Solar System, Kepler and Universal Gravitation

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Solar System, Kepler and Universal Gravitation

• Physics 12 Adv

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Ptolemaic System

• Geocentric
• Very complicated as sun and moon orbited Earth
  but other planets both orbited the Earth and
  completed a Epicycle in their orbital path

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Copernican System

• This system, proposed by Nicholas Copernicus in
  1543, simplified the mathematics as the Sun
  became the centre of the Solar System
• This was rejected by the clergy and is most
  famous as a result of the persecution of Galileo
  Galilei for supporting this system

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Tychonic System

• An intermediate system where the moon and Sun
  orbit the Earth but other planets orbit the Sun
• This system never gained widespread acceptance
  but Tycho Brahe was responsible for contributing
  a significant amount of detailed information
  regarding the Solar System

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Astronomy as Science

• Before the advent of modern telescopes, Tycho
  Brahe was able to develop instruments that were
  precise to 1/30 of a degree without magnification
• As a result, he was able to accurately catalogue
  over 700 stars was well as detailed information
  about our Solar System
• Astronomy continued to evolve as a science as the
  ability to machine high quality lenses was refined

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Keplers Laws

• Brahe invited Kepler to be one of his assistants
  in 1600 which gave Kepler access to Brahes
  detailed records
• Kepler was able to develop three empirical
  relationships to describe heavenly bodies
• Today, these relationships are known as Keplers
  Laws

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1st Law Planets move in elliptical orbits with
the Sun at one Focus
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2nd Law A Planet will sweep out an equal area
in equal time intervals
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3rd Law The ratio of the radius cubed to the
period squared will be the same for any two
objects orbiting the same object
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Kepler, Halley and Newton

• When Kepler published his equations, they were
  not based on an understanding of why the universe
  behaved in this manner, rather it simply
  described how it behaved
• Sir Edmond Halley had described a relationship
  between gravity and the square of the distance
  between objects but couldnt make it predict
  orbits
• He approached Newton about how to apply this
  concept

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Newtons Law of Universal Gravitation

• Newton immediately answered Halley with the fact
  that orbits must be elliptical despite the fact
  that this answer was purely intuitive, it led to
  an article called De Motu (on motion)
• Newton later expanded this into one of the most
  famous works in scientific literature,
  Philosophiae Naturalis Principia Mathematica
  (often just called Principia)

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Newtons Law of Universal Gravitation

• Fg is the gravitational force
• G is the Universal Gravitational Constant
• m1 and m2 are the masses of the objects
• r is the distance between the objects
• Note, Newton did not measure G but it is now
  known to be
• 6.67x10-11Nm2/kg2

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Newtons Universal Law of Gravitation and
Keplers First Law

• Newton had shown in his original article De Motu
  and later in Principia that the inverse square
  nature of gravity would lead to elliptical not
  circular orbits

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Newtons Universal Law of Gravitation and
Keplers Second Law

• Even though planets are moving in elliptical
  orbits, the concepts of circular motion mostly
  apply
• Determine the speed of the Earth using the
  following data and assume that centripetal force
  is equal to gravitational force
• Suns Mass 1.99x1030kg
• Earths Mass 5.98x1024kg
• Distance Earth to Sun (aphelion) 152,171,522 km
  
• Distance Earth to Sun (perihelion) 147,166,462
  km

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Newtons Universal Law of Gravitation and
Keplers Second Law

• Earths speed
• Aphelion 2.95x104m/s
• Perihelion 3.00x104m/s
• This would indicate that Keplers Second Law is
  also supported by Newtons Universal Law of
  Gravitation

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Newtons Universal Law of Gravitation and
Keplers Third Law

• Since Keplers Third Law is a ratio of the
  orbtial radius cubed to the orbital period
  squared, we should be able to apply Newtons
  Universal Law of Gravitation to the planetary
  motion to determine the value of the constant
• Set the centripetal force equation equal to
  Newtons Universal Law of Gravitation
• Replace the velocity expression using orbital
  radius and orbital period

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Newtons Universal Law of Gravitation and
Keplers Third Law

• This is called Newtons version of Keplers Third
  Law

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Mass of the Sun and Planets

• Henry Cavendish developed a torsion balance to
  determine the value of the Universal Gravitation
  Constant (approximately 70 years after Newtons
  death)
• Using his apparatus, he was able to determine a
  value of G 6.75x10-11Nm2/kg2 which is within 1
  of the currently accepted answer
• Using G, he was then able to determine the mass
  of the Sun and other planets

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Practice Problems

• Page 580
• Questions 1-8
• Page 586
• Questions 9-14