AST 101, Week 6 Notes
AST 101, Week 6 Notes 101
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This 5 page Class Notes was uploaded by Bethany Marsfelder on Monday October 17, 2016. The Class Notes belongs to 101 at Syracuse University taught by Professor Walter Freeman in Fall 2016. Since its upload, it has received 34 views. For similar materials see Our Corner of the Universe in Astronomy at Syracuse University.
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Date Created: 10/17/16
October 4, 2016 AST 101: Professor Freeman Lecture: Kepler’s laws of orbital motion Textbook Pages: 65-75 Lecture Tutorials: 21-28 “And yet it moves.” – Galileo (attributed), on the Earth The geocentric, Ptolemaic model The heliocentric, Copernican model The planets and everything else go around Earth is one of many planets all orbiting the Earth Sun Inelegant system of “epicycles” needed to get Apparent motion = motion of Earth and planets right motion of planets Everything moved in circles (elegant per No (or very small) epicycles Greeks) Earth and humanity at center (theologically Matched Galileo’s observations: moons of not challenging) Jupiter, phases of Venus Very accurate predictions Less accurate than Ptolemaic model Science doesn’t attempt to explain “only a little”; if Nature is based on physical laws, they should predict everything Can we refine the Copernican model? o Different circular orbits? o Epicycles again? o Different shapes? o We have to think about the earth and the planets Is the data wrong? o Measurements of sky were “good enough” for navigation, but were not ever intended for natural philosophy/determining the truth of things Tycho Brahe o Danish nobleman and astronomer, 1546-1601 o Greatest naked-eye astronomer o Built a fancy castle called Uraniborg to do research, taxed peasants to pay for it o Lost a big chunk of his nose in a duel and got a brass replacement made o Had better facial hair than any of us o Had a pet moose; drank too much beer and died o Observed a supernova in the constellation Cassiopeia o Old worldview: world beyond man/the Moon is eternal/unchanging o …nope, no observed parallax in the supernova; it’s very far away o Didn’t observe parallax in distant stars o Two options: Earth doesn’t move The stars are very far away o Made the wrong choice (former), proposed another model of the Solar System o Made high-precision observations of the motions of the planets and stars, had a correction for atmosphere-bending light o Measurements accurate to 1/60 of a degree o His assistant, Kepler, finally got it right Johannes Kepler o Copernican, tried to improve Copernicus’ model Data very precise But, no matter how he rearranged the circle, there was at least 8/60 degree error for Mars Wanted it to be precise We need to reexamine our ideas at what “elegance” looks like Kepler didn’t want to discover how the planets moved, he wanted to know why. Led to a complete reformation of astronomy “There is a force in the Earth which causes the Moon to move” Kepler’s Laws o 1. The planets move in ellipses, with the Sun at one focus o 2. The line joining the planet and the Sun sweeps out equal areas in equal times Alternatively: Within its orbit, a planet’s speed is inversely proportional to its distance from the Sun o 3. The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of the ellipse. Kepler’s First Law o An ellipse is a stretched circle; the curve around two points so that the sum of the distances to these points is a constant. o A circle is an ellipse with both foci at the same point. o Terms Focus – one of the two aforementioned points Semimajor axis – largest distance from the center to the edge Eccentricity – how stretched out an ellipse is o Ideas The two foci always lie along the major (or “wide”) axis The closer together the foci, the less eccentric If both foci are exactly at the middle, you get a circle Both foci lie inside the ellipse Kepler’s Second Law o In an eccentric orbit, a planet travels fastest when it’s nearest to the Sun Apehelion – the furthest point of an object to the Sun Perihelion – the closest point of an object to the Sun ex: Halley’s Comet spends most of its time far from the son and only a little time near the Sun, so it moves quickly when it’s near aphelion (only about six months) and slowly at perihelion (the rest of its 76-year orbit) Kepler’s Third Law o The square of a planet’s orbital period is proportional to the cube of its semimajor axis. o Or, in actual English: If a planet is further from the Sun, it takes longer to go around If the distance is doubled, the time required to go around more than doubles. Additional o Kepler’s First Law This law tells us that a planet’s distance from the Sun varies during its orbit. It is closest at the point called perihelion (Greek; “near the Sun”) and furthest at aphelion (Greek; “away from the Sun”) The average of a planet’s perihelion and aphelion distances is the length of its semimajor axis (also known as the average distance from the Sun) o Kepler’s Second Law As a planet moves around its elliptical orbit, it moves faster when it is nearer the Sun and slower when it is farther from the Sun, with its speed varying in such a way that the planet’s path sweeps out equal areas in equal times. Equal areas swept in equal times are short and wide near perihelion, which moves that the planet moves a longer distance in the same amount of time and hence is going faster Equal areas swept in equal times are long and narrow near aphelion, which means the planet moves a shorter distance in the same amount of time and hence is going slower o Kepler’s Third Law More distant planets orbit the Sun at slower average speeds. Or, in math speak, p^2 = a^3 Where p is the planet’s orbital period in years Where a is its average distance from the Sun (semimajor axis) in astronomical units Because this relates orbital distance to orbital time (period), we can use the law to calculate a planet’s average orbital speed October 6, 2016 AST 101: Professor Freeman Lecture: Finishing up Kepler; Newton’s law of gravity Textbook Pages: 83-90 Lecture Tutorials: 29-32 Intro o Fusion of astronomy and natural mathematics = science o Industrial revolution: Universe is a complicated machine o “Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things.” Newton, Rules for methodizing the Apocalypse (“apocalypse” also means revealing) o “We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances.” Newton, Philosophiae Naturalis Principia Mathematica Kepler’s Laws, Summarized o 1. Planets travel in elliptical orbits, with the Sun at one focus o 2. The line going from the Sun to the planet sweeps out equal areas in equal times o 3. The time that a planet takes to go around the Sun increases as the 3/2 power of the long axis of the ellipse. (If it takes larger to go around if a planet is further from the Sun) (If distance doubled, time more than doubles) Example Problem/Kepler’s Third Law Planet Axis Length Orbital period Saturn 10 AU 30 yr Uranus 20 AU ? Earth I AU 1 year o 10^3/2 = approximately 30 o 20 3/2 = approximately 89.4 89.4 yr = ? o “square cube” – 3/2 power Kepler figured out what had eluded everyone else: a precise description of the orbits of the planets. But, he thought that there was more that the planet’s orbits were caused by the interplay of more fundamental agents. o In other words that they were consequences of something. o Turns out, he was right, and these are gravity and inertia Natural laws vs. their consequences o The world around us is super diverse, and some things look simple: The motions of the stars Near-perfect-spheres of the planets and moons Elliptical motions of the planets The colors in a rainbow o Others, though, are maddeningly complex: Seismic waves and earthquakes The colors in the Sun The weather The diversity of rocks on Earth Even the simplest living things …language, culture, music, art, and all the creations of humankind… o The laws of the Universe are simple and elegant (physics). The things the Universe builds out of them are often complex (astromechanics) Isaac Newton (1642-1727 or 1726) finally figured out laws that eluded Kepler o Forces cause objects to change their velocities o Calculus – the mathematics of changes o Gravity is such a force o The mathematical description of gravity o Principles of optics o The mathematics of cooling o …and much more Newton’s biggest idea o F = ma o F/m = a The law of gravity (“it’s a weightier subject.”) o What Kepler suspected all along o All objects attract all other objects with a force that is: Proportional to the product of their masses Inversely proportional to the distance between them squared o In symbols: F = (Gm1*m2)/r^2 o “every action has an equal and opposite reaction” Force of Earth pulling on the moon is the same as the force of the moon pulling on Earth
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