View source for Main sequence

{{short description|Continuous band of stars that appears on plots of stellar color versus brightness}}
{{For|the racehorse|Main Sequence (horse)}}
[[File:HRDiagram.png|thumb|upright=1.4|A [[Hertzsprung–Russell diagram]] plots the [[luminosity]] (or [[absolute magnitude]]) of a star against its [[color index]] (represented as B−V). The main sequence is visible as a prominent diagonal band from upper left to lower right. This plot shows 22,000 stars from the [[Hipparcos Catalog]] together with 1,000 low-luminosity stars (red and white dwarfs) from the [[Gliese Catalogue of Nearby Stars]].]]

In [[astronomy]], the '''main sequence''' is a classification of [[star]]s which appear on plots of stellar [[color index|color]] versus [[absolute magnitude|brightness]] as a continuous and distinctive band. Stars on this band are known as '''main-sequence stars''' or [[dwarf star]]s, and positions of stars on and off the band are believed to indicate their physical properties, as well as their progress through several types of star life-cycles. These are the most numerous true stars in the universe and include the [[Sun]]. Color-magnitude plots are known as [[Hertzsprung–Russell diagram]]s after [[Ejnar Hertzsprung]] and [[Henry Norris Russell]]. 

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 main-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/>

In [[Potsdam]] in 1906, the Danish astronomer [[Ejnar Hertzsprung]] noticed that the reddest stars—classified as K and M in the Harvard scheme—could be divided into two distinct groups. These stars are either much brighter than the Sun or much fainter. To distinguish these groups, he called them "giant" and "dwarf" stars. The following year he began studying [[star cluster]]s; large groupings of stars that are co-located at approximately the same distance. For these stars, he published the first plots of color versus [[luminosity]]. These plots showed a prominent and continuous sequence of stars, which he named the Main Sequence.<ref name=brown/>

At [[Princeton University]], [[Henry Norris Russell]] was following a similar course of research. He was studying the relationship between the spectral classification of stars and their actual brightness as corrected for distance—their [[absolute magnitude]]. For this purpose, he used a set of stars that had reliable [[parallax]]es and many of which had been categorized at Harvard. When he plotted the spectral types of these stars against their absolute magnitude, he found that dwarf stars followed a distinct relationship. This allowed the real brightness of a dwarf star to be predicted with reasonable accuracy.<ref name=obs36/>

Of the red stars observed by Hertzsprung, the dwarf stars also followed the spectra-luminosity relationship discovered by Russell. However, giant stars are much brighter than dwarfs and so do not follow the same relationship. Russell proposed that "giant stars must have low density or great surface brightness, and the reverse is true of dwarf stars". The same curve also showed that there were very few faint white stars.<ref name=obs36/>

In 1933, [[Bengt Strömgren]] introduced the term Hertzsprung–Russell diagram to denote a luminosity-spectral class diagram.<ref name=zfa7/> This name reflected the parallel development of this technique by both Hertzsprung and Russell earlier in the century.<ref name=brown/>

As evolutionary models of stars were developed during the 1930s, it was shown that, for stars with the same composition, the star's mass determines its luminosity and radius. Conversely, when a star's chemical composition and its position on the main sequence are known, the star's mass and radius can be deduced. This became known as the [[Vogt–Russell theorem]]; named after [[Heinrich Vogt (astronomer)|Heinrich Vogt]] and Henry Norris Russell. It was subsequently discovered that this relationship breaks down somewhat for stars of the non-uniform composition.<ref name=schatzman33/>

A refined scheme for [[stellar classification]] was published in 1943 by [[William Wilson Morgan]] and [[Philip Childs Keenan]].<ref name=keenan_morgan43/> The MK classification assigned each star a spectral type—based on the Harvard classification—and a luminosity class.  The Harvard classification had been developed by assigning a different letter to each star based on the strength of the hydrogen spectral line before the relationship between spectra and temperature was known.  When ordered by temperature and when duplicate classes were removed,  the [[spectral type]]s of stars followed, in order of decreasing temperature with colors ranging from blue to red, the sequence O, B, A, F, G, K, and M. (A popular [[mnemonic]] for memorizing this sequence of stellar classes is "Oh Be A Fine Girl/Guy, Kiss Me".) The luminosity class ranged from I to V, in order of decreasing luminosity. Stars of luminosity class V belonged to the main sequence.<ref name=tnc/>

