Initial mass functions - progenax — differentiable initial conditions for N-body simulations

Initial mass functions — model cards¶

Machine-generated from the provenance registry (ADR-0034): every model’s sources (with public DOI/arXiv/ADS pointers), governing equations, parameter meanings, code entry points, and the validation tests that pin them. The hand-authored theory pages hold the derivations; these cards are the citable single source of truth.

Salpeter / Kroupa piecewise power law¶

✅ verified · 3 equations · 2 sources · API: PowerLawIMF

N-segment piecewise power-law IMF with continuity enforced at the breaks: Salpeter (1955) is the 1-segment classic, Kroupa (2001) the canonical multi-segment form with the observed low-mass turnover. Exact analytic CDF/PPF per segment (expm1-stable through alpha = 1, audit S4).

Use it for	Not for
✓ literature-standard Salpeter/Kroupa populations and comparisons	✗ gradient-based IMF inference as the production default — the segment breaks are C^0 kinks; prefer the SMOOTH Maschberger form (house default)
✓ exact segment-wise closed-form sampling (no solver)	✗ smooth low-mass turnover shapes (use Chabrier or Maschberger)

Parameters¶

Name	Meaning	Units	Typical range	Code
`exponents`	per-segment slopes alpha_i (dN/dm proportional to m^-alpha_i); Salpeter 2.35, Kroupa (0.3, 1.3, 2.3)	dimensionless	0.3–2.7	`PowerLawIMF(exponents=[...])`
`breakpoints`	interior segment breaks (Kroupa: 0.08, 0.5 Msun)	Msun	0.08–1.0	`PowerLawIMF(breakpoints=[...])`
`m_min`	lower mass limit	Msun	0.01–1	`PowerLawIMF(m_min=...)`
`m_max`	upper mass limit	Msun	100–150	`PowerLawIMF(m_max=...)`

Equations¶

\xi(m) \;=\; \xi_0\,m^{-\alpha},\qquad \alpha = 2.35

(1)

Symbols: $\xi(m)$ : IMF in LINEAR mass, dN/dm (the log-mass slope is alpha - 1 = 1.35) [1 / Msun]. Assumes: valid for m >~ 1 Msun (no low-mass turnover). Derivation: theory page.

N(<m) / N \;=\; \frac{m^{1-\alpha} - m_{\min}^{1-\alpha}}{m_{\max}^{1-\alpha} - m_{\min}^{1-\alpha}}

(2)

Symbols: $N$ : cumulative count fraction [dimensionless]. Assumes: closed-form; inverted analytically for sampling; the alpha = 1 (and mean-mass alpha = 2) removable singularities use the expm1-stable kernels (progenax.numerics, audit S4) — gradients FD-exact AT the singular points. Derivation: theory page.

\xi(m) \;\propto\; \begin{cases} m^{-\alpha_0}, & 0.01 \le m < 0.08\,M_\odot & (\alpha_0 = 0.3) \\ m^{-\alpha_1}, & 0.08 \le m < 0.50\,M_\odot & (\alpha_1 = 1.3) \\ m^{-\alpha_2}, & 0.50 \le m < 1.00\,M_\odot & (\alpha_2 = 2.3) \\ m^{-\alpha_3}, & 1.00 \le m \le m_{\max} & (\alpha_3 = 2.3) \end{cases}

(3)

Symbols: $\alpha_i$ : per-segment slope; continuity coefficients enforced at each break [dimensionless]. Assumes: alpha_2 = alpha_3 = 2.3, so the 3-segment merge used by PowerLawIMF.kroupa() is exact. Derivation: theory page.

Sources¶

Salpeter (1955) — the original xi(m) proportional to m^-2.35 (scanned source; slope ledger-verified) (Eq. 5 + the luminosity-function fit (dN/dlogm slope 1.35 = linear-mass 2.35))
Kroupa (2001) — the canonical 4-segment broken power law with alpha = (0.3, 1.3, 2.3, 2.3) (Eq. 2)

Code & validation¶

code: PowerLawIMF (src/progenax/imf/power_law.py)
validation: tests/validation/test_imf_physics.py::TestSalpeterSlope::test_salpeter_high_mass_slope
validation: tests/validation/test_imf_physics.py::TestKroupaBreakpoints::test_kroupa_segment_slopes
validation: tests/validation/test_imf_physics.py::TestMeanMassAccuracy::test_mean_mass_resolution_converged

Chabrier (2003) lognormal + power law¶

✅ verified · 1 equation · 2 sources · API: ChabrierIMF

Lognormal low-mass body joined continuously to a Salpeter-like power-law tail at 1 Msun — the single-star (disk) IMF of Chabrier (2003). Smoother at low mass than a broken power law; the inverse CDF needs a fixed-iteration Newton solve.

