TheInfoList

Stellar nucleosynthesis is the creation (nucleosynthesis) of
chemical element In chemistry, an element is a pure Chemical substance, substance consisting only of atoms that all have the same numbers of protons in their atomic nucleus, nuclei. Unlike chemical compounds, chemical elements cannot be broken down into simp ...
s by
nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or m ...

reactions within stars. Stellar nucleosynthesis has occurred since the original creation of
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the che ...

,
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining") ...

and
lithium Lithium (from el, λίθος, lithos, lit=stone) is a with the Li and  3. It is a soft, silvery-white . Under , it is the least dense metal and the least dense solid element. Like all alkali metals, lithium is highly and flammable, a ...

during the
Big Bang The Big Bang is the prevailing of the from the through its subsequent large-scale evolution. The model describes how the from an initial state of high and , and offers a comprehensive explanation for a broad range of observed phenomen ...

. As a predictive theory, it yields accurate estimates of the observed abundances of the elements. It explains why the observed abundances of elements change over time and why some elements and their isotopes are much more abundant than others. The theory was initially proposed by
Fred Hoyle Sir Fred Hoyle FRS FRS may also refer to: Government and politics * Facility Registry System, a centrally managed Environmental Protection Agency database that identifies places of environmental interest in the United States * Family Resource ...
in 1946, who later refined it in 1954. Further advances were made, especially to nucleosynthesis by
neutron capture Neutron capture is a nuclear reaction In nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion a ...
of the elements heavier than iron, by
Margaret Margaret is a female first name, derived via French (''Marguerite (given name), Marguerite'') and Latin (''Margarita'') from grc, μαργαρίτης (''margarítēs'') meaning "pearl". The Greek is borrowed from Indo-Iranian languages, Persia ...
and
Geoffrey Burbidge Geoffrey Ronald Burbidge FRS (24 September 1925 – 26 January 2010) was an English astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studi ...
,
William Alfred Fowler William Alfred Fowler (August 9, 1911 – March 14, 1995) was an American nuclear physicist, later astrophysicist, who, with Subrahmanyan Chandrasekhar won the 1983 Nobel Prize in Physics ) , image = Nobel Prize.png , alt ...
and Hoyle in their famous 1957 B2FH paper, which became one of the most heavily cited papers in astrophysics history. because of changes in their composition (the abundance of their constituent elements) over their lifespans, first by burning hydrogen (
main sequence In astronomy, the main sequence is a continuous and distinctive band of stars that appears on plots of stellar Color index, color versus Absolute magnitude, brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams afte ...
star), then
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...
(
horizontal branch The horizontal branch (HB) is a stage of stellar evolution that immediately follows the red giant branch in stars whose masses are similar to the Sun's. Horizontal-branch stars are powered by helium fusion in the core (via the triple-alpha process) ...
star), and progressively burning higher elements. However, this does not by itself significantly alter the abundances of elements in the universe as the elements are contained within the star. Later in its life, a low-mass star will slowly eject its atmosphere via
stellar wind A stellar wind is a flow of gas ejected from the upper atmosphere of a star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown d ...
, forming a
planetary nebula A planetary nebula (PN, plural PNe), is a type of emission nebula An emission nebula is a nebula " from the Eagle Nebula. Evidence from the Spitzer Telescope suggests that the pillars may already have been destroyed by a supernova explos ...

, while a higher–mass star will eject mass via a sudden catastrophic event called a
supernova A supernova ( plural: supernovae or supernovas, abbreviations: SN and SNe) is a powerful and luminous stellar explosion. This transient astronomical event occurs during the last stellar evolution, evolutionary stages of a massive star or when a ...