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 stellar mass objects}}
[[File:Hot and brilliant O stars in star-forming regions.jpg|thumb|left|upright=1.2|Hot and brilliant [[O-type main-sequence star]]s in star-forming regions. These are all regions of star formation that contain many hot young stars including several bright stars of spectral type O.<ref>{{cite news |title=The Brightest Stars Don't Live Alone |newspaper=ESO Press Release |url=https://www.eso.org/public/news/eso1230/ |access-date=27 July 2012}}</ref>]]
When a [[protostar]] is formed from the [[Jeans instability|collapse]] of a [[giant molecular cloud]] of gas and dust in the local [[interstellar medium]], the initial composition is homogeneous throughout, consisting of about 70% hydrogen, 28% helium, and trace amounts of other elements, by mass.<ref name=asr34_1/> The initial mass of the star depends on the local conditions within the cloud. (The mass distribution of newly formed stars is described empirically by the [[initial mass function]].)<ref name=science295_5552/> During the initial collapse, this [[pre-main-sequence star]] generates energy through gravitational contraction. Once sufficiently dense, stars begin converting hydrogen into helium and giving off energy through an [[exothermic]] [[nuclear fusion]] process.<ref name=tnc/>

When nuclear fusion of hydrogen becomes the dominant energy production process and the excess energy gained from gravitational contraction has been lost,<ref name=science293_5538/> the star lies along a [[curve]] on the [[Hertzsprung–Russell diagram]] (or HR diagram) called the standard main sequence. Astronomers will sometimes refer to this stage as "zero-age main sequence", or ZAMS.<ref name=zams_sao/><ref name=Hansen1999/> The ZAMS curve can be calculated using computer models of stellar properties at the point when stars begin hydrogen fusion. From this point, the brightness and surface temperature of stars typically increase with age.<ref name=clayton83/>

A star remains near its initial position on the main sequence until a significant amount of hydrogen in the core has been consumed, then begins to evolve into a more luminous star. (On the HR diagram, the evolving star moves up and to the right of the main sequence.) Thus the main sequence represents the primary hydrogen-burning stage of a star's lifetime.<ref name=tnc/>

== Classification ==
Main sequence stars are divided into the following types:

