Glossary

Factors of ten, the natural way to compare quantities that differ enormously. The Hierarchy of Cosmic Structure lays the Universe out on one axis from $10^6$ m (Ceres, 940 km across) to about $10^{27}$ m (the observable universe, $8.8\times 10^{26}$ m across), some twenty-two orders of magnitude. The relative strengths of the four interactions span about thirty-eight.

In the folios: The Hierarchy of Cosmic Structure › From planets to the cosmic web; Subatomic Matter › The four interactionsSee also Logarithmic scale, Observable universe, Fundamental interactions

An axis on which equal steps mean equal factors rather than equal differences. The folios use one wherever a quantity spans many decades: sizes from Ceres to the observable universe, distances from the Sun to the Oort cloud, particle masses from below 1 eV to the top quark, and the energy scale on which the strong coupling runs.

In the folios: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Belts, clouds and visitors; Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › Inside nucleons: QCDSee also Orders of magnitude

The energy an electron gains crossing a potential difference of one volt, the working unit of atomic, nuclear and particle physics. Chemistry happens at a few eV (13.6 eV ionises hydrogen), nuclear physics at MeV, and the heaviest known particles weigh in at about 100 GeV. The LHC collides protons at 13.6 TeV.

$1\text{eV}=1.602176634\times 10^{-19}\text{J}$ exactly (SI 2019)

In the folios: Subatomic Matter › Atoms and quantum probability; Mass in the Quantum World › Mass is rest energySee also Mass in energy units, Energy as temperature, Natural units

Because $E_0=m{c}^2$, particle masses are quoted as rest energies divided by $c^2$: the electron is 0.511 MeV/$c^2$ ($9.109\times 10^{-31}$ kg), the proton 938.27 MeV/$c^2$, the Higgs boson 125.2 GeV/$c^2$. Working in natural units drops the $c^2$, which is why the Subatomic Matter folio writes simply "0.511 MeV".

$1\text{GeV}/c^2=1.783\times 10^{-27}\text{kg}$; $1\text{u}=931.494\text{MeV}/c^2$

In the folios: Mass in the Quantum World › Particles are field excitations; Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › The Standard ModelSee also Electronvolt, Natural units, Rest energy

A convention of particle physics in which the speed of light and the reduced Planck constant are set to one. Energy, momentum and mass then share one unit, usually the GeV, and lengths and times become inverse energies. Converting back needs only the product $\hslash c$.

$\hslash c=197.327\text{MeV fm}$, so $1{\text{GeV}}^{-1}=0.1973\text{fm}$

See also Reduced Planck constant, Mass in energy units, Femtometre

$10^{-15}$ m, the size scale of nucleons and nuclei, also called the fermi after Enrico Fermi. The proton's charge radius is 0.84 fm, the nuclear force reaches 1–2 fm, and a lead nucleus is about 7 fm in radius.

$1\text{fm}=10^{-15}\text{m}$

In the folio: Subatomic Matter › The nuclear scale; Subatomic Matter › Inside nucleons: QCDSee also Nuclear radius, Proton radius

$10^{-10}$ m, roughly the diameter of an atom and the length of a chemical bond, named after the Swedish physicist Anders Ångström. The folio's atom "about $10^{-10}$ m across" is one ångström; hydrogen's Bohr radius is 0.529 Å.

$1\text{A˚}=10^{-10}\text{m}=100\text{pm}$

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Atom, Bohr radius

The speed in vacuum of every massless particle and of gravitational waves, and the conversion factor between mass and energy. Since 1983 its value has been fixed by definition, and the metre, still one of the SI base units, is defined from it and the second.

$c=299792458{\text{m s}}^{-1}$ exactly

In the folios: Mass in the Quantum World › Mass is rest energy; Subatomic Matter › Particles are excitations of quantum fieldsSee also Massless particle, Rest energy, GW170817

Planck's constant divided by $2\pi$: the quantum of angular momentum, and the conversion between energy and angular frequency ($E=\hslash \omega$) and between momentum and wavenumber ($p=\hslash k$). Spins are measured in units of $\hslash$.

$\hslash =h/2\pi =1.054571817\times 10^{-34}\text{J s}=6.582\times 10^{-22}\text{MeV s}$

In the folio: Mass in the Quantum World › Particles are field excitations; Mass in the Quantum World › Mass sets a quantum clockSee also Planck–Einstein relation, de Broglie relation, Spin, Natural units

The magnitude of the electron's charge, and the unit in which particle charges are quoted: the electron $-e$ (written −1 in the folio's table), the proton $+e$, the up quark $+\frac{2}{3}e$, the down quark $-\frac{1}{3}e$. Fractional charges are never seen on their own, because quarks are confined in hadrons whose charges are whole numbers.

$e=1.602176634\times 10^{-19}\text{C}$ exactly

In the folios: Mass in the Quantum World › Particles are field excitations; Subatomic Matter › The Standard ModelSee also Quark, Colour confinement, Fine-structure constant

The distance at which one astronomical unit subtends one arcsecond, so a star with a parallax of $p$ arcseconds lies $1/p$ parsecs away. It is the working unit of distance in astronomy; the nearest star, Proxima Centauri, is 1.30 pc away.

$1\text{pc}=3.086\times 10^{16}\text{m}\approx 3.26$ light-years $\approx 206265$ AU

In the folio: Cosmology and Astrophysics › Measuring the cosmosSee also Parallax, Kiloparsec, megaparsec, gigaparsec, Light-year

$10^3$, $10^6$ and $10^9$ parsecs. Rough landmarks: the Milky Way's disc is about 30 kpc across, the Andromeda galaxy is 0.78 Mpc away, galaxy clusters span a few Mpc, the sound horizon is 147 Mpc, and the observable universe has a comoving radius of about 14.3 Gpc.

$1\text{Mpc}=3.086\times 10^{22}\text{m}$

In the folio: Cosmology and Astrophysics › Measuring the cosmosSee also Parsec, Sound horizon, Observable universe

The distance light travels in a Julian year. Common in popular writing, rarely used in research, where the parsec is preferred.

The Hierarchy of Cosmic Structure quotes its sizes and distances in light-years, as popular astronomy does.

$1\text{ly}=9.461\times 10^{15}\text{m}=0.3066\text{pc}$

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Our cosmic addressSee also Parsec

The mean Earth–Sun distance, fixed by definition since 2012. It is the baseline of the parallax that defines the parsec.

$1\text{au}=1.495978707\times 10^{11}\text{m}$ exactly

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Parsec, Parallax

One sixtieth of an arcminute, $1/3600$ of a degree. Milliarcseconds (mas) and microarcseconds (μas) measure the parallaxes Gaia records and the black-hole images the Event Horizon Telescope resolves: the ring around M87* is 42 μas across, the apparent size of an orange on the Moon.

$1^{\prime \prime }=4.848\mu \text{rad}$

See also Parallax, Black-hole shadow

Thousands, millions and billions of years. The folio's timelines read naturally in these: the CMB was released at 380 kyr, the first stars formed at roughly 100–200 Myr, and the universe is 13.8 Gyr old.

$1\text{yr}=3.156\times 10^7\text{s}$ (Julian year); $1\text{Gyr}=10^9\text{yr}$

In the folio: Cosmology and Astrophysics › Cosmic historySee also Cosmic time, Age of the universe

The mass of the Sun, the unit for stars, black holes and galaxies. The symbol ☉ always means "of the Sun": $L_{\odot }$, $R_{\odot }$, $M_{\odot }$. A galaxy like the Milky Way holds about $10^{12}{M}_{\odot }$ including its dark halo; a supermassive black hole, $10^6$–$10^{10}{M}_{\odot }$.

$M_{\odot }=1.989\times 10^{30}\text{kg}$

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Solar luminosity, Solar radius

The Sun's total radiated power, used as the unit on the luminosity axis of the Hertzsprung–Russell diagram. Stars span roughly $10^{-4}$ to $10^6{L}_{\odot }$.

$L_{\odot }=3.828\times 10^{26}\text{W}$ (IAU nominal value)

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Hertzsprung–Russell diagram, Mass–luminosity relation

The Sun's radius, the natural unit for stellar sizes. A white dwarf is about $0.01R_{\odot }$, Earth-sized; a red supergiant can exceed $1000R_{\odot }$.

$R_{\odot }=6.957\times 10^8\text{m}$

In the folio: The Hierarchy of Cosmic Structure › StarsSee also White dwarf, Supergiant

The units for rocky planets. Earth is 12,742 km in mean diameter, and exoplanet sizes are given in Earth radii: a super-Earth or sub-Neptune has one to four $R_{\oplus }$.

$R_{\oplus }=6371\text{km}$ (mean), $M_{\oplus }=5.972\times 10^{24}\text{kg}$

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Planets of other starsSee also Earth, Super-Earths and sub-Neptunes

The unit for giant planets and brown dwarfs: about 318 Earth masses, or roughly a thousandth of the Sun's. Brown dwarfs span about 13 to 80 $M_J$.

$M_J=1.898\times 10^{27}\text{kg}\approx 9.5\times 10^{-4}{M}_{\odot }$

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Brown dwarf, Jupiter

The unit of the Hubble constant: recession speed gained per unit distance. It is an inverse time in disguise, and $H_0=67.4$ in these units is $2.18\times 10^{-18}{\text{s}}^{-1}$, whose inverse, the Hubble time, is 14.5 Gyr.

$1{\text{km s}}^{-1}{\text{Mpc}}^{-1}=3.241\times 10^{-20}{\text{s}}^{-1}$

In the folio: Cosmology and Astrophysics › What the Universe is made ofSee also Hubble constant, Hubble time and Hubble distance

The Hubble constant divided by 100 km s⁻¹ Mpc⁻¹, so $h\approx 0.674$. Quantities whose value depends on $H_0$ are often quoted with explicit powers of $h$ (such as ${\Omega }_b{h}^2$ or $8h^{-1}\text{Mpc}$), which keeps them independent of the still-disputed value.

$H_0=100h{\text{km s}}^{-1}{\text{Mpc}}^{-1}$

See also Hubble constant, Hubble tension

A subscript 0 means "today": $H_0$, $T_0$, $q_0$, ${\rho }_{\text{crit},0}$. The scale factor is normalised so that $a_0=1$.

See also Scale factor, Hubble constant

Cosmologists date events by redshift rather than by time, because redshift is what is observed and needs no model, while converting it to an age requires one. In Planck ΛCDM: $z\approx 1090$ is 380 kyr, $z=14.44$ is about 280 Myr, $z=6$ about 0.9 Gyr, $z=2$ about 3.3 Gyr, $z=1$ about 5.9 Gyr, and $z=0.3$ about 10.3 Gyr after the Big Bang.

In the folio: Cosmology and Astrophysics › Cosmic historySee also Cosmological redshift, Lookback time

The early universe is labelled by the thermal energy $k_{B}T$ of its particles: the electroweak crossover at 160 GeV, the QCD crossover at 155 MeV, neutrino decoupling at 1 MeV. In the radiation era a useful rule of thumb links this to time.

$1\text{eV}/k_B=11605\text{K}$; $t\sim 1\text{s}\times {\left(1\text{MeV}/k_{B}T\right)}^2$

In the folio: Cosmology and Astrophysics › Cosmic historySee also Radiation era, Electroweak crossover, QCD crossover, Electronvolt

A relative precision of $10^{-6}$. COBE's FIRAS showed the CMB spectrum to be a blackbody to about 50 ppm, the most perfect thermal spectrum measured in nature.

In the folio: Cosmology and Astrophysics › The cosmic microwave backgroundSee also FIRAS, Blackbody spectrum

A dense nucleus surrounded by electrons described by wavefunctions, the whole about $10^{-10}$ m across. The nucleus is about $10^{-5}$ of the atom's diameter yet holds over 99.9% of its mass; electrons add only about 0.03% to the mass of ordinary matter (0.05% for hydrogen). Rutherford's scattering experiments established this nuclear atom in 1911.

In the folios: Subatomic Matter › Atoms and quantum probability; Mass in the Quantum World › From quantum fields to everyday massSee also Atomic nucleus, Wavefunction, Atomic orbital, Rutherford scattering

The complex amplitude that describes a quantum state. Its squared magnitude gives probabilities and its phase governs interference. For an electron in an atom, $\psi (\mathbf{r})$ is not a path but a spread-out amplitude: in the folio's words, clouds show where an electron is likely to be found, not a path.

In the folios: Subatomic Matter › Atoms and quantum probability; Mass in the Quantum World › Mass sets a quantum clockSee also Probability density, Born rule, Quantum phase, Atomic orbital

The probability per unit volume of finding the particle at $\mathbf{r}$. Integrated over all space it gives one. The folio's 1s, 2p and 3d images are computed $|\psi {|}^2$ in a plane through the nucleus, each scaled separately, so their brightnesses cannot be compared with one another.

$\rho (\mathbf{r})=|\psi (\mathbf{r}){|}^2$, $\int \rho d^{3}r=1$

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Born rule, Wavefunction

The rule, proposed by Max Born in 1926, that the squared magnitude of a quantum amplitude is a probability. It is how a wavefunction connects to what a detector registers.

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Probability density, Wave–particle duality

A one-electron wavefunction of definite energy and angular momentum. The label gives the principal quantum number $n$ and a letter for the orbital angular momentum $l$: s, p, d, f for $l=0,1,2,3$, letters inherited from nineteenth-century spectroscopy (sharp, principal, diffuse, fundamental). The 1s orbital is spherical; each of the usual real 2p orbitals has two lobes and one nodal plane; and four of the five real 3d orbitals, including the one the folio shows, have four lobes and two nodal planes, while $3d_{z^2}$ has two lobes and a ring, its two nodal surfaces being cones.

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Quantum numbers, Node, Probability density

The labels of an electron's state in an atom: principal $n=1,2,3,\dots$ (mainly the energy), orbital $l=0,\dots ,n-1$ (the shape), magnetic $m_l=-l,\dots ,l$ (the orientation) and spin $m_s=\pm \frac{1}{2}$. The Pauli exclusion principle allows no two electrons in an atom to share all four, which is what builds the periodic table.

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Atomic orbital, Pauli exclusion principle, Spin

A surface on which a wavefunction, and so the probability density, is zero. A hydrogen orbital has $n-1$ nodes in all: $l$ angular nodes (planes or cones) and $n-l-1$ radial ones (spheres). The folio's 1s has none, 2p one and 3d two.

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Atomic orbital

The allowed energies of hydrogen's electron, negative because the electron is bound. The ground state lies 13.6 eV below the ionisation threshold, and the levels crowd together as $n$ grows. A transition between two levels emits or absorbs a photon of exactly the energy difference: the spectral lines from which astronomers read composition and redshift.

$E_n=-\frac{13.6\text{eV}}{n^2}$

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Rydberg energy, Bohr radius, Recombination

The ionisation energy of hydrogen, in the limit of an infinitely heavy nucleus, and the natural unit of atomic binding. It is set by the electron's mass and the fine-structure constant.

$\text{Ry}=\frac{1}{2}{\alpha }^2{m}_e{c}^2=13.606\text{eV}$

See also Hydrogen energy levels, Fine-structure constant

The most probable distance of the electron from the nucleus in hydrogen's ground state, and the natural length of atomic physics.

