10. Equilibria of galactic disks¶
In Chapter 7, we discussed how to compute the gravitational potential and the gravitational field of a general disk mass distribution. In Chapter 8, we used the kinematics of tracers on circular orbits or nearly-circular orbits to learn about the mass distributions of disk galaxies. This led to the important conclusion that we require large quantities of dark matter to explain the kinematics of galactic rotation in disk galaxies. In this chapter, we will go beyond circular or nearly-circular orbits in disk galaxies and we will investigate the dynamical equilibrium distribution of stars in disk galaxies, assuming axisymmetry as in the previous chapters. This will allow us to understand the galactic rotation of stars in the solar neighborhood, including older stars that have orbits that deviate significantly from circular rotation. We will also discuss the dynamical equilibrium state of stars in the direction perpendicular to the disk plane, a dimension which is similarly absent in the circular-rotation approximation of Chapter 8 that assumes that all bodies orbit in the disk’s mid-plane. The framework that we will develop will also let us constrain the presence of dark matter at the Sun’s position, which is difficult to do from the rotation curve alone.
We will investigate the properties of the equilibrium states of axisymmetric galactic disks using the same general tools that we first introduced in Chapter 5: the Jeans equations (Chapter 5.4) to connect the density profile to the first and second moments of the kinematics and the Jeans theorem (Chapter 5.5) to build full phase-space distribution functions of galactic disks. While the Jeans equations generalize to a slightly more complicated form in the axisymmetric case than the spherical Jeans equations of Chapter 5.4.2, we will see that building phase-space models using the Jeans theorem is significantly more complicated than it was in the spherical case.
The Jeans theorem tells us that equilibrium phase-space models of galactic disks are functions of the integrals of motion. As shown in Chapter 9.1, orbits in a static, axisymmetric gravitational potential have two classical integrals of motion: the specific energy \(E\) and the component of the specific angular momentum parallel to the symmetry axis (conventionally the \(z\) axis, \(L_z\)). However, we also saw in Chapter 9.2 that these orbits typically appear to conserve a third, non-classical integral (we will discuss this in more detail in Chapter 13). While we may be tempted to ignore this third integral and work with equilibrium models \(f \equiv f(E,L_z)\), a simple observation about the kinematics of stars in galactic disks demonstrates that this cannot produce realistic models: the vertical velocity dispersion \(\sigma_z\) of stars near the Sun is typically about half of the radial velocity dispersion \(\sigma_R\) (in cylindrical coordinates; e.g., Wielen 1974; Mackereth et al. 2019), varying between \(\approx0.4\) to \(\approx0.6\) depending on the age and angular momentum of the star. The same holds in external galaxies (Bottema 1993). If the distribution function only depends on \(E\) and \(L_z\), however, these dispersions should be equal to each other, because the vertical and radial velocity only appear in \(E\) and they do so in the same way (in the combination \(v_R^2/2+v_z^2/2\)). Thus, the phase-space distribution of stars near the Sun must depend on a third integral that, at a given \(E\) and \(L_z\) distinguishes between stars with greater or smaller vertical oscillations. The third integral is typically denoted as \(I_3\) and the distribution function therefore has to be \(f \equiv f(E,L_z,I_3)\).
For most realistic galactic potentials, we cannot write down an expression for the third integral and this makes the task of writing down equilibrium disk models \(f(E,L_z,I_3)\) challenging. To make headway in this chapter, we will follow two related approaches. First, we will describe equilibrium models for axisymmetric, razor-thin disks. Defining \(I_3\) such that \(I_3 = 0\) describes an orbit with zero vertical oscillation, such models have \(f(E,L_z,I_3) = f(E,L_z)\,\delta(I_3)\) (using somewhat sloppy notation, we use the same ‘\(f\)’ here for the three- and two-integral distribution function). Because these models are only non-zero within the mid-plane and/or for zero velocity perpendicular to the mid-plane, we do not need to be able to compute the third integral. Of course, we can only use these models to learn about the dynamical state within the mid-plane, not that perpendicular to it, but we will see that they are nevertheless useful for describing stars in the Milky Way.
To add a non-zero vertical extension to these models, we will make use of the insights about the basic properties of orbits in disk mass distributions in Chapter 9. There we looked at orbits with relatively high angular momenta, which are on disk orbits that are close to circular but not necessarily “nearly” circular (which we define here as so close to circular that the epicycle approximation provides a very good description of their orbits). We demonstrated that for such orbits the motion in the disk plane approximately separates from the motion perpendicular to the disk plane. We can usefully decompose orbits in the disk into their planar component and their vertical component (at the mean radius of the orbit) and still understand the basic orbital properties. Because all of the orbits approximately separate in planar and vertical components, the distribution of orbits can also be approximately separated into planar and vertical components, with little coupling between the two. As discussed in Chapter 9.2, we can split the Hamiltonian into a planar and vertical part and radial \(E_R\) and vertical \(E_z\) contributions to the energy are separately conserved. We can then identify the third integral as \(I_3 = E_z\) and build models using this third integral. A simple way to do this is to combine the functional form of razor-thin models, \(g(E,L_z)\), that we will discuss first in this chapter, with the functional form of one-dimensional models \(h(E_z)\) for the dynamics perpendicular to the mid-plane that we will discuss in Section 10.4. By allowing the parameters of \(h\) to depend on the guiding-center radius through \(L_z\), we can arrive at three-dimensional distribution functions \(f(E_R,L_z,E_z) = g(E_R,L_z)\times h(E_z;L_z)\) that accurately describe the dynamics in the plane and the radially-dependent dynamics perpendicular to the plane (e.g., Kuijken & Dubinski 1995).
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