Discovering the Universe Ninth Edition

Neil F. Comins • William J. Kaufmann III Discovering the Universe Ninth Edition CHAPTER 2 Gravitation and the Motion of the Planets

WHAT DO YOU THINK? • What makes a theory scientific? • What is the shape of Earth’s orbit around the Sun? • Do the planets orbit the Sun at constant speeds? • Do all of 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 does an object’s mass differ when measured on Earth and on the Moon? • Do astronauts orbiting Earth feel the force of gravity from our planet?

In this chapter you will discover… • what makes a theory scientific • the scientific discoveries that revealed that Earth is not at the center of the universe, as previously believed • Copernicus’s argument that the planets orbit the Sun • why the direction of motion of each planet on the celestial sphere sometimes appears to change • that Kepler’s determination of the shapes and other properties of planetary orbits depended on the careful observations of his mentor Tycho Brahe • how Isaac Newton formulated an equation to describe the force of gravity and how he thereby explained why the planets and moons remain in orbit

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. A theory can NEVER be proven – only disproven.

General Motion of the Planets • Planets, to the eyeball, look virtually the same as stars. • Planets rise in the East and set in the West, same as stars. • Over the course of a night, the relative motion of the planet on the celestial sphere is hardly noticeable. • But, over days, the planets DO shift their position relative to the fixed stars. • Planets move through the zodiac. They don’t follow the ecliptic precisely, but never stray far from it. • Their motion is complicated; they sometimes move in different directions around the zodiac.

The retrograde motion of Mars as it would be seen in a series of images taken on the same photographic plate every 3 days.

To help visualize this motion on the celestial sphere, astronomers often plot the position of Mars (or another body in retrograde motion) on a star chart. From Jan 25 through Apri13 of 2012, Mars undergoes retrograde motion as seen from Earth. The retrograde path is sometimes a loop north (shown here) or south of the normal path, and sometimes an S-shaped path above or below it.

Retrograde Motion –Geometrical Explanation c. 150 AD Each planet revolves around an epicycle, which in turn revolves around a deferent centered approximately on Earth. As seen from Earth, the speed of the planet on the epicycle alternately (a) adds to or (b) subtracts from the speed of the epicycle on the deferent, thus producing alternating periods of direct and retrograde motions.

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

Nicolaus Copernicus (1473–1543) Copernicus, the youngest of four children, was born in Torun, Poland. He pursued his higher education in Italy, where he received a doctorate in canon law and studied medicine. Copernicus developed a heliocentric theory of the known universe and just before his death in 1543 published this work under the title De Revolutionibus Orbium Coelestium. His revolutionary theory was flawed in that he assumed that the planets had circular orbits around the Sun. This was corrected by Johannes Kepler.

Tycho Brahe (1546–1601) Tycho was born to nobility in the Danish city of Knudstrup, which is now part of Sweden. At age 20, he lost part of his nose in a duel and wore a metal replacement thereafter. In 1576, the Danish king Frederick II built Tycho an astronomical observatory that Tycho named Uraniborg (after Urania, Greek muse of astronomy). Tycho rejected both Copernicus’s heliocentric theory and the Ptolemaic geocentric system. He devised a halfway theory called the Tychonic system. According to Tycho’s theory, Earth is stationary, with the Sun and Moon revolving around it, while all the other planets revolve around the Sun. Tycho died in 1601.

Johannes Kepler (1571–1630) .Kepler was educated in Germany, where he spent 3 years studying mathematics, philosophy, and theology. In 1596, Kepler published a booklet in which he attempted to mathematically predict the planetary orbits. Although his theory was altogether wrong, its boldness and originality attracted the attention of Tycho Brahe, whose staff Kepler joined in 1600. Kepler deduced his three laws from Tycho’s observations. Inset portrait: Tycho Brahe

Galileo Galilei (1564–1642) Born in Pisa, Italy, Galileo studied medicine and philosophy at the University of Pisa. He abandoned medicine in favor of mathematics. He held the chair of mathematics at the University of Padua, and eventually returned to the University of Pisa as a professor of mathematics. There Galileo formulated his famous law of falling bodies: All objects fall with the same acceleration regardless of their weight. In 1609 he constructed a telescope and made a host of discoveries that contradicted the teachings of Aristotle and the Roman Catholic Church. He summed up his life’s work on motion, acceleration, and gravity in the book Dialogues Concerning the Two Chief World Systems, published in 1632.

