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Day 4 Chapter 2 Gravitation and the Motion of the Planets Science is the key to understanding • Science: a body of knowledge and a process of learning about nature (called the scientific method). • Knowledge is acquired by observations and experiments. • Scientific method is a process for gaining more knowledge, that can be tested and accepted by everyone. • Scientific theory is an explanation of observations or experimental results that can be described quantitatively and tested. • The theory must make testable predictions that can be verified by new observations or experiments, and can possibly be refuted. • Theories can be modified and should be the simplest version that explains the observations (Occam’s razor). • Observe, hypothesize, predict, test, modify, economize. The Copernican Revolution • The model of the Greeks (attributed primarily to Ptolemy) had the Earth at the center of everything. • Copernicus proposed a model of the solar system with the Sun at the center (hence a solar system). • Galileo made a telescope and used it to view the sky, and saw that the phases of Venus refuted the geocentric model of the Greeks. • Tycho Brahe and Johannes Kepler developed a more detailed heliocentric model with elliptical orbits. Retrograde motion of a planet occurs over several weeks, and involves motion to the west, as compared to prograde (direct) motion, which is to the east (relative to the stars of the ”celestial sphere”). The Geocentric Model of planetary motion (Greek philosophy) The Geocentric Model does explain retrograde motion, using concepts like deferent and epicycle. However, it does not predict the motion with much accuracy, and does not predict phases of Venus (seen with a telescope). Ptolemy’s Model of planetary motion used deferents (big circles) and epicycles (little circles centered on a point that moves on the deferent). This involved up to 80 circles to describe 7 objects! Occam’s Razor says: Simplify this! Nicholas Copernicus and his Heliocentric model of the Solar System explained this in a simpler way with the Sun at the center. Retrograde motion is seen in this model, using Earth and Mars as the example. Retrograde Motion of Mars as seen from Earth Galileo Galilei and the Birth of Modern Astronomy Galileo built a telescope in 1609 and looked at the sky. Four objects: The Moon The Sun Jupiter Venus (and much more) Galileo looked at the Moon and saw mountains, craters, valleys, and topography like you might find on the Earth. The Moon was perhaps an object like the Earth! By projecting an image of the Sun, he could see imperfections on the Sun. Sunspots could be seen to move from east to west on the Sun and he deduced that the Sun rotated about once a month. Galilean Moons of Jupiter Small point of light could be seen near Jupiter. By observation during several weeks he deduced that these were moons and that they revolved around Jupiter. Perhaps this planet was like the Earth, with several moons of its own. It also seemed like a miniature model of the heliocentric solar system. Venus Phases in the Heliocentric model These are consistent with the observations in a telescope. Venus Phases in the Geocentric model are obviously wrong as soon as you observe with a telescope. This refutes Ptolemy! Geocentric vs. heliocentric theories 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). After Copernicus and Galileo, two major figures changed the way we come to understand the Universe: Kepler’s laws of planetary motion Newton’s laws of mechanics Further development of the heliocentric theory 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. Tycho Brahe obtained data over a period of 21 years that were later used by his assistant Johannes Kepler to determine that planetary orbits are NOT circles, but are ellipses. Johannes Kepler and the Laws of Planetary Motion 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 resulted in three major statements about the characteristics of planetary orbits: Kepler’s three laws of planetary motion. Kepler’s three laws of planetary motion • Orbital paths of the planets are ellipses. • An imaginary line connecting the planet with the Sun sweeps out equal areas of the ellipse in equal intervals of time. • The square of a planet’s orbital period is proportional to the cube of its semi-major axis. • Kepler published this in 1609, the same year that Galileo built his first telescope. Kepler’s 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. 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 GeoGebra demonstration: http://people.ucalgary.ca/~louro/geogebra/ellipse.html Some Properties of Planetary Orbits Kepler’s 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 Second Law: equal areas in equal time This also means higher speed at closer distances. Another graphic on Kepler’s Second Law: The Astronomical Unit is about 150,000,000 km Kepler’s 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 Third Law: P2 (in years) = a3 (in a.u.) Basically, it means that large orbits have long periods. 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. 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_period s_sim.html http://webphysics.davidson.edu/physlet_resources/bu_semester1/c17_solar_s im.html Newton’s Laws of Physics • • • • • • First law: inertia Second law: F = ma or acceleration = force / mass Third law: Action and Reaction means that forces occur in pairs. These can be used to show that orbits should obey Kepler’s 3 laws. Isaac Newton developed a quantitative and explanatory theory of mechanics, explaining the motion of objects resulting from forces. Newton’s First Law: The law of inertia. An object will continue in it’s motion without change of velocity unless it is acted on by a net external force. Newton’s Second Law: F = ma The 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-reaction For 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. Newton also developed the universal law of gravity. 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 The Sun’s gravity causes planets to move on a path called an orbit. These orbits obey Kepler’s Laws. Newton’s Laws explain Kepler’s Laws • Newton’s Laws account for all three of Kepler’s Laws. • The orbits of the planets are ellipses, but it is also possible to have orbits which are parabolas or hyperbolas. (conic sections) • Edmond Halley predicted a comet would return in 1758 and every 76 years after that. (seen in 1910, 1986, and will return in 2061) Halley’s comet has an elliptical orbit extending out past Neptune. • William Herschel discovered Uranus in 1781 by accident. • After 50 years it was seen to deviate from an elliptical orbit, and a calculation led to the discovery of Neptune in 1846. • To be precise, elliptical orbits would only occur if there were only the Sun and one planet. There are 8 planets and other objects which cause deviations from the perfect elliptical orbit. The first exam is on Thursday, Feb. 4 (next week!) We will have about 30 minutes of class before the exam. Then you will take the exam (which uses a Scantron). The exam is multiple choice and true/false questions. Coverage is Chapters 1 and 2 in your textbook. To review, look at the chapter summaries, my day notes, and a study guide that I will post this weekend.