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Equation summary: linear motion

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. v = 2 p r/P C = 2 p r L=mvr F c = mv 2 /r. v = speed r = radius P = period C = circumference L = angular momentum m = mass

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Equation summary: linear motion

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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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