Isaac Newton (1642–1727) Newton delighted in constructing mechanical devices, such as sundials, model windmills, a water clock, and a mechanical carriage. He received a bachelor’s degree in 1665 from the University of Cambridge. While there, he began developing the mathematics that later became calculus (developed independently by the German Gottfried Leibniz). While pursuing experiments in optics, Newton constructed a reflecting telescope and also discovered that white light is actually a mixture of all colors. His major work on forces and gravitation was the tome Philosophiae Naturalis Principia Mathematica (1687). In 1704, Newton published his second great treatise, Opticks, in which he described his experiments and theories about light and color. Upon his death in 1727, Newton was buried in Westminster Abbey, the first scientist to be so honored.

We define special positions of the planets in their orbits depending upon 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 Earth also orbits the Sun. Visualize Earth on the inside for the synodic period of Mars, etc.

Mercury has an especially eccentric orbit around the Sun. As seen from Earth, the angle of Mercury at greatest elongation ranges from 18° to 28°. In contrast, Venus’s orbit is nearly circular, with both greatest elongations of 47°.

Parallax The apparent change in the location of an object due to the difference in location of the observer is called parallax.

If … the supernova were nearby. Actual Observation: Supernova is in the Heavens When a new “star” appeared in the sky in 1572, Tycho Brahe reasoned that the distance of the object may be determined by direct measurements by examining the amount of 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 unchanging.

A Parsec The parsec, a unit of length commonly used by astronomers, is equal to 3.26 light-years. The parsec is defined as the distance at which 1 AU perpendicular to the observer’s line of sight makes an angle of 1 arcsec. One parsec = 308,000 AU

An ellipse can be drawn with a pencil, a loop of string, and two thumbtacks, as shown. If the string is kept taut, the pencil traces out an ellipse. The two thumbtacks are located at the two foci of the ellipse.

Eccentricity – amount of deviation from circular 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 in a planet’s orbit farthest from the Sun is called aphelion.

Kepler’s first law: The orbit of a planet about the Sun is an ellipse with the Sun at one focus. Kepler’s second law: A line joining the planet and the Sun sweeps out equal areas in equal intervals of time.

Kepler’s Third Law The square of the orbital period equals the cube of the semimajor axis P2 = a3 A Demonstration

First observed By Galileo The appearance (phase) of Venus changes as it moves along its orbit. The number below each view is the angular diameter (d) of the planet as seen from Earth, in arcseconds. The phases correlate with the planet’s angular size and its angular distance from the Sun, as seen from Earth. These observations clearly support the idea that Venus orbits the Sun.

In 1610, Galileo discovered four “stars” that move back and forth across Jupiter. He concluded that they are four moons that orbit Jupiter just as our Moon orbits Earth. The observations shown were made by Jesuits in 1620 of Jupiter and its four visible moons.

This is a photograph of the four Galilean satellites with an overexposed Jupiter. Each satellite would be bright enough to be seen with the unaided eye were it not overwhelmed by the glare of Jupiter.

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 and is inversely proportional to its mass. FNET= m a or a = FNET/m • 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.

Rate of increase of speed Rate of decrease of speed Rate of turning In general, how fast velocity vector is changing Special case: for an object moving in a circle at constant speed: a = v2/R, toward center of circle What is “Acceleration”

Angular Momentum and Torque (a) When a force acts through an object’s rotation axis or toward its center of mass, the force does not exert a torque on the object. (b) When a force acts in some other direction, then it exerts a torque, causing the body’s angular momentum to change.

Conservation of Angular Momentum As this skater brings her arms and outstretched leg in, she must spin faster to conserve angular momentum.

NEWTON’S LAW OF GRAVITATION • G is a constant showing the strength of gravity; m1 and m2 are masses; and r is the distance between the centers of the objects • Using the law of gravitation, Newton was able to derive Kepler’s laws of motion • Watching the apple fall made Newton think “why?”

Why Does a Satellite Orbit? 1. Imagine dropping a rock. It falls 5 m is 1 second. 2. Imagine throwing the rock horizontally at fastball speed (40 m/s) from a position on a tower. The rock still drops 5 m in 1 second, but it moves 40 m horizontally. 3. Now, crank the speed up to 8000 m/s. In one second the rock drops 5 m and moves 8000 m horizontally. The difference is that Earth’s curvature is not negligible. In 8 km, the Earth curves by 5 m, so the rock is at its original distance above the surface. 4. Presto! An orbit. 5. Of course, it needs to be above the atmosphere so that the horizontal speed stays at 8 km/s.

