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

Eqn summary: circular motion v = 2  r/P C = 2  r L=mvr F c = mv 2 /r v = speed r = radius P = period C = circumference L = angular momentum m = mass F c = Centrifugal Force

Eqn summary: gravity F g = mg F g = G Mm/d 2 g = GM/R 2 W = mgd v esc = Sqrt[2GM/R] F g = 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 v esc = escape speed

Kepler’s 3rd law M+m = a 3 /P 2 M+m = combined mass (in M sun ) a = semi-major axis (in au) P = Period (in years)

Eqn summary: waves & light  f = s E = h f F E = E/At =  T 4 max =2.8 x 10 6 /T Waves = wavelength f = frequency s= speed of wave (= c for light) Photon of light E = Energy h = Planck’s constant F E = radiative flux (energy E per area A and time t) T = temperature (in K) max = wavelength of maximum intensity (in nm)

Doppler shift ex#1:  = 500 nm ; obs = 550 nm  = obs -  =50 nm v = (50/500) c = c/10 = +30,000 km/s  Redshift => moving away from obs. ex#2:  = 500 nm ; obs = 495 nm  = obs -  = -5 nm v = (-5/500) c = -c/100 = km/s  Blueshift => moving toward obs.

Angle, distance, parallax, resolution a = 360 o s/2  d a arcsec = s au /d pc a = 2.5 x 10 5 arcsec ( /D) a = angular size s = linear size d = distance a arcsec = parallax angle in arcsec s au = separation size in au d pc = distance in parsec (pc) a = diffraction limit resolution for telescope of diameter D and light of wavelength

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

Luminosity & magnitude Luminosity: L =  T 4 4  R 2 – or: L/L sun = (T/T sun ) 4 (R/R sun ) 2 App. Brightness = L/4  D 2 Abs. Magnitude = log(L/L sun ) App. Mag. = Abs. Mag. + 5 log(D/10pc)

Gravitational Escape Grav. Escape energy: E g = GMm/R Kinetic energy: E k = mV 2 /2 E k =E g => Escape speed: V esc = Sqrt[2GM/R] For Sun: V esc ~= 600 km/s Thus: V esc ~= 600 km/s Sqrt[M/R] (where M,R in solar units)

Black Holes V esc = Sqrt[2GM/R] R = 2GM/(V esc ) 2 For black hole, V esc = c = speed of light R BH = 2GM/c 2 (a.k.a. “Schwarzschild radius”) For object with mass of Sun, R BH = 3 km Thus in general, R BH = 3 km (M/M sun )

Orbital motion centrifugal accel. a = v 2 /r gravity g ~ M / r 2 v 2 ~ M/r when a=g

Galactic Rotation speed vs. mass V 2 ~ M/r V ~ Sqrt[M/r] M(r) ~ V 2 r V ~ 

Hubble Law & Redshfit Doppler shift:V = c  / o – V= velocity – c = speed of light –  = - o =Doppler shift from rest wavelength o Hubble’s law:V = H o d –d= distance; H o = Hubble’s constant

“Drake Equation” N HP = number of habitable planets in the Galaxy f life = fraction of habitable planets which actually contain life f civ = fraction of life-planets where a civilization has at some time arisen Current number of civilizations in our Galaxy = N HP x f life x f civ x f now f now = fraction of civilizations which exist now