= Solution
\b[A <spiral density wave> is an organized <density> and gravitational-potential perturbation through which stars and <gas> can move; it is not generally a permanently labelled set of stars.] <Differential rotation> winds a material arm rapidly. In a wave description, a perturbation has phase $m\phi+\int k\,dr-\omega t$, with <pattern speed> $\Omega_p=\omega/m$ distinct from the local stellar angular speed.
For a thin fluid disk the local tightly wound dispersion relation is
$$
(\omega-m\Omega)^2=\kappa^2+c_s^2k^2-2\pi G\Sigma|k|.
$$
Rotation and <pressure> resist self-gravity. A collisionless stellar disk requires its stellar response function instead of simply replacing $c_s$ by a dispersion; the familiar local axisymmetric stability parameter is $Q_\star=\sigma_R\kappa/(3.36G\Sigma)$. This is distinct from the <gas> $Q_g=c_s\kappa/(\pi G\Sigma)$.
Waves exchange <angular momentum> at <Lindblad resonances> and near <corotation radius>. <Swing amplification> can strengthen a leading disturbance as shear turns it into a trailing one. Bars or companions can force a coherent spiral; recurrent self-excited disturbances also occur. Thus a single indefinitely steady wave is one model, not a description established for every <spiral galaxy>. <Gas> compression and shocks along arms can organize <star formation>, while gravitational torques redistribute <angular momentum>. Arm brightness, recent <star formation> and stellar <mass> <density> need not trace precisely the same pattern.<Spirals in Galaxies by Jerry A. Sellwood and Karen L. Masters>[Sections4–6: wave dynamics, recurrent versus quasi-steady patterns, and <gas> response]{ref}
\Image[/past-exam-of-the-formation-of-structure-in-the-universe-course-of-the-university-of-cambridge/2015/paper-1-spiral-pattern.png]
{title=Differentially orbiting stars can cross a spiral density pattern with its own angular speed}
{source=paper-1-spiral-pattern.py}
Back to article page
