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Transcript

The Motion of the Planets 7/17/06 ISP 209 -- 3B 1 The Motion of the Planets 7/17/06 ISP 209 -- 3B 2 The Solar System Precisely how do the planets (including Earth) move around the sun? What are the fundamental laws of nature that govern this motion? 7/17/06 ISP 209 -- 3B 3 Kepler’s Laws of Planetary Motion Johannes Kepler – a contemporary of Galileo – studied the data on astronomical observations of the planets, seeking to describe accurately their motion around the sun. After 20 years of work, he deduced three empirical laws of planetary motion. 1. A planet moves on an elliptical orbit with the sun at one focal point of the ellipse. 2. The radial line – the line from the sun to the planet – sweeps out equal areas in equal times. 3. The square of the period is proportional to the cube of the semi-major axis. 7/17/06 ISP 209 -- 3B 4 Ellipse geometry An ellipse with semi-major axis a and eccentricity e. This ellipse has a = 1 and e = 0.5 . Hint: perihelion+aphelion = 2a 7/17/06 ISP 209 -- 3B 5 The motion of the planets Diagram of a planet revolving around the sun. The eccentricity e is grossly exaggerated ― real orbits are very close to circular. In fact there are nine planets. The center of mass of the solar system is fixed (). To a first approximation the center of mass is at the Sun. () actually it moves around the 7/17/06 ISP 209 -- 3B center of the galaxy 6 Centripetal acceleration For an object in circular motion, the centripetal acceleration is a = v 2/r . (Christian Huygens) Example. Determine the string tension if a mass of 5 kg is whirled around your head on the end of a string of length 1 m with period of revolution 0.5 s. Answer : 790 N 7/17/06 ISP 209 -- 3B 7 Three concepts --– • Centripetal acceleration • Centripetal force • Centrifugal force 7/17/06 ISP 209 -- 3B v ar r 2 8 Centripetal force and “centrifugal force” View from above View from rear 7/17/06 ISP 209 -- 3B 9 Isaac Newton (1642 – 1727) Newton solved the premier scientific problem of his day – to explain why the planets move as they do. To solve this problem he developed … • the three laws of motion, • the theory of universal gravitation, • calculus, a branch of mathematics. Newton quote: “If I have been able to see farther than others it is because I stood on the shoulders of giants.” -- in a letter to Robert Hooke He could be referring to Galileo and Kepler. 7/17/06 ISP 209 -- 3B 10 Circular Orbits (a pretty good approximation for all the planets because the eccentricities are small.) (velocity) Centripetal acceleration v2 ar r (acceleration) Newton’s second law mv GMm r r2 2 There is a subtle approximation here: we are approximating the center of mass position by the position of the sun. This is a good approximation. 7/17/06 ISP 209 -- 3B 11 mv GMm 2 r r 2 Circular Orbits The planetary mass m cancels out. The speed is then GM v r Period of revolution Time = distance / speed i.e., Period = circumference / speed 2 r r 4 r 2 T 2 r , or T v GM GM 2 3 Kepler’s third law: T 2 r 3 7/17/06 ISP 209 -- 3B 12 Generalization to elliptical orbits (and the true center of mass!) 2 3 4 a T G( M m) 2 2 3 4 a GM where a is the semi-major axis of the ellipse The calculation of elliptical orbits is difficult mathematics. The story of Newton and Halley Many applications ... 7/17/06 ISP 209 -- 3B 13 Kepler’s second law -- The Law of Equal Areas Perihelion : fastest Aphelion : slowest Newton: Angular momentum is conserved—in fact that’s true for any central force—so the areal rate is constant. 7/17/06 ISP 209 -- 3B 14 Westminster Abbey 7/17/06 ISP 209 -- 3B 15 Here is buried Isaac Newton, Knight, who by a strength of mind almost divine, and mathematical principles peculiarly his own, explored the course and figures of the planets, the paths of comets, the tides of the sea, the dissimilarities in rays of light, and, what no other scholar has previously imagined, the properties of the colors thus produced. Diligent, sagacious and faithful, in his expositions of nature, antiquity and the holy Scriptures, he vindicated by his philosophy the majesty of God mighty and good, and expressed the simplicity of the Gospel in his manners. Mortals rejoice that there has existed such and so great an ornament of the human race! He was born on 25th December, 1642, and died on 20th March 1727. Newton’s monument in Westminster Abbey. 7/17/06 ISP 209 -- 3B 16 Comets 7/17/06 ISP 209 -- 3B 17 Ocean Tides -- an effect due to the gravity of the moon and the sun Bay of Fundy 7/17/06 ISP 209 -- 3B 18 If you have ever lived near the ocean, you have observed the tides at the beach. 7/17/06 ISP 209 -- 3B 19 Newton: Ocean tides are due to the gradient of the gravitational force 7/17/06 ISP 209 -- 3B 20 Newton said of himself… “ I know not what I appear to the world, but to myself I seem to have been only like a boy playing on the sea-shore, and diverting myself in now and then finding a smoother pebble or a prettier sea-shell, whilst the great ocean of truth lay all undiscovered before me.” 7/17/06 ISP 209 -- 3B 21 Quiz question 7/17/06 ISP 209 -- 3B Consider the star (S), which has two planets (P1 and P2). P2 is twice as far from S as P1. The period of revolution P1 is 1 year. What is the period of P2? 22 Newton’s figure to explain planetary orbits Newton’s theory has stood the test of time. We use the same theory today for planets, moons, satellites, etc. 7/17/06 ISP 209 -- 3B 23 Here is the general formula. We can always neglect m (mass of the satellite) compared to M (mass of the center). 4 a 4 a T G (M m ) GM 2 2 3 2 3 Applications • Earth and Sun • Other planets • Moon and Earth • Artificial satellites • Exploration of the solar system Some of these are covered in the CAPA problems. 7/17/06 ISP 209 -- 3B 24 Comets Comet Hale-Bopp 7/17/06 ISP 209 -- 3B 25