\magnification=1095 \hsize 16 true cm \vsize 23 true cm \hoffset 1 true cm \baselineskip 12pt \rm ~ \vskip 9 true cm \centerline{\bf Temporary Satellite Capture and Orbital Evolution of} \medskip \centerline{\bf Comet P/Helin-Roman-Crockett} \vskip 2 true cm \noindent{\bf Gonzalo Tancredi,~~~Mats Lindgren,~~~Hans Rickman} \bigskip\noindent {\obeylines Astronomiska Observatoriet, Uppsala University Box 515 S-75120 Uppsala, Sweden} \vfill\eject \baselineskip 18pt \def\lta{\raise.4ex\hbox{$<$}\llap{\lower.6ex\hbox{$\sim$}}} \def\gta{\raise.4ex\hbox{$>$}\llap{\lower.6ex\hbox{$\sim$}}} \rm {\sl 1. INTRODUCTION} \vskip 15mm Close encounters of minor bodies with Jupiter have been studied recently for two main purposes. The first is to analyse transfer mechanisms between different kinds of heliocentric orbit like those responsible for cometary capture or, more generally, the delivery of objects into Earth-approaching orbits from more remote regions. The second is to investigate the capture hypothesis for the origin of the outer Jovian satellites. For the problem of cometary origins, the close encounters play an important r\^ole since they govern the dynamical replenishment of the short-period comet population, sending comets from the outer to the inner solar system and vice versa. Indeed, most of the observed short-period comets experienced a close encounter with Jupiter (minimum distance to the planet $<$ 0.5 AU) during the last 400 years (Carusi {\it et al.} 1985). These encounters often mark the captures of the objects from orbits with longer periods and larger perihelion distances. Quite a few examples of transitions from perihelion-tangent to aphelion-tangent orbits are known (Carusi {\it et al.} 1987). In more rare cases the encounters have led to temporary satellite captures (TSC) which are of interest as possible routes by which stable captures might occur e.g. in the presence of a dissipative medium. Carusi {\it et al.} (1985) found 17 events where comets of the Jupiter family experience TSC in the sense of having temporarily an elliptic jovicentric orbit (Carusi and Valsecchi 1979). Nearly all the nine comets in question have values of the Tisserand parameter $T\approx 3$, indicating the possibility of low-velocity encounters with Jupiter whereby the jovicentric orbit can easily become elliptic. Figure 1 shows that three of the comets form a fairly distinct little group in the parametric plane of aphelion ($Q$) and perihelion ($q$) distances. Their distinctive property is to have large-$q$, low-eccentricity orbits near the 3/2 resonance with Jupiter ("quasi-Hilda type" motion; see Kres\'ak 1979), and the three members are: P/Oterma, P/Smirnova-Chernykh and P/Gehrels 3. However, the jovicentric orbit may more or less occasionally pass through a brief elliptical stage without Jupiter ever having full control of the motion. Thus a more restrictive definition of TSC may be applied (Rickman and Malmort 1981), whereby the object must perform a whole revolution around the planet in some sense. With such a definition there used to be only one comet fulfilling all requirements: P/Gehrels 3. This object was discovered shortly after a deep and long-lasting TSC, which sent the comet from an orbit totally outside the planet into one completely inside it (Rickman 1979). A new member of the above-mentioned group was discovered on Jan. 2, 1989: comet P/Helin-Roman-Crockett (hereafter: HRC). The peculiar dynamical properties of all the earlier known members, including their recent and complicated encounters with Jupiter, make it interesting to investigate if comet HRC shares a similar behaviour. We have thus performed numerical integrations of the motion of comet HRC using a model and algorithm to be described in section 2. Section 3 presents the results for a close encounter with Jupiter preceding discovery of the comet, showing that it experienced a TSC even with the most restrictive definition. In section 4 we present results concerning the long-term evolution of the orbit of HRC and its similarities with that of other quasi-Hilda type comets, and in section 5 we discuss the possible interpretations of these similarities. \vfill\eject {\sl 3. RESULTS: ENCOUNTER WITH JUPITER} \vskip 15mm Marsden (1989) noted that HRC had a close encounter with Jupiter shortly before its discovery. We thus performed backward integrations of the orbit of the comet, starting from the osculating elements derived from observations in early 1989. For