noindent The physical evolution is modelled in deterministic terms. The nucleus is taken to be a homogeneous sphere, whose surface is partly fresh and outgassing, and partly covered by dust. We distinguish between different dust mantles according to the $q$-ranges in which they form. Following calculations of dust mantling by Rickman {\it et al.\/} (1990) and the statistics of nongravitational effects vs.\ previous orbital stability by Rickman {\it et al.\/} (1991), we assume that: \parindent 10pt \item{$\bullet$} during the time a comet smoothly changes its $q$ its previously mantled areas remain unaffected, but the fresh area gradually becomes mantled; \item{$\bullet$} when a jump occurs into an orbit with a lower $q$, some of the previously formed mantles are removed and the corresponding area returns to the fresh ({\sl i.e.\/} `active') state; \item{$\bullet$} an upward jump in $q$ has no effect on previously formed mantles. \parindent 20pt \noindent In addition, outgassing from the active surface leads to mass loss, presumably producing holes resembling craters ({\sl cf.\/} P/Halley) or leading to changes in the gross shape of the nucleus. Here we neglect such complications and assume that the nuclei remain spherical, gradually shrinking in size whilst possibly becoming more completely covered by dust. Physical processing may also bring the comets to an end state. If dust mantling proceeds to a stage where only a tiny fraction of the surface is active, we regard the object as no longer observable as an active comet and transfer it to the category of dormant comets. We choose a fraction less than 0.0001 of the surface area as the limit for dormancy, since no active comet has been detected as active with a fraction smaller than 0.01 \% (Rickman 1991) and the upper limits to possible activity in Near Earth Asteroids lie in the same range (Luu and Jewitt 1992). If the fraction of free sublimating area is greater than 0.0001, the comet could still be inactive at some points of the orbit because {\sl e.g.} the active spots are not insolated, but changing the insolation pattern would produce detectable activity. These comets are thus considered as active. The dormant phase is a temporary end state, from which the comet may be revived were it to perform a downward jump in $q$ such that part of the dust mantle were blown off. In this way comets may undergo one or more transitions between the active and dormant states. If dynamical ejection occurs when a comet is dormant, then obviously dust mantling represents a definitive end state, which we call the asteroidal one. If mass loss proceeds far enough for the nucleus to disappear (shrinkage to zero mass and radius), we refer to this as the meteoroidal end state, since then the comet would be physically dispersed into a meteor stream. Finally, if dynamical ejection occurs while a comet is still active, we refer to the cometary end state. The rule for reactivation and the disappearance by shrinking the nucleus to zero are obviously oversimplications of much more complex processes. It was argued by Kres\'ak (1987) that several non detection of otherwise bright passages of short period comets could be explained by periods of dormancy and later reactivations. Some of these examples were not correlated with downward jumps; other processes, possibly depending on the specific characteristics of the objetcs, are requird to explain the reactivation. Outbursts that temporary enhance the mass loos as well as splittings that accelerate the disappearance are not included in the model. Although all these processes can be modeled by assigning {\sl e.g.} a probabilitity function for such events, the poor knowledge and small number statistics of these phenomena prevent any reliable modelling. Denoting the mantling rate by $f'(q)$, the fraction $f_a$ of the surface which remains active evolves according to $${df_a\over dt}=-f'(q) f_a \eqno(1)$$ The mantling rate function $f'(q)$ is a model parameter. We define it as a function of the initial nuclear radius and keep it constant throughout the simulation in spite of possible shrinkage of the nucleus. Considering that the mantles formed in different $q$-ranges have different stability properties (Rickman {\it et al.} 1990), we classify the mantles with respect to $q$. The range of perihelion distance of JF comets (0~to 5~AU) is divided into 10 bins of equal width (0.5~AU). We denote each $q$-bin by an index $i$ ($i=1,\dots ,10$). Calling $f_i$ the fraction of the surface area covered by the dust mantle formed in the $i$th $q$-bin, the state of the nuclear surface is then described by the vector ${\bf f} =\bigl\{f_i;i=1,\dots ,10\bigr\}$, and the active fraction obeys the condition: $f_a=1-\sum_1^{10} f_i$. When a downward jump occurs into the $j$th $q$-bin, we assume that specific fractions $b_1, b_2, \dots$ of the mantles previously formed in $q$-bins $j\!+\!1$, $j\!