Transcript Equation summary

Equation summary: linear motion
• v = d/t
v=speed (or velocity)
d=distance
t=time
• p = mv
p=momentum
m=mass
• a = v/t
a=acceleration
• F = ma
F=Force
Eqn summary: circular motion
• v = 2pr/P
• C = 2pr
v = speed
r = radius
P = period
C = circumference
• L=mvr
L = angular momentum
m = mass
• Fc = mv2/r
Fc = Centrifugal Force
Eqn summary: gravity
• Fg = mg
• Fg = G
Mm/d2
Fg = force of gravity (weight)
m = mass
g = acceleration of gravity
d = distance from mass m to M
• g = GM/R2
• W = mgd
• vesc = Sqrt[2GM/R]
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
Waves
• lf=s
• E=hf
• FE = E/At = s T4
• lmax=2.8 x 106/T
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)
Doppler shift
Dl
v
=
l c
• 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 105 arcsec (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
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 = 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
• 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
when a=g
v2 ~ M/r
Galactic Rotation speed vs. mass
• V2 ~ M/r
• V ~ Sqrt[M/r]
• M(r) ~ V2 r
• V ~ Dl
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 x flife x fciv x fnow
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