Distribution functions for spheroids

Abstract

Galaxy models comprising several components (including dark matter) that are bound by the self-consistently generated gravitational field are readily constructed from distribution functions (DFs) that are analytic functions of the action integrals J. We explain why such models have unphysical velocity distributions unless the DFs of hot components satisfy certain conditions as J_φ-> 0. We show how DFs for both isotropic and radially biased spherical systems can be constructed with specified f(J). We show how to construct DFs for flattened systems with significant velocity anisotropy. Construction of self-consistent models rather than populations that are confined by an external potential leads to the conclusion that radially-biased spherical systems are generically unstable to quadrupolar perturbations. Chaos is likely key to maintenance of these constraints during adiabatic disc growth.

Figures (10)

• Figure 1: A halo with a DF given by equations (\ref{['eq:Posti']}) and (\ref{['eq:defcJ0']}) trapped in the potential of a similar body flattened to axis ratio $c/a=0.5$. The lower panel shows the distributions of $v_R$ in black and $v_\phi$ in cyan at $(R,z)=(5,0)\,\mathrm{kpc}$. The red dashed curve shows the Gaussian with the dispersion of the $v_\phi$ distribution.
• Figure 2: The distributions of radial and azimuthal velocities at $(R,z)=(5,0)\,\mathrm{kpc}$ when the same population as that shown in Fig. \ref{['fig:oldDF_squash']} is confined by the equivalent spherical potential.
• Figure 3: Left panel: two orbits of the same energy in a perfect ellipsoid at $L_z=0$. The red (loop) orbit has $J_z>J_{z\rm crit}$ while the blue (box) orbit has $J_z<J_{z\rm crit}$. Right panel: two orbits that differ from those in the left panel in having small but non-zero $L_z$ and hence must be in the meridional plane.
• Figure 4: Each black curve shows how $\Omega_\phi/\Omega_r$ varies with circularity $c$ at a fixed energy in a dark-halo like potential (upper panel) or a Plummer potential (lower panel). The red lines show $y=x$.
• Figure 5: Top row: a population of test particles with DF $f(h_0[{\bf J}])$ trapped in a spherical double power-law potential. In the left panel the red curve shows the density distribution that generates the potential and the black curve shows the density of test particles. The square marks the point at which ${\rm d}\ln\rho/{\rm d}\ln r=-2$. The middle panel shows the radial and tangential velocity dispersions in this isotropic model. In the right panel black and overplotted orange curves show the distribution of $v_R$ and $v_{\rm t}$ velocity components $10\,\mathrm{kpc}$ from the centre. The dashed red curve shows the Gaussian with the same standard deviation as $N(v_{\rm t})$. Centre row: the same diagnostics for a population of test particles with DF $f(h_2[{\bf J}])$ with $\beta_v=1$ trapped in the same potential. In the right panel the dashed red curve shows the Gaussian with the same standard deviation as he orange distribution of $v_{\rm t}$. Bottom row: as the middle row but for a tangentially biased population with $f(h_1[{\bf J}])$ with negative $\beta_v=-0.5$.