AST cheat sheet

Hubble-Lemaître law: v = H0·d, with H0 ≈ 70 km/s/Mpc; redshift z = (λobs − λem)/λem and, for small z, v ≈ cz.

Cosmological redshift relates scale factor by 1 + z = a0/a; for large z, recession velocity is not simply cz.

Luminosity distance dL = (1 + z)·comoving transverse distance for flat space, so flux f = L/(4π dL²).

Distance modulus: m − M = 5 log10(d/10 pc); extinction Aλ raises observed m and reddens spectra.

Friedmann equation: H² = (8πG/3)ρ − kc²/a² + Λc²/3; Ωm + ΩΛ + Ωk = 1.

Critical density ρc = 3H0²/(8πG) ≈ 9.2×10⁻²⁷ kg/m³ for H0 = 70 km/s/Mpc.

Blackbody: λmax·T = 2.898×10⁻³ m·K (Wien); bolometric flux F = σT⁴ (Stefan-Boltzmann).

Magnitudes are logarithmic: Δm = 2.5 log10(F1/F2); a factor of 100 in flux equals 5 magnitudes.

CMB temperature T0 ≈ 2.725 K; last scattering at z ≈ 1100, roughly 380,000 years after the Big Bang.

Hydrostatic equilibrium: dP/dr = −G M(r) ρ(r)/r²; pressure gradient balances gravity at every shell.

Virial theorem for gravitationally bound systems: 2K + U = 0, so total energy E = U/2 = −K.

Mass-luminosity relation on the main sequence: L ∝ M^≈3.5 for solar-type stars; lifetime t ∝ M/L ∝ M^−2.5.

Energy transport: radiation when opacity is low, convection when the temperature gradient exceeds the adiabatic gradient.

Spectral classification sequence OBAFGKM by decreasing surface temperature; luminosity class I–V encodes evolutionary stage.

Main sequence turnoff age is set by the mass at which core hydrogen exhaustion time equals cluster age.

Hertzsprung-Russell diagram: giants and supergiants sit at high L and low T; white dwarfs at low L and high T.

Chandrasekhar limit ≈ 1.4 M☉ for electron-degenerate cores; above it, collapse proceeds to a neutron star or black hole.

Nucleosynthesis: p-p chain dominates in Sun-like stars; CNO cycle dominates above roughly 1.3 M☉; iron peak ends exothermic fusion.

Hubble morphological sequence: ellipticals (E0–E7), lenticulars (S0), spirals (Sa–Sc), barred spirals (SBa–SBc), irregulars (Irr).

Spiral arms trace density waves in the disk, not fixed material; star formation concentrates in the arms.

Milky Way structure: thin disk, thick disk, central bulge/bar, stellar halo, dark-matter halo; Sun at ≈8 kpc from the center.

Rotation curve v(r) stays roughly flat at large radii, implying dark matter or modified gravity; v² = GM(r)/r.

Tully-Fisher relation: L ∝ vmax⁴ for spirals; Faber-Jackson relation: L ∝ σ⁴ for ellipticals.

Galaxy colors: blue indicates recent star formation (spirals, irregulars); red indicates old populations or dust (ellipticals).

Virial mass estimate for clusters: M ≈ 5σ²R/G; galaxy clusters also reveal hot intracluster gas in X-rays.

Metallicity gradients and abundance patterns trace chemical enrichment history and merger events.

Active galactic nuclei are powered by accretion onto supermassive black holes; broad lines come from fast-moving gas near the center.

Kepler's third law: P² ∝ a³ in solar units (P in years, a in AU); generalized form P² = 4π²a³/[G(M1 + M2)].

Terrestrial planets: small, rocky, high density, few or no moons; giant planets: large, low density, many moons and rings.

Solar composition is dominated by hydrogen and helium; terrestrial planets are depleted in volatiles.

Roche limit: a body is tidally disrupted inside roughly 2.44 Rplanet(ρplanet/ρbody)^(1/3).

