A self-consistent model for the spiral arms of galaxies

Abstract

In this thesis, we try to prove the stability of spiral arms using the density wave theory. This theory assumes that the spiral arms are not rigid structures, but instead they are regions with increased density that move through the galaxy like a wave [1]. The spiral structure, in this theory, are driven by the Lindblad resonances, which occur when the star has had an integer amount of radial oscillations for every encounter with the spiral structure. Here, we consider the standard Lindblad model for a two-armed galaxy. We consider the corotation resonance, where the star has the same orbital frequency as the spiral structure, and the Lindblad resonances where the star has completed two radial oscillations for every revolution it has made with respect to the spiral structure. If the star moves more slowly than the spiral structure, it is in the outer Lindblad resonance; if it moves faster than the spiral structure, we call it the inner Lindblad resonance. To derive a formula for the density of stars in a galaxy, we make a few approximations. We assume that the spiral arms are Archimedian spirals, even though spiral arms are generally shaped more like logarithmic spirals [2]. We also make linear approximations, as we approximate the Hamiltonian in the case without spiral arms with the linear Hamiltonian H0. To calculate the density of stars, we assume that the stars follow a certain distribution in phase-space. We then integrate this distribution to obtain the spatial density of the stars. This density shows a clear spiral structure at outer Lindblad resonance and corotation. However, the linear approximations mean that the distribution around inner Lindblad resonance does not show any spiral structure. This is because we linearized the unpertubed Hamiltonian, which means that we assumed that the radial and orbital frequencies are constant for all stars in this resonance. However, when calculating these frequencies using the non-linearized Hamiltonian, we show that this approximation does not work for inner Lindblad resonance, despite being a good assumption for the other resonances. So using this model, we have derived the outer part of the spiral structure. Even though the density in the structure does not differ much from the surrounding galaxy, the increased star formation [3] means that these arms stand out more in luminosity than in density.