the first set of integrations we used orbit A, which is based on an observational arc of three months (Jan. -- March 1989) with variants corresponding to even earlier unperturbed orbits. The second set of integrations focussed on orbit B, based on five months of observations (Jan. -- May 1989). In what follows we shall concentrate on orbit B, using the results for orbit A mainly to indicate the level of uncertainty. Using the dynamical model described in the previous section, we find that the comet was captured as a temporary satellite of Jupiter for an extended period of time independent of which of the starting orbits is used. Fig. 2({\it a,b}) shows the jovicentric path in a rotating frame for both orbits, and Fig. 3({\it a,b}) shows the corresponding evolutions of the jovicentric orbital energy. With the preferred orbit comet HRC remained gravitationally bound to Jupiter for 11.53 years (Dec. 1973 to July 1985) and it performed two revolutions in the prograde sense around the planet during that interval. The closest approach to Jupiter occurred on Aug. 15.25, 1976, to a minimum distance of 0.018 AU. This was the first approach, and after a long excursion into the outer parts of Jupiter's sphere of action, where the distance from the planet stayed between 0.20 and 0.33 AU for five years, a second approach to a minimum distance of 0.063 AU occurred on Aug. 9, 1983. After this approach the comet escaped from Jupiter, moving toward perihelion in Sept. 1988. Comparing with the results for orbit A, we find that the jovicentric path of the latter was very similar to the final revolution of the preferred orbit. The same approach to Jupiter occurred in 1983, but the preceding sojourn at larger distances in orbit A does not lead back to an earlier close encounter. With orbit A the comet was bound to Jupiter for 7.4 years, performing one revolution around the planet. We note that in both cases the geometry of initial approach was nearly the same but there is a time shift of four years between the two events. The fact that both orbits A and B yield TSC's which are nearly identical for more than seven years is an indication that this feature of the TSC is reliable. Indeed we have further confirmation in the results for a vast sample of starting orbits in the vicinity of B, to be discussed in section 5. In spite of chaotic behaviour developing over several revolutions around Jupiter, these agree to good approximation during the 7 yrs period in question. Moreover, the comet did not reach the orbits of the Galilean satellites, although it came well inside the outer jovian satellites. Under such circumstances the satellites of Jupiter should not have any appreciable influence on the orbital evolution of the comet (Carusi and Valsecchi 1979). In Table II we present the heliocentric orbital elements for orbits A and B just before and after the encounter. Some similarities between the pre-encounter orbits are obvious, as further discussed below, but there is naturally a discrepancy in the orientations of the orbits corresponding to the four years' difference in the time of approach to Jupiter. As a common feature, HRC had a pre-encounter orbit between Jupiter and Saturn, and the close encounter transferred it into a new orbit completely inside Jupiter. \vfill\eject {\sl 4. RESULTS: LONG-TERM ORBITAL EVOLUTION} \vskip 15mm As illustrated by the difference of orientation of the pre-encounter orbits in Table II, it is very difficult to make precise statements about the long-term evolution of an object making close encounters with Jupiter. This is of course a consequence of the chaotic nature of such an evolution (Everhart 1979; Carusi {\it et al.} ....) -- even if we consider very close initial conditions, we may find very different evolutions especially if the motion leads through a complicated encounter involving several close approaches to the planet (Carusi {\it et al.