10.1. The axisymmetric Jeans equations and the asymmetric drift¶
Similar to what we did for spherical systems in Chapter 5.4, to start exploring the steady-state dynamics of galactic disks, we first consider moments of the collisionless Boltzmann equation and construct the cylindrical Jeans equations for axisymmetric systems.
10.1.1. The axisymmetric Jeans equations¶
To derive the cylindrical Jeans equations, we start from the equilibrium collisionless Boltzmann Equation (5.33) for the distribution function \(f\). We write this equation in cylindrical coordinates by using the Hamiltonian in cylindrical coordinates, which is given in Equation (3.41). Setting \(m=1\), the equilibrium collisionless Boltzmann equation is then \begin{align}\label{eq-collisionless-boltzmann-equil-nonaxi-cyl} p_R\,\frac{\partial f}{\partial R} + \frac{p_\phi}{R^2}\,\frac{\partial f}{\partial \phi} + p_z\,\frac{\partial f}{\partial z} + \left(\frac{p_\phi^2}{R^3}-\frac{\partial \Phi}{\partial R}\right)\,\frac{\partial f}{\partial p_R}-\frac{\partial \Phi}{\partial \phi}\,\frac{\partial f}{\partial p_\phi} -\frac{\partial \Phi}{\partial z}\,\frac{\partial f}{\partial p_z}& = 0\,. \end{align} For an axisymmetric system in an axisymmetric potential, the derivatives of \(f\) and \(\Phi\) with respect to \(\phi\) vanish and this equation simplifies to \begin{align}\label{eq-collisionless-boltzmann-equil-cyl} p_R\,\frac{\partial f}{\partial R} + p_z\,\frac{\partial f}{\partial z} + \left(\frac{p_\phi^2}{R^3}-\frac{\partial \Phi}{\partial R}\right)\,\frac{\partial f}{\partial p_R}-\frac{\partial \Phi}{\partial z}\,\frac{\partial f}{\partial p_z}& = 0\,. \end{align} Multiplying this equation by \(p_R\) and integrating over the three components of the momentum \((p_R,p_\phi,p_z)\) using that \(\mathrm{d} p_R\,\mathrm{d} p_\phi\,\mathrm{d}p_z = R\,\mathrm{d}v_R\,\mathrm{d} v_\phi\,\mathrm{d}v_z\) (where \(v_\phi = R\,\dot{\phi}\)), using partial integration to deal with the partial derivative with respect to \(p_R\), and using that \(\int \mathrm{d} p_R\,\mathrm{d} p_\phi\,\mathrm{d}p_z\, \partial g/ \partial R = \partial / \partial R\big(\int \mathrm{d} p_R\,\mathrm{d} p_\phi\,\mathrm{d}p_z \,g\big)\), this becomes \begin{equation}\label{eq-jeans-radial-axi} \frac{\partial [\nu\,\overline{v_R^2}]}{\partial R} +\frac{\partial [\nu\,\overline{v_R\,v_z}]}{\partial z} + \nu\,\left(\frac{\partial \Phi}{\partial R}+\frac{\overline{v_R^2}-\overline{v_\phi^2}}{R}\right)=0\,. \end{equation}
This is the axisymmetric, radial Jeans equation. If we instead multiply Equation (10.2) by \(p_z\) and integrate over all of the momenta, we obtain \begin{equation}\label{eq-jeans-vertical-axi} \frac{\partial [\nu\,\overline{v_z^2}]}{\partial z} +\frac{1}{R}\,\frac{\partial [R\,\nu\,\overline{v_R\,v_z}]}{\partial R} + \nu\,\frac{\partial \Phi}{\partial z}=0\,, \end{equation} which is the axisymmetric, vertical Jeans equation. Multiplying Equation (10.2) by \(p_\phi\) produces a Jeans equation that relates \(\overline{v_R\,v_\phi}\) and \(\overline{v_z\,v_\phi}\), both of which are zero if the distribution function is a function of the energy \(E\) and \(L_z\) alone.
As discussed at the beginning of this chapter, in Chapter 9.2 we demonstrated that for orbits that do not travel to large heights above and below the mid-plane of a disk mass distribution, the motion in the radial direction approximately decouples from that in the vertical direction, in which case the Hamiltonian can be approximately written as \(H = H_{R,\mathrm{eff}}(R,p_R) + H_z(z,p_z)\), and three integrals of the motion can be chosen to be \((L_z,E_R,E_z)\), where \(E_R\) and \(E_z\) are the energies corresponding to the planar and vertical components of the motion. From the stronger version of the Jeans theorem from Chapter 5.5, it follows that the steady-state distribution function can only depend on these three integrals \((L_z,E_R,E_z)\), which in turn are only functions of the following combinations of the velocities: \(v_\phi\) (from \(L_z\)), \(v_R^2\) (from \(E_R\)), and \(v_z^2\) (from \(E_z\)). Therefore we have that in the separable limit, \(\overline{v_R^2} = \sigma_R^2\), \(\overline{v_z^2} = \sigma_z^2\), \(\overline{v_R\,v_z} = 0\), and \(\overline{v_\phi^2} = \overline{v_\phi}^2+\sigma_\phi^2\). The axisymmetric Jeans equations then simplify to \begin{align}\label{eq-diskequil-axijeans-decoupledradial-1} \frac{\partial [\nu\,\sigma_R^2]}{\partial R} + \nu\,\left(\frac{\partial \Phi}{\partial R}+\frac{\sigma_R^2-\sigma_\phi^2-\overline{v_\phi}^2}{R}\right) & = 0\,,\\ \frac{\partial [\nu\,\overline{v_z^2}]}{\partial z} + \nu\,\frac{\partial \Phi}{\partial z} & =0\,.\label{eq-diskequil-axijeans-decoupledvertical-1} \end{align} Because the distribution of stellar orbits approximately decouples to a much better level than individual orbits do, these equations approximately hold for most of the stars in the solar neighborhood (roughly those that reach maximum \(|z| \lesssim1\,\mathrm{kpc}\)).
