Foundations

For small sub-Hubble-scale perturbations of pressureless matter in a homogeneous expanding background, the Newtonian growing-density model obeys δ̈+2Hδ̇−4πGρ̄_mδ=0.

For independent Poisson source, sky and dark counts and independent read noise, an aperture-summed CCD measurement has variance S+B+D+nₚNᵣσᵣ².

Photometric counts need a calibrated band response and zero point before they represent an astronomical flux or magnitude.

Stellar opacity includes electron scattering, bound–bound transitions, photoionization/recombination and free–free processes.

In a nondegenerate ideal ionized mixture in local thermal equilibrium, ions and electrons contribute their number densities to gas pressure; isotropic thermal radiation adds a radiation-pressure term.

For an ideal spin-½ electron gas with k_BT much smaller than its kinetic Fermi energy, occupied momentum states extend to p_F=ℏ(3π²nₑ)¹ᐟ³.

Dynamical readjustment has scale t_dyn∼(Gρ̄)⁻¹ᐟ², while radiating a gravitational energy reservoir gives Kelvin–Helmholtz scale t_KH∼GM²/(RL).

For a Newtonian hydrostatic spherical star with negligible surface pressure and classical monatomic ideal-gas support, integrating pressure balance gives 2U_th+W=0, where W is gravitational binding energy.

A spherical Newtonian hydrostatic polytrope assumes P=Kρ¹⁺¹⁄ⁿ.

A dark opaque planet crossing a uniformly bright stellar disk removes light in proportion to its projected area while fully superimposed on the disk.

In a Newtonian isolated two-body Keplerian model, the host star's line-of-sight orbital velocity has semi-amplitude K=(2πG/P)^(1/3) Mp sin i/[(Mstar+Mp)^(2/3)√(1−e²)].

For a specified birth population, an initial mass function ξ(m) defines dN=ξ(m)dm on a stated initial-mass interval.

Cold noninteracting electron support and spherical Newtonian hydrostatic balance give a nonrelativistic n=3/2 polytrope with R proportional to μₑ^(−5/3)M^(−1/3).

Neutron-star pulsars have rotating magnetized stellar remnants whose emission can sweep past the observer.

Core-collapse supernovae involve loss of support in an evolved stellar core, rapid contraction and an explosion whose energy transfer involves dense matter and neutrinos.

Matter moving inward in a gravitational field can release binding energy if energy is dissipated.

For isotropic luminosity L and a constant flux-weighted opacity per mass κ, outward radiative acceleration is κL/(4πr²c).

In a circular two-point-mass binary's corotating frame, the effective potential per mass is Φ_eff=−GM₁/r₁−GM₂/r₂−Ω²(x²+y²)/2, with origin at the barycentre and Ω²=G(M₁+M₂)/a³.

Active galactic nuclei can radiate strongly as matter accretes around a central supermassive black hole.

In hot Big-Bang nucleosynthesis, neutron–proton conversion initially approaches chemical equilibrium.

Atomic recombination binds many previously free electrons and lowers photon-scattering opacity.

An ionization-bounded pure-hydrogen region in steady balance equates absorbed external ionizing photons to net recombinations: Q_H=∫α_B nₑnₚ dV.

Linear perturbations in a uniform Newtonian gas model obey ω²=c_s²k²−4πGρ₀.

Electron–ion acceleration produces continuum emission without capture.

A magnetically deflected electron radiates P=2σ_TcU_Bβ²γ²sin²α.

An electron in an isotropic soft-photon field loses net power P_IC=(4/3)σ_TcU_radβ²γ² in the Thomson limit.

The ppI branch builds deuterium and helium-3 before combining two helium-3 nuclei.

A steady planar normal shock conserves mass, momentum and total-enthalpy flux across its thin transition: [ρu]=[ρu²+P]=[ρu(h+u²/2)]=0.

The solar wind carries charged particles, including protons and electrons, and solar magnetic field into interplanetary space.

In charged-particle usage, cosmic rays include energetic nuclei and electrons from space.