Models of the Solar System

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Models of the Solar System

• Positions of planets change, whereas stars
  appear relatively fixed
• Greeks held on to the Geocentric model because
  they could not observe stars to change their
  positions, and therefore thought that the earth
  must be stationary
• Ptolemy, Aristotle and others refined the
  geocentric model
• But there were problems.such as the path
  reversal by Mars ? Retrograde motion

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Retrograde motion of Mars(path reversal seen in
the Sky)
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Epicycles Ptolemic Geocentric Model
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How do we know the Earth is spherical ?

• Stars differ from place to place
• Northern and southern hemispheres
• What kind of an object always has a round shadow ?

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Earth Shadow during Lunar Eclipse
Multiple Exposure Photograph
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Cyrene
Syene
Tropic of Cancer
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The Spherical Earth

• The ancient Greeks had deduced not only that the
  Earth is spherical but also measured its
  circumference !

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Eratostheness method to measure the
circumference of the earth
7º
At noon on summer solstice day the Sun is
directly overhead at Syene, but at an angle of 7o
at Alexandria

• Distance (Alexandria - Syene)
• -- ---------------------------------------
• 360 Circumference of the Earth

Sunlight
Alexandria
Answer 40,000 stadia 25,000 mi !
Syene
Earth
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Earth-Moon-Sun GeometryAristarchuss
determination of distances(Closer the S-E-M
angle to 90, the farther the Sun)
If we replace the moon with a planet, then can
determine relative distances, as done by
Copernicus
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Copernicus
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Copernican ModelInferior and Superior
Planets(orbits inside or outside the Earths
orbit)

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Configurations of Inferior Planets, Earth, and
the Sun
Earth
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Configurations of Superior Planets, Earth, and
the Sun
Opposition
Conjunction
Earth
Synodic (apparent) period one conjunction to
next (or one opposition to next)
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Synodic and Sidereal Orbital Periods

• Inferior planets are never at opposition
  superior planets can not be at inferior
  conjunction
• Copernican model of orbital periods
• Synodic period is the apparent orbital period of
  a planet, viewed from the earth, when the
  earth-planet-sun are in successive conjunction or
  opposition
• Sidereal (with respect to stars) period is the
  real orbital period around the Sun
• Synodic periods of outer planets (except Mars)
  are just over one year

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Apparent (Synodic) and true (Sidereal with
respect to stars) orbital periods of planets
differ due to Earths relative motion
Synodic periods of all outer planets (except
mars) are just over 1 year because their Sidereal
periods are very long and they are in opposition
again soon after an earth-year
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Earth-Venus-Sun
Inferior planets appear farthest away from the
Sun at greatest elongation
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Measurements of Distances to Planets
Angle of max elongation
P-E-S
P
90 deg
Earth
P-E-S
E
S
Sin (P-E-S) PS / ES ES 1 AU
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Copernicus first determined the relative
distances of planets
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Copernican Heliocentric Model(Retrograde motion
of Mars seen when Earth overtakes Mars
periodically)
Earth is closer to the Sun, therefore moves
faster than Mars
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Tycho The most accurate pre-telescopic observer
Tycho charted very accurately the movement of
Mars in the Sky, but still believed In the
Geocentric Universe
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Kepler Tychos assistant(used Tychos data to
derive Keplers Laws)
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Planetary Orbits

• The Copernican heliocentric model is essentially
  correct
• But it consisted of circular orbits which did not
  exactly fit observations of planetary positions
• Kepler realized, based on Tychos data of the
  orbit of Mars, that orbits are elliptical ?
  Keplers First Law
• However, the difference for Mars is tiny, to
  within the accuracy of drawing a circle with a
  thick pen !

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Keplers First LawAll planetary orbits are
elliptical, with the Sun at one focus
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Eccentricity ee distance between foci/major
axis AB / ab
a
A
B
b
A circle has e 0, and a straight line has e
1.0
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Keplers Second LawPlanetary radius sweeps
equal area triangles in equal time
It follows that the velocity of the planet must
vary according to distance from the Sun --
fastest at Perihelion and slowest at Aphelion
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Keplers Third Law P2 a3P Orbital Period, a
semi-major axis
What is the size a of the orbit of a comet with
the period P of 8 years?
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Keplers Laws

• Empirically derived from observational data
  largely from Tycho (e.g. observations of the
  positions of Mars in its orbit around the Sun)
• Theoretical explanation had to await Newtons
  discovery of the Law of Gravitation
• Universally valid for all gravitationally
  orbiting objects (e.g. stars around black holes
  before falling in)

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Galileo
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Galileos Discoveries With Telescope

• Phases of Venus
• - Venus displays phases like the Moon as it
  revolves around the Sun
• Mountains and seas on the Moon
• - Other objects in the sky are like the
  Earth (not therefore special)
• Milky Way is made of stars like the Sun
• Sunspots
• - Imperfections or blemishes in
  otherwise perfect heavenly objects
• 4 Galilean satellites of Jupiter
• - Objects in the sky revolve around other
  objects, not the Earth (i.e. other moons)
• All of these supported the Copernican System
• Galileo also conducted experiments on
  gravity
• Regardless of mass or weight objects fall at
  the same rate

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Phases of Venus
Venus is never too far from the Sun, therefore
can not be in opposition like the Moon. Changing
phases of Venus demonstrate that it orbits the
Sun like the Earth.
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Orbits and Motions

