Kepler

1
Keplers laws

• KEPLER's LAW(Johannes Kepler, 1571 - 1630)
• 1. The orbit of a planet around the Sun is an
  ellipse with the Sun at one focus.
• 2. A line joining a planet and the Sun sweeps out
  equal areas in equal times.
• 3. The squares of the revolution periods of the
  planets are proportional to the cubes of their
  distance from the Sun P2 K a3 (K
  proportionality constant)
• Notes
• Ellipse locus of points for which the sum of
  the distances from two fixed points (the focal
  points, or foci) is constant, 2a, where a
  semimajor axis of the ellipse
• distance in 3rd law is really semimajor axis
• a circle is a special case of an ellipse, where
  the semimajor and semimajor axes are equal a b
  r
• excentricity of ellipse (distance of focus
  from center) divided by (semimajor axis)
• excentricity of a circle 0
• excentricities of most planetary orbits very
  small (except Pluto)

2
GRAVITATION

• material bodies have a property called
  gravitational mass
• due to this property, they can exert a force
  (gravitational force) on other bodies that also
  have this property.
• the force is attractive and acts along the line
  connecting the two bodies.
• the force acts without physical contact between
  the bodies (action at a distance)
• Newton's law of gravitation
• Between any two objects there acts an attractive
  force that is proportional to the product of the
  two objects' masses and inversely proportional to
  the square of the distance between the two
  objects Fg G ? m1 ? m2/d2
• the proportionality constant G is called
  gravitational constant G 6.67x 10 -11 N
  kg -2 m2
• it turns out that gravitational mass is
  proportional to inertial mass -- set them equal
  (this fixes value of gravitational constant)
• distance d the distance between the centers of
  mass (centers of gravity) of the two objects
• note that gravitational force is a very weak
  force
• gravitational force between two football players
  (assume mass 100kg each) at 1 ft distance is
  about 7x 10-6 newtons ? 1.6 x10-6 pounds!

3
ACCELERATION OF GRAVITY

• gravitational force on a body on Earth's surface
  Fg G ? m ? M/R2,
• where m the bodys (gravitational) mass
• M the Earth's (gravitational) mass,
• R the Earth's radius (distance from Earth's
  center)
• acceleration of body due to gravitational force
• Newton's 2nd law acceleration
  force/(inertial mass) ag Fg/mi
  
• where mi inertial mass of body ? ag ( m/
  mi ) ? (G ? M/R2)
• it is customary to use the symbol g for the
  second factor g G ? M/R2 ,
• so we find the acceleration of a body under the
  influence of the Earth's gravity ag ( m/mi
  ) ? g
• the fact that all bodies fall the same way means
  that inertial mass gravitational mass
• finally, we have at the surface of the Earth,
  all bodies are subject to the same acceleration g
  due to Earth's gravitational force g G ?
  M/R2 ? 9.8 m/s2
  
• the weight w of a body at the surface of the
  Earth ( the gravitational force exerted on it by
  Earth) is w Fg m ? g
  

4
GRAVITATION IN THE UNIVERSE

• gravitational force is weak, but has infinite
  range, and is not compensated by any repulsive
  antigravity ?in spite of its weakness,
  plays dominant rôle in universe
• governs motion of planets, stars, galaxies,....
• instrumental in birth and death of stars
• star formation
• density fluctuation in interstellar gas/dust
  cloud can lead to run-away accumulation of matter
  due to gravitational attraction -- gravitational
  collapse falling together of
  matter due to gravitational attraction
• formation of protostar huge ball of gas
  (mainly hydrogen, some helium, traces of heavier
  stuff)
• further contraction of protostar ?
  increase of temperature and pressure in its
  center
• when temperature and pressure high enough,
  nuclear fusion process starts
• radiation pressure due to nuclear fusion stops
  gravitational collapse

5
Life and death of stars

• stable midlife star (e.g. Sun) (also called
  main sequence star)
• dynamical equilibrium between gravitational
  attraction and radiation pressure from nuclear
  fusion of hydrogen into helium
• star generates energy ? life becomes possible.
• death of stars
• when hydrogen is all used up nothing to balance
  gravitational attraction ? collapse
• further fate depends on mass of the star
• light star (like our sun) temperature in
  center not high enough to allow fusion of helium
  into heavier nuclei ? end as white dwarf
• heavy star fusion of helium into
  successively heavier elements possible, fusion
  stops when all is fused into iron then
  gravitational collapse, stopped by
  neutronization abrupt stop of
  collapse ? supernova explosion
• supernova remnants
• neutron stars, pulsars,
• black holes