2Kepler and the Physics of Planetary Motion Laws of Planetary MotionLaw 1 - Law of EllipsesLaw 2 - Law of Equal AreasLaw 3 - Harmonic Law (r3/T2 = C)Kepler’s laws provide a concise and simple description of the motions of the planets
12Kepler’s Third LawThe ratio of the average distance* from the Sun cubed to the period squared is the same constant value for all planets.* Semimajor axisr3 = CT2
13Summarizing Kepler’s Laws Kepler's First Law:Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal timesKepler's Third Law: The squares of the periods of the planets are proportional to the cubes of their semi-major axes or:r3 = CT2
145. 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
15Newton’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.
16Orbital PathsExtending Kepler’s Law #1, Newton found that ellipses were not the only orbital paths.possible orbital pathsellipse (bound)parabola (unbound)hyperbola (unbound)
17Newton’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.
18Newton’s Universal Law of Gravitation G=6.67 x m3/(kg s2)
19Newton’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:T2 = 42 r3 / G MIf 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.
20What 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.
21What 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.
22Combining 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.
24G and gGeosynchronous satellites orbit the Earth at an altitude of about 3.58 x 107 meters. Given that the Earth’s radius is 6.38 x 106 meters and its mass is x 1024 kg, what is the magnitude of the gravitational acceleration at the altitude of one of these satellites?
25Orbiting velocityThe 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 1024 kg and its radius is 6.38 x 106 meters.
26Gravity 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.
27Gravitational Potential Energy Gravitational PE is negative.PE increases as r decreases.Potential energy vs. separation distance
28Sample problem What is the minimum escape speed from Earth? KEat Earth’s surface = PEouter space½ mvesc2 = GmM/rvesc =
29Sample problem.Calculate the total energy of a satellite in circular orbit about Earth with a separation distance of r?
30The 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?