$a_0=\frac{\hslash }{m_{e}c\alpha }=5.29\times 10^{-11}\text{m}$

See also Bohr model, Ångström

Niels Bohr's 1913 picture of electrons on circular orbits with quantised angular momentum. It gets hydrogen's energy levels right and little else; the folio's "wavefunctions, not orbits" is the correction quantum mechanics made in 1925–26.

In the folio: Subatomic Matter › Atoms and quantum probabilitySee also Hydrogen energy levels, Wavefunction

The experiments of Hans Geiger and Ernest Marsden, interpreted by Ernest Rutherford in 1911, in which a few alpha particles fired at a thin gold foil bounced back through large angles. Only a tiny, dense, positively charged centre could turn them round: the discovery of the atomic nucleus.

In the folio: Subatomic Matter › Milestones; Subatomic Matter › Atoms and quantum probabilitySee also Atomic nucleus, Alpha particle

A field defined at every point of space and time whose excitations come in discrete quanta. Every species of elementary particle has one: the electron field, the up-quark field, the gluon field, the Higgs field. All electrons are exactly alike because they are excitations of one and the same field.

In the folios: Mass in the Quantum World › Particles are field excitations; Mass in the Quantum World › From quantum fields to everyday mass; Subatomic Matter › Particles are excitations of quantum fieldsSee also Quantum field theory, Field excitation, Elementary particle

The framework that unites quantum mechanics with special relativity by quantising fields. It explains why particles can be created and destroyed, why identical particles are exactly identical, and why every particle has an antiparticle. The Standard Model is a quantum field theory.

In the folio: Mass in the Quantum World › Particles are field excitationsSee also Quantum field, Standard Model, Antiparticle

One quantum of a field: a localised ripple with sharply defined charge, spin and mass. "One electron" is one quantum of the electron field, not a tiny hard ball. A field in its lowest-energy state, the vacuum, contains none.

In the folios: Mass in the Quantum World › Particles are field excitations; Subatomic Matter › Particles are excitations of quantum fieldsSee also Quantum field, Vacuum

A particle with no known internal structure: the quarks, leptons, gauge bosons and Higgs boson of the Standard Model. "Structure: none observed" in the folio's electron card means experiments have found it point-like down to the smallest distances they probe. Protons and neutrons are not elementary; they are bound states of quarks and gluons.

In the folios: Mass in the Quantum World › Particles are field excitations; Mass in the Quantum World › From quantum fields to everyday mass; Subatomic Matter › The Standard ModelSee also Standard Model, Hadron

Quanta propagate like waves and are detected like particles. An electron spreads, diffracts and interferes, yet each detection registers one whole quantum at a single place. In field theory there is no paradox: the field behaves as a wave, and its energy comes in indivisible quanta.

In the folio: Mass in the Quantum World › Particles are field excitationsSee also Field excitation, Born rule, Interference

A quantum's energy is proportional to the angular frequency of its wave. Planck introduced it for radiation in 1900 and Einstein applied it to light quanta in 1905; it holds for every quantum.

$E=\hslash \omega =h\nu$

In the folio: Mass in the Quantum World › Particles are field excitationsSee also de Broglie relation, Reduced Planck constant

A quantum's momentum is proportional to the wavenumber of its wave, so its wavelength is $h/p$. Louis de Broglie proposed in 1924 that matter has a wavelength too; electron diffraction confirmed it in 1927.

$p=\hslash k$, $\lambda =h/p$

In the folio: Mass in the Quantum World › Particles are field excitationsSee also Planck–Einstein relation, Wavenumber

The spatial counterpart of angular frequency, $k=2\pi /\lambda$, in radians per metre. In quantum mechanics it is momentum divided by $\hslash$.

See also de Broglie relation

The wavefunction of a free particle with definite momentum $p$ and energy $E$: the same amplitude everywhere, with a phase that advances in space at the rate $p/\hslash$ and in time at the rate $E/\hslash$.

$\psi (x,t)\propto e^{i(px-Et)/\hslash }$

In the folio: Mass in the Quantum World › Mass sets a quantum clockSee also Quantum phase, Compton angular frequency

The angle of a complex amplitude, drawn in the folio as an arrow turning as time passes. An overall phase cannot be observed; phase differences between paths or components produce interference.

In the folio: Mass in the Quantum World › Mass sets a quantum clockSee also Interference, Plane wave, Compton angular frequency

The adding of amplitudes before squaring them: where two contributions arrive in phase the probability is enhanced, and where they arrive out of phase it is suppressed. It is how phase, otherwise unobservable, shows itself, as in double-slit fringes and atom interferometers.

In the folio: Mass in the Quantum World › Mass sets a quantum clockSee also Quantum phase, Wave–particle duality

The rate at which the phase of a particle at rest advances: its rest energy divided by $\hslash$, the "quantum clock" that mass sets. It is the intrinsic time scale of the quantum evolution, not a mechanical vibration of anything. For the electron it is about $7.76\times 10^{20}$ rad s⁻¹.

${\omega }_C=\frac{m{c}^2}{\hslash }$

In the folio: Mass in the Quantum World › Mass sets a quantum clockSee also Compton wavelength, Rest energy, Quantum phase

The length $h/mc$ associated with a mass; its reduced form $\hslash /mc$ is the distance light travels while the Compton clock turns through one radian. It sets the scale below which a particle cannot be localised without creating pairs, and for a force carrier of mass $M$ it is the range of the force: $\hslash /Mc\approx 2.5\times 10^{-18}$ m for the W boson.

electron: $h/m_{e}c=2.43\times 10^{-12}\text{m}$, $\hslash /m_{e}c=3.86\times 10^{-13}\text{m}$

In the folios: Mass in the Quantum World › Mass sets a quantum clock; Subatomic Matter › Particles are excitations of quantum fieldsSee also Compton angular frequency, Range of a force

Intrinsic angular momentum, measured in units of $\hslash$, that a particle carries whatever its motion. The matter particles have spin ½, the force carriers spin 1, the Higgs boson spin 0, and the hypothetical graviton spin 2. Spin decides whether a particle is a fermion or a boson.

In the folios: Subatomic Matter › The Standard Model; Mass in the Quantum World › Particles are field excitationsSee also Fermion, Boson, Quantum numbers

A particle of half-integer spin, obeying Fermi–Dirac statistics and the Pauli exclusion principle. All the matter particles, quarks and leptons, are fermions, as is any composite of an odd number of them, such as a proton or neutron.

In the folio: Subatomic Matter › The Standard ModelSee also Boson, Pauli exclusion principle, Spin

A particle of integer spin, obeying Bose–Einstein statistics: any number can share one state, which is what makes lasers and classical fields possible. The gauge bosons (spin 1) carry the forces, and the Higgs boson (spin 0) is a scalar.

In the folio: Subatomic Matter › The Standard ModelSee also Fermion, Gauge boson, Scalar boson

No two identical fermions can occupy the same quantum state. It gives atoms their shell structure, makes matter take up space, and supplies the degeneracy pressure that holds up white dwarfs and neutron stars. Wolfgang Pauli formulated it in 1925.

See also Fermion, Quantum numbers, Degeneracy pressure

For every particle, a partner of the same mass and spin with opposite charges; the folio's "every fermion has an antiparticle". Paul Dirac's equation predicted the electron's in 1928–31, and Carl Anderson found it in cosmic rays in 1932. Some neutral particles, such as the photon, are their own antiparticles; whether neutrinos are is not known.

In the folios: Subatomic Matter › The Standard Model; Subatomic Matter › Milestones; Mass in the Quantum World › Mass, energy and conservationSee also Positron, Antimatter, Majorana neutrino

The electron's antiparticle, with charge $+e$ and the same mass, 0.511 MeV/$c^2$: the first antimatter discovered, by Anderson in 1932. Electron–positron annihilation and pair creation are the Mass folio's examples of matter turning into radiation and back.

In the folios: Mass in the Quantum World › Mass, energy and conservation; Subatomic Matter › MilestonesSee also Antiparticle, Annihilation, Pair creation

Matter made of antiparticles. It is produced in accelerators, in cosmic rays and in radioactive decay, and annihilates on meeting ordinary matter. The observable universe contains almost none, a puzzle the Standard Model cannot solve. Dark matter is not antimatter: antimatter would annihilate and shine.

In the folio: Subatomic Matter › Beyond the Standard Model; Subatomic Matter › Open problems; Subatomic Matter › Dark matterSee also Antiparticle, Baryogenesis, Matter–antimatter asymmetry

In quantum field theory, the state of lowest energy. It is not nothing: the fields fluctuate in it, and the Higgs field takes a non-zero value throughout it. "The vacuum" is a state of the fields, not empty space.

In the folio: Mass in the Quantum World › The Higgs field and elementary massesSee also Vacuum expectation value, Higgs field, Cosmological constant

A photon is a transverse excitation of the electromagnetic field: its electric and magnetic fields oscillate perpendicular to its motion and to each other, and in phase. Being massless, it has only two polarisation states, where a massive spin-1 particle such as the W or Z has three.

In the folio: Subatomic Matter › Particles are excitations of quantum fieldsSee also Photon, Massless particle, Polarisation

The mass of a particle or system, defined from its total energy and momentum in a way all observers agree on. It measures inertia, the resistance to being accelerated. For a system it is the total internal energy divided by $c^2$, so it includes the kinetic and binding energies of the parts.

$m^2{c}^4=E^2-|\mathbf{p}{|}^2{c}^2$

In the folio: Mass in the Quantum World › Mass is rest energy; Mass in the Quantum World › Mass, energy and conservationSee also Energy–momentum relation, Rest energy, Mass is not additive

The relativistic link between a particle's energy, momentum and mass. Drawn as the folio's right triangle, the hypotenuse is $E$, one leg is $pc$, and the other leg, $m{c}^2$, is the same for every observer. Plotted as energy against momentum, a massive particle follows a curve that starts at $m{c}^2$; a massless one follows the straight line $E=pc$.

$E^2=(pc{)}^2+(m{c}^2{)}^2$

In the folio: Mass in the Quantum World › Mass is rest energySee also Invariant mass, Rest energy, Massless particle

The energy of a particle at rest, $p=0$: Einstein's $E_0=m{c}^2$. One kilogram corresponds to $9\times 10^{16}$ J, about 21 megatonnes of TNT.

$E_0=m{c}^2$

In the folio: Mass in the Quantum World › Mass is rest energy; Mass in the Quantum World › From quantum fields to everyday massSee also Mass–energy equivalence, Invariant mass

Einstein's 1905 result that mass is a form of energy: a body that gives off energy $\Delta E$ loses mass $\Delta E/c^2$. It is why a bound helium nucleus is lighter than its parts, why a proton is a hundred times heavier than its valence quarks, and why stars lose mass as they shine.

In the folio: Mass in the Quantum World › Mass, energy and conservation; Mass in the Quantum World › From quantum fields to everyday massSee also Rest energy, Mass defect, Proton mass decomposition

A particle with $m=0$, whose energy is all momentum and which always moves at the speed of light. In the Standard Model the photon and gluon are massless; the graviton would be too.

$E=pc$

In the folios: Mass in the Quantum World › Mass is rest energy; Subatomic Matter › The Standard ModelSee also Photon, Gluon, Energy–momentum relation

An older usage in which "mass" grows with speed as $\gamma m$. Modern physics avoids it: the folios' $m$ is always the invariant mass, and what grows with speed is energy.

See also Invariant mass

A body's resistance to being accelerated, measured by its mass. That the same mass also sets the strength of gravity, the equivalence of inertial and gravitational mass, is the founding observation of general relativity.

In the folio: Mass in the Quantum World › Mass is rest energySee also Invariant mass, General relativity

Energy and momentum combined into one four-vector $(E/c,\mathbf{p})$. Its length is the same in every frame, and is the invariant mass.

$p^{\mu }{p}_{\mu }=(E/c{)}^2-|\mathbf{p}{|}^2=m^2{c}^2$

See also Invariant mass, Energy–momentum relation

In an isolated system, total energy and momentum do not change. Because the invariant mass is built from them, it is conserved too, even as rest energy turns into kinetic energy or radiation and back. A subsystem that emits energy loses mass.

$M^2{c}^4=E_{\text{tot}}^2-|{\mathbf{p}}_{\text{tot}}{|}^2{c}^2$ stays fixed

In the folio: Mass in the Quantum World › Mass, energy and conservationSee also Invariant mass, Annihilation, Pair creation

A composite system's mass is its total internal energy divided by $c^2$, not the sum of the masses of its parts. Binding lowers it (a helium-4 nucleus, an atom, a molecule), and confined motion and field energy raise it (a proton weighs about a hundred times its valence quarks). At everyday scales the corrections are tiny, which is why adding masses works on a laboratory balance.

In the folio: Mass in the Quantum World › From quantum fields to everyday mass; Mass in the Quantum World › Mass, energy and conservationSee also Mass defect, Proton mass decomposition, Chemical binding energy

The amount by which a bound system is lighter than its separated parts, equal to its binding energy divided by $c^2$. Helium-4 is 28.3 MeV/$c^2$ lighter than two free protons and two free neutrons, 0.75% of their mass; the energy leaves as gamma rays or kinetic energy when the nucleus forms.

$m({}^4\text{He})=2m_p+2m_n-B/c^2$, $B=28.3\text{MeV}$

In the folios: Mass in the Quantum World › Mass, energy and conservation; Mass in the Quantum World › From quantum fields to everyday mass; Subatomic Matter › The nuclear scaleSee also Binding energy per nucleon, Mass is not additive, Nuclear fusion

The few electronvolts released when atoms bond. A water molecule is lighter than its separate atoms by less than one part in $10^9$, far too little for any balance to detect, which is why chemistry appears to conserve mass exactly.

In the folio: Mass in the Quantum World › From quantum fields to everyday massSee also Mass defect, Mass is not additive

A particle and its antiparticle meeting and turning into other particles, usually photons. An electron and a positron at rest give two 511 keV photons, back to back. Each photon is massless, yet the pair, taken as one system, has invariant mass $2m_e$: mass is conserved even when every particle in the final state is massless.

$e^{-}+e^{+}\to \gamma +\gamma$

In the folio: Mass in the Quantum World › Mass, energy and conservationSee also Positron, Pair creation, Invariant mass, Electron–positron annihilation

Radiation turning into a particle and its antiparticle. Two photons can make an electron–positron pair if their energy in the centre-of-momentum frame is at least $2m_e{c}^2=1.022$ MeV. A single photon cannot do it in empty space, because energy and momentum could not both be conserved; it needs a nearby nucleus to take up momentum, which is how high-energy photons shower in a calorimeter.

$\gamma +\gamma \to e^{-}+e^{+}$, $E_{\text{cm}}\ge 2m_e{c}^2$

In the folios: Mass in the Quantum World › Mass, energy and conservation; Subatomic Matter › How detectors see particlesSee also Annihilation, Threshold energy, Particle shower

The frame in which a system's total momentum is zero. There, all its energy is the system's rest energy, so the energy available for making new particles is the invariant mass times $c^2$.

See also Invariant mass, Threshold energy

The least energy that can make a reaction with heavier products go: in the centre-of-momentum frame, the products' total rest energy. It is why ever heavier particles need ever more energetic colliders.

See also Centre-of-momentum frame, Pair creation, Collider

The dense, positively charged core of an atom, made of protons and neutrons: a few femtometres across, about $10^{-5}$ of the atom's diameter, yet over 99.9% of its mass. Its radius grows as the cube root of the number of nucleons, so all nuclei have about the same density.

