16.3. Jeans Anisotropic Modeling (JAM)¶
There are only a small number of analytic three-integral \(f(E,L_z,I_3)\) distribution functions for axisymmetric, elliptical-galaxy-like systems and these are not particularly realistic representations of actual elliptical galaxies. To perform three-integral steady-state dynamical modeling of elliptical galaxies, we therefore either need to use the Schwarzschild orbit-superposition technique or use the Jeans equations. Because Schwarzschild modeling is computationally-expensive and complex to implement and apply, Jeans modeling is preferred for uniform analyses of large samples of galaxy kinematics. However, while two-integral Jeans modeling has a unique solution for a given density, gravitational potential, and mean-rotation field, three-integral Jeans modeling does not have a unique solution. This is a manifestation of the fact that the Jeans equations in general do not close as discussed in Chapter 5.4.1: in general there is an infinite number of distribution functions in a given gravitational potential that give rise to the same density and mean rotation pattern and this remains the case even if we assume that the system is axisymmetric.
To use the Jeans equations, we therefore have to make a closure assumption to make the Jeans equations return a unique answer. Two-integral Jeans modeling provides an example of this: by assuming that the distribution function has the form \(f(E,L_z)\), we have that \(\overline{v_R^2} = \overline{v_z^2}\) and \(\overline{v_R\,v_z} =0\), and the two axisymmetric Jeans equations (10.3) and (10.4) then only have two unknowns: \(\sigma_z^2 = \sigma_R^2 = \overline{v_R^2} = \overline{v_z^2}\) on the one hand and \(\overline{v_\phi^2}\) on the other. We can therefore solve for all of the second moments. An additional assumption is necessary to separate \(\overline{v_\phi^2}\) into the contribution from ordered rotation, \(\overline{v_\phi}^2\), and that from the velocity dispersion, \(\sigma_\phi^2\), through \(\overline{v_\phi^2} = \overline{v_\phi}^2 + \sigma_\phi^2\). With this additional assumption, we can determine all first moments (keeping in mind that \(\overline{v_R} = \overline{v_z} = 0\)).
We can close the Jeans equations for three-integral models \(f(E,L_z,I_3)\) by making similar assumptions about the different components of the mean velocity and the velocity dispersion tensor. The idea behind Jeans anisotropic modeling (JAM; Cappellari 2008) is to assume that the system has a constant anisotropy \(\beta_z\) in the meridional plane, defined similarly to the radial anisotropy for spherical systems from Equation (5.46) \begin{equation}\label{eq-ellipmass-anisotropybetaz-jam} \beta_z \equiv 1-{\overline{v_z^2} \over \overline{v_R^2}} = 1-{\sigma_z^2 \over \sigma_R^2}\,, \end{equation} such that \(\overline{v_R^2} = (1-\beta_z)^{-1}\overline{v_z^2}\) (the second equality holds because \(\overline{v_R} = \overline{v_z} = 0\) for an axisymmetric system). Additionally, we assume that \(\overline{v_R\,v_z} =0\) similar to the two-integral case. The latter assumption is equivalent to assuming that the velocity-dispersion tensor is diagonal in the cylindrical coordinate frame. The two-integral \(f(E,L_z)\) models are then simply obtained by setting \(\beta_z=0\), while for \(\beta_z\neq0\), the dispersion tensor is anisotropic. With these assumptions and the boundary condition that \(\nu\,\overline{v_z^2} \rightarrow 0\) as \(|z| \rightarrow \infty\), the axisymmetric Jeans equations (10.3) and (10.4) have the following solution \begin{align}\label{eq-massellip-JAM-sigmaz} \sigma_z^2 = \overline{v_z^2}(R,z) & = {1\over \nu(R,z)}\int_z^\infty\mathrm{d}z'\,\nu(R,z'){\partial \Phi \over \partial z'}\,,\\ \sigma_R^2 = \overline{v_R^2}(R,z) & = (1-\beta_z)^{-1}\,\overline{v_z^2}(R,z)\,,\label{eq-massellip-JAM-sigmaR}\\ \overline{v_\phi^2}(R,z) & = (1-\beta_z)^{-1}\,\left[{R\over \nu(R,z)}{\partial \nu\overline{v_z^2} \over \partial R}+\overline{v_z^2}\right]+R\,{\partial \Phi \over \partial R}\,.\label{eq-massellip-JAM-vphi2} \end{align}
As before, to also determine the first moment \(\overline{v_\phi}\), we need an additional assumption about how \(\overline{v_\phi^2}\) separates into \(\overline{v_\phi}^2\) and \(\sigma_\phi^2\). There are different options for how to do this, but one obvious one is to use a parameter that characterizes the deviation from isotropy in the radial and tangential directions, by defining the tangential anisotropy \begin{equation}\label{eq-massellip-tangential-anisotropy} \gamma = 1 - {\sigma_\phi^2 \over \sigma_R^2}\,, \end{equation} such that then \begin{align}\label{eq-massellip-JAM-sigmaphi} \overline{v_\phi} & = \sqrt{\overline{v_\phi^2}(R,z) - (1-\gamma)\,\overline{v_R^2}(R,z)}\,;\quad \sigma^2_\phi = (1-\gamma)\,\overline{v_R^2}(R,z)\,. \end{align}
This closure of the axisymmetric Jeans equations forms the backbone of the JAM method. A practical application of this method further needs to deal with the deprojection of the observed surface-brightness profile, the calculation of the gravitational potential from the deprojected profile, and the calculation of the integrals in the previous sets of equations. A solution to all of these issues is to use the multi-Gaussian expansion (MGE) of the surface brightness described in Chapter 14.3.3. As we discussed there, a sum of Gaussians can represent almost any observed galaxy profile and is straightforward to deproject for any assumed inclination. The gravitational potential of the deprojected profile can be computed using the methods of Chapter 12.2. Furthermore, Cappellari (2008) demonstrated that the integrals in the expressions for the first and second moments above can be significantly simplified to a single numerical quadrature for each quantity and they are therefore quick to compute, and that the projection of the three-dimensional kinematics onto the line of sight can be computed in a similar way. By representing a central black hole or a dark matter halo as a single or sum of Gaussians as well, the MGE formalism can be used to build realistic models of galaxies from their innermost regions to their outskirts. Thus, the MGE formalism provides a quick way to realistically model the kinematics of observed external galaxies. As with all use of the Jeans equations, there is no guarantee that there exists a distribution function \(f(E,L_z,I_3)\) with the kinematics given by the solution of the Jeans equation, but the results from Schwarzschild modeling of SAURON galaxies demonstrate that consistent \(f(E,L_z,I_3)\) models with similar kinematics as the JAM models exist (e.g., Cappellari et al. 2007).
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