\font\rmS=cmr10 scaled\magstep1 \font\bfS=cmbx10 scaled\magstep1 \font\rmSS=cmr10 scaled\magstep2 \font\bfSSS=cmbx10 scaled\magstep3 \baselineskip 14pt \def\lta{~\raise.4ex\hbox{$<$}\llap{\lower.6ex\hbox{$\sim$}}~} \def\gta{~\raise.4ex\hbox{$>$}\llap{\lower.6ex\hbox{$\sim$}}~} \def\deg{$^\circ$} \def\fdg{$^\circ\!\!$} \def\am{$^{\rm\prime}$} \def\h{$^h$} \def\mfrac{$^m\!\!$} \rm \def\P{\noindent} % % % \centerline{\bfSSS Searching for Comets} \centerline{\bfSSS Encountering Jupiter: } \centerline{\bfSSS First Campaign \rm \footnote*{Based on observations collected at the European Southern Observatory, La Silla, Chile}} \vskip 3 true cm \centerline{\rmSS G. Tancredi$^{1,2}$ and M. Lindgren$^1$} \vskip 2 true cm \centerline{$^1$Astronomiska Observatoriet, Box 515, S-751 20 Uppsala, Sweden} \centerline{$^2$Depto. de Astronomia, Facultad de Ciencias, Tristan Narvaja 1674,} \centerline{Casilla de Correo 10773, 11200 Montevideo ,Uruguay} \vfill\eject \noindent {\bfS Abstract} \sl \medskip\noindent This paper reports results from a first search for previously undetected comets in the vicinity of Jupiter. Combining these with a model for the probability for finding a comet in this region we estimate the total number of comets in the Jupiter family. 36 Schmidt plates were obtained at ESO in March and April 1992. We searched the plates down to a limiting nuclear B--magnitude of 13.8 . No comet was found. This result together with a model for the probability of finding a comet close to Jupiter, yields an upper estimate of the number of objects in the Jupiter family. If we assume that the comets are inactive at $\sim$ 5 AU from the Sun, we get a conservative estimate of $N(H_N^B<13.8) < 210$. We discuss the possible brightening due to activity and present estimates including this effect. By assuming a certain magnitude distribution, we then compare our results with previous attempts to estimate the total size of the Jupiter family. Although our estimates are still higher than previous values, our results are independent of the distribution of comets with perihelion distance. Ongoing and future searches with the same technique will further constrain the population size of the Jupiter family. \vfill \eject \def\deg{$^\circ$} \rm \noindent {\bfS 1. Introduction} \medskip \noindent The first problem one face when searching for short--period comets is how to define a comet, or rather, how to distinguish between comets and asteroids. Traditionally, objects are classified as comets or asteroids depending on their outgassing activity. But nowadays this separating wall is falling down. As illustrated by Kres\'ak (1979), there is a distinction in the orbital characteristics (and consequently dynamics) between short-period comets and asteroids. The distinction is clearly seen in the semimajor axis--eccentricity phase space where there is a clear border line between the two groups corresponding to a Tisserand parameter $T$, of 3. On the basis of this fact, Kres\'ak formulated a new distinction criterion between asteroidal and cometary orbits: a cometary orbit has $T\lta3$ and is not protected from encountering Jupiter by a simple resonance libration. An orbit that does not meet this requirement is asteroidal. This fact that comets can encounter Jupiter, while asteroids can not, was used to develop a strategy for searching for comets in the vicinity of Jupiter (Tancredi and Lindgren, 1992, hereafter TL92). We illustrate the situation in Fig. 1. By looking in the direction of Jupiter we eliminate any confusion with objects from the only two groups of asteroids that can reach heliocentric distances comparable to Jupiter's: the Trojans and the Hildas. The Trojans never get closer than about 30\deg\ angular distance from the planet. Although some Hildas can be seen in the direction of Jupiter they are at their perihelion due to their mean motion resonance and alignment of the perihelion direction with Jupiter. Thus, to be certain not to confuse inactive comets with asteroids, the only direction to look is towards Jupiter. Furthermore, the number density of observed JF comets at any time in a Jupiter-fixed rotating-pulsating frame peaks around the planet, showing that there is a greater chance of finding these objects close to Jupiter than in any other part of the Solar System (TL92). To verify this, we use the numerical integrations for 20000 yr presented in section 4.1 to make a similar plot (Fig. 2.). Although the peak around Jupiter is slightly less pronounced than in TL92, where a shorter integration was used, the number