Chapter 2: The Copernican Revolution The Birth of Modern Science Ancient Astronomy Models of the Solar System Laws of Planetary Motion Newton’s Laws Laws of Motion Law of Gravitation

The universe is full of magical things, patiently waiting for our wits to grow sharper. Eden Philpotts

Cosmology: study of the structure and evolution of the universe Ancient civilizations universe = solar system + fixed stars Today universe = totality of all space, time, matter and energy

There are three principle means of acquiring knowledge…… There are three principle means of acquiring knowledge……. observation of nature, reflection, and experimentation. Observation collects facts; reflection combines them; experimentation verifies the result of that combination. Denis Diderot (1713 - 1784)

Scientific Method Gather data Form theory Test theory

Astronomy in Ancient Times Ancient people had a better, clearer chance to study the sky and see the patterns of stars (constellations) than we do today. Drew pictures of constellations; created stories to account for the figures being in the sky. Used stars and constellations for navigation. Noticed changes in Moon’s shape and position against the stars. Created accurate calendars of seasons.

Ancient Astronomy Stonehenge on the summer solstice. As seen from the center of the stone circle, the Sun rises directly over the "heel stone" on the longest day of the year. The Big Horn Medicine Wheel in Wyoming, built by the Plains Indians. Its spokes and rock piles are aligned with the rising and setting of the Sun and other stars.

Astronomy in Early Americas Maya Indians developed written language and number system. Recorded motions of Sun, Moon, and planets -- especially Venus. Fragments of astronomical observations recorded in picture books made of tree bark show that Mayans had learned to predict solar and lunar eclipses and the path of Venus. One Mayan calendar more accurate than those of Spanish.

Ancient Contributions to Astronomy Egyptians recorded interval of floods on Nile every 365 days noted Sirius rose with Sun when floods due invented sundials to measure time of day from movement of the Sun. Babylonians first people to make detailed records of movements of Mercury, Venus, Mars, Jupiter, Saturn only planets visible until telescope

Greek Astronomy Probably based on knowledge from Babylonians. Thales predicted eclipse of Sun that occurred in 585 B.C. Around 550 B.C., Pythagoras noted that the Evening Star and Morning Star were really the same body (actually planet Venus). Some Greek astronomers thought the Earth might be in the shape of a ball and that moonlight was really reflected sunlight.

Time Line Ancient Greeks Pythagoras 6th century B.C. Aristotle 348-322 B.C. Aristarchus 310-230 B.C. Hipparchus ~130 B.C. Ptolemy ~A.D. 140

Pythagorean Paradigm The Pythagorean Paradigm had three key points about the movements of celestial objects: the planets, Sun, Moon and stars move in perfectly circular orbits; the speed of the planets, Sun, Moon and stars in the circular orbits is perfectly uniform; the Earth is at the exact center of the motion of the celestial bodies.

Aristotle’s Universe: A Geocentric Model Aristotle proposed that the heavens were literally composed of concentric, crystalline spheres to which the celestial objects were attached and which rotated at different velocities, with the Earth at the center (geocentric). The figure illustrates the ordering of the spheres to which the Sun, Moon, and visible planets were attached.

Planetary Motion From Earth, planets appear to move wrt fixed stars and vary greatly in brightness. Most of the time, planets undergo direct motion - moving W to E relative to background stars. Occasionally, they change direction and temporarily undergo retrograde motion - motion from E to W -before looping back. (retrograde-move)

Planetary Motion: Epicycles and Deferents Retrograde motion was first explained as follows: the planets were attached, not to the concentric spheres themselves, but to circles attached to the concentric spheres, as illustrated in the adjacent diagram. These circles were called "Epicycles",and the concentric spheres to which they were attached were termed the "Deferents". (epicycle-move)

Motions of Mercury and Venus Mercury and Venus exhibit a special motion, not observed in the other planets. They always remain close to the Sun, first moving away from it, then pausing, and then moving toward it. Venus and Mercury can be seen in the morning and evening skies, but never at midnight (except in polar latitudes). Venus never gets >48o from the Sun; Mercury more distant than 28o.

