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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
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
eqn summary circular motion
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
eqn summary gravity
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
kepler s 3rd law
M+m = a3/P2

M+m = combined mass

(in Msun)

a = semi-major axis

(in au)

P = Period

(in years)

Kepler’s 3rd law
eqn summary waves light
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
doppler shift
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.
angle distance parallax resolution
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
dating from radioactive decay
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)
luminosity magnitude
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)
gravitational escape
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)

black holes
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)
orbital motion

centrifugal accel.

a = v2/r

gravity

g ~ M /r2

v2 ~ M/r

when a=g

Orbital motion

galactic rotation speed vs mass
Galactic Rotation speed vs. mass
  • V2 ~ M/r
  • V ~ Sqrt[M/r]
  • M(r) ~ V2 r
  • V ~ Dl
hubble law redshfit
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
drake equation

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