density in the Jupiter neighborhood is higher by 20--25\% than in any other place around the orbit of Jupiter. The lower value of the density increase based on the longer integration is due to the fact that there is a dependence of the frequency of encounters on perihelion distance (Tancredi and Rickman 1992, hereafter TR92), and since there is a peak in the temporal distribution of perihelion distances due to observational bias towards comets with low perihelion distances, we expect a lower value. The problem of identification is minimised by looking toward Jupiter opposition. The spatial distinction between asteroids and comets in the general direction of Jupiter coupled with the low relative encounter velocity of JF comets lead to a clear separation in velocity space between comets undergoing an encounter and asteroids that are in the same region of the sky. When observing close to Jupiter opposition the difference in velocities translate into two separate regions in the hourly motion phase space, as shown in TL92. From integrations of the observed JF comets we find that the projected relative velocity during a close encounter (distance $<0.5$ AU from the planet) to be typically between 4 and 5 arcsec hr$^{-1}$ (see TL92 and section 5 for further discussion). Thus, when observing Jupiter's vicinity, with the telescope tracking on the planet, asteroids show up as trails longer than 8 arcsec in a one hour exposure. A typical value for main belt asteroids is $17$ arcsec, while comets encountering Jupiter show up as much shorter trails (less than 7 arcsec). The stars appear as trails of $\sim$ 19 arcsec but in opposite direction compared to the asteroids (the exact value depends on Jupiter's orbital position at opposition). Below in section 2 we present the observations from this first part of the project based on this strategy, carried out at ESO and the results are presented in section 3. In section 4 we estimate the number of comets encountering Jupiter as a function of the $q$-distribution and the distance to the planet. The magnitude distribution of cometary nuclei and the limiting magnitude of our search are discussed in section 5. Constraints on the size of the total population of JF comets are estimated in section 6, and in section 7 we present the conclusions derived from our search. \bigskip \noindent {\bfS 2. Observations} \medskip\noindent {\bf 2.1. The plates} \medskip \noindent During nine nights around new moon in early March 1992 we obtained 27 plates from the European Southern Observatory 1.0/1.6 m Schmidt telescope. Table I shows the dates and plate center coordinates. Since the field of view of the ESO Schmidt is 5.4\deg, the total area covered in this search is about 16\deg.2 by 16\deg.2 . The following new moon period in early April 1992 we obtained an extra set of plates, consisting of one exposure of each region. The plate center coordinates for these ``follow--up'' observations were shifted $\sim$6\deg\ to the west to compensate for one month of motion of the objects found on the March plates. Table I also shows the data for the April plates. Because of the high declination (+9\deg) the observing conditions at ESO (latitude -29\deg) were not ideal. For example: the central region culminates at an altitude above the local horizon of only 50\deg. This leads to an average seeing of not much better than about 1.5 arcsec (measured on field stars). We will use this as a mean value for all the plates in the following calculations. The March plates were exposed for 90 minutes, and the April plates for 45 minutes. The emulsion used throughout is the Kodak IIIa-J, hypersensitized according to standard procedures. By using only a blue wavelength cut--off filter (GG385) instead of a bandpass filter we detected all the light from the objects, leading to a fainter limiting magnitude of the search. To facilitate the identification of the objects and at the same time optimize the limiting magnitude, we used the apparent motion of Jupiter as the tracking rate during the March observations (TL92). Sidereal tracking rate was used during the April observations. \vbox{ \bigskip\noindent {\bf 2.2 Limiting magnitude} \medskip} \noindent The problem of determining the limiting magnitude in a photographic search for objects that are not stationary during an exposure, is a subject not much discussed in the literature. A first order estimate can be obtained by determining effective exposure times for objects moving at different speeds by simply measuring how long time it takes for the object to move the