* [[O-type main-sequence star]] 
* [[B-type main-sequence star]]
* [[A-type main-sequence star]]
* [[F-type main-sequence star]]
* [[G-type main-sequence star]]
* [[K-type main-sequence star]]
* [[M-type main-sequence star]]
M-type (and, to a lesser extent, K-type)<ref>{{cite 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'' − ''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''<sub>eff</sub> by the [[Stefan–Boltzmann law]]:
: <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''<sub>eff</sub>. 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<sub>I</sub>'', 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|1=''ε'' = ''L''/''M''}}, the energy generation rate per unit mass, as ''ε'' is proportional to ''T<sub>I</sub>''<sup>15</sup>, where ''T<sub>I</sub>'' 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''<sup>0.78</sup>}}.<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. -->
{| class="wikitable"
|+ Table of main-sequence stellar parameters<ref name=zombeck/>
|-
! [[Stellar classification|Stellar <br/>class]]
! [[Radius]], <br/>''R''/{{solar radius|link=yes}}
! Mass, <br/>''M''/{{solar mass|link=yes}}
! Luminosity, <br/>''L''/{{solar luminosity|link=yes}}
! {{abbr|Temp.|Surface temperature}} <br/>([[kelvin|K]])
! Examples<ref name=simbad/>
|-
| O2 ||     12    ||     100   || 800,000 || class="mw-no-invert" style="background-color:#{{Color temperature|50000|hexval}}"|50,000
| style="text-align: left;" | [[BI 253]]
|-
| O6 || {{0}}9.8  || {{0}}35   || 180,000 || class="mw-no-invert" style="background-color:#{{Color temperature|38000|hexval}}"|38,000
| style="text-align:left;" | [[Theta1 Orionis C|Theta<sup>1</sup> Orionis C]]
|-
| B0 || {{0}}7.4  || {{0}}18   || {{0}}20,000  || class="mw-no-invert" style="background-color:#{{Color temperature|30000|hexval}}"|30,000
|style="text-align:left;"|[[Phi1 Orionis|Phi<sup>1</sup> Orionis]]
|-
| B5 || {{0}}3.8  || {{0|00}}6.5  || {{0|000,}}800     || class="mw-no-invert" style="background-color:#{{Color temperature|16400|hexval}}"|16,400
|style="text-align:left;"|[[Pi Andromedae|Pi Andromedae A]]
|-
| A0 || {{0}}2.5  || {{0|00}}3.2  || {{0|000,0}}80      || class="mw-no-invert" style="background-color:#{{Color temperature|10800|hexval}}"|10,800
|style="text-align:left;"|[[Alpha Coronae Borealis|Alpha Coronae Borealis A]]
|-
| A5 || {{0}}1.7  || {{0|00}}2.1  || {{0|000,0}}20      || class="mw-no-invert" style="background-color:#{{Color temperature|8620|hexval}}"|{{0}}8,620
|style="text-align:left;"|[[Beta Pictoris]]
|-
| F0 || {{0}}1.3  || {{0|00}}1.7  || {{0|000,00}}6       || class="mw-no-invert" style="background-color:#{{Color temperature|7240|hexval}}"|{{0}}7,240
|style="text-align:left;"|[[Gamma Virginis]]
|-
| F5 || {{0}}1.2  || {{0|00}}1.3 || {{0|000,00}}2.5     || class="mw-no-invert" style="background-color:#{{Color temperature|6540|hexval}}"|{{0}}6,540
|style="text-align:left;"|[[Eta Arietis]]
|-
| G0 || {{0}}1.05 || {{0|00}}1.10 || {{0|000,00}}1.26    || class="mw-no-invert" style="background-color:#{{Color temperature|5920|hexval}}"|{{0}}5,920
|style="text-align:left;"|[[Beta Comae Berenices]]
|-
| G2
| {{n/a|align=left|{{0}}1{{0|.00}}}}
| {{n/a|align=left|{{0|00}}1{{0|.00}}}}
| {{n/a|align=left|{{0|000,00}}1{{0|.00}}}}
| class="mw-no-invert" style="background-color:#{{Color temperature|5780|hexval}}"|{{0}}5,780
|style="text-align:left;"| '''[[Sun]]'''<ref name=bydef group=note>The Sun is a typical type G2V star.</ref>
|-
| G5 || {{0}}0.93 || {{0|00}}0.93 || {{0|000,00}}0.79    || class="mw-no-invert" style="background-color:#{{Color temperature|5610|hexval}}"|{{0}}5,610
|style="text-align:left;"|[[Alpha Mensae]]
|-
| K0 || {{0}}0.85 || {{0|00}}0.78 || {{0|000,00}}0.40    || class="mw-no-invert" style="background-color:#{{Color temperature|5240|hexval}}"|{{0}}5,240
|style="text-align:left;"|[[70 Ophiuchi|70 Ophiuchi A]]
|-
| K5 || {{0}}0.74 || {{0|00}}0.69 || {{0|000,00}}0.16    || class="mw-no-invert" style="background-color:#{{Color temperature|4410|hexval}}"|{{0}}4,410
|style="text-align:left;"|[[61 Cygni|61 Cygni A]]<ref name=apj129/>
|-
| M0 || {{0}}0.51 || {{0|00}}0.60 || {{0|000,00}}0.072   || class="mw-no-invert" style="background-color:#{{Color temperature|3800|hexval}}"|{{0}}3,800
|style="text-align:left;"|[[Lacaille 8760]]
|-
| M5 || {{0}}0.18 || {{0|00}}0.15 || {{0|000,00}}0.0027  || class="mw-no-invert" style="background-color:#{{Color temperature|3120|hexval}}"|{{0}}3,120
|style="text-align:left;"|[[EZ Aquarii|EZ Aquarii A]]
|-
| M8 || {{0}}0.11 || {{0|00}}0.08 || {{0|000,00}}0.0004  || class="mw-no-invert" style="background-color:#{{Color temperature|2650|hexval}}"|{{0}}2,650
| style="text-align:left;" |[[VB 10|Van Biesbroeck's star]]<ref name=recons/>
|-
| L1 || {{0}}0.09 || {{0|00}}0.07 || {{0|000,00}}0.00017 || class="mw-no-invert" style="background-color:#{{Color temperature|2200|hexval}}"|{{0}}2,200
| style="text-align:left;" | [[2MASS J0523−1403]]
|}

[[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 [[kelvin]], 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 {{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 {{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 {{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 {{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 {{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 {{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/>

Other factors that broaden the main sequence band on the HR diagram include uncertainty in the distance to stars and the presence of unresolved [[binary star]]s that can alter the observed stellar parameters. However, even perfect observation would show a fuzzy main sequence because mass is not the only parameter that affects a star's color and luminosity. Variations in chemical composition caused by the initial abundances, the star's [[stellar evolution|evolutionary status]],<ref name=apj128_3/> interaction with a [[binary star|close companion]],<ref name=tayler94/> [[stellar rotation|rapid rotation]],<ref name=mnras113/> or a [[stellar magnetic field|magnetic field]] can all slightly change a main-sequence star's HR diagram position, to name just a few factors. As an example, there are [[metallicity|metal-poor stars]] (with a very low abundance of elements with higher atomic numbers than helium) that lie just below the main sequence and are known as [[subdwarf]]s. These stars are fusing hydrogen in their cores and so they mark the lower edge of the main sequence fuzziness caused by variance in chemical composition.<ref name=cwcs13/>

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 {{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>

