Equation summary: linear motion

• v = d/t

• p = mv

• a = v/t

• F = ma

v=speed (or velocity)d=distancet=time

p=momentumm=mass

a=acceleration

F=Force

Eqn summary: circular motion

• v = 2r/P

• C = 2r

• L=mvr

• Fc = mv2/r

v = speedr = radiusP = period

C = circumference

L = angular momentumm = mass

Fc = Centrifugal Force

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

Kepler’s 3rd law

M+m = a3/P2

M+m = combined mass

(in Msun)

a = semi-major axis

(in au)

P = Period

(in years)

Eqn summary: waves & light

• f = s

• E = h f

• FE = E/At = T4

• max=2.8 x 106/T

Waves = wavelength

f = frequency

s= speed of wave (= c for light)

Photon of lightE = Energy

h = Planck’s constant

FE = 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.

€

=v

c

• ex#2: = 500 nm ; obs = 495 nm

• = obs-= -5 nm

• v = (-5/500) c = -c/100 = -3000 km/s• Blueshift => moving toward obs.

Angle, distance, parallax, resolution

• a = 360o s/2d

• aarcsec = sau/dpc

• a = 2.5 x 105 arcsec (/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

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: L = T4 4R2

– or: L/Lsun= (T/Tsun)4 (R/Rsun)2

• App. Brightness = L/4D2

• Abs. Magnitude = +5 - 2.5 log(L/Lsun)

• App. Mag. = Abs. Mag. + 5 log(D/10pc)

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

• 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/rwhen a=g

Galactic Rotation speed vs. mass

• V2 ~ M/r

• V ~ Sqrt[M/r]

• M(r) ~ V2 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 = Ho d

–d= distance; Ho= Hubble’s constant

“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

Current number of civilizations in our Galaxy

= NHP x flife x fciv x fnow

fnow = fraction of civilizations which exist now