Main sequence

A Hertzsprung-Russell diagram plots the actual brightness (or absolute magnitude) of a star against its color index (represented as B-V). The main sequence is visible as a prominent diagonal band that runs from the upper left to the lower right.

Main sequence is the name for a continuous and distinctive band of stars that appear on plots of stellar color versus brightness. These color-magnitude plots are known as Hertzsprung-Russell diagrams after their co-developers, Ejnar Hertzsprung and Henry Norris Russell. Stars on this band are known as main-sequence stars or dwarf stars.

After a star has formed, it generates energy at the hot, dense core region through the nuclear fusion of hydrogen atoms into helium. Hydrostatic equilibrium is maintained by a balance between the thermal radiation within the star and the pressure from the overlying mass. This balance occurs because the rate of energy generation at the core depends on the temperature and pressure. During this stage of the star's lifetime, it is located along the main sequence at a position determined primarily by its mass, but also based upon its chemical composition and other factors. In general, the more massive the star the shorter its lifespan on the main sequence. After the hydrogen fuel at the core has been consumed, the star evolves away from the main sequence on the HR diagram. The behavior of a star now depends on its mass, with stars below 0.23 solar masses becoming white dwarfs, while stars with up to 10 solar masses pass through a red giant stage.[1] More massive stars can explode as a Type II supernova.[2]

The main sequence is sometimes divided into upper and lower parts, based on the processes that stars use to generate energy. Stars below about 1.5 times the mass of the Sun (or 1.5 solar masses) 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 can instead use atoms of carbon, nitrogen and oxygen as intermediaries in the production of helium from hydrogen atoms.

Energy generated at the core is steadily transported toward the surface until it is radiated away at the photosphere. The energy is carried by either radiation and convection, with the latter occurring in regions with steeper temperature gradients, higher opacity or both. Main sequence stars with more than 10 solar masses undergo convection in the core region, which acts to stir up the helium ashes and maintain the proportion of fuel needed for fusion to occur. When core convection does not occur, a helium-rich core develops surrounded by an outer layer of hydrogen. Main sequence stars below 0.4 solar masses undergo convection throughout their mass.

Contents

History

In the early part of the twentieth century, information about the types and distances of stars became more readily available. The spectra of stars were shown to have distinctive features, which allowed them to be categorized. Annie Jump Cannon and Edward C. Pickering at Harvard College Observatory developed a method of categorization that became known as the Harvard classification scheme, published in the Harvard Annals in 1901.[3]

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 clusters; large groupings of stars that are co-located at approximately the same distance. He published the first plots of color versus luminosity for these stars. These plots showed a prominent and continuous sequence of stars, which he named the main sequence.[4]

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 parallaxes 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.[5]

Of the red stars observed by Hertzsprung, the dwarf stars also followed the spectra-luminosity relationship discovered by Russell. However, the giant stars are much brighter than dwarfs and so do not follow the same relationship. Russell proposed that the "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.[5]

In 1933, Bengt Strömgren introduced the term Hertzsprung-Russell diagram to denote a luminosity-spectral class diagram.[6] This name reflected the parallel development of this technique by both Hertzsprung and Russell earlier in the century.[4]

As evolutionary models of stars were developed during the 1930s, it was shown that, for stars of a uniform chemical composition, a relationship exists between a star's mass and its luminosity and radius. That is, for a given mass and composition is known, there is a unique solution determining the star's radius and luminosity. This became known as the Vogt-Russell theorem; named after Heinrich Vogt and Henry Norris Russell. By this theorem, once a star's chemical composition and its position on the main sequence is known, so too is the star's mass and radius. (However, it was subsequently discovered that the theorem breaks down somewhat for stars of non-uniform composition.)[7]

A refined scheme for stellar classification was published in 1943 by W. W. Morgan and P. C. Keenan.[8] The MK classification assigned each star a spectral type—based on the Harvard classification—and a luminosity class. For historical reasons, the spectral types 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.[9]

Formation

Main article: Star formation

When a protostar is formed from the 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.[10] During the initial collapse, this pre-main sequence star generates energy through gravitational contraction. Upon reaching a suitable density, energy generation is begun at the core using an exothermic nuclear fusion process that converts hydrogen into helium.[9]

HR-diag-no-text-2.svg

Hertzsprung-Russell Diagram
Spectral Type
Brown dwarfs
White dwarfs
Red dwarfs
Subdwarfs
Main sequence
("dwarfs")
Subgiants
Giants
Bright Giants
Supergiants
Hypergiants
absolute
magni-
tude
(MV)

Once nuclear fusion of hydrogen becomes the dominant energy production process and the excess energy gained from gravitational contraction has been lost,[11] 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.[12] 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.[13]

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.[9]

Properties

Main sequence stars have been extensively studied through stellar models, allowing their formation and evolutionary history to be relatively well understood. The position of the star on the main sequence provides information about its physical properties.

