Origins of a Sun-centered solar system p.37: and a historical summary: Recall from Crowe p.30 that Aristarchus (ca. 310 - ca. 230 BC) posited that the Earth orbits the Sun. However, Ptolemy's Earth-centered system (ca. 100-ca. 170 AD) held sway until the middle ages. Copernicus revived Aristarchus' model . Kepler built three phenomenological laws from Tycho's data that refined Copernicus' model. Galileo's observations provided evidence in support of the Keplerian/Copernican model and Galileo introduced the mathematical notion of an equation of motion, F=ma: forces accelerate masses). Newton's theoretical work quantified the model with a special equation of motion for gravitationally interacting masses, giving it stronger explanatory and predictive power.
Explanation of retrograde motions (of all planets) and bounded motions (of Mercury and Venus, Crowe, p.17) - without resort to epicycles - was one of the first great successes of the new model.
2-1 p.39: Nicolaus Copernicus devised the first comprehensive heliocentric cosmology but retained eccentric circular orbits and added even more epicycles than Ptolemy.
2-2 p.41: Tycho Brahe (1572) made astronomical observations that disproved ancient ideas about the heavens, such as novae and parallax. The heavens were not unchanging after all, as Chinese astronomers had acknowledged at least as early as 1054 (see K.282). Tycho developed a hybrid cosmology similar to an ancient Egyptian model (Crowe p.173),then partied himself to death.
2-3 p.42: Kepler's laws describe the orbital shapes, changing speeds, and lengths of planetary years, without explaining why. These phenomenological "laws" were laboriously pieced together by trial and error, and are embedded in an amazingly complex and mostly wrong masterpiece of many "laws", Kepler's Astronomia Nova (Crowe p.153+)
K.1(p.42): The orbit of a planet about the Sun is an ellipse with the Sun at one focus. Newton showed that this is a mathematical consequence for bounded solutions to his equation of motion: all pairs of heavenly bodies move along conic sections (circle, ellipse, parabola, hyperbola). The 3-body problem is not analytically solvable, in general.
K.2(p.43) A line joining a planet and the Sun sweeps out equal area in equal intervals of time. Newton explained why: conservation of angular momentum. One consequence is that the planet moves faster when it is closer to the sun.
K.3 (p.43): The square of a planet's sidereal period is proportional to the cube of the length of its orbit's semimajor axis. Kepler found this by phenomenology and numerology. Newton derived it simply from his equation of motion, as we will do in class.
2-4 p.44: Galileo's discoveries strongly supported a heliocentric cosmogony.
Moon's mountains; Venus' phases; Jupiter's moons
2-5, p.46: Newton formulated three laws that describe fundamental properties of physical reality, building on Galileo's notion of force and explaining Kepler's phenomenological laws more generally.
N.1 (p.46) A body remains at rest or moves in a straight line at a constant speed unless acted upon by an unbalanced outside force. For example, the gravitational pull of the sun accelerates planets, bending them away from straight-line motion through space, into orbits. Acceleration applies to changing direction or changing speed. A circular orbit of radius r at constant speed v has an acceleration a=v2/r.
N.2 (p.47) F=ma. Force = mass * acceleration. F and a are vectors in the same direction. Mass is a scalar quantity. For example, near the surface of the Earth, everything accelerates with a rate of change of speed of a=9.8 (m/s) per second, or g=9.8 m/s2. Our gravitational interaction with Earth gives us the sensation of weight. Weight = mass * g. Our mass is the same on the moon, but since the moon's gravity is less than earth's, we weigh less on the moon.
N.3 (p.47) Whenever one body exerts a force on a second body, the second body exerts an equal and opposite force on the first body. For example, I pull the Earth up toward me with a force as strong as the Earth's downward pull on me. But since I have a smaller mass than the Earth, I experience a greater acceleration than the Earth does.
Angular momentum (L=mR x v) is conserved in the absence of an external torque. R=perpendicular distance between turning axis and point of application of force; v=velocity; torque = perpendicular component of R times the applied force.
2-6, p.47: Newton's description of gravity accounts for Kepler's laws. Fr example, we can derive g, knowing the mass and size of the Earth. And we can calculate the distance to any planet by observing its period around the sun, even though we may not know the size of the planet. F=GmM/r2 where G=6.67 x 10-11 m3/(kg.s2 )and r is the distance between the centers of the two masses m and M. Same force on each mass, toward the other.
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Discussion Question: DQ#17 (K.53): Which planet would you expect to exhibit the greatet variation in apparent brightness as seen from Earth? Explain your answer.
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Deriving the mass of Earth: we measure g by timing free-fall of an arbitrary mass m in the absence of air friction (or by counting pulses as an object rolls down a plane inclined at a known, shallow angle, as Galileo did), measure G by the Cavendish experiment (1798), and find the radius R of Earth using Eratosthenes' method (Kaufmann's Universe, p.40). Then F= mg=GmM/R2 and M=gR2/G
Deriving the mass of Sun: we measure the period T of Earth's orbit and the distance R to the sun by parallax (Universe p.41, and Crowe, 28-31). Assuming a circular orbit, GmM/R2= ma = mv2/R. Since speed = distance/time, insert v=2Pi*R/T to find both M of sun and derive Kepler's third law.
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Maintained by E.J. Zita
Last modified: 23.Apr.98