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Transcript

Modern Astronomy AST101 Lecture 3 Jan. 29, 2002 The beginning of the modern age in Astronomy began with Nicholas Copernicus (1473 – 1543), a cleric with independent fortune. Copernicus suggested that the Sun is at the center of the universe (solar system), and that the Earth rotates on its axis once a day to give the apparent daily turn of the stars and the Sun. He however kept the notion of epicycles and deferents and the insistence on the primacy of circles. The Copernican Revolution removes man (and Earth) from the center of the Universe. Copernicus objected to the ‘ugliness’ of the Ptolemaic theory. Copernicus (Aesthetic arguments often play a role in science – a correct theory should have some ‘beauty’. This notion continues today; ‘beauty’ guides our models.) Besides a being more ‘sensible’ picture, are there observational advantages of the new ideas? Copernicus did give a more plausible explanation for the maximum angle between Venus (or Mercury) and Sun: Since Venus is closer to Sun on a smaller circle, it never deviates from the Sun by more than angle q. Can see full disk of Venus bright (when on opposite side of Sun) Venus Sun q Earth Venus is the ‘morning’ or ‘evening’ star – the brightest object in the sky. Which of the two positions of Venus in the diagram is ‘morning star’ and which is ‘evening star’? (Hint: In what sense does Earth rotate relative to its orbital motion shown by ?) Tycho Brahe (1546 – 1601) was an autocratic Danish nobleman who devoted years of his life to observing the positions of the planets in the sky. He developed observational tools and methods (with a grant from the Danish king that would now be worth $1.5M to build an observatory). Tycho observed a ‘Nova stella’ – new star – in the heavens in 1572 that we would now call a supernova. The appearance of something not previously present countered the old idea of the unchanging heavens. Tycho’s main accomplishment was the body of accurate measurements of planets’ location in the sky over 20 years,. This proved invaluable to the next generation of astronomers in understanding the planets’ orbits (and the laws of Physics). Tycho’s observatory Tycho’s observatory, Uraniborg on the island of Hven Galileo and Kepler – the foundations of modern science Galileo Galilei 1564 – 1642 Galileo was an Italian mathematician and philosopher who pioneered the use of experiments and observations to understand the world. He heard of the invention of the telescope in Holland, and built a rudimentary telescope that he turned on the heavens. Galileo also pioneered experiments in physics, demonstrating the rules that govern falling bodies. With Galileo came the beginning of the notion that Science is based on experiment: “If you can’t see something experimentally, you aren’t allowed to say it is true” Telescope observations of Galileo: 1. There are many more stars in the sky than can be seen with the naked eye. If this is so, how can we hold the opinion, as in the Middle Ages, that the . heavens are provided for the sole benefit of mankind? 2. Jupiter has four moons not observable to the naked eye. (actually now see at least 28 moons!) This is a shock to a geocentric view of the world – there are bodies that do not revolve around Earth! 3. Venus shows phases – from full to crescent. In the geocentric model, there are only crescent phases. Copernican system predicts all phases. 4. The moon has craters and mountains. The sun shows blemishes called sunspots that come and go. The sunspots reveal that the sun rotates on its axis. The heavenly bodies are not perfect orbs and have their own motions! Galileo arrogantly published his findings supporting the Copernican view and belittling the Catholic church. His book “Dialogues” featured a character Simplicio (a simpleton) who tries to defend the church’s geocentric doctrine. Galileo spent the rest of his life in house arrest. Check out “Galileo’s Daughter”, a recent best seller by Dava Sobel based on the letters between Galileo and his daughter. Johannes Kepler 1571 – 1630 Kepler was one of most interesting characters in scientific history – with one leg in Middle Ages and one in the Renaissance. As Tycho’s assistant in court of Rudolf, Holy Roman Emporer in Prague, Kepler inherited the extensive data collected by Tycho to guide his calculations. He believed in the Copernican model, and wanted to find the underlying cause or model of the motions of the planets. However, he was inclined to seek mystical explanations for the planet’s orbits and was enamored of the ancient Pythagorean philosophy. Read Arthur Koestler’s book The Sleepwalkers – how did Kepler span the divide between the Middle Ages and the Renaissance? The Music of the Spheres – Kepler likened the orbits of planets to strings that could be plucked, sounding