Baumgardt & Makino (2003)

Abstract (paraphrased)¶

Reports a large set of collisional N-body simulations (NBODY4 on GRAPE6, $N = 8192$ to $131\,072$ stars) of multimass King clusters evolving in an external Galactic tidal field under the combined action of stellar evolution, two-body relaxation and tidal stripping. The clusters move on circular or eccentric orbits in a logarithmic Galactic potential with circular velocity $V_G = 220\ \mathrm{km\,s^{-1}}$ . The main result is that cluster lifetimes scale as $T \sim T_{\rm rh}^x$ with $x \approx 0.75$ (weakly sub-linear in the relaxation time), and that preferential loss of low-mass stars over the tidal boundary turns initially rising mass functions into ones that decline towards low masses. For progenax, the load-bearing content is the initial setup: the tidal radius of the King model is set equal to the tidal radius of the external field (their Eq. 1).

The load-bearing equation: tidal (Jacobi) radius in a logarithmic potential¶

For an orbit in a logarithmic external potential $\phi(R_G) = V_G^2 \ln R_G$ with circular velocity $V_G$ , the cluster radii are adjusted so the King tidal radius equals the tidal radius of the external field, given by (Eq. 1, p. 229):

r_t = \left(\frac{G\, m_c}{2\, V_G^2}\right)^{1/3} R_G^{2/3},

(1)

where $m_c$ is the cluster mass, $V_G$ the circular velocity of the Galaxy, and $R_G$ the distance of the cluster from the Galactic Centre (for eccentric orbits, evaluated at perigalacticon). This is the singular-isothermal-halo / flat-rotation-curve tidal radius.

A corollary used qualitatively in the mass-loss discussion (text after Eq. 11, p. 233): as a cluster loses mass its tidal radius shrinks as

r_t \propto m_c^{1/3},

(2)

which follows directly from (1) at fixed orbit ( $R_G$ , $V_G$ ).

Other key relations (context, not used in `tidal.py`)¶

Quantity	Form	Location
Adopted IMF	two-stage power law, $\xi(m)\,dm \sim m^{-1.3}\,dm$ ( $m<0.5\,M_\odot$ ), $m^{-2.3}\,dm$ ( $m\ge 0.5\,M_\odot$ ), 0.1– $15\,M_\odot$	Eq. 2, p. 229
Relaxation time	$T_{\rm rh} \sim \dfrac{\sqrt{m_h}\,r_h^{3/2}}{\langle m\rangle\,\sqrt{G}\,\ln(\gamma N)}$	Eq. 3, p. 230
Dissolution-time scaling	$T_{\rm diss} = \beta \left[\dfrac{N}{\ln(\gamma N)}\right]^x \dfrac{R_G}{\rm kpc}\left(\dfrac{V_G}{220\,{\rm km\,s^{-1}}}\right)^{-1}$ , $x\approx 0.75$ ( $W_0=5$ ), $x\approx 0.82$ ( $W_0=7$ )	Eqs. 6–7, p. 232
Eccentricity correction	$T_{\rm diss}(\epsilon) = T_{\rm diss}(0)\,(1-\epsilon)$	Eq. 8, p. 232

These are recorded for completeness; progenax does not implement the lifetime/dissolution fitting formulae.

Code cross-reference¶

src/progenax/tidal.py cites Baumgardt & Makino (2003) in two places:

Module-level reference (line 8): “Baumgardt & Makino (2003) MNRAS 340, 227 — Tidal stripping.” Correct attribution and bibliographic data.
fill_factor_to_r_h docstring (line 197): the function computes $r_h = f \cdot r_J$ (fill factor $f = r_h/r_J$ ) and lists Baumgardt & Makino (2003) as the reference.

The function whose formula matches (1) is jacobi_radius_isothermal:

# src/progenax/tidal.py, jacobi_radius_isothermal
Omega = V_circ / R_galactic
r_J = (G * M_cluster / (2.0 * Omega**2)) ** (1.0 / 3.0)

Substituting $\Omega = V_{\rm circ}/R_G$ gives $r_J = (G\,M / 2)^{1/3}\,(R_G/V_{\rm circ})^{2/3} = (G\,M / 2 V_{\rm circ}^2)^{1/3} R_G^{2/3}$ — exactly BM03 Eq. 1 with $m_c \to M_{\rm cluster}$ , $V_G \to V_{\rm circ}$ , $R_G \to R_{\rm galactic}$ . That function’s docstring currently cites only Binney & Tremaine (2008) §8.3.1; the same formula is BM03 Eq. 1, so adding the Baumgardt & Makino (2003) citation there would make the provenance complete. (jacobi_radius, the point-mass $r_J = R\,(M/3M_{\rm gal})^{1/3}$ , is a different relation — factor 3, not 2 — correctly attributed to King 1962 / Binney & Tremaine Eq. 8.91, not to this paper.)

Verification verdict¶

Verdict: MATCH (formula) + DISCREPANCY (citation placement)

jacobi_radius_isothermal ⇒ MATCH. The implemented formula reproduces BM03 Eq. 1 exactly (verified algebraically; factor of 2 confirmed against p. 229). The paper’s attribution and journal/volume/page data in the module header are correct.
fill_factor_to_r_h ⇒ DISCREPANCY (citation, not formula). The trivial relation $r_h = f\,r_J$ is correct, but the “fill factor” / “filling factor” concept ( $r_h/r_J$ ) does not appear anywhere in Baumgardt & Makino (2003) (verified by reading the full text, §1–§3.5; the paper sets $r_t$ via Eq. 1 and discusses $r_t \propto m_c^{1/3}$ but never defines a half-mass-to-tidal-radius ratio). The fill-factor terminology is later community usage (e.g. Hénon-ratio / Roche-filling-factor literature), not this paper. The citation on fill_factor_to_r_h is therefore unsupported as written.
Phase 2 (Anna-gated) recommendation: add the Baumgardt & Makino (2003) citation to jacobi_radius_isothermal (where it belongs), and either re-attribute or drop the Baumgardt & Makino (2003) reference on fill_factor_to_r_h. No src/ edits were made in this task.

Use in progenax¶

progenax.tidal.jacobi_radius_isothermal — implements (1) (BM03 Eq. 1).
progenax.tidal.fill_factor_to_r_h — uses $r_h = f\,r_J$ ; the fill-factor concept is not from this paper (see verdict).
Tidal physics & substructure — tidal-radius and truncation theory.

Notes¶

Eq. 1’s factor of 2 (vs. 3 for a Keplerian point mass) is specific to the logarithmic / flat-rotation-curve potential; do not confuse it with the point-mass Jacobi radius in jacobi_radius.

References¶

Baumgardt, H., & Makino, J. (2003). Dynamical evolution of star clusters in tidal fields. Monthly Notices of the Royal Astronomical Society, 340, 227–246. 10.1046/j.1365-8711.2003.06286.x