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In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, i.e. the imbalance of matter ( baryons) and antimatter (antibaryons) in the observed universe.

One of the outstanding problems in modern physics is the predominance of matter over antimatter in the universe. The universe, as a whole, seems to have a nonzero positive baryon number density. Since it is assumed in cosmology that the particles we see were created using the same physics we measure today, it would normally be expected that the overall baryon number should be zero, as matter and antimatter should have been created in equal amounts. A number of theoretical mechanisms are proposed to account for this discrepancy, namely identifying conditions that favour symmetry breaking and the creation of normal matter (as opposed to antimatter). This imbalance has to be exceptionally small, on the order of 1 in every (≈) particles a small fraction of a second after the Big Bang. After most of the matter and antimatter was annihilated, what remained was all the baryonic matter in the current universe, along with a much greater number of bosons. Experiments reported in 2010 at Fermilab, however, seem to show that this imbalance is much greater than previously assumed. These experiments involved a series of particle collisions and found that the amount of generated matter was approximately 1% larger than the amount of generated antimatter. The reason for this discrepancy is not yet known.

Most grand unified theories explicitly break the baryon number symmetry, which would account for this discrepancy, typically invoking reactions mediated by very massive X bosons or massive Higgs bosons (H). The rate at which these events occur is governed largely by the mass of the intermediate X or H particles, so by assuming these reactions are responsible for the majority of the baryon number seen today, a maximum mass can be calculated above which the rate would be too slow to explain the presence of matter today. These estimates predict that a large volume of material will occasionally exhibit a spontaneous proton decay, which has not been observed. Therefore, the imbalance between matter and antimatter remains a mystery.

Baryogenesis theories are based on different descriptions of the interaction between fundamental particles. Two main theories are electroweak baryogenesis ( standard model), which would occur during the electroweak phase transition, and the GUT baryogenesis, which would occur during or shortly after the grand unification epoch. Quantum field theory and statistical physics are used to describe such possible mechanisms.

Baryogenesis is followed by primordial nucleosynthesis, when atomic nuclei began to form.

Unsolved problem in physics:

Why does the observable universe have more matter than antimatter?

The majority of ordinary matter in the universe is found in atomic nuclei, which are made of neutrons and protons. These nucleons are made up of smaller particles called quarks, and antimatter equivalents for each are predicted to exist by the Dirac equation in 1928. Since then, each kind of antiquark has been experimentally verified. Hypotheses investigating the first few instants of the universe predict a composition with an almost equal number of quarks and antiquarks. Once the universe expanded and cooled to a critical temperature of approximately , quarks combined into normal matter and antimatter and proceeded to annihilate up to the small initial asymmetry of about one part in five billion, leaving the matter around us. Free and separate individual quarks and antiquarks have never been observed in experiments-quarks and antiquarks are always found in groups of three ( baryons), or bound in quark-antiquark pairs ( mesons). Likewise, there is no experimental evidence that there are any significant concentrations of antimatter in the observable universe.

There are two main interpretations for this disparity: either the universe began with a small preference for matter (total baryonic number of the universe different from zero), or the universe was originally perfectly symmetric, but somehow a set of phenomena contributed to a small imbalance in favour of matter over time. The second point of view is preferred, although there is no clear experimental evidence indicating either of them to be the correct one.

In 1967, Andrei Sakharov proposed a set of three necessary conditions that a baryon-generating interaction must satisfy to produce matter and antimatter at different rates. These conditions were inspired by the recent discoveries of the cosmic background radiation and CP-violation in the neutral kaon system. The three necessary "Sakharov conditions" are:

Baryon number violation is a necessary condition to produce an excess of baryons over anti-baryons. But C-symmetry violation is also needed so that the interactions which produce more baryons than anti-baryons will not be counterbalanced by interactions which produce more anti-baryons than baryons. CP-symmetry violation is similarly required because otherwise equal numbers of left-handed baryons and right-handed anti-baryons would be produced, as well as equal numbers of left-handed anti-baryons and right-handed baryons. Finally, the interactions must be out of thermal equilibrium, since otherwise CPT symmetry would assure compensation between processes increasing and decreasing the baryon number.

Currently, there is no experimental evidence of particle interactions where the conservation of baryon number is broken perturbatively: this would appear to suggest that all observed particle reactions have equal baryon number before and after. Mathematically, the commutator of the baryon number quantum operator with the (perturbative) Standard Model hamiltonian is zero: . However, the Standard Model is known to violate the conservation of baryon number only non-perturbatively: a global U(1) anomaly.[14] To account for baryon violation in baryogenesis, such events (including proton decay) can occur in Grand Unification Theories (GUTs) and supersymmetric (SUSY) models via hypothetical massive bosons such as the X boson.

The second condition - violation of CP-symmetry - was discovered in 1964 (direct CP-violation, that is violation of CP-symmetry in a decay process, was discovered later, in 1999). Due to CPT symmetry, violation of CP-symmetry demands violation of time inversion symmetry, or T-symmetry.

In the out-of-equilibrium decay scenario, the last condition states that the rate of a reaction which generates baryon-asymmetry must be less than the rate of expansion of the universe. In this situation the particles and their corresponding antiparticles do not achieve thermal equilibrium due to rapid expansion decreasing the occurrence of pair-annihilation.

The Standard Model can incorporate baryogenesis, though the amount of net baryons (and leptons) thus created may not be sufficient to account for the present baryon asymmetry. There is a required one excess quark per billion quark-antiquark pairs in the early universe in order to provide all the observed matter in the universe. This insufficiency has not yet been explained, theoretically or otherwise.

Baryogenesis within the Standard Model requires the electroweak symmetry breaking to be a first-order cosmological phase transition, since otherwise sphalerons wipe off any baryon asymmetry that happened up to the phase transition. Beyond this, the remaining amount of baryon non-conserving interactions is negligible.

The phase transition domain wall breaks the P-symmetry spontaneously, allowing for CP-symmetry violating interactions to break C-symmetry on both its sides. Quarks tend to accumulate on the broken phase side of the domain wall, while anti-quarks tend to accumulate on its unbroken phase side. Due to CP-symmetry violating electroweak interactions, some amplitudes involving quarks are not equal to the corresponding amplitudes involving anti-quarks, but rather have opposite phase (see CKM matrix and Kaon); since time reversal takes an amplitude to its complex conjugate, CPT-symmetry is conserved in this entire process.

Though some of their amplitudes have opposite phases, both quarks and anti-quarks have positive energy, and hence acquire the same phase as they move in space-time. This phase also depends on their mass, which is identical but depends both on flavor and on the Higgs VEV which changes along the domain wall. Thus certain sums of amplitudes for quarks have different absolute values compared to those of anti-quarks. In all, quarks and anti-quarks may have different reflection and transmission probabilities through the domain wall, and it turns out that more quarks coming from the unbroken phase are transmitted compared to anti-quarks.

Thus there is a net baryonic flux through the domain wall. Due to sphaleron transitions, which are abundant in the unbroken phase, the net anti-baryonic content of the unbroken phase is wiped off as anti-baryons are transformed into leptons. However, sphalerons are rare enough in the broken phase as not to wipe off the excess of baryons there. In total, there is net creation of baryons (as well as leptons).

In this scenario, non-perturbative electroweak interactions (i.e. the sphaleron) are responsible for the B-violation, the perturbative electroweak Lagrangian is responsible for the CP-violation, and the domain wall is responsible for the lack of thermal equilibrium and the P-violation; together with the CP-violation it also creates a C-violation in each of its sides.

The central question to Baryogenesis is what causes the preference for matter over antimatter in the universe, as well as the magnitude of this asymmetry. An important quantifier is the asymmetry parameter, given by

where and refer to the number density of baryons and antibaryons respectively and is the number density of cosmic background radiation photons.

According to the Big Bang model, matter decoupled from the cosmic background radiation (CBR) at a temperature of roughly 3000 kelvin, corresponding to an average kinetic energy of 3000 K / () = 0.3 eV. After the decoupling, the total number of CBR photons remains constant. Therefore, due to space-time expansion, the photon density decreases. The photon density at equilibrium temperature T per cubic centimeter, is given by

,

with as the Boltzmann constant, ħ as the Planck constant divided by 2π and c as the speed of light in vacuum, and ζ(3) as Apéry's constant. At the current CBR photon temperature of 2.725 K, this corresponds to a photon density of around 411 CBR photons per cubic centimeter.

Therefore, the asymmetry parameter η, as defined above, is not the "best" parameter. Instead, the preferred asymmetry parameter uses the entropy density s,

because the entropy density of the universe remained reasonably constant throughout most of its evolution. The entropy density is

with p and ρ as the pressure and density from the energy density tensor , and as the effective number of degrees of freedom for "massless" particles at temperature T (in so far as holds),

,

A possible explanation for the cause of baryogenesis is the decay reaction of B-Mesogenesis. This phenomenon suggests that in the early universe, particles such as the B-meson decay into a visible Standard Model baryon as well as a dark antibaryon that is invisible to current observation techniques.[22] The process begins by assuming a massive, long-lived, scalar particle that exists in the early universe before Big Bang nucleosynthesis.[23] The exact behavior of is as yet unknown, but it is assumed to decay into b quarks and antiquarks in conditions outside of thermal equilibrium, thus satisfying one Sakharov condition. These b quarks form into B-mesons, which immediately hadronize into oscillating CP-violating states, thus satisfying another Sakharov condition.[24] These oscillating mesons then decay down into the baryon-dark antibaryon pair previously mentioned, , where is the parent B-meson, is the dark antibaryon, is the visible baryon, and is any extra light meson daughters required to satisfy other conservation laws in this particle decay.[22] If this process occurs fast enough, the CP-violation effect gets carried over to the dark matter sector. However, this contradicts (or at least challenges) the last Sakharov condition, since the expected matter preference in the visible universe is balanced by a new antimatter preference in the dark matter of the universe and total baryon number is conserved.[23]

B-Mesogenesis results in missing energy between the initial and final states of the decay process, which, if recorded, could provide experimental evidence for dark matter. Particle laboratories equipped with B-meson factories such as Belle and BaBar are extremely sensitive to B-meson decays involving missing energy and currently have the capability to detect the channel.[25][26] The LHC is also capable of searching for this interaction since it produces several orders of magnitude more B-mesons than Belle or BaBar, but there are more challenges from the decreased control over B-meson initial energy in the accelerator.[22]

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In physical cosmology, baryogenesis (also known as baryosynthesis) is the physical process that is hypothesized to have taken place during the early universe to produce baryonic asymmetry, i.e. the imbalance of matter ( baryons) and antimatter (antibaryons) in the observed universe. One of the outstanding problems in modern physics is the predominance of matter over antimatter in the universe.

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A Timeline of space

By Alexander Henderson

For Mr Sedlisky's freshman Earth Science Honors

4th Block

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The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature.[1] It was first proposed in 1927 by Roman Catholic priest and physicist Georges Lemaître. Various cosmological models of the Big Bang explain the evolution of the observable universe from the earliest known periods through its subsequent large-scale form.[2][3][4] These models offer a comprehensive explanation for a broad range of observed phenomena, including the abundance of light elements, the cosmic microwave background (CMB) radiation, and large-scale structure. The overall uniformity of the Universe, known as the flatness problem, is explained through cosmic inflation: a sudden and very rapid expansion of space during the earliest moments. However, physics currently lacks a widely accepted theory of quantum gravity that can successfully model the earliest conditions of the Big Bang.

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The universe began as a singularity filled with energy, a disturbance likely interrupted mutual particle annihilation, allowing for particles to exist creating charged particles, this causes columbic force to take effect pushing similarly charged particles away from each other, this begins a chain reaction beginning the expansion of the universe Dark energy probably exists at this time. This theory is heavily supported by the Cosmic Background Microwave Radiation, and the movement of objects compared to us showing expansive properties, See Red/Blushift.

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The doppler effect says that when an object moves away from you, waves emitted will have a lower frequency (Redshift), and that waves moving closer to you become more frequent (Blueshift). These principles state that the common movement of objects away from us causes the light emitted by those objects to become redder.

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With the mass of hydrogen rapidly populating the newly expanded universe, space is filled with (mostly) equal amounts of hydrogen atoms. these atoms slowly coalesce into limited areas of gravity that slowly grow as their gravity increases. Eventually the pressure on the core of the star becomes great enough that it kickstarts the fusion reaction, synthesising new elements.

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Star formation is the process by which dense regions within molecular clouds in interstellar space, sometimes referred to as "stellar nurseries" or " star-forming regions", collapse and form stars. As a branch of astronomy, star formation includes the study of the interstellar medium (ISM) and giant molecular clouds (GMC) as precursors to the star formation process, and the study of protostars and young stellar objects as its immediate products. It is closely related to planet formation, another branch of astronomy. Star formation theory, as well as accounting for the formation of a single star, must also account for the statistics of binary stars and the initial mass function. Most stars do not form in isolation but as part of a group of stars referred as star clusters or stellar associations.

Interstellar clouds

A spiral galaxy like the Milky Way contains stars, stellar remnants, and a diffuse interstellar medium (ISM) of gas and dust. The interstellar medium consists of 10 −4 to 10 ⁶ particles per cm ³ and is typically composed of roughly 70% hydrogen by mass, with most of the remaining gas consisting of helium. This medium has been chemically enriched by trace amounts of heavier elements that were ejected from stars as they passed beyond the end of their main sequence lifetime. Higher density regions of the interstellar medium form clouds, or diffuse nebulae, where star formation takes place. In contrast to spirals, an elliptical galaxy loses the cold component of its interstellar medium within roughly a billion years, which hinders the galaxy from forming diffuse nebulae except through mergers with other galaxies.

In the dense nebulae where stars are produced, much of the hydrogen is in the molecular (H ₂) form, so these nebulae are called molecular clouds. Herschel Space Observatory have revealed that filaments are truly ubiquitous in the molecular cloud. Dense molecular filaments, which are central to the star formation process, will fragment into gravitationally bound cores, most of which will evolve into stars. Continuous accretion of gas, geometrical bending, and magnetic fields may control the detailed fragmentation manner of the filaments. In supercritical filaments observations have revealed quasi-periodic chains of dense cores with spacing comparable to the filament inner width, and includes embedded protostars with outflows. Observations indicate that the coldest clouds tend to form low-mass stars, observed first in the infrared inside the clouds, then in visible light at their surface when the clouds dissipate, while giant molecular clouds, which are generally warmer, produce stars of all masses. These giant molecular clouds have typical densities of 100 particles per cm ³, diameters of 100 light-years (9.5×10 ¹⁴ km), masses of up to 6 million solar masses (), and an average interior temperature of 10 K. About half the total mass of the galactic ISM is found in molecular clouds and in the Milky Way there are an estimated 6,000 molecular clouds, each with more than 100,000 M☉. The nearest nebula to the Sun where massive stars are being formed is the Orion Nebula, 1,300 ly (1.2×10 ¹⁶ km) away. However, lower mass star formation is occurring about 400-450 light years distant in the ρ Ophiuchi cloud complex.

A more compact site of star formation is the opaque clouds of dense gas and dust known as Bok globules, so named after the astronomer Bart Bok. These can form in association with collapsing molecular clouds or possibly independently. The Bok globules are typically up to a light year across and contain a few solar masses. They can be observed as dark clouds silhouetted against bright emission nebulae or background stars. Over half the known Bok globules have been found to contain newly forming stars.

Cloud collapse

An interstellar cloud of gas will remain in hydrostatic equilibrium as long as the kinetic energy of the gas pressure is in balance with the potential energy of the internal gravitational force. Mathematically this is expressed using the virial theorem, which states that, to maintain equilibrium, the gravitational potential energy must equal twice the internal thermal energy. If a cloud is massive enough that the gas pressure is insufficient to support it, the cloud will undergo gravitational collapse. The mass above which a cloud will undergo such collapse is called the Jeans mass. The Jeans mass depends on the temperature and density of the cloud, but is typically thousands to tens of thousands of solar masses. During cloud collapse dozens to ten thousands of stars form more or less simultaneously which is observable in so-called embedded clusters. The end product of a core collapse is an open cluster of stars.

