Scientific Method 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 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. 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. Ancient Astronomy 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 1
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 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). Aristotle s Universe: A Geocentric Model 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) 2
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 3
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) 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) Stellar Parallax 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. 5 th - 10 th century Renaissance Copernicus (1473-1543) Tycho Brahe Kepler Galileo (1546-1601) (1571-1630) (1564-1642) Newton (1642-1727) 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 Galilei 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 MedicianStars in honor of Cosimo II de'medici, who had become Grand Duke of Tuscany in 1609. Galileo and Jupiter 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. 5 th - 10 th century Renaissance Copernicus (1473-1543) Tycho Brahe Kepler Galileo (1546-1601) (1571-1630) (1564-1642) Newton (1642-1727) 4
Tycho Brahe 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 ). Quadrant Sextant Johannes Kepler: Laws of Planetary Motion 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 As a planet moves in an elliptical orbit, the Sun is at one focus (F or F ) of the ellipse. 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. F C F r 5
The line that connects the planet s point of closest approach As a planet moves in an elliptical orbit, the Sun is at one to the Sun, the perihelion... focus (F or F ) of the ellipse and its point of greatest separation from the Sun, As a planet moves in an elliptical orbit, the Sun is at one the aphelion focus (F or F ) of the ellipse v perihelion is called the major axis of the ellipse. v perihelion F C F r F C F r 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. Kepler s 1st Law: Semi means half, and so the semi-major axis a is half the length of the major axis: a v F C F r a 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. 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. Orbit-anim 6
According to Kepler s Second Law, sweeps out equal areas in equal intervals of time. the line joining the object and the Sun... equal areas 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. 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 3 rd Law: For objects orbiting a common central body (e.g. the Sun) in approximately circular orbits, the orbital period squared is proportional to the orbital radius cubed. m v r M 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. a v F C F r a It turns out that Kepler s 3 rd 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 semi major radius axis cubed. cubed. 7
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. the Sun), a a 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. 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 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). Kepler s 3rd Law P 2 = a 3 or P 2 /a 3 = 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) 8
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 10 6 km Kepler s 3rd Law for the Planets P 2 = a 3 or P 2 /a 3 = 1 Planet a (A.U.) P(Earth years) P*P/a*a*a eccentricity Mercury 0.387 0.241 0.98 0.206 Venus 0.723 0.615 0.99 0.007 Earth 1 1 1.00 0.017 Mars 1.524 1.881 0.99 0.093 Jupiter 5.203 11.86 0.99 0.048 Saturn 9.537 29.42 0.99 0.054 Uranus 19.19 83.75 0.99 0.047 Neptune 30.07 163.7 0.99 0.009 Pluto 39.48 248 0.99 0.249 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? 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). 9
Why do the planets move according to Kepler s laws? Or, more generally, why do objects move as they do? 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? What is a natural state of motion for an object? Slope down, Slope up, No slope, speed increases. speed decreases. does speed change? Without friction, NO, the speed is constant! At rest? Moving with constant velocity? Inertia: Inertia and 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. 10
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. instantaneously Forces can act through contact (baseball bat making contact with the baseball), or or continuously at a distance. (gravity keeping the baseball from flying into space). 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 andits 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 11
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 F = m a F = m a smaller mass, yields greater acceleration. For a given mass, greater force yields greater acceleration. F = m a F = m a smaller force, yields smaller acceleration. 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 : 12
Both bodies feel the same force, but in opposite directions. 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: 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. distance between the masses increasing 13
Gravity and Weight 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. F g = m(gm Earth /r Earth2 ) = 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. F g = 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. 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. g = F/m = F/m 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/r 2. 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? 14
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 On all the these trajectories trajectories, which the make projectile complete is in free orbits, fall. the projectile is travelling sideways fast enough... 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. that as it falls, the Earth curves away underneath On all these trajectories, the projectile is in free fall. it, and the projectile completes entire orbits without ever hitting the Earth. 15
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. 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. On That all is these why they trajectories, experience the weightlessness: projectile is in free just fall. as a platform diver feels while diving down towards a pool, or a sky diver feels while 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). (M 1 + M 2 ) P 2 = a 3 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 = v 2 /r, for circular orbit of radius r F = ma = mv 2 /r mv 2 /r = F g = GMm/ r 2 v = (GM/r) 1/2 P = 2pr/v = 2p (r 3 /GM) 1/2 M = rv 2 /G If the distance to an object and the orbital period of the object are known, the mass can be calculated. 16