In the folios: Subatomic Matter › The nuclear scale; Subatomic Matter › Atoms and quantum probability; Mass in the Quantum World › From quantum fields to everyday massSee also Nucleon, Nuclear radius, Rutherford scattering

A proton or a neutron, the constituents of nuclei. The two have almost the same mass and, apart from electric charge, behave nearly alike under the strong force.

In the folio: Subatomic Matter › The nuclear scaleSee also Proton, Neutron, Mass number

The positively charged nucleon, two up quarks and a down quark (uud), with mass 938.27 MeV/$c^2$ and charge radius 0.84 fm. Its quarks' rest masses add up to only about 9 MeV/$c^2$, about 1% of the total; the rest is the energy of confined quarks and gluons. Rutherford identified it as the hydrogen nucleus present in other nuclei in 1919. No proton has been seen to decay. Limits are set channel by channel: for the most searched, $p\to e^{+}{\pi }^0$, Super-Kamiokande puts the partial lifetime above $2.4\times 10^{34}$ years (90% confidence).

In the folios: Subatomic Matter › Inside nucleons: QCD; Mass in the Quantum World › Most visible mass comes from QCD; Subatomic Matter › MilestonesSee also Neutron, Proton mass decomposition, Valence quark, Proton radius

The neutral nucleon (udd), 939.57 MeV/$c^2$, slightly heavier than the proton. A free neutron beta-decays with a mean lifetime of about 880 s, just under a quarter of an hour; bound in a stable nucleus it lasts indefinitely. James Chadwick discovered it in 1932.

In the folios: Subatomic Matter › Inside nucleons: QCD; Mass in the Quantum World › Most visible mass comes from QCD; Subatomic Matter › MilestonesSee also Proton, Beta decay, Neutron star

The number of nucleons in a nucleus, written as a superscript before the element's symbol: ⁴He has $A=4$, ⁵⁶Fe $A=56$, ²³⁵U $A=235$. The atomic number $Z$, the number of protons, fixes the element, and $N=A-Z$ counts the neutrons.

$A=Z+N$

In the folio: Subatomic Matter › The nuclear scaleSee also Isotope, Nucleon

Nuclei with the same number of protons, so the same element, but different numbers of neutrons. Deuterium (²H) and ordinary hydrogen (¹H) are isotopes, as are uranium-235 and uranium-238.

In the folio: Subatomic Matter › The nuclear scaleSee also Mass number, Deuteron

Nuclei are roughly uniform spheres whose volume grows in proportion to the number of nucleons, so their radius grows as $A^{1/3}$. The rule gives a sharp-edged radius, about 7.1 fm for lead-208. Measurements usually quote the root-mean-square charge radius instead, which for a uniform sphere is $\sqrt{3/5}\approx 0.77$ of the sharp radius, so the two must be converted before they are compared: for lead-208 the converted 5.5 fm matches the measured value. The rule is rough for the lightest nuclei; for helium-4 it gives an rms radius of 1.5 fm against a measured 1.68 fm.

$R\approx 1.2A^{1/3}\text{fm}$ (sharp surface); $r_{\text{rms}}=\sqrt{3/5}R$ for a uniform sphere

In the folio: Subatomic Matter › The nuclear scaleSee also Nuclear density, Femtometre

Because $R\propto A^{1/3}$, every nucleus has about the same density, roughly $2\times 10^{17}$ kg m⁻³: a teaspoonful would weigh about a billion tonnes. Neutron stars are matter at and above this density.

$\rho \approx m_N/\frac{4}{3}\pi (1.2\text{fm}{)}^3\approx 2.3\times 10^{17}{\text{kg m}}^{-3}$

In the folio: Subatomic Matter › The nuclear scaleSee also Nuclear radius, Neutron star

The energy needed to take a nucleus apart, divided by its number of nucleons. It rises steeply through the light nuclei, peaks near iron-56 at about 8.8 MeV (nickel-62 is fractionally higher), and falls slowly to about 7.6 MeV for uranium. Broadly, light nuclei therefore release energy by fusing and heavy ones by splitting, though individual reactions depend on the particular nuclei.

²H 1.11 MeV, ⁴He 7.07 MeV, ⁵⁶Fe 8.79 MeV, ²³⁵U 7.59 MeV per nucleon

In the folio: Subatomic Matter › The nuclear scaleSee also Nuclear fusion, Nuclear fission, Semi-empirical mass formula, Iron-peak elements

A formula for nuclear binding energy that treats the nucleus as a charged liquid drop: a volume term, less corrections for the surface, the protons' Coulomb repulsion, any imbalance of protons and neutrons, and a pairing term. Carl Friedrich von Weizsäcker proposed it in 1935. The folio's binding-energy curve is computed from it; the dots are measured values.

$B=a_{V}A-a_S{A}^{2/3}-a_C\frac{Z(Z-1)}{A^{1/3}}-a_A\frac{(A-2Z{)}^2}{A}\pm \delta$

In the folio: Subatomic Matter › The nuclear scaleSee also Binding energy per nucleon, Liquid-drop model

The picture of the nucleus as a drop of incompressible nuclear fluid, behind the semi-empirical mass formula and Bohr and Wheeler's 1939 account of fission. It explains the smooth trends in binding; the extra stability of "magic" numbers of nucleons needs the nuclear shell model.

See also Semi-empirical mass formula, Nuclear fission

The joining of light nuclei into heavier ones. Because binding energy per nucleon rises, broadly, towards the iron peak, many fusion reactions of light nuclei release energy; whether a particular one does depends on the masses before and after. Two helium-4 nuclei fusing into beryllium-8 absorb 92 keV, which is why stars make carbon by adding a third helium nucleus before the beryllium falls apart. It powers the Sun and the other stars, and made the light elements in the universe's first minutes.

In the folios: Subatomic Matter › The nuclear scale; Mass in the Quantum World › Mass, energy and conservationSee also Binding energy per nucleon, Proton–proton chain, Big Bang nucleosynthesis

The splitting of a heavy nucleus, such as uranium-235 after absorbing a neutron, into two mid-sized nuclei and a few neutrons, releasing about 200 MeV. It was discovered by Otto Hahn and Fritz Strassmann, and explained by Lise Meitner and Otto Frisch, in 1938–39.

In the folio: Subatomic Matter › The nuclear scaleSee also Binding energy per nucleon, Liquid-drop model

The attraction that binds nucleons into nuclei: a short-range leftover of the strong force between their quarks, much as the van der Waals force between neutral molecules is a leftover of electromagnetism. It is strongly attractive at about 1–2 fm, repulsive at shorter distances, and negligible beyond a few femtometres.

In the folio: Subatomic Matter › The nuclear scaleSee also Pion, Yukawa potential, Quantum chromodynamics

The lightest mesons, about 140 MeV/$c^2$ (charged) and 135 MeV/$c^2$ (neutral), made of up and down quarks and antiquarks. Hideki Yukawa predicted in 1935 that the nuclear force is carried by such a particle, and its mass sets the force's range at about $\hslash /m_{\pi }c\approx 1.4$ fm. Pions were found in cosmic rays in 1947.

In the folio: Subatomic Matter › The nuclear scaleSee also Meson, Residual strong force, Yukawa potential, Chiral symmetry breaking

The nucleus of deuterium, one proton and one neutron: the simplest compound nucleus, bound by only 2.22 MeV. Its fragility is why nucleosynthesis in the early universe had to wait until the photons had cooled enough not to break it apart.

In the folio: Subatomic Matter › The nuclear scaleSee also Deuterium abundance, Big Bang nucleosynthesis, Isotope

A helium-4 nucleus, two protons and two neutrons, unusually tightly bound (28.3 MeV). Emitted in alpha decay, it was Rutherford's probe of the atom; it is also the main product of fusion in stars and in the Big Bang.

In the folios: Subatomic Matter › The nuclear scale; Mass in the Quantum World › Mass, energy and conservation; Subatomic Matter › MilestonesSee also Mass defect, Rutherford scattering, Primordial helium fraction

A weak-interaction decay in which a neutron becomes a proton, emitting an electron and an electron antineutrino (or the reverse, emitting a positron and a neutrino). At the quark level a down quark turns into an up quark by emitting a virtual W boson. The continuous spread of electron energies led Pauli to propose the neutrino in 1930.

$n\to p+e^{-}+{\bar{\nu }}_e$; $d\to u+W^{-}\to u+e^{-}+{\bar{\nu }}_e$

In the folio: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The four interactionsSee also Weak interaction, W boson, Electron neutrino, KATRIN

The quantum field theory of the strong interaction, an $SU(3)$ gauge theory of quarks and gluons formulated in 1973. Unlike photons, its force carriers carry the charge they respond to, which makes the force grow with distance (confinement) and weaken at high energy (asymptotic freedom).

In the folios: Subatomic Matter › Inside nucleons: QCD; Mass in the Quantum World › Most visible mass comes from QCDSee also Colour charge, Gluon, Colour confinement, Asymptotic freedom

An elementary fermion that feels all four interactions, with a fractional electric charge ($+\frac{2}{3}$ or $-\frac{1}{3}$) and one of three colours. There are six flavours in three generations: up and down, charm and strange, top and bottom. Quarks are never seen alone; they are confined in hadrons. Murray Gell-Mann and George Zweig proposed them in 1964, and experiments at SLAC saw them inside protons in 1968.

In the folio: Subatomic Matter › Inside nucleons: QCD; Subatomic Matter › The Standard Model; Subatomic Matter › MilestonesSee also Quark flavour, Colour charge, Hadron, Quark model

The six types of quark: up, down, strange, charm, bottom and top. The strong and electromagnetic interactions preserve flavour; only the weak interaction changes it, as when a down quark becomes an up quark in beta decay. Leptons have flavours too: electron, muon and tau.

In the folio: Subatomic Matter › The Standard ModelSee also Quark, Weak interaction, CKM matrix

The charge of the strong interaction, which comes in three kinds called red, green and blue; antiquarks carry anticolours. The names are a metaphor, and nothing is coloured. Every observed hadron is colour-neutral: a baryon combines all three colours, a meson a colour and its anticolour.

In the folio: Subatomic Matter › Inside nucleons: QCD; Subatomic Matter › The four interactionsSee also Quantum chromodynamics, Gluon, Colour confinement

The massless spin-1 carrier of the strong force. There are eight, each carrying a combination of colour and anticolour, so gluons couple directly to one another. Photons have no such direct coupling; they can scatter off each other only weakly, through loops of virtual charged particles. Inside a proton they form a field whose energy is about a third of the proton's mass. Gluon jets were first seen at DESY in 1979.

In the folios: Subatomic Matter › Inside nucleons: QCD; Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The Standard Model; Mass in the Quantum World › Most visible mass comes from QCDSee also Quantum chromodynamics, Colour charge, Gluon jets, Flux tube

The fact that coloured particles, quarks and gluons, are never found in isolation. The energy of the field between two quarks grows with their separation, so pulling them apart makes new quark–antiquark pairs instead of a free quark. Confinement is established by experiment and by lattice calculation, but not yet proved from the theory's first principles.

In the folio: Subatomic Matter › Inside nucleons: QCD; Subatomic Matter › Particles are excitations of quantum fieldsSee also Flux tube, Hadronisation, Strong coupling

The weakening of the strong coupling at high energy, or short distance, so that quarks struck hard inside a proton behave almost as free particles. Its discovery by David Gross and Frank Wilczek, and by David Politzer, in 1973 (Nobel Prize 2004) made QCD calculable.

In the folio: Subatomic Matter › Inside nucleons: QCDSee also Strong coupling, Running coupling, Deep inelastic scattering

The strength of the strong interaction, the QCD counterpart of the fine-structure constant. It depends on the energy scale $Q$ of a process: 0.118 at the Z mass, 91 GeV, rising towards one below about 1 GeV, where QCD can no longer be calculated by expanding in powers of ${\alpha }_s$.

${\alpha }_s(M_Z)=0.118$; at one loop ${\alpha }_s(Q)=\frac{12\pi }{(33-2n_f)\ln (Q^2/{\Lambda }^2)}$

In the folio: Subatomic Matter › Inside nucleons: QCDSee also Running coupling, Asymptotic freedom, Fine-structure constant, QCD scale

A coupling whose effective value depends on the energy scale at which it is measured, because quantum fluctuations screen or antiscreen the charge. The electromagnetic coupling grows slowly with energy, from 1/137 to about 1/128 at the Z mass; the strong coupling falls. The folio's curve is computed at two loops with five quark flavours, the number light enough to take part between the bottom and top masses.

In the folio: Subatomic Matter › Inside nucleons: QCDSee also Strong coupling, Fine-structure constant, Asymptotic freedom

The energy, a few hundred MeV, at which the strong coupling becomes large. It sets the size of hadrons, about $\hslash c/\Lambda \approx 1$ fm, and most of their masses, and it emerges from the theory itself rather than from any input mass, a phenomenon called dimensional transmutation.

See also Strong coupling, Trace anomaly, Proton mass decomposition

The narrow tube into which the gluon field between separating quarks is squeezed, because gluons attract one another. Its energy per unit length, the string tension, is roughly constant, so the energy grows in proportion to distance.

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Mass in the Quantum World › Most visible mass comes from QCDSee also Cornell potential, Colour confinement, Hadronisation

A model of the potential energy between a heavy quark and its antiquark: Coulomb-like at short range, where single-gluon exchange dominates, and rising linearly at long range, where the flux tube dominates. It was introduced at Cornell in 1975 to explain charmonium, and the folio's gluon curve is computed from it.

$V(r)=-\frac{4}{3}\frac{{\alpha }_s\hslash c}{r}+\sigma r$, $\sigma \approx 0.9\text{GeV/fm}$

In the folio: Subatomic Matter › Particles are excitations of quantum fieldsSee also Flux tube, J/ψ meson

The conversion of quarks and gluons into hadrons. When a flux tube is stretched far enough, creating a new quark–antiquark pair costs less than stretching it further; repeated breaking turns a fast quark or gluon into a collimated jet of hadrons. The same word names the early universe's QCD crossover, when the quark–gluon plasma condensed into hadrons.

In the folio: Subatomic Matter › Particles are excitations of quantum fieldsSee also Flux tube, Jet, QCD crossover

A narrow spray of hadrons made when a quark or gluon from a high-energy collision hadronises. Jets are how quarks and gluons show up in detectors, mostly as clusters of energy in the calorimeters.

In the folio: Subatomic Matter › How detectors see particlesSee also Hadronisation, Gluon jets, Hadronic calorimeter

Three-jet events seen in 1979 at the PETRA electron–positron collider at DESY in Hamburg: a quark, an antiquark, and a gluon radiated by one of them, each hadronising into a jet. They were the first direct evidence for the gluon.

In the folio: Subatomic Matter › MilestonesSee also Gluon, Jet, DESY

Any composite particle of quarks and gluons bound by the strong force. Baryons have three valence quarks (the proton and neutron); mesons a quark and an antiquark (the pion, the J/ψ). Hundreds are known, almost all unstable.

In the folio: Subatomic Matter › Inside nucleons: QCD; Subatomic Matter › How detectors see particlesSee also Baryon number, Meson, Quark, Charged and neutral hadrons, Baryons

A hadron made of a quark and an antiquark, with integer spin. The pion is the lightest. The J/ψ (charm and anticharm) and the Υ (bottom and antibottom) revealed the charm and bottom quarks.