The majority of stars on a typical HR diagram lie along the main sequence curve. This line is so pronounced because both the spectral type and the luminosity depend only on a star's mass, at least to 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" life doing.[14] These main-sequence (and therefore "normal") stars are called dwarf stars. This is not because they are unusually small, but instead comes from their smaller radii and lower luminosity as compared to the other main category of stars, the giant stars.[15] White dwarfs are a different kind of star that are much smaller than main sequence stars—being roughly the size of the Earth. These represent the final evolutionary stage of many main sequence stars.[16]

The temperature of a star determines its spectral type, because the energy modifies the physical properties of the plasma in the photosphere. Both the temperature and composition determines the star's energy emission at different wavelengths. A key indicator of this energy distribution is given by the color index, B − V, which measures the star's magnitude in blue (B) and green-yellow (V) light by means of filters.[note 1] Thus, this difference in magnitude provides a measure of a star's temperature.

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 Teff by the Stefan–Boltzmann law:

\begin{smallmatrix}L = 4\pi \sigma R^2 T_{eff}^4\end{smallmatrix}

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.[17]

Stellar 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 V. The actual values for a star may vary by as much as 20–30% from the values listed below.[18]

Table of main sequence stellar parameters[19]
Stellar
Class		 Radius Mass Luminosity Temperature Examples[20]
R/R☉ M/M☉ L/L☉ K
O5 18 40 500,000 38,000 Zeta Puppis
B0 7.4 18 20,000 30,000 Phi1 Orionis
B5 3.8 6.5 800 16,400 Pi Andromedae A
A0 2.5 3.2 80 10,800 Alpha Coronae Borealis A
A5 1.7 2.1 20 8,620 Beta Pictoris
F0 1.4 1.7 6 7,240 Gamma Virginis
F5 1.2 1.29 2.5 6,540 Eta Arietis
G0 1.05 1.10 1.26 6,000 Beta Comae Berenices
G2  1.00[note 2]  1.00[note 2]  1.00[note 2] 5,920 Sun
G5 0.93 0.93 0.79 5,610 Alpha Mensae
K0 0.85 0.78 0.40 5,150 70 Ophiuchi A
K5 0.74 0.69 0.16 — 61 Cygni A
M0 0.63 0.47 0.063 3,920 Gliese 185[21]
M5 0.32 0.21 0.0079 3,120 EZ Aquarii A
M8 0.13 0.10 0.0008 — Van Biesbroeck's star[22]

Energy generation

See also: Stellar nucleosynthesis
This graph shows the logarithm of the relative energy output (ε) for the proton-proton (PP), CNO and triple-α fusion processes at different temperatures. 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.[23]

Astronomers divide the main sequence into upper and lower parts, based on the dominant type of fusion process at the core. Stars in the upper main sequence have sufficient mass to efficiently use the CNO cycle to fuse hydrogen into helium. This process uses atoms of carbon, nitrogen and oxygen as intermediaries in the 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.[24] At a stellar core temperature of 18 million kelvins, both fusion processes 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 masses, the upper main sequence consists of stars above this mass. Roughly speaking, stars of spectral class F or cooler belong to the lower main sequence, while class A stars or hotter are upper main sequence stars.[25] In the Sun, only 1.5% of the energy is generated by the CNO cycle,[26] while in stars with 1.8 solar masses or above generate almost their entire energy output through the CNO cycle.[27]

The apparent upper limit for a main sequence star is 120–200 solar masses.[28] 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.[29] The lower limit for sustained nuclear fusion is about 0.08 solar masses.[24]

Structure

Main article: Stellar structure
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.[30][23]

In massive stars (above 10 solar masses)[31] 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.[24] 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.[23]

Intermediate mass stars such as Sirius may transport energy primarily by radiation, with a small core convection region.[32] 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 masses) are convective throughout.[33] Thus the helium produced at the core is distributed across the star, producing a relatively uniform atmosphere and a proportionately longer main sequence lifespan.[23]

Luminosity-color variation

As non-fusing helium ash 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. To compensate, the core temperature and pressure slowly increase, which causes a net increase in the overall fusion rate (to support the greater density of the inner star). This produces a steady increase in the luminosity and radius of the star over time.[13] Thus, for example, the luminosity of the early Sun was only about 70% of its current value.[34] As a star ages this luminosity increase changes its position on the HR diagram. This effect results in a broadening of the main sequence band because stars are observed at random stages in their lifetime. That is, the main sequence band develops a thickness on the HR diagram; it is not simply a narrow line.[35]

