Elson, Fall & Freeman (1987) - progenax — differentiable initial conditions for N-body simulations

Abstract (paraphrased)¶

Surface-brightness profiles (from star counts + aperture photometry) for 10 young massive star clusters in the LMC (NGC 1818, 1831, 1866, 2004, 2156, 2157, 2159, 2164, 2172, 2214), ages $8\times10^6$ – $3\times10^8$ yr. At large radii the projected density falls off as a power law $\propto r^{-\gamma}$ with $2.2 \lesssim \gamma \lesssim 3.2$ and median $\gamma \approx 2.6$ — not the abrupt King-style tidal cutoff. The clusters do not appear tidally truncated; the authors suggest up to ~50% of the mass may lie in unbound halos.

The EFF formula (verified against the paper, p. 61)¶

EFF fit the surface brightness $\mu(r)$ (the figures plot $V$ mag arcsec $^{-2}$ ) with

\mu(r) = \mu_0 \left(1 + \frac{r^2}{a^2}\right)^{-\gamma/2} \qquad\text{(EFF Eq. 1)}

(1)

where $r$ is the projected radius, $\mu_0$ the central surface brightness, $a$ a scale length, and $\gamma$ the projected (surface-brightness) slope ( $\gamma \in [2.2, 3.2]$ , median 2.6; Table 1). “This form was chosen purely for mathematical convenience.” Integrating gives the enclosed projected luminosity (EFF Eq. 2):

L_p(r) = \frac{2\pi\mu_0 a^2}{\gamma - 2}\left[1 - \left(1 + \frac{r^2}{a^2}\right)^{1-\gamma/2}\right].

(2)

Surface brightness vs. 3-D density — progenax’s convention

EFF Eq. 1 is the projected surface brightness $\mu(r)$ , not the 3-D volume density. progenax.profiles.EFFProfile adopts the same functional form as the 3-D volume density $\rho(r) = \rho_0\,[1 + (r/a)^2]^{-\gamma/2}$ — the standard convention in N-body/IC codes (e.g. McLuster). Consequently progenax’s $\gamma$ is a 3-D density slope, distinct from EFF87’s observed surface slope: a 3-D density $\propto r^{-\gamma}$ deprojects/projects with a one-power offset (Abel), so an EFF87 surface slope $\gamma_{\rm EFF}$ corresponds to a 3-D slope $\approx \gamma_{\rm EFF}+1$ . At $\gamma = 5$ the 3-D form reduces exactly to Plummer. EFF has no built-in tidal truncation; progenax adds an optional truncation radius $r_t$ .

Use in progenax¶

EFF profile — derivation and progenax implementation
progenax.profiles.EFFProfile(a, gamma, r_t) — young-cluster 3-D density profile with optional tidal truncation; Eddington-inversion velocity DF in progenax.kinematics.EFFVelocityDF

Notes¶

The right choice for young massive clusters where the outer slope is a free observational parameter rather than a tidal cutoff. When comparing progenax’s $\gamma$ to EFF87 / observed surface-brightness fits, mind the surface-vs-3-D convention above.

References¶

Elson, R. A. W., Fall, S. M., & Freeman, K. C. (1987). The structure of young star clusters in the Large Magellanic Cloud. The Astrophysical Journal, 323, 54–78. 10.1086/165807