distance of a seeing disk. By comparing with exposures of point sources of known magnitudes, using the measured effective exposure time, it is then possible to estimate the magnitude correction due to trailing. A more exact method is to consider the expression for the intensity $i(x,y)$ of a source distributed as a gaussian point spread function with a FWHM of $d$ and defining $i(0,0) \equiv I$. Considering that the source is moving during a time $t$ with a constant speed $v$ along the $x$--axis, from $-\ell/2$ to $\ell/2$, where $\ell=vt$ is the length of the trail, the exposure $e(x,y)$ at $(x,y)=(0,0)$ is then $$ e_T(0,0) = \int\limits_{-t/2}^{t/2}i_T(x(\tau),0)~d\tau~=\Bigl\lbrace x={v\tau\over{d}}\Bigr\rbrace= \int\limits_{-t/2}^{t/2} I_T~{\rm exp}\Bigl(-{4\ln 2~{(v\tau)^2}\over{d^2}}\Bigr)~d\tau \eqno(1) $$ where the subscript $T$ denotes a trailed image. This is integrated to $$ e_T(0,0) = \sqrt{{\pi\over{\ln 2}}}~{I_T~ d\over{2v}}~{\rm erf} \Bigl({\sqrt{\ln 2}~\ell\over {d}}\Bigr)\simeq 1.06~I_T~{d\over{v}}~{\rm erf}\Bigl({0.83~\ell\over{d}}\Bigr)~~, \eqno(2) $$ where erf is the error function. The exposure $e_P(0,0)$ from a stationary source of known magnitude exposed using the same exposure time $t$ as for the moving object is given by $$ e_P(0,0) = \int\limits_{-t/2}^{t/2}i_P(0,0)~d\tau=~I_P~t \eqno(3) $$ where the subscript $P$ denotes a stationary source. Equating the exposures of the stationary source and the trailed image (i. e. the peak of the gaussian profile of the stationary source and the ``height of the ridge'' of the trail) one then finds that the difference in magnitude is $$ \Delta m = -2.5~\log\Bigl({\ell\over{1.06~d~{\rm erf}({0.83~\ell \over{d}})}}\Bigr) \eqno(4)$$ A plot of this dependence on trail length (expressed in units of the FWHM of the seeing disk, $\ell/d$) of the magnitude correction can be seen in Fig. 3. For long trails ($\ell\gta2.6d$) the corresponding erf value approaches 1 and in that case eq. (4) reduces to the same relation as for the simple concept of an effective exposure time described above (i.e. $\Delta m = -2.5\log({\ell\over d})~$), apart from the small factor 1.06 inside the log. To clarify the meaning of this calculation: consider the situation where one wants to find out if an object with known apparent magnitude $m_0$ and known apparent speed is visible. Equation (4) then tells us that the recording on the plate (``the height of the ridge'') would be the same as for a stationary object with magnitude $m_0+\Delta m$. For photographic emulsions, the expression (4) is valid as long as the exposure time is long enough to place both the trail and the point source on the linear part of the characteristic sensitometric curve for the emulsion used. Density measurements of the sky background show that in our 90 min. exposures we are in this linear part. A simple way of checking this theoretical calculation of the limiting magnitude of trailed objects is to compare directly with previously known objects moving with the same speed as the search objects. In general one can not expect this to be the case, but in this particular situation where we are looking in the vicinity of Jupiter we do have ``standard candles'' in the magnitude range of interest: the outer Jovian satellites. The apparent magnitude during the first week of March 1992 goes down to $B=21.1$ for satellite XIII Leda (Astronomical Almanac 1992). It turned out that the trail of Leda corresponds almost exactly to what we estimate as the faintest trailed object that we would have been able to find during the scanning of the plates. For $B=21.1$ and the speed of Leda (2.8 arcsec hr$^{-1}$), we get using eq. (4) a limiting magnitude for stationary objects of 22.1, in good agreement with exposures of photometric standard stars. We found that, on 90 min. exposures using sidereal tracking rate, in the fields F873-8 and F873-16 (Stobie {\it et al.} 1985) the faintest stars visible were of B--magnitude $\approx 22.1$. To believe in these estimates, remembering that we have not used a bandpass filter, we have to show that the difference in color between the objects we are looking for and the standard stars, is not too large. The colors of cometary nuclei have not been very well determined. However, as Hartmann {\it et al.