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

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4. On the initial assumptions and numerical integration

Assuming the creation of cometary nuclei in molecular clouds, it seems to be reasonable to suppose that their number density in an element of volume is roughly linearly proportional to the density of cloud material in this volume.

The mathematical cloud model described by Eqs. (10) and (11) extends to an infinite radius. In practice, we need to constrain the size of the cloud, when we construct the initial distribution of the hypothetical nuclei: since we terminate the numerical integration of motion of a given nuclei when the nebular mass inside reaches [FORMULA] (see Sect. 3), we consequently perform the numerical integration only for those hypothetical nuclei that are inside the infalling shock front at the beginning of integration. To evaluate the possible influence of the nuclei outside the shock front on the resulting distribution, we construct one distribution containing only the nuclei initially inside of shock front and another distribution containing also the nuclei initially outside of this front, in which their resulting distribution is identical to their initial distribution (no integration of their motion is performed). It is convenient to consider the nuclei outside the shock front up to a heliocentric distance at which their number density does not decrease below about one order of the number density at the distance of shock front in the collapse beginning, i.e. up to a distance, where [FORMULA] (in other words, up to [FORMULA] or [FORMULA] for the model with [FORMULA] or [FORMULA], respectively).

Concerning the velocity magnitudes, we assume a Maxwell-Boltzmann distribution with the most probable velocity [FORMULA] (the peak of the distribution).

Specifically, we assume, in the beginning of the collapse, that there are


hypothetical cometary nuclei in each element of volume, [FORMULA], of the considered cloud, which move in the direction of a given elementary space angle, [FORMULA], characterized by angles [FORMULA], [FORMULA], with the initial velocity ranging from [FORMULA] to [FORMULA]. [FORMULA] is a constant proportional to the total number of cometary nuclei in the cloud, [FORMULA] and [FORMULA] are the densities of material in the given element of volume and in the centre of the nebula, respectively.

Because of symmetries, we can analytically integrate through all the possible values [FORMULA], [FORMULA], and [FORMULA] and thus simplify Eq. (13) resulting in


In our particular case, we consider the grid points constructed putting [FORMULA], [FORMULA], and [FORMULA]. The quantity [FORMULA]AU is the radius of the shock front at the beginning of the collapse. The circular velocity at [FORMULA] at the beginning of the collapse, [FORMULA]m s-1, is chosen as a characteristic velocity scale. In the integration, the range of the individual variables is: [FORMULA], [FORMULA], and [FORMULA] (the upper limit, [FORMULA], was chosen to satisfy the requirement that the number of nuclei moving with this velocity has to be less than [FORMULA] of those moving with maximal velocity [FORMULA], i.e. the number of nuclei above the limit is negligible).

The most probable velocity, [FORMULA], is a free parameter in the integration. We perform a series of six integrations with [FORMULA] equal to 0.250, 0.375, 0.500, 0.625, 0.750, and 0.875[FORMULA].

The integration is performed using the Runge-Kutta method of numerical integration.

As we already mentioned (Sect. 3), no exact description of the end of the protosolar nebula collapse has been made, therefore we are compelled to terminate each integration prematurely: we terminate it when the mass contained within the interior of the sphere of radius equal to the distance of the nucleus from the centre the nebula reaches [FORMULA]. A further collapse above this mass was improbable. It appears that a nucleus spirals toward the centre during the collapse. When the collapse is artificially turned into an expansion (changing the sign of time), then the nucleus begins to spiral outward. Taking this into account, the premature termination of the integration implies that our resulting orbits should be regarded as minimum-distance orbits.

Apart from the above termination, we also terminate the integration, when the nucleus approaches the centre of the nebula at a distance less than [FORMULA]AU. Such a nucleus is regarded as incorporated in the protoplanetary disc. Moreover, we check whether the nucleus enters a region having density higher than [FORMULA]kg m-3 (critical density at which the material begins to be heated (Gaustad 1963) and the model of collapse used becomes inadequate). Eventually we check that the force of drag in the cloud material decelerating the nucleus does not exceed [FORMULA] of the gravitational force assuming the minimum acceptable mean density of the cometary nucleus of [FORMULA]kg m-3 (Rickman 1987; Rickman et al. 1987), i.e. the maximum efficiency of drag deceleration. Such a deceleration could debase the result of given numerical integration. It appears that neither of these two latter cases occur at [FORMULA] AU.

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Online publication: September 5, 2000