Epicycles, Deferents, and the Orbits of Mercury and Venus Special features of the orbits of Mercury and Venus modeled by requiring that the center of the epicycle of the planet be firmly attached to the line joining the Earth and Sun.

Epicycle/Deferent Modifications In actual models, the center of the epicycle moved with uniform circular motion, not around the center of the deferent, but around a point that was displaced by some distance from the center of the deferent. This modification predicted planetary motions that more closely matched the observed motions.

Further Modifiations In practice, even this was not enough to account for the detailed motion of the planets on the celestial sphere! In more sophisticated epicycle models further "refinements" were introduced: In some cases, epicycles were themselves placed on epicycles, as illustrated in the adjacent figure. The full Ptolemaic model required 80 different circles!!

Ptolemy 127-151 A.D. in Alexandria Accomplishments completion of a “geocentric” model of solar system that accurately predicts motions of planets by using combinations of regular circular motions invented latitude and longitude (gave coordinates for 8000 places) first to orient maps with NORTH at top and EAST at right developed magnitude system to describe brightness of stars that is still used today

Aristarchus 310-230 B.C. Applied geometry to find distance to Moon Directly measure angular diameter Calculate linear diameter using lunar eclipse relative distances and sizes of the Sun and Moon ratio of distances to Sun and Moon by observing angle between the Sun and Moon at first or third quarter Moon. Proposed that the Sun is stationary and that the Earth orbits the Sun and spins on its own axis once a day.

Hipparchus ~190-125 B.C. Often called “greatest astronomer of antiquity.” Contributions to astronomy improved on Aristarchus’ method for calculating the distances to the Sun and Moon, improved determination of the length of the year, extensive observations and theories of motions of the Sun and Moon, earliest systematic catalog of brighter stars , first estimate of precession shift in the vernal equinox.

Time Line Ancient Greeks Pythagoras 6th century B.C. Aristotle 348-322 B.C. Aristarchus 310-230 B.C. Hipparchus ~130 B.C. Ptolemy ~A.D. 140 Dark Ages A.D. 5th - 10th century Arabs translated books, planets positions China 1054 A.D. supernova; Crab Nebula

Heliocentric Model - Copernicus In 1543, Copernicus proposed that: the Sun, not the Earth, is the center of the solar system. Such a model is called a heliocentric system. Ordering of planets known to Copernicus in this new system is illustrated in the figure. Represents modern ordering of planets. (copernican-move)

Stellar Parallax Stars should appear to change their position with the respect to the other background stars as the Earth moved about its orbit. In Copernicus’ day, no stellar parallax was observed, so the Copernican model was considered to be only a convenient calculation tool for planetary motion. In 1838, Friedrich Wilhelm Bessel succeeded in measuring the parallax of the nearby, faint star 61 Cygni. ( penny at 4 miles)

Time Line Ancient Greeks Pythagoras 6th century B.C. Aristotle 348-322 B.C. Aristarchus 310-230 B.C. Ptolemy ~A.D. 140 Dark Ages A.D. 5th - 10th century Renaissance Copernicus (1473-1543) Tycho Brahe Kepler Galileo (1546-1601) (1571-1630) (1564-1642) Newton (1642-1727)

Galileo Galilei Galileo used his telescope to show that Venus went through a complete set of phases, just like the Moon. This observation was among the most important in human history, for it provided the first conclusive observational proof that was consistent with the Copernican system but not the Ptolemaic system.

Galileo and Jupiter Galileo observed 4 points of light that changed their positions with time around the planet Jupiter. He concluded that these were objects in orbit around Jupiter. Galileo called them the Medicea Siderea-the “Medician Stars” in honor of Cosimo II de'Medici, who had become Grand Duke of Tuscany in 1609.

