DSST Astronomy cheat sheet

Kepler's first law: planetary orbits are ellipses with the Sun at one focus; eccentricity e = c/a, where c is the focal distance from center and a is the semimajor axis.

Kepler's second law: a line from Sun to planet sweeps equal areas in equal times; orbital speed is greatest at perihelion and least at aphelion.

Kepler's third law: P^2 = a^3 when P is in years and a is in astronomical units (AU); more generally P^2 = 4π^2 a^3 / [G(M + m)].

Newton's law of gravitation: F = G m1 m2 / r^2, with G = 6.674×10^-11 N·m^2/kg^2; the force is attractive and acts along the line joining the centers.

Circular orbital speed: v = sqrt(GM/r); escape speed from a body of mass M and radius R is v_esc = sqrt(2GM/R) = sqrt(2) × v_circular at that radius.

Newton's form of Kepler's third law for the solar system: P^2 = a^3 / (M_total in solar masses); for binary stars, total mass M1 + M2 = a^3 / P^2 in solar units.

Tidal force scales as 1/r^3; differential gravity stretches a body along the line to the primary and compresses it perpendicular to that line.

Conservation laws: angular momentum L = m v r sin θ is constant in a central force field; orbital energy E = 1/2 m v^2 − GMm/r is constant, with E < 0 for bound orbits.

Roche limit: a satellite held together only by gravity is torn apart inside roughly 2.44 R_primary (density ratio)^(1/3); this explains ring systems.

Hierarchy of structure: planet–moon systems, star clusters, galaxies, galaxy groups and clusters, superclusters, and the cosmic web of filaments and voids.

Open clusters: young (10^7–10^9 yr), metal-rich, irregular, found in galactic disks, contain 10^2–10^3 stars; example: Pleiades.

Globular clusters: old (10^10 yr), metal-poor, spherical, orbit galactic halos, contain 10^5–10^6 stars; example: Omega Centauri.

Eclipsing binaries: orbital plane nearly edge-on to Earth; light curves yield radii and, combined with radial velocities, stellar masses.

Spectroscopic binaries: detected through periodic Doppler shifts of spectral lines; double-lined systems give mass ratio directly.

Visual binaries: resolvable as two stars; Kepler's third law applied to the relative orbit gives total mass.

Star clusters are used as cosmic laboratories because member stars share age and initial composition, producing clean Hertzsprung–Russell diagrams.

The Local Group contains the Milky Way, Andromeda (M31), Triangulum (M33), and dozens of dwarf galaxies; diameter roughly 3 Mpc.

Galaxy clusters contain hundreds to thousands of galaxies, hot intracluster gas (10^7–10^8 K, X-ray emitting), and dark matter; example: Coma Cluster.

Refracting telescope: objective lens; chromatic aberration limits aperture; practical maximum aperture about 1 m.

Reflecting telescope: concave primary mirror; no chromatic aberration; examples: Newtonian, Cassegrain, Ritchey–Chrétien.

Angular resolution (Rayleigh criterion): θ = 1.22 λ / D radians; larger aperture D or shorter wavelength λ gives finer resolution.

Light-gathering power scales as D^2; a 200 mm telescope collects 4 times the light of a 100 mm telescope.

Magnification = focal length of objective / focal length of eyepiece; useful magnification is limited by atmospheric seeing and aperture.

CCD detectors replaced photographic plates; quantum efficiency can exceed 80% versus a few percent for film.

Spectrographs disperse light with prisms or diffraction gratings; gratings give higher dispersion and linear wavelength calibration.

Radio telescopes need large dishes for resolution; interferometry (VLBI) combines widely separated antennas to synthesize an aperture equal to the baseline.

Adaptive optics uses a deformable mirror and a wavefront sensor (often a laser guide star) to correct atmospheric turbulence in real time.

Terrestrial planets: Mercury, Venus, Earth, Mars - small, rocky, high density, few or no moons, no ring systems.

Jovian planets: Jupiter, Saturn, Uranus, Neptune - large, low density, hydrogen/helium rich, many moons, ring systems.

Asteroid belt lies between Mars and Jupiter at 2.2–3.3 AU; Kuiper Belt lies beyond Neptune at about 30–50 AU.

Nebular hypothesis: solar system formed from a collapsing rotating cloud; conservation of angular momentum produced a flattened disk, and planetesimals accreted from dust.

Solar composition by mass: about 71% hydrogen, 27% helium, 2% heavier elements; energy from core hydrogen fusion via the proton–proton chain.

Sun's core temperature about 15 million K; photosphere about 5,800 K; sunspots are cooler regions (~4,000 K) tied to magnetic activity.

Venus has a runaway greenhouse effect: surface temperature about 730 K, atmospheric pressure about 90 times Earth's, dominated by CO2 and sulfuric acid clouds.

Mars shows evidence of past liquid water (valley networks, layered sediments); present atmosphere is thin CO2, about 0.6% of Earth's pressure.

Jupiter's Great Red Spot is a long-lived anticyclonic storm; Jupiter has four large Galilean moons: Io (volcanic), Europa (ice ocean), Ganymede (largest moon), Callisto.

Spectral classes in order of decreasing temperature: O, B, A, F, G, K, M; the Sun is G2.

Hertzsprung–Russell diagram plots luminosity versus temperature (or spectral class); main sequence runs diagonally from hot, luminous O stars to cool, dim M stars.

Mass–luminosity relation for main-sequence stars: L/L_sun ≈ (M/M_sun)^3.5; more massive stars are far more luminous and shorter-lived.

