Is luminosity really
mass to the 3.5?

“Luminosity scales as mass to the 3.5” is one of the most quoted numbers in stellar astrophysics — the empirical mass-luminosity relation, L ∝ M^α, that appears in every intro textbook's treatment of main-sequence stars. It's rarely checked against a large, uniform, modern sample of the stars the rule is supposedly about. Gaia DR3's FLAME pipeline fits both mass and luminosity to every star with good enough photometry and parallax, so this run pulls 24,676 of them — quality-flagged to FLAME's own best bin, restricted to logg ≥ 4.0 (the standard dwarf/giant boundary, since the classic exponent describes main-sequence stars, not evolved giants), and down-sampled by a fixed modulus on Gaia's source ID (a deterministic, unbiased cut, not a selection on mass or luminosity) from the roughly 617,000 stars that pass the quality filter alone.

The whole-sample slope is real, and steeper than advertised. log₁₀(L) regressed on log₁₀(M) across all 24,676 stars: +4.88, 95% CI [+4.86, +4.89] (R²=0.93), agreeing almost exactly with a 4,000-draw bootstrap (CI [+4.86, +4.89]). That interval excludes not just the textbook 3.5 but also 4.0 — the exponent most often cited for solar-type stars specifically. Nearly 40% steeper than the number in circulation, on a sample two orders of magnitude larger than most textbook calibration sets.

Split the sample and the slope itself moves. Stars under 2 M☉ — 24,513 of the 24,676, essentially the whole thing, since this Gaia sample tops out at 5.7 M☉ and only 163 stars clear the 2 M☉ line at all — fit +4.95, CI [+4.93, +4.97]: essentially the whole-sample number, since they are essentially the whole sample. The 163 stars at or above 2 M☉ fit a visibly shallower +4.02, CI [+3.86, +4.17] — closer to the textbook figure, but its interval still excludes 3.5 outright; a bootstrap check puts the slope roughly split on whether it clears 4.0 at all (59% of 4,000 resamples land above 4.0, essentially a coin flip on that specific threshold, even as both the OLS and bootstrap intervals cleanly exclude 3.5). A formal interaction test, run on the full sample with a mass-regime indicator and its own slope term, confirms the two fitted slopes are not the same line by chance: -0.94 log-units, 95% CI [-1.26, -0.61] — excludes zero (p=2.3e-08). The exponent genuinely flattens as mass climbs past the Sun's; it does not flatten all the way to 3.5 within the mass range this pull actually covers.

Forcing the textbook number onto the data costs real fit quality. Pin the slope to exactly 3.5 and re-solve only for the intercept: the residual sum of squares is 1,110, more than double the 529 the freely-fit slope achieves on the identical 24,676 stars — not a rounding difference, a fit that is measurably worse by the desk's own numbers. The free fit's residuals are tight (std 0.146 dex, 84.6% of stars land within 50% of their predicted luminosity, 98.7% within a factor of two) — this is not a noisy relationship struggling to find any slope; it is a clean relationship whose actual slope is not 3.5.

Named check, and the sample's own bias. Set M=1 (a Sun-mass star) in the whole-sample fit and it predicts L=1.17 L☉ — the Sun's own true luminosity is 1 L☉ by definition, so this is a +17% miss on the one star in the universe whose mass and luminosity are not in dispute. That is not the Sun being an outlier; it is this sample's own selection speaking. Gaia's FLAME masses and luminosities are derived from photometry and parallax fit against stellar-evolution model grids, run on whichever stars have the cleanest astrometry and spectrophotometry — a mix skewed toward the sample's more numerous F/G-type field stars, not a magnitude-matched solar-twin census. Named plainly as a limit, not smoothed over: the sample's own median mass sits below one solar mass, and the fit is dominated by that population, not tuned to reproduce the Sun exactly.