Marks et al. (2012)

Abstract (paraphrased)¶

Uses residual-gas-expulsion modelling of the low-mass present-day mass functions of Galactic globular clusters (GCs) to infer their birth conditions (mass, radius, density, metallicity) and the high-mass IMF slope $\alpha_3$ required to expel the gas. Finds the high-mass IMF must become top-heavy (lower $\alpha_3$) with increasing pre-cluster core density and decreasing metallicity. A Fundamental Plane in $(\alpha_3, \log\rho_{\rm cl}, {\rm [Fe/H]})$ captures both dependences. This is the empirical basis for the density/metallicity-dependent high-mass IMF used in IGIMF theory.

The physics: top-heaviness from the gas-expulsion energy budget (§2, verified)¶

The α₃ relations are not fitted ad hoc — they fall out of an energy argument. An embedded cluster forms with star-formation efficiency $\epsilon = M_{\rm ecl}/(M_{\rm ecl}+M_{\rm gas})$; the leftover gas must be expelled by feedback from the massive (O/B) stars. Marks et al. ask: what high-mass slope $\alpha_3$ delivers exactly enough energy to unbind the residual gas within a cluster crossing time?

Energy required. For a Plummer cluster ($r_h = 1.305\,r_{\rm pl}$) that expels its gas, the change in binding energy is (Eq. 5)

with $M_{\rm cl}=M_{\rm ecl}/\epsilon$ the cloud-core mass and $r_{h,\rm f}=r_{h,\rm i}\,M_{\rm cl}/M_{\rm ecl}$ the post-expulsion half-mass radius (Eq. 6, slow/adiabatic expulsion, Hills 1980).

Energy supplied. The radiative + mechanical power deposited by all stars is (Eqs. 8–9)

so the budget is dominated by massive stars and is negligible for low-mass stars. Integrated over a crossing time $\tau_{\rm cr}=\tfrac{2}{\sqrt G}M_{\rm cl}^{-1/2}r_h^{3/2}$ (Eq. 7), $\alpha_3$ is chosen so that $E_{\rm OB}^{\tau_M}(\alpha_3)=E_{\rm OB}^{\rm req}$.

Why it goes top-heavy with density. A denser, more massive cluster sits in a deeper potential with more gas to expel ($E_{\rm OB}^{\rm req}$ larger), so it needs more massive stars — a flatter (smaller) $\alpha_3$. Lower metallicity raises the Jeans mass, also favouring massive stars. Hence $\alpha_3$ decreases with $\rho_{\rm cl}$ and with decreasing [Fe/H] — exactly the trends fit below.

Inputs — what actually sets α₃ (and what does not)¶

The variation is driven by three environmental quantities only: the pre-cluster cloud-core density $\rho_{\rm cl}$, the metallicity [Fe/H], and the cluster mass ($M_{\rm cl}$/$M_{\rm ecl}$, which fix $\rho_{\rm cl}$ via the $r_h$–$M$ relation). It does not involve any turbulence statistic — neither the density-PDF width $\sigma_s$ nor the turbulent power-spectrum slope $\beta$. In progenax this means the environment-dependent IMF slopes are independent of cluster.turbulence.spectral_slope_from_mach: the gravoturbulent $\beta$ feeds only the experimental FDF spatial field, never the mass function.

The canonical IMF and the α₃ relations (verified against the paper)¶

The stellar IMF is the canonical multi-power-law (Eq. 2; Kroupa 2001), $\xi(m)\propto m^{-\alpha_i}$ with $\alpha_1=1.3$ (0.08–$0.5\,M_\odot$), $\alpha_2=\alpha_3=2.3$ ($\ge 0.5\,M_\odot$). The star-formation efficiency is $\epsilon = M_{\rm ecl}/(M_{\rm ecl}+M_{\rm gas})$, $0.1<\epsilon<0.5$ (Eq. 1).

1-D relations (Eq. 11, Table 3, p. 2251). Each environmental variable $\lambda$ gives

Fundamental Plane (Eqs. 13–14, p. 2252; range corrected by the 2014 erratum). With $\vartheta = 98°$,

with $\cos 98° = -0.139$, $\sin 98° = 0.990$. Density dominates metallicity in setting $\alpha_3$ (smaller scatter in Fig. 3 than Fig. 4).

Low-mass metallicity dependence (Eq. 12, p. 2251).

reproducing the Table 4 grid (e.g. ${\rm [Fe/H]}=-2 \Rightarrow \alpha_1=0.30,\ \alpha_2=1.30$).

Use in progenax¶

• Environment-dependent IMFs — the density/metallicity-dependent high-mass IMF.

• progenax.imf.environment.alpha3_marks_plane — Fundamental Plane (4).

• progenax.imf.environment.alpha3_marks_table3 — the four 1-D relations (3).

• progenax.imf.environment.lowmass_slopes_metallicity — Eq. 12 low-mass slopes.

• MARKS_COEFFICIENTS, MARKS_TABLE3_COEFFICIENTS — the tabulated constants (all verified exact).

Notes¶

The 2014 erratum. Marks et al. (2014) (MNRAS 442, 3315; verified against the published PDF) reports a missing minus sign in the range of validity of Eq. 14: the printed “$x' \ge 0.87$” should read “$x' \ge -0.87$” (the same typo also appears just before Eq. 15). The slope and intercept are unchanged; the erratum quotes the rounded form $\alpha_3 = -0.41\,x' + 1.94$. The authors state the correct (-0.87) range was used in their own analysis and codes, so the paper’s results and Table 1 are unaffected. progenax encodes the negative -0.87 threshold in both MARKS_COEFFICIENTS and JERABKOVA_COEFFICIENTS (the latter via Jeřábková et al. (2018) Eq. 6), and keeps this paper’s full-precision MNRAS slope/intercept $-0.4072/1.9383$ rather than the erratum’s rounded $-0.41/1.94$.

The radius–mass relation $r_h = 0.1\,(M_{\rm ecl}/M_\odot)^{0.13}$ pc used in the density chain comes from the companion paper Marks & Kroupa (2012), A&A 543, A8 (a different paper), not this one.