Calculation of a Circular Orbit The only force of consequence is Earth’s gravitation, so FNET = FGRAV. Equate Newton’s 2nd Law with FGRAV msat asat = G mEarth msat / r2 where “sat” refers to the satellite. The r is the distance from the CENTER of the Earth to the satellite. In circular motion, the acceleration is v2/r. Substitute: msat v2/r = G mEarth msat / r2 Simplify: v2 = G mEarth / r or v =sqrt(G mEarth / r) For an orbit around the Sun, replace mEarth by mSun The satellite (or Space Shuttle) is NOT free of gravity.

Weightlessness So, how do astronauts float around if they aren’t free of gravity?? They, along with the shuttle, are orbiting. You sense weight because of two factors: a. Earth’s gravity pulls on you b. The floor (or chair, etc) counters its pull. Factor (b) is missing for the astronauts. It’s like being in an elevator and the cable breaks. You would have a sense of weightlessness as you fall. (By the way, as a technological note, an elevator doesn’t actually fall if its cable breaks – thank you, Elijah Graves Otis.)

Conic Sections A conic section is any one of a family of curves obtained by slicing a cone with a plane, as shown. The orbit of one body around another can be an ellipse, a parabola, or a hyperbola. Circular orbits are possible because a circle is just an ellipse for which both foci are at the same point.

Neptune was discovered in 1845 by Adams and Leverrier independently by calculations on Uranus’ orbit. It wasn’t seen until 1846. Halley’s Comet Halley’s Comet orbits the Sun with an average period of about 76 years. Halley recognized its periodic nature in 1705.

Gravity Works at All Scales This figure shows a few of the effects of gravity: here on Earth in the solar system in our Milky Way Galaxy … and beyond. Gravity is universal.

PLANET ORBITS ARE ONLY SLIGHTLY ELLIPTICAL White Dashed lines: circles centered on the Sun Red: Mars’ orbit Blue: Earth’s orbit

Summary of Key Ideas

Science: Key to Comprehending the Cosmos • The ancient Greeks laid the groundwork for progress in science by stating that the universe is comprehensible. • The scientific method is a procedure for formulating theories that correctly predict how the universe behaves. • A scientific theory must be testable, that is, capable of being disproved. • Theories are tested and verified by observation or experimentation and result in a process that often leads to their refinement or replacement and to the progress of science. • Observations of the cosmos have led astronomers to discover some fundamental physical laws of the universe.

Origins of a Sun-centered Universe • Common sense (for example, Earth doesn’t appear to be moving) led early natural philosophers to devise a geocentric cosmology, which placed Earth at the center of the universe. • Copernicus’s heliocentric (Sun-centered) theory simplified the general explanation of planetary motions compared to the geocentric theory. • The heliocentric cosmology refers to motion of planets and smaller debris orbiting the Sun. Other stars do not orbit the Sun. • The sidereal orbital period of a planet is measured with respect to the stars, and determines the length of the planet’s year. A planet’s synodic period is measured with respect to the Sun as seen from the moving Earth (for example, from one opposition to the next).

Kepler’s and Newton’s Laws - I • Ellipses describe the paths of the planets around the Sun much more accurately than do the circles used in previous theories. Kepler’s three laws give important details about elliptical orbits. • The invention of the telescope led Galileo to new discoveries, such as the phases of Venus and the moons of Jupiter, that supported a heliocentric view of the universe. • Newton based his explanation of the universe on three assumptions, now called Newton’s laws of motion. These laws and his law of universal gravitation can be used to deduce Kepler’s laws and to describe most planetary motions with extreme accuracy.

Kepler’s and Newton’s Laws - II • The mass of an object is a measure of the amount of matter in it; weight is a measure of the force with which the gravity of a world pulls on an object’s mass when the two objects are at rest with respect to each other (or, equivalently, how much the object pushes down on a scale). • The path of one astronomical object around another, such as that of a comet around the Sun, is an ellipse, a parabola, or a hyperbola. Ellipses are bound orbits, while objects with parabolic and hyperbolic orbits fly away, never to return.

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

WHAT DID YOU THINK? • What makes a theory scientific? • A theory is an idea or set of ideas proposed to explain something about the natural world. A theory is scientific if it makes predictions that can be objectively tested and potentially disproved.

WHAT DID YOU THINK? • What is the shape of Earth’s orbit around the Sun? • All planets have elliptical orbits around the Sun.

WHAT DID YOU THINK? • Do the planets orbit the Sun at constant speeds? • No. The closer a planet is to the Sun in its elliptical orbit, the faster it is moving. The planet moves fastest at perihelion and slowest at aphelion.

WHAT DID YOU THINK? • Do all of the planets orbit the Sun at the same speed? • No. A planet’s speed depends on its average distance from the Sun. The closest planet moves fastest; the most distant planet moves slowest.

Discovering the Universe Ninth Edition