. The term
supernova nucleosynthesis Supernova nucleosynthesis is the nucleosynthesis Nucleosynthesis is the process that creates new atomic nuclei The atomic nucleus is the small, dense region consisting of proton A proton is a subatomic particle, symbol or , with a positive ...
is used to describe the creation of elements during the explosion of a massive star or white dwarf. The advanced sequence of burning fuels is driven by
gravitational collapse Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the centre of mass#Centre of gravity, centre of gravity. Gravitational collapse is a fundamenta ...
and its associated heating, resulting in the subsequent burning of
carbon Carbon (from la, carbo "coal") is a with the C and 6. It is lic and —making four s available to form s. It belongs to group 14 of the periodic table. Carbon makes up only about 0.025 percent of Earth's crust. Three occur naturally, ...
,
oxygen Oxygen is the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same ...
and
silicon Silicon is a chemical element with the Symbol (chemistry), symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic lustre, and is a Tetravalence, tetravalent metalloid and semiconductor. It is a member ...
. However, most of the nucleosynthesis in the mass range (from silicon to nickel) is actually caused by the upper layers of the star collapsing onto the core, creating a compressional
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound The speed of sound is the distance travelled per unit of time by a sound wave as it propagat ...
rebounding outward. The shock front briefly raises temperatures by roughly 50%, thereby causing furious burning for about a second. This final burning in massive stars, called ''explosive nucleosynthesis'' or
supernova nucleosynthesis Supernova nucleosynthesis is the nucleosynthesis Nucleosynthesis is the process that creates new atomic nuclei The atomic nucleus is the small, dense region consisting of proton A proton is a subatomic particle, symbol or , with a positive ...
, is the final epoch of stellar nucleosynthesis. A stimulus to the development of the theory of nucleosynthesis was the discovery of variations in the . The need for a physical description was already inspired by the relative abundances of the chemical elements in the solar system. Those abundances, when plotted on a graph as a function of the atomic number of the element, have a jagged sawtooth shape that varies by factors of tens of millions (see history of nucleosynthesis theory). This suggested a natural process that is not random. A second stimulus to understanding the processes of stellar nucleosynthesis occurred during the 20th century, when it was realized that the
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

released from nuclear fusion reactions accounted for the longevity of the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

as a source of heat and light.

# History

In 1920,
Arthur Eddington Sir Arthur Stanley Eddington (28 December 1882 – 22 November 1944) was an English astronomer, physicist, and mathematician. He was also a Philosophy of science, philosopher of science and a populariser of science. The Eddington limit, the nat ...

, on the basis of the precise measurements of atomic masses by and a preliminary suggestion by
Jean Perrin Jean Baptiste Perrin (30 September 1870 – 17 April 1942) was a French physicist who, in his studies of the Brownian motion of minute particles suspended in liquids, verified Albert Einstein’s explanation of this phenomenon and thereby confirme ...
, proposed that stars obtained their energy from
nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or m ...

of
hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the che ...

to form
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining") ...

and raised the possibility that the heavier elements are produced in stars. This was a preliminary step toward the idea of stellar nucleosynthesis. In 1928
George Gamow George Gamow (March 4, 1904 – August 19, 1968), born Georgiy Antonovich Gamov (russian: Георгий Антонович Гамов), was a Ukrainian-American theoretical physicist and cosmologist. He was an early advocate and developer of ...
derived what is now called the Gamow factor, a quantum-mechanical formula yielding the probability for two contiguous nuclei to overcome the electrostatic
Coulomb barrier The Coulomb barrier, named after Coulomb's law ''F'' between two point charges ''q''1 and ''q''2 is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them. Like cha ...
between them and approach each other closely enough to undergo nuclear reaction due to the
strong nuclear force In nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are also studied. Nuclear physics should not be confused with atomic physics, which ...
which is effective only at very short distances. In the following decade the Gamow factor was used by Atkinson and Houtermans and later by
Edward Teller Edward Teller ( hu, Teller Ede; January 15, 1908 – September 9, 2003) was a Hungarian-American Hungarian Americans (Hungarian language, Hungarian: ''amerikai magyarok'') are United States, Americans of Hungarian people, Hungarian ...
and Gamow himself to derive the rate at which nuclear reactions would occur at the high temperatures believed to exist in stellar interiors. In 1939, in a Nobel lecture entitled "Energy Production in Stars",
Hans Bethe Hans Albrecht Bethe (; July 2, 1906 – March 6, 2005) was a German-American German Americans (german: Deutschamerikaner, ) are Americans who have full or partial Germans, German ancestry. With an estimated size of approximately 43 million ...