+\!2, \dots$ are blown off (increasing bin numbers correspond to increasing $q$). Note that these fractions do not depend on $j$ but only on the number of bins between the current one and that of mantle formation. As a result of mantle blow-off, the mantled fractions change by $$\Delta f_k=-f_k b_{k-j}\; ;\quad k=j+1,\dots ,10\eqno(2)$$ and the active fraction increases by $$\Delta f_a=\sum_{k=j+1}^{10}f_k b_{k-j}\eqno(3)$$ Mass loss is modelled by considering a quantity $F(q)$ describing the gaseous mass loss per unit surface area in the active state, per unit time. $F(q)$ is an average of the gas flux per revolution (see below for further description). The mass $M$ evolves according to $${d M \over dt} = - 4 \pi R^2 (1+r_{dg}) F(q) f_a \eqno(4)$$ Here $R$ is the radius of the nucleus [$R=(3\ M/(4\ \pi \rho)^{1/3}$], $\rho$ is the density of the nucleus, and $r_{dg}$ is the dust/gas ratio of the outflowing material, such that allowance is made for the non-volatile mass loss as well as the gaseous one. Both $\rho$ and $r_{dg}$ are constant model parameters. Calling $M_0$ and $R_0$ the initial mass and radius, we define a mass fraction $m=M/M_0$. As we mentioned before, comets are classified as active or dormant depending on the fraction of mantled area. If the comet is active, however, this does not immediately imply that it would be detectable as an active object. We estimate the chances of detection of a comet by first computing its total magnitude at perihelion. Given the radius $R$, active fraction $f_a$ and gas flux at perihelion $F_q$, the total gas production rate $Q_w$ is given by $$Q_w = 4 \pi R^2 f_a F_q \eqno(5)$$ The gas production rate can be transformed into an heliocentric total magnitude with the aid of an empirical formula like the one proposed by Jorda {\it et al.} (1992), {\sl i.e.} $$m_h=128.1-4.17 \log Q_w \eqno(6)$$ The absolute total magnitude at perihelion is calculated from $$H_T=m_h - \alpha \log q \eqno(7)$$ where $\alpha$ is generally assumed to be 10. It has been known for a long time that cometary total magnitudes depart from the $10 \log r_h$ law at far heliocentric distances $r_h$. Analysing the lightcurves of P/Halley and other well observed short-period comets, as presented by Kam\'el (1992), we tentatively assume $$\alpha = \cases{ 10 & if q $\leq$ 2.5 AU \cr 15 & if q $>$ 2.5 AU \cr} \eqno(8)$$ The possibility that the nuclear magnitude could be brighter than the total magnitude for low-active comets is also considered but does not introduce significant changes in the results. Fern\'andez {\it et al.} (1992) found that the detection limit in absolute total magnitude can be approximately given by $$H_T^\ell = 15.2 - 3.0 q \eqno(9)$$ If the object at perihelion is brighter than $H_T^\ell$, we deem the object as detectable. \bigskip \noindent{\it Physical Parameters} \medskip \noindent The mantling rate function $f'(q)$ is taken from simulations of the dust mantling process by Rickman {\it et al.} (1990). The percent of stable mantle formed after 8 revolutions as a function of perihelion distance (presented in Fig. 7 of their paper) is then transformed into a mantling rate per revolution and plotted in Fig. 2. Two values of the nuclear radius were considered in their simulations: 1 and 5 km. For intermediate radii we use a linear interpolation between these two curves. Although the physics of the problem is too complex to allow an easy justification, we consider that the criterion for mantle stability ({\sl i.e.}, the core-mantle interface saturation pressure does not overcome the weight of the mantle per unit area) involves a pressure threshold directly proportional to $R$. Since there are several uncertain input parameters in Rickman {\it et al.}'s model ({\sl e.g.} spin orientation and rotation period of the nuclei, maximum grain radii, etc.), we consider possible variations of the mantling rates up to $\pm 50 \%$ of the values plotted in Fig. 2. This range of variations covers the expected outcomes for reasonable changes in the input parameters. The blow-off vector ${\bf b}=\bigl\{b_\ell ; \ell=1,\dots, 9 \bigr\}$ is a second set of model parameters, chosen independently of nuclear size among several alternatives intended to span the range that we consider reasonable. As in Paper I, we consider three different blow-off models, for which values of the vector {\bf b} are shown in Fig. 3. A common feature is that $b_\ell$ increases with $\ell$ and that the values chosen are always substantial ($b_\ell\gta0.5$) for $\ell>3$. The water fluxes are taken from the results of thermal models (see Froeschl\'e and Rickman 1986) for a rotating nucleus with its spin axis normal to the orbital plane. We have concentrated on latitudes not very far from the subsolar ({\sl i.e.} equatorial) one, arguing that a typical cometary nucleus might have its active surface concentrated around those latitudes which are subsolar near perihelion. The water flux at perihelion $F_q(q)$ is presented in Fig 4, by a dotted curve. Note that due to the particular spin configuration, no seasonal effects occur and the water flux is only a function of the heliocentric distance. To compute the average flux $F(q)$ over a revolution, we assume a fixed aphelion distance of 6 AU (typical of JF comets) and we integrate the flux from perihelion to aphelion. The results are plotted using a full-drawn curve in Fig 4. %\vfill\eject %. \vfill\eject \noindent{\bf 4. The combined evolution} \medskip \noindent For each of the 467 objects we compute the combined physico-dynamical evolution corresponding to three different initial radii: $R_o=1.5$, 3 and 5 km. We first analyse the results using the intermediate blow-off model (curve 2 in Fig. 3), and later we will discuss the differences with the others. We make parallel computations of the results to be presented in the following plots using the 467 simulated objects and the 135 ``real'' objects alone. Unless it is specifically said, no significant and/or systematic differences were found except for the better quality of the statistics in the larger sample. The simulations start with the comet outside the JF and a completely free-sublimating area ($f_a=1$). At intervals of the output time-step of the dynamical integrations (10 yr) we record a state vector ($q$, $m$, $f_a$). For each object, the simulation proceeds until the object is either dynamically ejected from the family or dispersed into a meteor stream. Out of the 467 objects we only get 33, 3 and 0 meteoroidal end states for the 1.5, 3 and 5 km radii, respectively. We first analyse the general characteristics of the evolution. Figure 5 presents a histogram of the distribution of comets as a function of the overall minimum perihelion distance $q_m$. The peak at $\sim 1$ AU reflects the observational bias of our sampled population, {\sl i.e.