## 5. Results and discussionThe distributions of perihelion distances, semi-major axes, and aphelion distances of the assumed grid of cometary orbits are displayed in Fig. 1, Fig. 2, and Fig. 3, respectively. Each figure consists of plots from (a) to (f), where each pair of adjoining plots show the corresponding distributions for two cloud models characterized by ratios equal to 1 and 1.954 respectively, but with the same value of the peak velocity , in the appropriate Maxwell-Boltzmann distribution of initial velocities of nuclei. Three such pairs of plots, for , 0.50, and , are given in each figure.
As stated in Sect. 4, the integration of motion of the hypothetical nuclei was also performed for the values of equal to 0.375, 0.625, and . For all the values of , some numerical characteristics are given in Table 1.
Hypothetical cometary nuclei inside the shock front were considered in the construction of the plots in Figs. 1-3. If analogous plots are constructed including also the nuclei outside the shock front (up to the distance where ; see Sect. 4), we observe an insignificant shift of the bars to larger abscissa values (the abscissa values for the peaks of the distributions are given in Table 1). In the distribution of perihelion distances (Fig. 1), the behaviour is not truncated at for and (plots from (c) to (f)), but has an exponential-like tail above . Therefore, ignoring the external nuclei has no significant influence on the conclusion about the large heliocentric distances of these nuclei after the collapse. Looking at the figures, it is clear that the nuclei remain at large heliocentric distances, if their initial velocity distribution lies within an appropriate range, i.e. if is a value from a certain interval. More specifically, a very low number of nuclei remains at the large distances at . On the other hand, the number of nuclei in a cometary cloud does not continue to increase if the peak exceeds the value of , because an increasingly significant fraction of the nuclei leaves the system along hyperbolic orbits (see Table 1). We can assume that the nuclei gained their velocities mainly due to
the gravitational perturbations by the protostars being formed in the
neighbourhood of the protosun, in a common association. Unfortunately,
it is difficult to appreciate the possible complex perturbations of
motion of the nuclei from the beginning of their creation in an
interstellar cloud to the beginning of the collapse of protosolar
cloud. To give an order of magnitude estimate of the velocity
dispersion, we adopt the concept of the birthplace of the solar system
proposed by Fernández (1997). In this concept, the Sun formed
within a molecular cloud and, perhaps, a star cluster. It is
reasonable to assume that some stars of the cluster formed earlier and
some later than the Sun. The stars which formed earlier influenced the
motion of the nuclei in the birth cloud of solar system. Following
Fernández, let us assume a cluster of stellar density
15 pc Fernández moreover assumed a few very close stellar
approaches. Such an approach of a star to the protosolar nebula could
last years, with
AU. In this case, the order of
one-impulse velocity change can be estimated to be
m s For an illustration of a possible real value of peak
, it might be worthwhile, in this
context, to mention the fact that stellar random perturbations of
orbits in the outer Oort cloud have scattered the near-aphelion
velocities by m s Duncan et al. (1987) showed that the semi-major axes of comets in the inner Oort cloud ranged from about to AU (from 3.5 to 4.3 in common-logarithm scale). The axes longer than a few thousand astronomical units are necessary the Oort cloud perturbers could move the nuclei to the dynamically active outer cloud. Our integration of cometary orbits shows that the distribution of semi-major axes follows the requirement by Duncan et al., if is equal or higher than about (and does not considerably exceed , of course). If , then roughly
to
of the cometary nuclei, being in
the protosolar nebula before its collapse, remained in the Oort cloud
after the collapse (the complement of the relative numbers of comets
with and those having hyperbolic
orbits - see Table 1). So, the number of comets in the nebula
before the collapse was a factor of from 2 to 5 higher than that in
the Oort cloud after the collapse. If the initial number of Oort cloud
comets is known, then we can estimate the total number as well as the
average number density of comets in the nebula before the collapse. If
we suppose an initial total number of cometary nuclei in the Oort
cloud of order , then the initial
number of these nuclei in the nebula is of order
to
, what corresponds with an average
number density of order to
nuclei per AU Figs. 1 to 3 give the appropriate distributions of the entire Oort cloud comet population in the beginning of the existence of the solar system. A majority of comets represents the inner cloud. As a consequence of the cloud perturbers (stars and massive interstellar clouds randomly passing the solar system, galactic disc and nucleus), the comets from the inner cloud have been pumped up to the outer cloud to form it during the entire subsequent period. This scenario of replenishment of the outer cloud presented by Duncan et al. (1987) is also assumed in our concept. After the formation of the Oort cloud, the formation of the solar system continued with the creation of the protosun and planetesimals in the protoplanetary disc. If we suppose a total number of cometary nuclei in the inner Oort cloud of order at the end of our integration (at the beginning of planet formation), then a significant number of these nuclei approached the centre of the system to nearer than AU (see Table 1), and they can be assumed to have become part of the protoplanetary disc due to the drag of the disc material. Since these nuclei ended up in a relatively dense region, they had to be altered in interactions with surrounding material. In this context, it is worthwhile to mention the conclusion by Delsemme (1991) on the probable origin of two populations of comets of different symmetry (the Oort cloud and the Kuiper-Edgeworth belt). We state that the comet-like bodies of Kuiper-Edgeworth belt cannot be regarded as bodies identical to the common comets because of their different birth-place on the outskirts of the protoplanetary disc. However, we can expect an essential similarity in the composition of both groups because of the identity of their original material and the similarity of creation conditions. As can be seen in Table 1, the number of nuclei with AU was about the same order as the number of nuclei in the Oort cloud. In other words, there were at least cometary nuclei, in the protoplanetary disc at the beginning of the accretion process, which should have accelerated the accretion. In the suggested comet origin scenario, we do not need to tune initial conditions of planet and comet origin theories. For example, the theory of planet formation by Greenberg et al. (1984) results in planetesimals in order of km in diameter in the Uranus-Neptune zone. However, this size interval does not correspond to the observed range of diameters of cometary nuclei, which are usually 2 orders of magnitude lower. No comet on a clearly interstellar trajectory has been observed passing through the region of planets. This fact constrains the number density of comets in interstellar space. Weissman (1990) estimated this density taking into account the primordial theory scenario of ejection of comets from the Uranus-Neptune region (it is estimated that between 3 and 50 times as many comets are ejected by protoplanets as are placed in the Oort cloud) and obtained a value which exceeds the upper limit by about 1.3 to 23 times. Looking at Table 1, the amount of nuclei escaping into interstellar space along hyperbolic orbits does not exceed about (in fact it is probably less than about ). This amount is much lower than that yielded by the primordial theory. Hence, the estimate of the space density of interstellar comets should considerably be reduced. © European Southern Observatory (ESO) 2000 Online publication: September 5, 2000 |