== Evolutionary tracks ==
{{Main|Stellar evolution}}
[[File:Evolutionary track 1m.svg|thumb|left|Evolutionary track of a star like the sun]]

When a main-sequence star has consumed the hydrogen at its core, the loss of energy generation causes its gravitational collapse to resume and the star evolves off the main sequence.  The path which the star follows across the HR diagram is called an evolutionary track.<ref name="Iben2012">{{cite book |author=Icko Iben |title=Stellar Evolution Physics |url=https://books.google.com/books?id=IU357EiecWwC&pg=PA1481 |date=29 November 2012 |publisher=Cambridge University Press |isbn=978-1-107-01657-6 |pages=1481–}}</ref>

[[File: Open cluster HR diagram ages.gif|right|thumb|upright=1.2|[[Hertzsprung–Russell diagram|H–R diagram]] for two open clusters: [[NGC 188]] (blue) is older and shows a lower turn off from the main sequence than [[Messier 67|M67]] (yellow). The dots outside the two sequences are mostly foreground and background stars with no relation to the clusters.]]

Stars with less than {{solar mass|0.23}}<ref name=romp69>{{cite journal |author1=Adams, Fred C. |author2=Laughlin, Gregory |title=A Dying Universe: The Long Term Fate and Evolution of Astrophysical Objects |journal=Reviews of Modern Physics |date=April 1997 |volume=69 |issue=2 |pages=337–372 |doi=10.1103/RevModPhys.69.337 |bibcode=1997RvMP...69..337A |arxiv=astro-ph/9701131 |s2cid=12173790}}</ref> are predicted to directly become [[white dwarf]]s when energy generation by nuclear fusion of hydrogen at their core comes to a halt, but stars in this mass range have main-sequence lifetimes longer than the current age of the universe, so no stars are old enough for this to have occurred.

In stars more massive than {{solar mass|0.23}}, the hydrogen surrounding the helium core reaches sufficient temperature and pressure to undergo fusion, forming a hydrogen-burning shell and causing the outer layers of the star to expand and cool. The stage as these stars move away from the main sequence is known as the [[subgiant branch]]; it is relatively brief and appears as a [[Hertzsprung gap|gap]] in the evolutionary track since few stars are observed at that point.

When the helium core of low-mass stars becomes degenerate, or the outer layers of intermediate-mass stars cool sufficiently to become opaque, their hydrogen shells increase in temperature and the stars start to become more luminous. This is known as the [[red-giant branch]]; it is a relatively long-lived stage and it appears prominently in H–R diagrams. These stars will eventually end their lives as white dwarfs.<ref name=pmss_atoe>{{cite web |author=Staff |date=12 October 2006 |url=http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_postmain.html |title=Post-Main Sequence Stars |publisher=Australia Telescope Outreach and Education |access-date=2008-01-08 |archive-url=https://web.archive.org/web/20130120215215/http://outreach.atnf.csiro.au/education/senior/astrophysics/stellarevolution_postmain.html |archive-date=20 January 2013 }}</ref><ref name=aaas141>{{cite journal |author1=Girardi, L. |author2=Bressan, A. |author3=Bertelli, G. |author4=Chiosi, C. |title=Evolutionary tracks and isochrones for low- and intermediate-mass stars: From 0.15 to 7 M<sub>sun</sub>, and from Z=0.0004 to 0.03 |journal=Astronomy and Astrophysics Supplement |date=2000 |volume=141 |issue=3 |pages=371–383 |doi=10.1051/aas:2000126 |arxiv=astro-ph/9910164 |bibcode=2000A&AS..141..371G |s2cid=14566232}}</ref>

The most massive stars do not become red giants; instead, their cores quickly become hot enough to fuse helium and eventually heavier elements and they are known as [[supergiant]]s. They follow approximately horizontal evolutionary tracks from the main sequence across the top of the H–R diagram.  Supergiants are relatively rare and do not show prominently on most H–R diagrams. Their cores will eventually collapse, usually leading to a [[supernova]] and leaving behind either a [[neutron star]] or [[black hole]].<ref name=sitko00>{{cite web |last=Sitko |first=Michael L. |date=24 March 2000 |url=http://www.physics.uc.edu/~sitko/Spring00/4-Starevol/starevol.html |title=Stellar Structure and Evolution |publisher=University of Cincinnati |access-date=2007-12-05  |archive-url=https://web.archive.org/web/20050326090756/http://www.physics.uc.edu/~sitko/Spring00/4-Starevol/starevol.html |archive-date=26 March 2005}}</ref>

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 ==
{{reflist|group=note}}

== References ==
{{reflist|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>