Use it for	Not for
✓ literature-standard Chabrier single-star populations	✗ the fastest / cleanest-gradient sampling (no closed-form inverse; ~30 Newton steps per draw — the SMOOTH closed-form Maschberger is the house default)
✓ smooth low-mass turnover with the canonical disk parameters	✗ system (unresolved-binary) IMF studies without switching to the Table-1 system parameters

Parameters¶

Name	Meaning	Units	Typical range	Code
`m_c`	lognormal characteristic mass (single-star disk value 0.079 Msun)	Msun	0.08 (single-star) – 0.22 (system)	`ChabrierIMF(m_c=...)`
`sigma`	lognormal width in log10 m (single-star disk 0.69)	dex	0.55–0.69	`ChabrierIMF(sigma=...)`
`alpha`	high-mass power-law slope (linear-mass convention)	dimensionless	2.3	`ChabrierIMF(alpha=...)`
`m_trans`	lognormal-to-power-law transition mass	Msun	1.0	`ChabrierIMF(m_trans=...)`

Equations¶

\xi(m) \;\propto\; \begin{cases} \dfrac{1}{m\,\ln 10}\,\exp\!\Bigl[-\dfrac{(\log_{10} m - \log_{10} m_c)^2}{2\sigma_{\log m}^2}\Bigr], & m \le 1\,M_\odot \\ m^{-\alpha_3}, & m > 1\,M_\odot \end{cases}

(4)

Symbols: $m_c$ : characteristic (peak) mass of the lognormal [Msun]; $\sigma_{\log m}$ : lognormal width in dex [dex]. Assumes: single-star DISK parameters (not the system IMF); continuity enforced at m_trans. Derivation: theory page.

Sources¶

Chabrier (2003) — the lognormal + power-law single-star (disk) IMF and its parameters (Table 1 (single-star disk: m_c = 0.079, sigma = 0.69); Eq. 17 form)
Maschberger (2013) — independent parameter cross-check: his Eq. 8 restates the Chabrier form with k1 = 0.158, log m_c = log 0.079, sigma = 0.69, tail m^-2.3 (Eq. 8, p. 1727)

Code & validation¶

code: ChabrierIMF (src/progenax/imf/chabrier.py)
validation: tests/validation/test_imf_physics.py::TestChabrierParameters::test_chabrier_characteristic_mass
validation: tests/validation/test_imf_physics.py::TestChabrierParameters::test_chabrier_pdf_continuous_at_mtrans
validation: tests/validation/test_imf_physics.py::TestChabrierParameters::test_chabrier_high_mass_slope

Maschberger (2013) L3 — the smooth house default¶

✅ verified · 2 equations · 1 source · API: Maschberger

The L3 IMF: one smooth analytic form that reproduces the Kroupa/Chabrier shape (Salpeter tail + low-mass turnover) with a CLOSED-FORM quantile function. progenax’s production default.

Use it for	Not for
✓ PREFERRED over Kroupa/Chabrier as the production default: SMOOTH (C^inf, no breakpoint kinks) — exactly what gradient-based inference (HMC / Fisher forecasting) wants	✗ matching a literature analysis that is DEFINED on Kroupa/Chabrier segments (use those forms for like-for-like comparisons)
✓ fast exact sampling: closed-form quantile, no solver, ~5x faster than Chabrier under vmap
✓ IMF-parameter inference (alpha, beta, mu are clean continuous knobs)

Parameters¶

Name	Meaning	Units	Typical range	Code
`alpha`	high-mass exponent (Salpeter-like tail)	dimensionless	2.3 (canonical single-star)	`Maschberger(alpha=...)`
`beta`	low-mass tapering exponent	dimensionless	1.4 (single-star; 2.0 system)	`Maschberger(beta=...)`
`mu`	scale parameter (sets the turnover; NOT the peak mass itself)	Msun	0.2	`Maschberger(mu=...)`