} (1987) showed, the available data points to cometary nuclei being rather red, resembling D type asteroids. Asteroids in this class have B-V colors in the range 0.7 to 0.8, which is the same as for faintest stars in the F873-8 and F873-16 fields that we could see. Thus, since the color only enters as a second order correction to the magnitude, we can safely say that we have reached a limiting magnitude of $B = 22.1$. \bigskip\noindent {\bf 2.3 Reductions} \medskip \noindent We scanned the plates, using ordinary light tables and magnifiers. Since the tracking rate used was non--sidereal, every light source appear as a trail. The stars along one direction and with the same length, and the solar system bodies with varying directions and lengths. Because of the large number of trails due to asteroids present in the field of view (200--300 per plate) we have to apply some kind of selection criteria for inclusion in the further processing of the data. The simple criterion for marking an object (a trail) was that the trail should be clearly shorter than the trails of the stars (cf. section 1), which resulted in that all trails shorter than 15 arcsec. were marked. This obviously means that we excluded many objects not of interest to this project. But the convenience of making quick decisions without having to do any quantitative measurements during the very tedious work of scanning the plates by eye, by far outweighs the small amount of extra work done when making the astrometric measurements and orbit determinations. The shortest trails we found were about 12 arcsec long, corresponding to an hourly motion of 8 arcsec which is just outside the region of ``slow--moving objects'' as defined in section 1. This forces us to suspect that we did not find a single comet approaching Jupiter. However, to confirm the suspicion we measured the positions of these trails and calculated the orbits. The procedure we followed was first to link the trails visually on different plates and then calculate a preliminary orbit, using either V\"ais\"al\"a type orbits (V\"ais\"al\"a, 1939 and Marsden, 1985) or the algorithm described by Neutsch (1981). After discarding misidentified trails and recalculating the orbits we were left with 20 orbits (table II). Table II shows the pertinent orbital elements along with information that indicate the accuracy of the determined orbits (i.e. the mean residuals of the positions), as well as the position of the object in its orbit. As can be seen, none of the objects is anywhere near Jupiter; in fact, the most distant object (T) has an aphelion distance of only 4.4 AU. Although interesting, since that object together with the objects Q, R and S, appears to be a new member of the Hilda group at the 3/2 mean motion resonance with Jupiter, it is not a comet undergoing a close encounter with the planet. That they are Hildas is verified by noting that they are all within 30\deg\ from perihelion (Table II). This result obviously confirms that we have found no Jupiter approaching object. \bigskip \noindent{\bfS 3. Results of the search for comets} \medskip \noindent To summarize the results of the search we can say that we found no slow moving object with a corresponding stationary object magnitude of $B<22.1$. What we mean by that is that the faintest objects a visual scan could detect would have a B--magnitude of 22.1 if untrailed. The uncertainty of this value is difficult to determine accurately, since the crucial part has to do with a subjective estimate of how faint trails could be found during the visual scan by eye of the plates. We estimate the uncertainty to be certainly less than 0.5 magnitudes, but probably not better than 0.2 magnitudes. It is worth to point out that at the time of our observations there was a known comet close to Jupiter: P/Maury. That we could not see it in our plates is not a surprise, since, for an estimated nuclear magnitude 15 (Kam\'el 1992), the object would be more than 1 mangnitude fainter than our limit. \bigskip \vbox{ \noindent {\bfS 4. Estimate of the number of encounters} \noindent{\bf 4.1 Method} \medskip \noindent The number of comets encountering Jupiter at a distance less than D at any time ($N_e(D)$~) is a function of the total number of JF comets ($N_{JF}$) and the probability for a comet to be close to the planet at any given time (the neighboring probability $p(D)$~): $$ N_e (D) = \sum_{j=1}^{N_{JF}} ~ p_j(D) ~ ~ . \eqno(5)$$ \noindent The neighboring probability is uniquely determined given the orbital elements of the comet and Jupiter. For small $D$ values it is zero if the comet's orbit does not cross Jupiter's orbit, or if it is locked in a mean motion resonance. Since we know very little about the exact distribution of the orbital elements of the JF population, we choose only one element to characterize the comet's orbit