<ref name=powell06>{{cite web |url=http://www.atlasoftheuniverse.com/hr.html |title=The Hertzsprung Russell Diagram |author=Richard Powell |date=2006 |work=An Atlas of the Universe |access-date=2009-10-29}}</ref>

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<ref name=longair06>{{cite book |first=Malcolm S. |last=Longair |date=2006 |title=The Cosmic Century: A History of Astrophysics and Cosmology |url=https://archive.org/details/cosmiccenturyhis0000long |url-access=registration |pages=[https://archive.org/details/cosmiccenturyhis0000long/page/25 25–26] |publisher=Cambridge University Press |isbn=978-0-521-47436-8}}</ref>

<ref name=brown>{{cite book |editor1-first=Laurie M. |editor1-last=Brown |editor2-first=Abraham |editor2-last=Pais |editor2-link=Abraham Pais |editor3-first=A. B. |editor3-last=Pippard |editor3-link=A. B. Pippard |page=1696 |date=1995 |title=Twentieth Century Physics |publisher=[[Institute of Physics]], [[American Institute of Physics]] |location=[[Bristol]]; New York |isbn=978-0-7503-0310-1 |oclc=33102501}}</ref>

<ref name=obs36>{{cite journal |last=Russell |first=H. N. |title="Giant" and "dwarf" stars |journal=The Observatory |date=1913 |volume=36 |pages=324–329 |bibcode=1913Obs....36..324R}}</ref>

<ref name=zfa7>{{cite journal |last=Strömgren |first=Bengt |title=On the Interpretation of the Hertzsprung-Russell-Diagram |journal=Zeitschrift für Astrophysik |date=1933 |volume=7 |pages=222–248 |bibcode=1933ZA......7..222S}}</ref>

<ref name=schatzman33>{{cite book |first=Evry L. |last=Schatzman |date=1993 |author2=Praderie, Francoise |title=The Stars |url=https://archive.org/details/stars0000scha |url-access=registration |pages=[https://archive.org/details/stars0000scha/page/96 96–97] |publisher=Springer |isbn=978-3-540-54196-7}}</ref>

<ref name=keenan_morgan43>{{cite book |first=W. W. |last=Morgan |author2=Keenan, P. C. |author3=Kellman, E.  |date=1943 |title=An atlas of stellar spectra, with an outline of spectral classification |publisher=The University of Chicago press |location=Chicago, Illinois |url=http://ned.ipac.caltech.edu/level5/ASS_Atlas/MK_contents.html |access-date=2008-08-12}}</ref>

<ref name=tnc>{{cite book |first=Albrecht |last=Unsöld |date=1969 |title=The New Cosmos |page=268 |publisher=Springer-Verlag New York Inc |isbn=978-0-387-90886-1}}</ref>

<ref name=asr34_1>{{cite journal |last=Gloeckler |first=George |author2=Geiss, Johannes |title=Composition of the local interstellar medium as diagnosed with pickup ions  |journal=Advances in Space Research |date=2004 |volume=34 |issue=1 |pages=53–60 |bibcode=2004AdSpR..34...53G |doi=10.1016/j.asr.2003.02.054}}</ref>

<ref name=science293_5538>{{cite journal |last=Schilling |first=Govert |title=New Model Shows Sun Was a Hot Young Star |journal=Science |date=2001 |volume=293 |issue=5538 |pages=2188–2189 |doi=10.1126/science.293.5538.2188 |pmid=11567116 |s2cid=33059330|doi-access=free }}</ref>

<ref name=zams_sao>{{cite encyclopedia |url=https://astronomy.swin.edu.au/cosmos/Z/Zero+Age+Main+Sequence |title=Zero Age Main Sequence |encyclopedia=The SAO Encyclopedia of Astronomy |publisher=Swinburne University |access-date=2007-12-09}}</ref>

<ref name=mss_atoe>{{cite web |title=Main Sequence Stars |publisher=Australia Telescope Outreach and Education |url=https://www.atnf.csiro.au/outreach/education/senior/astrophysics/stellarevolution_mainsequence.html |url-status=live |archive-url=https://web.archive.org/web/20211125235409/https://www.atnf.csiro.au/outreach/education/senior/astrophysics/stellarevolution_mainsequence.html |archive-date=2021-11-25}}</ref>

<ref name=moore06>{{cite book |first=Patrick |last=Moore |author-link=Patrick Moore |date=2006 |title=The Amateur Astronomer |publisher=Springer |isbn=978-1-85233-878-7}}</ref>

<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>

<ref name=zombeck>{{cite book |first=Martin V. |last=Zombeck |date=1990 |title=Handbook of Space Astronomy and Astrophysics |publisher=Cambridge University Press |edition=2nd |url=https://ads.harvard.edu/books/hsaa/toc.html |access-date=2007-12-06 |isbn=978-0-521-34787-7}}</ref>