analyzed the different possibilities for reactions by which hydrogen is fused into helium. He defined two processes that he believed to be the sources of energy in stars. The first one, the , is the dominant energy source in stars with masses up to about the mass of the Sun. The second process, the , which was also considered by Carl Friedrich von Weizsäcker in 1938, is more important in more massive main-sequence stars. These works concerned the energy generation capable of keeping stars hot. A clear physical description of the proton–proton chain and of the CNO cycle appears in a 1968 textbook. Bethe's two papers did not address the creation of heavier nuclei, however. That theory was begun by Fred Hoyle in 1946 with his argument that a collection of very hot nuclei would assemble thermodynamically into
iron Iron () is a chemical element In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behav ...

. Hoyle followed that in 1954 with a paper describing how advanced fusion stages within massive stars would synthesize the elements from carbon to iron in mass. Hoyle's theory was extended to other processes, beginning with the publication of the 1957 review paper "Synthesis of the Elements in Stars" by Burbidge, Burbidge, Fowler and Hoyle, more commonly referred to as the B2FH paper. This review paper collected and refined earlier research into a heavily cited picture that gave promise of accounting for the observed relative abundances of the elements; but it did not itself enlarge Hoyle's 1954 picture for the origin of primary nuclei as much as many assumed, except in the understanding of nucleosynthesis of those elements heavier than iron by neutron capture. Significant improvements were made by Alastair G. W. Cameron and by Donald D. Clayton. In 1957 Cameron presented his own independent approach to nucleosynthesis, informed by Hoyle's example, and introduced computers into time-dependent calculations of evolution of nuclear systems. Clayton calculated the first time-dependent models of the ''s''-process in 1961 and of the in 1965, as well as of the burning of silicon into the abundant alpha-particle nuclei and iron-group elements in 1968, and discovered radiogenic chronologies for determining the age of the elements.

# Key reactions

The most important reactions in stellar nucleosynthesis: *
Hydrogen Hydrogen is the chemical element with the Symbol (chemistry), symbol H and atomic number 1. Hydrogen is the lightest element. At standard temperature and pressure, standard conditions hydrogen is a gas of diatomic molecules having the che ...

fusion: **
Deuterium fusion Deuterium fusion, also called deuterium burning, is a nuclear fusion Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles (neutrons ...
** The
proton–proton chain lang=en, 250px, Scheme of the proton–proton branch I reaction The proton–proton chain, also commonly referred to as the p-p chain, is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates ...
** The *
Helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

fusion: ** The
triple-alpha process The triple-alpha process is a set of nuclear fusion reactions by which three helium-4 nuclei (alpha particles) are transformed into carbon. Triple-alpha process in stars Helium accumulates in the stellar core, cores of stars as a result of the ...
** The
alpha process The alpha process, also known as the alpha ladder, is one of two classes of nuclear fusion Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatom ...
* Fusion of heavier elements: ** Lithium burning: a process found most commonly in
brown dwarf A brown dwarf is a type of substellar object that has a mass between the most massive gas giant planets and the least massive stars, approximately 13 to 80 Jupiter mass, times that of Jupiter (). Unlike main sequence stars, brown dwarfs do not a ...

s **
Carbon-burning process The carbon-burning process or carbon fusion is a set of nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is the process by which tw ...
**
Neon-burning process The neon-burning process (nuclear decay) is a set of nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is the process by which two o ...
**
Oxygen-burning processThe oxygen-burning process is a set of nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is the process by which two or more atoms are ...
**
Silicon-burning process In , silicon burning is a very brief sequence of reactions that occur in massive s with a minimum of about 8–11 solar masses. burning is the final stage of fusion for massive stars that have run out of the fuels that power them for their long l ...
* Production of elements heavier than
iron Iron () is a chemical element In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behav ...