Proof of the Heliocentric Hypothesis In 1729, James Bradley (British Astronomer Royal) discovered a phenomenon called aberration of starlight while trying to observe stellar parallax. In one year, noted 20’’ shift in a star’s observed position from its true position. Information yields value for the speed of Earth through space (18.6 miles/sec).

Aberration of Starlight

Time Line Ancient Greeks Pythagoras 6th century B.C. Aristotle 348-322 B.C. Aristarchus 310-230 B.C. Ptolemy ~A.D. 140 Dark Ages A.D. 5th - 10th century Renaissance Copernicus (1473-1543) Tycho Brahe Kepler Galileo (1546-1601) (1571-1630) (1564-1642) Newton (1642-1727)

Tycho Brahe

Tycho Brahe Danish astronomer Quadrant Sextant Tycho Brahe Danish astronomer Studied a bright new star in sky that faded over time. In 1577, studied a comet in trying to determine its distance from Earth by observing from different locations, noted that there was no change in apparent position proposed comet must be farther from Earth than the Moon. Built instrument to measure positions of planets and stars to within one arc minute (1’).

Johannes Kepler: Laws of Planetary Motion

Kepler’s Firsts First to investigate the formation of pictures with a pin hole camera; First to explain the process of vision by refraction within the eye; First to formulate eyeglass designing for nearsightedness and farsightedness; First to explain the use of both eyes for depth perception. First to describe: real, virtual, upright and inverted images and magnification; First to explain the principles of how a telescope works; First to discover and describe the properties of total internal reflection. His book Stereometrica Doliorum formed the basis of integral calculus. First to explain that the tides are caused by the Moon. Tried to use stellar parallax caused by the Earth's orbit to measure the distance to the stars; the same principle as depth perception. Today this branch of research is called astrometry. First to suggest that the Sun rotates about its axis in Astronomia Nova. First to derive the birth year of Christ, that is now universally accepted. First to derive logarithms purely based on mathematics, independent of Napier's tables published in 1614. He coined the word "satellite" in his pamphlet Narratio de Observatis a se quatuor Iovis sattelitibus erronibus

Kepler: Elliptical orbits The amount of "flattening" of the ellipse is the eccentricity. In the following figure the ellipses become more eccentric from left to right. A circle may be viewed as a special case of an ellipse with zero eccentricity, while as the ellipse becomes more flattened the eccentricity approaches one. (eccentricity-anim)

Elliptical Orbits and Kepler’s Laws Some orbits in the Solar System cannot be approximated at all well by circles - for example, Pluto’s separation from the Sun varies by about 50% during its orbit! According to Kepler’s First Law, closed orbits around a central object under gravity are ellipses.

As a planet moves in an elliptical orbit, the Sun is at one focus (F or F’) of the ellipse.

The line that connects the planet’s point of closest approach to the Sun, the perihelion ... As a planet moves in an elliptical orbit, the Sun is at one focus (F or F’) of the ellipse perihelion v r F’ F C

… and its point of greatest separation from the Sun, the aphelion As a planet moves in an elliptical orbit, the Sun is at one focus (F or F’) of the ellipse is called the major axis of the ellipse. perihelion v r F’ F C aphelion

The only other thing we need to know about ellipses is how to identify the length of the “semi-major axis”, because that determines the period of the orbit. “Semi” means half, and so the semi-major axis a is half the length of the major axis: v r F’ F C a

Kepler’s 1st Law: The orbits of the planets are ellipses, with the Sun at one focus of the ellipse.

Kepler’s 2nd Law The line joining the planet to the Sun sweeps out equal areas in equal times as the planet travels around the ellipse. Orbit-anim

An object in a highly elliptical orbit travels very slowly when it is far out in the Solar System, … but speeds up as it passes the Sun.

According to Kepler’s Second Law, … the line joining the object and the Sun ...

… sweeps out equal areas in equal intervals of time.

That is, Kepler’s Second Law states that The line joining a planet and the Sun sweeps out equal areas in equal intervals of time.