} most of the observed comets have $q < 2$ AU and they are generally at present near the minimum perihelion distance of their evolution (Tancredi and Rickman 1992). Our results will thus be biassed by this fact, and the conclusions to be drawn mainly correspond to a sample of the JF that presents similar characteristics to the observed sample -- not necessarily to the total population. %\vskip 75 true mm %\noindent {\bf Fig. 5.--} Histogram of the distribution of overall minimum perihelion distance of the 467 objects attained during the dynamical integrations. %\vskip 5 true mm The distributions of a) total time span in the JF (a ``visit'') ({\it Total Lifetime}), b) total time in the active phase ({\it Active Lifetime}), c) total time in the dormant phase ({\it Dormant Lifetime}), and d) the interval until the last time the comet was active (several periods of dormancy could have occurred in between) ({\it End of Activity Lifetime}) are shown in Fig. 6 for the 3 km radius. The medians of the distributions are shown for each plot. Since these lifetimes have important variations depending on the overall minimum $q$, we classify the lifetimes according to $q_m$ and compute the median of the distribution in each $q_m$-bin (Fig. 7). %\vfill\eject %\topskip 100 true mm %\noindent {\bf Fig. 6.--} Histogram of the distributions of lifetimes. a) total time span in the JF (a ``visit''), b) total time in the active phase, c) total time in the dormant phase and d) the interval until the last time the comet was active. The $y$-axis corresponds to fractions of the total population. The medians for each distribution expressed in years are shown in the corresponding plot. %\vskip 105 true mm %\noindent {\bf Fig. 7.--} The lifetimes presented in Fig. 6 as a function of $q$. The medians for each distribution are shown. %\vfill\eject %\topskip=0pt The total lifetime can now be compared with the dynamical lifetime estimated by other authors ({\sl e.g.} Lindgren 1992), since we have very few cases of meteoroidal end state. A preliminary correction for differences in the $q_m$ distributions of the studied populations and the lifetime definitions, leads us to conclude that our estimated lifetimes are a factor less than two longer than Lindgren's estimates. This difference is mainly due to the fact that Lindgren's objects on the average have Tisserand parameters ($T$) closer to 3 than our ``real'' objects. Generally objects with $T \lta 3$ are more dynamically unstable than lower-$T$ objects and hence their lifetimes are shorter. The total time the comet is active is only $\sim 1/5$ of the total duration of a visit. The time until the comet ends its activity is slightly longer than half the total lifetime (the latter is, typically, the time it takes to reach the overall minimum perihelion distance). The comets thus tend to be active while still decreasing their $q$, but rapidly become dormant after reaching the overall minima. A comparison with the results for 1.5 and 5 km radii shows that the median of the total lifetime is slightly shorter for the 1.5 km case due to the greater contribution of the meteoroidal end state, but no significant difference is seen for the 5 km case. The active lifetime is anticorrelated with the radius (median of 3540 yr for 1.5 km and 1700 for 5 km), because the mantling rate increases with nuclear size. The time until the end of activity shows no significant differences among the three models. Table I and Fig 7 compare the results for the median of the lifetimes computed for the whole data set and the 135 ``real'' objects alone. We find that the lifetimes are 10 to 20 \% shorter for them, with the exception of the active lifetime where the values for the ``real'' sample are slightly higher (less than 10\% in all cases). The difference is due to a higher contribution of short total lifetimes in the ``real'' population, which have evolutions with frequent jumps . This confirms what we stated above: the observed sample seems to be slightly more chaotic on a short time scale compared to the fictitious objects. %\vskip 95 true mm %\noindent {\bf Fig. 8.--} The steady state distribution of comets with $q$ for the pure dynamical simulations. The full lines are the mean $values of the distributions and the dotted lines show the $\pm 3 \sigma$ errors. The dashed line corresponds to the distributions of comets with $q$ at the beguinning of the integrations ({\sl i.e.} the present distribution of observable JF comets). %\vskip 5 true mm Another piece of information extractable from our data is a steady state picture of the population. We randomly pick a state vector for each object and construct the distribution of the desired parameter using the 467 samples (or 135 samples for the ``real'' population). The same procedure is repeated 150 times to compute the distribution of the distributions, in particular the mean over all the distributions and the variance corresponding to this mean. Behind these ideas we assume that given a fictitious parent population, our 467 alternative evolutions are random samples of all possible evolutions. %\vskip 15 true cm %\noindent {\bf Fig. 9.