Tout, Pols, Eggleton & Han (1996)

Abstract (paraphrased)¶

Presents strictly analytic fitting formulae for the zero-age main-sequence (ZAMS) radius and luminosity of stars as functions of mass and metallicity. The fits are built from a grid of 600 ZAMS stellar models computed with the Eggleton evolution code and OPAL/Alexander opacities, spanning $0.1 \le M/M_\odot \le 100$ and $0.0001 \le Z \le 0.03$. Both relations take the form of rational functions in mass with strictly positive coefficients (so the denominators never vanish), and each coefficient is itself a low-order polynomial in $\log_{10}(Z/Z_\odot)$. The formulae are everywhere continuous and differentiable in both mass and metallicity — important for luminosity-function and number-count work, which depends on the gradient of the mass–luminosity relation.

The formulae (Eqs. 1–4, verified against the PDF, p. 258)¶

Luminosity (Eq. 1):

Radius (Eq. 2):

with $M \equiv M/M_\odot$. The non-integer powers ($M^{5.5}$, $M^{2.5}$, $M^{6.5}$, …) are deliberate: the authors found that pure integer powers could not fit the relations to the required accuracy, so a $\sqrt{M}$ factor was introduced in eight places. Each coefficient (except $\nu$) is a degree-4 polynomial in metallicity (Eqs. 3–4):

The coefficient $\nu$ is fixed at its $Z = 0.02$ optimum for all metallicities (PDF p. 258): some radius-fit coefficients varied by orders of magnitude across $Z$ and threatened to drive the rational function negative, so $\nu$ was held $Z$-independent to stabilise the fit. At solar metallicity ($\zeta = 0$) every coefficient equals its column-$a$ value.

Validity and accuracy (abstract + §3, verified)¶

• Mass range: $0.1 \le M/M_\odot \le 100$. (Extrapolation beyond gives “inaccurate but still reasonable” numbers.)

• Metallicity range: $0.0001 \le Z \le 0.03$. Metallicity extrapolation must be avoided — outside the fitted range the rational functions can go negative.

• Accuracy: luminosity errors generally $< 7.5\%$ ($< 3\%$ at solar $Z = 0.02$); radius errors $< 5\%$ ($< 1.2\%$ at solar).

Solar ZAMS anchors (computed from the verified coefficients at $M = 1$, $Z = 0.02$)¶

• $L(1\,M_\odot) = (\alpha + \beta)/(\gamma + 1 + \delta + \epsilon + \zeta + \eta) \approx 0.70\,L_\odot$

• $R(1\,M_\odot) = (\theta + \iota + \kappa + \lambda + \mu)/(\nu + \xi + o + 1 + \pi) \approx 0.89\,R_\odot$

These are the ZAMS Sun (slightly under-luminous and more compact than today’s $L = R = 1$); progenax’s ZAMS tests anchor on these values rather than present-day solar.

Use in progenax¶

• progenax.stellar — ZAMS $L(M,Z)$ (1) and $R(M,Z)$ (2); the verified literals live in _TOUT_L_COEFFS (Table 1), _TOUT_R_COEFFS and _TOUT_R_NU (Table 2).

• ZAMS relations validation — validation of the ZAMS relations against the paper’s accuracy claims and the solar ZAMS anchors above.

• Used by the binary / dynamical-mass workflows that need stellar luminosities and radii from the sampled masses (e.g. mass–luminosity conversion for binary populations and dynamical-mass diagnostics).

Notes¶

progenax.stellar clips the input metallicity to $[10^{-4},\,0.03]$ before evaluating the fits — this is the paper’s own explicit guidance (avoid $Z$ extrapolation), not an arbitrary guard. These ZAMS formulae are a present-day placeholder for the stellar physics that the planned startrax (Hurley+ SSE/BSE) and stellax (MESA-on-JAX) packages will eventually supply.