EFF profile - progenax — differentiable initial conditions for N-body simulations

The EFF profile¶

The Elson et al. (1987) (EFF) profile is the standard choice for young massive star clusters — typically $\lesssim 100$ Myr old, cluster mass $M_{\mathrm{cl}} \gtrsim 10^4\,\Msun$ — where the surface-brightness profile shows a power-law fall-off whose slope is not well captured by Plummer’s fixed $r^{-5}$ outer asymptote or by King’s exponential cutoff. EFF was originally fit to 10 LMC clusters of ages 10⁷–10⁹ yr Elson et al., 1987; it is now used routinely for young Galactic and LMC/SMC clusters where ages and dynamical states preclude King-like tidal truncation.

The density profile¶

The EFF profile is

\rho(r) \;=\; \rho_0\,\biggl[1 + \biggl(\frac{r}{a}\biggr)^{\!2}\,\biggr]^{-\gamma/2}

(1)

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with three parameters: central density $\rho_0$ , scale radius $a$ , and power-law slope $\gamma$ . The structural similarity to Plummer (which is exactly EFF at $\gamma = 5$ ) is intentional — EFF generalises Plummer by exposing the outer slope as a free parameter.

progenax’s

\gamma

is a 3-D density slope, not EFF87’s surface slope

Elson et al. (1987) Eq. 1 defines $\mu(r) = \mu_0(1+r^2/a^2)^{-\gamma/2}$ as the projected surface brightness (their $\gamma \in [2.2, 3.2]$ , median 2.6, is the surface slope). progenax adopts the same functional form as the 3-D volume density $\rho(r)$ above — the standard N-body/IC-code convention (e.g. McLuster). So progenax’s $\gamma$ is a 3-D density slope, offset by $\approx 1$ from EFF87’s observed surface slope (Abel projection of a 3-D power law $r^{-\gamma}$ gives a surface power law $r^{-(\gamma-1)}$ ). Thus EFF87’s median surface slope $\gamma_{\rm EFF}\approx 2.6$ corresponds to a progenax 3-D slope $\approx 3.6$ ; the mass-convergence thresholds in the table below ( $\gamma > 3$ ) are 3-D-density statements. See Elson, Fall & Freeman (1987).

EFF behaviour vs. $\gamma$ .

$\gamma$	Outer slope at $r \gg a$	Cluster type
2	$\rho \propto r^{-2}$	Mass diverges; not physically valid
3	$\rho \propto r^{-3}$	Mass logarithmically divergent at infinity; truncation needed
4	$\rho \propto r^{-4}$	Typical young massive cluster
5	$\rho \propto r^{-5}$	Plummer (special case)
6	$\rho \propto r^{-6}$	Rare; very compact clusters

The untruncated EFF needs

\gamma > 3

— but progenax’s EFF is always truncated

For the untruncated profile ( $r_t \to \infty$ ) the total-mass integral converges only for $\gamma > 3$ : at $\gamma = 3$ it diverges logarithmically and at $\gamma < 3$ polynomially. progenax’s EFFProfile, however, always carries a finite truncation radius $r_t$ (constructor argument, default $r_t = 10$ ), so the sampled mass $M(<r_t)$ is finite for any $\gamma$ — including the default $\gamma = 3$ and shallower slopes. There is therefore no $\gamma > 3$ validation and no ValueError in EFFProfile; the truncation does the work that finite mass would otherwise require. The closed-form $M_{\rm total}$ below ((3)) is the $r_t \to \infty$ limit and applies only when $\gamma > 3$ ; the actual sampled mass is the truncated $M(<r_t)$ .

Enclosed mass¶

For the density $\rho(r) = \rho_0\,(1 + r^2/a^2)^{-\gamma/2}$ the enclosed mass has no elementary closed form for general $\gamma$ ; the exact result is a hypergeometric function,

M(<r) \;=\; \frac{4\pi}{3}\,\rho_0\,r^3\; {}_2F_1\!\left(\tfrac{3}{2},\,\tfrac{\gamma}{2};\,\tfrac{5}{2};\,-\frac{r^2}{a^2}\right).

