Foundations

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.

A classical black-hole event horizon is a causal boundary: signals from its interior cannot reach distant future observers in the exterior.

A point mass in the weak-field geometric-optics limit deflects light by magnitude α̂=4GM/(c²b).

Before recombination, scattering tightly coupled photons and baryons.

A cohort life table follows individuals born together.

In a fixed density-independent age model, the growth factor λ satisfies Σ_{x≥1}λ^(−x)l_xm_x=1.

A simple internally colonized metapopulation tracks the occupied fraction p of equivalent habitat patches.

Failure to detect a species does not establish absence.

For a collection containing N individuals with N_i in each observed species, the expected species count in a uniformly drawn subset of n individuals without replacement is E[S_n]=Σ_i[1−C(N−N_i,n)/C(N,n)].

A genetic perturbation tests a causal prediction when the intended change and relevant phenotype are measured against appropriate controls.

Green chemistry evaluates product and process design to reduce hazardous inputs, releases and waste over a stated lifecycle.