1445 Introductory Astronomy I

1
1445 Introductory Astronomy I

• Chapter 2
• Gravity and Planetary Motion
• R. S. Rubins
  Fall, 2008

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Geocentric and Heliocentric Cosmologies

• Cosmology is 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 about 1800 years,
  both in science and religion, until Copernicus
  (1476 1543) reintroduced the heliocentric
  system.

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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.

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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 Earths
  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
  (directly overhead) at Syene, about 520 miles to
  the South.
• Since a circle corresponds to a rotation of 360o,
  he estimated the Earths circumference to be 520
  x (360/7) 27,000 mi i.e. a diameter of 8600 mi
  (actual value, 7970 mi).

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Eratosthenes Measurement
Vertical
Vertical
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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
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The Size of the Moon

• By observing the the Earths curved shadow on the
  Moon during a
• lunar eclipse, Aristarchus deduced that RM 0.3
  RE.

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The Distance of the Moon

• Knowing the Moons diameter DM 2 RM and the
  angle it subtends at the eye F (in radians),
  Aristarchus determined the distance of the moon R
  to be about
• R DM/F 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.

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The Next 1700 Years

• Apollonius (240 - 190 BCE) introduced the concept
  of epicycles (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.

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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, the epicycle.
• When his book was translated by Arabic scholars
  in about 800 CE, it was given the title Almagest,
  meaning the greatest compilation.

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Epicycle Direct Motion
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Epicycle Retrograde Motion
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Retrograde Motion of Mars 1
East
West

• Lie on your back, with your head pointing
  north.

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Retrograde Motion of Mars 2
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Nicolaus Copernicus (1473-1543)
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Copernicuss 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.

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Planetary Configurations
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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).

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Tycho Brahe (1546 -1601)

• Tycho Brahe was colorful character and a
  remarkable observational astronomer.
• Living before the invention of the telescope,
  Tycho Brahe 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.

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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 Brahes
  observations to the Copernican heliocentric
  system, using Ptolemys 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.

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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 Tychos 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 Keplers greatest achievement was the
  major psychological breakthrough he achieved
  through his rejection of the circular orbit
  dogma. (He was also excommunicated by the
  Church.)

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Johannes Kepler and Tycho Brahe
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Ellipses of Different Eccentricities

• The eccentricity e varies from 0 for a circle to
  1 for a straight line.

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Keplers First Law of Planetary Motion

• The orbit of a planet about the Sun is an
  ellipse, with the Sun at one focus.

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Keplers 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.

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Keplers Third Law of Planetary Motion

• The square of the sidereal period of a planet is
  proportional to its (Mean distance from the
  Sun)3 i.e. P2 a3.

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Conic Sections

• A conic section is a curve obtained by
  slicing a cone with a plane.

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Galileo Galilei (1564-1642) 1

• A contemporary of Kepler, Galileo is famous for
  his contributions to the physics of falling
  bodies, for making the first known telescopic
  observations of the night sky, and for
  publicizing the work of Copernicus, which lead to
  his trial, and subsequent punishment by the
  Church.
• He observed the following
• i. mountains on the Moon
• ii. dark spots on the Sun, which is clearly
  blemished
• iii. four moons orbiting Jupiter
• iv. the phases of Venus, which showed that
  Venus orbited
• the Sun, not the Earth
• v. a ring on Saturn
• vi. that 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.

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Galileo Galilei 2

• Galileo made important contributions to basic
  physics, paving the way for Newtons 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 Earths surface.
• This hypothesis disagreed with Aristotles 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 Newtons 1st Law of
  Motion, also known as the Law of Inertia.

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Galileo Galilei
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Observations of Jupiters Moons

• First observed by Galileo in 1610, these drawings
  of Jupiter
• and its 4 largest moons were made by Jesuits in
  1620.

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Jupiter and its Galilean Moons
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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.

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Isaac Newton (1642-1727)

• Newtons 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.
• Newtons laws of motion and gravity (1687), the
  greatest achievement of classical physics,
  explained all of mechanics, including Keplers
  laws.

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Sir Isaac Newton

• Nature and Natures laws lay hid in the Night
• God said, Let Newton be! and all was Light
• Alexander Pope,1727

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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
  Earths 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.

