Main sequence: Difference between revisions

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After condensation and ignition of a star, it generates [[thermal energy]] in its dense [[stellar core|core region]] through [[nuclear fusion]] of [[hydrogen]] into [[helium]]. During this stage of the star's lifetime, it is located on the main sequence at a position determined primarily by its mass but also based on its chemical composition and age. The cores of main-sequence stars are in [[hydrostatic equilibrium]], where outward thermal pressure from the hot core is balanced by the inward pressure of [[gravitational collapse]] from the overlying layers. The strong dependence of the rate of energy generation on temperature and pressure helps to sustain this balance. Energy generated at the core makes its way to the surface and is radiated away at the [[photosphere]]. The energy is carried by either [[radiation]] or [[convection]], with the latter occurring in regions with steeper temperature gradients, higher opacity, or both.

The main sequence is sometimes divided into upper and lower parts, based on the dominant process that a star uses to generate energy. The Sun, along with main sequence stars below about 1.5 times the [[solar mass|mass of the Sun]] ({{solar mass|1.5}}), primarily fuse hydrogen atoms together in a series of stages to form helium, a sequence called the [[proton–proton chain]]. Above this mass, in the upper main sequence, the nuclear fusion process mainly uses atoms of [[carbon]], [[nitrogen]], and [[oxygen]] as intermediaries in the [[CNO cycle]] that produces helium from hydrogen atoms. Main-sequence stars with more than two solar masses undergo convection in their core regions, which acts to stir up the newly created helium and maintain the proportion of fuel needed for fusion to occur. Below this mass, stars have cores that are entirely radiative with convective zones near the surface. With decreasing stellar mass, the proportion of the star forming a convective envelope steadily increases. The Mainmain-sequence stars below {{solar mass|0.4}} undergo convection throughout their mass. When core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen.

The more massive a star is, the shorter its lifespan on the main sequence. After the hydrogen fuel at the core has been consumed, the star [[stellar evolution|evolves]] away from the main sequence on the HR diagram, into a [[supergiant]], [[red giant]], or directly to a [[white dwarf]].

== History ==

{{Star nav}}

In the early part of the 20th century, information about the types and distances of [[star]]s became more readily available. The [[stellar spectrum|spectra]] of stars were shown to have distinctive features, which allowed them to be categorized. [[Annie Jump Cannon]] and [[Edward Charles Pickering]] at [[Harvard College Observatory]] developed a method of categorization that became known as the [[stellar classification|Harvard Classification Scheme]], published in the ''Harvard Annals'' in 1901.<ref name=longair06/>

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In April 2018, astronomers reported the detection of the most distant "ordinary" (i.e., main sequence) [[star]], named [[Icarus (star)|Icarus]] (formally, [[MACS J1149 Lensed Star 1]]), at 9 billion light-years away from [[Earth]].<ref name=" NA-20180402">{{cite journal |author=Kelly, Patrick L. |display-authors=etal |title=Extreme magnification of an individual star at redshift 1.5 by a galaxy-cluster lens |date=2 April 2018 |journal=[[Nature (journal) |Nature]] |volume=2 |issue=4 |pages=334–342 |doi=10.1038/s41550-018-0430-3 |arxiv=1706.10279 |bibcode=2018NatAs...2..334K |s2cid=125826925}}</ref><ref name=" SPC-20180402">{{cite web |last=Howell |first=Elizabeth |title=Rare Cosmic Alignment Reveals Most Distant Star Ever Seen |url=https://www.space.com/40171-cosmic-alignment-reveals-most-distant-star-yet.html |date=2 April 2018 |work=[[Space.com]] |access-date=2 April 2018}}</ref>

== Formation and evolution ==

{{Star formation}}

{{Main|Star formation|Protostar|Pre-main-sequence star|Stellar evolution#Main sequence}}

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* [[K-type main-sequence star]]

* [[M-type main-sequence star]]

M-type (and, to a lesser extent, K-type)<ref>{{Citecite journal |last1=Pettersen |first1=B. R. |last2=Hawley |first2=S. L. |date=1989-06-01 |title=A spectroscopic survey of red dwarf flare stars. |url=https://ui.adsabs.harvard.edu/abs/1989A&A...217..187P |journal=Astronomy and Astrophysics |volume=217 |pages=187–200 |bibcode=1989A&A...217..187P |issn=0004-6361}}</ref> main-sequence stars are usually referred to as [[Red dwarf|red dwarfs]].