• Orbits can not be circular since objects do NOT
  revolve around each other, but around their
  common center-of-mass
• The Earth and the Moon both revolve around each
  other
• This motion is in addition to Earths Rotation,
  Revolution, Precession

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The Earth-Moon Barycenter

• The earth and the moon both revolve around a
  common center of mass called the Barycenter
• The barycenter of Sun-planet systems lies inside
  the Sun
• As the earth is much more massive, the barycenter
  lies 1700 Km inside the earth
• Calculate its position O from
• M(E) x EO M (M) x MO

M
E
O
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Gravity

• Galileos observations on gravity led to Newtons
  Law of Gravitation and the three Laws of Motion
• Objects fall at the same rate regardless of mass
  because more massive objects have more inertia or
  resistance to motion
• Fgrav G (m1 x m2) / r2
• Force of gravity between two masses is
  proportional to the product of masses divided by
  distance squared ? inverse square law

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Newton Three Laws of Motion

1. Inertia
2. F ma
3. Action Reaction

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Newtons Laws of Motion

• Law of Inertia A body continues in state of
  rest or motion unless acted on by an external
  force Mass is a measure of inertia
• Law of Acceleration For a given mass m, the
  acceleration is proportional to the force applied
• F m a
• Law of Action equals Reaction For every action
  there is an equal and opposite reaction momemtum
  (mass x velocity) is conserved

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Velocity, Speed, Acceleration

• Velocity implies both speed and direction speed
  may be constant but direction could be changing,
  and hence accelerating
• Acceleration implies change in speed or
  direction or both
• For example, stone on a string being whirled
  around at constant speed direction is constantly
  changing therefore requires force

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Ball Swung around on a String
Same Speed, (in uniform circular motion) Changing
Direction (swinging around the circle)
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Donut Swung around on a String
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Conservation of momemtumaction equal reaction

• The momemtum (mv) is conserved before and after
  an event
• Rocket and ignited gases
• M(rocket) x V(rocket) m(gases) x v(gases)
• Two billiard balls
• m1 v1 m2 v2 m1 v1 m2 v2
• v1,v2 velocities before collision
• v1,v2 velocities after collision
• Example you and your friend (twice as heavy)
  on ice!

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Action Reaction
Equal and Opposite Force from the Table
Net Force is Zero, No Net Motion
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Acceleration due to gravity

• Acceleration is rate of change of velocity,
  speed or direction of motion, with time ? a v/t
• Acceleration due to Earths gravity a ? g
• g 9.8 m per second per second, or 32 ft/sec2
• Speed in free-fall
• T (sec) v (m/sec) v
  (ft/sec)
• 0 0
  0
• 1 9.8
  32
• 2 19.6
  64
• 3 29.4
  96
• 60 mi/hr 88 ft/sec (between 2 and 3
  seconds)

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Galileos experiment revisited

• What is your weight and mass ?
• Weight W is the force of gravity acting on a
  mass m causing acceleration g
• Using F m a, and the Law of Gravitation
• W m g G (m MEarth) /R2
• (R Radius of the Earth)
• The mass m of the falling object cancels out
  and does not matter therefore all objects fall
  at the same rate or acceleration
• g GM / R2
• i.e. constant acceleration due to gravity
  9.8 m/sec2

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Galileos experiment on gravity

• Galileo surmised that time differences between
  freely falling objects may be too small for human
  eye to discern
• Therefore he used inclined planes to slow down
  the acceleration due to gravity and monitor the
  time more accurately

v
Changing the angle of the incline changes the
velocity v
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g on the Moon

• g(Moon) G M(Moon) / R(Moon)2
• G 6.67 x 10-11 newton-meter2/kg2
• M(Moon) 7.349 x 1022 Kg
• R(Moon) 1738 Km
• g (Moon) 1.62 m/sec/sec
• About 1/6 of g(Earth) objects on the Moon
  fall at a rate six times slower than on the Earth

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Escape Velocity and Energy

• To escape earths gravity an object must have
  (kinetic) energy equal to the gravitational
  (potential) energy of the earth
• Kinetic energy due to motion
• K.E. ½ m v2
• Potential energy due to position and force
• P.E. G m M(Earth) / R
• (note the similarity with the Law of
  Gravitation)
• Minimum energy needed for escape K.E. P.E.
• ½ m v2 G m M / R
• Note that the mass m cancels out, and
• v (esc) 11 km/sec 7 mi/sec 25000 mi/hr
• The escape velocity is the same for all
  objects of mass m

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Object in orbit ? Continuous fall !
Object falls towards the earth at the same rate
as the earth curves away from it
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Angular Momentum
Conservation of angular momentum says that
product of radius r and momentum mv must be
constant ? radius times rotation rate (number of
rotations per second) is constant
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Angular Momentum

• All rotating objects have angular momentum
• L mvr acts perpendicular to the plane of
  rotation
• Examples helicopter rotor, ice skater, spinning
  top or wheel (experiment)
• Gyroscope (to stabilize spacecrafts) is basically
  a spinning wheel whose axis maintains its
  direction slow precession like the Earths axis
  along the Circle of Precession

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Conservation of Angular Momentum

• Very important in physical phenomena observed in
  daily life as well as throughout the Universe.
  For example,
• Varying speeds of planets in elliptical orbits
  around a star
• Jets of extremely high velocity particles, as
  matter spirals into an accretion disc and falls
  into a black hole

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Relativistic1 Jet From Black Hole
1. Relativistic velocities are close to the
speed of light
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