and average over the variation of the other elements. We use the perihelion distance ($q$) because the information about the JF is very $q$--dependent. The distribution information is complete only for comets with $q\lta1.5$ AU. } To estimate the neighboring probability we numerically integrated the whole sample of known JF comets (140 objects) for $\pm$ 10000 yr. A comet was considered to belong to the JF when the period became less than 20 yr., the Tisserand parameter $20.1$ AU, there is almost no dependence on $D$ for each $q$-bin. That is, the variations of the projected velocities at different distance to the planet are negligible. A mean value during the encounters was then computed for each $q$-bin and plotted in Fig. 7. As the mean relative velocities decrease with $q$, the limiting magnitude tend to increase and we can detect fainter objects. We will use a fixed limiting magnitude corresponding to the average projected relative velocity (4.0 arcsec hr$^{-1}$) over the range $1.5 < q < 4.5$ AU. Using eq. (4), we calculate a correction to the point source limiting magnitude for slow--moving objects of $\Delta m=-1.4$. The apparent limiting magnitude then becomes $B=20.7$, which in asteroid nomenclature is called the opposition magnitude $B(0)$. \bigskip \noindent {\bfS 6. A limit on the population size} \medskip \noindent In Section 3 we concluded that we found no comet encountering Jupiter brighter than the limiting magnitude of our search. The region covered by the plates is a square in the sky-plane of 16\deg.2 $\times$ 16\deg.2. The corresponding volume is less precicely defined, but since no object beyond the heliocentric distance 4 AU was found (Table II), we define the inner surface of the volume by the geocentric distance 3 AU (remembering that we are looking at the direction of opposition). The outer surface of the volume we define by a geocentric distance of 5.8 AU, since the outer edge of the peak of the Q--distribution of JF comets is at roughly 6.8 AU. The neighboring probability described in Section 4.2 is a cumulative probability of the distance to the planet. That is, all encounters with distances less than $D$ are considered. To compute the probability to find a comet in the ``truncated pyramid'' described above, we have to integrate the differential probability over this volume. The cumulative probability to find a comet within a distance $D$ is given by eq. (6). The probability to find one inside a volume $V$ is then $$ p_V = {2.5~ k \over 4 \pi } \int_V {1 \over \sqrt D} dV \ \ , \eqno(8)$$ \noindent where $k=0.014$. After numerical integration of this over the volume, we get $$ p_V = 1.00 k = 0.014 \ \ .\eqno(9)$$ \noindent Since we did not observe any object, we can by using this probability only derive an upper limit on the number of comets in the volume brighter than opposition magnitude $B(0) = 20.7$. Consider a population of $N$ objects that can have close encounters with Jupiter. Three types of objects contribute to this population: i) active comets ($N_a$), ii) dormant comets ($N_d$); and possibly iii) asteroids ($N_{ast}$) in cometary orbits (Milani et al. 1989, Hahn et al. 1991). The probability of finding $n$ objects at a given time at a distance less than $D$ to Jupiter is given by the Poisson distribution for $n$ events with a parameter $\lambda$ (total number of encounters at any time) given by $$\lambda=N_a p_a + N_d p_d + N_{ast} p_{ast}\ \ , \eqno(10)$$ \noindent where $p_a$, $p_d$ and $p_{ast}$ are the corresponding neighboring probability for each type of objects. With a 95\% certainty, we get $\lambda \le - \ln 0.05$. This inequality also holds for $N_a p_a + N_d p_d$ alone. Assuming that the dynamics of dormant comets can be described by the integrations discussed in section 4.1 (Tancredi 1993), we can say that $p_a = p_d$. Thus, for $p_a = 0.014$, $N_a + N_d < -\ln 0.05 / 0.014 = 210$. This inequality also holds for each of the types of objects individually, and since we do not know the ratio of contribution, we will have a twofold interpretation of the upper limit. First we consider the case where the objects we would have detected are the bare nuclei of the comets. The absolute nuclear magnitude is computed from the apparent magnitude ($m$) using $$H_N = m - 5 log (r \Delta) - \beta \alpha \ \ ,\eqno(11)$$ where $r$ and $\Delta$ are the heliocentric and geocentric distances of the object, respectively; $\alpha$ is the phase angle and $\beta$ the linear phase coefficient for which we adopt a value of 0.035 mag/deg (Jewitt 1991). About the opposition effect, we can not say very much, except that it is probably very small. The reason for this is that the only available investigations are of asteroids (e.g. Bowell {\it et al.