== Properties ==

The majority of stars on a typical HR diagram lie along the main-sequence curve. This line is pronounced because both the [[stellar classification|spectral type]] and the [[luminosity]] depends only on a star's mass, at least to [[order of approximation|zeroth-order approximation]], as long as it is fusing hydrogen at its core—and that is what almost all stars spend most of their "active" lives doing.<ref name=mss_atoe/>

The temperature of a star determines its [[spectral type]] via its effect on the physical properties of [[plasma (physics)|plasma]] in its [[photosphere]]. A star's energy emission as a function of wavelength is influenced by both its temperature and composition. A key indicator of this energy distribution is given by the [[color index]], {{nowrap|''B''&nbsp; −&nbsp; ''V''}}, which measures the star's [[apparent magnitude|magnitude]] in blue (''B'') and green-yellow (''V'') light by means of filters.<ref group=note>By measuring the difference between these values, eliminates the need to correct the magnitudes for distance. However, this can be affected by [[Extinction (astronomy)|interstellar extinction]].</ref> This difference in magnitude provides a measure of a star's temperature.

== Dwarf terminology ==

Main-sequence stars are called dwarf stars,<ref name=smith91/><ref name=powell06/> but this terminology is partly historical and can be somewhat confusing. For the cooler stars, dwarfs such as [[red dwarf]]s, [[orange dwarf]]s, and [[yellow dwarf star|yellow dwarf]]s are indeed much smaller and dimmer than other stars of those colors. However, for hotter blue and white stars, the difference in size and brightness between so-called "dwarf" stars that are on the main sequence and so-called "giant" stars that are not, becomes smaller. For the hottest stars the difference is not directly observable and for these stars, the terms "dwarf" and "giant" refer to differences in [[spectral line]]s which indicate whether a star is on or off the main sequence. Nevertheless, very hot main-sequence stars are still sometimes called dwarfs, even though they have roughly the same size and brightness as the "giant" stars of that temperature.<ref name=moore06/>

The common use of "dwarf" to mean the main sequence is confusing in another way because there are dwarf stars that are not main-sequence stars. For example, a [[white dwarf]] is the dead core left over after a star has shed its outer layers, and is much smaller than a main-sequence star, roughly the size of [[Earth]]. These represent the final evolutionary stage of many main-sequence stars.<ref name=wd_sao/>

== Parameters ==

[[File:Morgan-Keenan spectral classification.svg|thumb|right|upright=1.2|Comparison of main sequence stars of each spectral class]]

By treating the star as an idealized energy radiator known as a [[black body]], the luminosity ''L'' and radius ''R'' can be related to the [[effective temperature]] ''T''_eff by the [[Stefan–Boltzmann law]]:

: <math>L = 4 \pi \sigma R^2 T_\text{eff}^4</math>

:<math>L = 4 \pi \sigma R^2 T_\text{eff}^4</math>

where ''σ'' is the [[Stefan–Boltzmann constant]]. As the position of a star on the HR diagram shows its approximate luminosity, this relation can be used to estimate its radius.<ref name=ohrd/>

The mass, radius, and luminosity of a star are closely interlinked, and their respective values can be approximated by three relations. First is the Stefan–Boltzmann law, which relates the luminosity ''L'', the radius ''R'' and the surface temperature ''T''_eff. Second is the [[mass–luminosity relation]], which relates the luminosity ''L'' and the mass ''M''. Finally, the relationship between ''M'' and ''R'' is close to linear. The ratio of ''M'' to ''R'' increases by a factor of only three over 2.5 [[orders of magnitude]] of ''M''. This relation is roughly proportional to the star's inner temperature ''T_I'', and its extremely slow increase reflects the fact that the rate of energy generation in the core strongly depends on this temperature, whereas it has to fit the mass-luminosity relation. Thus, a too-high or too-low temperature will result in stellar instability.