In triggered star formation, one of several events might occur to compress a molecular cloud and initiate its gravitational collapse. Molecular clouds may collide with each other, or a nearby supernova explosion can be a trigger, sending shocked matter into the cloud at very high speeds. (The resulting new stars may themselves soon produce supernovae, producing self-propagating star formation.) Alternatively, galactic collisions can trigger massive starbursts of star formation as the gas clouds in each galaxy are compressed and agitated by tidal forces. The latter mechanism may be responsible for the formation of globular clusters.

A supermassive black hole at the core of a galaxy may serve to regulate the rate of star formation in a galactic nucleus. A black hole that is accreting infalling matter can become active, emitting a strong wind through a collimated relativistic jet. This can limit further star formation. Massive black holes ejecting radio-frequency-emitting particles at near-light speed can also block the formation of new stars in aging galaxies. However, the radio emissions around the jets may also trigger star formation. Likewise, a weaker jet may trigger star formation when it collides with a cloud.

As it collapses, a molecular cloud breaks into smaller and smaller pieces in a hierarchical manner, until the fragments reach stellar mass. In each of these fragments, the collapsing gas radiates away the energy gained by the release of gravitational potential energy. As the density increases, the fragments become opaque and are thus less efficient at radiating away their energy. This raises the temperature of the cloud and inhibits further fragmentation. The fragments now condense into rotating spheres of gas that serve as stellar embryos.

Complicating this picture of a collapsing cloud are the effects of turbulence, macroscopic flows, rotation, magnetic fields and the cloud geometry. Both rotation and magnetic fields can hinder the collapse of a cloud. Turbulence is instrumental in causing fragmentation of the cloud, and on the smallest scales it promotes collapse.

A protostellar cloud will continue to collapse as long as the gravitational binding energy can be eliminated. This excess energy is primarily lost through radiation. However, the collapsing cloud will eventually become opaque to its own radiation, and the energy must be removed through some other means. The dust within the cloud becomes heated to temperatures of 60-100 K, and these particles radiate at wavelengths in the far infrared where the cloud is transparent. Thus the dust mediates the further collapse of the cloud.

During the collapse, the density of the cloud increases towards the center and thus the middle region becomes optically opaque first. This occurs when the density is about . A core region, called the first hydrostatic core, forms where the collapse is essentially halted. It continues to increase in temperature as determined by the virial theorem. The gas falling toward this opaque region collides with it and creates shock waves that further heat the core.

When the core temperature reaches about 2000 K, the thermal energy dissociates the H ₂ molecules. This is followed by the ionization of the hydrogen and helium atoms. These processes absorb the energy of the contraction, allowing it to continue on timescales comparable to the period of collapse at free fall velocities. After the density of infalling material has reached about 10 −8 g / cm ³, that material is sufficiently transparent to allow energy radiated by the protostar to escape. The combination of convection within the protostar and radiation from its exterior allow the star to contract further. This continues until the gas is hot enough for the internal pressure to support the protostar against further gravitational collapse-a state called hydrostatic equilibrium. When this accretion phase is nearly complete, the resulting object is known as a protostar.

Accretion of material onto the protostar continues partially from the newly formed circumstellar disc. When the density and temperature are high enough, deuterium fusion begins, and the outward pressure of the resultant radiation slows (but does not stop) the collapse. Material comprising the cloud continues to "rain" onto the protostar. In this stage bipolar jets are produced called Herbig-Haro objects. This is probably the means by which excess angular momentum of the infalling material is expelled, allowing the star to continue to form.

When the surrounding gas and dust envelope disperses and accretion process stops, the star is considered a pre-main-sequence star (PMS star). The energy source of these objects is gravitational contraction, as opposed to hydrogen burning in main sequence stars. The PMS star follows a Hayashi track on the Hertzsprung-Russell (H-R) diagram. The contraction will proceed until the Hayashi limit is reached, and thereafter contraction will continue on a Kelvin-Helmholtz timescale with the temperature remaining stable. Stars with less than 0.5 M☉ thereafter join the main sequence. For more massive PMS stars, at the end of the Hayashi track they will slowly collapse in near hydrostatic equilibrium, following the Henyey track.

Finally, hydrogen begins to fuse in the core of the star, and the rest of the enveloping material is cleared away. This ends the protostellar phase and begins the star's main sequence phase on the H-R diagram.

The stages of the process are well defined in stars with masses around 1 M☉ or less. In high mass stars, the length of the star formation process is comparable to the other timescales of their evolution, much shorter, and the process is not so well defined. The later evolution of stars is studied in stellar evolution.

Protostar

Protostar outburst - HOPS 383 (2015).

Key elements of star formation are only available by observing in wavelengths other than the optical. The protostellar stage of stellar existence is almost invariably hidden away deep inside dense clouds of gas and dust left over from the GMC. Often, these star-forming cocoons known as Bok globules, can be seen in silhouette against bright emission from surrounding gas. Early stages of a star's life can be seen in infrared light, which penetrates the dust more easily than visible light. Observations from the Wide-field Infrared Survey Explorer (WISE) have thus been especially important for unveiling numerous galactic protostars and their parent star clusters. Examples of such embedded star clusters are FSR 1184, FSR 1190, Camargo 14, Camargo 74, Majaess 64, and Majaess 98.

The structure of the molecular cloud and the effects of the protostar can be observed in near-IR extinction maps (where the number of stars are counted per unit area and compared to a nearby zero extinction area of sky), continuum dust emission and rotational transitions of CO and other molecules; these last two are observed in the millimeter and submillimeter range. The radiation from the protostar and early star has to be observed in infrared astronomy wavelengths, as the extinction caused by the rest of the cloud in which the star is forming is usually too big to allow us to observe it in the visual part of the spectrum. This presents considerable difficulties as the Earth's atmosphere is almost entirely opaque from 20μm to 850μm, with narrow windows at 200μm and 450μm. Even outside this range, atmospheric subtraction techniques must be used.

X-ray observations have proven useful for studying young stars, since X-ray emission from these objects is about 100-100,000 times stronger than X-ray emission from main-sequence stars. The earliest detections of X-rays from T Tauri stars were made by the Einstein X-ray Observatory. For low-mass stars X-rays are generated by the heating of the stellar corona through magnetic reconnection, while for high-mass O and early B-type stars X-rays are generated through supersonic shocks in the stellar winds. Photons in the soft X-ray energy range covered by the Chandra X-ray Observatory and XMM-Newton may penetrate the interstellar medium with only moderate absorption due to gas, making the X-ray a useful wavelength for seeing the stellar populations within molecular clouds. X-ray emission as evidence of stellar youth makes this band particularly useful for performing censuses of stars in star-forming regions, given that not all young stars have infrared excesses. X-ray observations have provided near-complete censuses of all stellar-mass objects in the Orion Nebula Cluster and Taurus Molecular Cloud.

The formation of individual stars can only be directly observed in the Milky Way Galaxy, but in distant galaxies star formation has been detected through its unique spectral signature.

Initial research indicates star-forming clumps start as giant, dense areas in turbulent gas-rich matter in young galaxies, live about 500 million years, and may migrate to the center of a galaxy, creating the central bulge of a galaxy.

On February 21, 2014, NASA announced a greatly upgraded database for tracking polycyclic aromatic hydrocarbons (PAHs) in the universe. According to scientists, more than 20% of the carbon in the universe may be associated with PAHs, possible starting materials for the formation of life. PAHs seem to have been formed shortly after the Big Bang, are widespread throughout the universe, and are associated with new stars and exoplanets.

In February 2018, astronomers reported, for the first time, a signal of the reionization epoch, an indirect detection of light from the earliest stars formed - about 180 million years after the Big Bang.

An article published on October 22, 2019, reported on the detection of 3MM-1, a massive star-forming galaxy about 12.5 billion light-years away that is obscured by clouds of dust.At a mass of about 10 10.8 solar masses, it showed a star formation rate about 100 times as high as in the Milky Way.

Notable pathfinder objects

MWC 349 was first discovered in 1978, and is estimated to be only 1,000 years old.

VLA 1623 - The first exemplar Class 0 protostar, a type of embedded protostar that has yet to accrete the majority of its mass. Found in 1993, is possibly younger than 10,000 years.

L1014 - An extremely faint embedded object representative of a new class of sources that are only now being detected with the newest telescopes. Their status is still undetermined, they could be the youngest low-mass Class 0 protostars yet seen or even very low-mass evolved objects (like brown dwarfs or even rogue planets).

GCIRS 8* - The youngest known main sequence star in the Galactic Center region, discovered in August 2006. It is estimated to be 3.5 million years old.

Stars of different masses are thought to form by slightly different mechanisms. The theory of low-mass star formation, which is well-supported by observation, suggests that low-mass stars form by the gravitational collapse of rotating density enhancements within molecular clouds. As described above, the collapse of a rotating cloud of gas and dust leads to the formation of an accretion disk through which matter is channeled onto a central protostar. For stars with masses higher than about 8 M☉, however, the mechanism of star formation is not well understood.

Massive stars emit copious quantities of radiation which pushes against infalling material. In the past, it was thought that this radiation pressure might be substantial enough to halt accretion onto the massive protostar and prevent the formation of stars with masses more than a few tens of solar masses. Recent theoretical work has shown that the production of a jet and outflow clears a cavity through which much of the radiation from a massive protostar can escape without hindering accretion through the disk and onto the protostar. Present thinking is that massive stars may therefore be able to form by a mechanism similar to that by which low mass stars form.

There is mounting evidence that at least some massive protostars are indeed surrounded by accretion disks. Several other theories of massive star formation remain to be tested observationally. Of these, perhaps the most prominent is the theory of competitive accretion, which suggests that massive protostars are "seeded" by low-mass protostars which compete with other protostars to draw in matter from the entire parent molecular cloud, instead of simply from a small local region.

Another theory of massive star formation suggests that massive stars may form by the coalescence of two or more stars of lower mass.

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21. Majaess, D. (2013). ^ Discovering protostars and their host clusters via WISE, ApSS, 344, 1 (VizieR catalog)

22. Camargo et al. (2015). ^ New Galactic embedded clusters and candidates from a WISE Survey, New Astronomy, 34

23. ^ "Young Star-Forming Clump in Deep Space Spotted for First Time".

24. ^ Hoover, Rachel (February 21, 2014). "Need to Track Organic Nano-Particles Across the Universe? NASA's Got an App for That". NASA.

25. ^ Gibney, Elizabeth (February 28, 2018). "Astronomers detect light from the Universe's first stars - Surprises in signal from cosmic dawn also hint at presence of dark matter". Nature. doi:10.1038/d41586-018-02616-8.

26. ^ Andre, Philippe; Ward-Thompson, Derek; Barsony, Mary (1993). "Submillimeter continuum observations of Rho Ophiuchi A - The candidate protostar VLA 1623 and prestellar clumps". The Astrophysical Journal. 406: 122-141. Bibcode:1993ApJ...406..122A. doi:10.1086/172425. ISSN 0004-637X.

27. ^ I. A. Bonnell; M. R. Bate; C. J. Clarke; J. E. Pringle (1997). "Accretion and the stellar mass spectrum in small clusters". Monthly Notices of the Royal Astronomical Society. 285 (1): 201-208. Bibcode:1997MNRAS.285..201B. doi:10.1093/mnras/285.1.201.

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Star formation is the process by which dense regions within molecular clouds in interstellar space, sometimes referred to as "stellar nurseries" or " star-forming regions", collapse and form stars. As a branch of astronomy, star formation includes the study of the interstellar medium (ISM) and giant molecular clouds (GMC) as precursors to the star formation process, and the study of protostars and young stellar objects as its immediate products.

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Fusion is the interaction between two atoms resulting in one more massive atom with byproducts of released energy, the process occurs more commonly in atoms with lower masses. It is how we achieve elements above helium naturally.

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Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manifested as either the release or absorption of energy. This difference in mass arises due to the difference in nuclear binding energy between the atomic nuclei before and after the reaction. Nuclear fusion is the process that powers active or main-sequence stars and other high-magnitude stars, where large amounts of energy are released.

A nuclear fusion process that produces atomic nuclei lighter than iron-56 or nickel-62 will generally release energy. These elements have a relatively small mass and a relatively large binding energy per nucleon. Fusion of nuclei lighter than these releases energy (an exothermic process), while the fusion of heavier nuclei results in energy retained by the product nucleons, and the resulting reaction is endothermic. The opposite is true for the reverse process, called nuclear fission. Nuclear fusion uses lighter elements, such as hydrogen and helium, which are in general more fusible; while the heavier elements, such as uranium, thorium and plutonium, are more fissionable. The extreme astrophysical event of a supernova can produce enough energy to fuse nuclei into elements heavier than iron.

In 1920, Arthur Eddington suggested hydrogen-helium fusion could be the primary source of stellar energy. Quantum tunneling was discovered by Friedrich Hund in 1927, and shortly afterwards Robert Atkinson and Fritz Houtermans used the measured masses of light elements to demonstrate that large amounts of energy could be released by fusing small nuclei. Building on the early experiments in artificial nuclear transmutation by Patrick Blackett, laboratory fusion of hydrogen isotopes was accomplished by Mark Oliphant in 1932. In the remainder of that decade, the theory of the main cycle of nuclear fusion in stars was worked out by Hans Bethe. Research into fusion for military purposes began in the early 1940s as part of the Manhattan Project. Self-sustaining nuclear fusion was first carried out on 1 November 1952, in the Ivy Mike hydrogen (thermonuclear) bomb test.

While fusion was achieved in the operation of the hydrogen bomb (H-bomb), for it to be a useful energy source controlled, sustained fusion is required. Research into developing controlled fusion inside fusion reactors has been ongoing since the 1930s, but the technology is still in its developmental phase.

The US National Ignition Facility, which uses laser-driven inertial confinement fusion, was designed with a goal of break-even fusion; the first large-scale laser target experiments were performed in June 2009 and ignition experiments began in early 2011. On 13 December 2022, the United States Department of Energy announced that on 5 December 2022, they had successfully accomplished break-even fusion, "delivering 2.05 megajoules (MJ) of energy to the target, resulting in 3.15 MJ of fusion energy output."

Prior to this breakthrough, controlled fusion reactions had been unable to produce break-even (self-sustaining) controlled fusion. The two most advanced approaches for it are magnetic confinement (toroid designs) and inertial confinement (laser designs). Workable designs for a toroidal reactor that theoretically will deliver ten times more fusion energy than the amount needed to heat plasma to the required temperatures are in development (see ITER). The ITER facility is expected to finish its construction phase in 2025. It will start commissioning the reactor that same year and initiate plasma experiments in 2025, but is not expected to begin full deuterium-tritium fusion until 2035.

Private companies pursuing the commercialization of nuclear fusion received $2.6 billion in private funding in 2021 alone, going to many notable startups including but not limited to Commonwealth Fusion Systems, Helion Energy Inc., TAE Technologies Inc. and Zap Energy Inc.

Similarly, Canadian-based General Fusion, which is developing a magnetized target fusion nuclear energy system, aims to build its demonstration plant by 2025.

The release of energy with the fusion of light elements is due to the interplay of two opposing forces: the nuclear force, a manifestation of the strong interaction, which holds protons and neutrons tightly together in the atomic nucleus; and the Coulomb force, which causes positively charged protons in the nucleus to repel each other. Lighter nuclei (nuclei smaller than iron and nickel) are sufficiently small and proton-poor to allow the nuclear force to overcome the Coulomb force. This is because the nucleus is sufficiently small that all nucleons feel the short-range attractive force at least as strongly as they feel the infinite-range Coulomb repulsion. Building up nuclei from lighter nuclei by fusion releases the extra energy from the net attraction of particles. For larger nuclei, however, no energy is released, because the nuclear force is short-range and cannot act across larger nuclei.