10.1.2. The asymmetric drift¶
Returning to the general case of the radial Jeans equation in Equation (10.3), we write \(\overline{v_\phi^2} = \overline{v_\phi}^2+\sigma_\phi^2\) and evaluate the equation at \(z=0\), where \(\partial \Phi / \partial R = v_c^2(R)/R\). Under the assumption that \(\partial \nu / \partial z = 0\) (good for a symmetric stellar distribution around the mid-plane that smoothly turns over at \(z=0\)), we can write this as \begin{equation} v_c-\overline{v_\phi} = \frac{\overline{v_R^2}}{v_c+\overline{v_\phi}}\,\left[\frac{\sigma_\phi^2}{\overline{v_R^2}}-1-\frac{\partial \ln [\nu\,\overline{v_R^2}]}{\partial \ln R} -\frac{R}{\overline{v_R^2}}\,\frac{\partial \overline{v_R\,v_z}}{\partial z}\right] \,. \end{equation} For stars that are on close-to-circular orbits, \(\overline{v_R^2} \ll v_c^2\) and thus \(v_c-\overline{v_\phi} \ll v_c\) from this equation (the part in square brackets is a number of order one). Then we can approximate \(v_c+\overline{v_\phi} \approx 2\,v_c\), which gives the Stromberg asymmetric drift equation \begin{equation}\label{eq-stromberg} v_c-\overline{v_\phi} = \frac{\overline{v_R^2}}{2\,v_c}\,\left[\frac{\sigma_\phi^2}{\overline{v_R^2}}-1-\frac{\partial \ln [\nu\,\overline{v_R^2}]}{\partial \ln R} -\frac{R}{\overline{v_R^2}}\,\frac{\partial \overline{v_R\,v_z}}{\partial z}\right] \,. \end{equation} This equation demonstrates that for stars that are on close-to-circular orbits, the difference between the circular velocity \(v_c\) and the mean rotational velocity \(\overline{v_\phi}\) is proportional to the radial velocity dispersion (\(\sigma_R^2 \approx \overline{v_R^2}\) for any reasonable disk distribution function). In particular, Equation (10.8) shows that the mean rotational velocity is not necessarily equal to \(v_c\) when \(\overline{v_R^2} > 0\).
To get a better sense of what Equation (10.8) implies, we can use further approximations that are appropriate for nearly-circular orbits. For such orbits the radial and vertical motion decouples and \(\overline{v_R\,v_z} \approx 0\), as discussed above, and \(\overline{v_R^2} = \sigma_R^2\). In Chapter 9.3.2, we derived the relation \(\sigma_\phi^2 / \sigma_R^2 = B/(B-A)\), where \(A\) and \(B\) are the Oort constants. Furthermore, for typical disk populations (although not all), both \(\nu\) and \(\sigma_R^2\) decrease with increasing \(R\) and we can approximate each as having an exponential dependence: \(\nu(R) \approx e^{-R/h_R}\) and \(\sigma_R^2 \approx e^{-R/h_{2\sigma_R}}\); therefore \(\partial \ln [\nu\,\overline{v_R^2}]/\partial \ln R = -R\,(1/h_R+1/h_{2\sigma_R})\). Using all of these approximations, Equation (10.8) becomes \begin{equation} v_c-\overline{v_\phi} = \frac{\sigma_R^2}{2\,v_c}\,\left[\frac{A}{B-A}+R\,\left(\frac{1}{h_R}+\frac{1}{h_{2\sigma}}\right)\right]\,. \end{equation} When looking at a sample of stars near the Sun, the second term in the square brackets in this equation is typically positive (but not always) and is larger than the first term is negative. The difference \(v_c-\overline{v_\phi}\) is then positive and the mean rotational velocity is less than the circular velocity. This is a first manifestation of the general phenomenon of asymmetric drift in disk galaxies: the distribution of rotational velocities at a given point in a disk galaxy is asymmetric around \(v_\phi = v_c(R)\) and therefore the mean rotational velocity is different from the circular velocity. Later in this chapter, we will see that the distribution of \(v_\phi\) is also skewed, not just simply shifted to a different mean. This phenomenon makes it difficult to relate the observed rotational velocity of a set of stars to the circular velocity: in the previous chapter we assumed that gas is on circular orbits such that its observed velocity directly tells us about the circular velocity; for stars we cannot simply generalize this to use the mean observed velocity without taking into account the asymmetric drift.
We will look the Stromberg asymmetric drift equation in the context of the solar neighborhood further in Section 10.3.2 below.
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