Albedo and equilibrium temperature: Teq = Tstar·(Rstar/2a)^(1/2)·(1 − A)^(1/4).

Planetary atmospheres: Jeans escape favors loss of light, hot species; greenhouse gases raise surface temperature.

Orbital resonances and tidal locking shape satellite systems; the Moon is tidally locked with synchronous rotation.

Asteroid belt lies between Mars and Jupiter; Kuiper belt and Oort cloud are icy reservoirs of comets.

Telescope resolution: θ ≈ 1.22 λ/D (radians); larger aperture improves both resolution and light-gathering power.

Light-gathering power scales as D²; integration time to reach a given signal-to-noise scales inversely with D².

Seeing from atmospheric turbulence typically limits ground-based resolution to about 1 arcsecond without adaptive optics.

CCD detectors: quantum efficiency, read noise, dark current, and gain determine signal-to-noise; shot noise scales as √N.

Spectrograph resolving power R = λ/Δλ; higher R separates closely spaced spectral lines.

Photometric filters (e.g., UBVRI or ugriz) define passbands; color indices are magnitude differences between bands.

Interferometry combines apertures to reach angular resolution set by baseline, not individual aperture size.

Radio astronomy uses large wavelengths; dishes must be large to achieve modest angular resolution.

Space telescopes avoid atmospheric absorption and seeing, enabling UV, X-ray, and infrared observations.

Newtonian gravity: F = G M1 M2 / r²; gravitational parameter μ = G(M1 + M2).

Vis-viva equation: v² = μ(2/r − 1/a), linking speed, distance, and semimajor axis.

Orbital energy: ε = −μ/(2a); bound orbits have negative energy, parabolic zero, hyperbolic positive.

Kepler's laws: elliptical orbits with the focus at the primary, equal areas in equal times, P² ∝ a³.

Escape velocity: vesc = √(2μ/r); circular orbital speed: vcirc = √(μ/r).

Angular momentum h = r × v is conserved in a central force field; orbit plane stays fixed.

Perturbations from a third body cause precession, nodal regression, and resonance locking (e.g., Kozai-Lidov cycles).

Tidal forces scale as 1/r³; they drive orbital circularization and spin-down over time.

Two-body problem reduces to a one-body problem with reduced mass; center of mass moves uniformly.

ISM phases: cold molecular clouds (T ≈ 10–20 K), warm neutral medium, warm ionized medium, and hot ionized gas.

Jeans mass: MJ ∝ T^(3/2) ρ^(−1/2); cold, dense clouds collapse more readily.

Molecular clouds are traced by CO emission and dust extinction; H₂ is hard to detect directly.

Star formation occurs in dense clumps; the initial mass function is roughly a power law at high masses.

HII regions around massive stars are ionized by UV photons and glow in recombination lines such as Hα.

Dust reddens and extinguishes starlight; extinction increases toward shorter wavelengths.

Supernovae and stellar winds inject energy, drive turbulence, and enrich the ISM with heavy elements.

Magnetic fields and turbulence slow collapse; ambipolar diffusion allows neutral gas to slip past ions.

Newton's laws: F = ma; action-reaction pairs; inertia defines resistance to acceleration.

Conservation laws: energy, linear momentum, and angular momentum hold in isolated systems.

Thermodynamics: ideal gas law P = nkT; adiabatic index γ = 5/3 for monatomic gas.

Radiation: Planck spectrum, Wien displacement, Stefan-Boltzmann law; photon energy E = hc/λ.

Special relativity: Lorentz factor γ = 1/√(1 − v²/c²); time dilation and length contraction at high speed.

Doppler effect: Δλ/λ ≈ v/c for nonrelativistic radial motion; blueshift for approach, redshift for recession.

Gravitational lensing: light bends by angle α = 4GM/(c²b) for a point mass; Einstein ring when source, lens, observer align.

Quantum basics: energy levels quantized; spectral lines arise from transitions with E = hν.

Plasma physics: charged particles respond to electromagnetic fields; Debye length sets screening scale.