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1445 Introductory Astronomy I. Chapter 2 Gravity and Planetary Motion R. S. Rubins Fall, 2009. Geocentric and Heliocentric Cosmologies. Cosmology is the study of the origin and structure of the Universe.
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1445 Introductory Astronomy I Chapter 2 Gravity and Planetary Motion R. S. Rubins Fall, 2009
Geocentric and Heliocentric Cosmologies • Cosmologyis the study of the origin and structure of the Universe. • In geocentric cosmology, which was the prevailing cosmology up to the 16th century, the Earth was considered to be at the center of the universe. • In heliocentric cosmology, the planets were assumed to orbit the Sun, which was at the center of the universe. • Apart from the ideas of Heracleides (c. 350 BCE) and deductions of Aristarchus (c. 270 BCE), geocentric ideas held sway for about1800 years, both in science and religion, until Copernicus (1476 – 1543) reintroduced the heliocentric system.
The Ancient Greeks • Plato (428 - 348 BCE) asserted that heavenly motion must be in perfect circles, which was an idea which hindered science for about 2000 years. • Aristotle (384 - 322 BCE) argued for a geocentric universe. • Heraclaides (c. 350 BCE) is credited with suggesting that the “heavenly” objects orbit the Sun and not the Earth, and also that the Earth rotates around its north-south axis. • However, Aristarchus of Samos (c. 270 BCE), the philosopher most strongly associated with the heliocentric universe, based his deductions on his trigonometric estimates of the relative sizes of the Sun, Moon and Earth. • Although his book “On the Sizes and Distances of Sun and Moon” became famous, his heliocentric philosophy was harshly criticized. • His status as a philosopher was insufficient to overcome the prejudice against the heliocentric universe for over 1800 years.
The Earth is Round • Aristotle (ca. 350 BCE) deduced that the Earth was spherical from the following observations: • the curved shadow of the Earth during a lunar eclipse; • the change in position of the constellations with latitude; • the disappearance of the hull of a ship first when it crosses the horizon. • Eratosthenes (ca 200 BCE) determined the Earth’s radius by comparing angles made by the Sun with the vertical at the summer solstice. This angle was about 7o at Alexandria, Egypt and 0o (directlyoverhead) at Syene, about 520 miles to the South. • Since a circle corresponds to a rotation of 360o, he estimated the Earth’s circumference to be 520 x (360/7) ≈ 27,000 mi; i.e. a diameter of 8600 mi (actual value, 7970 mi).
Eratosthenes’ Measurement Vertical Vertical
The Size and Distance of the Sun 1 • By measuring the Sun-Earth-Moon angle when the Moon is exactly half lit, Aristarchus (ca. 270 BCE) calculated that the Sun is about 20 times further from Earth than is the Moon (correct answer 390). 23½o
The Size of the Moon By observing the the Earth’s curved shadow on the Moon during a lunar eclipse, Aristarchus deduced that RM≈ 0.3 RE.
The Distance of the Moon • Knowing the Moon’s diameter DM = 2 RM and the angle it subtends at the eye Φ(in radians), Aristarchus determined the distance of the moon R to be about R = DM/Φ ≈ 250,000 mi. • Since the Sun has a much larger diameter than the Earth, Aristarchus proposed that the Earth must orbit the Sun, and not vice-versa.
The Next 1700 Years • Apollonius (240 - 190 BCE) introduced the concept ofepicycles (circles upon circles) to explain the retrograde motions of planets. • Hipparcus (190 - 120 BCE) developed Apollonius’ theory into a method that could predict planetary positions. • In medieval times, the triumphs of Greek thought were preserved and enhanced in the Arab world. • In about 140 CE, Ptolemy used a geocentric model to perfect a method of predicting planetary motions, which remained in use for 1500 years. • Over 1300 years later, the Polish astronomer , Copernicus (1476 -1543), was able to remove the complexities of the Ptolomaic system, by reviving the heliocentric model.