: **
Neutron The neutron is a subatomic particle, symbol or , which has a neutral (not positive or negative) charge, and a mass slightly greater than that of a proton. Protons and neutrons constitute the nuclei of atoms. Since protons and neutrons behav ...

capture: *** The
r-process In nuclear astrophysics Nuclear astrophysics is an interdisciplinary part of both nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions. Other forms of nuclear matter are als ...

*** The
s-process The slow neutron-capture process, or ''s''-process, is a series of reactions in nuclear astrophysics Nuclear astrophysics is an interdisciplinary part of both nuclear physics Nuclear physics is the field of physics that studies atomic nuclei an ...
** Proton capture: *** The
rp-process Image:Rapid Proton Capture.svg, 280px, Nucleosynthesis of proton-rich nuclei by rapid proton capture The rp-process (rapid proton capture process) consists of consecutive proton captures onto Seed nucleus, seed nuclei to produce heavier elements. ...
*** The
p-process The term p-process (''p'' for proton A proton is a subatomic particle, symbol or , with a positive electric charge of +1''e'' elementary charge and a mass slightly less than that of a neutron. Protons and neutrons, each with masses of approxima ...
**
Photodisintegration Photodisintegration (also called phototransmutation, or a photonuclear reaction) is a nuclear process in which an atomic nucleus The atomic nucleus is the small, dense region consisting of s and s at the center of an , discovered in 1911 by ...

## Hydrogen fusion

Hydrogen fusion Nuclear fusion is a reaction Reaction may refer to a process or to a response to an action, event, or exposure: Physics and chemistry *Chemical reaction A chemical reaction is a process that leads to the IUPAC nomenclature for organic tr ...
(nuclear fusion of four protons to form a
helium-4 Helium-4 () is a stable isotope of the element helium. It is by far the more abundant of the two naturally occurring isotopes of helium, making up about 99.99986% of the helium on Earth. Its nucleus is identical to an alpha particle, and consists ...

nucleus) is the dominant process that generates energy in the cores of
main-sequence In astronomy, the main sequence is a continuous and distinctive band of stars that appears on plots of stellar Color index, color versus Absolute magnitude, brightness. These color-magnitude plots are known as Hertzsprung–Russell diagrams aft ...
stars. It is also called "hydrogen burning", which should not be confused with the
chemical A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which ...

combustion of hydrogen in an
oxidizing Redox (reduction–oxidation, pronunciation: or ) is a type of chemical reaction A chemical reaction is a process that leads to the chemical transformation of one set of chemical substance A chemical substance is a form of matter ...
atmosphere. There are two predominant processes by which stellar hydrogen fusion occurs:
proton–proton chain lang=en, 250px, Scheme of the proton–proton branch I reaction The proton–proton chain, also commonly referred to as the p-p chain, is one of two known sets of nuclear fusion reactions by which stars convert hydrogen to helium. It dominates ...
and the carbon–nitrogen–oxygen (CNO) cycle. Ninety percent of all stars, with the exception of
white dwarfs A white dwarf, also called a degenerate dwarf, is a stellar core remnant composed mostly of electron-degenerate matter Degenerate matter is a highly dense state of fermionic matter in which the Pauli exclusion principle exerts significant ...
, are fusing hydrogen by these two processes. In the cores of lower-mass main-sequence stars such as the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

, the dominant energy production process is the . This creates a helium-4 nucleus through a sequence of reactions that begin with the fusion of two protons to form a
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes The term stable isotope has a meaning similar to stable nuclide, but is preferably used when speaking of nuclides of a specific elemen ...

