Foundations

The minimal Standard Model has gauge group SU(3)_C×SU(2)_L×U(1)_Y with strong and electroweak gauge sectors.

In a unitary light-neutrino mixing model |ν_α⟩=Σ_iU_αi*|ν_i⟩.

A nonrelativistic neutron of mass m and kinetic energy E scattered elastically from a free stationary nucleus of mass M retains E′/E=(η²+1+2ηcosθ_cm)/(η+1)², η=M/m.

Fission can emit neutrons that induce further fissions, while capture without fission and leakage remove neutrons from the chain.

In a dilute nondegenerate gas of independently distributed reacting nuclei at common temperature T, reaction events per volume/time are R_ab=n_an_b⟨σv⟩/(1+δ_ab), where δ_ab=1 for the same species and 0 otherwise.

For repulsively charged nuclei with κ=Z_aZ_be²/(4πε₀)>0 and relative speed v, the Sommerfeld parameter is η=κ/(ħv) and the leading low-energy Coulomb penetration exponential is exp(−2πη)=exp[−√(E_G/E)], E_G=2μ(πκ/ħ)².

For a real periodic one-dimensional potential of period a, define V_G=a⁻¹∫₀ᵃV(x)e^(−iGx)dx, G=2π/a, and ε_k=ħ²k²/(2m).

For a periodic chain of N orthonormal localized orbitals, one per site with spacing a, take H_nm=ε₀δ_nm−t(δ_(n+1,m)+δ_(n−1,m)).

Geological history is organized into nested eons, eras, periods, epochs and ages.

Accretion, impacts, gravitational redistribution and radioactive decay contributed to early-Earth heating.

Oxygenic photosynthesis supplied oxygen while chemical reactions with reduced material consumed it.

An isotope delta compares a specified sample isotope ratio with the same ratio in a stated reference.

Gravity supplies both downslope shear and normal loading on a slope.

Coastal landforms reflect sediment supply, wave/current transport, erosion and deposition.

An ascending air parcel expands as pressure falls and can cool toward saturation.

For an isotropic source in transparent Euclidean space, bolometric flux decreases as luminosity divided by the area of a sphere.

A magnitude difference encodes a flux ratio on a logarithmic scale: brighter objects have smaller magnitudes.

Continuum shape and spectral-line patterns constrain temperature, composition and the physical conditions in a stellar atmosphere.

A CCD records photo-generated charge in pixels over an exposure.

A circular aperture has a diffraction angular scale proportional to wavelength divided by diameter.

Dust absorbs and scatters radiation along a line of sight.

A distance ladder transfers geometric calibrations to luminosity-based indicators such as Cepheids and standardized supernovae.

Along a ray, local emission adds and extinction removes specific intensity.

For a spherical quasistatic Newtonian star, enclosed mass increases with shell volume and density, while the pressure gradient balances gravity due to enclosed mass.

In an optically thick stellar region close to local thermodynamic equilibrium, radiative diffusion relates luminosity to the temperature gradient through density and an appropriate mean opacity.

A stellar shell's luminosity divergence accounts for local energy production, change in specific internal energy and compression or expansion work.

A displaced parcel that adjusts pressure rapidly but exchanges little heat can continue rising when it becomes less dense than its environment.

The proton–proton chains and CNO cycles convert hydrogen into helium through different reaction sequences; CNO nuclei participate in a catalytic cycle.

An HR diagram compares luminosity with effective temperature or a related spectral classification.

A resolved relative orbit and period can constrain a binary's total mass through Newtonian two-body dynamics.