See also Hadron, Pion, J/ψ meson

A quantity counting quarks minus antiquarks, divided by three: $+1$ for a proton or neutron, $-1$ for their antiparticles, 0 for mesons and leptons. It is conserved in every reaction yet observed, which is why the proton is stable. In the Standard Model it is only an accidental symmetry: rare electroweak processes called sphalerons, never observed but expected to have been frequent in the hot early universe, change baryon and lepton number together while conserving their difference. The universe's excess of baryons requires some violation of it.

See also Hadron, Baryons, Baryogenesis

The quarks that give a hadron its quantum numbers: uud for the proton, udd for the neutron. In the folios' drawings the coloured spheres are valence quarks; the colour stands for colour charge and the letter for flavour.

In the folios: Subatomic Matter › Inside nucleons: QCD; Mass in the Quantum World › Most visible mass comes from QCDSee also Sea quarks, Parton, Proton

Quark–antiquark pairs constantly created and annihilated by the gluon field inside a hadron. They carry no net quantum numbers but share in its momentum, spin and mass; even the proton contains strange sea quarks.

In the folios: Subatomic Matter › Inside nucleons: QCD; Mass in the Quantum World › Most visible mass comes from QCDSee also Valence quark, Parton, Sigma terms

Richard Feynman's 1969 name for the point-like constituents seen inside protons in high-energy scattering, now identified with quarks and gluons. In a hard collision at the LHC it is partons, each carrying a fraction of a proton's momentum, that actually collide.

See also Deep inelastic scattering, Valence quark, Sea quarks

The scattering of high-energy electrons, or neutrinos, off protons with a large transfer of momentum, breaking the proton up. Experiments at SLAC from 1968 found electrons rebounding from hard, point-like objects inside it: the first evidence that quarks are real (Nobel Prize 1990 to Jerome Friedman, Henry Kendall and Richard Taylor).

In the folio: Subatomic Matter › MilestonesSee also Parton, Quark, SLAC

The 1964 proposal by Gell-Mann and, independently, Zweig that the many hadrons then known are built from three kinds of quark: up, down and strange. It explained the patterns of hadron masses and charges, although the quarks were at first widely regarded as bookkeeping.

In the folio: Subatomic Matter › MilestonesSee also Quark, Hadron, Deep inelastic scattering

The root-mean-square radius of the proton's charge, 0.84 fm. Measurements with muonic hydrogen in 2010 gave a smaller value than older electron data, the "proton radius puzzle"; most newer electron and hydrogen measurements agree with the smaller value.

In the folio: Subatomic Matter › Inside nucleons: QCDSee also Proton, Femtometre

Because quarks are confined, their masses cannot be weighed directly; they are parameters of QCD that depend on the scheme and energy scale in which they are defined. Light-quark masses are quoted in the $\overline{\text{MS}}$ scheme at 2 GeV (up 2.16, down 4.70, strange 93.5 MeV), charm and bottom at their own mass scale (1.27 and 4.18 GeV), and top from direct reconstruction (172.6 GeV). These "current" masses come from Higgs couplings; the "constituent" masses of about 300 MeV used in simple quark models include QCD binding energy.

In the folios: Subatomic Matter › The Standard Model; Mass in the Quantum World › The Higgs field and elementary masses; Mass in the Quantum World › Most visible mass comes from QCDSee also MS-bar scheme, Yukawa coupling, Proton mass decomposition, Chiral symmetry breaking

The modified minimal-subtraction scheme, the standard prescription for removing infinities from QCD calculations, and so for defining quark masses and couplings. Quantities defined in it depend on a chosen energy scale, often 2 GeV. Different schemes split the proton's energy between quarks and gluons differently, which is why the folio calls that split scheme-dependent.

In the folio: Mass in the Quantum World › Most visible mass comes from QCDSee also Quark mass, Proton mass decomposition

QCD computed numerically on a grid of points in space and time, the only first-principles method where the strong coupling is large. It reproduces hadron masses to about a percent, gives the folio's decomposition of the proton's mass, and by 2025 had brought the prediction for the muon's magnetic moment into line with experiment.

In the folios: Mass in the Quantum World › Most visible mass comes from QCD; Subatomic Matter › Beyond the Standard ModelSee also Quantum chromodynamics, Proton mass decomposition, Muon g−2

A lattice-QCD breakdown (Yang and colleagues, 2018) of where the proton's 938 MeV comes from: quark masses, the sigma terms, about 9%; the kinetic and interaction energy of the quarks about 33%; the gluon field's energy about 37%; and the trace anomaly about 23%. Rounded, they sum to 102%; the uncertainties are a few points, and the split between quarks and gluons depends on the renormalisation scheme. The headline is robust: Higgs-given quark masses supply only about a tenth.

In the folio: Mass in the Quantum World › Most visible mass comes from QCDSee also Sigma terms, Trace anomaly, Energy–momentum tensor, Mass is not additive

The contributions of the up, down and strange quark masses, sea quarks included, to a hadron's mass. For the proton they total about 9%, although the valence quarks' masses alone add up to only about 1%.

In the folio: Mass in the Quantum World › Most visible mass comes from QCDSee also Proton mass decomposition, Sea quarks, Quark mass

A purely quantum contribution to hadron masses. Without quark masses classical QCD contains no scale at all, but quantising it introduces one, ${\Lambda }_{\text{QCD}}$, through the running of the coupling. This breaking of scale invariance contributes about a quarter of the proton's mass.

In the folio: Mass in the Quantum World › Most visible mass comes from QCDSee also QCD scale, Running coupling, Proton mass decomposition

The field-theory quantity recording the density and flow of energy and momentum, and the source of gravity in general relativity. The proton-mass decomposition splits the proton's energy, $⟨T^{00}⟩$, into quark and gluon pieces.

In the folio: Mass in the Quantum World › Most visible mass comes from QCDSee also Proton mass decomposition, Gravity

The QCD vacuum is filled with a condensate of quark–antiquark pairs, which gives the nearly massless light quarks a large effective mass inside hadrons. The pions are so light because they are the almost massless excitations this breaking leaves behind.

See also Pion, Quark mass, Proton mass decomposition

A state of matter in which quarks and gluons are not confined in hadrons, above about 155 MeV ($1.8\times 10^{12}$ K). It filled the universe for its first ten microseconds or so, and is recreated briefly in collisions of heavy nuclei at the LHC and RHIC, where it behaves as an almost perfect liquid.

See also QCD crossover, Energy as temperature

A meson of a charm quark and its antiquark, 3.097 GeV/$c^2$, discovered at the same time at Brookhaven (Samuel Ting) and SLAC (Burton Richter) in November 1974. Its long lifetime for so heavy a particle could only be explained by a new, fourth quark; the episode is called the November Revolution.

In the folio: Subatomic Matter › MilestonesSee also Charm quark, Meson, Cornell potential

The quantum field theory of the known elementary particles and of the strong, electromagnetic and weak interactions. It has the gauge symmetry $SU(3)\times SU(2)\times U(1)$, three generations of quarks and leptons, and a Higgs field that breaks the electroweak symmetry. Its predictions agree with experiment, some to better than a part in a billion, but it omits gravity, dark matter and neutrino masses, and has about 19 free parameters.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › Beyond the Standard ModelSee also Free parameters, Gauge theory, Generation, Beyond the Standard Model

A field theory whose interactions follow from a local symmetry: demanding that a phase or rotation of the fields may vary from point to point requires a force-carrying field. Electromagnetism follows from $U(1)$, the weak interaction from $SU(2)$, QCD from $SU(3)$. Each generator of the symmetry is a gauge boson: one photon, three weak bosons, eight gluons.

See also Gauge boson, Standard Model

A spin-1 force carrier: the photon, the eight gluons, and the W⁺, W⁻ and Z. The folio's "four kinds of force carrier" are the photon, gluon, W and Z.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › The four interactionsSee also Gauge theory, Photon, Gluon, W boson, Z boson

A spin-0 particle. The Higgs boson is the only elementary scalar known.

In the folio: Subatomic Matter › The Standard ModelSee also Higgs boson, Spin

One of three families of matter particles, each a copy of the others except in mass: (up, down, electron, electron neutrino), (charm, strange, muon, muon neutrino) and (top, bottom, tau, tau neutrino). Ordinary matter is made entirely of the first. Measurements of the Z boson show there are exactly three light neutrinos, but why there are three generations is unknown.

In the folios: Subatomic Matter › The Standard Model; Mass in the Quantum World › What we still do not knowSee also Quark, Lepton, Z boson, Fermion mass hierarchy

An elementary fermion that does not feel the strong force: the charged electron, muon and tau, and their three neutrinos. The charged leptons feel electromagnetism and the weak force; neutrinos only the weak force and gravity.

In the folio: Subatomic Matter › The Standard ModelSee also Electron, Muon, Tau, Neutrino

The lightest quark: charge $+\frac{2}{3}$, mass about 2.16 MeV/$c^2$. Two up quarks and a down quark make a proton.

In the folio: Subatomic Matter › The Standard ModelSee also Down quark, Proton, Quark mass

Charge $-\frac{1}{3}$, mass about 4.70 MeV/$c^2$. The proton has one and the neutron two; in beta decay a down quark turns into an up quark.

In the folio: Subatomic Matter › The Standard ModelSee also Up quark, Neutron, Beta decay

The second-generation partner of the down quark: charge $-\frac{1}{3}$, mass about 93.5 MeV/$c^2$. It is named after the "strange", unexpectedly long-lived particles, such as kaons, found in cosmic rays in the late 1940s.

In the folio: Subatomic Matter › The Standard ModelSee also Quark flavour, Sea quarks

Charge $+\frac{2}{3}$, mass about 1.27 GeV/$c^2$. Sheldon Glashow, John Iliopoulos and Luciano Maiani proposed it in 1970 to explain why certain decays are so rare; it was found in 1974 in the J/ψ meson.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › MilestonesSee also J/ψ meson, Strange quark

Charge $-\frac{1}{3}$, mass about 4.18 GeV/$c^2$; also called beauty. Leon Lederman's group at Fermilab discovered it in 1977 in the Υ (upsilon) meson.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › MilestonesSee also Top quark, Fermilab

The heaviest known elementary particle: charge $+\frac{2}{3}$, mass about 172.6 GeV/$c^2$, almost as heavy as a gold atom. It decays in about $5\times 10^{-25}$ s, too quickly to form hadrons. The CDF and D0 experiments at Fermilab's Tevatron discovered it in 1995. Its Yukawa coupling is very close to one.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › MilestonesSee also Yukawa coupling, Fermilab, Fermion mass hierarchy

The lightest charged lepton: charge $-e$, spin ½, mass 0.511 MeV/$c^2$ ($9.109\times 10^{-31}$ kg), with no internal structure found. J. J. Thomson identified it in cathode rays in 1897, the first elementary particle discovered.

In the folios: Subatomic Matter › The Standard Model; Mass in the Quantum World › Particles are field excitations; Subatomic Matter › MilestonesSee also Lepton, Positron, Quantum field

A heavier copy of the electron, 105.7 MeV/$c^2$ (207 electron masses), living 2.2 μs on average. Carl Anderson and Seth Neddermeyer found it in cosmic rays in 1936, prompting Isidor Rabi's "Who ordered that?". Muons pass through almost everything, so they are the charged particles seen outside a detector's calorimeters.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › How detectors see particles; Subatomic Matter › MilestonesSee also Lepton, Muon chambers, Muon g−2

The heaviest charged lepton, 1.777 GeV/$c^2$, about 3500 electron masses; heavy enough to decay into hadrons, it lives only $2.9\times 10^{-13}$ s. Martin Perl's group at SLAC discovered it in 1975.

In the folio: Subatomic Matter › The Standard ModelSee also Lepton, Tau neutrino, SLAC

A neutral lepton that feels only the weak force and gravity, so it passes through matter almost unhindered: some 65 billion neutrinos from the Sun cross every square centimetre of Earth each second. There are three flavours. Pauli proposed the neutrino in 1930, and Clyde Cowan and Frederick Reines first detected it in 1956, at a nuclear reactor. The Standard Model made neutrinos massless; their oscillations show they are not. In a detector a neutrino escapes unseen, inferred from missing momentum.

In the folios: Subatomic Matter › The Standard Model; Subatomic Matter › How detectors see particles; Subatomic Matter › Milestones; Mass in the Quantum World › What we still do not knowSee also Electron neutrino, Muon neutrino, Tau neutrino, Neutrino oscillation, Neutrino mass, Missing transverse momentum

The neutrino of the first generation, produced in beta decay and in the Sun. Measurements of the tritium beta spectrum limit its effective mass to below 0.45 eV (KATRIN, 2025).

In the folio: Subatomic Matter › The Standard ModelSee also Neutrino, Neutrino mass, KATRIN

The neutrino produced with muons. Leon Lederman, Melvin Schwartz and Jack Steinberger showed at Brookhaven in 1962 that it differs from the electron neutrino, the first sign that neutrinos come in kinds. Oscillations tie its mass to the same scale as the others'.

In the folio: Subatomic Matter › The Standard ModelSee also Neutrino, Neutrino oscillation

The third-generation neutrino, and the last Standard Model fermion to be observed directly: the DONUT experiment at Fermilab saw its interactions in 2000.

In the folio: Subatomic Matter › The Standard Model; Subatomic Matter › MilestonesSee also Neutrino, Tau, Fermilab

Neutrinos have mass, but very little. Oscillations measure differences of squared masses, which require at least two of the three to be heavier than about 0.009 and 0.05 eV. The direct limit on the effective electron-neutrino mass is 0.45 eV (KATRIN 2025), at least a million times below the electron's, and cosmology bounds the sum of all three more tightly still. The absolute scale, the ordering of the masses and whether neutrinos are their own antiparticles are open, and the minimal Standard Model has no mechanism to give them mass at all.

$m_{\beta }={\left(\sum _i|U_{ei}{|}^2{m}_i^2\right)}^{1/2}<0.45\text{eV}$ (90% confidence)

In the folios: Subatomic Matter › The Standard Model; Subatomic Matter › Beyond the Standard Model; Subatomic Matter › Open problems; Mass in the Quantum World › The Higgs field and elementary masses; Mass in the Quantum World › What we still do not knowSee also Neutrino oscillation, KATRIN, Mass ordering, Majorana neutrino, Seesaw mechanism, Cosmic neutrino background

Neutrinos are produced and detected in flavour states (${\nu }_e$, ${\nu }_{\mu }$, ${\nu }_{\tau }$) that are mixtures of states of definite mass. As they travel, the phases of the mass components advance at different rates, so the flavour mix changes periodically with distance over energy. Super-Kamiokande's 1998 atmospheric-neutrino result, followed by SNO's solar measurements, established it and so proved that neutrinos have mass (Nobel Prize 2015 to Takaaki Kajita and Arthur McDonald).

$P({\nu }_{\alpha }\to {\nu }_{\beta })={\sin }^{2}2\theta {\sin }^2\left(\frac{\Delta m^2{c}^{4}L}{4\hslash cE}\right)$ (two flavours)

In the folio: Subatomic Matter › Milestones; Subatomic Matter › Beyond the Standard ModelSee also Neutrino mass, Super-Kamiokande, Quantum phase, PMNS matrix

The $3\times 3$ Pontecorvo–Maki–Nakagawa–Sakata matrix relating neutrino flavour states to mass states: three mixing angles, all measured, and a CP-violating phase that is not yet well determined. Unlike the quarks' mixing, the neutrinos' angles are large.