Equations¶

\xi(m) \;\propto\; \biggl(\frac{m}{\mu}\biggr)^{\!-\alpha}\, \biggl[\,1 + \biggl(\frac{m}{\mu}\biggr)^{\!1-\alpha}\,\biggr]^{-\beta}

(5)

Symbols: $\mu$ : scale parameter [Msun]; $\alpha$ : high-mass exponent [dimensionless]; $\beta$ : low-mass tapering exponent [dimensionless]. Assumes: alpha != 1, beta != 1 (the L3 parameterization); m >> mu: Salpeter tail; m << mu: turnover. Derivation: theory page.

m(u) \;=\; \mu\,\biggl[\biggl(\frac{P_{\min} + u\,(P_{\max} - P_{\min})}{C}\biggr)^{\!1/(1-\beta)} - 1\biggr]^{1/(1-\alpha)}

(6)

Symbols: $u$ : uniform draw in [0, 1] [dimensionless]; $C$ : mu/[(1-beta)(1-alpha)]: converts the primitive P to Maschberger’s auxiliary G, making this ALGEBRAICALLY IDENTICAL to his Table-1 quantile [Msun]. Assumes: closed-form: no Newton solve, no rejection — fully vmap-compatible and cleanly differentiable. Derivation: theory page.

Sources¶

Maschberger (2013) — the L3 pdf, auxiliary function G(m), closed-form quantile, and canonical parameters (Table 1 rows 1-4 (G, CDF, pdf, quantile; alpha = 2.3, beta = 1.4, mu = 0.2), p. 1726 — PDF-verified 2026-07-11)

Code & validation¶

code: Maschberger (src/progenax/imf/smooth.py)
validation: tests/validation/test_imf_physics.py::TestMaschbergerProperties::test_maschberger_peak_mass
validation: tests/validation/test_imf_physics.py::TestMaschbergerProperties::test_maschberger_high_mass_salpeter
validation: tests/validation/test_imf_physics.py::TestMeanMassAccuracy::test_mean_mass_resolution_converged

Tapered power law (Parravano+2011 STPL)¶

✅ verified · 1 equation · 2 sources · API: TaperedPowerLaw

Power law with an exponential low-mass taper — the Smoothed Two-Power Law of Parravano, McKee & Hollenbach (2011): power law at both ends, smoothly joined. Smooth like Maschberger but WITHOUT a closed-form inverse (fixed-iteration Newton ppf).

Use it for	Not for
✓ the PMH11 galactic-disk IMF shape specifically	✗ the production default (Maschberger has the same smoothness PLUS a closed-form quantile)
✓ smooth turnover with an independently tunable taper sharpness beta

Parameters¶

Name	Meaning	Units	Typical range	Code
`alpha`	power-law slope (LINEAR-mass convention; PMH11’s log-mass Gamma = alpha - 1)	dimensionless	2.3	`TaperedPowerLaw(alpha=...)`
`m_peak`	taper/turnover mass (PMH11’s m_ch)	Msun	0.3	`TaperedPowerLaw(m_peak=...)`
`beta`	taper sharpness (PMH11’s gamma + Gamma exponent)	dimensionless	2.0	`TaperedPowerLaw(beta=...)`

Equations¶

f(m) \;\propto\; m^{-\alpha}\,\Bigl(1 - e^{-(m/m_{\mathrm{peak}})^{\beta}}\Bigr)

(7)

Symbols: $m_{\mathrm{peak}}$ : turnover mass (PMH11 m_ch) [Msun]; $\beta$ : taper sharpness [dimensionless]. Assumes: linear-mass convention: PMH11 write the log-mass slope Gamma = alpha - 1 and taper exponent gamma + Gamma = beta.