(2)

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The total (untruncated) mass converges only for $\gamma > 3$ , where the integral evaluates to a ratio of Gamma functions,

M_{\mathrm{total}} \;=\; \lim_{r\to\infty} M(<r) \;=\; 4\pi\,\rho_0\,a^3\,\frac{\sqrt{\pi}}{4}\, \frac{\Gamma\!\big(\tfrac{\gamma-3}{2}\big)}{\Gamma\!\big(\tfrac{\gamma}{2}\big)}.

(3)

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As $\gamma \to 3^{+}$ the factor $\Gamma\!\big(\tfrac{\gamma-3}{2}\big)$ diverges: a shallow outer slope makes the profile so extended that its untruncated mass is infinite. This is exactly why a finite truncation radius $r_t$ is required for shallow $\gamma$ — the sampled mass is always the truncated $M(<r_t)$ .

Two special cases are worth keeping as sanity anchors:

$\gamma = 5$ is identical to a Plummer sphere of scale radius $a$ (both are $\rho \propto (1+r^2/a^2)^{-5/2}$ ), so the hypergeometric collapses to the elementary Plummer form $M(<r) = \tfrac{4\pi}{3}\rho_0 a^3\,\hat r^{3}\,(1+\hat r^2)^{-3/2}$ with $\hat r = r/a$ , giving $M_{\mathrm{total}} = \tfrac{4\pi}{3}\rho_0 a^3$ and the Plummer half-mass radius $r_h = a/\sqrt{2^{2/3}-1} \approx 1.305\,a$ .
$\gamma = 4$ integrates elementarily to $M(<r) = 2\pi\rho_0 a^3\big[\arctan\hat r - \hat r/(1+\hat r^2)\big]$ , with $M_{\mathrm{total}} = \pi^2\rho_0 a^3$ .

Sampling¶

EFFProfile is constructed from the scale radius $a$ directly (EFFProfile(a, gamma, r_t)) — it does not take $r_h$ . Because (2) has no elementary inverse, EFFProfile.sample_positions builds a numerical cumulative-mass table: it evaluates $4\pi r^2\rho(r)$ on a square-stretched grid $r = r_t\,u^2$ (which concentrates points in the core), forms the running trapezoidal integral, normalises it to a CDF, and draws radii by inverse-CDF interpolation (jnp.interp) of uniform deviates. The sampled mass is therefore the truncated $M(<r_t)$ .

Sampling stays differentiable in $a$ , $\gamma$ , and $r_t$ : the density enters the grid analytically, so gradients flow through the tabulated CDF and the interpolation. This is the practical advantage of EFF over King for fitting a cluster’s outer slope — $\gamma$ can be inferred via HMC alongside $r_h$ and the IMF parameters, whereas the King concentration $W_0$ requires re-solving the Poisson ODE at each gradient evaluation.

Velocities: Eddington inversion (no closed form)¶

Elson et al. (1987) is a surface-brightness model only — the paper fits the projected light profile $\mu(r)$ (Eq. 1) and its enclosed projected luminosity (Eq. 2). It gives no distribution function and no velocity-dispersion formula: there is no “EFF Eq. 7” for $\sigma_r^2(r)$ . Velocities in progenax are therefore not read from a closed form. Instead, treating (1) as the 3-D volume density, EFFVelocityDF builds the exact isotropic ergodic DF $f(E)$ by Eddington inversion of the (truncated) EFF density against its self-consistent potential $\Psi(r)$ ,

f(\mathcal{E}) \;=\; \frac{1}{\sqrt{8}\,\pi^2} \left[\int_0^{\mathcal{E}} \frac{\mathrm{d}^2\rho}{\mathrm{d}\Psi^2}\frac{\mathrm{d}\Psi}{\sqrt{\mathcal{E}-\Psi}} + \frac{1}{\sqrt{\mathcal{E}}}\left(\frac{\mathrm{d}\rho}{\mathrm{d}\Psi}\right)_{\!\Psi=0}\right],

(4)

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evaluated numerically on a tabulated grid at initialisation; speeds are then drawn per particle from $g(v)\propto v^2 f(\Psi(r) - v^2/2)$ by inverse-CDF. The construction is differentiable in $\gamma$ , $a$ , and $r_t$ through the tabulated inversion. Because the EFF density is empirical (not derived from a DF), a sharply truncated EFF is only approximately stationary: mild truncation (e.g. $\gamma = 5$ , which is exactly Plummer) is virial to $\sim1\%$ , whereas the steep $\gamma = 3$ default left strongly truncated is a few percent sub-virial — intrinsic to truncating an empirical profile, not a DF error. For a strict lowered-DF equilibrium use the King model.

