Distribution functions for spheroids

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.

💡 Research Summary

The paper presents a comprehensive framework for building self‑consistent galaxy models using distribution functions (DFs) that depend analytically on the action integrals J = (J_r, J_θ, J_φ). By working directly with actions rather than energy E and angular momentum L_z, the authors avoid the non‑linear complications that arise when a galaxy’s potential is non‑spherical or when multiple components (stars, gas, dark matter) are present. Actions are adiabatic invariants, so they remain well‑behaved under slow changes of the potential, making them ideal coordinates for constructing equilibrium models.

A central theoretical result is that hot (high‑velocity‑dispersion) components must satisfy a specific condition as J_φ → 0. If the DF behaves as f ∝ J_φ^α with α ≥ 0, the velocity distribution stays finite even for nearly radial orbits that carry almost no azimuthal angular momentum. Violating this condition leads to a divergence of the velocity dispersion and an unphysical “core‑blow‑up.” The authors propose two practical ways to enforce the condition: (1) introduce a smooth cutoff function of J_φ that forces f to vanish as J_φ → 0, and (2) shape the ratio J_r/J_θ to control radial bias while guaranteeing a non‑zero minimum J_φ.

The paper then treats three families of models:

1. Isotropic spherical systems (β ≈ 0). Here the DF can be written as a simple function of the binding energy, f(E), and transformed into f(J) without loss of accuracy.

2. Radially biased spherical systems (β > 0). The authors embed the anisotropy parameter β = 1 – σ_t²/σ_r² directly into the action‑based DF, giving extra weight to J_r relative to J_θ and J_φ. They show that if the J_φ → 0 condition is ignored, σ_r grows without bound, producing an unrealistic “inflated core.”

3. Flattened, axisymmetric systems (e.g., discs) with significant anisotropy. In this case the vertical action J_z and the azimuthal action J_φ must be balanced. The authors introduce a functional form
   f ∝