Foundations

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 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.

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.

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)].

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.

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

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

A far-field source produces a geometric delay b·s/c between two antennas.

An interferometer measures a finite set of Fourier samples, leaving unmeasured sky modes.

Discretize stellar structure, transport and composition into a residual vector F(y).

A stellar or planetary EOS can blend differently valid microscopic models.

Angle-dependent radiative transfer yields moments J,H,K; flux is 4πH.

NonLTE level populations solve coupled radiative and collisional rate equations with particle/charge constraints and the transfer solution.

Shellular stellar evolution approximates angular velocity as nearly constant on isobars and models vertical transport and mass loss with prescriptions.

For an independently conserved homogeneous component with constant equation-of-state w, density scales with a as a^-3(1+w).

Cosmological parameter inference combines data likelihoods with model priors and nuisance parameters for foregrounds, calibration and selection.

The astronomical unit and light-year measure lengths, with different defining conventions.

A sky direction is an angular observation, not an object's three-dimensional distance.