Abstract
It is easy to assume that heavier stars shine longer because they have more fuel. This paper counts, in a model where fuel and luminosity are both powers of mass, how lifetime changes with mass and how sensitive it is to the exponent. No new theorem or law is claimed.
First, heavier stars burn out sooner: with the Sun at $10$, mass $0.5$ gives $56.568542$, $2$ gives $1.767767$, $10$ gives $0.031623$ and $20$ gives $\mathbf{0.005590}$ (billion years) (Section 2).
Second, and this is the core. The difference in luminosity far exceeds the difference in fuel: between masses $0.5$ and $20$, fuel differs $40$ times but luminosity $404771.540502$ times, and lifetime $\mathbf{10119.288513}$ times (Section 2).
Third, lifetime estimates are sensitive to the luminosity exponent: the lifetime of a mass-$10$ star is $0.100000$ at exponent $3.0$, $0.031623$ at $3.5$ and $0.010000$ at $4.0$ (Section 3).
Fourth, a $0.5$ difference in exponent matters more for heavier stars. It changes the lifetime $\mathbf{3.162278}$-fold for mass $10$ and $4.472136$-fold for mass $20$ (Section 3).
Fifth, the formula matches stepped burning: burning the fuel of a mass-$5$ star in steps took $0.178890$, within $10^{-4}$ of the formula’s $0.178885$ (Section 2).
Sixth, the separator is whether one asks about the ratio of masses or the difference in the power’s exponent: lifetime is set by the $-2.5$ power of mass, and a small change in exponent moves estimates for heavy stars more than the mass ratio itself suggests (Section 4).
Keywords: stellar lifetime, mass-luminosity relation, main sequence, power law, exponent sensitivity.
Scope (what is not claimed)
No new theorem or law is claimed: that main-sequence luminosity grows roughly as a power of mass, that lifetime is estimated as fuel divided by luminosity, and that heavier stars live shorter, are all known in astronomy. No real stars are claimed: the luminosity exponent $3.5$, the solar lifetime of $10$ billion years, and a burnt share independent of mass are model values matching no star. Internal structure is not entered: differences in fusion reactions, convection and mass loss are not included. Colour is not treated: the relation of colour to surface temperature and the luminosity--temperature diagram are not treated. Relation to earlier papers: Paper 835 showed that a lifetime’s heartbeats are independent of body mass only because two powers cancel. This paper stands beside it. There the powers cancelled and body mass vanished. Here fuel’s first power and luminosity’s $3.5$ power do not cancel, and mass to the $-2.5$ remains. Paper 497 showed that Kelvin’s calculation was never refuted; what was replaced was the reservoir. This paper stands beside it too. There the Sun’s lifetime estimate moved with the assumed reservoir. Here it moves with the assumed luminosity exponent. What is added is laying out luminosity and lifetime for six masses, measuring the lifetime ratio $10119.288513$ between masses $0.5$ and $20$, measuring lifetime sensitivity to the luminosity exponent, and placing the separator on mass ratio versus exponent difference.
1. Notation and the Question
Setting: a star’s fuel is proportional to its mass $M$ (Sun $=1$), and its luminosity grows as mass to the power $p$. Lifetime is fuel divided by luminosity, with the solar lifetime set to $10$ (billion years).
$$L\propto M^{p},\qquad t=10\times\frac{M}{M^{p}}=10\,M^{1-p},\qquad p=3.5\ \Rightarrow\ t=10\,M^{-2.5}$$Lifetime was obtained from the formula and as the time for fuel, reduced by the luminosity in steps of $10^{-5}$, to run out.
Question: do heavy stars with more fuel shine longer?
2. Mass and Lifetime
Computation 2.1: with $p=3.5$, luminosity and lifetime were computed for each mass.
| mass | luminosity | lifetime (billion years) |
|---|---|---|
| $0.5$ | $0.088388$ | $56.568542$ |
| $1$ | $1.000000$ | $10.000000$ |
| $2$ | $11.313708$ | $1.767767$ |
| $5$ | $279.508497$ | $0.178885$ |
| $10$ | $3162.277660$ | $0.031623$ |
| $20$ | $35777.087640$ | $0.005590$ |
Observation 2.2 this is the core. The difference in luminosity far exceeds the difference in fuel. Between masses $0.5$ and $20$, fuel differs only $40$ times, but luminosity differs $404771.540502$ times, so lifetime differs $10119.288513$ times, and heavier stars burn out sooner. Burning a mass-$5$ star’s fuel in steps took $0.178890$, matching the formula’s $0.178885$.