nucleus (one proton plus one neutron) along with an ejected positron and neutrino. In each complete fusion cycle, the proton–proton chain reaction releases about 26.2 MeV. The proton–proton chain reaction cycle is relatively insensitive to temperature; a 10% rise of temperature would increase energy production by this method by 46%, hence, this hydrogen fusion process can occur in up to a third of the star's radius and occupy half the star's mass. For stars above 35% of the Sun's mass, the
energy flux Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context: # Total rate of energy transfer (not per unit area); SI units: W = J⋅s−1. # Specific quantity, Specific rat ...
toward the surface is sufficiently low and energy transfer from the core region remains by
radiative heat transfer Thermal radiation is electromagnetic radiation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space an ...
, rather than by convective heat transfer. As a result, there is little mixing of fresh hydrogen into the core or fusion products outward. In higher-mass stars, the dominant energy production process is the
CNO cycle The CNO cycle (for carbon–nitrogen–oxygen; sometimes called Bethe–Weizsäcker cycle after Hans Albrecht Bethe and Carl Friedrich von Weizsäcker) is one of the two known sets of nuclear fusion, fusion nuclear reaction, reactions by which s ...

, which is a
catalytic cycleIn chemistry Chemistry is the scientific discipline involved with Chemical element, elements and chemical compound, compounds composed of atoms, molecules and ions: their composition, structure, properties, behavior and the changes they undergo ...

that uses nuclei of carbon, nitrogen and oxygen as intermediaries and in the end produces a helium nucleus as with the proton–proton chain. During a complete CNO cycle, 25.0 MeV of energy is released. The difference in energy production of this cycle, compared to the proton–proton chain reaction, is accounted for by the energy lost through
neutrino A neutrino ( or ) (denoted by the Greek letter ) is a fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. T ...

emission. The CNO cycle is very temperature sensitive, a 10% rise of temperature would produce a 350% rise in energy production. About 90% of the CNO cycle energy generation occurs within the inner 15% of the star's mass, hence it is strongly concentrated at the core. This results in such an intense outward energy flux that
convective Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity Homogeneity and heterogeneity are concepts often used in the sciences and statistics Statistics ...
energy transfer becomes more important than does
radiative transfer Radiative transfer is the physical phenomenon of energy transfer in the form of electromagnetic radiation. The propagation of radiation through a medium is affected by absorption, emission, and scattering Scattering is a term used in physics to d ...
. As a result, the core region becomes a
convection zone A convection zone, convective zone or convective region of a star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, near ...
, which stirs the hydrogen fusion region and keeps it well mixed with the surrounding proton-rich region. This core convection occurs in stars where the CNO cycle contributes more than 20% of the total energy. As the star ages and the core temperature increases, the region occupied by the convection zone slowly shrinks from 20% of the mass down to the inner 8% of the mass. Our Sun produces on the order of 1% of its energy from the CNO cycle. The type of hydrogen fusion process that dominates in a star is determined by the temperature dependency differences between the two reactions. The proton–proton chain reaction starts at temperatures about , making it the dominant fusion mechanism in smaller stars. A self-maintaining CNO chain requires a higher temperature of approximately , but thereafter it increases more rapidly in efficiency as the temperature rises, than does the proton–proton reaction. Above approximately , the CNO cycle becomes the dominant source of energy. This temperature is achieved in the cores of main-sequence stars with at least 1.3 times the mass of the
Sun The Sun is the star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many othe ...

. The Sun itself has a core temperature of about . As a main-sequence star ages, the core temperature will rise, resulting in a steadily increasing contribution from its CNO cycle.

