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Comprehensive Overview of Motion, Gravity, Waves, and Astrophysics Equations

This document summarizes key equations and concepts related to linear motion, circular motion, waves, and astrophysics. From foundational principles like speed, momentum, and gravitational forces to advanced topics such as escape velocity and the Doppler effect, it enables a deep understanding of classical mechanics and cosmology. Additionally, it covers Kepler's laws, radioactive decay, and the Drake Equation, providing insight into the dynamics of planetary systems and the potential for extraterrestrial life.

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Comprehensive Overview of Motion, Gravity, Waves, and Astrophysics Equations

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  1. v = d/t p = mv a = v/t F = ma v=speed (or velocity) d=distance t=time p=momentum m=mass a=acceleration F=Force Equation summary: linear motion

  2. v = 2pr/P C = 2pr L=mvr Fc = mv2/r v = speed r = radius P = period C = circumference L = angular momentum m = mass Fc = Centrifugal Force Eqn summary: circular motion

  3. Fg = mg Fg = G Mm/d2 g = GM/R2 W= mgd vesc = Sqrt[2GM/R] Fg = force of gravity (weight) m = mass g = acceleration of gravity d = distance from mass m to M G = gravitation constant M = mass (e.g. of planet) R = radius (e.g. of planet) W = Work against gravity vesc = escape speed Eqn summary: gravity

  4. M+m = a3/P2 M+m = combined mass (in Msun) a = semi-major axis (in au) P = Period (in years) Kepler’s 3rd law

  5. l f = s E = h f FE = E/At = s T4 lmax=2.8 x 106/T Waves l = wavelength f = frequency s= speed of wave (= c for light) Photon of light E = Energy h = Planck’s constant FE = radiative flux (energy E per area A and time t) T = temperature (in K) lmax = wavelength of maximum intensity (in nm) Eqn summary: waves & light

  6. Doppler shift • ex#1: l = 500 nm ; lobs = 550 nm • Dl = lobs-l =50 nm • v = (50/500) c = c/10 = +30,000 km/s • Dl>0 => Redshift => moving away from obs. • ex#2: l = 500 nm ; lobs = 495 nm • Dl = lobs-l = -5 nm • v = (-5/500) c = -c/100 = -3000 km/s • Dl<0 => Blueshift => moving toward obs.

  7. a = 360o s/2pd aarcsec = sau/dpc a = 2.5 x 105arcsec (l/D) a = angular size s = linear size d = distance aarcsec = parallax angle in arcsec sau = separation size in au dpc = distance in parsec (pc) a = diffraction limit resolution for telescope of diameter D and light of wavelength l Angle, distance, parallax, resolution

  8. Dating from Radioactive Decay • If element “x” decays with half-life th, then: • # of x at time t: Nx(t) = Nx(t=0) 2-t/th • & fraction of x at t: fx(t) = Nx(t)/Nx(t=0) = 2-t/th • To solve for time t, take Log of both sides: • Log(fx) = - (t/th) Log(2) = (t/th) Log(1/2) => • t = th Log(fx)/Log(1/2)

  9. Luminosity & magnitude • Luminosity: L = sT4 4pR2 • or: L/Lsun= (T/Tsun)4 (R/Rsun)2 • App. Brightness = L/4pD2 • Abs. Magnitude = +5 - 2.5 log(L/Lsun) • App. Mag. = Abs. Mag. + 5 log(D/10pc)

  10. Gravitational Escape • Grav. Escape energy: Eg = GMm/R • Kinetic energy: Ek = mV2/2 • Ek=Eg => Escape speed: Vesc = Sqrt[2GM/R] • For Sun: Vesc ~= 600 km/s • Thus: Vesc ~= 600 km/s Sqrt[M/R] (where M,R in solar units)

  11. Black Holes • Vesc = Sqrt[2GM/R] • R = 2GM/(Vesc)2 • For black hole, Vesc = c = speed of light • RBH = 2GM/c2 (a.k.a. “Schwarzschild radius”) • For object with mass of Sun, RBH = 3 km • Thus in general, RBH = 3 km (M/Msun)

  12. centrifugal accel. a = v2/r gravity g ~ M /r2 v2 ~ M/r when a=g Orbital motion

  13. Galactic Rotation speed vs. mass • V2 ~ M/r • V ~ Sqrt[M/r] • M(r) ~ V2 r • V ~ Dl

  14. Hubble Law & Redshfit • Doppler shift: V = c Dl/lo • V= velocity • c = speed of light • Dl = l-lo=Doppler shift from rest wavelength lo • Hubble’s law: V = Ho d • d= distance; Ho= Hubble’s constant

  15. Current number of civilizations in our Galaxy = NHP xflife xfciv xfnow “Drake Equation” NHP = number of habitable planets in the Galaxy flife = fraction of habitable planets which actually contain life fciv = fraction of life-planets where a civilization has at some time arisen fnow = fraction of civilizations which exist now

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