} ....; Rickman and Malmort 1981). For HRC our possibilities of characterizing the orbital evolution before the recent encounter are obviously limited. On the other hand, the forward evolution shows very little difference between nearby starting conditions until the next encounter, which will occur around 2075. In Fig. 3 we present the evolutions of perihelion distance ($q$), aphelion distance ($Q$), inclination ($i$) and Tisserand parameter ($T$) during the period 1900-2100 for orbit B. Throughout the interval the inclination remains low and the Tisserand parameter stays slightly above 3, except for temporary excursions during the encounters. From 1987 to 2070 even the details appear reliable such as, e.g., the little jumps in $q$ experienced around 2010 and 2050. These correspond to relatively distant encounters with Jupiter. The first one occurs approximately two jovian revolutions after the recent encounter due to the fact that the comet is currently near the 3/2 resonance. As a result of the perturbation the comet moves closer to the 4/3 resonance, so the following encounter occurs approximately three jovian revolutions later. The encounter found around 2075 is a particularly deep and long-lasting one, especially in orbit B. Fig. 4 shows the jovicentric path during this encounter and the distance from Jupiter as a function of time for orbit B as an indication of the complex dynamics that HRC can meet. However, no confidence can be placed on the details since the uncertainty of the present orbital elements and the possible action of nongravitational forces during the coming century make the detailed geometry of initial approach to Jupiter in 2070 quite uncertain. In the case of orbit B the encounter will transfer the comet back to the region just outside Jupiter, into an orbit similar to the one it had before the recent encounter. In other cases the comet would stay inside Jupiter until further close encounters occur later on. As a tentative conclusion, comet HRC shares the behaviour of the other quasi-Hilda type comets in that it repeatedly switches between the regions just outside and just inside Jupiter's orbit. The region near the 3/2 and 4/3 resonances seems preferred for the interior sojourns. \vfill\eject \def\deg{$^\circ\!\!$} \def\H{\hfil} {\bf Table 1.} Initial orbital elements for P/Helin--Roman--Crockett used in the integrations. Orbit A is unperturbed, but orbit B includes planetary perturbations. In the former case, the epoch is an average date for the period of observations, but in the latter case it is the epoch of osculation. $a$ is the semi--major axis; $e$ is the eccentricity; $q$ is the perihelion distance; $Q$ is the aphelion distance; $i$ is the inclination; $\Omega$ is the longitude of the ascending node; $\omega$ is the argument of perihelion; and $M$ is the mean anomaly at the epoch. $i$, $\Omega$ and $\omega$ are referred to the ecliptic and equinox of 1950.0 (\#\#\# CHECK \#\#\#). \vskip 10 mm \settabs 6 \columns \+& &\hfil{\sl Orbit A}&\hfil{\sl Orbit B}&\cr \vskip 3 mm \+&\hfil Epoch (JD) &\hfil2447562.5&\hfil2447400.5&\cr \+&\hfil $a$ (AU) &\hfil4.0406108&\hfil4.0423790&\cr \+&\hfil $e$ &\hfil0.1410927&\hfil0.1414364&\cr \+&\hfil $q$ (AU) &\hfil3.4705101&\hfil3.4706395&\cr \+&\hfil $Q$ (AU) &\hfil4.6107115&\hfil4.6141185&\cr \+&\hfil $i$ &\hfil4\deg.23763&\hfil4\deg.23372&\cr \+&\hfil $\Omega$ &\hfil91\deg.41474&\hfil91\deg.38876&\cr \+&\hfil $\omega$ &\hfil9\deg.62176&\hfil10\deg.19224&\cr \+&\hfil $M$ &\hfil17\deg.92936&\hfil357\deg.85905&\cr \vfill\eject {\bf Table 2.} Orbital elements for orbit A and B of P/Helin--Roman--Crockett The elements are given at epochs when the distance from Jupiter was about 2 AU, before and after the temporary satellite capture (TSC). The longitude of perihelion is denoted by $\varpi$. \vskip 10 mm \settabs 4 \columns \+&Before TSC\H&&After TSC\H&\cr \settabs 6 \columns \+&\H{\sl Orbit A}\H&\H{\sl Orbit B}\H&&\H{\sl Orbit A}\H&\H{\sl Orbit B}\H&\cr \vskip 3 mm \+\H Epoch\H&\H Jan. 28 1975\H&\H May 19 1971\H&&\H July 24 1987\H&\H July 24 1987\H&\cr \+\H$e$\H& \H0.1915779\H& \H0.2466365\H&& \H0.1438814\H& \H0.1438422\H&\cr \+\H$q$ (AU)\H& \H5.7593302\H& \H5.5903564\H&& \H3.4725931\H& \H3.4728647\H&\cr \+\H$Q$ (AU)\H& \H8.4889941\H& \H9.2507040\H&& \H4.6398184\H& \H4.6398095\H&\cr \+\H$i$\H& \H1\deg.94621\H& \H2\deg.47991\H&& \H4\deg.23173\H& \H4\deg.22805\H&\cr \+\H$\varpi$\H& \H74\deg.92299\H& \H335\deg.55742\H&& \H100\deg.27071\H& \H100\deg.852\H&\cr \vfill\eject \settabs 8 \columns {\bf Table 3.} Comparison between the orbital elements of P/Helin--Roman--Crockett (HRC) and P/Gehrels 3. The elements are given at epochs corresponding to distances of about 2 AU from Jupiter, after the respective TSC. \vskip 10 mm \+&&\H HRC\H&&\H P/Gehrels 3\H&\cr \vskip 3 mm \+&\H Epoch\H&\H July 24 1987\H&&\H May 14 1976\H&\cr \+&\H$a$ (AU)\H&\H4.0563372\H&&\H4.0491897\H&\cr \+&\H$e$\H&\H0.1438422\H&&\H0.1540510\H&\cr \+&\H$i$\H&\H4\deg.22805\H&&\H1\deg.09837\H&\cr \+&\H$\varpi$\H&\H100\deg.852\H&&\H113\deg.58057\H&\cr \end