## Helium fusion

Main sequence stars accumulate helium in their cores as a result of hydrogen fusion, but the core does not become hot enough to initiate helium fusion. Helium fusion first begins when a star leaves the red giant branch after accumulating sufficient helium in its core to ignite it. In stars around the mass of the sun, this begins at the tip of the red giant branch with a
helium flash A helium flash is a very brief thermal runaway nuclear fusion of large quantities of helium into carbon through the triple-alpha process in the core of low mass stars (between 0.8 solar masses () and 2.0 ) during their red giant phase (the Sun is p ...

from a
degenerate Degeneracy may refer to: Science * Codon degeneracy * Degeneracy (biology), the ability of elements that are structurally different to perform the same function or yield the same output * Degeneration (medical) ** Degenerative disease, a diseas ...
helium core, and the star moves to the
horizontal branch The horizontal branch (HB) is a stage of stellar evolution that immediately follows the red giant branch in stars whose masses are similar to the Sun's. Horizontal-branch stars are powered by helium fusion in the core (via the triple-alpha process) ...
where it burns helium in its core. More massive stars ignite helium in their core without a flash and execute a
blue loop In the field of stellar evolution, a blue loop is a stage in the life of an evolved star where it changes from a cool star to a hotter one before cooling again. The name derives from the shape of the evolutionary track on a Hertzsprung–Russell ...
before reaching the
asymptotic giant branch The asymptotic giant branch (AGB) is a region of the Hertzsprung–Russell diagram populated by evolved cool luminous stars. This is a period of stellar evolution undertaken by all low- to intermediate-mass stars (0.6–10 solar masses) late in the ...
. Such a star initially moves away from the AGB toward bluer colours, then loops back again to what is called the
Hayashi track #REDIRECT Hayashi track The Hayashi track is a luminosity–temperature relationship obeyed by infant stars of less than in the pre-main-sequence phase (PMS phase) of stellar evolution. It is named after Japanese astrophysicist Chushiro Hayashi. ...
. An important consequence of blue loops is that they give rise to classical
Cepheid variable A Cepheid variable () is a type of star A star is an astronomical object consisting of a luminous spheroid of plasma (physics), plasma held together by its own gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the ...
s, of central importance in determining distances in the
Milky Way The Milky Way is the galaxy A galaxy is a gravitation Gravity (), or gravitation, is a natural phenomenon by which all things with mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and con ...

and to nearby galaxies. Despite the name, stars on a blue loop from the red giant branch are typically not blue in colour but are rather yellow giants, possibly Cepheid variables. They fuse helium until the core is largely
carbon Carbon (from la, carbo "coal") is a with the C and 6. It is lic and —making four s available to form s. It belongs to group 14 of the periodic table. Carbon makes up only about 0.025 percent of Earth's crust. Three occur naturally, ...

and
oxygen Oxygen is the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same ...

. The most massive stars become supergiants when they leave the main sequence and quickly start helium fusion as they become
red supergiant Red is the color at the long wavelength end of the visible spectrum of light, next to Orange (colour), orange and opposite Violet (color), violet. It has a dominant wavelength of approximately 625–740 nanometres. It is a primary color in the ...
s. After the helium is exhausted in the core of a star, it will continue in a shell around the carbon-oxygen core. In all cases, helium is fused to carbon via the triple-alpha process, i.e., three helium nuclei are transformed into carbon via . This can then form oxygen, neon, and heavier elements via the alpha process. In this way, the alpha process preferentially produces elements with even numbers of protons by the capture of helium nuclei. Elements with odd numbers of protons are formed by other fusion pathways.