--} The steady state distributions of comets with $q$ for the three radii and for the combined population with a mass index $s=0.7$. From top to bottom, the curves correspond to total number of comets (both active and dormant), number of active comets and number of detectable ones. %The full lines are the mean values of the distributions and the dotted lines show the $\pm 3 \sigma$ errors. On top of each plot the ratio between total number of active comets over total number of comets %({\tt act/tot}) and the ratio between total number of detectable over total number of active comets ({\tt det/act}) are presented. %\vskip 5 true mm To understand the combined physical and dynamical evolution, we first analyse the pure dynamical evolution of our population of objects, by running independent simulations where the physical modelling is completely neglected. In Fig. 8 we present a steady state distribution of comets over perihelion distance (full line). Although the $q$-distribution has evolved from the initial one (dotted line in the same plot) in that we get a larger number of comets with high $q$, we still see a peak at relative low $q$. It seems that for the observed sample of ``real'' comets and our variants the ways into and out of the family are much quicker than the intervening residence in low-$q$ orbits. This reinforces the point that our results would correspond to an observed-like sample of the JF and not to the total population. In Fig. 9 we present the $q$-distribution of the number of objects for the three initial radii: 1.5, 3 and 5 km. From top to bottom, the curves correspond to total number of comets (both active and dormant), number of active comets and number of detectable ones. The full lines are the mean values of the distributions and the dotted lines show the $\pm 3 \sigma$ errors. We combine the populations using a mass power law distribution with index 0.7, since a value within the range 0.5--0.75 was suggested by Bailey (1990b). For each $q$-bin and initial radius, we use the corresponding mean mass when computing the weight factor for its contribution to the total population. The overall shape of the total number of comets shows no changes with radius and corresponds to the pure dynamical case, but the contributions of active and detectable comets present remarkable differences. The ratio between active and total numbers is larger for the smallest radius and the difference is most pronounced for low-$q$ orbits. The mantling rate for larger comets is higher (see Fig. 2) and thus we get a higher number of dormant objects. While for the 1.5 km radius, the peak in the distribution of active comets also lies in the 1--2 AU range, for 5 km the peak is shifted towards higher $q$ values. For $q\gta1.5$ AU most of the active comets are not detectable with the present techniques. The combined population pretty much resembles the 1.5 km case. The shapes of the distributions for the sample of 135 ``real'' comets show a similar appearance to the corresponding distributions for the whole sample, but there are noticeable differences in the absolute values and the relative positions of the curves. Table II compares the ratio {\tt act/tot} and {\tt det/act} for the different radii and the combined population. As we would expect from the comparison of the lifetimes, we obtain a slightly higher ratio of number of active to total number of comets for the ``real'' population. The relative differences are still in the 10-20\% level of the total population values. %\vfill\eject Figure 10 shows the fraction of active area $f_a$ of active comets for the three radii as a function of $q$. The full lines correspond to the mean over the means of all 150 distributions and the dotted parallel lines to the $\pm 3 \sigma$ errors (the errors do not express the spread between individual values for the comets in a certain $q$-range, but the variance respect to the mean over the 150 distributions). Small comets are highly active at low $q$ ($f_a \sim 0.3$ at $q\lta 1$ AU) since at that point they reach their minimum $q$ by jumping from higher $q$-ranges and partially blowing off their mantles. As the mantling rate in low-$q$ orbits for small radii is negligible (see Fig. 2), the newly free sublimating area is maintained during the stay at low $q$. For intermediate and large nuclei $f_a$ is generally smaller than 10 \% for $q$ between 1 and 4 AU. All the models show a minimum in the fraction of active area at $q\sim 1.5$ AU. We mentioned before that comets are generally active on their way to the overall minimum $q$, and become definitively dormant on the way out. Since most of our objects have $q_m$ in the range 1 to 2 AU, the fraction of active area at large $q$ is computed mainly from the objects on their way in, and the decreasing trend in $f_a$ from 5 to 1 AU reflects the increasing mantle of the comets on their way to $q_m$. After reaching $q_m$, they soon become completely mantled and generally do not contribute anymore to the active sample. The mass fractions $m$ are presented in Fig. 11. Full lines correspond to the mean mass of the comets in the active state and dashed lines to the dormant phase. Lines at $\pm 3 \sigma$ are drawn parallel to the previous ones (the same significance of these errors holds for $m$ as for $f_a$). Dormant comets are more massive than active ones at low $q$ but the opposite occurs at high $q$. If the comet is active at small $q$, the mass is being greatly reduced; after passing the overall minimum $q_m$, the comet becomes dormant and initiates its retreat from the family, visiting high-$q$ regions with a mass much smaller than the active comets that are yet on their way in. %\vfill\eject From the absolute total magnitudes computed by eqs. (5--8), we have constructed cumulative distributions, and the results for the three radii are presented in Fig. 12 (a--c). The full lines correspond to the cumulative distribution of the whole active population while dashed lines are for the sample of comets with $q < 2 $ AU. A straight line corresponding to a slope of 0.4 is shown for comparison. The index 0.4 was found by Fern\'andez {\it et al.