Astron. Astrophys. 319, 547-560 (1997)

4. Conclusions

We have presented a comparison of the numerical solutions for the standard isothermal collapse test using two independent second-order hydrodynamic codes. One code uses specialized finite-difference techniques on a radially moving, spherical coordinate grid (code FD) and the other is based on the method of smoothed particle hydrodynamics (SPH) coupled with the hierarchical tree algorithm of Hernquist (1987) for the calculation of the gravitational forces (code TREESPH). Two model calculations were run with each code differing only in the initial resolution.

The results indicate that up to the time the calculations could be compared the outcome of the evolution is the formation of a protostellar binary system in agreement with previous calculations of the standard test case. We find that the intermediate forms of the evolution, as calculated with both codes, are very similar and much less sensitive to resolution than was previously found by Monaghan & Lattanzio 1986 (ML). Contrary to previous first-order calculations, here the protostellar binary forms by direct fragmentation, i.e., without the occurrence of an intermediate bar configuration. The disappearance of the intermediate bar was also observed in recent second-order FD calculations by Myhill & Boss (1993), suggesting that this aspect of the solution is a result of the decreased numerical diffusion associated with the new second-order schemes.

Our results show that the basic differences encountered by ML between the finite difference methods and the SPH approach for this problem are actually mitigated. The high-resolution calculation with code FD no longer results in a run-away collapse of the fragments, and as for the TREESPH calculations, the binary companions show a clear tendency to fall towards each other. The fragments are seen to undergo rapid collapse until a certain value of the maximum density is reached. The major differences between the FD and the TREESPH calculations are just observed to occur during this short phase of the evolution. The fragments formed from the FD calculations appear to possess a lower rotational energy and so they collapse on a relatively shorter time-scale with respect to the TREESPH fragments, which must dispose of their excess of rotational energy. These differences in the rotational properties influence the details of the accretion process, which then is seen to determine the observed differences in the final shape of the fragments. The resulting properties of the clumps seem to indicate that their outer regions are very close to equilibrium and that only a small amount of mass at the centre of each fragment may continue to collapse. The results of the long term evolution with code TREESPH confirm this view and show that the central collapse of the fragments may eventually stop. The subsequent evolution is controlled by the strong interaction between them, which halts their collapse to higher densities and forces the binary system to become close enough to eventually coalesce. However, complete coalescence of the fragments is not guaranteed by the present calculations, and as shown by Bate et al. (1995), they could even survive and sub-fragment in the later evolution, leading to the formation of a multiple protostellar system. These trends suggest that the protostellar binary system could be a transient feature of the overall evolution. Although the calculations with code FD were not continued for such a long time in the evolution, the separation distance between the FD fragments was seen to be decreasing at a rate similar to that found in the TREESPH calculations, and so we may expect the gravitational interaction between the FD fragments to become equally important in the long term evolution.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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