Sources¶

Parravano et al. (2011) — the STPL form psi(m) proportional to m^-Gamma [1 - exp(-(m/m_ch)^(gamma+Gamma))] — PDF-verified 2026-07-11 (Eq. 1, p. 2 (form historically first proposed by Paresce & De Marchi 2000))
Maschberger (2013) — the same form in linear-mass notation, as his comparison Eq. 12 (Eq. 12, p. 1727)

Code & validation¶

code: TaperedPowerLaw (src/progenax/imf/smooth.py)
validation: tests/unit/imf/test_smooth.py::TestTaperedPowerLaw::test_taper_suppresses_low_mass
validation: tests/unit/imf/test_smooth.py::TestTaperedPowerLaw::test_full_cdf_ppf_roundtrip

Truncated IMF (operator)¶

✅ verified · 1 equation · 1 source · API: TruncatedIMF

Wrapper applying hard m_min / m_max cutoffs to any base IMF, with the normalization recomputed on the truncated support. An operator card: the physics lives in the wrapped base IMF.

Use it for	Not for
✓ cluster-mass-dependent upper limits (m_max(M_cl))	✗ changing the SHAPE of the IMF (only the support is modified)
✓ regime studies with forced mass windows

Parameters¶

Name	Meaning	Units	Typical range	Code
`base_imf`	any IMF implementing the shared protocol	—	—	`TruncatedIMF(base_imf=...)`
`m_min`	hard lower cutoff	Msun	0.01–1	`TruncatedIMF(m_min=...)`
`m_max`	hard upper cutoff	Msun	5–150	`TruncatedIMF(m_max=...)`

Equations¶

\xi_{\mathrm{trunc}}(m) \;=\; \frac{\xi(m)\,\mathbf{1}[m_{\min} \le m \le m_{\max}]}{\int_{m_{\min}}^{m_{\max}} \xi(m^\prime)\,\mathrm{d}m^\prime}

(8)

Symbols: $\mathbf{1}[\cdot]$ : indicator of the truncated support [dimensionless]. Assumes: shape inherited from the base IMF; support and normalization replaced.

Sources¶

Marks et al. (2012) — the m_max(M_cl) motivation (Weidner-Kroupa relation) for cluster-dependent truncation (Section 2 context)

Code & validation¶

code: TruncatedIMF (src/progenax/imf/truncated.py)
validation: tests/unit/imf/test_imf_core.py::TestCDFProperties::test_cdf_at_m_max[TruncatedChabrier]
validation: tests/unit/imf/test_imf_core.py::TestCDFProperties::test_cdf_at_m_min[TruncatedChabrier]

Environment-dependent IMF (Marks+2012 / Jerabkova+2018)¶

✅ verified · 2 equations · 2 sources · API: BirthEnvironment

The functional alpha_3(environment) relations: the high-mass IMF slope as a function of birth-cloud density and metallicity (the Marks+2012 fundamental-plane relation, erratum-corrected, as restated by Jerabkova+2018). progenax’s BirthEnvironment + env_to_imf_params — NOT a galaxy-wide IGIMF integration.

Use it for	Not for
✓ top-heavy IMFs in dense / metal-poor birth environments (per-cluster alpha_3)	✗ galaxy-wide IGIMF calculations (out of scope; no IGIMF class exists — audit R7)
✓ environment-dependent population synthesis inputs to the binary engine	✗ sub-solar-mass slope changes (only alpha_3, and optionally alpha_1/alpha_2 via metallicity, respond)

Parameters¶

Name	Meaning	Units	Typical range	Code
`rho_cl`	birth-cloud (pre-cluster core) density	Msun / pc^3	1e2–1e7	`BirthEnvironment(rho_cl=...)`
`Fe_H`	metallicity [Fe/H]	dex	-2 – +0.5	`BirthEnvironment(Fe_H=...)`

Equations¶

\alpha_3(\hat x) \;=\; \begin{cases} -0.4072\,\hat x + 1.9383, & \hat x \ge -0.87 \\ 2.3, & \hat x < -0.87 \end{cases}

(9)

Symbols: $\hat x$ : fundamental-plane coordinate (density-metallicity combination) [dimensionless]. Assumes: erratum-corrected threshold -0.87 (Jerabkova 2018 Eq. 6); canonical 2.3 below threshold. Derivation: theory page.