3. Luminosity Exponent and Lifetime
Computation 3.1: the luminosity exponent was varied and the lifetime of a mass-$10$ star computed.
| luminosity exponent $p$ | lifetime at mass $10$ (billion years) |
|---|---|
| $3.0$ | $0.100000$ |
| $3.5$ | $0.031623$ |
| $4.0$ | $0.010000$ |
Observation 3.2 lifetime estimates are sensitive to the luminosity exponent. Between exponents $3.0$ and $3.5$, the lifetime of a mass-$10$ star differs $3.162278$ times ($\sqrt{10}$): for mass $20$ it differs $4.472136$ times ($\sqrt{20}$). A $0.5$ difference in exponent enters lifetime as mass to the $0.5$ power, so it matters more for heavier stars.
4. The Separator
Observation 4.1 the separator is whether one asks about the ratio of masses or the difference in the power’s exponent. Asking about the mass ratio, heavy stars with more fuel seem to shine longer: asking about the exponents, the difference $-2.5$ between fuel’s first power and luminosity’s $3.5$ power sets lifetime, and a small change in that exponent moves estimates for heavy stars several times over.
Observation 4.2 placed beside Papers 835 and 497. 835 showed that two powers cancel exactly so that a lifetime’s heartbeats do not depend on body mass. Here the powers of fuel and luminosity do not cancel, and mass to the $-2.5$ remains in lifetime. 497 showed that the Sun’s lifetime estimate moves with the assumed reservoir; here it moves with the assumed luminosity exponent.
Observation 4.3 what is not said. Real stellar lifetimes and luminosities, internal structure, mass loss, and colour--temperature relations are not treated. What is shown is that heavier stars burn out sooner, and the estimate is sensitive to the luminosity exponent.
5. Census and Closing
- lifetime $t=10M^{1-p}$ (Section 1): fuel over luminosity.
- $56.568542$ at mass $0.5$ (Computation 2.1): light stars last long.
- $\mathbf{0.005590}$ at mass $20$ (Computation 2.1): heavy stars are brief.
- luminosity ratio $404771.540502$ (Observation 2.2): fuel only $40$ times.
- lifetime ratio $\mathbf{10119.288513}$ (Observation 2.2): the core.
- stepped burning $0.178890$ (Observation 2.2): matches the formula.
- $0.100000$ at exponent $3.0$ (Computation 3.1): mass $10$.
- $0.010000$ at exponent $4.0$ (Computation 3.1): a tenth.
- $\mathbf{3.162278}$ times for a $0.5$ difference (Observation 3.2): $4.472136$ at mass $20$.
- mass ratio, or exponent difference (Observation 4.1): the separator; Papers 835, 497.
Closing: Heavier stars have more fuel yet burn out sooner, and their estimated lifetimes are sensitive to the exponent of luminosity. If fuel is proportional to mass and luminosity grows as the $3.5$ power, lifetime goes as the $-2.5$ power, and masses $0.5$ and $20$ differ in lifetime $10119.288513$ times, because their fuel differs only $40$ times while their luminosity differs $404771.540502$ times. Changing the luminosity exponent from $3.0$ to $3.5$ alone changes the lifetime of mass $10$ by $3.162278$ times and of mass $20$ by $4.472136$ times: a difference in exponent enters as a power of mass, mattering more for heavier stars. Placed among the earlier papers: beside Paper 835’s “two powers cancel exactly” and Paper 497’s “the estimate moves with the assumed reservoir”, we placed powers that do not cancel and a lifetime that moves with the assumed exponent. To be honest: the luminosity exponent and burnt share are model values, and neither internal structure nor mass loss is included.
References
- Carroll, B. W., & Ostlie, D. A. (2007). An Introduction to Modern Astrophysics (2nd ed.). Pearson Addison-Wesley.
- Yamagishi, Y. (2026). Kelvin’s Calculation Was Never Refuted; What Was Replaced Was the Reservoir [Paper 497]. YOI: https://www.yamagishiyuuki.com/yoi/497; DOI: 10.5281/zenodo.22760194
- Yamagishi, Y. (2026). A Lifetime’s Heartbeats Are Independent of Body Mass Only Because Two Powers Cancel [Paper 835]. YOI: https://www.yamagishiyuuki.com/yoi/835; DOI: 10.5281/zenodo.23015605
Reference
Yamagishi, Y. (2026). Heavier Stars Have More Fuel Yet Burn Out Sooner, and Their Estimated Lifetimes Are Sensitive to the Exponent of Luminosity [Paper 892]. YOI: https://www.yamagishiyuuki.com/yoi/892
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 892. This paper is a preprint and has not been peer reviewed.