} 1989). Assuming the same behavior as for asteroids with very low albedo, we expect that the brightening of cometary nuclei at small phase angles is negligible. Using Jupiter's heliocentric and geocentric distances at the time of the observations ($r=5.4$ AU and $\Delta =4.4$ AU) as a mean for the encountering comets, and a mean value for the phase angle of 1\deg, we find a magnitude correction of $-$6.9. For an opposition magnitude of 20.7, we get an absolute limiting magnitude of $H_N^B<13.8$. By using a typical geometrical albedo of 0.03 (Jewitt 1991) for cometary nuclei the corresponding nuclear radius is 9 km. Although most of the observed comets have absolute nuclear blue magnitudes between 15 and 18, there are a few (certain) cases of extreme brightness such as P/Neujmin 1 ($H_N^B=13.3$) and P/Oterma ($H_N^B=13.7$) (Fern\'andez {\it et al.} 1992). In fact, P/Neujmin 1 is one of the few well--studied cases (Jewitt and Meech 1988). There are others, generally with high perihelion distances, but not as reliable. We then conclude that the total number of Jupiter family comets both in the active and inactive phase brighter than absolute nuclear magnitude 13.8 is less than 210. This estimate can be transformed into an estimate of the total size of the JF population if a certain magnitude distribution is assumed. Figure 8 shows the corresponding upper limits at different magnitudes for three power law indices of the nuclear magnitude distribution ($s=$0.3, 0.4 and 0.5). On the upper horizontal axis we show the corresponding nuclear radius for a geometrical albedo of 0.03. For an absolute nuclear magnitude of $H_N^B=18$ (corresponding to a nuclear radius of 1.3 km) the upper limits for the combined and individual populations of active and dormant JF comets are 3900, 10000 and 27000 for $s=$ 0.3, 0.4 and 0.5, respectively. A second alternative is to consider that a known fraction of the comets is active at distances larger than 4 AU from the Sun, and the observed magnitudes hence ``contaminated'' by a coma. To investigate this we looked at the data available on observations of JF comets at large heliocentric distances. From the Comet Light Curve Catalogue/Atlas (CLICC/A, Kam\'el 1992) we extracted all (19) the observations of JF comets made at larger heliocentric distances than 4 AU. We then added data for two comets published later (Luu and Jewitt 1992, Mueller 1992), as well as own observations of 6 nuclei not yet published (two of them already included in CLICC/A). We ended up with 11 out of 25 objects classified as inactive. Based on this fact, we then argue that roughly half of the comets appear nuclear at distances $\ge$ 4 AU and the rest is brightened by typically 1.5 mag. There is of course an observational bias towards active comets, (i.e. where the activity makes an otherwise too faint comet brigter than the limiting magnitude), since unsuccessful observations seldom get reported as limits on the magnitude of the undetected object. Define $x$ to be the fraction of comets that appear to be $\Delta H$ brighter than their respective nuclear magnitudes $H$. The distribution of the absolute magnitude ($\nu(H)$) of the observed objects is then related to the distribution of absolute nuclear magnitudes $N(H)$ by \vbox{ $$\nu(H) = (1 - x) \ N(H) \ + \ x \ N(H+\Delta H)\eqno(12). $$ Using eq. (7), we get $$\eqalignno{\nu(H) & = (1 - x) \ N(H) \ + \ x \ 10^{s \Delta H} N(H)\cr & = \left( 1 + (10^{s \Delta H} - 1) \ x \right) \ N(H) \cr & \equiv \beta N(H) & (13). }$$ } \noindent For $x=0.5$, $\Delta H=1.5$ mag. and $s=$0.3, 0.4 and 0.5 we get $\beta=$1.9, 2.5 and 3.3, respectively. The upper limits of the number of active JF comets brighter than nuclear magnitude 13.8 is then 110, 80 and 60 for the three indices respectively. In Fig 8 we show the corresponding upper limits at different magnitudes for the three power law indices. For an absolute nuclear magnitude of $H_N^B=18$ the upper limits for the population of active JF comets are 2000, 4000 and 8000 for $s=$ 0.3,0.4 and 0.5, respectively. For the different distribution indices we then have a range of possible upper limits, depending on the activity level of JF comets at 5 AU from the Sun. \bigskip \noindent {\bfS 7. Discussion} \medskip \noindent As mentioned above, our knowledge about the number distribution of comets is limited to objects with small perihelion distance. The