Fusion powers stars and produces virtually all elements in a process called nucleosynthesis. The Sun is a main-sequence star, and, as such, generates its energy by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses 620 million metric tons of hydrogen and makes 616 million metric tons of helium each second. The fusion of lighter elements in stars releases energy and the mass that always accompanies it. For example, in the fusion of two hydrogen nuclei to form helium, 0.645% of the mass is carried away in the form of kinetic energy of an alpha particle or other forms of energy, such as electromagnetic radiation.

It takes considerable energy to force nuclei to fuse, even those of the lightest element, hydrogen. When accelerated to high enough speeds, nuclei can overcome this electrostatic repulsion and be brought close enough such that the attractive nuclear force is greater than the repulsive Coulomb force. The strong force grows rapidly once the nuclei are close enough, and the fusing nucleons can essentially "fall" into each other and the result is fusion and net energy produced. The fusion of lighter nuclei, which creates a heavier nucleus and often a free neutron or proton, generally releases more energy than it takes to force the nuclei together; this is an exothermic process that can produce self-sustaining reactions.[citation needed ]

Energy released in most nuclear reactions is much larger than in chemical reactions, because the binding energy that holds a nucleus together is greater than the energy that holds electrons to a nucleus. For example, the ionization energy gained by adding an electron to a hydrogen nucleus is -less than one-millionth of the released in the deuterium- tritium (D-T) reaction shown in the adjacent diagram. Fusion reactions have an energy density many times greater than nuclear fission; the reactions produce far greater energy per unit of mass even though individual fission reactions are generally much more energetic than individual fusion ones, which are themselves millions of times more energetic than chemical reactions. Only direct conversion of mass into energy, such as that caused by the annihilatory collision of matter and antimatter, is more energetic per unit of mass than nuclear fusion. (The complete conversion of one gram of matter would release of energy.)

An important fusion process is the stellar nucleosynthesis that powers stars, including the Sun. In the 20th century, it was recognized that the energy released from nuclear fusion reactions accounts for the longevity of stellar heat and light. The fusion of nuclei in a star, starting from its initial hydrogen and helium abundance, provides that energy and synthesizes new nuclei. Different reaction chains are involved, depending on the mass of the star (and therefore the pressure and temperature in its core).

Around 1920, Arthur Eddington anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper The Internal Constitution of the Stars. At that time, the source of stellar energy was unknown; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to Einstein's equation . This was a particularly remarkable development since at that time fusion and thermonuclear energy had not yet been discovered, nor even that stars are largely composed of hydrogen (see metallicity). Eddington's paper reasoned that:

1. The leading theory of stellar energy, the contraction hypothesis, should cause the rotation of a star to visibly speed up due to conservation of angular momentum. But observations of Cepheid variable stars showed this was not happening.

2. The only other known plausible source of energy was conversion of matter to energy; Einstein had shown some years earlier that a small amount of matter was equivalent to a large amount of energy.

3. Francis Aston had also recently shown that the mass of a helium atom was about 0.8% less than the mass of the four hydrogen atoms which would, combined, form a helium atom (according to the then-prevailing theory of atomic structure which held atomic weight to be the distinguishing property between elements; work by Henry Moseley and Antonius van den Broek would later show that nucleic charge was the distinguishing property and that a helium nucleus, therefore, consisted of two hydrogen nuclei plus additional mass). This suggested that if such a combination could happen, it would release considerable energy as a byproduct.

4. If a star contained just 5% of fusible hydrogen, it would suffice to explain how stars got their energy. (it is now known that most 'ordinary' stars contain far more than 5% hydrogen.)

5. Further elements might also be fused, and other scientists had speculated that stars were the "crucible" in which light elements combined to create heavy elements, but without more accurate measurements of their atomic masses nothing more could be said at the time.

All of these speculations were proven correct in the following decades.

The primary source of solar energy, and that of similar size stars, is the fusion of hydrogen to form helium (the proton-proton chain reaction), which occurs at a solar-core temperature of 14 million kelvin. The net result is the fusion of four protons into one alpha particle, with the release of two positrons and two neutrinos (which changes two of the protons into neutrons), and energy. In heavier stars, the CNO cycle and other processes are more important. As a star uses up a substantial fraction of its hydrogen, it begins to synthesize heavier elements. The heaviest elements are synthesized by fusion that occurs when a more massive star undergoes a violent supernova at the end of its life, a process known as supernova nucleosynthesis.

A substantial energy barrier of electrostatic forces must be overcome before fusion can occur. At large distances, two naked nuclei repel one another because of the repulsive electrostatic force between their positively charged protons. If two nuclei can be brought close enough together, however, the electrostatic repulsion can be overcome by the quantum effect in which nuclei can tunnel through coulomb forces.

When a nucleon such as a proton or neutron is added to a nucleus, the nuclear force attracts it to all the other nucleons of the nucleus (if the atom is small enough), but primarily to its immediate neighbors due to the short range of the force. The nucleons in the interior of a nucleus have more neighboring nucleons than those on the surface. Since smaller nuclei have a larger surface-area-to-volume ratio, the binding energy per nucleon due to the nuclear force generally increases with the size of the nucleus but approaches a limiting value corresponding to that of a nucleus with a diameter of about four nucleons. It is important to keep in mind that nucleons are quantum objects. So, for example, since two neutrons in a nucleus are identical to each other, the goal of distinguishing one from the other, such as which one is in the interior and which is on the surface, is in fact meaningless, and the inclusion of quantum mechanics is therefore necessary for proper calculations.

The electrostatic force, on the other hand, is an inverse-square force, so a proton added to a nucleus will feel an electrostatic repulsion from all the other protons in the nucleus. The electrostatic energy per nucleon due to the electrostatic force thus increases without limit as nuclei atomic number grows.

The net result of the opposing electrostatic and strong nuclear forces is that the binding energy per nucleon generally increases with increasing size, up to the elements iron and nickel, and then decreases for heavier nuclei. Eventually, the binding energy becomes negative and very heavy nuclei (all with more than 208 nucleons, corresponding to a diameter of about 6 nucleons) are not stable. The four most tightly bound nuclei, in decreasing order of binding energy per nucleon, are , , , and . Even though the nickel isotope, , is more stable, the iron isotope is an order of magnitude more common. This is due to the fact that there is no easy way for stars to create through the alpha process.

An exception to this general trend is the helium-4 nucleus, whose binding energy is higher than that of lithium, the next heavier element. This is because protons and neutrons are fermions, which according to the Pauli exclusion principle cannot exist in the same nucleus in exactly the same state. Each proton or neutron's energy state in a nucleus can accommodate both a spin up particle and a spin down particle. Helium-4 has an anomalously large binding energy because its nucleus consists of two protons and two neutrons (it is a doubly magic nucleus), so all four of its nucleons can be in the ground state. Any additional nucleons would have to go into higher energy states. Indeed, the helium-4 nucleus is so tightly bound that it is commonly treated as a single quantum mechanical particle in nuclear physics, namely, the alpha particle.

The situation is similar if two nuclei are brought together. As they approach each other, all the protons in one nucleus repel all the protons in the other. Not until the two nuclei actually come close enough for long enough so the strong nuclear force can take over (by way of tunneling) is the repulsive electrostatic force overcome. Consequently, even when the final energy state is lower, there is a large energy barrier that must first be overcome. It is called the Coulomb barrier.

The Coulomb barrier is smallest for isotopes of hydrogen, as their nuclei contain only a single positive charge. A diproton is not stable, so neutrons must also be involved, ideally in such a way that a helium nucleus, with its extremely tight binding, is one of the products.

Using deuterium-tritium fuel, the resulting energy barrier is about 0.1 MeV. In comparison, the energy needed to remove an electron from hydrogen is 13.6 eV. The (intermediate) result of the fusion is an unstable ⁵He nucleus, which immediately ejects a neutron with 14.1 MeV. The recoil energy of the remaining ⁴ He nucleus is 3.5 MeV, so the total energy liberated is 17.6 MeV. This is many times more than what was needed to overcome the energy barrier.

The reaction cross section (σ) is a measure of the probability of a fusion reaction as a function of the relative velocity of the two reactant nuclei. If the reactants have a distribution of velocities, e.g. a thermal distribution, then it is useful to perform an average over the distributions of the product of cross-section and velocity. This average is called the 'reactivity', denoted . The reaction rate (fusions per volume per time) is times the product of the reactant number densities:

If a species of nuclei is reacting with a nucleus like itself, such as the DD reaction, then the product must be replaced by .

increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of 10- 100 keV. At these temperatures, well above typical ionization energies (13.6 eV in the hydrogen case), the fusion reactants exist in a plasma state.

The significance of as a function of temperature in a device with a particular energy confinement time is found by considering the Lawson criterion. This is an extremely challenging barrier to overcome on Earth, which explains why fusion research has taken many years to reach the current advanced technical state.[20]

If matter is sufficiently heated (hence being plasma) and confined, fusion reactions may occur due to collisions with extreme thermal kinetic energies of the particles. Thermonuclear weapons produce what amounts to an uncontrolled release of fusion energy. Controlled thermonuclear fusion concepts use magnetic fields to confine the plasma.

Inertial confinement fusion (ICF) is a method aimed at releasing fusion energy by heating and compressing a fuel target, typically a pellet containing deuterium and tritium.

Inertial electrostatic confinement is a set of devices that use an electric field to heat ions to fusion conditions. The most well known is the fusor. Starting in 1999, a number of amateurs have been able to do amateur fusion using these homemade devices. Other IEC devices include: the Polywell, MIX POPS and Marble concepts.

Beam-beam or beam-target fusion

Accelerator-based light-ion fusion is a technique using particle accelerators to achieve particle kinetic energies sufficient to induce light-ion fusion reactions. Accelerating light ions is relatively easy, and can be done in an efficient manner-requiring only a vacuum tube, a pair of electrodes, and a high-voltage transformer; fusion can be observed with as little as 10 kV between the electrodes. The system can be arranged to accelerate ions into a static fuel-infused target, known as beam-target fusion, or by accelerating two streams of ions towards each other, beam-beam fusion.

The key problem with accelerator-based fusion (and with cold targets in general) is that fusion cross sections are many orders of magnitude lower than Coulomb interaction cross-sections. Therefore, the vast majority of ions expend their energy emitting bremsstrahlung radiation and the ionization of atoms of the target. Devices referred to as sealed-tube neutron generators are particularly relevant to this discussion. These small devices are miniature particle accelerators filled with deuterium and tritium gas in an arrangement that allows ions of those nuclei to be accelerated against hydride targets, also containing deuterium and tritium, where fusion takes place, releasing a flux of neutrons. Hundreds of neutron generators are produced annually for use in the petroleum industry where they are used in measurement equipment for locating and mapping oil reserves.

A number of attempts to recirculate the ions that "miss" collisions have been made over the years. One of the better-known attempts in the 1970s was Migma, which used a unique particle storage ring to capture ions into circular orbits and return them to the reaction area. Theoretical calculations made during funding reviews pointed out that the system would have significant difficulty scaling up to contain enough fusion fuel to be relevant as a power source. In the 1990s, a new arrangement using a field-reverse configuration (FRC) as the storage system was proposed by Norman Rostoker and continues to be studied by TAE Technologies as of 2021. A closely related approach is to merge two FRC's rotating in opposite directions, which is being actively studied by Helion Energy. Because these approaches all have ion energies well beyond the Coulomb barrier, they often suggest the use of alternative fuel cycles like p- that are too difficult to attempt using conventional approaches.

Muon-catalyzed fusion is a fusion process that occurs at ordinary temperatures. It was studied in detail by Steven Jones in the early 1980s. Net energy production from this reaction has been unsuccessful because of the high energy required to create muons, their short 2.2 µs half-life, and the high chance that a muon will bind to the new alpha particle and thus stop catalyzing fusion.

Some other confinement principles have been investigated.

Antimatter-initialized fusion uses small amounts of antimatter to trigger a tiny fusion explosion. This has been studied primarily in the context of making nuclear pulse propulsion, and pure fusion bombs feasible. This is not near becoming a practical power source, due to the cost of manufacturing antimatter alone.

Pyroelectric fusion was reported in April 2005 by a team at UCLA. The scientists used a pyroelectric crystal heated from −34 to 7 °C (−29 to 45 °F), combined with a tungsten needle to produce an electric field of about 25 gigavolts per meter to ionize and accelerate deuterium nuclei into an erbium deuteride target. At the estimated energy levels, the D-D fusion reaction may occur, producing helium-3 and a 2.45 MeV neutron. Although it makes a useful neutron generator, the apparatus is not intended for power generation since it requires far more energy than it produces. D-T fusion reactions have been observed with a tritiated erbium target.

Nuclear fusion-fission hybrid (hybrid nuclear power) is a proposed means of generating power by use of a combination of nuclear fusion and fission processes. The concept dates to the 1950s, and was briefly advocated by Hans Bethe during the 1970s, but largely remained unexplored until a revival of interest in 2009, due to the delays in the realization of pure fusion.

Project PACER, carried out at Los Alamos National Laboratory (LANL) in the mid-1970s, explored the possibility of a fusion power system that would involve exploding small hydrogen bombs (fusion bombs) inside an underground cavity. As an energy source, the system is the only fusion power system that could be demonstrated to work using existing technology. However it would also require a large, continuous supply of nuclear bombs, making the economics of such a system rather questionable.

Bubble fusion also called sonofusion was a proposed mechanism for achieving fusion via sonic cavitation which rose to prominence in the early 2000s. Subsequent attempts at replication failed and the principal investigator, Rusi Taleyarkhan, was judged guilty of research misconduct in 2008.

At the temperatures and densities in stellar cores, the rates of fusion reactions are notoriously slow. For example, at solar core temperature ( T ≈ 15 MK) and density (160 g/cm ³), the energy release rate is only 276 μW/cm ³-about a quarter of the volumetric rate at which a resting human body generates heat. Thus, reproduction of stellar core conditions in a lab for nuclear fusion power production is completely impractical. Because nuclear reaction rates depend on density as well as temperature and most fusion schemes operate at relatively low densities, those methods are strongly dependent on higher temperatures. The fusion rate as a function of temperature (exp(− E/ kT)), leads to the need to achieve temperatures in terrestrial reactors 10-100 times higher than in stellar interiors: T ≈ .

In artificial fusion, the primary fuel is not constrained to be protons and higher temperatures can be used, so reactions with larger cross-sections are chosen. Another concern is the production of neutrons, which activate the reactor structure radiologically, but also have the advantages of allowing volumetric extraction of the fusion energy and tritium breeding. Reactions that release no neutrons are referred to as aneutronic.

To be a useful energy source, a fusion reaction must satisfy several criteria. It must:

Be exothermic

This limits the reactants to the low Z (number of protons) side of the curve of binding energy. It also makes helium the most common product because of its extraordinarily tight binding, although and also show up.

Involve low atomic number (Z) nuclei

This is because the electrostatic repulsion that must be overcome before the nuclei are close enough to fuse is directly related to the number of protons it contains - its atomic number.[citation needed ]

Have two reactants

At anything less than stellar densities, three-body collisions are too improbable. In inertial confinement, both stellar densities and temperatures are exceeded to compensate for the shortcomings of the third parameter of the Lawson criterion, ICF's very short confinement time.

Have two or more products

This allows simultaneous conservation of energy and momentum without relying on the electromagnetic force.

Conserve both protons and neutrons

The cross sections for the weak interaction are too small.

Few reactions meet these criteria. The following are those with the largest cross sections:

For reactions with two products, the energy is divided between them in inverse proportion to their masses, as shown. In most reactions with three products, the distribution of energy varies. For reactions that can result in more than one set of products, the branching ratios are given.

Some reaction candidates can be eliminated at once. The D- ⁶Li reaction has no advantage compared to - because it is roughly as difficult to burn but produces substantially more neutrons through - side reactions. There is also a - reaction, but the cross section is far too low, except possibly when T i > 1 MeV, but at such high temperatures an endothermic, direct neutron-producing reaction also becomes very significant. Finally there is also a - reaction, which is not only difficult to burn, but can be easily induced to split into two alpha particles and a neutron.

