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

3. Description of the collapse of the protosolar nebula

To describe the motion of a cometary nucleus inside the collapsing protosolar nebula, we need to solve its equation of motion:

where m is mass of the nucleus, is its radius vector, is the second time derivative of this vector, G is gravitational constant, and is mass inside the sphere of radius . To integrate this equation, we have to know mass . This mass can be obtained from the equations describing the collapse of the nebula.

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 r in time t, u is outward radial flow velocity, P is pressure, is isothermal sound speed equal to , and k is Boltzmann's constant. In agreement with other authors (e.g. Penston 1969; Larson 1969), we put the temperature of the gaseous sphere K. The mean molecular weight, µ, of a typical molecular cloud is ( is the mass of a hydrogen atom), approximately.

Following Shu (1977), Whitworth & Summers re-wrote these equations with the help of the only dimensionless variable x defined by

and the dimensionless quantities w, y, z defined by

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 x in sonic point , the model at the second border is characterized by .

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 r from the centre, where .

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
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