problem to estimate the true number distribution for the whole population has been approached using both observational and theoretical considerations. Dynamical studies by Froeschl\'e and Rickman (1980) and Fern\'andez (1984) show that there is a steep increase of the number of comets with $q$. Perhaps with a flattening distribution at high $q$ values. From high-precision numerical integrations of the whole sample of JF comets over 4400 years, Nakamura and Yoshikawa (1992) estimate that there should be 20 times more comets with $q$ in the range 3.5 to 5 AU than in the range 1 to 2 AU, reinforcing the idea of a steep increase. Regarding observational evidence we have the detailed analysis of Roemer's photographic nuclear magnitudes data set by Shoemaker and Wolfe (1982). In the $q$-range 1.1 to 1.7 AU, where they claim to have near completeness of the discoveries, they found a flat distribution of the number of comets with $q$. For higher values of $q$ their sample suffers from incompleteness and they draw no conclusion. They extrapolate the flat trend to a distance of 5.5 AU, and get a total number of 1400 JF comets down to blue nuclear magnitude $H_N^B=18$. Several comets with $q$ in the range 1.1 to 1.7 AU have since then been discovered and the estimate should therefore be revised upwards. In contrast, a more recent analysis of a large amount of observational material by Fern\'andez {\it et al.} (1992) points towards a steep increase of the number distribution with $q$, at least up to $q \approx 2.5$. Estimates at larger distances were also in this case prevented by incompleteness of the sample. Nevertheless, they estimated that the total number of JF comets brighter than absolute total magnitude $H_{T,10}=11$ is $\approx 4000$. Although the magnitude limits applied in both estimates are different, but nevertheless comparable, the factor of 3 of discrepancy between the two results illustrates the large uncertainties that exist on our knowledge of the true size of the JF population. Regarding the population of dormant and/or extinct comets, the only attempt to estimate its size was done by Shoemaker {\it et al.} (1986). They considered that the 9 Jupiter approaching asteroids discovered by 1986 are extinct comets. They assumed for this population the same magnitude and flat perihelion distribution found for cometary nuclei. A number of 1460 extinct comets up to $H_N^B=18$ was estimated, very similar to the number of active objects. Comparing with these previous estimates of the size of JF population, the upper limits found in this work (assuming no activity at 5 AU) seem still to be too high. However, remember that the upper limits also hold for both active and dormant comets. Little is known about the total number of active JF comets and we know almost nothing about the combined population. If the active comets contribute significantly to the population of objects encountering Jupiter, the upper limits set by our search are comparable to previous estimates. The big advantage of our estimate is due to the fact that the number of comets encountering Jupiter is not dependent on the particular shape of the number distribution with $q$. To estimate the total size of the JF population we do not make use of an uncertain $q$-distribution, but we do make use of an ill--defined nuclear magnitude distribution as the other estimates did. Another interesting conclusion is that the comets discovered by this strategy would have a $q$-distribution that would reflect the true $q$-distribution of the whole JF population. For example, if the population of JF comets consists of 3 to 4 thousand members up to nuclear blue magnitude 18, and if a region of similar size to the one observed by us could be covered with a limiting magnitude $B\sim25$ for stationary objects, about 50 new comets would be found. A better knowledge of the activity level at large heliocentric distances and of the size distribution of JF comets can make our strategy a powerful, reliable and fast tool to estimate the size of the JF population as well as its true $q$-distribution. \bigskip \noindent {\bfS 8. Future work} \medskip \noindent Figure 5 in TL92 shows that the half--duration of encounters are typically less than one year. Since the time between encounters never is shorter, most of the comets that are encountering Jupiter at one opposition will generally not be close to the planet at the next. New comets will have replenished the almost constant number of Jupiter vicinity visitors. A new search during next Jupiter opposition will hence