In addition to the fusion reactions, the following reactions with neutrons are important in order to "breed" tritium in "dry" fusion bombs and some proposed fusion reactors:

The latter of the two equations was unknown when the U.S. conducted the Castle Bravo fusion bomb test in 1954. Being just the second fusion bomb ever tested (and the first to use lithium), the designers of the Castle Bravo "Shrimp" had understood the usefulness of ⁶Li in tritium production, but had failed to recognize that ⁷Li fission would greatly increase the yield of the bomb. While ⁷ Li has a small neutron cross-section for low neutron energies, it has a higher cross section above 5 MeV. The 15 Mt yield was 150% greater than the predicted 6 Mt and caused unexpected exposure to fallout.

To evaluate the usefulness of these reactions, in addition to the reactants, the products, and the energy released, one needs to know something about the nuclear cross section. Any given fusion device has a maximum plasma pressure it can sustain, and an economical device would always operate near this maximum. Given this pressure, the largest fusion output is obtained when the temperature is chosen so that is a maximum. This is also the temperature at which the value of the triple product nTτ required for ignition is a minimum, since that required value is inversely proportional to (see Lawson criterion). (A plasma is "ignited" if the fusion reactions produce enough power to maintain the temperature without external heating.) This optimum temperature and the value of at that temperature is given for a few of these reactions in the following table.

Note that many of the reactions form chains. For instance, a reactor fueled with and creates some , which is then possible to use in the - reaction if the energies are "right". An elegant idea is to combine the reactions (8) and (9). The from reaction (8) can react with in reaction (9) before completely thermalizing. This produces an energetic proton, which in turn undergoes reaction (8) before thermalizing. Detailed analysis shows that this idea would not work well,[citation needed ] but it is a good example of a case where the usual assumption of a Maxwellian plasma is not appropriate.

Neutronicity, confinement requirement, and power density

Any of the reactions above can in principle be the basis of fusion power production. In addition to the temperature and cross section discussed above, we must consider the total energy of the fusion products E fus, the energy of the charged fusion products E ch, and the atomic number Z of the non-hydrogenic reactant.

Specification of the - reaction entails some difficulties, though. To begin with, one must average over the two branches (2i) and (2ii). More difficult is to decide how to treat the and products. burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. The - reaction is optimized at a much higher temperature, so the burnup at the optimum - temperature may be low. Therefore, it seems reasonable to assume the but not the gets burned up and adds its energy to the net reaction, which means the total reaction would be the sum of (2i), (2ii), and (1):

For calculating the power of a reactor (in which the reaction rate is determined by the D-D step), we count the - fusion energy per D-D reaction as E fus = (4.03 MeV + 17.6 MeV) × 50% + (3.27 MeV) × 50% = 12.5 MeV and the energy in charged particles as E ch = (4.03 MeV + 3.5 MeV) × 50% + (0.82 MeV) × 50% = 4.2 MeV. (Note: if the tritium ion reacts with a deuteron while it still has a large kinetic energy, then the kinetic energy of the helium-4 produced may be quite different from 3.5 MeV, so this calculation of energy in charged particles is only an approximation of the average.) The amount of energy per deuteron consumed is 2/5 of this, or 5.0 MeV (a specific energy of about 225 million MJ per kilogram of deuterium).

Another unique aspect of the - reaction is that there is only one reactant, which must be taken into account when calculating the reaction rate.

With this choice, we tabulate parameters for four of the most important reactions

The last column is the neutronicity of the reaction, the fraction of the fusion energy released as neutrons. This is an important indicator of the magnitude of the problems associated with neutrons like radiation damage, biological shielding, remote handling, and safety. For the first two reactions it is calculated as . For the last two reactions, where this calculation would give zero, the values quoted are rough estimates based on side reactions that produce neutrons in a plasma in thermal equilibrium.

Of course, the reactants should also be mixed in the optimal proportions. This is the case when each reactant ion plus its associated electrons accounts for half the pressure. Assuming that the total pressure is fixed, this means that particle density of the non-hydrogenic ion is smaller than that of the hydrogenic ion by a factor 2/( Z + 1). Therefore, the rate for these reactions is reduced by the same factor, on top of any differences in the values of . On the other hand, because the - reaction has only one reactant, its rate is twice as high as when the fuel is divided between two different hydrogenic species, thus creating a more efficient reaction.

Thus there is a "penalty" of 2/( Z + 1) for non-hydrogenic fuels arising from the fact that they require more electrons, which take up pressure without participating in the fusion reaction. (It is usually a good assumption that the electron temperature will be nearly equal to the ion temperature. Some authors, however, discuss the possibility that the electrons could be maintained substantially colder than the ions. In such a case, known as a "hot ion mode", the "penalty" would not apply.) There is at the same time a "bonus" of a factor 2 for - because each ion can react with any of the other ions, not just a fraction of them.

We can now compare these reactions in the following table.

The maximum value of is taken from a previous table. The "penalty/bonus" factor is that related to a non-hydrogenic reactant or a single-species reaction. The values in the column "inverse reactivity" are found by dividing by the product of the second and third columns. It indicates the factor by which the other reactions occur more slowly than the - reaction under comparable conditions. The column " Lawson criterion" weights these results with E ch and gives an indication of how much more difficult it is to achieve ignition with these reactions, relative to the difficulty for the - reaction. The next-to-last column is labeled "power density" and weights the practical reactivity by E fus. The final column indicates how much lower the fusion power density of the other reactions is compared to the - reaction and can be considered a measure of the economic potential.

Bremsstrahlung losses in quasineutral, isotropic plasmas

The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with electrons that in aggregate neutralize the ions' bulk electrical charge and form a plasma. The electrons will generally have a temperature comparable to or greater than that of the ions, so they will collide with the ions and emit x-ray radiation of 10-30 keV energy, a process known as Bremsstrahlung.

The huge size of the Sun and stars means that the x-rays produced in this process will not escape and will deposit their energy back into the plasma. They are said to be opaque to x-rays. But any terrestrial fusion reactor will be optically thin for x-rays of this energy range. X-rays are difficult to reflect but they are effectively absorbed (and converted into heat) in less than mm thickness of stainless steel (which is part of a reactor's shield). This means the bremsstrahlung process is carrying energy out of the plasma, cooling it.

The ratio of fusion power produced to x-ray radiation lost to walls is an important figure of merit. This ratio is generally maximized at a much higher temperature than that which maximizes the power density (see the previous subsection). The following table shows estimates of the optimum temperature and the power ratio at that temperature for several reactions:

The actual ratios of fusion to Bremsstrahlung power will likely be significantly lower for several reasons. For one, the calculation assumes that the energy of the fusion products is transmitted completely to the fuel ions, which then lose energy to the electrons by collisions, which in turn lose energy by Bremsstrahlung. However, because the fusion products move much faster than the fuel ions, they will give up a significant fraction of their energy directly to the electrons. Secondly, the ions in the plasma are assumed to be purely fuel ions. In practice, there will be a significant proportion of impurity ions, which will then lower the ratio. In particular, the fusion products themselves must remain in the plasma until they have given up their energy, and will remain for some time after that in any proposed confinement scheme. Finally, all channels of energy loss other than Bremsstrahlung have been neglected. The last two factors are related. On theoretical and experimental grounds, particle and energy confinement seem to be closely related. In a confinement scheme that does a good job of retaining energy, fusion products will build up. If the fusion products are efficiently ejected, then energy confinement will be poor, too.

The temperatures maximizing the fusion power compared to the Bremsstrahlung are in every case higher than the temperature that maximizes the power density and minimizes the required value of the fusion triple product. This will not change the optimum operating point for - very much because the Bremsstrahlung fraction is low, but it will push the other fuels into regimes where the power density relative to - is even lower and the required confinement even more difficult to achieve. For - and -, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For -, - and - the Bremsstrahlung losses appear to make a fusion reactor using these fuels with a quasineutral, isotropic plasma impossible. Some ways out of this dilemma have been considered but rejected. This limitation does not apply to non-neutral and anisotropic plasmas; however, these have their own challenges to contend with.

In a classical picture, nuclei can be understood as hard spheres that repel each other through the Coulomb force but fuse once the two spheres come close enough for contact. Estimating the radius of an atomic nuclei as about one femtometer, the energy needed for fusion of two hydrogen is:

This would imply that for the core of the sun, which has a Boltzmann distribution with a temperature of around 1.4 keV, the probability hydrogen would reach the threshold is , that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics.

The probability that fusion occurs is greatly increased compared to the classical picture, thanks to the smearing of the effective radius as the de Broglie wavelength as well as quantum tunneling through the potential barrier. To determine the rate of fusion reactions, the value of most interest is the cross section, which describes the probability that particles will fuse by giving a characteristic area of interaction. An estimation of the fusion cross-sectional area is often broken into three pieces:

where is the geometric cross section, T is the barrier transparency and R is the reaction characteristics of the reaction.

is of the order of the square of the de Broglie wavelength where is the reduced mass of the system and is the center of mass energy of the system.

T can be approximated by the Gamow transparency, which has the form: where is the Gamow factor and comes from estimating the quantum tunneling probability through the potential barrier.

R contains all the nuclear physics of the specific reaction and takes very different values depending on the nature of the interaction. However, for most reactions, the variation of is small compared to the variation from the Gamow factor and so is approximated by a function called the astrophysical S-factor, , which is weakly varying in energy. Putting these dependencies together, one approximation for the fusion cross section as a function of energy takes the form:

More detailed forms of the cross-section can be derived through nuclear physics-based models and R-matrix theory.

The Naval Research Lab's plasma physics formulary gives the total cross section in barns as a function of the energy (in keV) of the incident particle towards a target ion at rest fit by the formula:

Bosch-Hale also reports a R-matrix calculated cross sections fitting observation data with Padé rational approximating coefficients. With energy in units of keV and cross sections in units of millibarn, the factor has the form:

DT(1)

DD(2ii)

DHe³(3)

THe⁴

31.3970

68.7508

31.3970

34.3827

A4

0

A5

0

0

B1

0

0

B2

0

0

B3

0

0

B4

0

0

0

Applicable Energy Range [keV]

0.5-5000

0.3-900

0.5-4900

0.5-550

2.0

2.2

2.5

1.9

where

In fusion systems that are in thermal equilibrium, the particles are in a Maxwell-Boltzmann distribution, meaning the particles have a range of energies centered around the plasma temperature. The sun, magnetically confined plasmas and inertial confinement fusion systems are well modeled to be in thermal equilibrium. In these cases, the value of interest is the fusion cross-section averaged across the Maxwell-Boltzmann distribution. The Naval Research Lab's plasma physics formulary tabulates Maxwell averaged fusion cross sections reactivities in .

For energies the data can be represented by:

with T in units of keV.

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Nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles ( neutrons or protons). The difference in mass between the reactants and products is manifested as either the release or absorption of energy.

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 Basic ExampleFusion Element Chain in Moderately Sized Stars

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 Basic ExampleFusion Element Chain in Moderately Sized Stars

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Logarithm of the relative energy output (ε) of proton-proton (PP), CNO and 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.

Stellar nucleosynthesis is the creation (nucleosynthesis) of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. 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 in 1946, who later refined it in 1954. Further advances were made, especially to nucleosynthesis by neutron capture of the elements heavier than iron, by Margaret and Geoffrey Burbidge, William Alfred Fowler and Fred Hoyle in their famous 1957 , which became one of the most heavily cited papers in astrophysics history.

Stars evolve because of changes in their composition (the abundance of their constituent elements) over their lifespans, first by burning hydrogen ( main sequence star), then helium ( horizontal branch 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, forming a planetary nebula, while a higher-mass star will eject mass via a sudden catastrophic event called a supernova. The term supernova nucleosynthesis 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 and its associated heating, resulting in the subsequent burning of carbon, oxygen and silicon. However, most of the nucleosynthesis in the mass range A = 28-56 (from silicon to nickel) is actually caused by the upper layers of the star collapsing onto the core, creating a compressional shock wave 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, is the final epoch of stellar nucleosynthesis.

A stimulus to the development of the theory of nucleosynthesis was the discovery of variations in the abundances of elements found in the universe. 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 released from nuclear fusion reactions accounted for the longevity of the Sun as a source of heat and light.

In 1920, Arthur Eddington proposed that stars obtained their energy from nuclear fusion of hydrogen to form helium and also raised the possibility that the heavier elements are produced in stars.

In 1920, Arthur Eddington, on the basis of the precise measurements of atomic masses by F.W. Aston and a preliminary suggestion by Jean Perrin, proposed that stars obtained their energy from nuclear fusion of hydrogen to form helium 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 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 between them and approach each other closely enough to undergo nuclear reaction due to the strong nuclear force 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 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 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 proton-proton chain reaction, is the dominant energy source in stars with masses up to about the mass of the Sun. The second process, the carbon-nitrogen-oxygen cycle, 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. 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 . 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 r-process 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.

Cross section of a supergiant showing nucleosynthesis and elements formed.

A version of the periodic table indicating the origins - including stellar nucleosynthesis - of the elements.

The most important reactions in stellar nucleosynthesis:

Hydrogen fusion:

Helium fusion:

Fusion of heavier elements:

Production of elements heavier than iron:

Hydrogen fusion (nuclear fusion of four protons to form a helium-4 nucleus) is the dominant process that generates energy in the cores of main-sequence stars. It is also called "hydrogen burning", which should not be confused with the chemical combustion of hydrogen in an oxidizing atmosphere. There are two predominant processes by which stellar hydrogen fusion occurs: proton-proton chain and the carbon-nitrogen-oxygen (CNO) cycle. Ninety percent of all stars, with the exception of white dwarfs, are fusing hydrogen by these two processes.

In the cores of lower-mass main-sequence stars such as the Sun, the dominant energy production process is the proton-proton chain reaction. This creates a helium-4 nucleus through a sequence of reactions that begin with the fusion of two protons to form a deuterium 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 toward the surface is sufficiently low and energy transfer from the core region remains by radiative heat transfer, 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, which is a catalytic cycle 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 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 energy transfer becomes more important than does radiative transfer. As a result, the core region becomes a convection zone, 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. The 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 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.

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 from a degenerate helium core, and the star moves to the horizontal branch where it burns helium in its core. More massive stars ignite helium in their core without a flash and execute a blue loop before reaching the asymptotic giant branch. Such a star initially moves away from the AGB toward bluer colours, then loops back again to what is called the Hayashi track. An important consequence of blue loops is that they give rise to classical Cepheid variables, of central importance in determining distances in the Milky Way 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 and oxygen. The most massive stars become supergiants when they leave the main sequence and quickly start helium fusion as they become red supergiants. After the helium is exhausted in the core of a star, helium fusion 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.

The reaction rate density between species A and B, having number densities n, is given by:

where k is the reaction rate constant of each single elementary binary reaction composing the nuclear fusion process:

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 , where is the de Broglie wavelength. Thus semi-classically the cross section is proportional to .

However, since the reaction involves quantum tunneling, there is an exponential damping at low energies that depends on Gamow factor E G, giving an Arrhenius equation:

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 and the relation:

where is the reduced mass.

Since this integration has an exponential damping at high energies of the form and at low energies from the Gamow factor, the integral almost vanished everywhere except around the peak, called Gamow peak,[36]: 185  at E ₀, where:

Thus:

The exponent can then be approximated around E ₀ as:

And the reaction rate is approximated as:

Values of S( E ₀) are typically , but are damped by a huge factor when involving a beta decay, due to the relation between the intermediate bound state (e.g. diproton) half-life and the beta decay half-life, as in the proton-proton chain reaction. Note that typical core temperatures in main-sequence stars give kT of the order of keV.

Thus, the limiting reaction in the CNO cycle, proton capture by , has S( E ₀) ~ S(0) = 3.5keV·b, while the limiting reaction in the proton-proton chain reaction, the creation of deuterium 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.

1. Hoyle, F. (1946). "The synthesis of the elements from hydrogen". Monthly Notices of the Royal Astronomical Society. 106 (5): 343-383. Bibcode:1946MNRAS.106..343H. doi:10.1093/mnras/106.5.343.

