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Astron. Astrophys. 361, 369-378 (2000)

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5. Results and discussion

The 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 [FORMULA] equal to 1 and 1.954 respectively, but with the same value of the peak velocity [FORMULA], in the appropriate Maxwell-Boltzmann distribution of initial velocities of nuclei. Three such pairs of plots, for [FORMULA], 0.50, and [FORMULA], are given in each figure.

[FIGURE] Fig. 1a-f. The distribution of perihelion distances, q, of the cometary nuclei sample considered. The left-hand plots (a , c , e ) correspond to the model of protosolar cloud collapse with central and sonic-point initial density ratio, [FORMULA], equal to 1, the right-hand plots (b , d , f ) correspond to the model with [FORMULA]. The first pair of plots (a , b ) is constructed assuming the initial velocity distribution peak, [FORMULA], equal to [FORMULA]m s-1, the second pair (c , d ) for peak [FORMULA]m s-1, and the third pair (e , f ) for peak [FORMULA]m s-1. The values in abscissa are expressed in common-logarithm scale.

[FIGURE] Fig. 2a-f. The distribution of semi-major axes, a, of the cometary nuclei sample considered. The distribution is constructed for the same models of cloud collapse and Maxwell-Boltzmann distributions of initial velocity as in Fig. 1.

[FIGURE] Fig. 3a-f. The distribution of aphelion distances, Q, of the cometary nuclei sample considered. The distribution is constructed for the same models of cloud collapse and Maxwell-Boltzmann distributions of initial velocity as in Fig. 1.

As stated in Sect. 4, the integration of motion of the hypothetical nuclei was also performed for the values of [FORMULA] equal to 0.375, 0.625, and [FORMULA]. For all the values of [FORMULA], some numerical characteristics are given in Table 1.


Table 1. Some values characterizing the distributions of perihelion distances (q), semi-major axes (a), and aphelion distances (Q) of hypothetical cometary nuclei at the end of the protosolar cloud collapse for two models of the collapse and six Maxwell-Boltzmann distributions of the initial velocity of the nuclei. Each model is characterized by the ratio of central to sonic-point initial density ratio, [FORMULA]. A Maxwell-Boltzmann distribution of initial velocity with peak [FORMULA] is assumed. In the range of heliocentric distances from [FORMULA] to [FORMULA]AU considered for the numerical integration, the interval where the relative number of nuclei exceeds [FORMULA], and the maximum of the appropriate distribution are given. (The appropriate values are expressed in common-logarithm scale as indicated in the first column in brackets.) Moreover, the relative number of nuclei having final perihelion distances less than [FORMULA]AU and the relative number of nuclei in hyperbolic orbits, for each model (in per cent unit) are given. We considered the distribution of only those cometary nuclei that were present inside the shock front at the beginning of the protosolar nebula collapse. Below each ordinary value, in parentheses, a corresponding value is given, if the nuclei outside the shock front are also included (see second paragraph of Sect. 4).

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 [FORMULA]; 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 [FORMULA] for [FORMULA] and [FORMULA] (plots from (c) to (f)), but has an exponential-like tail above [FORMULA]. 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 [FORMULA] is a value from a certain interval. More specifically, a very low number of nuclei remains at the large distances at [FORMULA]. On the other hand, the number of nuclei in a cometary cloud does not continue to increase if the peak exceeds the value of [FORMULA], 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 [FORMULA]15 pc-3, rms relative velocity [FORMULA]1 km s-1, and stellar flux in the protosolar nebula's neighbourhood [FORMULA]15 stars pc[FORMULA]Myr-1. Then the mean separation between cluster stars is [FORMULA] AU. The mean impulsive change in the comet's velocity relative to the centre of nebula is given by [FORMULA]. The mean mass of an approaching star, [FORMULA], can roughly be assumed to be one solar mass. Before the protosolar nebula collapse, the highest number of cometary nuclei were at its border at distances [FORMULA] AU. The typical minimum distance of the approaching star causing the impulse, [FORMULA], can be identified approximately with the mean separation [FORMULA]. Thus, we obtain a one-impulse velocity change [FORMULA] m s-1 (0.006 [FORMULA]). Assuming the stellar flux [FORMULA]15 stars pc[FORMULA]Myr-1, we find that [FORMULA]8 stars cross the circular area of radius [FORMULA] per 1 Myr. According to Fernández, the nebular galactic environment in which the solar system formed could have persisted for at most a few [FORMULA] years. If the Sun formed as one of the last stars of the cluster, then the nuclei in the protosolar nebula could have been exposed to [FORMULA](8 Myr-1)[FORMULA](a few [FORMULA]years), i.e. roughly a hundred individual impulses. That means that the velocity dispersion from the mean approaches of cluster stars could have been expected to be from about 1.6 to 160 m s-1 (0.006 to 0.6 [FORMULA]).

Fernández moreover assumed a few very close stellar approaches. Such an approach of a star to the protosolar nebula could last [FORMULA] years, with [FORMULA] AU. In this case, the order of one-impulse velocity change can be estimated to be [FORMULA] m s-1 (0.9 [FORMULA]). Both the above estimates imply that the velocities of cometary nuclei in the birth cloud of the solar system could actually be dispersed to the expected magnitude by the neighbouring stars.

For an illustration of a possible real value of peak [FORMULA], 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 [FORMULA]m s-1 during the last 4.5 billion years (Delsemme 1985). If the original mean near-aphelion velocity (before the scattering) was negligible in comparison with the latter (the Gauss distribution velocity can be identified with the Maxwell one), then the value found by Delsemme corresponds to [FORMULA]m s- 1 = 171 m s-1 ([FORMULA]). We have to emphasize the indicative character of this illustration, however, because it is necessary to realize that the value [FORMULA]m s-1 is related to the nuclei in the outer Oort cloud accelerated by stellar perturbations during a long period, whilst [FORMULA] is related to the entire cometary spherical halo and initial velocity distribution.

Duncan et al. (1987) showed that the semi-major axes of comets in the inner Oort cloud ranged from about [FORMULA] to [FORMULA]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 [FORMULA] is equal or higher than about [FORMULA] (and does not considerably exceed [FORMULA], of course).

If [FORMULA], then roughly [FORMULA] to [FORMULA] 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 [FORMULA] 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 [FORMULA], then the initial number of these nuclei in the nebula is of order [FORMULA] to [FORMULA], what corresponds with an average number density of order [FORMULA] to [FORMULA] nuclei per AU3.

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 [FORMULA] 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 [FORMULA]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 [FORMULA] AU was about the same order as the number of nuclei in the Oort cloud. In other words, there were at least [FORMULA] 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 [FORMULA]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 [FORMULA] (in fact it is probably less than about [FORMULA]). 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.

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Online publication: September 5, 2000