Jeřábková et al. (2018) - progenax — differentiable initial conditions for N-body simulations

Abstract (paraphrased)¶

Builds a comprehensive grid of galaxy-wide IMFs (gwIMF) under the Integrated Galactic IMF (IGIMF) theory, for metallicities ${\rm [Fe/H]}\in(-3,1)$ and SFRs 10^-5– $10^{5}\,M_\odot\,{\rm yr}^{-1}$ . The gwIMF is the sum of the IMFs of all embedded clusters forming in a galaxy over $\delta t\approx 10$ Myr; each embedded cluster’s stellar IMF varies with its density and metallicity following Marks et al. (2012). The resulting gwIMF is bottom-light + top-heavy at high SFR and bottom-heavy at high metallicity. Three IGIMF variants are defined (IGIMF1/2/3); IGIMF3 lets the full IMF vary.

The IGIMF framework — how the galaxy-wide IMF is assembled (§3, verified)¶

The IGIMF idea: a galaxy does not form stars in one event but as a population of embedded clusters, and the galaxy-wide IMF (gwIMF) is the sum of all their stellar IMFs over a star-formation epoch $\delta t \approx 10$ Myr (the molecular-cloud free-fall/cycle time):

\xi_{\rm gwIMF}(m) = \int_{M_{\rm ecl,min}}^{M_{\rm ecl,max}({\rm SFR})} \xi_{\star}\!\big(m \,\big|\, \rho_{\rm cl}(M_{\rm ecl}),\,{\rm [Fe/H]}\big)\; \xi_{\rm ecl}(M_{\rm ecl})\; dM_{\rm ecl}.

(1)

Two ingredients:

The embedded-cluster mass function (ECMF) — how many clusters of each mass form (Eqs. 1–2): a power law $\xi_{\rm ecl}(M_{\rm ecl})\propto M_{\rm ecl}^{-\beta}$ with a galaxy-SFR-dependent slope $\beta = -0.106\,\log_{10}{\rm SFR} + 2$ (Weidner+2004), and an upper truncation $M_{\rm ecl,max}({\rm SFR})$ from $M_{\rm tot}={\rm SFR}\cdot\delta t$ (Eq. 3). Higher SFR → more, more-massive clusters → top-heavier gwIMF.
The embedded-cluster stellar IMF — the multi-power-law (Eq. 4) whose slopes vary with the cluster’s own $(\rho_{\rm cl}, {\rm [Fe/H]})$ following Marks et al. (2012) (below).

The IGIMF variants differ in what is allowed to vary: IGIMF1 (high-mass only), IGIMF2 (adds a metallicity-dependent ECMF), IGIMF3 (full stellar-IMF variation). progenax implements the embedded-cluster stellar-IMF mapping (the integrand), not the galaxy integral.

The embedded-cluster α₃ and the density parameter x (verified, §3.2–3.3)¶

The stellar IMF in an embedded cluster is the multi-power-law (Eq. 4) with high-mass slope (Eq. 6; attributed to Marks et al. 2012 + its 2014 erratum)

\alpha_3 = \begin{cases} 2.3, & x < -0.87\\ -0.41\,x + 1.94, & x \ge -0.87\end{cases}

(2)

with the density/metallicity parameter (Eq. 7)

x = -0.14\,{\rm [Fe/H]} + 0.99\,\log_{10}\!\Big(\tfrac{\rho_{\rm cl}}{10^6\,M_\odot\,{\rm pc}^{-3}}\Big),

(3)

where (Eq. 8) $\rho_{\rm cl}=3M_{\rm cl}/(4\pi r_h^3)$ , $r_h/{\rm pc}=0.1\,(M_{\rm ecl}/M_\odot)^{0.13}$ (Marks & Kroupa 2012), and $\epsilon = M_{\rm ecl}/M_{\rm cl} = 0.33$ . The low-mass slopes follow $\alpha_i = \alpha_{i,c}+\Delta\alpha\,{\rm [Fe/H]}$ (Eq. 10, $\Delta\alpha\approx0.5$ ). The embedded-cluster mass function (ECMF) is a power law of slope $\beta=-0.106\log_{10}{\rm SFR}+2$ (Eqs. 1–2).

Jeřábková et al. also give a concise mass-based form of the density parameter (Eq. 9, verified against the PDF, p. 6) that folds the whole density chain into a function of the embedded-cluster mass $M_{\rm ecl}$ alone:

x = -0.14\,{\rm [Fe/H]} + 0.6\log_{10}\!\Big(\tfrac{M_{\rm ecl}}{10^6\,M_\odot}\Big) + 2.83.

(4)

Warning

The Eq. 9 “2.83” intercept reflects Jeřábková’s 4π density convention, not progenax’s. Eq. 9 is derived from Eq. 7 plus $\log_{10}\rho_{\rm ecl} = 0.61\log_{10}M_{\rm ecl} + 2.08$ and $\log_{10}\rho_{\rm cl} = 0.61\log_{10}M_{\rm ecl} + 2.85$ (PDF p. 6). progenax instead uses the internally-consistent 8π half-mass density of Marks & Kroupa (2012) (which reproduces Marks 2012 Table 1 exactly), under which the consistent intercept is 0.2161, not 2.83. The printed 2.83 is verified to be exactly what the paper prints — progenax’s departure from it is the deliberate 8π-convention fix documented in the environment chapter, not a transcription error.

Use in progenax¶

Environment-dependent IMFs — the IGIMF embedded-cluster α₃ mapping.
progenax.imf.environment.alpha3_jerabkova_rho / x_jerabkova_rho — Eqs. 6–7 directly.
progenax.imf.environment.alpha3_jerabkova_generalized / alpha3_jerabkova_mecl — the mass-based form (x from $M_{\rm ecl}$ via the density chain).
JERABKOVA_COEFFICIENTS — the verified constants (-0.14, 0.99, -0.41, 1.94, -0.87).

Notes¶

Density-convention note (resolved — progenax follows the 8π definition)

Eq. 8 writes $\rho_{\rm cl}=3M_{\rm cl}/(4\pi r_h^3)$ , but this is internally inconsistent with Jeřábková’s own numerical relation $\log_{10}\rho_{\rm ecl}=0.61\log_{10}M_{\rm ecl}+2.08$ , which corresponds to the 8π half-mass density. The authoritative definition is Marks & Kroupa (2012) (A&A 543, A8, p. 2): $\rho_{\rm ecl}=3M_{\rm ecl}/(8\pi r_h^3)$ . progenax follows this 8π convention ( $\rho_{\rm cl}=\rho_{\rm ecl}/\epsilon=3M_{\rm cl}/(8\pi r_h^3)$ ), which reproduces Marks (2012) Table 1 exactly (NGC 104: $3\cdot9.40\times10^6/(8\pi\cdot0.49^3)=9.54\times10^6$ = the tabulated $\rho_{\rm cl}$ ) — the actual calibration basis for the α₃–ρ relation. The code is therefore consistent with the source; Jeřábková’s Eq. 8 “4π” is the inconsistency, not progenax.

References¶

Jeřábková, T., Kroupa, P., Dabringhausen, J., Hilker, M., & Bekki, K. (2018). Impact of metallicity and star formation rate on the time-dependent, galaxy-wide stellar initial mass function. Astronomy and Astrophysics, 620, A39. 10.1051/0004-6361/201833055