} (1992) to fit the observed cumulative magnitude distribution of comets with $q< 1.5$ AU. The three cumulative distributions show a rapid increase at bright magnitudes, which slows down in different magnitude ranges. If the comets would not change their sizes or their fractions of active area, the distributions would have rectangular shapes. The departure from this shape is a measure of the scatter in $f_a$ and $m$ for the different radii. The distributions of the combined populations with a wide range of mass power law indices (0.2, 0.7 and 1.2) are shown in Fig. 12 (d), (e) and (f), respectively. The small bumps in the dashed curves of Figs. 12 (e) and (f) at $H_T \sim 7$ correspond to the rapid drop-off of the $R_o=1.5$ km distribution and are artifacts due to our discrete number of initial radii. Although we could claim that the best agreement between the 0.4 slope and the distribution of the combined population with $q<2$ AU corresponds to the mass index $s=0.7$, there seems to be a wide range of mass indices that could possibly fit the observed distribution as well. It seems that the observed distribution of total absolute magnitude largely echoes the size distribution of the observed population, but the effect of the spread in the fraction of active area has a non-negligible influence on the combined magnitude distribution involving different radii. The estimates of the mass power law index from the total magnitude distribution thus have to be taken with caution, especially if a very narrow range of mass index is given. \bigskip \noindent{\it Variation of physical parameters} \medskip \noindent Two set of variations of the input physical parameters are considered: i) the mantling rate $f'(q)$ is changed by $\pm$ 50\% of the values presented in Fig. 2; ii) the fraction of mantle that blow-off as a consequence of a downward jump is varied according to Model 1 and 3 in Fig. 3. If the perihelion distance of the object is kept constant from $t=0$ to $t=T$, an initial fully active comet ($f_{a_O}=1$) would have a fraction of active area ($f_a$) at the end of the period given by the integration of eq. (1), $$\ln f_a = - \ f'(q) \ T \eqno(10)$$ \noindent Considering that the comet becomes dormant if $f_a$ reduces below certain limit, the active lifetime ($T_{act}$) will be inversely proportional to the mantling rate. We thus expect that a change of $f'$ by +50\% (-50\%) would produce a variation of $T_{act}$ by a factor of 2/3 (2 respectively). These variations are almost confirmed in the simulations, where we obtain the corresponding reduction of $T_{act}$ when we increase $f'$, but $T_{act}$ increases by only a factor of $\sim 1.5$ when $f'$ is reduced. Although we find some variations on the other lifetimes in the expected directions ({\sl e.g.} when $f'$ is increased we obtain a longer dormant lifetime and a shorter end of activity lifetime), the differences respect to the standard values are relatively much smaller (less than 10 \%). Noticeable changes are also observed in the steady state distributions when we introduce variations of $f'$. The ratio between the number of active to total number of comets ({\tt act/tot}) for the combined population decreases from 0.37 to 0.28 when $f'$ is increased by 50\% and increases to 0.52 when $f'$ is decreased by 50\%. Remarkably, the contribution of dormant comets to the total population is still as high as $\sim$ 50\% for the slowest mantling process. A prediction of the simulations that could be tested with an analysis of the distribution of active JF comets and dormant candidates; a study that it is out of the scope of this paper, regarding the complexity of such comparison. No significant differences are observed in the mean fraction of active areas when $f'$ is increased by 50\%. For the simulations with a reduction of $f'$ the values of $f_a$ increase and the curves in Fig. 10 are shifted upward by $\sim $ 0.05. Finally we consider the comparison between results with different blow-off models. No significant differences are observed in the lifetimes and the steady state distribution of number of comets, not even in the ratio between active and dormant comets. The fractions of active area slightly increase with greater blow-off fractions (differences less than 20 \% with respect to blow-off model 2 in the $q$ range 1 to 5 AU). For $q<1$ AU the differences in $f_a$ get larger. The peak at low $q$ almost disappears for model 1 and is enhanced for model 3. Comets reach the region with $q<1$ AU by small jumps, so the big difference in the blow-off fraction for small $\Delta q$ between the different models explains the different $f_a$ in this $q$ range. \vfill\eject %\topskip 175 true mm %\noindent {\bf Fig. 12.--} The cumulative distributions of absolute total magnitudes computed for the three radii and for the combined popualtions with mass power law indices: 0.2, 0.7 and 1.2. The full lines correspond to the cumulative distribution of the whole active population while dashed lines are for the sample of comets with $q < 2 $ AU. A straight line corresponding to a slope of 0.4 is shown for comparison. %\topskip 0 mm %\vfill\eject \noindent{\bf 5. Discussion} \medskip \noindent We have improved on our earlier study (Paper I) by modelling the dynamics of the JF in a more realistic way. The effects of medium- and long-term trapping into resonances are better represented than in the previous stochastic approach. The dynamical characteristics of the sample population are known accurately and their long-term evolutions are possible to trace. Concerning the evolution of the whole JF population, the major shortcoming of our modelling is that we still know very little about the general characteristics of