# Reaction rate

The reaction rate density between species ''A'' and ''B'', having number densities ''n''''A'',''B'' is given by: ::$r = n_A \, n_B \, k$ where k is the
reaction rate constant In chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with thermodynamics, which deals with the dir ...
of each single elementary binary reaction composing the
nuclear fusion 400 px, The nuclear binding energy curve. The formation of nuclei with masses up to iron-56 releases energy, as illustrated above. Nuclear fusion is a nuclear reaction, reaction in which two or more atomic nuclei are combined to form one or m ...

process: ::$k = \langle \sigma\left(v\right)\,v \rangle$ here, σ(''v'') is the cross-section at relative velocity ''v'', and averaging is performed over all velocities. Semi-classically, the cross section is proportional to $\pi\,\lambda^2$, where $\lambda = h/p$ is the
de Broglie wavelength Matter waves are a central part of the theory of quantum mechanics Quantum mechanics is a fundamental Scientific theory, theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic ...
. Thus semi-classically the cross section is proportional to $\frac$. However, since the reaction involves
quantum tunneling In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
, there is an exponential damping at low energies that depends on Gamow factor ''E''G, giving an
Arrhenius equation In physical chemistry Physical chemistry is the study of macroscopic scale, macroscopic and particulate phenomena in chemistry, chemical systems in terms of the principles, practices, and concepts of physics such as Motion (physics), motion, ene ...
: ::$\sigma\left(E\right) = \frac e^$ where ''S''(''E'') depends on the details of the nuclear interaction, and has the dimension of an energy multiplied for a cross section. One then integrates over all energies to get the total reaction rate, using the
Maxwell–Boltzmann distribution In physics (in particular in statistical mechanics), the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann. It was first defined and used for describing particle speeds i ...
and the relation : ::$\frac = n_A n_B \int_0^\frac \, e^ 2\sqrt e^ \,\sqrtdE$ where $m_\text = \frac$ is the
reduced massIn physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spac ...

. Since this integration has an exponential damping at high energies of the form $\sim e^$ and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak, at ''E''0, where: ::$\frac \left\left( -\sqrt - \frac\right\right) \, = \, 0$ Thus: ::$E_0 = \left\left(\frackT \sqrt\right\right)^\frac$ The exponent can then be approximated around ''E''0 as: ::$e^ \approx e^ \exp\left\left(-\frac\right\right)$ And the reaction rate is approximated as: ::$\frac \approx n_A \, n_B \, \frac\, \sqrt \frac e^$ Values of ''S''(''E''0) are typically , but are damped by a huge factor when involving a
beta decay In , beta decay (''β''-decay) is a type of in which a (fast energetic or ) is emitted from an , transforming the original to an of that nuclide. For example, beta decay of a transforms it into a by the emission of an electron accompanie ...

, due to the relation between the intermediate bound state (e.g. )
half-life Half-life (symbol ''t''1⁄2) is the time required for a quantity to reduce to half of its initial value. The term is commonly used in nuclear physics Nuclear physics is the field of physics Physics is the natural science that studies ...
and the beta decay half-life, as in the . Note that typical core temperatures in
main-sequence star In astronomy Astronomy (from el, ἀστρονομία, literally meaning the science that studies the laws of the stars) is a natural science that studies astronomical object, celestial objects and celestial event, phenomena. It uses m ...
s give ''kT'' of the order of keV. Thus, the limiting reaction in the
CNO cycle The CNO cycle (for carbon–nitrogen–oxygen; sometimes called Bethe–Weizsäcker cycle after Hans Albrecht Bethe and Carl Friedrich von Weizsäcker) is one of the two known sets of nuclear fusion, fusion nuclear reaction, reactions by which s ...

, proton capture by , has ''S''(''E''0) ~ ''S''(0) = 3.5keV·b, while the limiting reaction in the , the creation of
deuterium Deuterium (or hydrogen-2, symbol or deuterium, also known as heavy hydrogen) is one of two stable isotopes The term stable isotope has a meaning similar to stable nuclide, but is preferably used when speaking of nuclides of a specific elemen ...

from two protons, has a much lower ''S''(''E''0) ~ ''S''(0) = 4×10−22keV·b. Incidentally, since the former reaction has a much higher Gamow factor, and due to the relative abundance of elements in typical stars, the two reaction rates are equal at a temperature value that is within the core temperature ranges of main-sequence stars.