2. Hoyle, F. (1954). "On Nuclear Reactions Occurring in Very Hot STARS. I. The Synthesis of Elements from Carbon to Nickel". The Astrophysical Journal Supplement Series. 1: 121. Bibcode:1954ApJS....1..121H. doi:10.1086/190005.

3. Burbidge, E. M.; Burbidge, G. R.; Fowler, W.A.; Hoyle, F. (1957). "Synthesis of the Elements in Stars" (PDF). Reviews of Modern Physics. 29 (4): 547-650. Bibcode:1957RvMP...29..547B. doi:10.1103/RevModPhys.29.547.

4. ^ Clayton, D. D. (1968). Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press.

5. ^ Selle, D. (October 2012). "Why the Stars Shine" (PDF). Guidestar. Houston Astronomical Society. pp. 6-8. Archived (PDF) from the original on 2013-12-03.

6. Clayton, D. D. (1968). ^ Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. p. 365.

7. ^ Cameron, A. G. W. (1957). Stellar Evolution, Nuclear Astrophysics, and Nucleogenesis (PDF) (Report). Atomic Energy of Canada Limited. Report CRL-41.

8. ^ Clayton, D. D.; Fowler, W. A.; Hull, T. E.; Zimmerman, B. A. (1961). "Neutron capture chains in heavy element synthesis". Annals of Physics. 12 (3): 331-408. Bibcode:1961AnPhy..12..331C. doi:10.1016/0003-4916(61)90067-7.

9. Jones, Lauren V. (2009), Stars and galaxies, Greenwood guides to the universe, ABC-CLIO, pp. 65-67, ISBN 978-0-313-34075-8

10. Seeds, M. A., ^ Foundations of Astronomy (Belmont, CA: Wadsworth Publishing Company, 1986), p. 245.

11. ^ Böhm-Vitense, Erika (1992), Introduction to Stellar Astrophysics, vol. 3, Cambridge University Press, pp. 93-100, ISBN 978-0-521-34871-3

12. ^ Reiners, Ansgar; Basri, Gibor (March 2009). "On the magnetic topology of partially and fully convective stars". Astronomy and Astrophysics. 496 (3): 787-790. arXiv:. Bibcode:2009A&A...496..787R. doi:10.1051/0004-6361:200811450. S2CID 15159121.

13. de Loore, Camiel W. H.; Doom, C. (1992), Structure and evolution of single and binary stars, Astrophysics and space science library, vol. 179, Springer, pp. 200-214, ISBN 978-0-7923-1768-5

14. ^ Jeffrey, C. Simon (2010), Goswami, A.; Reddy, B. E. (eds.), "Principles and Perspectives in Cosmochemistry", Astrophysics and Space Science Proceedings, Springer, 16: 64-66, Bibcode:2010ASSP...16.....G, doi:10.1007/978-3-642-10352-0, ISBN 978-3-642-10368-1

15. ^ "Neutrinos yield first experimental evidence of catalyzed fusion dominant in many stars". phys.org.

16. ^ Choppin, G. R., Liljenzin, J.-O., Rydberg, J., & Ekberg, C., Radiochemistry and Nuclear Chemistry (Cambridge, MA: Academic Press, 2013), p. 357.

17. ^ Agostini, M.; Altenmüller, K.; Appel, S.; Atroshchenko, V.; Bagdasarian, Z.; Basilico, D.; Bellini, G.; Benziger, J.; Biondi, R.; Bravo, D.; Caccianiga, B. (25 November 2020). "Experimental evidence of neutrinos produced in the CNO fusion cycle in the Sun". Nature. 587 (7835): 577-582. arXiv:. Bibcode:2020Natur.587..577B. doi:10.1038/s41586-020-2934-0. ISSN 1476-4687. PMID 33239797. S2CID 227174644.

18. ^ Reid, I. Neill; Hawley, Suzanne L. (2005), New light on dark stars: red dwarfs, low-mass stars, brown dwarfs, Springer-Praxis books in astrophysics and astronomy (2nd ed.), Springer, p. 108, ISBN 978-3-540-25124-8 .

19. ^ Salaris, Maurizio; Cassisi, Santi (2005), Evolution of stars and stellar populations, John Wiley and Sons, pp. 119-123, ISBN 978-0-470-09220-0

20. ^ Schuler, S. C.; King, J. R.; The, L.-S. (2009), "Stellar Nucleosynthesis in the Hyades Open Cluster", The Astrophysical Journal, 701 (1): 837-849, arXiv:, Bibcode:2009ApJ...701..837S, doi:10.1088/0004-637X/701/1/837, S2CID 10626836

21. Karttunen, H., Kröger, P., Oja, H., Poutanen, M., & Donner, K. J., eds., ^ Fundamental Astronomy (Berlin/Heidelberg: Springer, 1987), p. 250.

22. ^ "University College London astrophysics course: lecture 7 - Stars" (PDF). Archived from the original (PDF) on January 15, 2017.

23. ^ Adelberger, Eric G.; Austin, Sam M.; Bahcall, John N.; Balantekin, A. B.; Bogaert, Gilles; Brown, Lowell S.; Buchmann, Lothar; Cecil, F. Edward; Champagne, Arthur E.; de Braeckeleer, Ludwig; Duba, Charles A. (1998-10-01). "Solar fusion cross sections". Reviews of Modern Physics. 70 (4): 1265-1291. arXiv:astro-ph/9805121. Bibcode:1998RvMP...70.1265A. doi:10.1103/RevModPhys.70.1265. ISSN 0034-6861. S2CID 16061677.

24. ^ Adelberger, E. G. (2011). "Solar fusion cross sections. II. Theppchain and CNO cycles". Reviews of Modern Physics. 83 (1): 195-245. arXiv:. Bibcode:2011RvMP...83..195A. doi:10.1103/RevModPhys.83.195. S2CID 119117147.

25. Goupil, M., Belkacem, K., Neiner, C., Lignières, F., & Green, J. J., eds., ^ Studying Stellar Rotation and Convection: Theoretical Background and Seismic Diagnostics (Berlin/Heidelberg: Springer, 2013), p. 211.

# data[].attributes.attachment_link.description
Stellar nucleosynthesis is the creation (nucleosynthesis) of chemical elements by nuclear fusion reactions within stars. Stellar nucleosynthesis has occurred since the original creation of hydrogen, helium and lithium during the Big Bang. As a predictive theory, it yields accurate estimates of the observed abundances of the elements.

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The Zoomed out timeline works as a table of contents

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The Zoomed out timeline works as a table of contents

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How to play: Use your arrow keys to move the tiles. When two tiles with can be fused ( check the fusion table below ) they fuse!

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This Source was used to generate pictures of an atomic model for use.

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Build an atom out of protons, neutrons, and electrons, and see how the element, charge, and mass change. Then play a game to test your ideas!

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Image taken from Wikimedia Commons, created by Viktor Hahn.

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Color: Blue

Wavelength: 470 nm

Formula

Where b is Wien's constant, W is wavelength, and T is temperature in kelvin

T = b/W

T = 2.8977719/470 nm

6165 K = 2.8977719/470 nm

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Elemental Fusion List (See Elemental Models)

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Elemental Fusion List (See Elemental Models)

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Hydrogen = N/A a single proton and electron make 1 hydrogen

Helium = Hydrogen 2 + Hydrogen

Li = Mostly originated from the big bang, Hydrogen + Helium would theoretically work but was unlikely

Beryllium = Helium + Helium

Carbon = Beryllium + Helium

Oxygen = Carbon + Helium

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using the dead material of other stars, new stars form beginning their fusion dance again.

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The Big Bang | Evidence | University of Western Australia

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The Big Bang | Evidence | University of Western Australia

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star emission spectrum (Example: Proxima Centauri)

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star emission spectrum (Example: Proxima Centauri)

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When a large enough cloud of gas forms, it will begin to condense into millions of stars rotating around a central point this rotation creates two arms from the initial sphere, afterwards the types of galaxies diverge, and they fall into SB and S types, where S types gain more arms as their mass is pulled inward, and how SB simply elongates its arms.

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Large amounts of space dust accumulate in an area

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Space dust begins to accumulate into dense bodies whose gravity pulls more mass in, eventually, the bodies will have all become one and reach stellar mass, This is a protostar. There is high pressure throughout the star due to gravity.

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Slowly an accretion disc begins to form around the protostar comprised of quick-moving gas that is channeled into a protostar's core, then the process of fusion is initiated as gasses are fused into higher weight matereal.

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As a stars gravity exceeds its ability to resist its gravity, the star will collapse into a supernova, this event is caused by the rapid pull from the main star and the force of resistance after this event, expelling all amounts of gas and forming heavy elements in the process (e.g. Iron 52, )

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Bing AI Image generator | Powered by Dall-E 3

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After detonation elements are sent flying across the universe, each one headed somewhere with a purpose, eventually your elements found this solar system maybe after millions of supernovae and gravity slingshots it has come to be you!

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Fission is the process by which an atom is split into lesser atoms, commonly found in stars as radioactive decay caused by unstable atoms.

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Iron 56 comes from Nickel 56, in the process of radioactive decay

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Hubble's Law states that objects moving away from us move at a rate scaled by their distance. This means that objects further away move away faster, this is used to justify universal expansion. This is proven by redshift which shows that the color of a star shifts red if it is moving away from us. Hubble's law formula is v = H*D, where V is the velocity, H is Hubbles constant, and D is the object's Proper Distance.

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An analogy for explaining Hubble's law, using raisins in a rising loaf of bread in place of galaxies. If a raisin is twice as far away from a place as another raisin, then the farther raisin would move away from that place twice as quickly.

Hubble's law, also known as the Hubble-Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible spectrum.

Hubble's law is considered the first observational basis for the expansion of the universe, and today it serves as one of the pieces of evidence most often cited in support of the Big Bang model. The motion of astronomical objects due solely to this expansion is known as the Hubble flow. It is described by the equation , with H ₀ the constant of proportionality-the Hubble constant-between the "proper distance" D to a galaxy, which can change over time, unlike the comoving distance, and its speed of separation v, i.e. the derivative of proper distance with respect to the cosmological time coordinate. (See Comoving and proper distances § Uses of the proper distance for discussion of the subtleties of this definition of velocity.)

The Hubble constant is most frequently quoted in ( km/ s)/ Mpc, thus giving the speed in km/s of a galaxy 1 megaparsec (3.09×10 ¹⁹ km) away, and its value is about 70 (km/s)/Mpc. However, crossing out units reveals that H ₀ is a unit of frequency (SI unit: ) and the reciprocal of H ₀ is known as the Hubble time. The Hubble constant can also be interpreted as the relative rate of expansion. In this form H ₀ = 7%/Gyr, meaning that at the current rate of expansion it takes a billion years for an unbound structure to grow by 7%.

Although widely attributed to Edwin Hubble, the notion of the universe expanding at a calculable rate was first derived from general relativity equations in 1922 by Alexander Friedmann. Friedmann published a set of equations, now known as the Friedmann equations, showing that the universe might be expanding, and presenting the expansion speed if that were the case. Then Georges Lemaître, in a 1927 article, independently derived that the universe might be expanding, observed the proportionality between recessional velocity of, and distance to, distant bodies, and suggested an estimated value for the proportionality constant; this constant, when Edwin Hubble confirmed the existence of cosmic expansion and determined a more accurate value for it two years later, came to be known by his name as the Hubble constant. Hubble inferred the recession velocity of the objects from their redshifts, many of which were earlier measured and related to velocity by Vesto Slipher in 1917. Though the Hubble constant H ₀ is constant at any given moment in time, the Hubble parameter H, of which the Hubble constant is the current value, varies with time, so the term constant is sometimes thought of as somewhat of a misnomer.

Three steps to the Hubble constant[18]

A decade before Hubble made his observations, a number of physicists and mathematicians had established a consistent theory of an expanding universe by using Einstein field equations of general relativity. Applying the most general principles to the nature of the universe yielded a dynamic solution that conflicted with the then-prevalent notion of a static universe.

In 1912, Vesto M. Slipher measured the first Doppler shift of a " spiral nebula" (the obsolete term for spiral galaxies) and soon discovered that almost all such nebulae were receding from Earth. He did not grasp the cosmological implications of this fact, and indeed at the time it was highly controversial whether or not these nebulae were "island universes" outside the Milky Way galaxy.

In 1922, Alexander Friedmann derived his Friedmann equations from Einstein field equations, showing that the universe might expand at a rate calculable by the equations. The parameter used by Friedmann is known today as the scale factor and can be considered as a scale invariant form of the proportionality constant of Hubble's law. Georges Lemaître independently found a similar solution in his 1927 paper discussed in the following section. The Friedmann equations are derived by inserting the metric for a homogeneous and isotropic universe into Einstein's field equations for a fluid with a given density and pressure. This idea of an expanding spacetime would eventually lead to the Big Bang and Steady State theories of cosmology.

In 1927, two years before Hubble published his own article, the Belgian priest and astronomer Georges Lemaître was the first to publish research deriving what is now known as Hubble's law. According to the Canadian astronomer Sidney van den Bergh, "the 1927 discovery of the expansion of the universe by Lemaître was published in French in a low-impact journal. In the 1931 high-impact English translation of this article, a critical equation was changed by omitting reference to what is now known as the Hubble constant." It is now known that the alterations in the translated paper were carried out by Lemaître himself.

Before the advent of modern cosmology, there was considerable talk about the size and shape of the universe. In 1920, the Shapley-Curtis debate took place between Harlow Shapley and Heber D. Curtis over this issue. Shapley argued for a small universe the size of the Milky Way galaxy, and Curtis argued that the universe was much larger. The issue was resolved in the coming decade with Hubble's improved observations.

Edwin Hubble did most of his professional astronomical observing work at Mount Wilson Observatory, home to the world's most powerful telescope at the time. His observations of Cepheid variable stars in "spiral nebulae" enabled him to calculate the distances to these objects. Surprisingly, these objects were discovered to be at distances which placed them well outside the Milky Way. They continued to be called nebulae, and it was only gradually that the term galaxies replaced it.

Fit of redshift velocities to Hubble's law.[25] Various estimates for the Hubble constant exist. The HST Key H ₀ Group fitted type Ia supernovae for redshifts between 0.01 and 0.1 to find that H ₀ = 71 ± 2 (statistical) ± 6 (systematic) km⋅s−1⋅Mpc−1,[26] while Sandage et al. find H ₀ = 62.3 ± 1.3 (statistical) ± 5 (systematic) km⋅s−1⋅Mpc−1.[27]

The parameters that appear in Hubble's law, velocities and distances, are not directly measured. In reality we determine, say, a supernova brightness, which provides information about its distance, and the redshift z = ∆ λ/ λ of its spectrum of radiation. Hubble correlated brightness and parameter z.

Combining his measurements of galaxy distances with Vesto Slipher and Milton Humason's measurements of the redshifts associated with the galaxies, Hubble discovered a rough proportionality between redshift of an object and its distance. Though there was considerable scatter (now known to be caused by peculiar velocities-the 'Hubble flow' is used to refer to the region of space far enough out that the recession velocity is larger than local peculiar velocities), Hubble was able to plot a trend line from the 46 galaxies he studied and obtain a value for the Hubble constant of 500 (km/s)/Mpc (much higher than the currently accepted value due to errors in his distance calibrations; see cosmic distance ladder for details).

Hubble diagram

Hubble's law can be easily depicted in a "Hubble diagram" in which the velocity (assumed approximately proportional to the redshift) of an object is plotted with respect to its distance from the observer. A straight line of positive slope on this diagram is the visual depiction of Hubble's law.

After Hubble's discovery was published, Albert Einstein abandoned his work on the cosmological constant, which he had designed to modify his equations of general relativity to allow them to produce a static solution, which he thought was the correct state of the universe. The Einstein equations in their simplest form model either an expanding or contracting universe, so Einstein's cosmological constant was artificially created to counter the expansion or contraction to get a perfect static and flat universe. After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "biggest mistake". On its own, general relativity could predict the expansion of the universe, which (through observations such as the bending of light by large masses, or the precession of the orbit of Mercury) could be experimentally observed and compared to his theoretical calculations using particular solutions of the equations he had originally formulated.

