arXiv subjects

Tomoaki Matsumoto

Publications and source records attributed to Tomoaki Matsumoto.

95 records · Page 6Linked to original sources

arXiv2002

A Fast Algorithm for Solving the Poisson Equation on a Nested Grid

We present a numerical method for solving the Poisson equation on a nested grid. The nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus our numerical method is suitable for computing the gravity of a centrally condensed object. It consists of two parts: the difference scheme for the Poisson equation on the nested grid and the multi-grid iteration algorithm. It has three advantages: accuracy, fast convergence, and scalability. First it computes the gravitational potential of a close binary accurately up to the quadraple moment, even when the binary is resolved only in the fine grids. Second residual decreases by a factor of 300 or more by each iteration. We confirmed experimentally that the iteration converges always to the exact solution of the difference equation. Third the computation load of the iteration is proportional to the total number of the cells in the nested grid. Thus our method gives a good solution at the minimum expense when the nested grid is large. The difference scheme is applicable also to the adaptive mesh refinement in which cells of different sizes are used to cover a domain of computation.

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arXiv2000

Stability of Toomre-Hayashi Disk

We investigate stability of Toomre-Hayashi model for a self-gravitating, rotating gas disk. The rotation velocity, $ v_ϕ$, and sound speed, $ c _{\rm s} $, are spatially constant in the model. We show that the model is unstable against an axisymmetric perturbation irrespectively of the values of $ v_ϕ$ and $ c_{\rm s} $. When the ratio, $ v_ϕ/ c_{\rm s} $, is large, the model disk is geometrically thin and unstable against an axisymmetric perturbation having a short radial wavelength, i.e., unstable against fragmentation. When $ v_ϕ/ c_{\rm s} $ is smaller than 1.20, it is unstable against total collapse. Toomre-Hayashi model of $ 1.7 \la v_ϕ/ c_{\rm s} \la 3 $ was thought to be stable against axisymmetric perturbations in earlier studies in which only radial motion was taken into account. Thus, the instability of Toomre-Hayashi model having a medium $ v_ϕ/ c_{\rm s} $ is due to not change in the surface density but that in the disk height. We also find that the singular isothermal perturbation is stable against non-spherical peturbations.

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arXiv2000

Growth of a Vortex Mode during Gravitational Collapse Resulting in Type II Supernova

We investigate stability of a gravitationally collapsing iron core against non-spherical perturbation. The gravitationally collapsing iron core is approximated by a similarity solution for dynamically collapsing polytropic gas sphere. We find that the similarity solution is unstable against non-spherical perturbations. The perturbation grows in proportion to $ (t - t_0) ^{-σ} $ while the the central density increases in proportion to $ (t - t_0) ^{-2} $. The growth rate is $ σ= 1/3 + \ell (γ- 4/3) $, where $ γ$ and $ \ell $ denote the polytropic index and the parameter $ \ell $ of the spherical harmonics, $ Y_{\ell} ^m (θ, ϕ) $, respectively. The growing perturbation is dominated by vortex motion. Thus it excites global convection during the collapse and may contribute to material mixing in a type II supernova.

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arXiv1999

Stability of Dynamically Collapsing Gas Sphere

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 = K ρ^γ$. We examine three types of perturbations: bar ($ \ell = 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 - t_0 | ^{-σ} $ while the central density increases in proportion to $ ρ_c \propto (t - t_0) ^{-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$.

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arXiv1994

Fragmentation of a Magnetized Filamentary Molecular Cloud Rotating around its Axis

The dynamical instability of a self-gravitating magnetized filamentary cloud was investigated while taking account of rotation around its axis. The filamentary cloud of our model is supported against self-gravity in part by both a magnetic field and rotation. The density distribution in equilibrium was assumed to be a function of the radial distance from the axis, $ ρ_0 (r) \, = \, ρ_{\rm c} \, ( 1 \, + \, r ^2 / 8 H ^2 ) ^{-2} $, where $ ρ_{\rm c} $ and $ H $ are model parameters specifying the density on the axis and the length scale, respectively; the magnetic filed was assumed to have both longitudinal ($z$-) and azimuthal ($φ$-) components with a strength of $ B _0 (r) \, \propto \, \sqrt{ ρ_0 (r) } $. The rotation velocity was assumed to be $ v _{0φ} \, = \, Ω_{\rm c} \, r \, (1 \, + \, r ^2 / 8 H ^2 ) ^{-1/2} $. We obtained the growth rate and eigenfunction numerically for (1) axisymmetric $ ( m \, = \, 0 ) $ perturbations imposed on a rotating cloud with a longitudinal magnetic field, (2) non-axisymmetric $ ( m \, = \, 1 ) $ perturbations imposed on a rotating cloud with a longitudinal magnetic filed, and (3) axisymmetric perturbations imposed on a rotating cloud with a helical magnetic field. The fastest growing perturbation is an axisymmetric one for all of the model clouds studied. Its wavelength is $ λ_{\rm max} \, \le \, 11.14 \, H $ for a non-rotating cloud without a magnetic field, and is shorter

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