Dejonghe (1987)

Abstract (paraphrased)¶

A one-parameter family of anisotropic distribution functions all consistent with the same Plummer mass density, illustrating the DF indeterminacy of a given density. The family parameter $q$ tunes the orbital structure: $q = 0$ is the isotropic model, $0 < q \le 2$ radially anisotropic (“radial clusters”), $q < 0$ tangential. Moments, energy distributions, and observable line profiles are all analytic — making the paper a premier source of exact oracles for Plummer-family kinematics.

What progenax uses (all PDF-verified 2026-07-10/11)¶

Model units $G = M = a = 1$ throughout ($\psi = (1+r^2)^{-1/2}$, Eq. 13).

• Eq. 14–16 — the isotropic chain. $\rho = \tfrac{3}{4\pi}(1+r^2)^{-5/2}$, $\rho(\psi) = \tfrac{3}{4\pi}\psi^5$, and the isotropic DF $F(E) = \tfrac{3}{7\pi^3}(2E)^{7/2}$ — since $3 \cdot 2^{7/2} = 24\sqrt{2}$, this independently confirms progenax’s Plummer DF coefficient $24\sqrt{2}/(7\pi^3)$ (theory page).

• Eq. 17 — the intrinsic dispersion. $\sigma^2(r) = \tfrac16 (1+r^2)^{-1/2}$, i.e. $\sigma_r^2 = GM/(6\sqrt{r^2+a^2})$ — the PlummerVelocityDF moment oracle.

• Eqs. 22a/22b/23 — the anisotropic moments. $\sigma_r^2 = \tfrac{1}{6-q}(1+r^2)^{-1/2}$ and Binney’s $\beta(r) = \tfrac{q}{2}\,\tfrac{r^2}{1+r^2}$ (Eq. 23) — an OM-like monotone anisotropy profile.

• Eq. 21 (p. 18) — the Merritt bridge. The limiting member $q = 2$ is identical to the Merritt (1985) model with $r_a = 1$: an independent cross-check between the two anisotropic constructions progenax implements (see the om_anisotropy model card).

• Eq. 43 (p. 24) — THE projected-dispersion oracle. Via the Binney & Mamon (1982) projection integral (his Eq. 42),

  $$\sigma_p^2(r_p) \;=\; \frac{3\pi}{32}\,\frac{1}{6-q}\, \frac{1}{\sqrt{1+r_p^2}}\left(3 - \frac{5q}{4}\,\frac{r_p^2}{1+r_p^2}\right).$$

  (1)

  At $q = 0$ this reduces to $\sigma_p^2 = \tfrac{3\pi}{64}(1+r_p^2)^{-1/2}$ — restoring units, $\sigma_{\rm los}^2(R) = \tfrac{3\pi}{64}\,GM/\sqrt{a^2+R^2}$, the tight absolute oracle used by tests/validation/test_dispersion_physics.py for project_dispersion. Formerly cited there as a “standard result”; now source-verified. Dejonghe notes all members pass through $\sigma_p^2 = 3\pi\sqrt{3/5}/64$ at $r_p = \sqrt{2/3}$ — a family-invariant point.

Connections in progenax¶

• progenax.kinematics.dispersion.project_dispersion — the LOS/PM projection whose Plummer anchor is Eq. 43 ($q=0$); the projection kernel itself is Eq. 42 = Binney & Mamon (1982).

• plummer_df model card — cites Eqs. 16/17/43.

• Merritt (1985) note — the $q=2 \leftrightarrow r_a=1$ bridge.