For circular orbits around one particular mass - e. g For circular orbits around one particular mass - e.g. the Sun - we know that the period of the orbit (the time for one complete revolution) depended only on the radius r - this is Kepler’s 3rd Law: M For objects orbiting a common central body (e.g. the Sun) in approximately circular orbits, r r m v the orbital period squared is proportional to the orbital radius cubed.

Let’s see what determines the period for an elliptical orbit: For elliptical orbits, the period depends not on r, but on the semi-major axis a instead. v r F’ F C a a

It turns out that Kepler’s 3rd Law applies to all elliptical orbits, not just circles, if we replace “orbital radius” by “semi major axis”: For objects orbiting a common central body (e.g. the Sun) the orbital period squared is proportional to the orbital radius cubed. the orbital period squared is proportional to the semi major axis cubed.

So as all of these elliptical orbits have the same semi-major axis a, so they have the same period.

So if each of these orbits is around the same massive object (e. g So if each of these orbits is around the same massive object (e.g. the Sun),

So if each of these orbits is around the same massive object (e. g So if each of these orbits is around the same massive object (e.g. the Sun), then as they all have the same semi-major axis length a,

So if each of these orbits is around the same massive object (e. g So if each of these orbits is around the same massive object (e.g. the Sun), then as they all have the same semi-major axis length a, then, by Kepler’s Third Law, they have the same orbital period.

Ellipses and Orbits Ellipse animation

Kepler’s 3rd Law The ratio of the squares of the revolution periods (P) for two planets is equal to the ratio of the cubes of their semi-major axes (a). P2 = a3 or P2/a3 = 1 where P is the planet’s sidereal orbital period (in Earth years) and a is the length of the semi-major axis (in astronomical units)

Astronomical Unit One astronomical unit is the semi-major axis of the Earth’s orbit around the Sun, essentially the average distance between Earth and the Sun. abbreviation: A.U. one A.U. ~ 150 x 106 km

Kepler’s 3rd Law for the Planets P2 = a3 or P2/a3 = 1

Planetary Motions The planets’ orbits (except Mercury and Pluto) are nearly circular. The further a planet is from the Sun, the greater its orbital period. Although derived for the six innermost planets known at the time, Kepler’s Laws apply to all currently known planets. Do Kepler’s laws apply to comets orbiting the Sun? Do they apply to the moons of Jupiter?

Chapter 2 Homework Text, page 58. Problem # 1 - accuracy of Tycho Brahe’s observations Use equation on page 26 relating unknown diameter (uncertainty in position) to angular diameter (1’ = 1 arc minute), and distance to object (distance to Moon, Sun, Saturn from Earth). distance to Moon - p. 198 distance to Sun - 1 A.U. distance from Sun to Saturn at perihelion p. A-5, Table 3A Problem # 6 - elliptical orbit of Halley’s comet

Kepler’s Laws 1st Law: Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse. 2nd Law: The straight line joining a planet and the Sun sweeps out equal areas in equal intervals of time. 3rd Law: The squares of the periods of revolution of the planets are in direct proportion to the cubes of the semi-major axes of their orbits.

What’s important so far? Through history, people have used the scientific method: observe and gather data, form theory to explain observations and predict behavior test theory’s predictions. Greeks produced first surviving, recorded models of universe: geocentric (Earth at center of universe), other celestial objects in circular orbits about Earth, and move with constant speed in those orbits. Geocentric models require complicated combinations of deferents and epicycles to explain observed motion of planets. Ptolemaic model required 80 such combinations. Copernicus revived heliocentric model of solar system, but kept circular, constant speed orbits.