See also Neutrino oscillation, CKM matrix, CP violation

Whether the third neutrino mass state is the heaviest (normal ordering) or the lightest (inverted ordering). Oscillation experiments such as JUNO, DUNE and Hyper-Kamiokande aim to settle it.

See also Neutrino mass, Neutrino oscillation

The quantum of the electromagnetic field: massless, spin 1, two polarisations, coupling to electric charge with strength $\alpha \approx 1/137$. It is its own antiparticle and carries no charge, so, unlike gluons, photons do not couple to one another directly (they scatter off each other only through loops of virtual charged particles, first seen at the LHC in 2017), and the force has infinite range. In a detector a photon leaves no track and showers in the electromagnetic calorimeter.

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The Standard Model; Subatomic Matter › The four interactions; Subatomic Matter › How detectors see particles; Mass in the Quantum World › Mass is rest energySee also Photon polarisation, Electromagnetic interaction, Fine-structure constant, Massless particle

The charged carrier of the weak force, 80.37 GeV/$c^2$. It changes quark and lepton flavour, as in beta decay. Its mass, acquired through electroweak symmetry breaking, confines the force to about $10^{-18}$ m. The UA1 and UA2 experiments at CERN discovered it in 1983 (Nobel Prize 1984 to Carlo Rubbia and Simon van der Meer).

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The Standard Model; Subatomic Matter › The four interactions; Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › MilestonesSee also Weak interaction, Z boson, Electroweak symmetry breaking, Beta decay

The neutral carrier of the weak force, 91.19 GeV/$c^2$, which acts without changing flavour. Discovered at CERN in 1983, months after the W. Millions of Z bosons made at CERN's LEP collider in 1989–2000 tested the Standard Model precisely and showed there are three light neutrino species. Its mass is the reference scale for the strong coupling.

In the folios: Subatomic Matter › The Standard Model; Subatomic Matter › The four interactions; Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › MilestonesSee also W boson, Weak interaction, Generation, Strong coupling

The quantum of the Higgs field, a ripple about its vacuum value: spin 0, no charge, 125.2 GeV/$c^2$. ATLAS and CMS at the LHC announced its discovery on 4 July 2012, 48 years after it was predicted. Its couplings to other particles grow in proportion to their masses, as predicted, which has been checked to about 10%.

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The Standard Model; Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › MilestonesSee also Higgs field, Higgs mechanism, Large Hadron Collider, Hierarchy problem

The hypothetical quantum of gravity: massless, spin 2. None has been detected. A single graviton interacts so weakly that detecting one is extraordinarily difficult; recent proposals suggest that massive quantum sensors could register single-graviton exchange with a passing gravitational wave, although even that would not on its own prove that gravity is quantised. Gravitational waves behave as classical waves made of very many.

In the folio: Subatomic Matter › The four interactions; Subatomic Matter › Beyond the Standard ModelSee also Gravity, Quantum gravity, Gravitational waves

The constants the Standard Model cannot predict and takes from measurement: six quark masses, three charged-lepton masses, three gauge couplings, four quark-mixing parameters, the Higgs mass and vacuum value, and the QCD vacuum angle θ, 19 in all. Neutrino masses and mixing add seven more (three masses, three angles, one phase), making 26, or 28 if neutrinos are their own antiparticles.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Standard Model, CKM matrix, PMNS matrix, Strong CP problem

The $3\times 3$ Cabibbo–Kobayashi–Maskawa matrix giving the strength of weak transitions between quark flavours: up to down is strong, up to bottom very weak. Its single complex phase is the Standard Model's source of CP violation among quarks (Nobel Prize 2008 to Makoto Kobayashi and Toshihide Maskawa).

See also Quark flavour, CP violation, PMNS matrix

An asymmetry between the laws for particles and those for their mirror-imaged antiparticles. Discovered in kaon decays in 1964 by James Cronin and Val Fitch, it is needed to make more matter than antimatter, but the Standard Model has far too little to explain the matter in the universe.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Baryogenesis, CKM matrix, Strong CP problem, Matter–antimatter asymmetry

The four ways particles act on one another: strong, electromagnetic, weak and gravitational. At nuclear distances their relative strengths are roughly 1, $10^{-2}$, $10^{-6}$ and $10^{-38}$, and their ranges about $10^{-15}$ m, infinite, $10^{-18}$ m and infinite. Such comparisons are only orders of magnitude, since the strengths depend on distance, energy and the particles involved.

In the folio: Subatomic Matter › The four interactionsSee also Strong interaction, Electromagnetic interaction, Weak interaction, Gravity, Range of a force

The force carried by eight gluons and acting on colour charge, so on quarks and gluons; described by QCD. It binds quarks into hadrons and, as a residual force, nucleons into nuclei. Its effective range of about $10^{-15}$ m comes from confinement, not from any mass of the gluons.

In the folio: Subatomic Matter › The four interactions; Subatomic Matter › Inside nucleons: QCDSee also Quantum chromodynamics, Residual strong force, Colour confinement

The force carried by the photon and acting on electric charge, with infinite range. Described by quantum electrodynamics, it holds atoms and molecules together and is responsible for chemistry, light and almost everything of everyday experience.

In the folio: Subatomic Matter › The four interactionsSee also Quantum electrodynamics, Photon, Fine-structure constant, Coulomb potential

The force carried by the W⁺, W⁻ and Z bosons, acting on all fermions and the only one that changes flavour. It drives beta decay and the first step of fusion in the Sun. It is weak at low energies only because its carriers are heavy, which confines it to about $10^{-18}$ m; above about 100 GeV it is as strong as electromagnetism. It distinguishes left from right, as Chien-Shiung Wu showed in 1957.

Fermi constant $G_F/(\hslash c{)}^3=1.166\times 10^{-5}{\text{GeV}}^{-2}$

In the folio: Subatomic Matter › The four interactions; Subatomic Matter › Particles are excitations of quantum fieldsSee also W boson, Z boson, Beta decay, Electroweak theory

The attraction that acts on all energy and momentum, the weakest interaction by far: between two protons it is about $10^{36}$ times weaker than their electric repulsion. It still dominates on large scales, because mass comes in only one sign and adds up. General relativity describes it classically; there is no tested quantum theory of it.

In the folios: Subatomic Matter › The four interactions; Subatomic Matter › Beyond the Standard Model; Subatomic Matter › Open problems; Mass in the Quantum World › What we still do not knowSee also General relativity, Graviton, Quantum gravity, Energy–momentum tensor

Einstein's 1915 theory of gravity as the curvature of spacetime by energy and momentum. It has passed every test from the Solar System to merging black holes and underlies the Friedmann equations of cosmology. It is a classical theory. Treated as a quantum field theory it works at low energies, but there is no complete, tested quantum theory of gravity.

In the folio: Subatomic Matter › The four interactionsSee also Gravity, Friedmann equations, Gravitational waves, Quantum gravity

The quantum field theory of charged particles and photons, completed by Richard Feynman, Julian Schwinger, Sin-Itiro Tomonaga and Freeman Dyson around 1948. Its prediction of the electron's magnetic moment agrees with measurement to better than a part in a billion, the most precise test of any theory.

In the folio: Subatomic Matter › The four interactionsSee also Electromagnetic interaction, Fine-structure constant, Photon

The dimensionless strength of electromagnetism, about 1/137 at low energies and about 1/128 at the Z mass. It sets the size of atoms and the splitting of their spectral lines, from which it takes its name.

$\alpha =\frac{e^2}{4\pi {\epsilon }_0\hslash c}\approx \frac{1}{137.036}$

In the folio: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The four interactionsSee also Running coupling, Quantum electrodynamics, Strong coupling

The unification of the electromagnetic and weak interactions as an $SU(2)\times U(1)$ gauge theory broken by the Higgs field, by Sheldon Glashow (1961), Steven Weinberg (1967) and Abdus Salam (1968); Nobel Prize 1979. It predicted the Z boson and weak neutral currents, seen at CERN in 1973, and the W and Z masses. Above about 100 GeV the two forces are aspects of one.

In the folios: Subatomic Matter › The four interactions; Mass in the Quantum World › The Higgs field and elementary massesSee also Electroweak symmetry breaking, W boson, Z boson, Electroweak crossover

In quantum field theory an interaction between particles is described as the exchange of quanta of a force field: photons for electromagnetism, gluons for the strong force, W and Z bosons for the weak force, and perhaps gravitons for gravity. The exchanged quanta are virtual, and the carrier's mass sets the force's range.

In the folio: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The four interactionsSee also Gauge boson, Range of a force, Virtual particle

An intermediate quantum in a calculation that does not satisfy $E^2=(pc{)}^2+(m{c}^2{)}^2$ and cannot be observed directly. It is the language of Feynman diagrams, a picture of a field's response rather than a particle in flight.

See also Force carrier, Energy–momentum relation

A force carried by a massless field has a $1/r$ potential and reaches indefinitely far; one carried by a field of mass $M$ is cut off exponentially beyond the carrier's reduced Compton wavelength, about $2.5\times 10^{-18}$ m for the W and 1.4 fm for the pion. The strong force is short-ranged for a different reason, confinement.

$V(r)\propto \frac{e^{-r/\lambda }}{r}$, $\lambda =\frac{\hslash }{Mc}$

In the folio: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The four interactionsSee also Yukawa potential, Coulomb potential, Compton wavelength

The $1/r$ potential energy between two charges, the dashed "massless photon" curve in the folio's weak-force panel; the force falls as $1/r^2$.

$V(r)=\frac{q_1{q}_2}{4\pi {\epsilon }_{0}r}$

In the folio: Subatomic Matter › Particles are excitations of quantum fieldsSee also Range of a force, Electromagnetic interaction

The potential of a force with a massive carrier, $e^{-r/\lambda }/r$, which Hideki Yukawa proposed in 1935 for the nuclear force carried by the pion. The folio's solid curve for the W and Z has this form.

$V(r)=-g^2\frac{e^{-r/\lambda }}{r}$

In the folio: Subatomic Matter › Particles are excitations of quantum fields; Subatomic Matter › The nuclear scaleSee also Range of a force, Pion, Residual strong force

A theory that would describe gravity, and with it spacetime, in quantum terms, needed at the Planck scale, in the earliest universe and inside black holes. String theory and loop quantum gravity are leading candidates, and none has been tested. General relativity treated as a quantum field theory works at low energies but fails near the Planck energy.

In the folios: Subatomic Matter › Beyond the Standard Model; Subatomic Matter › Open problems; Mass in the Quantum World › What we still do not knowSee also Planck scale, Graviton, Planck era

The energy, about $1.22\times 10^{19}$ GeV, and length, $1.6\times 10^{-35}$ m, built from $\hslash$, $c$ and $G$, at which quantum effects of gravity become strong. The Higgs boson is about $10^{17}$ times lighter than the Planck mass: the hierarchy problem.

$m_P=\sqrt{\hslash c/G}=2.18\times 10^{-8}\text{kg}$, ${\ell }_P=\sqrt{\hslash G/c^3}=1.6\times 10^{-35}\text{m}$

In the folios: Subatomic Matter › Open problems; Mass in the Quantum World › What we still do not knowSee also Planck era, Hierarchy problem, Quantum gravity

A scalar field, a doublet of complex fields, filling space, whose lowest-energy state is not at zero: in the vacuum it has a value set by $v\approx 246$ GeV everywhere. Through it the W and Z bosons and the fermions acquire mass, and its quantum is the Higgs boson. It does not give the proton most of its mass.

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Mass in the Quantum World › The Higgs field and elementary massesSee also Higgs potential, Vacuum expectation value, Higgs mechanism, Higgs boson

The energy density of the Higgs field as a function of its value, shaped like a sombrero. Its centre, $\phi =0$, is symmetric but a hilltop, not the lowest energy; the minimum is a circular trough at $|\phi |=v/\sqrt{2}$. The field settles somewhere in the trough, and in doing so breaks the symmetry. The Subatomic Matter folio draws a slice through it.

$V(\phi )=\lambda {\left(|\phi {|}^2-\frac{v^2}{2}\right)}^2$

In the folios: Subatomic Matter › Particles are excitations of quantum fields; Mass in the Quantum World › The Higgs field and elementary massesSee also Spontaneous symmetry breaking, Vacuum expectation value, Higgs self-coupling

The value a field takes in the vacuum. For the Higgs field $|⟨\phi ⟩|=v/\sqrt{2}$, with $v\approx 246$ GeV fixed by the strength of the weak interaction; it sets the scale of every elementary mass.

$v=(\sqrt{2}{G}_F{)}^{-1/2}\approx 246.22\text{GeV}$

In the folios: Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › Particles are excitations of quantum fieldsSee also Higgs potential, Weak interaction, Yukawa coupling

A symmetry of the laws that the lowest-energy state does not share. A ball balanced on the hat's central peak is symmetric; in the trough it has chosen a direction. The equations keep the symmetry while the vacuum hides it, as in a magnet, whose atomic laws have no preferred direction although its magnetisation does.

In the folios: Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › Particles are excitations of quantum fieldsSee also Electroweak symmetry breaking, Higgs potential

The spontaneous breaking of $SU(2)\times U(1)$ down to the $U(1)$ of electromagnetism by the Higgs vacuum. Three of the Higgs field's four components become the extra polarisation states of the W⁺, W⁻ and Z, which become massive; the photon, belonging to the unbroken combination, stays massless, and the gluon, which the Higgs does not touch, stays massless too. In the early universe it happened at about 160 GeV, as a smooth crossover.

$m_W=\frac{1}{2}gv\approx 80.4\text{GeV}$, $m_Z=m_W/\cos {\theta }_W\approx 91.2\text{GeV}$

In the folio: Mass in the Quantum World › The Higgs field and elementary massesSee also Electroweak crossover, Higgs mechanism, Weak mixing angle

The angle describing how the photon and the Z boson are mixtures of the $SU(2)$ and $U(1)$ gauge fields. It relates the W and Z masses.

$\cos {\theta }_W=m_W/m_Z$, ${\sin }^2{\theta }_W\approx 0.22$–$0.23$ (depending on the scheme)

See also Electroweak symmetry breaking, Z boson

The way a gauge boson acquires mass through spontaneous symmetry breaking without spoiling the gauge symmetry. It was proposed in 1964 by François Englert and Robert Brout, by Peter Higgs, and by Gerald Guralnik, Carl Hagen and Tom Kibble; Englert and Higgs shared the 2013 Nobel Prize after the boson's discovery.

In the folios: Mass in the Quantum World › The Higgs field and elementary masses; Subatomic Matter › MilestonesSee also Electroweak symmetry breaking, Higgs boson, Yukawa coupling

The strength with which a fermion couples to the Higgs field; its mass is this coupling times the vacuum value. It is about $2.9\times 10^{-6}$ for the electron and 0.99 for the top quark. The couplings are free parameters with no known explanation. They are named after Hideki Yukawa, whose coupling of nucleons to pions had the same form.

