Stability of Dynamically Collapsing Gas Sphere

T. Hanawa and T. Matsumoto Stability of Dynamically Collapsing Gas Sphere 1 document page1 Stability of Dynamically Collapsing Gas Sphere Tomoyuki Hanawa Department of Astrophysics, Nagoya University, Chikusa-ku, Nagoya 464-8602 E-mail(TH): [email protected] and Tomoaki Matsumoto Faculty of Liberal Arts, Hosei University, Fujimi, Chiyoda-ku, Tokyo 102-8160 E-mail(TH): [email protected] We discuss stability of dynamically collapsing gas spheres. We use a similarity solution for a dynamically collapsing sphere as the unperturbed state. In the similarity solution the gas pressure is approximated by a polytrope of P = Kρ . We examine three types of perturbations: bar (` = 2) mode, spin-up mode, and Ori-Piran mode. When γ < 1.097, it is unstable against bar-mode. It is unstable against spin-up mode for any γ. When γ < 0.961, the similarity solution is unstable against Ori-Piran mode. The unstable mode grows in proportion to |t − t0|−σ while the central density increases in proportion to ρc ∝ (t − t0)−2 in the similarity solution. The growth rate, σ is obtained numerically as a function of γ for bar mode and Ori-Piran mode. The growth rate of the bar mode is larger for a smaller γ. The spin-up mode has the growth rate of σ = 1/3 for any γ. Gravitation — Hydrodynamics — ISM: clouds — Stars: formation Introduction As well as the majority of nearby pre-main sequence stars, most young stars have their companions (see, e.g., Mathieu 1994). Since even young protostars have their companions, fragmentation is more plausible for formation of multiple star systems than capture and other processes. Fragmentation during the star formation has been paid much attention and a number of numerical simulations have been performed for clarifying it. Nonetheless the results are not conclusive. Some early simulations have only limited spatial resolution and their results on fragmentation are unreliable (Truelove et al. 1997). A gas cloud fragments artificially in numerical simulations of low resolution. If we exclude the artificial fragmentation, fragmentation proceeds but only slowly in numerical simulations. Fragmentation competes with the total collapse of a cloud. A cloud collapses and the density increases before it fragments. Only when an initial model was very much elongated, the simulation showed fragmentation of a cloud at a relatively low density. This initial model is, however, not applicable to a roundish dense core containing a binary. Fragmentation has been studied not only by numerical simulations but also by linear stability analyses. While numerical simulations are advantageous at handling nonlinearity, linear stability analyses are helpful for elucidating underlying physics. In this sense these two technologies are complementary. Most linear stability analyese thus far, however, deal with the stability of an equilibrium cloud against small perturbation. The competition with the total collapse could not be taken account into them. Silk, Suto (1988) tried to compute the stability of a similarity solution for the collapse of an isothermal gas cloud but could not obtain a self-consistent solution. Only recently Hanawa, Matsumoto (1999) succeeded in analyzing the stability of a collapsing isothermal cloud against non-spherical perturbations. They found that a collapsing isothermal cloud is unstable against bar mode (` = 2) perturbation. The bar mode grows in proportion to ρc where ρc denotes the central density. When it grows, a collapsing gas cloud may change its shape from sphere to filament. This result is consistent with recent three-dimensional simulations by Truelove et al. (1998) and Matsumoto, Hanawa (1999) in which a filament forms in a collapsing cloud. The former showed that the filament fragments to form a binary. Thus the bar mode is interesting for understanding fragmentation during the collapse. In this paper we extend the linear stability analyses of Hanawa, Matsumoto (1999) for collapsing nonisothermal spheres. For simplicity we employ the polytropic relation, P = Kρ , for the model cloud. The polytrope model can describe temperature change during the collapse in the most simple form. The polytrope model of γ ' 4/3 can be applied also to a collapsing iron core resulting Type II supernova. In section 2 we present the similarity solution for gravitationally collapsing sphere of a polytropic gas cloud. This section is essentially the review of Yahil (1983) and Suto, Silk (1991). We summarize their results to investigate the stability. In section 3 we examine the stability of the similarity solution against bar mode. The growth rate of the perturbation is larger when γ is smaller. The bar mode is degenerate and its growth rate is independent of the azimuthal wavenumber, m. The shape of the core suffering the bar mode depends on m