What’s important so far? continued Without use of a telescope, Tycho Brahe made very accurate measurements of the positions of celestial objects. Johannes Kepler inherited Brahe’s data and determined three empirical laws governing the motion of orbiting celestial objects. 1st Law: Each planet moves around the Sun in an orbit that is an ellipse, with the Sun at one focus of the ellipse. 2nd Law: The straight line joining a planet and the Sun sweeps out equal areas in equal intervals of time. 3rd Law: The squares of the periods of revolution of the planets are in direct proportion to the cubes of the semi-major axes of their orbits. Galileo used a telescope to observe the Moon and planets. The observed phases of Venus validated the heliocentric model proposed by Copernicus. Also discovered 4 moons orbiting Jupiter, Saturn’s rings, named lunar surface features, studied sunspots, noted visible disk of planets (stars - point sources).

according to Kepler’s laws? why do objects move as they do? Why do the planets move according to Kepler’s laws? Or, more generally, why do objects move as they do?

How do you describe motion? A piece of paper and a rubber ball are dropped from the same height, at the same time. Predict which will hit the ground first. The piece of paper is crushed into a ball, approximately the same size as the rubber ball. The paper ball and the rubber ball are dropped from the same height, at the same time. A wooden block and piece of paper have the same area. They are dropped at the same time from the same height. Describe the motion of the block and of the paper.

Historical Views of Motion Aristotle: two types of motion natural motion violent motion Galileo discredited Aristotelian view of motion Animations: Air resistance Free-fall

Galileo: Why do objects move as they do? Slope up, Slope down, No slope, speed increases. speed decreases. does speed change? Without friction, NO, the speed is constant!

What is a “natural” state of motion for an object? Moving with constant velocity? At rest?

Inertia and Mass Inertia: Mass: a body’s resistance to a change in its motion. Mass: a measure of an object’s inertia or, loosely, a measure of the total amount of matter contained within an object.

Newton’s First Law Called the law of inertia. Since time of Aristotle, it was assumed that a body required some continual action on it to remain in motion, unless that motion were a part of natural motion of object. Newton’s first law simplifies concept of motion.

Animation: collision-1st-law

FORCES and MOTION An object will remain (a) at rest or (b) moving in a straight line at constant speed until (c) some net external force acts on it.

What if there is an outside influence? To answer this question, Newton invoked the concept of a FORCE acting on a body to cause a change in the motion of the body.

Forces can act through contact instantaneously or continuously (baseball bat making contact with the baseball), or continuously (gravity keeping the baseball from flying into space). or at a distance.

Velocity and Acceleration Velocity: describes the change in position of a body divided by the time interval over which that change occurs. Velocity is a vector quantity, requiring both the speed of the body and its direction. Acceleration: The rate of change of the velocity of a body, any change in the body’s velocity: speeding up, slowing down, changing direction. Animation: circularmotion

Newton’s Second Law: F = ma Relates net external force F applied to object of mass m to resulting change in motion of object, acceleration a.

If there is a NET FORCE on an object, how much will the object accelerate? For a given force, greater mass (greater inertia), yields smaller acceleration. F = m a smaller mass, yields greater acceleration. F = m a For a given mass, greater force yields greater acceleration. F = m a smaller force, yields smaller acceleration. F = m a

Questions - Newton’s Laws of Motion Consider a game of kick-the-can played with two cans --- one empty and one filled with concrete. Which can has greater mass? If someone came along and kicked the two cans with exactly the same force, which can would have a greater acceleration? Explain in terms of Newton’s laws of motion. What does Newton’s 3rd law predict about the effect on the foot that kicks the two cans?

How’s That? Newton’s Laws of Motion Inertia F = ma Action/reaction penny/cup/paper F = ma chair/empty/person Action/reaction hand-to-hand

Newton and Gravitation Newton’s three laws of motion enable calculation of the acceleration of a body and its motion, BUT must first calculate the forces. Celestial bodies do not touch ------ do not exert forces on each other directly. Newton proposed that celestial bodies exert an attractive force on each other at a distance, across empty space. He called this force “gravitation.”

Isaac Newton discovered that two bodies share a gravitational attraction, where the force of attraction depends on both their masses:

Both bodies feel the same force, but in opposite directions.

