## 3. Description of the collapse of the protosolar nebulaTo describe the motion of a cometary nucleus inside the collapsing protosolar nebula, we need to solve its equation of motion: where The collapse of a spherically symmetric, isothermal gaseous sphere has most recently and most completely been described by Whitworth & Summers (1985). They followed the previous work by Penston (1969), Larson (1969), Shu (1977), and Hunter (1977). The equations describing the collapse can be written as: where is density of the gaseous
sphere at distance Following Shu (1977), Whitworth & Summers re-wrote these
equations with the help of the only dimensionless variable and the dimensionless quantities The new equations have the form Whitworth & Summers found a whole two-parameter continuum of solutions of equations describing the collapse. These solutions can be divided qualitatively, with respect to the initial density behaviour, into three groups representing: (i) a centrally rarefied cloud, (ii) a cloud with flat central density behaviour, (iii) a cloud with a centrally peaked density. Quantitatively, a given cloud can be characterized by the ratio of central and shock-front-(sonic point)-distance densities, . According to Whitworth & Summers, an infinite number of types of density behaviour outside the shock front can occur for one given type of behaviour inside the front. We empirically find and utilize the fact that the appropriate numerical integration always results in the same behaviour outside the shock front (using whatever reasonable integration step), if the integration is performed from centre to edge of the cloud and no integration step starts exactly (or almost exactly) in the appropriate sonic point. (Such a start would be incorrect because the sonic point is a singular point.) In nature, only centrally flat or mildly peaked clouds can be considered in the given context, because a collapse of a matter into a relatively rarefied space is improbable or, on the other hand, supposing a highly centrally peaked cloud, we would avoid the problem of how a denser region appears in a more or less uniform gaseous interstellar environment anyway. It is clear that the model of a cloud with initially flat
() behaviour of density around the
cloud centre (the solution found independently by Penston 1969 and
Larson 1969) represents the first border of cloud models acceptable in
the context of our problem. This model has already been considered in
our first study (Neslusan
1999). As was demonstrated by Whitworth & Summers (1985,
Table 1), there is no solution of Eqs. (10) and (11) in the
range of from 1.955 to 3755. Hence,
the value appears to be natural as
the second border of acceptability. To avoid numerical difficulties in
numerical integration at the true border, we assume value
as the border, in practice. Whilst
the cloud model at the first border is characterized by the value
In both models considered, the beginning of the nebula collapse is considered to be the moment when the mass inside the shock front just equals ( is solar mass). The radius of the spherical shock front is at that moment. Unfortunately, the end of the collapse is not well defined. The
collapse described with Eqs. (10) and (11) would continue during
an infinite period. In nature, the real collapse of protosolar nebula
was terminated by a wind of corpuscular particles and radiation
emitted by the protosun as it came into being. Alternatively, the
accretion of the protosun could be terminated by the disruption of the
appropriate star forming environment by a nearby hot star (Hestler et
al. 1996). Since no final evolution of the outer parts of the
protosolar nebula has been exactly described, we considered it best to
terminate our numerical integration for a given cometary nucleus, when
it is located at such distance The rotation of the protosolar nebula is not considered because it is not significant during the period investigated. An additional orbital momentum coming from the rotation could only enlarge the final orbits, therefore neglecting the rotation does not act against the main goal of our endeavour. We note that the motion of a nucleus inside the shock front can be described analytically, in the form of an infinite power series, for the flat density model (Neslusan 1997). The analytical solution can be utilized in checking the numerical integrability of the problem and can also accelerate the computations. © European Southern Observatory (ESO) 2000 Online publication: September 5, 2000 |