$m_f=\frac{y_{f}v}{\sqrt{2}}$

In the folio: Mass in the Quantum World › The Higgs field and elementary massesSee also Fermion mass hierarchy, Vacuum expectation value, Free parameters

The charged fermions' masses span more than five orders of magnitude, from the electron's 0.511 MeV/$c^2$ to the top quark's 172.6 GeV/$c^2$, a factor of about 340,000, and the neutrinos are at least a million times lighter again. Masses rise from generation to generation, but nothing in the Standard Model explains the pattern, or why there are three generations.

In the folio: Mass in the Quantum World › The Higgs field and elementary masses; Mass in the Quantum World › What we still do not knowSee also Yukawa coupling, Generation, Neutrino mass

The coefficient that sets the depth of the Higgs potential, fixed by the boson's mass at about 0.13. Measuring it directly, through the production of Higgs-boson pairs, is a main goal of the high-luminosity LHC, since it decides the shape of the potential and the long-term stability of the vacuum.

$m_H=\sqrt{2\lambda }v$

See also Higgs potential, Higgs boson

Why the Higgs boson, at 125 GeV, is about $10^{17}$ times lighter than the Planck scale. Quantum corrections to a scalar's mass grow with the highest energy scale in the theory, so a light Higgs needs either new physics near the TeV scale to cancel them, supersymmetry being the favourite, or an extraordinarily fine cancellation. The LHC has so far found no such new particles.

In the folios: Subatomic Matter › Open problems; Mass in the Quantum World › What we still do not knowSee also Planck scale, Supersymmetry, Higgs boson

An instrument built in layers around the collision point: a tracker in a magnetic field, an electromagnetic calorimeter, a hadronic calorimeter, the magnet, and muon chambers outside. Each species leaves its own signature: a photon no track and an electromagnetic shower; an electron a curved track and an electromagnetic shower; a charged hadron a track and a hadronic shower; a neutral hadron only a hadronic shower; a muon a track through everything; a neutrino nothing at all.

In the folio: Subatomic Matter › How detectors see particlesSee also Tracker, Electromagnetic calorimeter, Hadronic calorimeter, Muon chambers, Missing transverse momentum

The place inside a detector where the beams cross and collisions happen. Tracks are traced back to it, and to secondary vertices millimetres away that betray the decays of bottom hadrons and tau leptons.

In the folio: Subatomic Matter › How detectors see particlesSee also Tracker, Particle detector

The innermost layers, made of silicon pixels and strips at the LHC, which record the points where charged particles pass and join them into tracks. In the solenoid's field, which runs along the beam, tracks spiral: the radius of curvature in the plane across the beam gives the transverse momentum, the track's angle to the beam gives the rest, and the direction of bending gives the sign of the charge. Neutral particles leave no track.

$p_T[\text{GeV}/c]\approx 0.3|q/e|B[\text{T}]r[\text{m}]$, with $r$ the radius in the plane perpendicular to $B$

In the folio: Subatomic Matter › How detectors see particlesSee also Detector magnet, Particle detector

A large magnet that bends charged tracks so their momenta can be measured. CMS is built around a superconducting solenoid of 3.8 T; ATLAS uses a 2 T solenoid inside large toroids.

In the folio: Subatomic Matter › How detectors see particlesSee also Tracker, ATLAS and CMS

A detector that measures a particle's energy by absorbing it: a particle entering dense material starts a shower of secondaries whose total signal is proportional to its energy.

In the folio: Subatomic Matter › How detectors see particlesSee also Electromagnetic calorimeter, Hadronic calorimeter, Particle shower

A cascade in dense material. A fast electron radiates photons, each photon makes an electron–positron pair, and the process repeats until the energy per particle is too low. Electromagnetic showers are compact, a few radiation lengths (0.89 cm in the lead tungstate of the CMS crystals); hadronic showers, driven by nuclear collisions, are broader and deeper (an interaction length is about 17 cm of iron).

In the folio: Subatomic Matter › How detectors see particlesSee also Pair creation, Electromagnetic calorimeter, Hadronic calorimeter

The layer that absorbs and measures electrons and photons. CMS's uses about 76,000 lead-tungstate crystals and was decisive in seeing the Higgs boson decay to two photons. Electrons and photons are told apart by whether a track points to the shower.

In the folio: Subatomic Matter › How detectors see particlesSee also Particle shower, Calorimeter, Higgs boson

The thicker layer, of dense metal interleaved with detectors, that absorbs hadrons through their nuclear interactions and measures the energy of jets and of neutral hadrons.

In the folio: Subatomic Matter › How detectors see particlesSee also Particle shower, Jet, Charged and neutral hadrons

Charged hadrons, such as pions, kaons and protons, leave a track and a hadronic shower; neutral ones, such as neutrons and long-lived neutral kaons, leave only the shower. Together they make up most of a jet.

In the folio: Subatomic Matter › How detectors see particlesSee also Hadron, Hadronic calorimeter, Jet

The outermost layers. A muon, about 200 times heavier than an electron, radiates little and does not feel the strong force, so it passes through the calorimeters and iron, the only charged particle that does. Its track outside identifies and measures it.

In the folio: Subatomic Matter › How detectors see particlesSee also Muon, Particle detector

The momentum imbalance perpendicular to the beams. Before a collision the momentum across the beam is zero, so if the visible products do not balance, something invisible may have carried off the difference: a neutrino, or a new particle such as dark matter. Mismeasured jets, detector noise and particles escaping the detector's coverage can fake an imbalance, and searches must control them first. Only the transverse part is used because the colliding partons' momenta along the beam are not known.

In the folio: Subatomic Matter › How detectors see particles; Subatomic Matter › Searching for dark matter particlesSee also Transverse momentum, Neutrino, Collider searches for dark matter

The component of a particle's momentum perpendicular to the beam axis. High transverse momentum marks the hard, head-on collisions that make heavy particles.

See also Missing transverse momentum

Short-lived particles decay long before reaching any detector (the Higgs boson within about $10^{-22}$ s), so they are found by combining the energies and momenta of their decay products into an invariant mass. Genuine decays pile up in a peak at the parent's mass above a smooth background: the Higgs boson appeared as a bump near 125 GeV in the two-photon and four-lepton spectra.

$M^2{c}^4={\left(\sum E_i\right)}^2-{\left|\sum {\mathbf{p}}_i\right|}^2{c}^2$

In the folio: Subatomic Matter › How detectors see particlesSee also Invariant mass, Higgs boson

A machine that accelerates two beams in opposite directions and collides them head-on, so that all their energy is available to make new particles; firing one beam at a fixed target wastes most of it on recoil. The LHC, Tevatron, LEP and PETRA are colliders.

In the folio: Subatomic Matter › How detectors see particles; Subatomic Matter › Searching for dark matter particlesSee also Large Hadron Collider, Threshold energy, Centre-of-momentum frame

Any physics not contained in the Standard Model. Established facts call for it: neutrinos have mass; dark matter exists, and most explanations of it need new particles; matter outnumbers antimatter; and gravity still lacks a quantum theory. The folio's proposals include supersymmetry, axions, sterile neutrinos, extra Higgs bosons and the graviton.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Neutrino mass, Dark matter, Matter–antimatter asymmetry, Quantum gravity, Supersymmetry

The observable universe is made of matter: antimatter galaxies would betray themselves by gamma rays where they met matter, and none are seen. The small primordial excess that survived, about one baryon per $1.6\times 10^9$ photons, needs more CP violation than the Standard Model provides.

In the folio: Subatomic Matter › Beyond the Standard Model; Subatomic Matter › Open problemsSee also Baryogenesis, CP violation, Antimatter, Baryon-to-photon ratio

A proposed symmetry pairing every fermion with a boson and every boson with a fermion: squarks, sleptons, gluinos and so on. It would cancel the quantum corrections behind the hierarchy problem, help the forces unify at high energy, and, in versions where the lightest superpartner is stable and neutral (R-parity conservation), supply a dark-matter candidate. The LHC has found no superpartners. In simplified models it excludes gluinos up to about 2.3 TeV and light-flavour squarks up to about 1.8 TeV, but limits depend on the assumed spectrum and decays and are much weaker for the electroweak partners and for compressed spectra.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Hierarchy problem, WIMP, Large Hadron Collider

Many extensions of the Standard Model, supersymmetry among them, have more than one Higgs doublet, and so predict further neutral and charged scalar bosons. None has been seen, and the 125 GeV boson so far behaves as the Standard Model's single Higgs.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Higgs boson, Supersymmetry

A hypothetical neutrino with no weak interaction, felt only through gravity and its mixing with ordinary neutrinos. Light ones have been proposed to explain anomalies in short-baseline experiments; keV-mass ones are dark-matter candidates that would show up as kinks in beta-decay spectra or as X-ray lines from their decay. KATRIN's planned TRISTAN upgrade is designed to search the tritium spectrum for them.

In the folio: Subatomic Matter › Searching for dark matter particles; Subatomic Matter › Beyond the Standard ModelSee also Neutrino mass, KATRIN, Dark matter, Seesaw mechanism

A neutrino that is its own antiparticle, which is possible because it carries no electric charge. If neutrinos are Majorana particles, lepton number is not conserved and neutrinoless double beta decay becomes possible, although its rate could be too small to observe.

In the folios: Subatomic Matter › Beyond the Standard Model; Mass in the Quantum World › What we still do not knowSee also Antiparticle, Neutrinoless double beta decay, Seesaw mechanism

A hypothetical decay in which two neutrons in a nucleus become two protons, emitting two electrons and no neutrinos. Seeing it would show that neutrinos are Majorana particles. If it proceeds by the exchange of light neutrinos, its rate measures the effective mass $m_{\beta \beta }=|{\sum }_i{U}_{ei}^2{m}_i|$, in which phases can partly cancel, through nuclear matrix elements that are uncertain; this differs from the $m_{\beta }$ of beta-decay experiments, and a null result does not rule out Majorana neutrinos. It has not been seen; in xenon-136, for example, its half-life exceeds $10^{26}$ years.

See also Majorana neutrino, Beta decay

A proposed explanation of the neutrinos' tiny masses: if ordinary neutrinos mix with very heavy sterile partners of mass $M$, the light masses come out small because $M$ is large. With couplings near one, $M$ would be about $10^{14}$–$10^{15}$ GeV.

$m_{\nu }\sim \frac{(yv{)}^2}{M}$

See also Neutrino mass, Sterile neutrino, Majorana neutrino

QCD allows a term, with an angle θ, that would violate CP symmetry and give the neutron an electric dipole moment. None is found: experiments limit the moment to below $1.8\times 10^{-26}e$ cm, so θ is below about $10^{-10}$. Why it is so small is unexplained; the axion, proposed after Roberto Peccei and Helen Quinn's 1977 idea, would make it small dynamically.

In the folio: Subatomic Matter › Open problems; Subatomic Matter › Searching for dark matter particlesSee also Axion, CP violation, Free parameters

The muon's magnetic moment differs from the value $g=2$ of Dirac's theory because of quantum loops. Heavy virtual particles affect the muon more than the electron, so a discrepancy would point to new physics, and for two decades measurement exceeded prediction by four to five standard deviations. In 2025 Fermilab's final result reached 127 parts per billion, and the theory update based on lattice-QCD calculations of the hadronic contribution agreed with it, largely dissolving the hint.

$a_{\mu }=\frac{g-2}{2}\approx 1.1659207\times 10^{-3}$

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Lattice QCD, Hadronic vacuum polarisation, Quantum electrodynamics, Fermilab

The contribution of virtual quarks and gluons to the muon's magnetic moment, the largest source of theoretical uncertainty. Estimates from electron–positron collision data and from lattice QCD disagreed; the lattice values, now adopted, bring theory into line with experiment.

In the folio: Subatomic Matter › Beyond the Standard ModelSee also Muon g−2, Lattice QCD

The Sun and everything bound to it: eight planets and their moons, dwarf planets, asteroids, comets and dust, out to the Oort cloud tens of thousands of astronomical units away. The planets lie within 30 au; the heliosphere, the solar wind's bubble, which is not a sphere, reaches about 120 au in the two directions the Voyagers crossed it; and the Sun's gravity holds the Oort cloud out to perhaps 100,000 au or more. Where the Solar System ends depends on which of these boundaries is meant. It formed 4.6 Gyr ago from a collapsing cloud of gas and dust.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › Our cosmic addressSee also Sun, Planet, Heliosphere, Oort cloud

Our star: a G2V main-sequence star 1.39 million km across, 109 times Earth, holding 99.86% of the Solar System's mass. It is 4.6 Gyr old, about halfway through its hydrogen-burning life, with a surface temperature of 5772 K. G stars are about 7.6% of main-sequence stars.

In the folios: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Stars; Cosmology and Astrophysics › Stars forge the elementsSee also Solar mass, Solar luminosity, Solar radius, Proton–proton chain, Spectral type

By the International Astronomical Union's 2006 Resolution B5, a body that orbits the Sun, is massive enough for its gravity to pull it round, and has cleared the neighbourhood of its orbit. Eight qualify. A body that meets the first two conditions but not the third is a dwarf planet. The definition covers only the Solar System; planets of other stars are usually distinguished from brown dwarfs by mass, below about 13 Jupiter masses.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › DiscoveriesSee also Dwarf planet, Exoplanet, International Astronomical Union, Brown dwarf

A round body orbiting the Sun that has not cleared its orbital zone. The IAU recognises five: Ceres, Pluto, Eris, Haumea and Makemake. The category was created in 2006, after the discovery of Eris.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Planet, Ceres, Pluto, Eris

The largest body in the asteroid belt, 940 km across, and the smallest dwarf planet. Giuseppe Piazzi discovered it on 1 January 1801, the first asteroid found, and it was counted a planet for half a century. NASA's Dawn spacecraft orbited it from 2015 to 2018. It is the smallest object on the folio's scale of sizes.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › DiscoveriesSee also Asteroid belt, Dwarf planet

Our planet, 12,742 km across and 1 au from the Sun: the starting point of the folio's cosmic address.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Our cosmic addressSee also Earth radius and Earth mass, Astronomical unit, Cosmic address

Earth's satellite, 3,475 km across, about a quarter of Earth's diameter, and 384,400 km away on average. It probably formed from the debris of a giant impact about 4.5 Gyr ago.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic webSee also Earth

The largest planet, about 140,000 km across (eleven Earths) and 318 Earth masses: a gas giant made mostly of hydrogen and helium. Galileo discovered its four large moons in 1610.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › DiscoveriesSee also Jupiter mass, Galilean moons, Gas giant

Io, Europa, Ganymede and Callisto, discovered by Galileo Galilei in January 1610 with a small telescope: the first bodies seen to orbit another planet, and strong evidence against an Earth-centred cosmos. Ganymede is the largest moon in the Solar System, bigger than the planet Mercury.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Jupiter

The seventh planet, discovered by William Herschel in 1781, the first planet found with a telescope. It is an ice giant whose spin axis is tipped almost into the plane of its orbit.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Neptune

The eighth and outermost planet, about 30 au from the Sun, so the planets' orbits span some 60 au. Johann Galle found it in 1846 within a degree of where Urbain Le Verrier had predicted from irregularities in the orbit of Uranus, John Couch Adams having made a similar calculation: the first planet found by mathematics.