A better approximation is to take {{nowrap begin}}|1=''ε'' = ''L''/''M''{{nowrap end}}, the energy generation rate per unit mass, as ''ε'' is proportional to ''T_I''¹⁵, where ''T_I'' is the core temperature. This is suitable for stars at least as massive as the Sun, exhibiting the [[CNO cycle]], and gives the better fit {{nowrap|''R'' ∝ ''M''^0.78}}.<ref>{{cite web |title=A course on stars' physical properties, formation and evolution |publisher=University of St. Andrews |url=http://www-star.st-and.ac.uk/~kw25/teaching/stars/STRUC4.pdf |access-date=2010-05-18 |archive-date=2020-12-02 |archive-url=https://web.archive.org/web/20201202003201/http://www-star.st-and.ac.uk/~kw25/teaching/stars/STRUC4.pdf |url-status=dead }}</ref>

=== Sample parameters ===

The table below shows typical values for stars along the main sequence. The values of [[luminosity]] (''L''), [[radius]] (''R''), and [[mass]] (''M'') are relative to the Sun—a dwarf star with a spectral classification of G2 &nbsp;V. The actual values for a star may vary by as much as 20–30% from the values listed below.<ref name=siess00/>{{Why|date=December 2023}}

<!-- Please include a solid reference if you add additional values to this table. -->

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[[File:Representative lifetimes of stars as a function of their masses.svg|thumb|upright=1.35|Representative lifetimes of stars as a function of their masses]]

== Energy generation ==

{{See also|Stellar nucleosynthesis}}

[[File:Nuclear energy generation.svg|right|upright=1.5|thumb|[[Logarithm]] of the relative energy output (ε) of [[proton–proton chain|proton–proton]] (PP), [[CNO cycle|CNO]] and [[triple-alpha process|triple-α]] fusion processes at different temperatures (''T''). The dashed line shows the combined energy generation of the PP and CNO processes within a star. At the Sun's core temperature, the PP process is more efficient.]]

All main-sequence stars have a core region where energy is generated by nuclear fusion. The temperature and density of this core are at the levels necessary to sustain the energy production that will support the remainder of the star. A reduction of energy production would cause the overlaying mass to compress the core, resulting in an increase in the fusion rate because of higher temperature and pressure. Likewise, an increase in energy production would cause the star to expand, lowering the pressure at the core. Thus the star forms a self-regulating system in [[hydrostatic equilibrium]] that is stable over the course of its main-sequence lifetime.<ref name=brainerd/>

Main-sequence stars employ two types of hydrogen fusion processes, and the rate of energy generation from each type depends on the temperature in the core region. Astronomers divide the main sequence into upper and lower parts, based on which of the two is the dominant fusion process. In the lower main sequence, energy is primarily generated as the result of the [[proton–proton chain]], which directly fuses hydrogen together in a series of stages to produce helium.<ref name=hannu/> Stars in the upper main sequence have sufficiently high core temperatures to efficiently use the [[CNO cycle]] (see chart). This process uses atoms of [[carbon]], [[nitrogen]], and [[oxygen]] as intermediaries in the process of fusing hydrogen into helium.

At a stellar core temperature of 18 million [[Kelvinkelvin]], the PP process and CNO cycle are equally efficient, and each type generates half of the star's net luminosity. As this is the core temperature of a star with about 1.5 {{solar mass|1.5}}, the upper main sequence consists of stars above this mass. Thus, roughly speaking, stars of spectral class F or cooler belong to the lower main sequence, while A-type stars or hotter are upper main-sequence stars.<ref name=clayton83/> The transition in primary energy production from one form to the other spans a range difference of less than a single solar mass. In the Sun, a one solar-mass star, only 1.5% of the energy is generated by the CNO cycle.<ref name=apj555/> By contrast, stars with 1.8 {{solar mass|1.8}} or above generate almost their entire energy output through the CNO cycle.<ref name=maurizio05/>

The observed upper limit for a main-sequence star is 120–200 {{solar mass|120–200}}.<ref name=apj620_1/> The theoretical explanation for this limit is that stars above this mass can not radiate energy fast enough to remain stable, so any additional mass will be ejected in a series of pulsations until the star reaches a stable limit.<ref name=apj162/> The lower limit for sustained proton-proton nuclear fusion is about 0.08 {{solar mass|0.08}} or 80 times the mass of [[Jupiter]].<ref name=hannu/> Below this threshold are sub-stellar objects that can not sustain hydrogen fusion, known as [[brown dwarf]]s.<ref name=apj406_1/>

== Structure ==

{{Main|Stellar structure}}

[[File:Solar internal structure.svg|right|upright=1.0|thumb|This diagram shows a cross-section of a Sun-like star, showing the internal structure.]]