<ref name=simbad>{{cite web |url=http://simbad.u-strasbg.fr/simbad/ |title=SIMBAD Astronomical Database |publisher=Centre de Données astronomiques de Strasbourg |access-date=2008-11-21}}</ref>

<ref name=apj129>{{cite journal |author1=Luck, R. Earle |author2=Heiter, Ulrike |title=Stars within 15 Parsecs: Abundances for a Northern Sample |journal=The Astronomical Journal |date=2005 |volume=129 |issue=2 |pages=1063–1083 |bibcode=2005AJ....129.1063L  |doi=10.1086/427250 |doi-access=free}}</ref>

<ref name=recons>{{cite web |author=Staff |date=1 January 2008 |url=http://www.chara.gsu.edu/RECONS/TOP100.posted.htm |title=List of the Nearest Hundred Nearest Star Systems |publisher=Research Consortium on Nearby Stars |access-date=2008-08-12  |archive-url=https://web.archive.org/web/20120513202710/http://www.chara.gsu.edu/RECONS/TOP100.posted.htm |archive-date=13 May 2012}}</ref>

<ref name=brainerd>{{cite web |last=Brainerd |first=Jerome James |date=16 February 2005 |url=https://www.astrophysicsspectator.com/topics/stars/MainSequence.html |title=Main-Sequence Stars |publisher=The Astrophysics Spectator |access-date=2007-12-04}}</ref>

<ref name=hannu>{{cite book |first=Hannu |last=Karttunen |date=2003 |title=Fundamental Astronomy |publisher=Springer |isbn=978-3-540-00179-9}}</ref>

<ref name=clayton83>{{cite book |first=Donald D. |last=Clayton |date=1983 |title=Principles of Stellar Evolution and Nucleosynthesis |url=https://archive.org/details/principlesofstel0000clay |url-access=registration |publisher=University of Chicago Press |isbn=978-0-226-10953-4}}</ref>

<ref name=apj555>{{cite journal |author1=Bahcall, John N. |author2=Pinsonneault, M. H. |author3=Basu, Sarbani |title=Solar Models: Current Epoch and Time Dependences, Neutrinos, and Helioseismological Properties |journal=The Astrophysical Journal |year=2003 |volume=555 |issue=2 |pages=990–1012 |bibcode=2001ApJ...555..990B |doi=10.1086/321493 |arxiv=astro-ph/0212331 |s2cid=13798091}}</ref>

<ref name=maurizio05>{{cite book |first=Maurizio |last=Salaris |author2=Cassisi, Santi |date=2005 |title=Evolution of Stars and Stellar Populations |url=https://archive.org/details/evolutionofstars0000sala |url-access=registration |page=[https://archive.org/details/evolutionofstars0000sala/page/128 128] |publisher=John Wiley and Sons |isbn=978-0-470-09220-0}}</ref>

<ref name=apj620_1>{{cite journal |last=Oey |first=M. S. |author2=Clarke, C. J. |title=Statistical Confirmation of a Stellar Upper Mass Limit |journal=The Astrophysical Journal |date=2005 |volume=620 |issue=1 |pages=L43–L46 |bibcode=2005ApJ...620L..43O |doi=10.1086/428396 |arxiv=astro-ph/0501135 |s2cid=7280299}}</ref>

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<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>

<ref name=aaa102_1>{{cite journal |author1=Bressan, A. G. |author2=Chiosi, C. |author3=Bertelli, G. |title=Mass loss and overshooting in massive stars |journal=Astronomy and Astrophysics |date=1981 |volume=102 |issue=1 |pages=25–30 |bibcode=1981A&A...102...25B}}</ref>

<ref name=lockner06>{{cite web |last=Lochner |first=Jim |author2=Gibb, Meredith |author3=Newman, Phil |date=6 September 2006 |url=http://imagine.gsfc.nasa.gov/docs/science/know_l2/stars.html |title=Stars |publisher=NASA |access-date=2007-12-05  |archive-url=https://web.archive.org/web/20141119192647/http://imagine.gsfc.nasa.gov/docs/science/know_l2/stars.html |archive-date=2014-11-19}}</ref>

<ref name=sp74>{{cite journal |last=Gough |first=D. O. |title=Solar interior structure and luminosity variations |journal=Solar Physics |date=1981 |volume=74 |issue=1 |pages=21–34 |bibcode=1981SoPh...74...21G |doi=10.1007/BF00151270 |s2cid=120541081}}</ref>

<ref name=padmanabhan01>{{cite book |first=Thanu |last=Padmanabhan |date=2001 |title=Theoretical Astrophysics |publisher=Cambridge University Press |isbn=978-0-521-56241-6}}</ref>