In the folio: The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › DiscoveriesSee also Uranus, Kuiper belt, Planet Nine

Discovered by Clyde Tombaugh at Lowell Observatory in 1930; 2,377 km across. Counted the ninth planet until 2006, it is now a dwarf planet and one of the largest Kuiper belt objects. New Horizons flew past it in 2015.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Dwarf planet, Kuiper belt, Eris

A dwarf planet discovered in 2005 by Mike Brown, Chad Trujillo and David Rabinowitz, almost the same size as Pluto (2,326 km) and about a quarter more massive. Its discovery forced the IAU to define "planet" in 2006. It orbits between about 38 and 98 au from the Sun.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Dwarf planet, Pluto, Planet, Trans-Neptunian object

A small rocky or metallic body orbiting the Sun, left over from the formation of the planets. Well over a million have been catalogued, most of them in the main belt; near-Earth asteroids, whose orbits approach Earth's, are tracked to assess the risk of impact.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Asteroid belt, Near-Earth object, Minor Planet Center

The ring of asteroids between Mars and Jupiter, 2.2–3.3 au from the Sun. Jupiter's gravity kept its material from forming a planet, and its total mass is only about 3% of the Moon's.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Asteroid, Ceres

An asteroid or comet whose orbit brings it within 1.3 au of the Sun, and so close to Earth's orbit. Tens of thousands are known; surveys aim to find nearly all those larger than 140 m, and NASA's DART mission showed in 2022 that an impact can change an asteroid's orbit.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Asteroid, Vera C. Rubin Observatory

An icy body that, near the Sun, releases gas and dust into a glowing coma and tails. Short-period comets have orbits under 200 years, like Halley's at 76. Most of them are Jupiter-family comets, with periods under about 20 years, such as 67P/Churyumov–Gerasimenko, thought to come from the Kuiper belt and the scattered disc beyond it; Halley-type comets, Halley's included, and the long-period comets are thought to have fallen in from the Oort cloud.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Halley's Comet, Oort cloud, Kuiper belt

The best-known short-period comet, returning about every 76 years. Edmond Halley predicted its 1758 return, the first proof that comets orbit the Sun. It last passed the Sun in 1986, when the Giotto spacecraft visited it, and returns in 2061.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Comet

A disc of icy bodies beyond Neptune, about 30–50 au from the Sun, including Pluto, Haumea, Makemake and Arrokoth, which New Horizons visited in 2019. The first member after Pluto and Charon, 1992 QB1 (now Albion), was found by David Jewitt and Jane Luu in 1992. With the scattered disc beyond it, it is thought to be the source of the Jupiter-family comets, the commonest short-period kind.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › DiscoveriesSee also Pluto, Comet, Trans-Neptunian object

Any Solar System body orbiting beyond Neptune: Kuiper belt objects, scattered-disc objects such as Eris, and detached objects such as Sedna whose orbits never approach Neptune. The apparent clustering of the most distant orbits underlies the Planet Nine hypothesis.

In the folio: The Hierarchy of Cosmic Structure › Still being mappedSee also Kuiper belt, Eris, Planet Nine

A vast, roughly spherical cloud of icy bodies, inferred from the orbits of long-period comets and proposed by Jan Oort in 1950. It may begin about 2,000 au from the Sun and reach 100,000 to 200,000 au, a large fraction of the way to the nearest stars. It has never been observed directly; how far it extends and how many comets it holds are open.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › From planets to the cosmic web; The Hierarchy of Cosmic Structure › Still being mappedSee also Comet, Heliosphere, Astronomical unit

The bubble blown into interstellar gas by the solar wind, the Sun's outflow of charged particles. Its boundary is the heliopause.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › Our cosmic addressSee also Heliopause, Voyager 1 and 2, Interstellar medium

The edge of the heliosphere, where the solar wind meets interstellar gas, about 120 au from the Sun in the two directions measured: Voyager 1 crossed it in August 2012 at 121 au and Voyager 2 in November 2018 at 119 au.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitorsSee also Heliosphere, Voyager 1 and 2

A body passing through the Solar System on an unbound orbit from interstellar space, given an "I" designation. Three were known by the end of 2025: 1I/ʻOumuamua (2017), the comet 2I/Borisov (2019) and the comet 3I/ATLAS (2025). The Rubin Observatory is expected to find many more.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › DiscoveriesSee also ʻOumuamua, 3I/ATLAS, Vera C. Rubin Observatory

The first interstellar object, found by the Pan-STARRS telescope in Hawaii in October 2017. It was elongated or flattened, a few hundred metres long, showed no visible coma, and accelerated slightly more than gravity alone allows, probably because of outgassing that could not be seen. The name is Hawaiian for a scout.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › DiscoveriesSee also Interstellar object

The third interstellar object, a comet discovered on 1 July 2025 by the ATLAS survey telescope in Chile. It arrived faster than either predecessor, about 58 km s⁻¹ before the Sun's pull, and passed closest to the Sun, at 1.36 au, in late October 2025. It may be older than the Solar System.

In the folio: The Hierarchy of Cosmic Structure › Belts, clouds and visitors; The Hierarchy of Cosmic Structure › DiscoveriesSee also Interstellar object

A proposed undiscovered planet of perhaps five to ten Earth masses, on a distant orbit hundreds of astronomical units out, invoked by Konstantin Batygin and Mike Brown in 2016 to explain the clustered orbits of the most distant trans-Neptunian objects. It has not been found, and the clustering may be an artefact of where surveys have looked.

In the folio: The Hierarchy of Cosmic Structure › Still being mappedSee also Trans-Neptunian object, Vera C. Rubin Observatory, Neptune

A planet orbiting a star other than the Sun. The NASA Exoplanet Archive listed 6,080 confirmed in January 2026, having passed 6,000 in 2025: by its categories 2,051 Neptune-like, 2,036 gas giants, 1,765 super-Earths and 228 terrestrial and other planets. The counts reflect what detection methods find easily, not what is commonest. The first were found around a pulsar in 1992, and around a Sun-like star in 1995.

In the folio: The Hierarchy of Cosmic Structure › Planets of other stars; The Hierarchy of Cosmic Structure › DiscoveriesSee also Transit method, Radial-velocity method, Hot Jupiter, Super-Earths and sub-Neptunes, NASA Exoplanet Archive

Finding a planet by the slight dimming of its star as it crosses in front: the depth gives the planet's size relative to the star. Jupiter would dim the Sun by about 1%, Earth by less than 0.01%. Most known exoplanets were found this way, by Kepler and TESS.

depth $\approx (R_p/R_{\star }{)}^2$

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Exoplanet, Radial-velocity method, Kepler and TESS

Finding a planet by the periodic Doppler shift in its star's spectrum as both orbit their common centre of mass; it gives a minimum mass. Jupiter moves the Sun at about 12.5 m s⁻¹, Earth at only 9 cm s⁻¹. 51 Pegasi b was found this way.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Transit method, 51 Pegasi b

Photographing a planet itself, separated from its star's glare by coronagraphs and adaptive optics, or with JWST. It works best for young, hot, giant planets far from their stars; fewer than 100 have been imaged.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Exoplanet, JWST

The brief brightening of a background star when a foreground object passes almost exactly in front of it, its gravity acting as a lens. A planet adds a short extra blip, so microlensing finds planets far from their stars and rogue planets with no star at all. The Roman Space Telescope will survey the Galactic bulge for them.

In the folio: The Hierarchy of Cosmic Structure › Planets of other stars; The Hierarchy of Cosmic Structure › Still being mappedSee also Rogue planet, Gravitational lensing, Nancy Grace Roman Space Telescope

A gas giant orbiting very close to its star, in a few days, and heated above 1000 K; 51 Pegasi b takes 4.2 days. Easy to find, they dominated early discoveries, but only about 1% of Sun-like stars have one. Many are thought to have formed farther out and moved inwards, either through the gas disc or by the tidal shrinking of a highly eccentric orbit; formation in place is also studied.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also 51 Pegasi b, Gas giant

The first planet found around a Sun-like star, by Michel Mayor and Didier Queloz in 1995 using the radial-velocity method (Nobel Prize 2019): about half Jupiter's mass, on a 4.2-day orbit, 50 light-years away. That such a planet existed at all, against expectations, launched the field.

In the folio: The Hierarchy of Cosmic Structure › Planets of other stars; The Hierarchy of Cosmic Structure › DiscoveriesSee also Hot Jupiter, Radial-velocity method

Planets between about one and four Earth radii: the commonest kind known, yet absent from our own system. The smaller super-Earths are often consistent with mostly rocky compositions; the larger sub-Neptunes usually need a substantial layer of low-density material, such as a hydrogen–helium envelope or water. A dip in the population near 1.8 Earth radii, the radius valley, whose position shifts with orbital period, separates them. An individual planet's composition needs its mass, and so its density, not its size alone.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Earth radius and Earth mass, Neptune-like planet, TRAPPIST-1

NASA's category for planets similar in size to Neptune or Uranus, with hydrogen–helium atmospheres around ice and rock; at 2,051 in January 2026, the largest category of confirmed exoplanets.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Gas giant, Super-Earths and sub-Neptunes

A large planet made mainly of hydrogen and helium, like Jupiter and Saturn; 2,036 confirmed exoplanets were in this category in January 2026.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Jupiter, Hot Jupiter

A rocky planet of Earth's size or smaller, like Mercury, Venus, Earth and Mars. Only 228 confirmed exoplanets were in NASA's "terrestrial and other" category in January 2026, because small planets are the hardest to detect.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Super-Earths and sub-Neptunes, TRAPPIST-1

A system of several planets packed close to their star, often all within what would be Mercury's orbit; common around red dwarfs. TRAPPIST-1 is the archetype.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also TRAPPIST-1, Red dwarf

An ultracool red dwarf 40 light-years away with seven Earth-sized planets, all orbiting closer than Mercury does to the Sun, with periods from 1.5 to 19 days; several are in the temperate zone. They were found in 2016–17 with the TRAPPIST telescope and Spitzer, and JWST is probing their atmospheres.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Compact planetary system, Red dwarf, Terrestrial planet

A planet bound to no star, either ejected from its system or formed alone, found by microlensing and by infrared imaging of young clusters. How many drift between the stars is open; they may be as numerous as stars, and the Roman Space Telescope is expected to find hundreds.

In the folio: The Hierarchy of Cosmic Structure › Planets of other stars; The Hierarchy of Cosmic Structure › Still being mappedSee also Gravitational microlensing, Nancy Grace Roman Space Telescope

The nearest known exoplanet, orbiting the red dwarf Proxima Centauri 4.2 light-years away: at least 1.07 Earth masses, on an 11.2-day orbit within the star's habitable zone, discovered by the radial-velocity method in 2016. Flares from its star may strip its atmosphere.

In the folio: The Hierarchy of Cosmic Structure › Planets of other starsSee also Proxima Centauri, Red dwarf, Radial-velocity method

The first confirmed exoplanets, found in 1992 by Aleksander Wolszczan and Dale Frail around the pulsar PSR B1257+12 from tiny variations in the arrival times of its pulses. Three planets orbit it, the smallest about twice the mass of the Moon.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Pulsar, Exoplanet

A self-gravitating ball of plasma held up by the pressure of nuclear fusion in its core, from about 0.08 to more than 100 solar masses. Its mass sets its luminosity, colour, lifetime and fate. Small, cool red dwarfs vastly outnumber the bright stars that dominate the night sky.

In the folio: The Hierarchy of Cosmic Structure › Stars; The Hierarchy of Cosmic Structure › From planets to the cosmic webSee also Main sequence, Spectral type, Red dwarf, Brown dwarf

Stellar luminosity plotted against surface temperature. By a convention from spectral classification, temperature increases to the left. Most stars lie on the main sequence; giants and supergiants lie above it, white dwarfs below.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Main sequence, Effective temperature, Spectral type

The temperature of a blackbody that would emit the star's flux per unit area. It sets the star's colour, from about 3000 K (red) to over 30,000 K (blue-white); the Sun's is 5772 K.

$L=4\pi R^2\sigma T_{\text{eff}}^4$

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › StarsSee also Hertzsprung–Russell diagram, Spectral type

The classification of stars by their spectral lines, which in effect orders them by temperature: O hottest, M coolest, each subdivided 0–9. A luminosity class follows: V for main sequence, III for giants, I for supergiants. The Sun is G2V.

The Hierarchy of Cosmic Structure's census of main-sequence stars by class: O above 30,000 K (0.00003%), B 10,000–30,000 K (0.1%), A 7,500–10,000 K (0.6%), F 6,000–7,500 K (3%), G 5,200–6,000 K (7.6%), K 3,700–5,200 K (12%) and M 2,400–3,700 K (76%). These are widely quoted rough figures for stars near the Sun, rounded, so they do not sum exactly to 100%.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Effective temperature, Hertzsprung–Russell diagram, Sun, Red dwarf

The band on the HR diagram of stars fusing hydrogen into helium in their cores, where a star spends about 90% of its life. Its position on the band is set almost entirely by its mass.

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › StarsSee also Mass–luminosity relation, Main-sequence lifetime

On the main sequence, luminosity rises steeply with mass. The exponent varies from about 2.3 for low-mass stars to about 1 for the most massive; 3.5 is the classic average for stars like the Sun.

$L\propto M^{3.5}$

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Main-sequence lifetime

Fuel grows with mass but luminosity grows much faster, so massive stars burn out first: a $10M_{\odot }$ star lasts about 30 Myr, the Sun about 10 Gyr, and the smallest red dwarfs trillions of years.

$t_{\text{MS}}\approx 10\text{Gyr}(M/M_{\odot }{)}^{-2.5}$

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Mass–luminosity relation, Red dwarf

A small, cool main-sequence star of about 0.08–0.6 $M_{\odot }$ (spectral type M). The commonest stars, about three-quarters of the total. Their lifetimes far exceed the present age of the universe, so none has yet died.

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › StarsSee also Main-sequence lifetime, Proxima Centauri, TRAPPIST-1

An object between the largest planets and the smallest stars, about 13 to 80 Jupiter masses: heavy enough to fuse deuterium for a while, but below about 0.075 $M_{\odot }$, too light to sustain hydrogen fusion, so it cools and fades for ever. The first were confirmed in 1995.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Jupiter mass, Red dwarf, Star

A star that has exhausted the hydrogen in its core and swollen to tens or hundreds of solar radii while hydrogen burns in a shell. The Sun will become one in about 5 Gyr. The brightest red giants mark the tip of the red giant branch.

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › StarsSee also Tip of the red giant branch, Asymptotic giant branch, Arcturus

An evolved massive star, with luminosity up to about $10^6{L}_{\odot }$ and, for red supergiants, radii above $1000R_{\odot }$. Betelgeuse is an example. Many massive supergiants end in core collapse, which may produce a visible supernova or a failed explosion with direct collapse to a black hole.

The Hierarchy folio draws Betelgeuse at about 750 solar radii, within the spread of current estimates; placed at the Sun, it would reach beyond the asteroid belt. Blue supergiants such as Rigel are much smaller but far hotter.

