Anisotropy ansatz for the Jeans equations: oblate galaxies

TL;DR

This work extends the $b$-anisotropy framework for the axisymmetric Jeans equations to oblate ellipsoidal galaxies, clarifying how the velocity-dispersion closure $\sigma_R^2 = b(z) \sigma_z^2$ interacts with the kinematic fields before solving the equations. By detailing the $B$, $C$, and $D$ fields and establishing constraints on $b(z)$ through sign-definite regions, the authors show that more flattened stellar densities can sustain larger $b$-anisotropy, while the specific density profile plays a minor role. The analysis spans one-component ellipsoidal Sérsic and $\gamma$-models and two-component systems with DM halos, with analytic insight from power-law toy-models that reproduce the trends. The findings provide practical guidelines for model building and data interpretation in axisymmetric galaxies, including a qualitative link to empirical $b$-limits and JAM results on velocity anisotropy. Overall, flattening is the key driver of allowable $b$-anisotropy, with DM halos offering modest augmentation of the permissible range.

Abstract

In the solution of the Jeans equations for axisymmetric galaxy models the ``$b$-ansatz" is often adopted to prescribe the relation between the vertical and radial components of the velocity dispersion tensor, and close the equations. However, $b$ affects the resulting azimuthal velocity fields quite indirectly, so that the analysis of the model kinematics is usually performed after numerically solving the Jeans equations, a time consuming approach. In a previous work we presented a general method to determine the main properties of the kinematical fields resulting in the $b$-ansatz framework before solving the Jeans equations; results were illustrated by means of disk galaxy models. In this paper we focus more specifically on realistic ellipsoidal galaxy models. It is found that how and where $b$ affects the galaxy kinematical fields is mainly dependent on the flattening of the stellar density distribution, moderately on the presence of a Dark Matter halo, and much less on the specific galaxy density profile. The main trends revealed by the numerical exploration, in particular the fact that more flattened systems can support larger $b$-anisotropy, are explained with the aid of simple ellipsoidal galaxy models, for which most of the analysis can be conducted analytically. The obtained results can be adopted as guidelines for model building and in the interpretation of observational data.

Figures (8)

• Figure 1: The ansatz-independent regions $\mathscr{B}^{\pm}$ and $\mathscr{C}^{\pm}$ for the ellipsoidal one and two-component models in Sections 3 and 4. For each model $\mathscr{B}^{-}$ is the part of the plane below $\mathscr{B}^0$ (red lines), and $\mathscr{C}^-$ is the part of the plane above $\mathscr{C}^0$ (green lines); the stellar distribution flattening increases from left to right panels, with the axial ratio decreasing as $q=0.75$, $0.5$, and $0.25$. The different line styles identifies different values of the parameters of the models. Top panels: one-component Sérsic model of index $n=2$, $4$, and $6$; $R$ and $z$ are in units of $R_{\mathrm{e}}$, the semi-major axis of the effective isophote. Middle panels: one-component $\gamma$-models for $\gamma=1.5$, $2$, and $2.5$; $R$ and $z$ are in units of $a_*$. Bottom panels: ellipsoidal stellar dV model of total mass $M_*$in the potential of a SIS of circular velocity $v_{\mathrm{h}}$, for different values of the parameter $\mathscr{R} =v_{\mathrm{h}}^2R_{\mathrm{e}}/(GM_*)$; $R$ and $z$ are in units of $R_{\mathrm{e}}$.
• Figure 2: The critical values $b_1$ (black), $b_0$ (blue), $b_2$ (green), and $b_{\mathrm{M}}$ (red), as a function of the axial ratio $q$ of the stellar component (rounder models for increasing $q$), for the same one-component Sérsic models (top left), $\gamma$-models (top right), and dV models in a SIS DM halo (bottom left) shown in Figure \ref{['f:fig1']}. Bottom right: the critical $b$ curves for the $\gamma=2$ power-law ellipsoidal stellar model in a dominant SIS potential (thick continuous lines, Appendix \ref{['sec:PL_SIS']}), and in a dominant monopole potential (thin dashed lines, Appendix \ref{['sec:pl_mon']}). For simplicity, the $b_2$ green lines are shown only for the dV, the $\gamma=2$, and the dV+SIS ($\mathscr{R} = 0.5$) models. As discussed in Section 5, the magenta line represents the limit on the $b$-anisotropy as a function of galaxy flattening determined by Cappellari2007MNRAS.379..418C.
• Figure 3: From top to bottom, maps in the $(R,z)$-plane of $\overline{v_{\varphi}^2}$, $\Delta_R$, and $\Delta$ for the dV model with $q=0.5$; all the fields are in units of $GM_*/R_{\mathrm{e}}$, $R$ and $z$ in units of $R_{\mathrm{e}}$. Columns correspond (from left to right) to $b=0.5$, $1$, and $3.5$. Red and green curves represent the $\mathscr{B}^0$ and $\mathscr{C}^0$ lines, as in Figure \ref{['f:fig1']}; black ellipses are equally spaced stellar isodensity contours. In the white regions, the value of the field is negative.
• Figure 4: Same as in Figure \ref{['f:fig3']}, but with an increased flattening of the stellar distribution, corresponding to the axial ratio of $q=0.25$.
• Figure 5: The $\mathscr{B}^{\pm}$ regions of the one and two-component dV models for two flattenings of the stellar distribution (red and blue lines), and three values of the DM-to-stellar mass ratios for each flattening (lines of increasing thickness). As in Figure \ref{['f:fig1']}, where these curves are present, $\mathscr{B}^{-}$ for each models lies below the corresponding line. $R$ and $z$ are in units of $R_{\mathrm{e}}$. The figure illustrates how stellar flattening affects the size of the regions more than the DM presence, even for dominant halos.