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Gravitation and the Waltz of the Planets

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In a heliocentric system, the Earth is one of the planets orbiting the Sun ... Using data collected by Tycho Brahe, Kepler deduced three laws of planetary motion: ... – PowerPoint PPT presentation

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Title: Gravitation and the Waltz of the Planets


1
Gravitation and the Waltz of the Planets
  • Chapter Four

2
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

3
  • 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

8
A planet undergoes retrograde motion as seen from
Earth when the Earth and the planet pass each
other
9
A planets synodic period is measured with
respect to the Earth and the Sun (for example,
from one opposition to the next)
10
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
13
Parallax Shift
14
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

15
Keplers First Law
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Keplers Second Law
18
Keplers Third Law
  • P2 a3
  • P planets sidereal period, in years
  • a planets semimajor axis, in AU

19
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).

20
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

21
  • 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

22
  • There is a correlation between the phases of
    Venus and the planets angular distance from the
    Sun

23
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

24
  • 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

27
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

28
Newtons description of gravity accounts for
Keplerslaws and explains the motions of the
planets and other orbiting bodies
29
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)

30
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

31
Orbits may be any of a family of curves called
conic sections
32
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.

33
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

34
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 .

35
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

36
Gravitational forces between two objectsproduce
tides
37
The Origin of Tidal Forces
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