In 1931, Einstein went to Mount Wilson Observatory to thank Hubble for providing the observational basis for modern cosmology.

The cosmological constant has regained attention in recent decades as a hypothetical explanation for dark energy.

A variety of possible recessional velocity vs. redshift functions including the simple linear relation v = cz; a variety of possible shapes from theories related to general relativity; and a curve that does not permit speeds faster than light in accordance with special relativity. All curves are linear at low redshifts. See Davis and Lineweaver.[32]

The discovery of the linear relationship between redshift and distance, coupled with a supposed linear relation between recessional velocity and redshift, yields a straightforward mathematical expression for Hubble's law as follows:

where

is the recessional velocity, typically expressed in km/s.

H ₀ is Hubble's constant and corresponds to the value of (often termed the Hubble parameter which is a value that is time dependent and which can be expressed in terms of the scale factor) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the universe for a given comoving time.

is the proper distance (which can change over time, unlike the comoving distance, which is constant) from the galaxy to the observer, measured in mega parsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just v = dD/dt).

Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted and is not established except for small redshifts.

For distances D larger than the radius of the Hubble sphere r HS , objects recede at a rate faster than the speed of light ( See Uses of the proper distance for a discussion of the significance of this):

Since the Hubble "constant" is a constant only in space, not in time, the radius of the Hubble sphere may increase or decrease over various time intervals. The subscript '0' indicates the value of the Hubble constant today. Current evidence suggests that the expansion of the universe is accelerating ( see Accelerating universe), meaning that for any given galaxy, the recession velocity dD/dt is increasing over time as the galaxy moves to greater and greater distances; however, the Hubble parameter is actually thought to be decreasing with time, meaning that if we were to look at some fixed distance D and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.

Redshift can be measured by determining the wavelength of a known transition, such as hydrogen α-lines for distant quasars, and finding the fractional shift compared to a stationary reference. Thus, redshift is a quantity unambiguous for experimental observation. The relation of redshift to recessional velocity is another matter. For an extensive discussion, see Harrison.

Redshift velocity

The redshift z is often described as a redshift velocity, which is the recessional velocity that would produce the same redshift if it were caused by a linear Doppler effect (which, however, is not the case, as the shift is caused in part by a cosmological expansion of space, and because the velocities involved are too large to use a non-relativistic formula for Doppler shift). This redshift velocity can easily exceed the speed of light. In other words, to determine the redshift velocity v rs, the relation:

is used. That is, there is no fundamental difference between redshift velocity and redshift: they are rigidly proportional, and not related by any theoretical reasoning. The motivation behind the "redshift velocity" terminology is that the redshift velocity agrees with the velocity from a low-velocity simplification of the so-called Fizeau-Doppler formula.

Here, λ o, λ e are the observed and emitted wavelengths respectively. The "redshift velocity" v rs is not so simply related to real velocity at larger velocities, however, and this terminology leads to confusion if interpreted as a real velocity. Next, the connection between redshift or redshift velocity and recessional velocity is discussed. This discussion is based on Sartori.

Recessional velocity

Suppose R(t) is called the scale factor of the universe, and increases as the universe expands in a manner that depends upon the cosmological model selected. Its meaning is that all measured proper distances D(t) between co-moving points increase proportionally to R. (The co-moving points are not moving relative to each other except as a result of the expansion of space.) In other words:

where t ₀ is some reference time. If light is emitted from a galaxy at time t e and received by us at t ₀, it is redshifted due to the expansion of space, and this redshift z is simply:

Suppose a galaxy is at distance D, and this distance changes with time at a rate . We call this rate of recession the "recession velocity" v r:

We now define the Hubble constant as

and discover the Hubble law:

From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity contributed by the expansion of space and (ii) the distance to an object; the connection between redshift and distance is a crutch used to connect Hubble's law with observations. This law can be related to redshift z approximately by making a Taylor series expansion:

If the distance is not too large, all other complications of the model become small corrections, and the time interval is simply the distance divided by the speed of light:

or

According to this approach, the relation is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure.

Strictly speaking, neither v nor D in the formula are directly observable, because they are properties now of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.

For relatively nearby galaxies (redshift z much less than unity), v and D will not have changed much, and v can be estimated using the formula where c is the speed of light. This gives the empirical relation found by Hubble.

For distant galaxies, (1 + v (or D) cannot be calculated from z without specifying a detailed model for how H changes with time. The redshift is not even directly related to the recession velocity at the time the light set out, but it does have a simple interpretation: z) is the factor by which the universe has expanded while the photon was travelling towards the observer.

In using Hubble's law to determine distances, only the velocity due to the expansion of the universe can be used. Since gravitationally interacting galaxies move relative to each other independent of the expansion of the universe, these relative velocities, called peculiar velocities, need to be accounted for in the application of Hubble's law. Such peculiar velocities give rise to redshift-space distortions.

The parameter is commonly called the "Hubble constant", but that is a misnomer since it is constant in space only at a fixed time; it varies with time in nearly all cosmological models, and all observations of far distant objects are also observations into the distant past, when the "constant" had a different value. "Hubble parameter" is a more correct term, with denoting the present-day value.

Another common source of confusion is that the accelerating universe does not imply that the Hubble parameter is actually increasing with time; since , in most accelerating models increases relatively faster than , so H decreases with time. (The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times.)

On defining the dimensionless deceleration parameter , it follows that

From this it is seen that the Hubble parameter is decreasing with time, unless ; the latter can only occur if the universe contains phantom energy, regarded as theoretically somewhat improbable.

However, in the standard Lambda cold dark matter model (Lambda-CDM or ΛCDM model), will tend to −1 from above in the distant future as the cosmological constant becomes increasingly dominant over matter; this implies that will approach from above to a constant value of ≈ 57 (km/s)/Mpc, and the scale factor of the universe will then grow exponentially in time.

The mathematical derivation of an idealized Hubble's law for a uniformly expanding universe is a fairly elementary theorem of geometry in 3-dimensional Cartesian/Newtonian coordinate space, which, considered as a metric space, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply stated, the theorem is this:

Any two points which are moving away from the origin, each along straight lines and with speed proportional to distance from the origin, will be moving away from each other with a speed proportional to their distance apart.

In fact, this applies to non-Cartesian spaces as long as they are locally homogeneous and isotropic, specifically to the negatively and positively curved spaces frequently considered as cosmological models (see shape of the universe).

An observation stemming from this theorem is that seeing objects recede from us on Earth is not an indication that Earth is near to a center from which the expansion is occurring, but rather that every observer in an expanding universe will see objects receding from them.

The age and ultimate fate of the universe can be determined by measuring the Hubble constant today and extrapolating with the observed value of the deceleration parameter, uniquely characterized by values of density parameters (ΩM for matter and ΩΛ for dark energy). A "closed universe" with ΩM > 1 and ΩΛ = 0 comes to an end in a Big Crunch and is considerably younger than its Hubble age. An "open universe" with ΩM ≤ 1 and ΩΛ = 0 expands forever and has an age that is closer to its Hubble age. For the accelerating universe with nonzero ΩΛ that we inhabit, the age of the universe is coincidentally very close to the Hubble age.

The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called deceleration parameter , which is defined by

In a universe with a deceleration parameter equal to zero, it follows that H = 1/ t, where t is the time since the Big Bang. A non-zero, time-dependent value of simply requires integration of the Friedmann equations backwards from the present time to the time when the comoving horizon size was zero.

It was long thought that q was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the universe less than 1/ H (which is about 14 billion years). For instance, a value for q of 1/2 (once favoured by most theorists) would give the age of the universe as 2/(3 H). The discovery in 1998 that q is apparently negative means that the universe could actually be older than 1/ H. However, estimates of the age of the universe are very close to 1/ H.

The expansion of space summarized by the Big Bang interpretation of Hubble's law is relevant to the old conundrum known as Olbers' paradox: If the universe were infinite in size, static, and filled with a uniform distribution of stars, then every line of sight in the sky would end on a star, and the sky would be as bright as the surface of a star. However, the night sky is largely dark.

Since the 17th century, astronomers and other thinkers have proposed many possible ways to resolve this paradox, but the currently accepted resolution depends in part on the Big Bang theory, and in part on the Hubble expansion: in a universe that existed for a finite amount of time, only the light of a finite number of stars has had enough time to reach us, and the paradox is resolved. Additionally, in an expanding universe, distant objects recede from us, which causes the light emanated from them to be redshifted and diminished in brightness by the time we see it.

Instead of working with Hubble's constant, a common practice is to introduce the dimensionless Hubble constant, usually denoted by h and commonly referred to as "little h", then to write Hubble's constant H ₀ as h × 100 km⋅ s−1⋅ Mpc−1, all the relative uncertainty of the true value of H ₀ being then relegated to h. The dimensionless Hubble constant is often used when giving distances that are calculated from redshift z using the formula . Since H ₀ is not precisely known, the distance is expressed as:

In other words, one calculates 2998×z and one gives the units as or

Occasionally a reference value other than 100 may be chosen, in which case a subscript is presented after h to avoid confusion; e.g. h ₇₀ denotes , which implies .

This should not be confused with the dimensionless value of Hubble's constant, usually expressed in terms of Planck units, obtained by multiplying H ₀ by 1.75 × 10 −63 (from definitions of parsec and ), for example for H ₀ = 70, a Planck unit version of 1.2 × 10 −61 is obtained.

A value for measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating"[46] (although the Hubble factor is still decreasing with time, as mentioned above in the Interpretation section; see the articles on dark energy and the ΛCDM model).

Start with the Friedmann equation:

where is the Hubble parameter, is the scale factor, G is the gravitational constant, is the normalised spatial curvature of the universe and equal to −1, 0, or 1, and is the cosmological constant.

If the universe is matter-dominated, then the mass density of the universe can be taken to include just matter so

where is the density of matter today. From the Friedmann equation and thermodynamic principles we know for non-relativistic particles that their mass density decreases proportional to the inverse volume of the universe, so the equation above must be true. We can also define (see density parameter for )

therefore:

Also, by definition,

where the subscript 0 refers to the values today, and . Substituting all of this into the Friedmann equation at the start of this section and replacing with gives

If the universe is both matter-dominated and dark energy-dominated, then the above equation for the Hubble parameter will also be a function of the equation of state of dark energy. So now:

where is the mass density of the dark energy. By definition, an equation of state in cosmology is , and if this is substituted into the fluid equation, which describes how the mass density of the universe evolves with time, then

If w is constant, then

implying:

Therefore, for dark energy with a constant equation of state w, . If this is substituted into the Friedman equation in a similar way as before, but this time set , which assumes a spatially flat universe, then (see shape of the universe)

If the dark energy derives from a cosmological constant such as that introduced by Einstein, it can be shown that . The equation then reduces to the last equation in the matter-dominated universe section, with set to zero. In that case the initial dark energy density is given by[47]

and

If dark energy does not have a constant equation-of-state w, then

and to solve this, must be parametrized, for example if , giving[48]

The Hubble constant has units of inverse time; the Hubble time t H is simply defined as the inverse of the Hubble constant,[52] i.e.

This is slightly different from the age of the universe, which is approximately 13.8 billion years. The Hubble time is the age it would have had if the expansion had been linear, and it is different from the real age of the universe because the expansion is not linear; it depends on the energy content of the universe (see § Derivation of the Hubble parameter).

We currently appear to be approaching a period where the expansion of the universe is exponential due to the increasing dominance of vacuum energy. In this regime, the Hubble parameter is constant, and the universe grows by a factor e each Hubble time:

Likewise, the generally accepted value of 2.27 Es−1 means that (at the current rate) the universe would grow by a factor of in one exasecond.

Over long periods of time, the dynamics are complicated by general relativity, dark energy, inflation, etc., as explained above.

The Hubble length or Hubble distance is a unit of distance in cosmology, defined as - the speed of light multiplied by the Hubble time. It is equivalent to 4,420 million parsecs or 14.4 billion light years. (The numerical value of the Hubble length in light years is, by definition, equal to that of the Hubble time in years.) The Hubble distance would be the distance between the Earth and the galaxies which are currently receding from us at the speed of light, as can be seen by substituting into the equation for Hubble's law, v = H ₀ D.

The Hubble volume is sometimes defined as a volume of the universe with a comoving size of The exact definition varies: it is sometimes defined as the volume of a sphere with radius or alternatively, a cube of side Some cosmologists even use the term Hubble volume to refer to the volume of the observable universe, although this has a radius approximately three times larger.

Value of the Hubble constant, including measurement uncertainty for recent surveys[53]

The value of the Hubble constant is estimated by measuring the redshift of distant galaxies and then determining the distances to them by some other method than Hubble's law. This approach forms part of the cosmic distance ladder for measuring distances to extragalactic objects. Uncertainties in the physical assumptions used to determine these distances have caused varying estimates of the Hubble constant.

Multiple methods have been used to determine the Hubble constant. "Late universe" measurements using calibrated distance ladder techniques have converged on a value of approximately 73 (km/s)/Mpc. Since 2000, "early universe" techniques based on measurements of the cosmic microwave background have become available, and these agree on a value near 67.7 (km/s)/Mpc. (This is accounting for the change in the expansion rate since the early universe, so is comparable to the first number.) As techniques have improved, the estimated measurement uncertainties have shrunk, but the range of measured values has not, to the point that the disagreement is now highly statistically significant. This discrepancy is called the Hubble tension.

The cause of the Hubble tension discrepancy is unknown, and there are many possible proposed solutions. The most conservative is that there is an unknown systematic error affecting either early-universe or late-universe observations. Although intuitively appealing, this explanation requires multiple unrelated effects regardless of whether early-universe or late-universe observations are incorrect, and there are no obvious candidates. Furthermore, any such systematic error would need to affect multiple different instruments, since both the early-universe and late-universe observations come from different telescopes.

Alternatively, it could be that the observations are correct, but some unaccounted-for effect is causing the discrepancy. If the cosmological principle fails (see " Violations of the cosmological principle" in the " Lambda-CDM model" article), then the existing interpretations of the Hubble constant and the Hubble tension have to be revised, which might resolve the Hubble tension. In particular, we would need to be located within a very large void, up to about a redshift of 0.5, for such an explanation to not be in tension with supernovae & baryon acoustic oscillation observations. Yet another possibility is that the uncertainties in the measurements could have been underestimated.

The most exciting possibility is new physics beyond the ΛCDM model. There are very many theories in this category, e.g. replacing general relativity with modified Newtonian dynamics could potentially resolve the tension, as can a dark energy component in the early universe, dark energy with a time-varying equation of state, or dark matter that decays into dark radiation. A problem faced by all these theories is that both early-universe and late-universe measurements rely on multiple independent lines of physics, and it is difficult to modify any of those lines while preserving their successes elsewhere. The scale of the challenge can be seen from how some authors have argued that new early-universe physics alone is not sufficient; while other authors argue that new late-universe physics alone is also not sufficient. Nonetheless, astronomers are trying, with interest in the Hubble tension growing strongly in the past several years.

The observations of astronomer Walter Baade led him to define distinct " populations" for stars (Population I and Population II). The same observations led him to discover that there are two types of Cepheid variable stars. Using this discovery, he recalculated the size of the known universe, doubling the previous calculation made by Hubble in 1929. He announced this finding to considerable astonishment at the 1952 meeting of the International Astronomical Union in Rome.

For most of the second half of the 20th century, the value of was estimated to be between 50 and 90 (km/s)/Mpc.

The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs, who claimed the value was around 100, and Allan Sandage, who claimed the value was near 50. In one demonstration of vitriol shared between the parties, when Sandage and Gustav Andreas Tammann (Sandage's research colleague) formally acknowledged the shortcomings of confirming the systematic error of their method in 1975, Vaucouleurs responded "It is unfortunate that this sober warning was so soon forgotten and ignored by most astronomers and textbook writers". In 1996, a debate moderated by John Bahcall between Sidney van den Bergh and Gustav Tammann was held in similar fashion to the earlier Shapley-Curtis debate over these two competing values.

