Plummer (1911)

Abstract (paraphrased)¶

Seeks a physical basis for the spatial distribution of stars in globular clusters from star counts. Plummer adopts Schuster’s (1883) closed-form solution of the polytropic (convective-equilibrium) gas sphere for $\gamma = 1.2$ — equivalently the $n = 5$ polytrope — as the space-density law, and shows it reproduces the projected star counts of ω Centauri and other clusters.

The density law (verified against the paper, §5–6)¶

Plummer carefully distinguishes the space density $\phi(r)$ (stars per volume), the cylinder counts $F(r)$, and the projected counts $f(r)$ / $\Sigma(r)$ (perpendicular to the line of sight). Schuster’s $\gamma = 1.2$ polytrope solution (his Eq. 11) gives the space density

i.e. the canonical $\rho(r) = \rho_0\,[1 + (r/a)^2]^{-5/2}$. The projected (surface) count is the different $f(r) = \tfrac{4}{3}N(1+r^2)^{-2}$, $\sigma(r) = \tfrac{4}{3}\pi N r^2 (1+r^2)^{-1}$ (Eq. 13). The model is fit to ω Centauri (his Table I) and a second cluster (Table II) with good agreement to the counts $\Sigma(r) = 3540\,r/\sqrt{1+r^2}$.

Use in progenax¶

• Plummer profile — derivation and progenax implementation (half-mass→scale-radius $a = r_h\sqrt{2^{2/3}-1} \approx 0.766\,r_h$)

• Plummer velocity distribution functions — the matched Eddington-inversion DF $f(E) \propto (-E)^{7/2}$

• progenax.profiles.PlummerProfile — production-default spatial profile

• progenax.kinematics.PlummerVelocityDF — matched equilibrium velocity DF

Notes¶

The simplest self-consistent cluster equilibrium with a closed-form density, potential, and DF — a research tool (production ICs) and the canonical pedagogical example of Eddington inversion. The $n=5$ polytrope is the unique polytrope with finite mass yet infinite extent.