Gravitational collapse of filamentary clouds

We consider gravitational collapse of a filamentary cloud under the assumption that it is axisymmetric and uniform along the axis. The pressure is approximated by a polytrope of P = K rho(gamma). We found a similarity solution for the collapse when the polytropic index lies in the range 0 < gamma < 1. According to the similarity solution, the collapse consits of two phases. In the first phase the filament becomes denser and thiner. The density at the center (rho(c)) increases in proportion to (t(0) - t)(-2), where t(0) denotes the epoch at the end of the first phase. Meanwhile, the filament diameter (FWHM) decreases in proportion to (t(0) - t)((2 - gamma)). Th, line density of the central filament of rho > 0.1 rho(c) vanishes at t = t(0), since it is proportional to the central density and the square of the diameter [alpha (t(0) - t)(2(1 - gamma))]. In the second phase the central filament grows in mass by accretion. The collapse in the second phase is similar to the inside-out collapse of a spherical gas cloud. When the polytropic index gamma is closer to unity, the collapse is slower. When gamma = 0.999, the collapse is 12-times slower than the dynamical one. We also found from numerical simulations having various initial conditions that the collapse of a filamentary cloud approaches asymptotically the similarity solution irrespectively of the initial condition.