********Useful facts and formulas**************

 Radius of Earth = 6.4 X 10^6 m

 Radius of Moon = 1.7 X 10^6 m

 Radius of Sun = 7 X 10^{8} m

 Mass of Earth = 6 X 10^{24} kg

 Mass of Sun = 2 X 10^{30} kg

 Mass of Moon = 7 X 10^{22} kg

 Mean Radius of Mars' orbit = 1.5 AU

 Temperature of Sun = 6000 K

 Lifetime of Sun = 10^{10} years

 1 day ~ 1 X 10^5 sec

 1 month ~ 3 X 10^6 sec

 1 year ~ 3 X 10^7 sec

 1 Astronomical Unit = 1.5 X 10^{11} m

 1 light year = 1 X 10^{16} m

 1 parsec ~ 3 light years = 200,000 AU

 1 Megaparsec (Mpc) = 10^6 pc

 Radius of Moon's orbit around the Earth = 4 X 10^{8} m

 Luminosity of Sun = 4 X 10^{26} Joules/sec

 Newton's constant G = 2/3 X 10^{-10} m^3 s^{-2} kg^{-1}

 Planck's constant h = 2/3 X 10^{-33} Joule sec

 Speed of light c= 3 X 10^8 m/sec

 1 Joule = 1 kilogram/meter^2/second^2 (a unit of energy)

 One electron volt = 1.6 X 10^{-19} Joules. 

 Stephan-Boltzmann Constant sigma = 5.6 X 10^{-8} W/m^2/K^4.

 Boltzmann's constant k = 1.4 X 10^{-23} Joules/K.  

 Mass of proton ~ Mass of neutron ~ Mass of Hydrogen
atom = 1/6 X 10^{-26} kg

 0 degrees Centigrade corresponds to 273 kelvins

 Hubble Constant ~ 70 km/s/Mpc

 pi ~ 3

   The angle subtended by an object in radians is given by its diameter, 
divided by its distance (the small-angle formula). 

 1 radian ~ 60 degrees ~ 200,000 arcsec.  A full circle
covers 360 degrees, or roughly 6 radians.

   The circumference of a circle of radius r is 2 pi r; its area
is pi r^2. 

    The surface area of a sphere of radius r is 4pi r^2; its
volume is 4/3 pi r^3. 

    The acceleration required to keep an object in a circle of radius
r at uniform speed v is a = v^2/r. 

   The gravitational force between two objects of mass M and m
separated by a distance r is G M m/r^2. 

   Kepler's Third Law states that for orbits around a body of mass M,
the period squared is proportional to the radius of the orbit cubed,
divided by the mass M.  If period is measured in years, the radius in
AU, and the mass in solar masses, the constant of proportionality is
unity.   In physical units, Newton's form
of Kepler's Third Law is P^2 = (4 pi^2 a^3)/(GM). 

   The gravitational potential energy associated with raising a mass m
   a height h in a uniform gravitational field g is mgh.  The
   gravitational potential energy of a mass m a distance R from a mass
   M is -GMm/R.  

   The escape speed from an object of radius R and mass M is 
     (2GM/R)^(1/2).  The orbital speed of a satellite a distance r
     from the center of the object is (GM/r)^(1/2). 

   The kinetic energy of a body of mass m moving at a speed v is 
     1/2 m v^2. 

   The kinetic energy of each particle of a gas at temperature T is 
     3/2 k T.

   The energy per unit time emitted by a blackbody of surface area A
and temperature T is equal to sigma A T^4. 

 The wavelength lambda and frequency nu of a photon are
related as lambda nu = c, where c is the speed of light.  The
energy of a photon is proportional to its frequency, E=h nu.

   The blackbody spectrum of an object of temperature T peaks at a
wavelength lambda = 2.9/T millimeter, if T is measured in Kelvin.  

   The brightness of a distant object is proportional to its luminosity
times the inverse square of its distance to us.

   The equilibrium temperature of a planet of albedo a distance d away
   from a star of surface temperature T_{star} and radius r_{star} is
   T_{planet} = T_{star}(1-A)^{1/4} (r_{star}/2d)^{1/2}. 

 The parallax due to the Earth's orbit around the Sun, of a star 1
parsec away, is one arcsecond.  Parallax is inversely proportional to
distance.

   The luminosity of a main sequence star is proportional to its mass
to the 3.5 power. 

 The Sun is 25,000 light years from the center of the Milky Way, and
makes a full orbit once every 2.5 X 10^8 years.  The mass of
the Milky Way within the Sun's orbit is 10^{11} solar masses. 

 The Doppler shift in the wavelength lambda of a light wave
emitted from an object moving at speed v along the line of sight is
Delta lambda/lambda = v/c.   If the speed is close to the speed of
light, one should use the more exact formula: 
Delta lambda/lambda = ((1 + v/c)/(1 - v/c))^{1/2} - 1

 Hubble's Law: The recession velocity of a galaxy is equal to its
distance times the Hubble Constant H. 

 The age of the universe since the Big Bang is roughly the inverse of
the Hubble Constant.  This gives roughly 13 billion years. 

 The critical density of the universe is 3 H_0^2/8 pi G, or
roughly 10^{-26}kg/m^3. 

 Energy E and mass m are equivalent: E = m c^2.