I. Kepler' Laws of Planetary Orbits

Kepler's 1st Law: The orbits of planets are ellipses with the sun at one focus.

• ellipses are "squashed circles" (Fig. 4-20)
• focus (foci) are points where "string is held"
• sun lies at one focus (nothing at other)
• semi-major axis (a) = half of long axis,
• eccentricity (e) = "squashedness":

Kepler's 2nd Law: A line connecting a planet with the sun sweeps over equal areas in equal intervals of time.

• for a circle, sun/planet distance is constant: •
• for an ellipse, sun/planet distance varies: (Fig. 4-20c)
• in general, orbital speed of a planet varies as moves along elliptical orbit.

Kepler's Third Law: A planet's orbital period (P) is related to its average distance from the sun (a) by P² = a³.

• P is time (in years) it takes to complete 1 orbit.
• a is semi-major axis of the orbital ellipse in AU.
• 1 Astronomical Unit (AU) = distance between Earth and Sun.
• examples:
  • What is Period of a planet whose average distance from the sun is 1 AU?
  • What is Period of a planet whose average distance from the sun is 4 AU?
• the more distant a planet is from the sun, the longer it takes to complete one orbit.
• simply by observing orbital periods, Kepler figured out distances to all planets (in AU).

  II. Galileo's Experiments with Motion (1600-1640)

• rolled spheres down inclined ramp & measured speed
• found:
  • objects do not fall at constant speed, they accelerate
  • 9.8 m/s (32 ft/s) faster for every second they fall (Fig. 5-3)
  • rate of acceleration does not depend on weight of object
• studied how spheres rolled on a flat surface:
  • push an object, keeps rolling.
• experiments flawed: air resistance & rolling friction:
  • recognized effects & accounted for them

III. Newton's Law of Gravity (1670-1720)

• mathematician and theorist
• used Galileo's observations to develop 3 "laws of motion"

Newton's 1st Law: An object at rest remains at rest, and an object in motion remains in motion at a constant speed and direction, unless the object is acted on by a force.

• objects don't spontaneously move
  • need a force (push)
• once moving, the object won't deviate unless acted on by a force
• an expression of inertia

Mass = the amount of matter making up an object

• sum total of all protons, neutrons, electrons in object.

Velocity = speed in a specific direction.

• speed is distance traveled in certain time
• v = d / t
• units can be mi/hr, km/sec

Acceleration = change in velocity

• speed up
• slow down
• turn (change direction)

Newton's 2nd Law: An object of mass m accelerates by an amount a when acted on by a force F according to F = ma.

• the acceleration is in same direction as force
• for a given mass, larger force produces larger acceleration
• for a given force, a larger mass accelerates less
• cause and effect -- F and a.

Mass and Weight:

• mass is amount of material
• weight reflects force of gravity pulling on object:
  • W = F_g = ma = m * 9.8 m/s
• in deep space, no gravity
  • an object has mass but no weight
• on moon, a = 1.6 m/s, so given object weighs less than on Earth.

Newton's 3rd Law: To every action, there is an equal and opposite reaction.

• all forces occur in oppositely directed pairs.

Newton realized Earth's gravity holds moon in its orbit • 

Newton recognized that forces come in pairs:

• Moon / Earth
• Moon+Earth / Sun
• Planets / Sun
• Sun / Stars

Concluded gravity is mutual and universal:

• every atom in the universe attracts every other atom!

Newton's equation for force of gravity:

• M and m are masses of two objects,
• r is distance between them
• G is gravitational constant: a number,
• "-" means it pulls objects together (decreases r).
• units of gravity are "Newtons" (N, if kg/m/sec)
• strength decreases with the square of the distance: 1/r²

Later experiments measurred G:

• G = 6.67 x 10^-11 N m²/kg²
• small number, so force is weak unless M is huge.

• Read Sec. 5-2 (pp.83-89) -- How gravity shapes orbits.