Bianchini et al. (2016)

Why progenax holds this: it supplies the physical interpretation of the mass-dependence of velocity dispersion that our Engine A multimass models (Gieles & Zocchi 2015, Peuten et al. 2017) produce. Crucially, its Appendix A derives the σ(m) relation directly from the GZ15 lowered-isothermal multimass DF — so it is the bridge that lets us state, with provenance, that progenax’s standard model already captures equipartition saturation, and quantify it.

The σ(m) fitting function (§3.1, eqs 3–4 — verified)¶

Globular clusters never reach full energy equipartition; two-body relaxation drives them only to partial equipartition, with a mass-dependent local slope. A single power law σ ∝ m^{−η} (Trenti & van der Marel 2013) fits only a restricted mass range, so Bianchini introduces an exponential fitting function valid across the whole range (eq 3):

equivalently σ² ∝ exp(−m/m_eq). The local slope (eq 4) is η(m) = −d ln σ / d ln m = ½ (m/m_eq) for m ≤ m_eq and ½ for m > m_eq. The piecewise m^{−1/2} branch above m_eq is imposed so the slope cannot exceed ½ (which would unphysically exceed equipartition) and to match the asymptotic limits of the analytic multimass DF models (App. A).

Meaning of m_eq. It is the mass above which the cluster has reached full equipartition (σ ∝ m^{−1/2}); below it, equipartition is only partial. Smaller m_eq ⇒ closer to global equipartition. Typical fitted values are m_eq ≳ 1 M_⊙, i.e. only stars/remnants above ~1 M_⊙ are in equipartition.

The fit is performed via a Gaussian likelihood over the observed (m_i, v_i) (eq 5), and white dwarfs are excluded (recent mass-loss leaves their kinematics inconsistent with their present mass).

Appendix A — σ(m) derived FROM the GZ15 multimass DF (the bridge — verified)¶

This is the part progenax relies on. Bianchini shows the exponential is not ad hoc: it is the low-mass Taylor expansion of the central velocity dispersion of a GZ15 multimass component. The component central dispersion, in corrected form (eq A1 — see the typo note below):

with m̄ the central-density-weighted mean mass and δ the GZ15 equipartition index (δ=½ ⇒ m_j s_j² = m̄ σ² constant).

Expanding for low mass (μ_j ≪ 1, δ=½) to second order (eq A2) and matching against the expansion of the exponential gives (eq A3):

and the high-mass limit (μ_j ≫ 1) gives σ̂_{1d,j0} ∼ μ_j^{−δ} ∝ m^{−1/2} (GZ15 §3.2.1) — the equipartition branch. Matching the linear terms of A2 ↔ A3 identifies the equipartition mass as a derived quantity:

i.e. m_eq is fixed by the mean mass m̄, the truncation order g, and the central concentration Ŵ₀ = φ̂₀ — it is not a free input to the DF.

The standard multimass model (μ_j = m_j/m̄, δ=½, no extra parameter) therefore already produces the Bianchini saturation. progenax uses (4) to validate that its differentiable Engine A model reproduces the equipartition relation analytically (see the multimass theory page). The LIMEPY code’s meq (in μ_j=(m_j+meq)/m̄) is a separate phenomenological knob that adds extra softening to decouple equipartition from g/Ŵ₀; it is not the (4) physics.

m_eq ↔ relaxation state (§4–6, the headline result — verified)¶

Fitting m_eq across Monte-Carlo cluster simulations (Downing et al. 2010) at matched time-snapshots, Bianchini finds a tight correlation between the degree of equipartition and dynamical age: with n_eq ≡ T_age/T_rc (cluster age in core relaxation times), clusters older than ~20 core relaxation times reach a maximum degree of equipartition (more concentrated / older ⇒ smaller m_eq). Consequences: (i) m_eq measured kinematically (HST proper motions) is a proxy for a cluster’s relaxation state; (ii) knowing T_rc photometrically predicts the σ(m) trend (incl. the unobservable low-mass / remnant regime); (iii) deviations from the tight m_eq–n_rel relation flag peculiar histories (post-core-collapse, IMBH, accretion). Binaries and dark remnants follow the same σ(m) as single stars (except recently-formed white dwarfs).

The Spitzer (1969) instability — heavy stars sinking and forming a self-gravitating subsystem that decouples and never equipartitions — is the physical origin of the LIMEPY code’s zeta “decoupling” knob (deferred in progenax).

Use in progenax¶

• Derived-m_eq validation ((4)): a zero-new-parameter check that our Engine A σ(m) matches Bianchini eq 3 with the derived m_eq.

• Honest provenance for the deferred meq/zeta knobs: Bianchini motivates them physically but does not define the code’s (m_j+meq) form — that is a code heuristic, documented as such (see Peuten et al. 2017).