Stellar structure balance: hydrostatic equilibrium (pressure gradient vs. gravity) and thermal equilibrium (energy generation vs. radiation).

Energy generation: proton–proton chain dominates in stars like the Sun; CNO cycle dominates in stars more massive than about 1.3 M_sun.

Main-sequence lifetime: t ≈ 10^10 (M/M_sun)^−2.5 years; a 10 M_sun star lives roughly 3×10^7 years.

Low-mass stars end as white dwarfs (supported by electron degeneracy pressure), possibly after a planetary-nebula phase.

High-mass stars end as core-collapse supernovae (Type II), leaving neutron stars or black holes; Type Ia supernovae come from white dwarfs in binary systems.

Stellar remnants: white dwarf mass limit about 1.4 M_sun (Chandrasekhar); neutron star mass limit about 2–3 M_sun; black hole Schwarzschild radius R_s = 2GM/c^2.

Milky Way is a barred spiral galaxy; diameter about 100,000 light-years; the Sun lies about 26,000 light-years from the center.

Galactic components: thin disk (young stars, gas, dust), thick disk, central bulge, and stellar halo with globular clusters.

Differential rotation: inner regions orbit faster than outer regions; the rotation curve is measured via 21 cm hydrogen emission and stays roughly flat at large radii, implying dark matter.

Sun's orbital speed around the galaxy is about 220 km/s; one galactic year is roughly 230 million years.

Interstellar medium phases: molecular clouds (cold, dense, star-forming), neutral hydrogen (H I), ionized H II regions, and hot coronal gas.

Dust extinction reddens starlight because shorter wavelengths scatter more; observed color must be corrected for interstellar reddening.

Spiral arms are density waves, not fixed material; star formation concentrates in arms where gas clouds are compressed.

Galactic center hosts the radio source Sagittarius A*, a supermassive black hole of about 4 million solar masses.

Population I stars are young, metal-rich, disk stars; Population II stars are old, metal-poor, halo and globular-cluster stars.

Hubble's law: v = H0 × d, with H0 about 70 km/s/Mpc; recession velocity is proportional to distance for galaxies beyond the Local Group.

Cosmological redshift: z = (λ_observed − λ_emitted)/λ_emitted; for small z, v ≈ c z.

Big Bang evidence: cosmic microwave background (T ≈ 2.725 K), primordial light-element abundances, and observed galaxy recession.

CMB was emitted at recombination (z ≈ 1,100, about 380,000 years after the Big Bang) when the universe became transparent.

Friedmann equation governs expansion: H^2 = (8πG/3)ρ − k c^2/a^2 + Λc^2/3; critical density ρ_c = 3H0^2/(8πG).

Density parameter Ω = ρ/ρ_c; Ω_total = 1 corresponds to a flat universe; current measurements give Ω_matter ≈ 0.3 and Ω_Λ ≈ 0.7.

Dark energy drives accelerated expansion; the accelerating phase began when the universe was several billion years old.

Nucleosynthesis in the first few minutes produced about 75% hydrogen and 25% helium by mass, plus trace deuterium and lithium.

Age of the universe from current cosmological parameters is about 13.8 billion years.

Parallax: p (arcseconds) = 1 / d (parsecs); a star at 1 pc has a parallax of 1 arcsecond; 1 pc = 3.26 light-years = 3.086×10^16 m.

Stellar parallax is limited by atmospheric seeing and spacecraft precision; Gaia measures parallaxes to microarcsecond levels for bright stars.

Standard candles: objects of known intrinsic luminosity L; apparent brightness b = L / (4π d^2), so d = sqrt(L / 4π b).

Distance modulus: m − M = 5 log10(d) − 5, with d in parsecs; m is apparent magnitude, M is absolute magnitude.

Cepheid variables follow a period–luminosity relation; longer periods mean greater luminosity, enabling distance measurement in nearby galaxies.

Type Ia supernovae are standardizable candles visible at cosmological distances; they revealed accelerated expansion.

Tully–Fisher relation uses galaxy rotation width versus luminosity for spiral galaxies; Faber–Jackson uses velocity dispersion versus luminosity for ellipticals.

Redshift-based distances require a cosmological model; Hubble's law is reliable only for large-scale distances where peculiar velocities are small.

Magnitude scale: a difference of 5 magnitudes equals a brightness ratio of 100; each magnitude is a factor of about 2.512.

Habitable zone: range of orbital distances where liquid water can exist on a planet's surface; location depends on stellar luminosity and spectral type.

Drake equation: N = R* × f_p × n_e × f_l × f_i × f_c × L; estimates of N span many orders of magnitude because several factors are poorly constrained.

SETI searches for narrow-band radio signals (for example, the 21 cm hydrogen line) and optical laser pulses; no confirmed extraterrestrial signal has been detected.

Fermi paradox: given the age and size of the galaxy, if intelligent life were common we would expect evidence of it; proposed resolutions include rare-Earth hypotheses and short civilization lifetimes.

Extremophiles on Earth (thermophiles, halophiles, psychrophiles) expand the range of conditions considered potentially habitable.

Biosignatures include atmospheric oxygen and ozone together, methane in disequilibrium, and spectral features of chlorophyll-like pigments.

Mars and icy moons Europa and Enceladus are leading targets in the search for past or present life because of evidence for liquid water.

Panspermia proposes that life or its precursors can travel between bodies on meteorites; it does not explain the origin of life itself.

Rare-Earth hypothesis argues that complex life requires many unusual conditions, such as a large moon, plate tectonics, and a Jupiter-like planet to reduce impact rates.