Gravitation and the Waltz of the Planets

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Gravitation and the Waltz of the Planets

• Chapter Four

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Ancient astronomers invented geocentric modelsto
explain planetary motions

• Like the Sun and Moon, the planets move on the
  celestial sphere with respect to the background
  of stars
• Most of the time a planet moves eastward in
  direct (prograde) motion, in the same direction
  as the Sun and the Moon, but from time to time it
  moves westward in retrograde motion

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• Ancient astronomers believed the Earth to be at
  the center of the universe
• They invented a complex system of epicycles and
  deferents to explain the direct and retrograde
  motions of the planets on the celestial sphere

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Nicolaus Copernicus devised the first
comprehensive heliocentric model

• Copernicuss heliocentric (Sun-centered) theory
  simplified the general explanation of planetary
  motions
• In a heliocentric system, the Earth is one of the
  planets orbiting the Sun
• The sidereal period of a planet, its true orbital
  period, is measured with respect to the stars

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A planet undergoes retrograde motion as seen from
Earth when the Earth and the planet pass each
other
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A planets synodic period is measured with
respect to the Earth and the Sun (for example,
from one opposition to the next)
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Sidereal and Synodic Orbital periods

• For Inferior Planets
• 1/P 1/E 1/S
• For Superior Planets
• 1/P 1/E 1/S
• P Sidereal Period of the planet
• S Synodic Period of planet
• E Earths Sidereal Period (1 year)

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Tycho Brahes astronomical observations disproved
ancient ideas about the heavens
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Parallax Shift
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Johannes Kepler proposed elliptical pathsfor the
planets about the Sun

• Using data collected by Tycho Brahe, Kepler
  deduced three laws of planetary motion
• the orbits are ellipses
• a planets speed varies as it moves around its
  elliptical orbit
• the orbital period of a planet is related to the
  size of its orbit

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Keplers First Law
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Keplers Second Law
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Keplers Third Law

• P2 a3
• P planets sidereal period, in years
• a planets semimajor axis, in AU

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Ellipse Relations

• An ellipse is a conic section whose eccentricity,
  e, is 0 e lt 1. The circle is an ellipse with e
  0.
• The relation between the semi-major (a) and
  semi-minor (b) axes is
• b2 a2(1 - e2).
• The point in the orbit where the planet is
  closest to the Sun is the perihelion and the
  associated perihelion distance,
• dp a(1 - e)
• The aphelion is the point in the orbit furthest
  from the Sun and the aphelion distance,
• da a(1 e).

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Galileos discoveries with a telescope
stronglysupported a heliocentric model

• The invention of the telescope led Galileo to new
  discoveries that supported a heliocentric model
• These included his observations of the phases of
  Venus and of the motions of four moons around
  Jupiter

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• One of Galileos most important discoveries with
  the telescope was that Venus exhibits phases like
  those of the Moon
• Galileo also noticed that the apparent size of
  Venus as seen through his telescope was related
  to the planets phase
• Venus appears small at gibbous phase and largest
  at crescent phase

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• There is a correlation between the phases of
  Venus and the planets angular distance from the
  Sun

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Geocentric

• To explain why Venus is never seen very far from
  the Sun, the Ptolemaic model had to assume that
  the deferents of Venus and of the Sun move
  together in lockstep, with the epicycle of Venus
  centered on a straight line between the Earth and
  the Sun
• In this model, Venus was never on the opposite
  side of the Sun from the Earth, and so it could
  never have shown the gibbous phases that Galileo
  observed

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• In 1610 Galileo discovered four moons, now called
  the Galilean satellites, orbiting Jupiter

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Isaac Newton formulated three laws that
describefundamental properties of physical
reality

• Isaac Newton developed three principles, called
  the laws of motion, that apply to the motions of
  objects on Earth as well as in space
• These are
• the law of inertia a body remains at rest, or
  moves in a straight line at a constant speed,
  unless acted upon by a net outside force
• F m x a (the force on an object is directly
  proportional to its mass and acceleration)
• the principle of action and reaction whenever
  one body exerts a force on a second body, the
  second body exerts an equal and opposite force on
  the first body

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Newtons Law of Universal Gravitation

• F gravitational force between two objects
• m1 mass of first object
• m2 mass of second object
• r distance between objects
• G universal constant of gravitation
• If the masses are measured in kilograms and the
  distance between them in meters, then the force
  is measured in newtons
• Laboratory experiments have yielded a value for G
  of
• G 6.67 1011 newton m2/kg2

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Newtons description of gravity accounts for
Keplerslaws and explains the motions of the
planets and other orbiting bodies
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Mass vs Weight

• Mass is an intrinsic quantity and for a given
  object is invariant of position. It is measured
  in kg.
• Weight by contrast is the response of mass to
  the local gravitational field. It is a force and
  measured in Newtons
• Thus while you would have the same mass on the
  earth and its Moon, your weight is different.
• W(eight) m(ass) x g(ravitational acceleration)

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Orbits

• The law of universal gravitation accounts for
  planets not falling into the Sun nor the Moon
  crashing into the Earth
• Paths A, B, and C do not have enough horizontal
  velocity to escape Earths surface whereas Paths
  D, E, and F do.
• Path E is where the horizontal velocity is
  exactly what is needed so its orbit matches the
  circular curve of the Earth

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Orbits may be any of a family of curves called
conic sections
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Energy

• Kinetic energy refers to the energy a body of
  mass m1 has due to its speed v Ek ½ m1
  v2 (where energy is measured in Joules, J).
• Potential energy is energy due to the position of
  m1 a distance r away from another body of mass
  m2, Ep -G m1 m2 / r.
• The total energy, E, is a sum of the kinetic plus
  potential energies E Ek Ep.
• A body whose total energy is lt 0, orbits a more
  massive body in a bound, elliptical orbit (e lt
  1).
• A body whose total energy is gt 0, is in an
  unbound, hyperbolic orbit (e gt 1) and escapes to
  infinity.
• A body whose total energy is exactly 0 just
  escapes to infinity in a parabolic orbit (e 1)
  with zero velocity.

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Escape velocity

• The velocity that must be acquired by a body to
  just escape, i.e., to have zero total energy, is
  called the escape velocity. By setting Ek Ep
  0, we find
• v2escape 2 G m2 / r

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Velocity

• A body of mass m1 in a circular orbit about a
  (much) more massive body of mass m2 orbits at a
  constant speed or the circular velocity, vc where
  
• v2c G m2 / r
• (This is derived by equating the gravitational
  force with the centripetal force, m1 v2 / r ).
• Note that v2escape is 2 v2c .

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Keplers Third Law a la Newton

• P2 (4 x p2 x a3)/(G x (m1 m2))
• P Sidereal orbital period (seconds)
• A Semi-major axis planet orbit (kilometers)
• m mass of objects (planets, etc. kilograms)
• G Gravitational constant
• 6.673 x 10-11 N-m2/kg2

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Gravitational forces between two objectsproduce
tides
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The Origin of Tidal Forces
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