the Greek and Medieval pentatonic scale (the black keys of the piano) Saturn Jupiter Mars Mercury Venus The ratio of circumferences of the planet’s orbits were about right to give the pentatonic scale. (Kepler had to invent the math to allow him to calculate the tones.) Kepler also thought the ‘Five Perfect Solids’ of Pythagoras and Plato could be the basis for the planetary orbits – He tried to inscribe and circumscribe the spheres containing the orbits in nested Platonic solids. The size of the spheres that allowed the nesting were about right for the known planets. The perfect Platonic Solids 8 triangles 6 squares 4 triangles 12 pentagons 20 triangles Calculating this model was a tour de force in solid geometry! Kepler’s model of the 5 perfect solids Although the Harmony of the Spheres, and the Perfect Solids came close to reproducing the orbits, Tycho’s data was too good, and Kepler was too honest, to ignore the discrepancies. He then set out to find a more complete and accurate representation of the known planet orbits using painstaking calculations of the orbits found by Tycho. After about 30 years, he wrote his conclusions in the form of 3 Laws (buried in a mass of mystic mumbo jumbo … how did Newton find the pearls of truth?) Kepler’s Laws 1. The planets move in ellipses, with the Sun at one focus. 2. The line from the Sun to the moving planet sweeps out equal areas in equal times. 3. The square of the planet’s orbital period (P) is proportional to the cube of the semi-major axis of the ellipse. The planets move in ellipses, with the Sun at one focus. Planet Focii Major axis Sun Minor axis 1. center Semimajor axis = a Elliptical motion is a major departure from the Ptolemic model based on circles! Drawing an ellipse Ellipse is the set of points for which the sum of distances to 2 fixed points (the focii) is held constant. Eccentricity e is ratio of distance between focii and length of major axis. An ellipse with e = 0 is a circle (the two focii coincide). Draw your own ellipses, varying the separation between focii from zero to ½ the major axis. 2. The line from the Sun to the moving planet sweeps out equal areas in equal times. The light blue shaded areas represent the motion of planet in the same fixed time interval. The areas of all the light shaded sectors are the same. Definition: The point on the orbit at nearest approach to the Sun is perihelion. The point furthest from the Sun is aphelion. Kepler’s 2nd law tells us that the planet does not move with uniform speed – another major departure from the Ptolemaic model. Does planet move faster at perihelion or aphelion? 3. The square of the planet’s orbital period (P) is proportional to the cube of the semi-major axis of the ellipse. In symbols: If P = period (time to revolve one full turn) and a = semi-major axis: P2 ~ a3 “~“ means “proportional to” ) Equivalently: P2 =k a3 The constant of proportionality k depends (mainly) on the mass of the sun, so the relation holds for all planets in a given solar system. Same relation for another solar system, but with a different value for k (see update by Newton on what the constant k means) The value of the constant of proportionality for our solar system can be fixed using the Earths orbit: Earth’s period is 1 year (the definition of year) and semi-major axis is 1 AU (the definition of AU). Thus for our solar system: P2 =kSS a3 P2 = 1 (yr2) P2 = a3 ; a3 = 1 (AU)3 so kss = 1 in these units (if P is in years and a is in AU) for our solar system Ratio relation: For any two planets in the same system, with periods P1 and P2 and semi-major axes a1 and a2 : P12 = k a13 (a) P22 = k a23 (b) Divide (b) by (a): The k’s cancel and we get: (P2/P1)2 = (a2/a1)3 Example: Mars orbits the Sun every 1.881 years. Predict the size of its orbit (that is, find the semi-major axis of Mars orbit). P2(Mars) = 1.8812 = 3.538 yr2 Using P2 = a3 = 3.528, 3 a = 3.538 = 1.524 AU (Doing the cube root requires a good calculator! You can try it in reverse to show that 1.5243 = 3.538 ) Direct observation of Mars orbit gives a = 1.524 AU, so prediction and observation agree. Check this calculation for another planet using the data in Table 2.1 of the text Example: In some other other planetary system, we see two planets. The first planet revolves around its star every 2 years and has a semi-major axis of 3 AU. The second revolves around the star every 16 years. What is the size of the orbit of the second planet? Let P1 = period of planet 1 = 2 yr P2 = period of planet 2 = 16 yr a1 = semi-major axis for planet 1 = 3 AU a2 = semi-major axis for planet 2 (unknown) Since this is a different planetary system, the constant of proportionality is different from our solar system. However, we can still use: (P2/P1)2 = (a2/a1)3 Thus (P2/P1)2 = (16/2)2 = 82 = 64 = (a2/a1)3 . 3 Then 64 = 4 = (a2/a1), giving a2 = 4 a1 = 4x3 = 12 AU