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<ref name=tayler94>{{cite book |first=Roger John |last=Tayler |date=1994 |title=The Stars: Their Structure and Evolution |publisher=Cambridge University Press |isbn=978-0-521-45885-6}}</ref>

<ref name=mnras113>{{cite journal |last=Sweet |first=I. P. A. |author2=Roy, A. E. |title=The structure of rotating stars |journal=[[Monthly Notices of the Royal Astronomical Society]] |date=1953 |volume=113 |issue=6 |pages=701–715 |bibcode=1953MNRAS.113..701S |doi=10.1093/mnras/113.6.701 |doi-access=free}}</ref>

<ref name=cwcs13>{{cite conference |last=Burgasser |first=Adam J. |author2=Kirkpatrick, J. Davy |author3=Lépine, Sébastien  |title=Spitzer Studies of Ultracool Subdwarfs: Metal-poor Late-type M, L and T Dwarfs |work=Proceedings of the 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun |page=237 |publisher=Dordrecht, D. Reidel Publishing Co |date=5–9 July 2004 |location=Hamburg, Germany |bibcode=2005ESASP.560..237B |url=https://ui.adsabs.harvard.edu/abs/2005ESASP.560..237B/ |access-date=2007-12-06}}</ref>

<ref name=green04>{{cite book |first=S. F. |last=Green |author2=Jones, Mark Henry |author3=Burnell, S. Jocelyn |date=2004 |title=An Introduction to the Sun and Stars |publisher=Cambridge University Press |isbn=978-0-521-54622-5}}</ref>

<ref name=rit_ms>{{cite web |last=Richmond |first=Michael W. |date=10 November 2004 |url=http://spiff.rit.edu/classes/phys230/lectures/star_age/star_age.html |title=Stellar evolution on the main sequence |publisher=Rochester Institute of Technology |access-date=2007-12-03}}</ref>

<ref name="prialnik00">{{cite book |first=Dina |last=Prialnik |date=2000 |title=An Introduction to the Theory of Stellar Structure and Evolution |publisher=Cambridge University Press |isbn=978-0-521-65937-6}}</ref>

<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 {{val|8|e=14|ul=J|upl=kg}} while helium fusion produces {{val|8|e=13|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>

<ref name=Hansen1999>{{citation
 |title=Stellar Interiors: Physical Principles, Structure, and Evolution
 |series=Astronomy and Astrophysics Library
 |first1=Carl J. |last1=Hansen |first2=Steven D. |last2=Kawaler
 |publisher=Springer Science & Business Media
 |year=1999 |isbn=978-0-387-94138-7 |page=39
 |url=https://books.google.com/books?id=m-_6LYuUbUkC&pg=PA39}}</ref>
}}

==Further reading==

===General===
* Kippenhahn, Rudolf, ''100 Billion Suns'', Basic Books, New York, 1983.

