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Line 179: − = Kepler's Laws = − == Concept flow == − − *The key concept to understand here is that gravitational forces play an important role in planetary motion. − − *Three laws of planetary motion that describe the motion of the planets have been postulated based on detailed astronomical observations − − == Laws of Planetary Motion == − − We now know that satellites are continually falling towards the Earth following a curved path whose curvature is greater than that of the curvature of the Earth. The Moon is just such a satellite that moves around the Earth. In a similar way, all the planets that move around the Sun are satellites of the Sun. The motion described in such a situation is not strictly circular - it is elliptical. − − Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky. − − 1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.<br><br> − − 2. The Law of Areas: Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time.<br><br> − − 3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.<br><br> − − Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.<br><br> − === The Law of Orbits === − − All planets move in elliptical orbits, with the sun at one focus. An ellipse is a closed curve such that the sum of the distances from any point P on the curve to two fixed points (called the foci, F1 and F2) remains constant. <br><br> − '''Orbit eccentricity''' − − The semi major axis of the ellipse is a and represents the planet's average distance from the Sun. The eccentricity, “e” is defined so that “ea” is the distance from the centre to either focus. A circle is a special case of an ellipse where the two foci coincide. The Earth and most of the other planets have nearly circular orbits. For Earth, “e” = 0.017.<br><br> − − − − [[Image:Gravitation%20for%20wiki_html_m75870f64.gif]] <br> − − === The Law of Equal Areas === − − Kepler's second law states that each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal periods of time. − − This can be shown to be true using the law of conservation of angular momentum. − − [[Image:Gravitation%20for%20wiki_html_m4a5ede30.png]] <br> − − If “v” is the velocity of the planet, in time “dt” the planet moves a distance vdt and sweeps out an area equal to the area of a triangle of base “r” and altitude vdt sinα. − − Hence dA = ½ (r) (“v” x “dt” x sinα) − − dA/ dt = ½ rv sinα − − The magnitude of the angular momentum of the planet about the Sun is L = mvr sinα. − − dA/ dt = (½)L/m − − Because the angular momentum is conserved, the rate of change of area covered is constant. This means that the planets move with different velocities depending upon their position in the orbits.<br><br> − − === The Law of Periods === − − The ratio of the squares of the periods of any two planets revolving about the Sun is equal to the ratio of the cubes of their semi-major axes. − − Can you derive this? − − [[Image:Gravitation%20for%20wiki_html_m290004f7.gif]] <br> − − G m1 Ms / r12 = m1 (v12)/ r1 − − v1 = 2πr1/T1 − − Substituting and rearranging we get − − T12/ r13 = 4π2 / G Ms − − Deriving this for another planet, we can arrive at the third law.<br><br>
Significance of Gravitational Force (view source)
Revision as of 11:17, 4 December 2013
, 11:17, 4 December 2013→Kepler's Laws
[http://phet.colorado.edu/sims/gravity-and-orbits/gravity-and-orbits_en.jnlp Click to run PhET simulation]
[http://phet.colorado.edu/sims/gravity-and-orbits/gravity-and-orbits_en.jnlp Click to run PhET simulation]
5. www.hyperphysics.com - From Classical Mechanics to General Relativity - This is a good description of the geometry of Newtonian gravity and how to move from classical mechanics to relativity.
5. www.hyperphysics.com - From Classical Mechanics to General Relativity - This is a good description of the geometry of Newtonian gravity and how to move from classical mechanics to relativity.