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Kepler’s Laws

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Kepler and the Physics of Planetary Motion Laws of Planetary Motion Law 1 - Law of Ellipses Law 2 - Law of Equal Areas Law 3 - Harmonic Law (r 3 /T 2 = C) Kepler’s laws provide a concise and simple description of the motions of the planets

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Kepler’s First Law

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The Law of Ellipses: The planets move in elliptical orbits with the Sun at one focus.

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Major Axis Minor Axis aphelion perihelion Focus Points Center 90° e=0 perfect circle e=1 flat line Semi-major Axis = ½ Major Axis The Ellipse

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perihelion aphelion Center Verifying Kepler’s 1st P1P1 L1L1 L2L2 P2P2 L3L3 L4L4 L 1 + L 2 = L 3 + L 4 ??

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Kepler’s Second Law As a planet orbits the Sun, it moves in such a way that a line drawn from the Sun to the planet sweeps out equal areas in equal time intervals.

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Lunar Orbit of Explorer 35 Moon Points represent satellite positions separated by equal time intervals. periluna apoluna

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Verifying Kepler’s 2 nd Equal area in equal time. Area = ½ base X height A2A2 A 1 = A 2 ?? A1A1 base height

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Kepler’s Third Law The ratio of the average distance* from the Sun cubed to the period squared is the same constant value for all planets. * Semimajor axis r 3 = C T2T2

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Summarizing Kepler’s Laws Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal times Kepler's First Law: Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus. Kepler's Third Law: The squares of the periods of the planets are proportional to the cubes of their semi-major axes or: r 3 = C T2T2

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5. Universal Laws of Motion “ If I have seen farther than others, it is because I have stood on the shoulders of giants.” Sir Isaac Newton (1642 – 1727) Physicist

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Newton’s Universal Law of Gravitation Isaac Newton discovered that it is gravity that plays the vital role of determining the motion of the planets - concept of action at a distance. Gravity is the force that results in centripetal acceleration of the planets.

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Orbital Paths Extending Kepler’s Law #1, Newton found that ellipses were not the only orbital paths. possible orbital paths ellipse (bound) parabola (unbound) hyperbola (unbound)

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Newton’s Universal Law of Gravitation Between every two objects there is an attractive force, the magnitude of which is directly proportional to the mass of each object and inversely proportional to the square of the distance between the centers of the objects.

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Newton’s Universal Law of Gravitation G=6.67 x m 3 /(kg s 2 )

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Newton’s Version of Kepler’s Third Law Using calculus, Newton was able to derive Kepler’s Third Law from his own Law of Gravity. In its most general form: T 2 = 4 2 r 3 / G M If you can measure the orbital period of two objects (T) and the distance between them (r), then you can calculate the mass of the central object, M.

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What have we learned? What is the universal law of gravitation? The force of gravity is directly proportional to the product of the objects’ masses and declines with the square of the distance between their centers (Inverse Square Law). What types of orbits are possible according to the law of gravitation? Objects may follow bound orbits in the shape of ellipses (or circles) and unbound orbits in the shape of parabolas or hyperbolas.

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What have we learned? How can we determine the mass of distant objects? Newton’s version of Kepler’s third law allows us to calculate the mass of a distant object if it is orbited by another object, and we can measure the orbital distance and period.

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Combining Newton’s and Kepler’s Laws, we can.... Determine the mass of an unknown planet. Determine the escape and orbiting velocities for a satelite. Determine the acceleration due to gravity on a planet. You should be able to derive equations for the above determinations.

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Derivations Escape velocity Orbiting velocity “g” Kepler’s “C”

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G and g Geosynchronous satellites orbit the Earth at an altitude of about 3.58 x 10 7 meters. Given that the Earth’s radius is 6.38 x 10 6 meters and its mass is 5.97 x kg, what is the magnitude of the gravitational acceleration at the altitude of one of these satellites?

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Orbiting velocity The International Space Station orbits the Earth at an average altitude of 362 kilometers. Assume that its orbit is circular, and calculate its orbital speed. The Earth’s mass is 5.97 x kg and its radius is 6.38 x 10 6 meters.

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Gravity is a source of energy Because gravity is a force, it can be associated with potential energy: Recall: Solving, the formula for gravitational PE is: The minus sign indicates that PE decreases as the masses get closer together.

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Gravitational Potential Energy Gravitational PE is negative. PE increases as r decreases. Potential energy vs. separation distance

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Sample problem What is the minimum escape speed from Earth? KE at Earth’s surface = PE outer space ½ mv esc 2 = GmM/r v esc =

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Sample problem. Calculate the total energy of a satellite in circular orbit about Earth with a separation distance of r?

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The change in a system’s energy equals work. How much work is required to move the satellite to an orbit with a separation distance of 2r?

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