===Technical===
* {{cite book |last=Arnett |first=David |title=Supernovae and Nucleosynthesis |publisher=[[Princeton University Press]] |location=Princeton |year=1996}}
* {{cite book |last=Bahcall |first=John N. |title=Neutrino Astrophysics |url=https://archive.org/details/neutrinoastrophy0000bahc |url-access=registration |publisher=[[Cambridge University Press]] |location=Cambridge |year=1989 |isbn=978-0-521-37975-5}}
* {{cite journal |last1=Bahcall |first1=John N. |last2=Pinsonneault |first2=M.H. |last3=Basu |first3=Sarbani |title=Solar Models: Current Epoch and Time Dependences, Neutrinos, and Helioseismological Properties |journal=The Astrophysical Journal |volume=555 |pages=990–1012 |number=2 |year=2001 |arxiv=astro-ph/0010346 |doi=10.1086/321493 |bibcode=2001ApJ...555..990B |s2cid=13798091}}
* {{cite book |editor1-last=Barnes |editor1-first=C. A. |editor2-last=Clayton |editor2-first=D. D. |editor3-last=Schramm |editor3-first=D. N. |title=Essays in Nuclear Astrophysics |publisher=Cambridge University Press |location=Cambridge |year=1982}}
* {{cite book |last1=Bowers |first1=Richard L. |last2=Deeming |first2=Terry |title=Astrophysics I: Stars |publisher=Jones and Bartlett |location=Boston |year=1984}}
* {{cite book |title=An Introduction to Modern Astrophysics |first1=Bradley W. |last1=Carroll  |first2=Dale A. |last2=Ostlie  |name-list-style=amp |date=2007 |publisher=Pearson Education Addison-Wesley |location=San Francisco |isbn=978-0-8053-0402-2}}
* {{cite journal |last1=Chabrier |first1=Gilles |last2=Baraffe |first2=Isabelle |title=Theory of Low-Mass Stars and Substellar Objects |journal=Annual Review of Astronomy and Astrophysics |volume=38 |pages=337–377 |year=2000 |arxiv=astro-ph/0006383 |doi=10.1146/annurev.astro.38.1.337 |bibcode=2000ARA&A..38..337C |s2cid=59325115}}
* {{cite book |last=Chandrasekhar |first=S. |author-link=Subramanyam Chandrasekhar |title=An Introduction to the study of stellar Structure |publisher=Dover |location=New York |year=1967}}
* {{cite book |last=Clayton |first=Donald D. |title=Principles of Stellar Evolution and Nucleosynthesis |url=https://archive.org/details/principlesofstel0000clay |url-access=registration |publisher=University of Chicago |location=[[Chicago]] |year=1983 |isbn=978-0-226-10952-7}}
* {{cite book |last1=Cox |first1=J. P. |last2=Giuli |first2=R. T. |title=Principles of Stellar Structure |publisher=Gordon and Breach |location=[[New York City]] |year=1968}}
* {{cite journal |last1=Fowler |first1=William A. |author-link1=William A. Fowler |last2=Caughlan |first2=Georgeanne R. |author-link2=Georgeanne R. Caughlan |last3=Zimmerman |first3=Barbara A. |title=Thermonuclear Reaction Rates, I |journal=Annual Review of Astronomy and Astrophysics |volume=5 |page=525 |year=1967 |doi=10.1146/annurev.aa.05.090167.002521 |bibcode=1967ARA&A...5..525F}}
* {{cite journal |last1=Fowler |first1=William A. |last2=Caughlan |first2=Georgeanne R. |last3=Zimmerman |first3=Barbara A. |title=Thermonuclear Reaction Rates, II |journal=Annual Review of Astronomy and Astrophysics |volume=13 |page=69 |year=1975 |doi=10.1146/annurev.aa.13.090175.000441 |bibcode=1975ARA&A..13...69F}}
* {{cite book |last1=Hansen |first1=Carl J. |last2=Kawaler |first2=Steven D. |last3=Trimble |first3=Virginia |title=Stellar Interiors: Physical Principles, Structure, and Evolution, Second Edition |publisher=Springer-Verlag |location=New York |year=2004}}
* {{cite journal |last1=Harris |first1=Michael J. |last2=Fowler |first2=William A. |last3=Caughlan |first3=Georgeanne R. |last4=Zimmerman |first4=Barbara A. |title=Thermonuclear Reaction Rates, III |journal=Annual Review of Astronomy and Astrophysics |volume=21 |page=165 |year=1983 |doi=10.1146/annurev.aa.21.090183.001121 |bibcode=1983ARA&A..21..165H}}
* {{cite journal |last1=Iben |first1=Icko Jr |title=Stellar Evolution Within and Off the Main Sequence |journal=Annual Review of Astronomy and Astrophysics |volume=5 |page=571 |year=1967 |doi=10.1146/annurev.aa.05.090167.003035 |bibcode=1967ARA&A...5..571I}}
* {{cite journal |last1=Iglesias |first1=Carlos A. |last2=Rogers |first2=Forrest J. |title=Updated Opal Opacities |journal=The Astrophysical Journal |volume=464 |page=943 |year=1996 |doi=10.1086/177381 |bibcode=1996ApJ...464..943I}}
* {{cite book |last1=Kippenhahn |first1=Rudolf |last2=Weigert |first2=Alfred |title=Stellar Structure and Evolution |publisher=[[Springer-Verlag]] |location=Berlin |year=1990}}
* {{cite journal |last1=Liebert |first1=James |last2=Probst |first2=Ronald G. |title=Very Low Mass Stars |journal=Annual Review of Astronomy and Astrophysics |volume=25 |page=437 |year=1987 |doi=10.1146/annurev.aa.25.090187.002353 |bibcode=1987ARA&A..25..473L}}
* {{cite book |last1=Novotny |first1=Eva |title=Introduction to Stellar Atmospheres and Interior |publisher=[[Oxford University Press]] |location=New York City |year=1973}}
* {{cite book |last1=Padmanabhan |first1=T. |title=Theoretical Astrophysics |publisher=Cambridge University Press |location=Cambridge |year=2002}}
* {{cite book |last=Prialnik |first=Dina |title=An Introduction to the Theory of Stellar Structure and Evolution |publisher=Cambridge University Press |location=Cambridge |year=2000|bibcode=2000itss.book.....P }}
* {{cite book |last=Shore |first=Steven N. |title=The Tapestry of Modern Astrophysics |publisher=John Wiley and Sons |location=Hoboken |year=2003}}

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