This is worth thinking about - for example, drop a pen to the floor This is worth thinking about - for example, drop a pen to the floor. Newton’s laws say that the force with which the pen is attracting the Earth is equal and opposite to the force with which the Earth is attracting the pen, even though the pen is much lighter than the Earth!

Newton also worked out that if you keep the masses of the two bodies constant, the force of gravitational attraction depends on the distance between their centers: mutual force of attraction

For any two particular masses, the gravitational force between them depends on their separation as: distance between the masses increasing magnitude of the gravitational force between 2 fixed masses as the separation between the masses is increased, the gravitational force of attraction between them decreases quickly.

Gravity and Weight Fg = m(GmEarth/rEarth2) = mg The weight of an object is a measure of the gravitational force the object feels in the presence of another object. For example on Earth, two objects with different masses will have different weights. Fg = m(GmEarth/rEarth2) = mg What is the weight of the Earth on us?

Mass and Weight Mass A measure of the total amount of matter contained within an object; a measure of an object’s inertia. Weight The force due to gravity on an object. Weight and mass are proportional. Fg = mg where m = mass of the object and g = acceleration of gravity acting on the object

Free Fall If the only force acting on an object is force of gravity (weight), object is said to be in a state of free fall. A heavier body is attracted to the Earth with more force than a light body. Does the heavier object free fall faster? NO, the acceleration of the body depends on both the force applied to it and the mass of the object, resisting the motion. g = F/m = F/m

Newton’s Law of Gravitation We call the force which keeps the Moon in its orbit around the Earth gravity. Sir Isaac Newton’s conceptual leap in understanding of the effects of gravity largely involved his realization that the same force governs the motion of a falling object on Earth - for example, an apple - and the motion of the Moon in its orbit around the Earth.

Your pen dropping to the floor and a satellite in orbit around the Earth have something in common - they are both in freefall.

Planets, Apples, and the Moon Some type of force must act on planet; otherwise it would move in a straight line. Newton analyzed Kepler’s 2nd Law and saw that the Sun was the source of this force. From Kepler’s 3rd Law, Newton deduced that the force varied as 1/r2. The force must act through a distance, and Newton knew of such a force - the one that makes an apple accelerate downward from the tree to the Earth as the apple falls. Could this force extend all the way to the Moon?

To see this, let’s review Newton’s thought experiment Is it possible to throw an object into orbit around the Earth?

On all these trajectories, the projectile is in free fall under gravity. (If it were not, it would travel in a straight line - that’s Newton’s First Law of Motion.)

If the ball is not given enough “sideways” velocity, its trajectory intercepts the Earth ... that is, it falls to Earth eventually.

On the trajectories which make complete orbits, the projectile is travelling “sideways” fast enough ... On all these trajectories, the projectile is in free fall. On all these trajectories, the projectile is in free fall.

… that as it falls, the Earth curves away underneath it, and the projectile completes entire orbits without ever hitting the Earth. On all these trajectories, the projectile is in free fall.

Gravity and Orbits The Sun’s inward pull of gravity on the planet competes with the planet’s tendency to continue moving in a straight line.

“One had to be Newton to see that the Moon is falling, “One had to be Newton to see that the Moon is falling, when every one sees that it doesn’t.” Paul Valery French poet and philosopher, 1871-1945

Navigating in Space Newton's law of universal gravitation combined with Kepler's three laws explain planetary orbits. They also suggested the possibility of placing artificial satellites in orbit around the Earth or sending space probes to the planets. According to Newton's laws of motion and gravitation, if an object moves fast enough, its path will match the curvature of the Earth, and it will never hit the ground. It goes into orbit. Circular orbital velocity for a low Earth orbit is ~ 5 miles/sec. If the object's velocity is > 5 miles/sec, but < 7 miles/sec, its orbit will be an ellipse. Velocities >7 miles/sec reach escape velocity, and the object moves in a curved path that does not return to Earth.

The effect of launch speed on the trajectory of a satellite. Required launch speed for Earth satellites is: ~8 km/s (17,500 mph) for circular orbit just above atmosphere, ~11 km/s (25,000 mph) to escape from Earth.

