Solar System, Kepler and Solar System, Kepler and Universal GravitationUniversal Gravitation

Physics 12 AdvPhysics 12 Adv

Ptolemaic SystemPtolemaic 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

Copernican SystemCopernican 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

Tychonic SystemTychonic 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

Astronomy as ScienceAstronomy 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

Kepler’s LawsKepler’s Laws

• Brahe invited Kepler to be one of his assistants in 1600 which gave Kepler access to Brahe’s detailed records

• Kepler was able to develop three empirical relationships to describe heavenly bodies

• Today, these relationships are known as Kepler’s Laws

11stst Law – Planets move in elliptical Law – Planets move in elliptical orbits with the Sun at one Focusorbits with the Sun at one Focus

22ndnd Law – A Planet will sweep out Law – A Planet will sweep out an equal area in equal time an equal area in equal time

intervalsintervals

33rdrd Law – The ratio of the radius Law – The ratio of the radius cubed to the period squared will be cubed to the period squared will be

the same for any two objects the same for any two objects orbiting the same objectorbiting the same object

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Kepler, Halley and NewtonKepler, 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 couldn’t make it predict orbits

• He approached Newton about how to apply this concept

Newton’s Law of Universal Newton’s Law of Universal GravitationGravitation

• 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)

Newton’s Law of Universal Newton’s Law of Universal GravitationGravitation

• 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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Newton’s Universal Law of Newton’s Universal Law of Gravitation and Kepler’s First LawGravitation and Kepler’s 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

Newton’s Universal Law of Gravitation Newton’s Universal Law of Gravitation and Kepler’s Second Lawand Kepler’s 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:– Sun’s Mass – 1.99x1030kg– Earth’s Mass – 5.98x1024kg– Distance Earth to Sun (aphelion) – 152,171,522 km – Distance Earth to Sun (perihelion) – 147,166,462

km

Newton’s Universal Law of Newton’s Universal Law of Gravitation and Kepler’s Second Gravitation and Kepler’s Second

LawLaw• Earth’s speed:

– Aphelion – 2.95x104m/s– Perihelion – 3.00x104m/s

• This would indicate that Kepler’s Second Law is also supported by Newton’s Universal Law of Gravitation

Newton’s Universal Law of Newton’s Universal Law of Gravitation and Kepler’s Third LawGravitation and Kepler’s Third Law

• Since Kepler’s Third Law is a ratio of the orbtial radius cubed to the orbital period squared, we should be able to apply Newton’s Universal Law of Gravitation to the planetary motion to determine the value of the constant

• Set the centripetal force equation equal to Newton’s Universal Law of Gravitation

• Replace the velocity expression using orbital radius and orbital period

Newton’s Universal Law of Newton’s Universal Law of Gravitation and Kepler’s Third LawGravitation and Kepler’s Third Law

• This is called Newton’s version of Kepler’s Third Law

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Mass of the Sun and PlanetsMass of the Sun and Planets

• Henry Cavendish developed a torsion balance to determine the value of the Universal Gravitation Constant (approximately 70 years after Newton’s 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

Practice ProblemsPractice Problems

• Page 580– Questions 1-8

• Page 586– Questions 9-14