A. Coordinate systems¶
A.1. Generalities¶
We use a variety of coordinate systems in these notes, which we briefly introduce here. Because most stellar systems are either close-to-spherical or have a disk-like geometry, the two main coordinate systems that we use are spherical coordinates and cylindrical coordinates. Our convention will be that the spherical coordinates are \((r,\phi,\theta)\) for the (radial, azimuthal, and polar) coordinate, with \(\theta\) measured from the pole. These coordinates are given in terms of the Cartesian coordinates \((x,y,z)\) as \begin{align}\label{eq-sphere-coords} x &= r\,\sin \theta\, \cos \phi\,;\quad y = r\,\sin \theta\, \sin \phi\,;\quad z = r\,\cos \theta\,, \end{align} and \begin{align} r &= \sqrt{x^2 + y^2 + z^2}\,;\quad \phi = \mathrm{atan2}(y,x)\,;\quad \theta = \pi/2 - \mathrm{atan}(z/\sqrt{x^2+y^2})\,, \end{align} where \(\mathrm{atan2}(y,x)\) is the arctangent function with two arguments, which returns the phase of the complex number \(x+iy\). We will typically define velocities in the spherical coordinate system to be \((v_r,v_\phi,v_\theta) = (\dot{r},r\,\sin \theta\,\dot{\phi},r\,\dot{\theta})\), where the dot denotes the time derivative. The relation between the velocity \((v_x,v_y,v_z)\) in Cartesian coordinates and that in spherical coordinates is then \begin{align}\label{eq-vel-spher} v_x & = v_r\,\sin\theta\,\cos\phi-v_\phi\,\sin \phi+v_\theta\,\cos\theta\,\cos\phi\\ v_y & = v_r\,\sin\theta\,\sin\phi+v_\phi\,\cos\phi+v_\theta\,\cos\theta\,\sin\phi\\ v_z & = v_r\,\cos\theta\phantom{\,\sin\phi+v_\phi\,\cos\phi}-v_\theta\,\sin\theta\,. \end{align} The volume element is given by \begin{equation} \mathrm{d} \vec{x} = r^2\,\sin\theta\,\mathrm{d}r\,\mathrm{d}\phi\,\mathrm{d}\theta\,. \end{equation} The gradient \(\nabla\) in spherical coordinates is given by \begin{equation}\label{eq-gradient-spherical} \nabla = \vec{\hat{e}}_r\,\frac{\partial}{\partial r} + \frac{\vec{\hat{e}}_\theta}{r}\,\frac{\partial}{\partial \theta} + \frac{\vec{\hat{e}}_\phi}{r\,\sin\theta}\,\frac{\partial}{\partial \phi}\,, \end{equation} where \((\vec{\hat{e}}_r,\vec{\hat{e}}_\phi,\vec{\hat{e}}_\theta)\) are the unit vectors for spherical coordinates. The Laplacian \(\nabla^2\) is given by \begin{equation}\label{eq-laplacian-spherical} \nabla^2 = \frac{1}{r^2}\,\frac{\partial}{\partial r}\left(r^2\,\frac{\partial}{\partial r}\right) +\frac{1}{r^2\sin\theta}\,\frac{\partial}{\partial \theta}\left(\sin\theta\,\frac{\partial}{\partial \theta}\right) +\frac{1}{r^2\sin^2\theta}\,\frac{\partial^2}{\partial \phi^2}\,, \end{equation}
To describe the dynamics of disk galaxies, we will use cylindrical coordinates. Our convention is that cylindrical coordinates are \((R,\phi,z)\) for the (radial, azimuthal, and vertical) coordinate. These coordinates are given in terms of the Cartesian coordinates as \begin{align} x & = R\,\cos \phi\,;\quad y = R\,\sin \phi\,;\quad z = z\,, \end{align} and \begin{align} R &= \sqrt{x^2 + y^2}\,;\quad \phi = \mathrm{atan2}(y,x)\,;\quad z = z\,. \end{align} We will typically define velocities in the cylindrical coordinate system to be \((v_R,v_\phi,v_z) = (\dot{R},R\,\dot{\phi},\dot{z})\). The relation between the velocity \((v_x,v_y,v_z)\) in Cartesian coordinates and that in spherical coordinates is then \begin{align}\label{eq-vel-cyl} v_x & = v_R\,\cos\phi-v_\phi\,\sin \phi\,;\quad v_y = v_R\,\sin\phi+v_\phi\,\cos\phi\,;\quad v_z = v_z\,. \end{align} The volume element is given by \begin{equation} \mathrm{d} \vec{x} = R\,\mathrm{d}R\,\mathrm{d}\phi\,\mathrm{d}z\,. \end{equation} The gradient \(\nabla\) in cylindrical coordinates is given by \begin{equation}\label{eq-gradient-cylindrical} \nabla = \vec{\hat{e}}_R\,\frac{\partial}{\partial R} + \frac{\vec{\hat{e}}_\phi}{R}\,\frac{\partial}{\partial \phi}+\vec{\hat{e}}_z\,\frac{\partial}{\partial z}\,, \end{equation} where \((\vec{\hat{e}}_R,\vec{\hat{e}}_\phi,\vec{\hat{e}}_z)\) are the unit vectors for cylindrical coordinates. The Laplacian \(\nabla^2\) for cylindrical coordinates given by \begin{equation}\label{eq-laplacian-cylindrical} \nabla^2 = \frac{1}{R}\,\frac{\partial}{\partial R}\left(R\,\frac{\partial}{\partial R}\right) + \frac{1}{R^2}\,\frac{\partial^2}{\partial \phi^2}+\frac{\partial^2}{\partial z^2}\,. \end{equation}
Polar coordinates are the two-dimensional coordinate system corresponding to the \((R,\phi)\) part of cylindrical coordinates. In particular, the Laplacian in polar coordinates is simply \begin{equation}\label{eq-laplacian-polar} \nabla^2 = \frac{1}{R}\,\frac{\partial}{\partial R}\left(R\,\frac{\partial}{\partial R}\right) + \frac{1}{R^2}\,\frac{\partial^2}{\partial \phi^2}\,. \end{equation}
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