Kepler’s laws

1. 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)

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 body’s (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