In the folios: Cosmology and Astrophysics › Stars forge the elements; The Hierarchy of Cosmic Structure › Stars; The Hierarchy of Cosmic Structure › From planets to the cosmic webSee also Core-collapse supernova, Betelgeuse, Rigel

Two or more stars orbiting their common centre of mass. About half of Sun-like stars have companions, more of the massive stars and fewer of the red dwarfs. Binaries give the only direct measurements of stellar masses, and close ones produce Type Ia supernovae, X-ray binaries and gravitational-wave sources.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Alpha Centauri, Cygnus X-1, Type Ia supernova, Sirius

One of the hottest and most luminous stars visible to the naked eye, an O-type supergiant with a surface near 40,000 K, roughly 1,000 light-years away. The folio uses it as its example of the rare O stars.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Spectral type, Supergiant

The blue-white supergiant of type B8 at Orion's foot, about 79 solar radii and roughly 860 light-years away, shining with on the order of $10^5{L}_{\odot }$. Being a supergiant, it is not a main-sequence star, as the folio's asterisk notes.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Supergiant, Spectral type

The brightest star in the night sky, a binary 8.6 light-years away. Sirius A is an A-type main-sequence star of about two solar masses and 25 $L_{\odot }$; its companion Sirius B is the nearest white dwarf.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Sirius B, Spectral type

An F-type star 11.5 light-years away in Canis Minor, about 1.5 solar masses and beginning to leave the main sequence, with a white-dwarf companion of its own.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Spectral type

The nearest star system, about 4.37 light-years away: the Sun-like G star α Centauri A, the K star α Centauri B (the folio's K example), and the red dwarf Proxima Centauri, which orbits the pair at about 13,000 au.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Proxima Centauri, Binary star

A red dwarf of type M5.5 and about 0.12 solar masses, the nearest star to the Sun at 4.24 light-years, and too faint to see with the naked eye. It flares often and hosts the planet Proxima Centauri b.

In the folio: The Hierarchy of Cosmic Structure › Stars; The Hierarchy of Cosmic Structure › Planets of other starsSee also Red dwarf, Proxima Centauri b, Alpha Centauri

An orange giant of type K1.5 III, 37 light-years away and about 25 solar radii, the brightest star north of the celestial equator. It has left the main sequence, a preview of the Sun's own future.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Red giant

The red supergiant at Orion's shoulder, roughly 550 light-years away, with a radius estimated at 700 to 1,000 solar radii. Its "Great Dimming" of 2019–20 was caused by a dust cloud from an outburst on its surface. It is expected to explode as a core-collapse supernova within about 100,000 years.

In the folio: The Hierarchy of Cosmic Structure › StarsSee also Supergiant, Core-collapse supernova

The fusion sequence that powers the Sun, converting four protons into helium-4. About 2% of the energy leaves as neutrinos. Stars above about 1.3 $M_{\odot }$ instead rely mainly on the CNO cycle, in which carbon, nitrogen and oxygen catalyse the same net reaction. Hans Bethe and Charles Critchfield calculated the pp chain in 1938; Bethe, and independently Carl Friedrich von Weizsäcker, described the CNO cycle in 1938–39.

$4p+2e^{-}\to {}^4\text{He}+2{\nu }_e+26.73\text{MeV}$, the total including the annihilation of the two positrons the chain emits (24.69 MeV before it)

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Stellar nucleosynthesis, Nuclear fusion, Weak interaction

The building of elements heavier than helium inside stars and their explosions. The 1957 review by Burbidge, Burbidge, Fowler and Hoyle ("B²FH") set out the processes. The folio's table lists the main sources: low-mass stars for carbon, nitrogen and s-process elements; massive stars and their supernovae for oxygen to calcium; white-dwarf supernovae for iron-peak elements; neutron-star mergers for r-process elements.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also s-process, r-process, Alpha elements, Iron-peak elements

Slow neutron capture: nuclei absorb neutrons slowly enough that unstable isotopes beta-decay before the next capture, climbing the valley of stability. It runs in asymptotic-giant-branch stars and makes about half of the elements heavier than iron, including barium and lead.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Asymptotic giant branch, r-process

Rapid neutron capture: in extremely neutron-rich explosive environments nuclei absorb neutrons faster than they can decay, then decay back to stability, making the heaviest elements, including europium, platinum, gold and uranium. Neutron-star mergers were confirmed as a site by the kilonova of GW170817.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Kilonova, Neutron star, s-process

The last fusion stage of low- and intermediate-mass stars, with helium and hydrogen burning in alternating shells around a carbon–oxygen core. These stars shed their envelopes, returning carbon, nitrogen and s-process elements to space, and leave a white dwarf inside a planetary nebula ("dying low-mass stars" in the folio).

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also s-process, White dwarf, Red giant, Planetary nebula

The explosive death of a star above about 8 $M_{\odot }$: usually its iron core, which can no longer release energy by fusion, collapses (at the low-mass end an O–Ne–Mg core can instead collapse by electron capture) to a neutron star or black hole, and the rebound and neutrino heating blow off the envelope. Nearly all of the $\sim 10^{46}$ J released leaves as neutrinos, as observed from SN 1987A. These supernovae supply most oxygen, magnesium, silicon and calcium.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Alpha elements, Neutron star, Type Ia supernova, Supernova remnant, Betelgeuse

Elements built up by adding helium nuclei: oxygen, neon, magnesium, silicon, sulphur, calcium. They come mostly from massive stars and core-collapse supernovae, so their abundance relative to iron dates a star's gas.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Core-collapse supernova, Iron-peak elements

Chromium to nickel, around the maximum of nuclear binding energy per nucleon. Type Ia supernovae make much of them, about 0.6 $M_{\odot }$ of nickel-56 per explosion, which decays through cobalt-56 to iron-56 and powers the supernova's light.

In the folio: Cosmology and Astrophysics › Stars forge the elementsSee also Type Ia supernova, Alpha elements, Binding energy per nucleon

Generations of stars classified by metal content: Population I, metal-rich and young, like the Sun; Population II, metal-poor and old, in the halo and globular clusters; Population III, the hypothetical metal-free first stars.

In the folio: The Hierarchy of Cosmic Structure › Star clustersSee also First stars, Metals, Globular cluster

A stellar remnant held up against gravity by quantum degeneracy pressure (white dwarfs, neutron stars) or not held up at all (black holes).

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Degeneracy pressure, White dwarf, Neutron star, Black hole

Pressure from the Pauli exclusion principle: fermions packed densely must occupy high-momentum states. It persists at zero temperature, which is why a white dwarf or neutron star needs no fuel to support itself.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also White dwarf, Neutron star, Chandrasekhar limit, Pauli exclusion principle

The remnant core of a low- or intermediate-mass star, supported by electron degeneracy. Typically $0.6M_{\odot }$ in a body the size of Earth, with density near $10^9$ kg m⁻³; it slowly cools over billions of years.

The Sun's eventual fate. The nearest, Sirius B, is 8.6 light-years away.

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Chandrasekhar limit, Type Ia supernova, Degeneracy pressure, Sirius B, Planetary nebula

The white-dwarf companion of Sirius, about one solar mass in a body slightly smaller than Earth. Friedrich Bessel inferred it in 1844 from Sirius's wobble, and Alvan Graham Clark first saw it in 1862. It was not the first white dwarf seen: William Herschel had recorded 40 Eridani B in 1783, and that star was also the first recognised as a white dwarf, in 1910; Sirius B's spectrum showed in 1915 that it was another. At 8.6 light-years it is the nearest white dwarf.

In the folio: The Hierarchy of Cosmic Structure › Stellar remnants; The Hierarchy of Cosmic Structure › DiscoveriesSee also White dwarf, Sirius

The maximum mass of a white dwarf, about 1.4 $M_{\odot }$, above which electron degeneracy pressure, its electrons now relativistic, cannot resist gravity.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also White dwarf, Type Ia supernova

The collapsed core of a massive star, supported by neutron degeneracy and nuclear forces: about 1.2–2.1 $M_{\odot }$ in a sphere about 12 km in radius, at the density of an atomic nucleus. The maximum mass of a cold, non-rotating neutron star is called the TOV limit. It depends on the poorly known nuclear equation of state; model-dependent estimates, for example from the 2017 neutron-star merger, put it near 2.2–2.3 $M_{\odot }$, and rotation can support somewhat more.

nuclear saturation density $\approx 2.7\times 10^{17}{\text{kg m}}^{-3}$

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Pulsar, Kilonova, Degeneracy pressure

A rotating, magnetised neutron star whose radio beams sweep past Earth, giving pulses with clock-like regularity. Discovered by Jocelyn Bell Burnell in 1967. Millisecond pulsars, spun up by companions, rival atomic clocks.

More than 3,000 are known.

In the folios: Cosmology and Astrophysics › Milestones; The Hierarchy of Cosmic Structure › Stellar remnants; The Hierarchy of Cosmic Structure › DiscoveriesSee also Neutron star, Pulsar timing array, Magnetar, Pulsar planets

A neutron star with a magnetic field of $10^{10}$–$10^{11}$ T, about a thousand times that of an ordinary pulsar: the strongest magnets known. The decay of the field powers bursts of X-rays and gamma rays, and about 30 are known. A giant flare from one in 2004 disturbed Earth's ionosphere from 50,000 light-years away, and a Galactic magnetar produced a fast radio burst in 2020.

In the folio: The Hierarchy of Cosmic Structure › Stellar remnantsSee also Neutron star, Pulsar

A gravitational-wave detector the size of the Galaxy: decades of precise timing of dozens of millisecond pulsars, searching for correlated arrival-time shifts caused by passing waves with periods of years. In 2023 several arrays reported evidence for a nanohertz background, most likely from pairs of supermassive black holes.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Pulsar, Gravitational waves

A region of spacetime from which nothing, not even light, escapes, bounded by the event horizon. Stellar-mass black holes form from massive stars; supermassive ones of $10^6$–$10^{10}{M}_{\odot }$ sit at the centres of galaxies.

Schwarzschild radius $r_s=2GM/c^2\approx 2.95\text{km}(M/M_{\odot })$

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Black-hole shadow, Gravitational waves, Primordial black holes, Stellar-mass black hole, Supermassive black hole

A black hole formed by the collapse of a massive star, of about 3 to 100 solar masses. They are found as X-ray binaries feeding on a companion, like Cygnus X-1, as dark companions revealed by Gaia's astrometry, like Gaia BH3, and in gravitational-wave mergers.

In the folio: The Hierarchy of Cosmic Structure › Stellar remnantsSee also Black hole, Cygnus X-1, Gaia BH3, GW150914

A black hole of millions to billions of solar masses, found at the centres of most large galaxies: Sagittarius A* in the Milky Way has 4 million, M87* 6.5 billion. Gas falling into them powers active galaxies and quasars. How they grew so large so early is an open question.

In the folio: The Hierarchy of Cosmic Structure › Stellar remnants; The Hierarchy of Cosmic Structure › GalaxiesSee also M87* and Sgr A*, Active galactic nucleus, Quasar

A dormant black hole of 33 solar masses, about 1,900 light-years away in Aquila, found in 2024 in Gaia data from the wobble of the old star orbiting it. When announced, it was the most massive stellar black hole known in the Milky Way.

In the folio: The Hierarchy of Cosmic Structure › Stellar remnantsSee also Stellar-mass black hole, Gaia

A bright X-ray source in Cygnus, about 7,200 light-years away, identified in 1971 as a binary in which a blue supergiant feeds a compact object too massive to be a neutron star: the first strong black-hole candidate. Its black hole is now estimated at about 21 solar masses.

In the folio: The Hierarchy of Cosmic Structure › DiscoveriesSee also Stellar-mass black hole, Binary star

The dark central region seen against glowing gas around a black hole, bounded by the orbit of light. For a non-spinning hole it appears about 5.2 Schwarzschild radii across. The Event Horizon Telescope imaged it for M87*, a $6.5\times 10^9{M}_{\odot }$ black hole, in 2019, and for Sgr A* in 2022.

shadow diameter $=2\sqrt{27}GM/c^2\approx 5.2r_s$

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › DiscoveriesSee also M87* and Sgr A*, Event Horizon Telescope

The two black holes imaged so far. M87* is at the centre of the giant elliptical galaxy M87, 16.8 Mpc away. Sagittarius A* is at the centre of the Milky Way, $4\times 10^6{M}_{\odot }$ and 8.2 kpc away. The asterisk is part of the name.

In the folios: Cosmology and Astrophysics › Compact objects and gravitational waves; The Hierarchy of Cosmic Structure › Stellar remnantsSee also Black-hole shadow

Ripples in spacetime geometry that travel at the speed of light, radiated by accelerating masses with a changing quadrupole moment. Ground-based detectors hear compact binaries at tens to thousands of hertz; pulsar timing arrays listen at nanohertz.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Strain, Inspiral, merger and ringdown, LIGO–Virgo–KAGRA

The fractional change in length a passing wave produces. Detected signals have $h\sim 10^{-21}$, which over LIGO's 4 km arms is a change of a few $10^{-18}$ m, hundreds of times smaller than a proton.

$h\sim \Delta L/L$

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Gravitational waves

The three stages of a compact-binary signal. In the inspiral the orbit shrinks as waves carry energy away, so frequency and amplitude rise (a "chirp") at a rate set by the chirp mass. Merger is the plunge and coalescence. In the ringdown the remnant black hole settles, oscillating at frequencies fixed by its mass and spin.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Gravitational waves, GW150914

The first gravitational waves detected, on 14 September 2015: two black holes of 36 and 29 $M_{\odot }$ merged 410 Mpc away, radiating about $3M_{\odot }{c}^2$, some $5\times 10^{47}$ J, in a fraction of a second. Event names give the date as GW + YYMMDD.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Inspiral, merger and ringdown, LIGO–Virgo–KAGRA

The merger of two neutron stars on 17 August 2017, 40 Mpc away. A short gamma-ray burst arrived 1.7 s later, showing that gravitational waves travel at the speed of light to within about one part in $10^{15}$. A kilonova followed. It was the first standard-siren measurement of $H_0$.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Kilonova, Gamma-ray burst, Standard siren, r-process

The optical and infrared glow, lasting days, from radioactive decay of freshly made r-process nuclei thrown out in a neutron-star merger, about a thousand times brighter than a classical nova.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also GW170817, r-process

A brief flash of gamma rays from a relativistic jet. Short bursts, under about two seconds, usually come from compact-object mergers, and long ones usually from the collapse of rapidly rotating massive stars, but duration alone does not settle the origin: GRB 211211A lasted about 50 seconds yet was followed by a kilonova. Afterglows, host galaxies and any accompanying supernova, kilonova or gravitational waves decide.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also GW170817

The network of ground-based laser-interferometer detectors: the two LIGO observatories in the US (4 km arms), Virgo in Italy (3 km) and KAGRA in Japan (underground and cryogenic). Several detectors locate a source on the sky and separate signals from local noise.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also Gravitational waves, GWTC-4.0

Version 4.0 of the LVK Gravitational-Wave Transient Catalog (2025), which extends the list of detected compact-binary mergers through the first part of the fourth observing run, to 218 candidates.

In the folio: Cosmology and Astrophysics › Compact objects and gravitational wavesSee also LIGO–Virgo–KAGRA

A group of stars born together from one cloud, and so, to a first approximation, of one age and composition: natural laboratories for stellar evolution. Globular clusters break the simple picture, most holding several populations of stars of differing composition. They come as loose open clusters and dense globular clusters.

In the folio: The Hierarchy of Cosmic Structure › Star clusters; The Hierarchy of Cosmic Structure › From planets to the cosmic webSee also Open cluster, Globular cluster

Tens to thousands of stars, loosely held and usually young. Most drift apart within a few hundred million years, though a few survive for billions, such as NGC 6791 at about 8 Gyr. Thousands are known in the Milky Way's disc; the Pleiades is the best known.