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CHAPTER 2: Gravitation and the Waltz of the Planets

For the Quiz Know the basics of the Geocentric and Heliocentric theories, and be able to explain planetary motion in terms of each. Know the basics of the Geocentric and Heliocentric theories, and be able to explain planetary motion in terms of each. Be able to state Kepler’s laws and understand their meaning. Be able to state Kepler’s laws and understand their meaning. Be able to define “aphelion” and “perihelion.” Be able to define “aphelion” and “perihelion.” Be able to state Newton’s laws of motion, and understand their meaning. Be able to state Newton’s laws of motion, and understand their meaning. Be able to state Newton’s law of gravitation and understand its meaning. Be able to state Newton’s law of gravitation and understand its meaning. Define synodic and siderial period of a planet. Define synodic and siderial period of a planet. Know the important observations Galileo made with his telescope, and why those observations were important. Know the important observations Galileo made with his telescope, and why those observations were important.

WHAT DO YOU THINK? What makes a theory scientific? What makes a theory scientific? What is the shape of the Earth’s orbit around the Sun? What is the shape of the Earth’s orbit around the Sun? Do the planets orbit the Sun at constant speeds? Do the planets orbit the Sun at constant speeds? Do all the planets orbit the Sun at the same speed? Do all the planets orbit the Sun at the same speed? How much force does it take to keep an object moving in a straight line at a constant speed? How much force does it take to keep an object moving in a straight line at a constant speed? How does an object’s mass differ when measured on the Earth and on the Moon? How does an object’s mass differ when measured on the Earth and on the Moon?

The scientific method is used to develop new scientific theories. Scientific theories are accepted when they make testable predictions that can be verified using new observations and experiments.

Early models of the universe attempted to explain the motion of the five visible planets against the background of “fixed” stars. The main problem was that the planets do not move uniformly against the background of stars, but instead appear to stop, move backward, then move forward again. This backward motion is referred to as retrograde motion.

Ptolemy explained this motion using a geocentric (Earth- centered) model of the solar system in which the planets orbited the Earth indirectly, by moving on epicycles which in turn orbited the Earth.

Nicolaus Copernicus developed the first heliocentric (sun-centered) model of the solar system. In this model, the retrograde motion of Mars is seen when the Earth passes Mars in its orbit around the Sun. 1473-1543

We define special positions of the planets in their orbits depending where they appear in our sky. For example, while at a conjunction, a planet will appear in the same part of the sky as the Sun, while at opposition, a planet will appear opposite the Sun in our sky.

However, the cycle of these positions (a synodic period) is different from the actual orbital period of the planet around the Sun (a sidereal period) because both the Earth and the planet orbit around the Sun.

When a new “star” appeared in the sky during the 16th century, a Danish astronomer named Tycho Brahe (1546-1601) reasoned that the distance of the object may be determined by measuring the amount of parallax. The apparent change in the location of an object due to the difference in location of the observer is called parallax.

Because the parallax of the “star” was too small to measure, Tycho knew that it had to be among the other stars, thus disproving the ancient belief that the “heavens” were fixed and unchangeable.

After Tycho Brahe’s death, Johannes Kepler (pictured here with Tycho in the background) used Tycho’s observations to deduce the three laws of planetary motion. 1571-1630

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #1. The orbit of a planet around the Sun is an ellipse with the Sun at one focus.

The amount of elongation in a planet’s orbit is defined as its orbital eccentricity. An orbital eccentricity of 0 is a perfect circle while an eccentricity close to 1.0 is nearly a straight line. In an elliptical orbit, the distance from a planet to the Sun varies. The point in a planet’s orbit closest to the Sun is called perihelion, and the point farthest from the Sun is called aphelion.

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #2: A line joining the planet and the Sun sweeps out equal areas in equal intervals of time. Planet moves faster in its orbit when closer to the Sun. Planet moves slower in its orbit when farther away from the Sun.

KEPLER’S THREE LAWS OF PLANETARY MOTION LAW #3: The square of a planet’s sidereal period around the Sun is directly proportional to the cube of its semi-major axis. This law relates the amount of time for the planet to complete one orbit around the Sun to the planet’s average distance from the Sun. If we measure the orbital periods (P) in years and distances (a) in astronomical units, then the law mathematically can be written as P 2 = a 3.

Galileo was the first to use a telescope to examine celestial objects. His discoveries supported a heliocentric model of the solar system. Galileo discovered that Venus, like the Moon, undergoes a series of phases as seen from Earth. In the Ptolemaic (geocentric) model, Venus would be seen in only new or crescent phases. However, as Galileo observed, Venus is seen in all phases, which agrees with the Copernican model as shown. 1564-1642

Galileo also discovered moons in orbit around the planet Jupiter. This was further evidence that the Earth was not the center of the universe.

Isaac Newton formulated three laws to describe the fundamental properties of physical reality. NEWTON’S THREE LAWS OF MOTION LAW #1: A body remains at rest or moves in a straight line at constant speed unless acted upon by a net outside force. LAW #2: The acceleration of an object is proportional to the force acting on it. LAW #3: Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body. 1642-1727

Newton also discovered that gravity, the force that causes objects to fall to the ground on Earth, is the same force that keeps the Moon in its orbit around the Earth. NEWTON’S LAW OF UNIVERSAL GRAVITATION Two objects attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. With his laws, Newton was able to derive Kepler’s three laws, as well as predict other possible orbits.

Newton’s laws were applied to other objects in our solar system. Using Newton’s methods, Edmund Halley worked out the details of a comet’s orbit and predicted its return. Deviations from Newton’s Laws in the orbit of the planet Uranus led to the discovery of the eighth planet, Neptune.

acceleration angular momentum aphelion astronomical unit configuration (of a planet) conjunction conservation of angular momentum cosmology ellipse elongation focus (of an ellipse) force Galilean moons (satellites) gravity heliocentric cosmology hyperbola inferior conjunction Kepler’s laws kinetic energy law of equal areas law of inertia light-year mass model momentum Newton’s laws of motion Occam’s razor opposition parabola parallax parsec perihelion physics potential energy retrograde motion scientific method scientific theory semimajor axis (of an ellipse) sidereal period superior conjunction synodic period universal constant of gravitation universal law of gravitation velocity weight work Key Terms