The Planets: Retrograde Motion • Planets (from the Greek for wanderer) were so-called because they changed their positions on a nightly basis with respect to the “fixed stars”. • Viewed from the northern hemisphere, the usual motion of the planets, known as direct motion, is to the east with respect to the fixed stars. • Occasionally, retrograde motion - a reverse motion to the west with respect to the fixed stars - is observed. • Ptolemy (ca. 140 CE) represented planetary motion by superimposing two circles, a large one known as the deferent (guiding circle), and a small one, theepicycle. • When his book was translatedby Arabic scholars in about 800 CE, it was given the title Almagest, meaning “the greatest compilation”.
Retrograde Motion of Mars 1 • Lie on your back, with your head pointing north. East West
Copernicus’s System 1543 Surrounding the Sun, starting on the outside are the following: I. fixed stars; II. Saturn; III. Jupiter; IIII. Mars; V. Earth and Moon; VI. Venus; VII. Mercury.
Giordano Bruno (1548 – 1600) • Throughout history, there have been thinkers, who have been far ahead of their time, such as Heraclaides and Aristarchus, who introduced the heliocentric universe nearly 1900 years before Copernicus. • Giordano Bruno was such a thinker; he wrote: “In space there are countless constellations, suns and planets; we see only the suns because they give light; the planets remain invisible, for they are small and dark. There are also numberless earths circling around their suns, no worse and no less than this globe of ours.” • Bruno was burnt at the stake in Rome for his views, but over 400 years later, in 1995, the first extrasolar planet was discovered orbiting the star 51 Pegasi by Swiss astronomers.
Synodic Period of Mercury • The siderial period (the time for one complete circle) of Mercury is 0.24 y. • The synodic period, which is measured between successive inferior conjunctions, is 0.32 y (or 116 days).
Tycho Brahe (1546 -1601) • Tycho Brahe was colorful character and a remarkable observational astronomer. • Living before the invention of the telescope, Tycho made the most precise naked-eye planetary observations of all time. • His planetary measurements were accurate to within 1 arcsecond (less than the thickness of a finger nail held at arms length). • In 1600, he hired Johannes Kepler as his apprentice, and, on his death-bed a year later, begged Kepler to make sense of his observations. • He moved to Austria, and in 1599, as court astrologer and mathematician to the emperor, he made the observations that lead Kepler to deduce his famous laws of planetary motion.
Johannes Kepler (1571-1630) 1 • Kepler was a mystic who believed that numbers and simple geometrical shapes, in particular circles, governed the structure of the universe. • He was for many years the imperial court astrologer and mathematician, and since the emperor rarely paid his salary, made money by selling astrological material. • He spent many years trying to fit Brahe’s observations to the Copernican heliocentric system, using Ptolemy’s idea of circles and epicycles. • Kepler was a prolific writer, who wrote perhaps the first science fiction novel Somnium, a dream of a journey to the Moon, in which the concept of a gravitational force was anticipated.
Johannes Kepler (1571-1630) 2 • He labored for several years, but could not overcome a difference between theory and observation of up to 8 arc-minutes (one quarter the angular diameter of the Moon) between theory and observation. • Trusting Tycho’s careful work, Kepler was forced to abandon his deeply-held belief in circular orbits, and work on the hypothesis that planetary orbits were elliptical. • In 1609, Kepler published his laws of planetary motion. • Possibly Kepler’s greatest achievement was the major psychological breakthrough he achieved through his rejection of the circular orbit dogma. (He was also excommunicated by the Church.)
Ellipses of Different Eccentricities • The eccentricity e varies from 0 for a circle to 1 for a straight line.
Kepler’s First Law of Planetary Motion The orbit of a planet about the Sun is an ellipse, with the Sun at one focus.
Kepler’s Second Law of Planetary Motion • The line joining a planet to the Sun sweeps out equal areas in equal times. • Thus, a planet moves fastest when closest to the Sun.
Kepler’s Third Law of Planetary Motion • The square of the sidereal periodof a planetis proportional to its (Mean distance from the Sun)3 ; i.e.P2 = a3.
Conic Sections A conic section is a curve obtained by slicing a cone with a plane.
