https://teddyrookbook.com/physics/astrophysics/Recent content in Astrophysics on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/physics/astrophysics/magnitude-equations/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/magnitude-equations/Magnitudes compare fluxes on a log scale: m₁ − m₂ = −2.5 log(F₁/F₂). Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/distance-modulus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/distance-modulus/A star's absolute magnitude is its magnitude seen from 10 pc, and m − M = 5 log(d/10 pc). Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/extinction/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/extinction/Dust dims and reddens starlight, adding A to the distance modulus: m − M = 5 log(d/10 pc) + A. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/color-index/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/color-index/The color index B − V compares two magnitudes, and a smaller value means a hotter star. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/planck-function/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/planck-function/B_λ(T) gives a blackbody's spectrum; its peak gives Wien's law and its area gives σT⁴. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/effective-temperature/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/effective-temperature/Use L = 4πR²σT_eff⁴ with real values to find the radii of dwarfs, giants and supergiants. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/boltzmann-excitation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/boltzmann-excitation/The fraction of atoms in an excited level follows (g_b/g_a)e^(−ΔE/kT), which is why hydrogen lines peak in A stars. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/ionization-sequence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/ionization-sequence/Rising temperature ionizes each element in turn, so the spectral classes are a temperature sequence, not a sequence of composition. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/visual-binaries/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/visual-binaries/Parallax turns angular separation into a, Kepler's third law gives M₁ + M₂, and the center of mass splits it. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/spectroscopic-binaries/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/spectroscopic-binaries/Doppler shifts trace each star's orbit, and their speeds give the mass ratio m₁/m₂ = v₂/v₁. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/eclipsing-binaries/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/eclipsing-binaries/The timing of eclipses gives the stars' radii, and the depths compare their surface temperatures. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/exoplanet-transits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/exoplanet-transits/A transit's depth (R_p/R_*)² gives the planet's size, and its period gives the orbit. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/exoplanet-wobble/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/exoplanet-wobble/A planet's pull sets its star's velocity amplitude, which gives the planet's mass times sin i. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/planet-temperature/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/planet-temperature/Balancing absorbed starlight against emitted heat gives T = T_*(1 − A)^(1/4)√(R_*/(2d)). Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/hydrostatic-equilibrium/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/hydrostatic-equilibrium/In a star, pressure balances gravity layer by layer: dP/dr = −Gm(r)ρ/r². Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/central-conditions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/central-conditions/Order-of-magnitude estimates give P_c ~ GM²/R⁴ and T_c ~ GMm_p/(kR), millions of kelvins for the Sun. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/virial-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/virial-theorem/For a gas ball held by its own gravity, 2K + U = 0, so losing energy makes it contract and heat up. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/kelvin-helmholtz/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/kelvin-helmholtz/Contraction alone could power the Sun for only the Kelvin–Helmholtz time GM²/(RL), about 30 million years. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/fusion-chains/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/fusion-chains/Four protons become helium-4 and release 26.7 MeV; the CNO cycle takes over in hotter, heavier stars. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/fusion-tunneling/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/fusion-tunneling/The Coulomb barrier is far higher than kT in the Sun's core, so protons fuse only by tunneling. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/energy-transport/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/energy-transport/Photons random-walk outward, so N steps of length ℓ cover only √N·ℓ, and the trip out of the Sun takes on the order of 100,000 years. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/mass-luminosity/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/mass-luminosity/On the main sequence L ∝ M^3.5 roughly, so lifetimes go as M^(−2.5). Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/eddington-limit/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/eddington-limit/Radiation pressure on electrons balances gravity at L_Edd ≈ 3.3 × 10⁴ (M/M_⊙) L_⊙. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/solar-neutrinos/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/solar-neutrinos/Neutrinos leave the core at once, letting us see fusion directly, and their changing flavors solved the solar neutrino problem. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/evolution-tracks/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/evolution-tracks/A star's path across the H–R diagram follows the changes in its core. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/degeneracy-pressure/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/degeneracy-pressure/The exclusion principle makes packed electrons push back even when they are cold. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/white-dwarf-mass-radius/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/white-dwarf-mass-radius/Degeneracy gives R ∝ M^(−1/3), so heavier white dwarfs are smaller. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/chandrasekhar-limit/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/chandrasekhar-limit/Electrons moving near light speed can hold up at most about 1.4 solar masses. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/core-collapse/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/core-collapse/An iron core collapses in under a second, and neutrinos carry off about 10⁴⁶ J. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/neutron-star-physics/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/neutron-star-physics/A collapsing core keeps its angular momentum and magnetic flux, so neutron stars spin fast with huge fields. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/pulsar-spindown/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/pulsar-spindown/A pulsar's lost rotational energy, dE/dt = Iωω̇, powers the nebula around it. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/schwarzschild-radius/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/schwarzschild-radius/r_s = 2GM/c², about 3 km for each solar mass. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/gravitational-redshift/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/gravitational-redshift/A clock at radius r runs slow by √(1 − r_s/r) compared with one far away. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/tidal-forces/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/tidal-forces/The difference in gravity across a body, about 2GMmΔr/r³, tears apart moons and stretches matter near black holes. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/accretion-power/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/accretion-power/Matter falling onto a compact object releases about GMṁ/R, far more per kilogram than fusion near a neutron star. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/gravitational-waves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/gravitational-waves/Orbiting compact objects radiate gravitational waves and spiral inward; LIGO first detected merging black holes in 2015. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/rotation-curves/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/rotation-curves/v(r) = √(GM(r)/r), so a flat rotation curve needs mass that grows in proportion to r. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/supermassive-black-holes/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/supermassive-black-holes/Stars orbiting the center of the Milky Way weigh its black hole at about 4 million suns. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/distance-ladder/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/distance-ladder/Parallax calibrates Cepheids, Cepheids calibrate type Ia supernovae, and each rung reaches farther. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/redshift-scale-factor/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/redshift-scale-factor/1 + z = λ_observed/λ_emitted = a(now)/a(then): light stretches as space expands. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/hubble-time/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/hubble-time/For small z, d ≈ cz/H₀, and 1/H₀ sets the age scale of the universe. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/friedmann-equation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/friedmann-equation/H² = 8πGρ/3 − kc²/a², so a flat universe has the critical density 3H²/(8πG). Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/cosmic-eras/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/cosmic-eras/Matter thins as a⁻³, radiation as a⁻⁴, and dark energy stays constant, so each ruled in turn. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/cmb/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/cmb/The background's temperature scales as 1/a: 2.725 K today, about 3000 K when atoms formed. Astrophysics · Physics Ladderhttps://teddyrookbook.com/physics/astrophysics/nucleosynthesis/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/physics/astrophysics/nucleosynthesis/In its first minutes the universe made about 25% helium by mass, matching what we measure. Astrophysics · Physics Ladder