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 stars 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. In addition to variations in chemical composition—both because of the initial abundances and the star's evolutionary status,[36] interaction with a close companion,[37] rapid rotation,[38] or a magnetic field can also change a main sequence star's position slightly on the HR diagram, to name just a few factors. As an example, there are stars that have a very low abundance of elements with higher atomic numbers than helium—known as metal-poor stars—that lie just below the main sequence. Known as subdwarfs, these stars are also fusing hydrogen in their core and so they mark the lower edge of the main sequence's fuzziness resulting from chemical composition.[39]

A nearly vertical region of the HR diagram, known as the instability strip, is occupied by pulsating variable stars. 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. Main sequence stars in this region experience only small changes in magnitude and so this variation is difficult to detect.[40]

Lifetime

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 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 first approximation, as the total energy produced divided by the star's luminosity.[41]

For a star with at least 0.5 solar masses, once 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. As the energy output of the helium fusion process per unit mass is only about a tenth the energy output of the hydrogen process, this stage will only last for about 10% of a star's total active lifetime. Thus, about 90% of the observed stars above 0.5 solar masses will be on the main sequence.[42] On average, main sequence stars are known to follow an empirical mass-luminosity relationship.[43] The luminosity (L) of the star is roughly proportional to the total mass (M) as the following power law:

\begin{smallmatrix}L\ \propto\ M^{3.5}\end{smallmatrix}

This relationship applies to main sequence stars in the range 0.1–50 solar masses.[44]

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,[45] for a total main sequence lifetime of roughly 1010 years. Hence:[46]

\begin{smallmatrix} \tau_{ms}\ \approx \ 10^{10} \text{years} \cdot \left[ \frac{M}{M_{\bigodot}} \right] \cdot \left[ \frac{L_{\bigodot}}{L} \right]\ =\ 10^{10} \text{years} \cdot \left[ \frac{M_{\bigodot}}{M} \right]^{2.5} \end{smallmatrix}

where M and L are the mass and luminosity of the star, respectively, \begin{smallmatrix}M_{\bigodot}\end{smallmatrix} is a solar mass, \begin{smallmatrix}L_{\bigodot}\end{smallmatrix} is the solar luminosity and \tau_{ms} is the star's estimated main sequence lifetime.

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.

This is a counter-intuitive result, as more massive stars have more fuel to burn and might be expected to last longer. Instead, 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.[47]

The exact mass-luminosity relationship depends on how efficiently energy can be transported from the core to the surface. A higher 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.[48] Note, however, that a sufficiently high opacity can result in energy transport via convection, which changes the conditions needed to remain in equilibrium.[49]

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.[50] For stars below 10 times the solar mass, the opacity becomes dependent on temperature, resulting in the luminosity varying approximately as the fourth power of the star's mass.[51] For very low mass stars, molecules in the atmosphere also contribute to the opacity. Below about 0.5 solar masses, 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.[52]

Evolutionary tracks

See also: Stellar evolution
This shows the Hertzsprung-Russell diagrams for two open clusters. NGC 188 (blue) is older, and shows a lower turn off from the main sequence than that seen in M67 (yellow).

Once a main sequence star consumes the hydrogen at its core, the loss of energy generation causes gravitational collapse to resume. For stars with less than 0.23 solar masses,[1] energy generation by nuclear fusion comes to a halt and it becomes a white dwarf. (The star now migrates to the lower left on the HR diagram.) For higher mass stars with up to 10 solar masses, the hydrogen surrounding the helium core reaches sufficient temperature and pressure to undergo fusion, forming a hydrogen-burning shell. In consequence of this change, the outer envelope of the star expands and decreases in temperature, turning it into a red giant. At this point the star is evolving off the main sequence and entering the giant branch. The path the star now follows across the HR diagram is called an evolutionary track.

The helium core of a red giant continues to collapse until it is entirely supported by electron degeneracy pressure—a quantum mechanical effect that restricts how closely matter can be compacted. For stars of more than about 0.5 solar masses,[53] the core can reach a temperature where it becomes hot enough to burn helium into carbon via the triple alpha process.[54][55] Stars with at least four solar masses can also fuse elements with higher atomic numbers. For stars with ten solar masses, this process can lead to a cataclysmic explosion as a Type II supernova.[2]

When a cluster of stars is formed at about the same time, the life span of these stars will depend on their individual masses. The most massive stars will leave the main sequence first, followed steadily in sequence by stars of ever lower masses. Thus the stars will evolve in order of their position on the main sequence, proceeding from the most massive at the left toward the right of the HR diagram. The current position where stars in this cluster are leaving the main sequence is known as the turn-off point. By knowing the main sequence lifespan of stars at this point, it becomes possible to estimate the age of the cluster.[56]

See also

Notes

  1. ↑ By measuring the difference between these values, this eliminates the need to correct the magnitudes for distance. However, see extinction.
  2. ↑ 2.0 2.1 2.2 The Sun is a typical type G2V star.

