Kepler’s 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)
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!
ACCELERATION OF GRAVITY

gravitational force on a body on Earth's surface:
Fg = G  m  M/R2,




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


where m = the body’s (gravitational) mass
M = the Earth's (gravitational) mass,
R = the Earth's radius (distance from Earth's
center)
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;
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