Because there is a temperature difference between the core and the surface, or [[photosphere]], energy is transported outward. The two modes for transporting this energy are [[radiation]] and [[convection]]. A [[radiation zone]], where energy is transported by radiation, is stable against convection and there is very little mixing of the plasma. By contrast, in a [[convection zone]] the energy is transported by bulk movement of plasma, with hotter material rising and cooler material descending. Convection is a more efficient mode for carrying energy than radiation, but it will only occur under conditions that create a steep temperature gradient.<ref name=brainerd/><ref name=aller91/>

In massive stars (above 10 {{solar mass|10}})<ref name=aaa102_1/> the rate of energy generation by the CNO cycle is very sensitive to temperature, so the fusion is highly concentrated at the core. Consequently, there is a high temperature gradient in the core region, which results in a convection zone for more efficient energy transport.<ref name=hannu/> This mixing of material around the core removes the helium ash from the hydrogen-burning region, allowing more of the hydrogen in the star to be consumed during the main-sequence lifetime. The outer regions of a massive star transport energy by radiation, with little or no convection.<ref name=brainerd/>

Intermediate-mass stars such as [[Sirius]] may transport energy primarily by radiation, with a small core convection region.<ref name=lockner06/> Medium-sized, low-mass stars like the Sun have a core region that is stable against convection, with a convection zone near the surface that mixes the outer layers. This results in a steady buildup of a helium-rich core, surrounded by a hydrogen-rich outer region. By contrast, cool, very low-mass stars (below 0.4 {{solar mass|0.4}}) are convective throughout.<ref name=science295_5552/> Thus the helium produced at the core is distributed across the star, producing a relatively uniform atmosphere and a proportionately longer main-sequence lifespan.<ref name=brainerd/>

== Luminosity-color variation ==

[[File: The Sun in white light.jpg|thumb|upright=1.0|The [[Sun]] is the most familiar example of a main-sequence star]]

As non-fusing helium accumulates in the core of a main-sequence star, the reduction in the abundance of hydrogen per unit mass results in a gradual lowering of the fusion rate within that mass. Since it is fusion-supplied power that maintains the pressure of the core and supports the higher layers of the star, the core gradually gets compressed. This brings hydrogen-rich material into a shell around the helium-rich core at a depth where the pressure is sufficient for fusion to occur. The high power output from this shell pushes the higher layers of the star further out. This causes a gradual increase in the radius and consequently luminosity of the star over time.<ref name=clayton83/> For example, the luminosity of the early Sun was only about 70% of its current value.<ref name=sp74/> As a star ages it thus changes its position on the HR diagram. This evolution is reflected in a broadening of the main sequence band which contains stars at various evolutionary stages.<ref name=padmanabhan01/>

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A nearly vertical region of the HR diagram, known as the [[instability strip]], is occupied by pulsating [[variable star]]s known as [[Cepheid variable]]s. These stars vary in magnitude at regular intervals, giving them a pulsating appearance. The strip intersects the upper part of the main sequence in the region of class ''A'' and ''F'' stars, which are between one and two solar masses. Pulsating stars in this part of the instability strip intersecting the upper part of the main sequence are called [[Delta Scuti variable]]s. Main-sequence stars in this region experience only small changes in magnitude, so this variation is difficult to detect.<ref name=green04/> Other classes of unstable main-sequence stars, like [[Beta Cephei variable]]s, are unrelated to this instability strip.

== 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 0.5 {{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 0.5 {{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 0.1–50 {{solar mass|0.1–50}}.<ref name=rolfs_rodney88/>

:<math>L\ \propto\ M^{3.5}</math>

This relationship applies to main-sequence stars in the range 0.1–50 {{solar mass}}.<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 10¹⁰ 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

:<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.