give a better constraint on the size of the JF population. If we do not find any comet during next opposition, the upper limit given above will be reduced by a factor of $\approx 0.7$ (assuming a typical duration of encounter of 1.4 yr); and we will be much closer to the population sizes estimated by other methods. At least if a power law index with $s<0.4$ is assumed. A second search is now being conducted (March-April 1993) and its results will be published elsewhere. During the course of this second search an extraordinary comet encountering Jupiter was discovered by C. and E. Shoemaker and D. Levy (comet P/Shoemaker-Levy 9 -- 1993e, IAUC 5725). The comet split in more than 10 detectable pieces possibly due to tidal forces in a close encounter with Jupiter in 1992. We looked unfruitfully for comet 1993e in all the plates and in particular in a broad region centered on the positions suggested by Yeomans and Chodas (orbit N\deg\ 24) in the 1993e Bulletin board. The non--detection agrees with the hypothesis that the splitting ocurred close to July 2 1992, during the close approach. At the time of our observations, 4 months before, only the bare nucleus would have been detectable. On March 6 1992, the object was 5.5 AU from the Sun and 4.5 AU from the Earth and the expected relative motion to Jupiter was 2.3 "/hr. A lower limit of the absolute blue nuclear magnitude is then B(1,0)=14.3. For a geometrical albedo of 0.03, the nuclear radius should be smaller than 7.2 km. The discovery of this object illustrates the importance of looking for comets at a very interesting phase of their dynamical and physical evolution. \vskip 20mm \noindent{\bfS Acknowledgments} \vskip 5mm This search could not have been made properly without the support from the staff of the ESO-Schmidt telescope. We express our sincere thanks for the effort they spent in carrying out an important part of this project. 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LII, No. 2, Helsinki. \vfill \eject \nopagenumbers \noindent{\bfS Figure Captions} \bigskip \noindent {\bf Fig. 1 --} An ecliptic x--y projection of the orbits of the planets Earth, Mars and Jupiter together with the positions of 4295 asteroids (small dots) and 122 Jupiter family comets (large dots) on the date of Jupiter opposition, February 29, 1992. \medskip \noindent {\bf Fig. 2 --} The distribution of observed short-period comets at any time in a rotating pulsating frame with the Sun at (0,0) and Jupiter at (1,0). The scale (in arbitrary units) is also shown. \medskip \noindent {\bf Fig. 3 --} The dependence of the correction to the limiting magnitude ($\Delta m$) due to the trailing of a point source. $v t$ is the length of the trail and $d$ the FWHM of the seeing disk. \medskip \noindent {\bf Fig. 4 --} Histograms of the number of comets as a function of perihelion distance at different instants in the $\pm$ 10000 yr integrations. a) backward integrations: full line at present, dashed line at -5000 yr in the past and dotted line at -10000 yr. b) forward integrations: full line at present, dashed line at +5000 yr into the future and dotted line at +10000 yr. \medskip \noindent {\bf Fig. 5 --} The neighboring probability ($p$) as a function of $q$ for encounters with $D < 0.5$ AU (lower curve) and 1 AU (upper curve). \medskip \noindent {\bf Fig. 6 --} The dependence of the neighboring probability on $D$ for different $q$-bins (thin full lines). The full-drawn thick line is the mean of the curves in the range $1.5 < q < 4.5$ AU. The dots correspond to a power law with index 2.5 and $p(D < 1 {\rm AU}) = 0.014$. \medskip \noindent {\bf Fig. 7 --} The projected relative velocity ($U_{proj}$) as a function of $q$ for different units of the projected velocity. The left $y$-axis shows $U_{proj}$ in units of Jupiter's orbital velocity ($U_J$). The hourly motion in arcseconds, corresponding to a geocentric distance of 4.4 AU at opposition is on the inner right $y$-axis. The outer right $y$-axis corresponds to magnitude corrections computed with eq. (4) for 90 min. exposures. \medskip \noindent {\bf Fig. 8 --} The upper limits of the cumulative number of JF comets as a function of the absolute nuclear magnitude and radius using a geometrical albedo of 0.03. The steepest lines correspond to a power law magnitude distribution with an index $s = 0.5$, the middle lines to $s = 0.4$ and the least steep lines to $s = 0.3$. The full lines correspond to estimates assuming that comets are inactive at 5.4 AU. The dashed lines correspond to estimates assuming coma contribution. \bye