Galileo Galilei (1564-1642) 1 • Galileo is famous for the following contributions to science: i. his contributions to the physics of falling bodies; ii. the first known telescopic observations of the night sky; iii. publicizing the work of Copernicus. • He observed the following: i. mountains on the Moon; ii. dark spots on the Sun (sunspots); iii. four moons orbiting Jupiter; iv. the phases of Venus, showing that Venus orbited the Sun; v. a ring on Saturn; vi. that the white band in the night sky, known as the milky way is composed of many stars too close and faint to be resolved as separate by the unaided eye; vii. that planets appear as disks, and the stars as points.
Galileo Galilei 2 • Galileo made important contributions to basic physics, paving the way for Newton’s monumental work. • From experiments on objects rolling or sliding down slopes, he deduced that, in the absence of air resistance, all objects would fall at the same rate at the Earth’s surface. • This hypothesis disagreed with Aristotle’s 2000 year-old idea that heavier objects fell faster. • Galileo realized that an object moving on a frictionless horizontal surface could circle the Earth forever. • This idea was the basis for Newton’s 1st Law of Motion, also known as the Law of Inertia.
Observations of Jupiter’s Moons First observed by Galileo in 1610, these drawings of Jupiter and its 4 largest moons were made by Jesuits in 1620.
The Phases of Venus • Galileo noted that the apparent size of Venus was always largest at the crescent phase, and smallest at the gibbous phase. • This observation was a direct verification of the heliocentric model.
Isaac Newton (1642-1727) • Newton’s greatest work was done at the age of 25, when Cambridge University was closed for 18 months because of the Great Plague, and he was forced to live at home. • In his major work on physics and mathematics, “The Principia”, published in 1687, Newton introduced the first great laws of physics: the Laws of Motion and the Law of Gravitation. The consequences of these laws are still being calculated today. • Starting with “common sense” ideas about space and time, his three Laws of Motion deal with the effects of force on physical objects. • To obtain his Law of Gravitation, Newton applied the Laws of Motion to both a falling object and the Moon’ orbit, realizing that each was due to the same force – gravity. • Newton’s laws of motion and gravity (1687), the greatest achievement of classical physics, explained all of mechanics, including Kepler’s laws.
Sir Isaac Newton Nature and Nature’s laws lay hid in the Night God said, Let Newton be! and all was Light Alexander Pope,1727
Inertia, Mass and Weight • Mass is the quantitative measure of inertia, which is the resistance of an object to a change of motion. • We normally find the mass of an object by measuring its weight, which is the force of the Earth’s gravity on it; i.e. Fgrav= weight = mg, where g = 9.8 m/s2 is the gravitational acceleration. • Weight depends on location, but mass is independent of location. • On the Moon, gravity is 1/6 th of its value on Earth, so that a “120 lb person” would weigh just 20 lb there. • In a coasting space vehicle, objects are weightless, but their masses remain unchanged, so that an elephant hitting you in space would have the same effect as on Earth.
Newton’s Laws of Motion 1a • First Law An object moves with constant velocity unless a net force acts to change either its speed or its direction. • This law is also known as the Law of Inertia, since an object in motion resists being slowed down or speeded up. • A coasting spaceship needs no fuel to keep moving.
Newton’s Laws of Motion 2a • Second Law A net force F gives an object of mass m an acceleration a according to the equation F = ma. In the photo, the pitcher’s arm gives the ball its acceleration. Remember that an acceleration is a change of velocity.
Centripetal acceleration An object moving at constant speed in a circle has a centripetal acceleration, given by ac = v2/r.
Centripetal Force 1 A centripetal force Fc, which is a force towards the center of the circle, is needed to produce a centripetal acceleration ac; i.e. Fc= mac. For astronomical objects, this force is gravity.
Centripetal Force 2 Without gravity, the satellite would move in a straight line. Gravity continuously forces it from its straight line, causing it to move in a circle. The gravitational force on the satellite always points towards the Earth’s center.
Newton’s Laws of Motion 3a • Third Law To every action there is an equal and opposite reaction. • The downward force with which the gas is expelled from the rocket is equal in magnitude to the upward force on the rocket.
Newton’s Law of Gravitation 1 • Every object in the Universe attracts every other object with a force proportional to the product of the masses divided by the square of their separation d; i.e. F = G m1m2 /d2, where G is the (universal )gravitational constant.
