Foundations

Oxidation states assign electrons formally rather than measuring atomic partial charge.

For a balanced redox reaction transferring n electrons per reaction event, the molar reaction Gibbs energy is ΔrG=−nFE at the same temperature and composition when E is its reversible cell potential.

Characteristic archaeal membrane lipids contain branched isoprenoid chains ether-linked to a glycerol-1-phosphate backbone, compared with the common bacterial/eukaryotic fatty-acyl ester-linked glycerol-3-phosphate model.

Plasmogamy joins fungal cytoplasm without immediately fusing the nuclei.

Diatoms provide a photosynthetic eukaryotic example with silica-rich cell walls.

Epistasis means that the phenotypic effect associated with a locus depends on another locus, including masking effects in a biochemical pathway.

For an X-linked locus without a corresponding Y allele in the standard XX/XY model, an XY individual is hemizygous, and reciprocal crosses can yield different genotype distributions.

A semicrystalline polymer contains ordered crystallites and amorphous material.

For an ideal linear step-growth model, the number-average length Xn counts monomer-derived units per product molecule, including unreacted monomers in the counted population.

For a classical gas in local thermodynamic equilibrium, level ratios within one ionization stage obey N_b/N_a=(g_b/g_a)exp[−(E_b−E_a)/(k_BT)].

An absorption line's wavelength equivalent width is W_λ=∫[1−F_λ/F_{λ,c}]dλ, using the local continuum.

In an expanding FLRW model with present scale factor one, the line-of-sight comoving distance is D_C=c∫₀ᶻ dz′/H(z′), and the lookback time is t_L=∫₀ᶻ dz′/[(1+z′)H(z′)].

For transparent geometrical photon propagation in FLRW cosmology, angular diameter distance is D_A=D_M/(1+z), and luminosity distance is D_L=(1+z)D_M=(1+z)²D_A.

Radial stellar pulsation has a characteristic gravitational/acoustic period scale proportional to √[R³/(GM)].

A spherical planet intercepts stellar power over its projected disk and emits thermal power over its surface.

At an exobase radius r_ex and temperature T_ex, λ=GMm/(r_ex k_BT_ex) compares a species' binding energy with thermal energy.

Tides arise from gravitational acceleration differences across an extended body.

In a smooth stationary axisymmetric potential, expanding about a stable circular guiding radius R_g gives small radial oscillations at epicyclic frequency κ and vertical oscillations at ν.

For a razor-thin rotating ideal fluid disk, local axisymmetric tightly wound perturbations have dispersion ω²=κ²−2πGΣ|k|+c_s²k².

In a well-mixed closed gas-plus-star system with initially zero metallicity, instantaneous recycling and constant yield p per mass locked in long-lived stars, gas metallicity follows Z=p ln(1/μ), where μ=M_gas/M_total.

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.