Navigating in Space: Transfer Orbits To send a spacecraft to another planet, it is launched into a transfer orbit around the Sun that touches both the Earth's orbit and the orbit of the planet. Once the spacecraft is in the transfer orbit, it coasts to the planet. The gravitational force of the Sun takes over and this part of the ride is free. But transfer orbits put constraints on space travel. The launch must occur when the planet and the Earth are in the correct relative positions in their orbits. This span of time is called a launch opportunity. During each launch opportunity, which can be a few weeks in duration, the spacecraft must be launched during a specific time of the day - launch window. If the spacecraft is headed for an inner planet (Mercury or Venus), the launch window occurs in the morning. For outer planets (Mars and beyond), the launch window occurs in the early evening.

Navigating in Space: Gravity Assist Another technique used by space navigators is called gravity assist. When a spacecraft passes very close to a planet, it can use the strong gravitational field of the planet to gain speed and change its direction of motion. According to Newton's laws of motion, the planet looses and equal amount of energy in the process, but because the mass of the planet is so much greater than the mass of the spacecraft, only the spacecraft is noticeably affected.

Apparent Weightlessness in Orbit This astronaut on a space walk is also in free fall. The astronaut’s “sideways” velocity is sufficient to keep him or her in orbit around the Earth.

Why do astronauts in the Space Shuttle in Earth orbit feel weightless? Let’s take a little time to answer the following question: Why do astronauts in the Space Shuttle in Earth orbit feel weightless?

Some common misconceptions which become apparent in answers to this question are: (a) there is no gravity in space, (b) there is no gravity outside the Earth’s atmosphere, or (c) at the Shuttle’s altitude, the force of gravity is very small.

In spacecraft (like the Shuttle) in Earth orbit, astronauts are in free fall, at the same rate as their spaceships. That is why they experience weightlessness: just as a platform diver feels while diving down towards a pool, or a sky diver feels while in free fall. On all these trajectories, the projectile is in free fall.

Newton’s Form of Kepler’s 3rd Law Newton generalized Kepler’s 3rd Law to include sum of masses of the two objects in orbit about each other (in terms of the mass of the Sun). (M1 + M2) P2 = a3 Observe orbital period and separation of a planet’s satellite, can compute the mass of the planet. Observe size of a double stars orbit and its orbital period, deduce the masses of stars in binary system. Planet and Sun orbit the common center of mass of the two bodies. The Sun is not in precise center of orbit.

Mass of Planets, Stars, and Galaxies By combining Newton’s Laws of Motion and Gravitation Law, the masses of astronomical objects can be calculated. a = v2/r , for circular orbit of radius r F = ma = mv2/r mv2/r = Fg = GMm/ r2 v = (GM/r)1/2 P = 2r/v = 2 (r3/GM)1/2 M = rv2/G If the distance to an object and the orbital period of the object are known, the mass can be calculated.

What’s important in the last half? Definitions and examples: inertia mass acceleration force gravity weight free fall orbits Newton’s Laws of Motion and how they relate to one another and to objects. Newton’s Law of Gravitation

Review 1. Briefly describe the geocentric model of the universe. Who developed the model? What are the model’s basic flaws? 2. What is the Copernican model of the solar system? Flaws in the Copernican model? 3. What discoveries of Galileo helped confirm views of Copernicus? 4. Briefly describe Kepler’s three laws of orbital motion. List two modifications made by Newton to Kepler’s laws. 5. What are Newton’s three laws of motion? 6. What is Newton’s law of gravity? What is gravity? How does the gravitational force vary with the mass of the two objects? with distance between centers? 7. Discuss orbiting objects and free-fall. 8. What is escape speed?

Exploring the Solar System Solar System Object Flyby Orbit Probe Lander Sample Return Human Mercury * Venus Moon Mars Jupiter Saturn Uranus Neptune Pluto Asteroid ? Comet