Implementation in progenax¶

from progenax.profiles import EFFProfile
from progenax.kinematics import EFFVelocityDF
from jaxstro.units import STELLAR

profile = EFFProfile(a=1.0, gamma=3.0, r_t=10.0)   # a = scale radius, r_t = truncation
df = EFFVelocityDF(a=1.0, gamma=3.0, r_t=10.0)      # match a, gamma, r_t

masses = jnp.ones(1000)
positions = profile.sample_positions(masses, key_pos)
velocities = df.sample_velocities(positions, masses, key_vel, G=STELLAR.G)

EFFProfile is a fully-vectorised Equinox module. EFFVelocityDF builds the exact isotropic ergodic DF $f(E)$ of the (truncated) EFF density by Eddington inversion at initialisation, then samples speeds per particle from a tabulated inverse-CDF; it is fully JIT-compatible and differentiable. For radial anisotropy, pass anisotropy_radius (Osipkov–Merritt; see Anisotropy and rotation).

When to use EFF over Plummer or King¶

Use EFF when…	…because
Modelling young massive clusters	Observed surface-brightness profiles match power-law outer fall-off, not Plummer’s $r^{-5}$ or King’s exponential cutoff
Fitting $\gamma$ as a free parameter	$\gamma$ enters EFF analytically; no ODE re-solve per gradient step (unlike King’s $W_0$ )
LMC/SMC cluster work	Elson et al. (1987) calibrated EFF on 10 LMC clusters of ages 10⁷–10⁹ yr
Pre-relaxation / “post-formation” clusters	No assumption of dynamical equilibrium with the host galaxy’s tidal field

Use Plummer instead when…	…because
You want closed-form everything	Plummer’s DF is analytic; EFF’s requires quadrature
The outer slope doesn’t matter	For dynamical-evolution studies the central concentration dominates; outer slope effects on relaxation are sub-leading
Computational speed is critical	Plummer is ~ $3\times$ faster than EFF per IC realisation

Use King instead when…	…because
Modelling old globular clusters	Tidal truncation is well-defined and observationally constrained
The truncation radius is physically meaningful	King’s $r_t$ is the self-consistent tidal boundary where the DF vanishes; EFF’s $r_t$ is an imposed cutoff on an otherwise power-law (formally infinite) density
The host-galaxy tidal field is the dominant boundary	King’s lowered-isothermal DF assumes tidal-field equilibrium

Domain of validity and limitations¶

$\gamma > 3$ is required only for the untruncated profile to have finite total mass. progenax’s EFFProfile always truncates at a finite $r_t$ , so the sampled mass is finite for any $\gamma$ (the default is $\gamma = 3$ ); the constructor performs no $\gamma$ validation.
Truncation is intrinsic, not optional. The profile is sampled on $r \in [0, r_t]$ by construction; choose $r_t$ to match the cluster. Further tidal trimming can be layered via apply_tidal_truncation.
Only isotropic equilibrium DF. Anisotropic versions of EFF exist in the literature Elson et al., 1987 but are not currently in progenax. For anisotropy, use the King DF with the Osipkov-Merritt extension (Anisotropy and rotation).
Single-mass only. Multi-mass populations need post-segregation via Mass segregation.

Implementation, validation & references¶

In code: src/progenax/profiles/eff.py (density, truncated mass, analytic inverse-CDF sampling); velocities come from the Eddington inversion in src/progenax/kinematics/eddington.py via src/progenax/kinematics/eff_df.py. See the EFFProfile API.
Validated in: EFF profile — the regression suite, including the $\gamma = 5$ reduction to Plummer and the Eddington-DF virial check.
Primary sources: Elson et al. (1987) (the model). The power-law outer slope motivates other young-cluster families (Wilson, Woolley) that Gieles & Zocchi (2015) generalises uniformly (see the lowered-model family); EFF remains the simplest member capturing the key observation — a power-law outer fall-off with a free slope. Full notes in the bibliography.

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
Gieles, M., & Zocchi, A. (2015). A family of lowered isothermal models. Monthly Notices of the Royal Astronomical Society, 454, 576–592. 10.1093/mnras/stv1848