this population of objects. We do not know how the dynamics of the observed sample differs from the rest of the population. More importantly, we know very little about the distribution of JF comets with $q$ and it is one aim of this project to improve on that. Only an assumption about the source population of the JF could enable us to assess the differences. The analysis of alternative source populations is a possible future step of develoment of our simulation technique. We thus emphasize again that the results presented here apply to a subsample of the total population to which the observed sample belongs and whose dynamical characteristics are shared by the known comets. Comparing the results presented in Paper I, we note that the steady state distribution of the total number of comets for a 5 km radius presented in Fig. 9 (c) largely resembles the corresponding one of Paper I, Fig 2 (b) (computed with the jump probabilities of the observed sample). However, the distribution of active comets shows considerable differences: while in Paper I the distributions of active and total number of comets look similar, in Fig. 9 (c) the peak at low perihelion distance seen in the total number distribution disappears for the active population. Since the physical modelling is almost the same and the dynamical models are extracted from a similar database ({\sl i.e.}, the numerical integrations of the observed sample of JF comets), the dynamical modelling itself seems to be the only pausible explanation for the differences. This shows that it is worth a lot to have a correct representation of the dynamics when we simulate the combined evolution. We confirm previous expectations (Rickman 1992) that the lifetime to cease the cometary activity should be comparable to half the dynamical lifetime; but in between we could have several periods of dormancy, typically half the time until the end of activity is partially in the active phase and the other half is entirely dormant. In this respect, comets like P/Encke that are supposed to be in low-$q$ stable orbits for hundreds of revolutions are intriguing; since according to our present modelling, if the comet is too large it would have become completely sealed off soon after it reached the present orbit, or if it is too small the mantling rate is not high enough to prevent total disintegration. In any case the comet should have reached the actual orbit heavily mantled, otherwise the total disintegration is certain. Obviously a possible recent reactivation after a dormancy period could save the picture, but we do not pretend with our general scheme to account for all particular cases. If one assumes that the orbital evolution found as a function of $q_m$ is typical for the entire population ({\sl i.e.}, we could reproduce the possible evolutions although we could not reproduce the total number distribution with $q_m$), the information given in Fig. 7 on the lifetimes as functions of $q_m$ may be used with different distributions of $q_m$ corresponding to different source regions. Noticeable differences are found between the lifetimes computed with the whole sample and the ``real'' population alone. The contribution of a sample of objects with relative short lifetimes in orbits with frequent jumps is the reason for the differences. This result denotes again the difficulties in properly model the evolution of JF comets, since, even considering a fictitious population that shares several dynamical characteristics with the observed one ({\sl e.g.} same Tisserand parameter distribution), detectable variations in the results are observed. We have shown that the dormant state of JF comets is an extremely important phase of their evolution in terms of time spent in this state as well as in number of objects relative to the active state. Furthermore, the dormant comets resulting from our simualtions share the same dynamical characteristics of active members of the JF, {\sl e.g.}, similar distributions of Tisserand parameter and frequent close encounters with Jupiter. Steady-state distributions of the number of active and dormant comets show that we should expect more dormant comets than active ones, and the dormant/active ratio should increase with increasing size. We also show that with the present observational techniques we are able to detect only a minor fraction of the population of active comets and we barely detect any comets with $q > 3$ AU. Fern\'andez {\it et al.} (1992) found that for $q<1.5$ AU one would expect $\sim 60^{+40}_{-20}$ comets with $H_T < 11$, compared to the 15--20 already discovered. The detection limit for this perihelion range computed with eq. (9) is $H_T^\ell \lta 11$, so our number of detectable comets should be roughly equivalent to the total number of JF comets with $q<1.5$ AU and $H_T < 11$ found by Fern\'andez {\it et al}. For the standard model, the number of detectable comets with $q<1.5$ AU is roughly 45 \% of the number of active comets and 17 \% of the number of active plus dormant ones in the same perihelion range. One may thus expect as many as 135 active comets and 210 dormant comets with $q<1.5$ AU. The shortcomings discussed above prevent us to extrapolate this result to the total number of active and dormant JF comets. The fractions of active area are in agreement with the knowledge gained from the relatively few estimates of this parameter (Rickman 1991) and confirm the expectation that $f_a$ should only be on the order of a few percent for JF comets. The spread in the fractions of active area is shown to have a considerable effect in the shape of the magnitude distribution, but the size distribution is what mainly governs the slope of it. A study involving a finer distribution of radius for the simulated population would give us a more precise answer to this problem. We explore variations of the mantling rate by $\pm$ 50\% of the standard values according to the expected uncertanties coming from Rickman {\it et al.