This previously wide variance in estimates was partially resolved with the introduction of the ΛCDM model of the universe in the late 1990s. With the ΛCDM model observations of high-redshift clusters at X-ray and microwave wavelengths using the Sunyaev-Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a value of around 70 for the constant.[citation needed ]

Landscape of H0 measurements around 2021 with Planck (2018) and SH0ES (2020) values highlighted in pink and cyan respectively.[55]

More recent measurements from the Planck mission published in 2018 indicate a lower value of 67.66±0.42 (km/s)/Mpc, although, even more recently, in March 2019, a higher value of 74.03±1.42 (km/s)/Mpc has been determined using an improved procedure involving the Hubble Space Telescope. The two measurements disagree at the 4.4σ level, beyond a plausible level of chance. The resolution to this disagreement is an ongoing area of active research.

In October 2018, scientists presented a new third way (two earlier methods, one based on redshifts and another on the cosmic distance ladder, gave results that do not agree), using information from gravitational wave events (especially those involving the merger of neutron stars, like GW170817), of determining the Hubble constant.

In July 2019, astronomers reported that a new method to determine the Hubble constant, and resolve the discrepancy of earlier methods, has been proposed based on the mergers of pairs of neutron stars, following the detection of the neutron star merger of GW170817, an event known as a dark siren. Their measurement of the Hubble constant is (km/s)/Mpc.

Also in July 2019, astronomers reported another new method, using data from the Hubble Space Telescope and based on distances to red giant stars calculated using the tip of the red-giant branch (TRGB) distance indicator. Their measurement of the Hubble constant is (km/s)/Mpc.

In February 2020, the Megamaser Cosmology Project published independent results that confirmed the distance ladder results and differed from the early-universe results at a statistical significance level of 95%. In July 2020, measurements of the cosmic background radiation by the Atacama Cosmology Telescope predict that the Universe should be expanding more slowly than is currently observed.

Estimated values of the Hubble constant, 2001-2020. Estimates in black represent calibrated distance ladder measurements which tend to cluster around 73 (km/s)/Mpc; red represents early universe CMB/BAO measurements with ΛCDM parameters which show good agreement on a figure near 67 (km/s)/Mpc, while blue are other techniques, whose uncertainties are not yet small enough to decide between the two.

Date published

Hubble constant
(km/s)/Mpc

Observer

Citation

Remarks / methodology

2023-07-13

SPT-3G

[86]

CMB TT/TE/EE power spectrum. Less than 1 σ discrepancy with Planck.

2022-12-14

S. Contarini et al.

[88]

Statistics of cosmic voids using BOSS DR12 data set (Preprint).[89]

2021-12-08

SH0ES

[92]

Cepheids- SN Ia distance ladder (HST+ Gaia EDR3+"Pantheon+"). 5 σ discrepancy with planck.

2021-09-17

W. Freedman

[93]

Tip of the red-giant branch (TRGB) distance indicator (HST+Gaia EDR3)

2020-12-16

Hubble Space Telescope and Gaia EDR3

[94]

Combining earlier work on red giant stars, using the tip of the red-giant branch (TRGB) distance indicator, with parallax measurements of Omega Centauri from Gaia EDR3.

2020-12-15

Hubble Space Telescope and Gaia EDR3

[95]

Combination of HST photometry and Gaia EDR3 parallaxes for Milky Way Cepheids, reducing the uncertainty in calibration of Cepheid luminosities to 1.0%. Overall uncertainty in the value for is 1.8%, which is expected to be reduced to 1.3% with a larger sample of type Ia supernovae in galaxies that are known Cepheid hosts. Continuation of a collaboration known as Supernovae, , for the Equation of State of Dark Energy (SHoES).

2020-12-04

E. J. Baxter, B. D. Sherwin

[96]

Gravitational lensing in the CMB is used to estimate without referring to the sound horizon scale, providing an alternative method to analyze the Planck data.

2020-11-25

P. Denzel et al.

[97]

Eight quadruply lensed galaxy systems are used to determine to a precision of 5%, in agreement with both "early" and "late" universe estimates. Independent of distance ladders and the cosmic microwave background.

2020-11-07

T. Sedgwick et al.

[98]

Derived from 88 0.02 < < 0.05 Type Ia supernovae used as standard candle distance indicators. The estimate is corrected for the effects of peculiar velocities in the supernova environments, as estimated from the galaxy density field. The result assumes Ω m = 0.3, Ω Λ = 0.7 and a sound horizon of 149.3 Mpc, a value taken from Anderson et al. (2014).[99]

2020-09-29

S. Mukherjee et al.

[100]

Gravitational waves, assuming that the transient ZTF19abanrh found by the Zwicky Transient Facility is the optical counterpart to GW190521. Independent of distance ladders and the cosmic microwave background.

2019-08-20

K. Dutta et al.

[104]

This is obtained analysing low-redshift cosmological data within ΛCDM model. The datasets used are type-Ia supernovae, baryon acoustic oscillations, time-delay measurements using strong-lensing, measurements using cosmic chronometers and growth measurements from large scale structure observations.

2019-07-16

Hubble Space Telescope

[81][82][83]

Distances to red giant stars are calculated using the tip of the red-giant branch (TRGB) distance indicator.

2019-07-08

The LIGO Scientific Collaboration and The Virgo Collaboration

[80]

Uses radio counterpart of GW170817, combined with earlier gravitational wave (GW) and electromagnetic (EM) data.

2019-03-18

Hubble Space Telescope

[74]

Precision HST photometry of Cepheids in the Large Magellanic Cloud (LMC) reduce the uncertainty in the distance to the LMC from 2.5% to 1.3%. The revision increases the tension with CMB measurements to the 4.4 σ level (P=99.999% for Gaussian errors), raising the discrepancy beyond a plausible level of chance. Continuation of a collaboration known as Supernovae, , for the Equation of State of Dark Energy (SHoES).

2018-11-06

Dark Energy Survey

[109]

Supernova measurements using the inverse distance ladder method based on baryon acoustic oscillations.

2018-04-27

Hubble Space Telescope and Gaia

[112][113]

Additional HST photometry of galactic Cepheids with early Gaia parallax measurements. The revised value increases tension with CMB measurements at the 3.8 σ level. Continuation of the SHoES collaboration.

2018-02-22

Hubble Space Telescope

[114][115]

Parallax measurements of galactic Cepheids for enhanced calibration of the distance ladder; the value suggests a discrepancy with CMB measurements at the 3.7 σ level. The uncertainty is expected to be reduced to below 1% with the final release of the Gaia catalog. SHoES collaboration.

2017-10-16

The LIGO Scientific Collaboration and The Virgo Collaboration

[116]

Standard siren measurement independent of normal "standard candle" techniques; the gravitational wave analysis of a binary neutron star (BNS) merger GW170817 directly estimated the luminosity distance out to cosmological scales. An estimate of fifty similar detections in the next decade may arbitrate tension of other methodologies.[117] Detection and analysis of a neutron star-black hole merger (NSBH) may provide greater precision than BNS could allow.[118]

2016-11-22

Hubble Space Telescope

[119]

Uses time delays between multiple images of distant variable sources produced by strong gravitational lensing. Collaboration known as Lenses in COSMOGRAIL's Wellspring (H0LiCOW).

2016-08-04

Cosmicflows-3

[120]

Comparing redshift to other distance methods, including Tully-Fisher, Cepheid variable, and Type Ia supernovae. A restrictive estimate from the data implies a more precise value of 75±2.

2016-07-13

SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS)

[121]

Baryon acoustic oscillations. An extended survey (eBOSS) began in 2014 and is expected to run through 2020. The extended survey is designed to explore the time when the universe was transitioning away from the deceleration effects of gravity from 3 to 8 billion years after the Big Bang.[122]

2015-02

Planck Mission

[124][125]

Results from an analysis of Planck's full mission were made public on 1 December 2014 at a conference in Ferrara, Italy. A full set of papers detailing the mission results were released in February 2015.

2013-03-21

Planck Mission

[53][127][128][129][130]

The ESA Planck Surveyor was launched in May 2009. Over a four-year period, it performed a significantly more detailed investigation of cosmic microwave radiation than earlier investigations using HEMT radiometers and bolometer technology to measure the CMB at a smaller scale than WMAP. On 21 March 2013, the European-led research team behind the Planck cosmology probe released the mission's data including a new CMB all-sky map and their determination of the Hubble constant.

2010

WMAP (7 years), combined with other measurements

[132]

These values arise from fitting a combination of WMAP and other cosmological data to the simplest version of the ΛCDM model. If the data are fit with more general versions, H ₀ tends to be smaller and more uncertain: typically around 67±4 (km/s)/Mpc although some models allow values near 63 (km/s)/Mpc.[133]

2006-08

Chandra X-ray Observatory

[136]

Combined Sunyaev-Zeldovich effect and Chandra X-ray observations of galaxy clusters. Adjusted uncertainty in table from Planck Collaboration 2013.[137]

2001-05

Hubble Space Telescope Key Project

[26]

This project established the most precise optical determination, consistent with a measurement of H ₀ based upon Sunyaev-Zel'dovich effect observations of many galaxy clusters having a similar accuracy.

before 1996

50 - 90 (est.)

[71]

mid 1970's

Gérard de Vaucouleurs

[72]

De Vaucouleurs believed he had improved the accuracy of Hubble's constant from Sandage's because he used 5x more primary indicators, 10x more calibration methods, 2x more secondary indicators, and 3x as many galaxy data points to derive his 100 ± 10.

early 1970s

55 (est.)

Allan Sandage and Gustav Tammann

[140]

1958

75 (est.)

Allan Sandage

[141]

This was the first good estimate of H ₀, but it would be decades before a consensus was achieved.

1956

180

Humason, Mayall and Sandage

[140]

1929

500

Edwin Hubble, Hooker telescope

[142][140][143]

1927

625

Georges Lemaître

[144]

First measurement and interpretation as a sign of the expansion of the universe.

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Hubble's law, also known as the Hubble-Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth.

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Note: As this assignment requires 4 elements I have included the first 4 elements from the silicon-burning cycle found in supernovae that are named as such due to the high silicon and sulfur contents after the oxygen-burning cycle. Y is an extra photon emitted primarily as gamma rays.

28Si+4He⇌32S+γ

32S+4He⇌36Ar+γ

36Ar+4He⇌40Ca+γ

40Ca+4He⇌44Ti+γ

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Supernova nucleosynthesis is the nucleosynthesis of chemical elements in supernova explosions.

In sufficiently massive stars, the nucleosynthesis by fusion of lighter elements into heavier ones occurs during sequential hydrostatic burning processes called helium burning, carbon burning, oxygen burning, and silicon burning, in which the byproducts of one nuclear fuel become, after compressional heating, the fuel for the subsequent burning stage. In this context, the word "burning" refers to nuclear fusion and not a chemical reaction.

During hydrostatic burning these fuels synthesize overwhelmingly the alpha nuclides (), nuclei composed of integer numbers of helium-4 nuclei. Initially, two helium-4 nuclei fuse into a single beryllium-8 nucleus. The addition of another helium 4 nucleus to the beryllium yields carbon-12, followed by oxygen-16, neon-20 and so on, each time adding 2 protons and 2 neutrons to the growing nucleus. A rapid final explosive burning is caused by the sudden temperature spike owing to passage of the radially moving shock wave that was launched by the gravitational collapse of the core. W. D. Arnett and his Rice University colleagues demonstrated that the final shock burning would synthesize the non-alpha-nucleus isotopes more effectively than hydrostatic burning was able to do, suggesting that the expected shock-wave nucleosynthesis is an essential component of supernova nucleosynthesis. Together, shock-wave nucleosynthesis and hydrostatic-burning processes create most of the isotopes of the elements carbon (), oxygen (), and elements with (from neon to nickel). As a result of the ejection of the newly synthesized isotopes of the chemical elements by supernova explosions, their abundances steadily increased within interstellar gas. That increase became evident to astronomers from the initial abundances in newly born stars exceeding those in earlier-born stars.

Elements heavier than nickel are comparatively rare owing to the decline with atomic weight of their nuclear binding energies per nucleon, but they too are created in part within supernovae. Of greatest interest historically has been their synthesis by rapid capture of neutrons during the r-process, reflecting the common belief that supernova cores are likely to provide the necessary conditions. However, newer research has proposed a promising alternative (see the r-process below). The r-process isotopes are approximately 100,000 times less abundant than the primary chemical elements fused in supernova shells above. Furthermore, other nucleosynthesis processes in supernovae are thought to be responsible also for some nucleosynthesis of other heavy elements, notably, the proton capture process known as the rp-process, the slow capture of neutrons (s-process) in the helium-burning shells and in the carbon-burning shells of massive stars, and a photodisintegration process known as the (gamma-process). The latter synthesizes the lightest, most neutron-poor, isotopes of the elements heavier than iron from preexisting heavier isotopes.

In 1946, Fred Hoyle proposed that elements heavier than hydrogen and helium would be produced by nucleosynthesis in the cores of massive stars. It had previously been thought that the elements we see in the modern universe had been largely produced during its formation. At this time, the nature of supernovae was unclear and Hoyle suggested that these heavy elements were distributed into space by rotational instability. In 1954, the theory of nucleosynthesis of heavy elements in massive stars was refined and combined with more understanding of supernovae to calculate the abundances of the elements from carbon to nickel. Key elements of the theory included:

the prediction of the excited state in the ¹²C nucleus that enables the triple-alpha process to burn resonantly to carbon and oxygen;

the thermonuclear sequels of carbon-burning synthesizing Ne, Mg and Na; and

oxygen-burning synthesizing silicon, aluminum, and sulphur.

The theory predicted that silicon burning would happen as the final stage of core fusion in massive stars, although nuclear science could not then calculate exactly how. Hoyle also predicted that the collapse of the evolved cores of massive stars was "inevitable" owing to their increasing rate of energy loss by neutrinos and that the resulting explosions would produce further nucleosynthesis of heavy elements and eject them into space.

In 1957, a paper by the authors E. M. Burbidge, G. R. Burbidge, W. A. Fowler, and Hoyle expanded and refined the theory and achieved widespread acclaim. It became known as the B²FH or BBFH paper, after the initials of its authors. The earlier papers fell into obscurity for decades after the more-famous B²FH paper did not attribute Hoyle's original description of nucleosynthesis in massive stars. Donald D. Clayton has attributed the obscurity also to Hoyle's 1954 paper describing its key equation only in words, and a lack of careful review by Hoyle of the B²FH draft by coauthors who had themselves not adequately studied Hoyle's paper. During his 1955 discussions in Cambridge with his co-authors in preparation of the B²FH first draft in 1956 in Pasadena, Hoyle's modesty had inhibited him from emphasizing to them the great achievements of his 1954 theory.

Thirteen years after the B²FH paper, W.D. Arnett and colleagues demonstrated that the final burning in the passing shock wave launched by collapse of the core could synthesize non-alpha-particle isotopes more effectively than hydrostatic burning could, suggesting that explosive nucleosynthesis is an essential component of supernova nucleosynthesis. A shock wave rebounded from matter collapsing onto the dense core, if strong enough to lead to mass ejection of the mantle of supernovae, would necessarily be strong enough to provide the sudden heating of the shells of massive stars needed for explosive thermonuclear burning within the mantle. Understanding how that shock wave can reach the mantle in the face of continuing infall onto the shock became the theoretical difficulty. Supernova observations assured that it must occur.

White dwarfs were proposed as possible progenitors of certain supernovae in the late 1960s, although a good understanding of the mechanism and nucleosynthesis involved did not develop until the 1980s. This showed that type Ia supernovae ejected very large amounts of radioactive nickel and lesser amounts of other iron-peak elements, with the nickel decaying rapidly to cobalt and then iron.