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Newtons 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.

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Newtons Laws of Motion 1b
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Newtons 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 pitchers arm gives the
  ball its acceleration.
• Remember that an acceleration is a change of
  velocity.

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Centripetal acceleration

• An object moving at
• constant speed in a
• circle has a centripetal
• acceleration, given by
• ac v2/r.

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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.

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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
• Earths center.

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Newtons 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.

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Newtons Laws of Motion 3b
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Newtons Laws of Motion 3c
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Newtons 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 gravitational constant.

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Newtons Law of Gravitation 2

• Newtons realized that an apple falls to the
  ground for the same reason as the Moon orbits the
  Earth - gravity.
• Using his Laws of Motion and a little guesswork,
  Newton compared the centripetal acceleration of
  the Moon in orbit about the Earth with the
  downward acceleration (g 9.8 m/s2) of a
  falling object near the Earths surface.
• Gravity was the first of the fundamental forces
  to be described mathematically the others we
  know about are the electromagnetic force
  (nineteenth century) and the strong and weak
  nuclear forces (twentieth century).

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Matter and Energy 1

• Physics deals with energy and matter.
• Energy, comes in many forms, such as
• kinetic energy (KE), the energy of
  motion
• potential energy (PE) of many types,
  so-called
• because it can be converted to kinetic
  energy.
• radiative energy, carried by EM waves.
• Matter is simply material, characterized by its
  mass. Einsteins famous equation, E mc2,
  indicates that matter is just a form of energy.

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Matter and Energy 2

• The KE of a moving object is ½ mv2.
• The thermal energy of a gas is the total KE of
  its molecules.
• The temperature of a gas (in K) is proportional
  to the average KE of its molecules.
• Heat is the energy transferred from one object to
  another because of a difference in temperatures.
• Potential energy has many forms, such as
  gravitational, elastic, electrical, chemical.
• Energy units are J (joules) and eV
  (electron-volts).

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Conservation of Energy 1

• The Law of Conservation of Energy states that,
  although the form of energy may change, the total
  quantity of energy remains constant.
• Example 1 When you drop a rock , its
  gravitational potential energy is converted to
  kinetic energy. When the rock hits the ground,
  its kinetic energy is transferred largely to
  thermal energy in the rock and ground.
• Example 2 When a positron meets its
  antiparticle, an electron, the two particles
  annihilate each other, converting their
  mass-energy to electromagnetic energy in the form
  of gamma rays.

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Conservation of Energy 2 Potential energy lost
Kinetic energy gained
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Conservation of Energy 3
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Astronomical Triumphs of Newtons Laws

• Newtons Laws not only explained the motions of
  the planets, given by Keplers Laws, but also the
  motions of moons and comets
• Newtons Laws indicated that an unknown
  planet was affecting the orbit of Uranus.
  Astronomers searched, and found Neptune.

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Limitations of Newtons Laws

• The limitations of Newtons theories became
  apparent in the twentieth century, when they were
  superseded by the revolutionary new theories of
• i. Special Relativity for objects traveling at
  very
• high speeds
• ii. Quantum Mechanics for the smallest
  particles
• iii. General Relativity for the behaviour of
  large
• masses.
• Newtons theories are still used widely, for
  example, in structural design and aerospace
  engineering.

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On Newtons Law of Gravitation 1

• the most impressive fact is that gravity is
  simple. It is simple to state the principles
  completely and not have left any vagueness for
  anybody to change the ideas of the law. It is
  simple, and therefore it is beautiful. It is
  simple in its pattern. I do not mean it is
  simple in its action - to follow how all those
  stars in a globular cluster move is quite beyond
  our ability.
• Richard Feynman in the The Character of
  Physical Law, 1965.

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On Newtons Law of Gravitation 2

• Finally comes the universality of the
  gravitational law, that Newton, in his mind,
  worrying about the solar system, was able to
  predict what would happen in an experiment of
  Cavendish, where Cavendishs little model.of two
  balls attracting, has to be expanded ten million
  million times to become the solar system. Then
  ten million million times larger again we find
  galaxies attracting each other by exactly the
  same law.
• Richard Feynman in the The Character of
  Physical Law, 1965.