Day 4

Chapter 2

part 2

Kepler’s Laws

Newton’s Laws

is a framework of ideas and assumptions which is used to explain some set of observations of the real world,

can be used to make predictions about observations, and

can be refuted (proved wrong) but never proved correct, because there is some possibility that future observations will contradict the theory. The theory should be testable.

Theories should be as simple as possible (but not simpler). This is called Occam’s Razor.

Scientific theory

Kepler’s laws of planetary

motion

Newton’s laws of mechanics

Both described the positions and movement of the Sun, Moon, and 5 visible planets, as seen without a telescope.

The geocentric theory was too complicated (80 circles!). (Occam’s razor could be invoked to seek a simpler way.)

Once the telescope was used to observe Venus, the geocentric theory could not explain the phases of Venus.

The heliocentric theory of Copernicus explained many of Galileo’s observations, but also used circular orbits.

More accurate measurements did not agree with the simple theory of Copernicus (circles had to be replaced by ellipses in the newer theory of planetary motion).

Geocentric vs. heliocentric theories

Tycho Brahe

More detailed observations were made by Tycho Brahe (commonly called Tycho, 1546 - 1601).

He made observations of a supernova in 1572 which convinced him that it was a distant star.

He received an island and built an observatory to measure planetary motion to high accuracy over a period of more than 20 years.

His observations were inherited by an assistant, Johannes Kepler, when Tycho died in 1601.

Further development of the heliocentric theory

Kepler used decades of Tycho’s observations in his mathematical calculations, to determine the shape of the planetary orbits, and the speed of the planets as they went around the Sun.

This massive effort (math by hand) took over 20 years, and resulted in three major statements about the characteristics of planetary orbits: Kepler’s three laws of planetary motion.

Kepler’s laws of planetary motion

Johannes Kepler and the Laws of Planetary Motion

Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus.

Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time.

Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

Kepler’s laws of planetary motion

An Ellipse can be drawn with string and TWO foci

For an ellipse,

r1 + r2 = 2a

The eccentricity is defined as:

e = c/a

A circle results when e = 0

Some Properties of Planetary Orbits

Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus.

Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time.

Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

Kepler’s laws of planetary motion

Kepler’s Second Law: equal areas in equal time

This also means higher speed at closer distances.

Another graphic on Kepler’s Second Law:

Kepler’s first law: The orbital paths of the planets are elliptical, with the Sun at one focus.

Kepler’s second law: An imaginary line connecting the Sun to any planet sweeps out equal areas of the ellipse in equal intervals of time.

Kepler’s third law: The square of the planet’s orbital period is proportional to the cube of its semimajor axis.

Kepler’s laws of planetary motion

The Astronomical Unit is about 150,000,000 km

Kepler’s Third Law: P2 (in years) = a3 (in a.u.)Basically, it means that large orbits have long periods.

Peripheral remark:

Real orbits have the center of mass as one focus

For the Sun and planets, this is

not a large effect.

For binary stars, the center of mass

may be near the middle of the line connecting them.

Isaac Newton developed a quantitative and explanatory theory of mechanics, explaining the motion of objects resulting from forces.

Newton’s First Law: law of inertiaAn object will remain at rest unless a force acts on it.

Once in motion, it will remain in motion unless another force acts to change its direction or speed.

Newton’s Second Law: F = maThe acceleration of a mass is proportional

to the total force acting upon it, and inversely proportional to the mass of the object.

Newton’s Third Law: action-reactionFor every force acting upon an object (action),

there is a force acting on another object (reaction) which has the same magnitude (size) but points (acts) in the opposite direction.

Gravity is a force between two objects that have mass.

Gravitational force varies with the distance between the objects.

It depends on the product of the two masses, i.e.,

m1 x m2

and on the inverse of the square of the distance between the masses

(assuming they are small

compared with the distance).

1/r2

Sun’s Gravity causes planets to move on a path called an orbit. These obey Kepler’s Laws.

Let’s review Kepler’s Laws.

Review:  see if you can tell what these are simulating: 

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_kepler2.html http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_periods_sim.html

http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_solar_sim.html

Both the geocentric and heliocentric models tried to explain the SAME observations. 

See if you see the similarity in these two simulations: 

(show solsys.mov)

Exam 1 is next Thursday

• Sept. 9, after some class discussion

• Closed book, no notes, no assistance

• Multiple choice and true/false

• Answer on a Scantron form (pencil)

• Results will be posted on WesternOnline over the weekend.