Equation%20summary:%20linear%20motion

1
Equation summary linear motion

• v d/t
• p mv
• a v/t
• F ma

• vspeed (or velocity)
• ddistance
• ttime
• pmomentum
• mmass
• aacceleration
• FForce

2
Eqn summary circular motion

• v 2pr/P
• C 2pr
• Lmvr
• Fc mv2/r

• v speed
• r radius
• P period
• C circumference
• L angular momentum
• m mass
• Fc Centrifugal Force

3
Eqn summary gravity

• 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

• Fg mg
• Fg G Mm/d2
• g GM/R2
• W mgd
• vesc Sqrt2GM/R

4
Keplers 3rd law

• Mm combined mass
• (in Msun)
• a semi-major axis
• (in au)
• P Period
• (in years)

• Mm a3/P2

5
Eqn summary waves light

• l f s
• E h f
• FE E/At s T4
• lmax2.8 x 106/T

• Waves
• l wavelength
• f frequency
• s speed of wave ( c for light)
• Photon of light
• E Energy
• h Plancks constant
• FE radiative flux
• (energy E per area A and time t)
• T temperature (in K)
• lmax wavelength of maximum intensity (in nm)

6
Doppler shift

• ex1 l 500 nm lobs 550 nm
• Dl lobs-l 50 nm
• v (50/500) c c/10 30,000 km/s
• Dlgt0 gt Redshift gt moving away from obs.

• ex2 l 500 nm lobs 495 nm
• Dl lobs-l -5 nm
• v (-5/500) c -c/100 -3000 km/s
• Dllt0 gt Blueshift gt moving toward obs.

7
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

8
Dating from Radioactive Decay

• If element x decays with half-life th, then
• of x at time t Nx(t) Nx(t0)
  2-t/th
• fraction of x at t fx(t) Nx(t)/Nx(t0)
  2-t/th
• To solve for time t, take Log of both sides
• Log(fx) - (t/th) Log(2) (t/th) Log(1/2) gt
• 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
• EkEg gt Escape speed Vesc Sqrt2GM/R
• For Sun Vesc 600 km/s
• Thus Vesc 600 km/s SqrtM/R
• (where M,R in solar
  units)

11
Black Holes

• Vesc Sqrt2GM/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
Orbital motion
13
Galactic Rotation speed vs. mass

• V2 M/r
• V SqrtM/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-loDoppler shift from rest wavelength lo
• Hubbles law V Ho d
• d distance Ho Hubbles constant

15
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