The papers of Hoyle (1946) and Hoyle (1954) and of B²FH (1957) were written by those scientists before the advent of the age of computers. They relied on hand calculations, deep thought, physical intuition, and familiarity with details of nuclear physics. Brilliant as these founding papers were, a cultural disconnect soon emerged with a younger generation of scientists who began to construct computer programs that would eventually yield numerical answers for the advanced evolution of stars and the nucleosynthesis within them.

A supernova is a violent explosion of a star that occurs under two principal scenarios. The first is that a white dwarf star, which is the remnant of a low-mass star that has exhausted its nuclear fuel, undergoes a thermonuclear explosion after its mass is increased beyond its Chandrasekhar limit by accreting nuclear-fuel mass from a more diffuse companion star (usually a red giant) with which it is in binary orbit. The resulting runaway nucleosynthesis completely destroys the star and ejects its mass into space. The second, and about threefold more common, scenario occurs when a massive star (12-35 times more massive than the sun), usually a supergiant at the critical time, reaches nickel-56 in its core nuclear fusion (or burning) processes. Without exothermic energy from fusion, the core of the pre-supernova massive star loses heat needed for pressure support, and collapses owing to the strong gravitational pull. The energy transfer from the core collapse causes the supernova display.

The nickel-56 isotope has one of the largest binding energies per nucleon of all isotopes, and is therefore the last isotope whose synthesis during core silicon burning releases energy by nuclear fusion, exothermically. The binding energy per nucleon declines for atomic weights heavier than , ending fusion's history of supplying thermal energy to the star. The thermal energy released when the infalling supernova mantle hits the semi-solid core is very large, about 10 ⁵³ ergs, about a hundred times the energy released by the supernova as the kinetic energy of its ejected mass. Dozens of research papers have been published in the attempt to describe the hydrodynamics of how that small one percent of the infalling energy is transmitted to the overlying mantle in the face of continuous infall onto the core. That uncertainty remains in the full description of core-collapse supernovae.[citation needed ]

Nuclear fusion reactions that produce elements heavier than iron absorb nuclear energy and are said to be endothermic reactions. When such reactions dominate, the internal temperature that supports the star's outer layers drops. Because the outer envelope is no longer sufficiently supported by the radiation pressure, the star's gravity pulls its mantle rapidly inward. As the star collapses, this mantle collides violently with the growing incompressible stellar core, which has a density almost as great as an atomic nucleus, producing a shockwave that rebounds outward through the unfused material of the outer shell. The increase of temperature by the passage of that shockwave is sufficient to induce fusion in that material, often called explosive nucleosynthesis. The energy deposited by the shockwave somehow leads to the star's explosion, dispersing fusing matter in the mantle above the core into interstellar space.

After a star completes the 230-300 oxygen burning process, its core is composed primarily of silicon and sulfur. If it has sufficiently high mass, it further contracts until its core reaches temperatures in the range of 2.7-3.5 billion K ( keV). At these temperatures, silicon and other isotopes suffer photoejection of nucleons by energetic thermal photons (γ) ejecting especially alpha particles ( ⁴ He). The nuclear process of silicon burning differs from earlier fusion stages of nucleosynthesis in that it entails a balance between alpha-particle captures and their inverse photo ejection which establishes abundances of all alpha-particle elements in the following sequence in which each alpha particle capture shown is opposed by its inverse reaction, namely, photo ejection of an alpha particle by the abundant thermal photons:

The alpha-particle nuclei ⁴⁴ Ti and those more massive in the final five reactions listed are all radioactive, but they decay after their ejection in supernova explosions into abundant isotopes of Ca, Ti, Cr, Fe and Ni. This post-supernova radioactivity became of great importance for the emergence of gamma-ray-line astronomy.

In these physical circumstances of rapid opposing reactions, namely alpha-particle capture and photo ejection of alpha particles, the abundances are not determined by alpha-particle-capture cross sections; rather they are determined by the values that the abundances must assume in order to balance the speeds of the rapid opposing-reaction currents. Each abundance takes on a stationary value that achieves that balance. This picture is called nuclear quasiequilibrium. Many computer calculations, for example, using the numerical rates of each reaction and of their reverse reactions have demonstrated that quasiequilibrium is not exact but does characterize well the computed abundances. Thus, the quasiequilibrium picture presents a comprehensible picture of what actually happens. It also fills in an uncertainty in Hoyle's 1954 theory. The quasiequilibrium buildup shuts off after ⁵⁶ Ni because the alpha-particle captures become slower whereas the photo ejections from heavier nuclei become faster. Non-alpha-particle nuclei also participate, using a host of reactions similar to

and its inverse which set the stationary abundances of the non-alpha-particle isotopes, where the free densities of protons and neutrons are also established by the quasiequilibrium. However, the abundance of free neutrons is also proportional to the excess of neutrons over protons in the composition of the massive star; therefore the abundance of ³⁷Ar, using it as an example, is greater in ejecta from recent massive stars than it was from those in early stars of only H and He; therefore ³⁷Cl, to which ³⁷ Ar decays after the nucleosynthesis, is called a "secondary isotope".

In interest of brevity, the next stage, an intricate photo-disintegration rearrangement, and the nuclear quasiequilibrium that it achieves, are referred to as silicon burning. The silicon burning in the star progresses through a temporal sequence of such nuclear quasiequilibria in which the abundance of ²⁸Si slowly declines and that of ⁵⁶Ni slowly increases. This amounts to a nuclear abundance change 2 ²⁸Si ≫ ⁵⁶Ni, which may be thought of as silicon burning into nickel ("burning" in the nuclear sense). The entire silicon-burning sequence lasts about one day in the core of a contracting massive star and stops after ⁵⁶ Ni has become the dominant abundance. The final explosive burning caused when the supernova shock passes through the silicon-burning shell lasts only seconds, but its roughly 50% increase in the temperature causes furious nuclear burning, which becomes the major contributor to nucleosynthesis in the mass range 28-60 .

After the final ⁵⁶Ni stage, the star can no longer release energy via nuclear fusion, because a nucleus with 56 nucleons has the lowest mass per nucleon of all the elements in the sequence. The next step up in the alpha-particle chain would be ⁶⁰Zn. However ⁶⁰Zn has slightly more mass per nucleon than ⁵⁶Ni, and thus would require a thermodynamic energy loss rather than a gain as happened in all prior stages of nuclear burning.

⁵⁶Ni (which has 28 protons) has a half-life of 6.02 days and decays via to ⁵⁶Co (27 protons), which in turn has a half-life of 77.3 days as it decays to ⁵⁶Fe (26 protons). However, only minutes are available for the ⁵⁶ Ni to decay within the core of a massive star.

This establishes ⁵⁶Ni as the most abundant of the radioactive nuclei created in this way. Its radioactivity energizes the late supernova light curve and creates the pathbreaking opportunity for gamma-ray-line astronomy. See SN 1987A light curve for the aftermath of that opportunity.

Clayton and Meyer have recently generalized this process still further by what they have named the secondary supernova machine, attributing the increasing radioactivity that energizes late supernova displays to the storage of increasing Coulomb energy within the quasiequilibrium nuclei called out above as the quasiequilibria shift from primarily ²⁸Si to primarily ⁵⁶ Ni. The visible displays are powered by the decay of that excess Coulomb energy.

During this phase of the core contraction, the potential energy of gravitational compression heats the interior to roughly three billion kelvins, which briefly maintains pressure support and opposes rapid core contraction. However, since no additional heat energy can be generated via new fusion reactions, the final unopposed contraction rapidly accelerates into a collapse lasting only a few seconds. At that point, the central portion of the star is crushed into either a neutron star or, if the star is massive enough, into a black hole.

The outer layers of the star are blown off in an explosion triggered by the outward moving supernova shock, known as a Type II supernova whose displays last days to months. The escaping portion of the supernova core may initially contain a large density of free neutrons, which may synthesize, in about one second while inside the star, roughly half of the elements in the universe that are heavier than iron via a rapid neutron-capture mechanism known as the r-process. See below.

Composite image of Kepler's supernova from pictures by the Spitzer Space Telescope, Hubble Space Telescope, and Chandra X-ray Observatory.

Stars with initial masses less than about eight times the sun never develop a core large enough to collapse and they eventually lose their atmospheres to become white dwarfs, stable cooling spheres of carbon supported by the pressure of degenerate electrons. Nucleosynthesis within those lighter stars is therefore limited to nuclides that were fused in material located above the final white dwarf. This limits their modest yields returned to interstellar gas to carbon-13 and nitrogen-14, and to isotopes heavier than iron by slow capture of neutrons (the s-process).

A significant minority of white dwarfs will explode, however, either because they are in a binary orbit with a companion star that loses mass to the stronger gravitational field of the white dwarf, or because of a merger with another white dwarf. The result is a white dwarf which exceeds its Chandrasekhar limit and explodes as a type Ia supernova, synthesizing about a solar mass of radioactive ⁵⁶Ni isotopes, together with smaller amounts of other iron peak elements. The subsequent radioactive decay of the nickel to iron keeps Type Ia optically very bright for weeks and creates more than half of all the iron in the universe.

Virtually all of the remainder of stellar nucleosynthesis occurs, however, in stars that are massive enough to end as core collapse supernovae. In the pre-supernova massive star this includes helium burning, carbon burning, oxygen burning and silicon burning. Much of that yield may never leave the star but instead disappears into its collapsed core. The yield that is ejected is substantially fused in last-second explosive burning caused by the shock wave launched by core collapse. Prior to core collapse, fusion of elements between silicon and iron occurs only in the largest of stars, and then in limited amounts. Thus, the nucleosynthesis of the abundant primary elements defined as those that could be synthesized in stars of initially only hydrogen and helium (left by the Big Bang), is substantially limited to core-collapse supernova nucleosynthesis.

r-process nucleosynthesis

A version of the periodic table indicating the main origin of elements found on Earth. All elements past plutonium (element 94) are man-made.

During supernova nucleosynthesis, the r-process creates very neutron-rich heavy isotopes, which decay after the event to the first stable isotope, thereby creating the neutron-rich stable isotopes of all heavy elements. This neutron capture process occurs in high neutron density with high temperature conditions.

In the r-process, any heavy nuclei are bombarded with a large neutron flux to form highly unstable neutron rich nuclei which very rapidly undergo beta decay to form more stable nuclei with higher atomic number and the same atomic mass. The neutron density is extremely high, about 10 ²²⁻²⁴ neutrons per cubic centimeter.

Initial calculations of an evolving r-process, showing the evolution of calculated results with time, also suggested that the r-process abundances are a superposition of differing neutron fluences. Small fluence produces the first r-process abundance peak near atomic weight but no actinides, whereas large fluence produces the actinides uranium and thorium but no longer contains the abundance peak. These processes occur in a fraction of a second to a few seconds, depending on details. Hundreds of subsequent papers published have utilized this time-dependent approach. The only modern nearby supernova, 1987A, has not revealed r-process enrichments. Modern thinking is that the r-process yield may be ejected from some supernovae but swallowed up in others as part of the residual neutron star or black hole.

Entirely new astronomical data about the r-process was discovered in 2017 when the LIGO and Virgo gravitational-wave observatories discovered a merger of two neutron stars that had previously been orbiting one another. That can happen when both massive stars in orbit with one another become core-collapse supernovae, leaving neutron-star remnants.

The localization on the sky of the source of those gravitational waves radiated by that orbital collapse and merger of the two neutron stars, creating a black hole, but with significant spun off mass of highly neutronized matter, enabled several teams to discover and study the remaining optical counterpart of the merger, finding spectroscopic evidence of r-process material thrown off by the merging neutron stars.

The bulk of this material seems to consist of two types: Hot blue masses of highly radioactive r-process matter of lower-mass-range heavy nuclei ( A < 140) and cooler red masses of higher mass-number r-process nuclei ( A > 140) rich in actinides (such as uranium, thorium, californium etc.). When released from the huge internal pressure of the neutron star, this neutron-rich ejecta expands and radiates detected optical light for about a week. Such duration of luminosity would not be possible without heating by internal radioactive decay, which is provided by r-process nuclei near their waiting points. Two distinct mass regions ( A < 140 and A > 140) for the r-process yields have been known since the first time dependent calculations of the r-process. Because of these spectroscopic features it has been argued that r-process nucleosynthesis in the Milky Way may have been primarily ejecta from neutron-star mergers rather than from supernovae.

1. ^ Woosley, S.E.; Arnett, W.D.; Clayton, D.D. (1973). "The Explosive burning of oxygen and silicon". The Astrophysical Journal Supplement Series. 26: 231-312. Bibcode:1973ApJS...26..231W. doi:10.1086/190282. hdl:. S2CID 222372611.

2. See Figures 1, 3, and 4 in Arnett & Clayton (1970) and Fig. 2, p. 241 in Woosley, Arnett & Clayton 1973

3. Hoyle, F. (1946). "The Synthesis of the Elements from Hydrogen". Monthly Notices of the Royal Astronomical Society. 106 (5): 343-383. Bibcode:1946MNRAS.106..343H. doi:10.1093/mnras/106.5.343.

4. ^ Nomoto, Ken'Ichi (1980). "White dwarf models for type I supernovae and quiet supernovae, and presupernova evolution". Space Science Reviews. 27 (3-4): 563. Bibcode:1980SSRv...27..563N. doi:10.1007/BF00168350. S2CID 120969575.

5. ^ Nomoto, K.; Thielemann, F.-K.; Yokoi, K. (1984). "Accreting white dwarf models of type I supernovae. III - Carbon deflagration supernovae". The Astrophysical Journal. 286: 644. doi:10.1086/162639.

6. ^ Clayton, Donald D. (1983) [1968]. "Chapter 6. Calculation of Stellar Structure". Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. ISBN 9780226109534. Archived from the original on 2023-01-13.

7. for example ^ Iben, I. Jr. (1967). "Stellar Evolution. VI. Evolution from the Main Sequence to the Red-Giant Branch for Stars of Mass 1 M☉, 1.25 M☉, and 1.5 M☉*". Astrophysical Journal. 147: 624. Bibcode:1967apj...147..624i. doi:10.1086/149040. which contains a description of helium burning.

8. ^ Woosley, S.E.; Weaver, T.A. (1995). "The evolution and explosion of massive stars. II. Explosive hydrodynamics and nucleosynthesis" (PDF). The Astrophysical Journal Supplement Series. 101: 181. Bibcode:1995ApJS..101..181W. doi:10.1086/192237. Archived (PDF) from the original on 2023-01-13.

9. Clayton, D.D. (1983). Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press. pp. 519-524. ISBN 0226109534 - via Archive.org.

10. ^ Clayton, D. D. (1968). "Chapter 7". Principles of Stellar Evolution and Nucleosynthesis. University of Chicago Press.

11. Clayton, D. D. (2003). Handbook of Isotopes in the Cosmos. Cambridge University Press.

12. François, P.; Matteucci, F.; Cayrel, R.; Spite, M.; Spite, F.; Chiappini, C. (2004). "The evolution of the Milky Way fromits earliest phases: Constraints on stellar nucleosynthesis". Astronomy & Astrophysics. 421 (2): 613-621. arXiv:astro-ph/0401499. Bibcode:2004A&A...421..613F. doi:10.1051/0004-6361:20034140. S2CID 16257700.

13. ^ Kasen, D.; Metzger, B.; Barnes, J.; Quataert, E.; Ramirez-Ruiz, E. (2017). "Origin of the heavy elements in binary neutron-star mergers from a gravitational-wave event". . 551 (7678): 80-84. arXiv:. Bibcode:2017Natur.551...80K. doi:10.1038/nature24453. PMID 29094687. S2CID 205261425. Archived from the original on 2023-01-13.

# data[].attributes.attachment_link.description
Supernova nucleosynthesis is the nucleosynthesis of chemical elements in supernova explosions. In sufficiently massive stars, the nucleosynthesis by fusion of lighter elements into heavier ones occurs during sequential hydrostatic burning processes called helium burning, carbon burning, oxygen burning, and silicon burning, in which the byproducts of one nuclear fuel become, after compressional heating, the fuel for the subsequent burning stage.