References

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  19. ↑ Zombeck, Martin V. (1990). Handbook of Space Astronomy and Astrophysics (2nd edition ed.). Cambridge University Press. ISBN 0-521-34787-4. http://ads.harvard.edu/books/hsaa/toc.html. Retrieved on 2007-12-06. 
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  39. ↑ Burgasser, Adam J.; Kirkpatrick, J. Davy; Lepine, Sebastien (July 5–9, 2004). "Spitzer Studies of Ultracool Subdwarfs: Metal-poor Late-type M, L and T Dwarfs". Proceedings of the 13th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun: p. 237, Hamburg, Germany: Dordrecht, D. Reidel Publishing Co.. Retrieved on 2007-12-06. 
  40. ↑ Green, S. F.; Jones, Mark Henry; Burnell, S. Jocelyn (2004). An Introduction to the Sun and Stars. Cambridge University Press. ISBN 0-521-54622-2. 
  41. ↑ Richmond, Michael W. (2004-11-10). "Stellar evolution on the main sequence". Rochester Institute of Technology. Retrieved on 2007-12-03.
  42. ↑ Arnett, David (1996). Supernovae and Nucleosynthesis: An Investigation of the History of Matter, from the Big Bang to the Present. Princeton University Press. ISBN 0-691-01147-8. —Hydrogen fusion produces 8×1018 erg/g while helium fusion produces 8×1017 erg/g.
  43. ↑ For a detailed historical reconstruction of the theoretical derivation of this relationship by Eddington in 1924, see: Lecchini, Stefano (2007). How Dwarfs Became Giants. The Discovery of the Mass-Luminosity Relation. Bern Studies in the History and Philosophy of Science. ISBN 3-9522882-6-8. http://www.amazon.de/Dwarfs-Giants-Discovery-Mass-Luminosity-Relation/dp/3952288268. 
  44. ↑ Rolfs, Claus E.; Rodney, William S. (1988). Cauldrons in the Cosmos: Nuclear Astrophysics. University of Chicago Press. pp. 46. ISBN 0226724573. 
  45. ↑ Sackmann, I.-Juliana; Boothroyd, Arnold I.; Kraemer, Kathleen E. (November 1993). "Our Sun. III. Present and Future". Astrophysical Journal 418: 457–468. doi:10.1086/173407. 
  46. ↑ Hansen, Carl J.; Kawaler, Steven D. (1994). Stellar Interiors: Physical Principles, Structure, and Evolution. Birkhäuser. pp. 28. ISBN 038794138X. 
  47. ↑ Laughlin, Gregory; Bodenheimer, Peter; Adams, Fred C. (1997). "The End of the Main Sequence". The Astrophysical Journal 482: 420–432. doi:10.1086/304125. 
  48. ↑ Imamura, James N. (1995-02-07). "Mass-Luminosity Relationship". University of Oregon. Retrieved on 2007-01-08.
  49. ↑ Clayton, Donald D. (1983). Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. ISBN 0-226-10953-4. 
  50. ↑ Prialnik, Dina (2000). An Introduction to the Theory of Stellar Structure and Evolution. Cambridge UniversityPress. ISBN 0-521-65937-X. 
  51. ↑ Rolfs, Claus E.; Rodney, William S. (1988). Cauldrons in the Cosmos: Nuclear Astrophysics. University of Chicago Press. ISBN 0-226-72457-3. 
  52. ↑ Kroupa, Pavel (2002). "The Initial Mass Function of Stars: Evidence for Uniformity in Variable Systems". Science 295 (5552): 82–91. doi:10.1126/science.1067524. PMID 11778039. http://www.sciencemag.org/cgi/content/full/295/5552/82. Retrieved on 2007-12-03. 
  53. ↑ Fynbo, Hans O. U. et al (2004). "Revised rates for the stellar triple-α process from measurement of 12C nuclear resonances". Nature 433: 136–139. doi:10.1038/nature03219. 
  54. ↑ Sitko, Michael L. (2000-03-24). "Stellar Structure and Evolution". University of Cincinnati. Retrieved on 2007-12-05.
  55. ↑ Staff (2006-10-12). "Post-Main Sequence Stars". Australia Telescope Outreach and Education. Retrieved on 2008-01-08.
  56. ↑ Krauss, Lawrence M.; Chaboyer, Brian (2003). "Age Estimates of Globular Clusters in the Milky Way: Constraints on Cosmology". Science 299 (5603): 65–69. doi:10.1126/science.1075631. PMID 12511641. 

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