}'s (1990) model. The active lifetimes shows changes approximately inversely proportional to the mantling rate in accordance with simple analytical estimates. Even for the slowest mantling rate, the dormant phase contributes $\sim$ 50\% to the total population of JF comets. In spite of the large differences between the considered blow-off models, reflecting the poor knowledge of this parameter, the results do not show important variations. Nevertheless, a better estimate of these quantities would be desirable for future simulations. The decrease of mass and the meteoroidal end state are treated by a simple approach. Phenomena like cometary splitting, break-off of large chunks and sudden brightness increase (outbursts) are not modelled, but they can be included into future simulations by assuming a certain probability for these events. The inclusion of these phenomena would affect the lifetimes estimated before, since they would tend to produce smaller and more active nuclei, changing also the combined steady state distribution. Finally a major future improvement will be to allow for the tendency of comets to revisit the Jupiter family several times before being finally ejected. Thus, on a steady-state picture, fresh comets with the full initial mass would be injected only in response to final ejections. Obviously, the comets residing in the Jupiter family at any time will then be less fresh than we have found here, and the number of dormant comets will be even larger. Evidently, the physico-dynamical modelling described here may still be subject to many improvements. Nevertheless the results presented in this paper encourage us to continue such efforts until we get a full picture of the physical and dynamical characteristics of the Jupiter family. \vfill\eject \noindent{\bf References} \parindent=0pt \bigskip \hangindent=10pt\hangafter=1 {\bf Bailey, M.E.} ``Short-period comets: Probes of the inner core'', in {\it Asteroids, Comets, Meteors III}, C.-I. Lagerkvist and H. Rickman (eds.),pp. 221-230, Uppsala Univ., 1990a \hangindent=10pt\hangafter=1 {\bf Bailey, M.}E. ``Cometary masses'', in {\it Baryonic dark Matter}, D. Lynden-Bell and G. Gilmore (eds.), pp. 7-35. Kluwer Ac. Press, Netherlands, 1990b \hangindent=10pt\hangafter=1 {\bf Everhart, E.} ``The origin of short-period comets'', {\it Astrophys. Letters} {\bf 10}, 131-135, 1972 \hangindent=10pt\hangafter=1 {\bf Fern\'andez, J. A., H. Rickman, and L. Kam\'el} ``The population size and distribution of perihelion distances of the Jupiter family'', in {\it Periodic comets}, J. A. Fern\'andez and H. Rickman (eds.), pp. 143--157, Universidad de la Rep\'ublica, Montevideo, Uruguay, 1992 \hangindent=10pt\hangafter=1 {\bf Froeschl\'e, C.} ``Chaotic behaviour of asteroidal and cometary orbits'', in {\it Asteroids, Comets, Meteors III}, C.-I. Lagerkvist and H. Rickman (eds.), pp. 63-76, Uppsala Univ., 1990. \hangindent=10pt\hangafter=1 {\bf Froeschl\'e, C. and H. Rickman} ``Model calculations of nongravitational forces on short-period comets. I. Low-obliquity case'', {\it Astron. Astrophys.} {\bf 170}, 145-160, 1986 \hangindent=10pt\hangafter=1 {\bf Jorda L., J. Crovisier, and D.W.E. Green.} ``The correlation between water production rates and visual magnitudes in comets'', in {\it Asteroids, Comets, Meteors 1991}, A. Harris, E. Bowell (eds.), pp. 285-288, Lunar Planet. Inst., Houston, 1992 \hangindent=10pt\hangafter=1 {\bf Kres\'ak, \v L.} ``Dormant phases in the aging of periodic comets'', {\it Astron. Astrophys.} {\bf 187}, 906-908, 1987 \hangindent=10pt\hangafter=1 {\bf Kam\'el, L.} `` The Comet Light Curve Atlas. (The comet light curve catalogue/atlas. III. The atlas)''. {\it Astron. Astrophys. Suppl. Ser.} {\bf 92,} 85--149, 1992 \hangindent=10pt\hangafter=1 {\bf Lindgren, M.} `Dynamical timescales in the Jupiter family', in {\it Asteroids, Comets, Meteors 1991}, A. Harris, E. Bowell (eds.), pp. 371-374, Lunar Planet. Inst., Houston, 1992 \hangindent=10pt\hangafter=1 {\bf Luu, J., and D. Jewitt} ``High resolution surface brightness profile of Near-Earth Asteroids'', {\it Icarus} {\bf 97}, 276-287, 1992. \hangindent=10pt\hangafter=1 {\bf Marsden, B.G., and G.V. Williams}, {\it Catalogue of Cometary Orbits, 7th Edition}, Minor Planet Center, Smithsonian Astrophys. Obs., 1992 \hangindent=10pt\hangafter=1 {\bf Rickman, H.} `On the properties of comets, asteroids, and terrestrial planet impactors', {\it Adv. Space Res.}, {\bf 11}, (6)7-(6)18, 1991 \hangindent=10pt\hangafter=1 {\bf Rickman, H.} ``Physico--Dynamical evolution of aging comets'', in {\it Interrelations between physics and dynamics for minor bodies in the solar system}, D. Benest, C. Froeschle (eds.), pp. 197-263, Ed. Frontieres, France, 1992 \hangindent=10pt\hangafter=1 {\bf Rickman, H., J.A. Fern\'andez, and B.\AA .S. Gustafson} ``Formation of stable dust mantles on short-period comet nuclei'', {\it Astron. Astrophys.} {\bf 237}, 524-535, 1990 \hangindent=10pt\hangafter=1 {\bf Rickman, H., C. Froeschl\'e, L. Kam\'el, and M.C. Festou} ``Nongravitational effects and the aging of periodic comets'', {\it Astron. J.} {\bf 102}, 1446-1463, 1991 \hangindent=10pt\hangafter=1 {\bf Rickman, H., M.E. Bailey, G. Hahn, G. Tancredi}, `Monte Carlo simulations of Jupiter family evolution', in {\it Proc. of the International Workshop on Periodic Comets}, J.A. Fern\'andez and H. Rickman (eds.), pp.55-64 , Universidad de la Rep\'ublica, Montevideo, Uruguay, 1992 \hangindent=10pt\hangafter=1 {\bf Tancredi, G. and H. Rickman} ``The evolution of Jupiter family comets over 2000 years'', in {\it Chaos, resonance and collective dynamical phenomena in the solar system, Proceedings of the IAU symposium 152}, S. Ferraz--Mello (ed.) pp. 269--274. Kluwer, 1992 \hangindent=10pt\hangafter=1 {\bf Valsecchi, G.B.