Editing Main sequence (section)

== Lifetime ==
[[File:Isochrone ZAMS Z2pct.png|upright=1.0|right|thumb|This plot gives an example of the mass-luminosity relationship for zero-age main-sequence stars. The mass and luminosity are relative to the present-day Sun.]]
The total amount of energy that a star can generate through nuclear fusion of hydrogen is limited by the amount of hydrogen fuel that can be consumed at the core. For a star in equilibrium, the thermal energy generated at the core must be at least equal to the energy radiated at the surface. Since the luminosity gives the amount of energy radiated per unit time, the total life span can be estimated, to [[order of approximation|first approximation]], as the total energy produced divided by the star's luminosity.<ref name=rit_ms/>

For a star with at least {{solar mass|0.5}}, when the hydrogen supply in its core is exhausted and it expands to become a [[red giant]], it can start to fuse [[helium]] atoms to form [[carbon]]. The energy output of the helium fusion process per unit mass is only about a tenth the energy output of the hydrogen process, and the luminosity of the star increases.<ref name="prialnik00"/> This results in a much shorter length of time in this stage compared to the main-sequence lifetime. (For example, the Sun is predicted to spend {{nowrap|130 million years}} burning helium, compared to about 12 billion years burning hydrogen.)<ref name=mnras386_1/> Thus, about 90% of the observed stars above {{solar mass|0.5}} will be on the main sequence.<ref name=arnett96/> On average, main-sequence stars are known to follow an empirical [[mass–luminosity relation]]ship.<ref name=lecchini07/> The luminosity (''L'') of the star is roughly proportional to the total mass (''M'') as the following [[power law]]:
: <math>L\ \propto\ M^{3.5}</math>
This relationship applies to main-sequence stars in the range {{solar mass|0.1–50}}.<ref name=rolfs_rodney88/>

The amount of fuel available for nuclear fusion is proportional to the mass of the star. Thus, the lifetime of a star on the main sequence can be estimated by comparing it to solar evolutionary models. The [[Sun]] has been a main-sequence star for about 4.5 billion years and it will become a red giant in 6.5 billion years,<ref name=apj418>{{cite journal |last=Sackmann |first=I.-Juliana |author2=Boothroyd, Arnold I. |author3=Kraemer, Kathleen E. |title=Our Sun. III. Present and Future |journal=Astrophysical Journal |date=November 1993 |volume=418 |pages=457–468 |doi=10.1086/173407 |bibcode=1993ApJ...418..457S|doi-access=free }}</ref> for a total main-sequence lifetime of roughly 10<sup>10</sup> years. Hence:<ref name=hansen_kawaler94>{{cite book |first=Carl J. |last=Hansen |author2=Kawaler, Steven D. |date=1994 |title=Stellar Interiors: Physical Principles, Structure, and Evolution |page=[https://archive.org/details/stellarinteriors00hans/page/28 28] |publisher=Birkhäuser |isbn=978-0-387-94138-7 |url-access=registration |url=https://archive.org/details/stellarinteriors00hans/page/28}}</ref>
: <math>\tau_\text{MS} \approx
  10^{10} \text{years} \left[ \frac{M}{M_\bigodot} \right] \left[ \frac{L_\bigodot}{L} \right] =
  10^{10} \text{years} \left[ \frac{M}{M_\bigodot} \right]^{-2.5}
</math>
where ''M'' and ''L'' are the mass and luminosity of the star, respectively, <math>M_\bigodot</math> is a [[solar mass]], <math>L_\bigodot</math> is the [[solar luminosity]] and <math>\tau_\text{MS}</math> is the star's estimated main-sequence lifetime.

Although more massive stars have more fuel to burn and might intuitively be expected to last longer, they also radiate a proportionately greater amount with increased mass. This is required by the stellar equation of state; for a massive star to maintain equilibrium, the outward pressure of radiated energy generated in the core not only must but ''will'' rise to match the titanic inward gravitational pressure of its  envelope. Thus, the most massive stars may remain on the main sequence for only a few million years, while stars with less than a tenth of a solar mass may last for over a trillion years.<ref name=apj482>{{cite journal |last=Laughlin |first=Gregory |author2=Bodenheimer, Peter |author3=Adams, Fred C. |title=The End of the Main Sequence |journal=The Astrophysical Journal |date=1997 |volume=482 |issue=1 |pages=420–432 |doi=10.1086/304125 |bibcode=1997ApJ...482..420L |doi-access=free}}</ref>

The exact mass-luminosity relationship depends on how efficiently energy can be transported from the core to the surface. A higher [[opacity (optics)|opacity]] has an insulating effect that retains more energy at the core, so the star does not need to produce as much energy to remain in [[hydrostatic equilibrium]]. By contrast, a lower opacity means energy escapes more rapidly and the star must burn more fuel to remain in equilibrium.<ref name=imamura07>{{cite web |last=Imamura |first=James N. |date=7 February 1995 |url=http://zebu.uoregon.edu/~imamura/208/feb6/mass.html |title=Mass-Luminosity Relationship |publisher=University of Oregon |access-date=8 January 2007  |archive-url=https://web.archive.org/web/20061214065335/http://zebu.uoregon.edu/~imamura/208/feb6/mass.html |archive-date=14 December 2006}}</ref> A sufficiently high opacity can result in energy transport via [[convection]], which changes the conditions needed to remain in equilibrium.<ref name=clayton83/>

In high-mass main-sequence stars, the opacity is dominated by [[electron scattering]], which is nearly constant with increasing temperature. Thus the luminosity only increases as the cube of the star's mass.<ref name="prialnik00"/> For stars below {{solar mass|10}}, the opacity becomes dependent on temperature, resulting in the luminosity varying approximately as the fourth power of the star's mass.<ref name=rolfs_rodney88>{{cite book |first=Claus E. |last=Rolfs |author2=Rodney, William S. |date=1988 |title=Cauldrons in the Cosmos: Nuclear Astrophysics |publisher=University of Chicago Press |isbn=978-0-226-72457-7}}</ref> For very low-mass stars, molecules in the atmosphere also contribute to the opacity. Below about {{solar mass|0.5}}, the luminosity of the star varies as the mass to the power of 2.3, producing a flattening of the slope on a graph of mass versus luminosity. Even these refinements are only an approximation, however, and the mass-luminosity relation can vary depending on a star's composition.<ref name=science295_5552>{{cite journal |last=Kroupa |first=Pavel |title=The Initial Mass Function of Stars: Evidence for Uniformity in Variable Systems |journal=Science |date=2002 |volume=295 |issue=5552 |pages=82–91 |url=https://www.science.org/doi/10.1126/science.1067524 |access-date=2007-12-03 |doi=10.1126/science.1067524 |pmid=11778039 |arxiv=astro-ph/0201098 |bibcode=2002Sci...295...82K |s2cid=14084249}}</ref>