\alpha_3(\log\rho_{\mathrm{cl}}, [\mathrm{Fe/H}]) \;=\; 0.0572\,[\mathrm{Fe/H}] - 0.4072\,\log_{10}\!\left(\frac{\rho_{\mathrm{cl}}}{10^6\,M_\odot\,\mathrm{pc}^{-3}}\right) + 1.9383

(10)

Symbols: $\rho_{\mathrm{cl}}$ : birth-cloud density [Msun / pc^3]. Assumes: the Marks 2012 Eq. 15 plane in physical variables. Derivation: theory page.

Sources¶

Marks et al. (2012) — the fundamental-plane alpha_3(x) relation and the Table-3 1-D relations (all ledger-verified cell-for-cell) (Eqs. 14-15, p. 2252 (slope -0.4072, intercept 1.9383); Table 3, p. 2251)
Jeřábková et al. (2018) — the erratum-corrected threshold x >= -0.87 and the restated relations (Eq. 6 (cites the Marks erratum); Eq. 9 (the 4pi form; see deviations))

Code & validation¶

code: BirthEnvironment (src/progenax/imf/environment/birth_environment.py)
code: env_to_imf_params (src/progenax/imf/environment/mapping.py)
validation: tests/unit/imf/test_environment.py::TestMarksCoefficients::test_coefficient_values
validation: tests/unit/imf/test_environment.py::TestGCValidation::test_marks_table1_gc

Schechter exponential-cutoff IMF¶

✅ verified · 1 equation · 1 source · API: Schechter

Power law with an exponential high-mass cutoff, f(m) proportional to m^-alpha exp(-m/m_star) — Schechter’s (1976) galaxy-luminosity-function form repurposed as an IMF for extreme environments / IGIMF-style truncation studies.

Use it for	Not for
✓ smooth (C^inf) high-mass suppression with a tunable cutoff scale m_star	✗ the production default (no closed-form inverse; Newton ppf — use Maschberger)
✓ top-light IMF experiments where a hard m_max is too blunt	✗ low-mass turnover shapes (no taper below m_star; combine with truncation)

Parameters¶

Name	Meaning	Units	Typical range	Code
`alpha`	power-law slope	dimensionless	2.3	`Schechter(alpha=...)`
`m_star`	exponential cutoff scale	Msun	10–100	`Schechter(m_star=...)`

Equations¶

f(m) \;\propto\; m^{-\alpha}\, e^{-m/m_\star}

(11)

Symbols: $m_\star$ : exponential cutoff mass [Msun]. Assumes: fixed-iteration Newton ppf (no closed-form inverse); differentiable and JIT-safe.

Sources¶

Schechter (1976) — the analytic form (introduced for the galaxy luminosity function; DOI verified live) (Eq. 2 (the Schechter form))

Code & validation¶

code: Schechter (src/progenax/imf/smooth.py)
validation: tests/unit/imf/test_smooth.py::TestSchechter::test_logpdf_large_mass_decays
validation: tests/unit/imf/test_smooth.py::TestSchechter::test_full_cdf_ppf_roundtrip

References¶

Salpeter, E. E. (1955). The luminosity function and stellar evolution. The Astrophysical Journal, 121, 161–167. 10.1086/145971
Kroupa, P. (2001). On the variation of the initial mass function. Monthly Notices of the Royal Astronomical Society, 322, 231–246. 10.1046/j.1365-8711.2001.04022.x
Chabrier, G. (2003). Galactic stellar and substellar initial mass function. Publications of the Astronomical Society of the Pacific, 115, 763–795. 10.1086/376392
Maschberger, T. (2013). On the function describing the stellar initial mass function. Monthly Notices of the Royal Astronomical Society, 429, 1725–1733. 10.1093/mnras/sts479
Parravano, A., McKee, C. F., & Hollenbach, D. J. (2011). An Initial Mass Function for Individual Stars in Galactic Disks. I. Constraining the Shape of the Initial Mass Function. The Astrophysical Journal, 726, 27. 10.1088/0004-637X/726/1/27
Marks, M., Kroupa, P., Dabringhausen, J., & Pawlowski, M. S. (2012). Evidence for top-heavy stellar initial mass functions with increasing density and decreasing metallicity. Monthly Notices of the Royal Astronomical Society, 422, 2246–2254. 10.1111/j.1365-2966.2012.20767.x
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
Schechter, P. (1976). An analytic expression for the luminosity function for galaxies. The Astrophysical Journal, 203, 297–306. 10.1086/154079