}. in the first round--table discussion: ``Dynamics of periodic comets'', in {\it Periodic comets,}, J. A. Fern\'andez and H. Rickman (eds.), pp. 97--111, Universidad de la Rep\'ublica, Montevideo, Uruguay, 1992. \hangindent=10pt\hangafter=1 {\bf Weissman, P.R., M.F. A'Hearn, L.A. McFadden, and H. Rickman}\break ``Evolution of comets into asteroids'', in {\it Asteroids II}, R.P. Binzel, T. Gehrels and M.S. Matthews (eds.), pp. 880-920, Univ. Arizona Press, Tucson, 1989. \vfill\eject \nopagenumbers \settabs 6 \columns \+&&{\bf Median Lifetimes (yr)}\cr \smallskip \+{\bf Population} &{\bf Total } &{\bf Dormant} &{\bf Active} &{\bf End of Activity}\cr \smallskip \hrule width14cm \smallskip \+467 objects &13820 &~~12510 &~2410 &~~~7500 \cr \+135 objects &11350 &~~10490 &~2590 &~~~6040 \cr \vskip -18mm \hskip 15mm \vrule height20mm \vskip 1cm {\sl\baselineskip=18pt Table I -- The median values of the lifetimes for the total population (467 objects) and the ``real'' one (135 objects). } \vskip 6cm \settabs 9 \columns \+&{\bf ~~~~~~~~R=1.5 km} &&{\bf ~~~~~~~~R=3 km} &&{\bf ~~~~~~~~R=5 km} &&{\bf ~~~~~~~~Combined}\cr \smallskip \+{\bf Population} &{\tt ~~act/tot} &{\tt ~~det/act} &{\tt ~~act/tot} &{\tt ~~det/act} &{\tt ~~act/tot} &{\tt ~~det/act} &{\tt ~~act/tot} &{\tt ~~det/act} \cr \smallskip \hrule \smallskip \+467 objects &~~~~~~0.40 &~~~~~~0.046 &~~~~~~0.28 &~~~~~~0.049 &~~~~~~0.21 &~~~~~~0.045 &~~~~~~0.37 &~~~~~~0.046 \cr \+135 objects &~~~~~~0.47 &~~~~~~0.057 &~~~~~~0.33 &~~~~~~0.057 &~~~~~~0.25 &~~~~~~0.053 &~~~~~~0.43 &~~~~~~0.057 \cr \vskip -18mm \hskip 13mm \vrule height20mm \vskip 1cm {\sl\baselineskip=18pt Table II -- The ratio between number of active to total number of comets {\tt act/tot} and number of detectable to number of active comets {\tt det/act} for the total population (467 objects) and the ``real'' one (135 objects). } \vfill\eject \nopagenumbers \medskip \noindent {\bf Fig. 1.--} Dynamical evolutions of comets P/Neujmin 2, P/Shoemaker 2 and \break P/Kearns-Kwee as computed from the numerical integrations explained in the text. \medskip \noindent {\bf Fig. 2.--} The mantling rate per revolution ($f'$), {\sl i.e.} the fraction of the active area that becomes mantle in one revolution. Full-drawn curve corresponds to a nucleus of 5 km radius, and the dotted curve to 1 km radius. \medskip \noindent {\bf Fig. 3.--} The fraction of mantle that blow-off as a function of the difference $\Delta q$ between the $q$-bin where the mantle was formed and the $q$ where the comet has jumped to. \medskip \noindent {\bf Fig. 4.--} The water flux ate perihelion (dotted curve) and the average water flux over a revolution as a function of $q$. See text for an account of the thermal model used to compute the two curves. \medskip \noindent {\bf Fig. 5.--} Histogram of the distribution of overall minimum perihelion distance of the 467 objects attained during the dynamical integrations. \medskip \noindent {\bf Fig. 6.--} Histogram of the distributions of lifetimes. a) total time span in the JF (a ``visit''), b) total time in the active phase, c) total time in the dormant phase and d) the interval until the last time the comet was active (see text). The $y$-axis corresponds to fractions of the total population. The medians of each distribution expressed in years are shown in the corresponding plot. \medskip \noindent {\bf Fig. 7.--} The medians for each $q_m$-bin of the lifetimes presented in Fig. 6 for the whole sample (full lines) and the ``real'' comets (dashed lines). \medskip \noindent {\bf Fig. 8.--} The steady state distribution of comets with $q$ for the pure dynamical simulations. The full lines are the mean values of the distributions and the dotted lines show the $\pm 3 \sigma$ errors. The dashed line corresponds to the distributions of comets with $q$ at the beguinning of the integrations ({\sl i.e.} the present distribution of observable JF comets). \medskip \noindent {\bf Fig. 9.--} The steady state distributions of comets with $q$ for the three radii and for the combined population with a mass index $s=0.7$. From top to bottom, the curves correspond to total number of comets (both active and dormant), number of active comets and number of detectable ones. The full lines are the mean values of the distributions and the dotted lines show the $\pm 3 \sigma$ errors. On top of each plot the ratio between total number of active comets over total number of comets ({\tt act/tot}) and the ratio between total number of detectable over total number of active comets ({\tt det/act}) are presented. \medskip \noindent {\bf Fig. 10.--} The fraction of active area $f_a$ of active comets for the three radii as a function of $q$. The full lines correspond to the mean over the means of all 150 distributions and the dotted parallel lines to the $\pm 3 \sigma$ errors. \medskip \noindent {\bf Fig. 11.--} The mass fractions $m$ as a function of $q$. Full lines correspond to the mean mass of the comets in the active state and dashed lines to the dormant phase. Lines at $\pm 3 \sigma$ are drawn parallel to the previous ones. \medskip \noindent {\bf Fig. 12.--} The cumulative distributions of absolute total magnitudes computed for the three radii and for the combined popualtions with mass power law indices: 0.2, 0.7 and 1.2. The full lines correspond to the cumulative distribution of the whole active population while dashed lines are for the sample of comets with $q < 2 $ AU. A straight line corresponding to a slope of 0.4 is shown for comparison. \bye