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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 10 {{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 0.5 {{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>

== Evolutionary tracks ==

{{Main|Stellar evolution}}

[[File:Evolutionary track 1m.svg|thumb|left|Evolutionary track of a star like the sun]]

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When a [[star cluster|cluster of stars]] is formed at about the same time, the main-sequence lifespan of these stars will depend on their individual masses. The most massive stars will leave the main sequence first, followed in sequence by stars of ever lower masses. The position where stars in the cluster are leaving the main sequence is known as the [[turnoff point]]. By knowing the main-sequence lifespan of stars at this point, it becomes possible to estimate the age of the cluster.<ref name=science299_5603>{{cite journal |last=Krauss |first=Lawrence M. |author2=Chaboyer, Brian |title=Age Estimates of Globular Clusters in the Milky Way: Constraints on Cosmology |journal=Science |date=2003 |volume=299 |issue=5603 |pages=65–69 |doi=10.1126/science.1075631 |pmid=12511641 |bibcode=2003Sci...299...65K |s2cid=10814581 }}</ref>

== See also ==

* [[Lists of astronomical objects]]

== Notes ==

{{Reflistreflist|group=note}}

== References ==

{{Reflistreflist|30em|refs=

<ref name=smith91>{{cite web |url=https://cass.ucsd.edu/archive/public/tutorial/HR.html |title=The Hertzsprung-Russell Diagram |author=Harding E. Smith |date=21 April 1999 |work=Gene Smith's Astronomy Tutorial |publisher=Center for Astrophysics & Space Sciences, University of California, San Diego |access-date=2009-10-29}}</ref>

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<ref name=wd_sao>{{cite encyclopedia |url=https://astronomy.swin.edu.au/cosmos/W/White+Dwarf |title=White Dwarf |encyclopedia=COSMOS—The SAO Encyclopedia of Astronomy |publisher=Swinburne University |access-date=2007-12-04}}</ref>

<ref name=siess00>{{cite web |last=Siess |first=Lionel |date=2000 |url=http://www.astro.ulb.ac.be/~siess/WWWTools/Isochrones |title=Computation of Isochrones |publisher=Institut d'astronomie et d'astrophysique, Université libre de Bruxelles |access-date=2007-12-06 |url-status=live |archive-url=https://web.archive.org/web/20140110092115/http://www.astro.ulb.ac.be/~siess/WWWTools/Isochrones |archive-date=2014-01-10}}—Compare, for example, the model isochrones generated for a ZAMS of 1.1 solar masses. This is listed in the table as 1.26 times the [[solar luminosity]]. At metallicity ''Z'' = 0.01 the luminosity is 1.34 times solar luminosity. At metallicity ''Z'' = 0.04 the luminosity is 0.89 times the solar luminosity.</ref>

<ref name=ohrd>{{cite web |url=http://astro.unl.edu/naap/hr/hr_background3.html |title=Origin of the Hertzsprung-Russell Diagram |publisher=University of Nebraska |access-date=2007-12-06}}</ref>

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|title=An expanded set of brown dwarf and very low mass star models

|journal=Astrophysical Journal

|date=1993 |volume=406 |issue=1 |pages=158–71
|bibcode=1993ApJ...406..158B |doi=10.1086/172427|doi-access=free
}}</ref>

<ref name=aller91>{{cite book |first=Lawrence H. |last=Aller |date=1991 |title=Atoms, Stars, and Nebulae |publisher=Cambridge University Press |isbn=978-0-521-31040-6}}</ref>

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<ref name=mnras386_1>{{cite journal |author1=Schröder, K.-P. |author2=Connon Smith, Robert |title=Distant future of the Sun and Earth revisited |journal=Monthly Notices of the Royal Astronomical Society |date=May 2008 |volume=386 |issue=1 |pages=155–163 |bibcode=2008MNRAS.386..155S |doi=10.1111/j.1365-2966.2008.13022.x |doi-access=free |arxiv=0801.4031 |s2cid=10073988}}</ref>

<ref name=arnett96>{{cite book |first=David |last=Arnett |date=1996 |title=Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present |publisher=Princeton University Press |isbn=978-0-691-01147-9}}—Hydrogen fusion produces 8×10^{{val|8|e=14&nbsp;[[joule|ul=J]]/[[kilogram|upl=kg]]}} while helium fusion produces 8×10^{{val|8|e=13&nbsp;|u=J/kg}}.</ref>

<ref name=lecchini07>For a detailed historical reconstruction of the theoretical derivation of this relationship by Eddington in 1924, see: {{cite book |first=Stefano |last=Lecchini |date=2007 |title=How Dwarfs Became Giants. The Discovery of the Mass-Luminosity Relation |publisher=Bern Studies in the History and Philosophy of Science |isbn=978-3-9522882-6-9}}</ref>