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Galactic Spiral Arms by Swing Amplification

Based on the swing amplification model of Julian and Toomre (1966), we investigate the formation and structure of stellar spirals in disk galaxies. We calculate the pitch angle, wavelengths, and amplification factor of the most amplified mode. We also obtain the fitting formulae of these quantities as a function of the epicycle frequency and Toomre's $Q$. As the epicycle frequency increases, the pitch angle and radial wavelength increases, while the azimuthal wavelength decreases. The pitch angle and radial wavelength increases with $Q$, while the azimuthal wavelength weakly depends on $Q$. The amplification factor decreases with $Q$ rapidly. In order to confirm the swing amplification model, we perform local $N$-body simulations. The wavelengths and pitch angle by the swing amplification model are in good agreement with those by $N$-body simulations. The dependence of the amplification factor on the epicycle frequency in $N$-body simulations is generally consistent with that in the swing amplification model. Using these results, we estimate the number of spiral arms as a function of the shear rate. The number of spiral arms increases with the shear rate if the disk to halo mass ratio is fixed.