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Sect. 6-5: Kepler’s Laws & Newton’s Synthesis. Johannes Kepler. German astronomer (1571 – 1630) Spent most of his career tediously analyzing huge amounts of observational data (most compiled by Tycho Brahe) on planetary motion (orbit periods, orbit radii, etc.)

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## Sect. 6-5: Kepler’s Laws & Newton’s Synthesis

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**Johannes Kepler**• German astronomer (1571 – 1630) • Spent most of his career tediously analyzing huge amounts of observational data (most compiled by Tycho Brahe) on planetary motion (orbit periods, orbit radii, etc.) • He used his analysis to develop “Laws” of Planetary Motion. “Laws” in the sense that they agree with observation, butnot true theoretical laws, such asNewton’s Laws of Motion & Newton’s Universal Law of Gravitation.**Kepler’s “Laws”**Kepler’s “Laws” are consistent with & are obtainable from Newton’s Laws • Kepler’s First Law All planets move in elliptical orbits with the Sun at one focus • Kepler’s Second Law The radius vector drawn from the Sun to a plane sweeps out equal areas in equal time intervals • Kepler’s Third Law The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the elliptical orbit**Math Review: Ellipses**• The points F1 & F2 are each a focus of the ellipse • Located a distance c from the center • Sum of r1and r2 is constant • Longest distance through center is the major axis,2a a is called the semimajor axis • Shortest distance through center is the minor axis,2b b is called the semiminor axis Typical Ellipse • Theeccentricity is defined as e = (c/a) • For a circle, e = 0 • The range of values of the eccentricity for ellipses is 0 < e < 1 • The higher the value of e, the longer and thinner the ellipse**Ellipses & Planet Orbits**• The Sun is at one focus • Nothing is located at the other focus • Aphelionis the point farthest away from the Sun • The distance for aphelion is a + c For an orbit around the Earth, this point is called the apogee • Perihelionis the point nearest the Sun • The distance for perihelion is a – c For an orbit around the Earth, this point is called the perigee**Kepler’s 1st Law**All planets move in elliptical orbits with the Sun at one focus • A circular orbit is a special case of an elliptical orbit The eccentricity of a circle is e = 0. • Kepler’s 1st Law can be shown(& was by Newton)to be a direct result of the inverse square nature of the gravitational force. Comes out of N’s 2nd Law + N’s Gravitation Law + Calculus • Elliptic (and circular) orbits are allowed for bound objects • A bound object repeatedly orbits the center • An unbound object would pass by and not return • These objects could have paths that are parabolas (e = 1) and hyperbolas (e > 1)**Kepler’s 1st LawEach planet’s orbit is an ellipse, with**the Sun at one focus.**Orbit Examples**• Fig. (a):Mercury’sorbit has the largest eccentricity of the planets. eMercury = 0.21 Note:Pluto’s eccentricity is ePluto = 0.25, but, as of 2006, it is officially no longer classified as a planet! • Fig. (b):Halley’s Comet’sorbit has high eccentricity eHalley’s comet = 0.97 • Remember that nothing physical is located at the second focus • The small dot**Kepler’s 2nd Law**The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals • Kepler’s 2nd Law can be shown (& was by Newton) to be a direct result of the fact that N’s Gravitation Law gives Conservation of Angular Momentum for each planet. • The Gravitational force produces no Torque (it is to the motion) so that Angular Momentum is conserved. (Neither torque nor angular momentum have been discussed yet.)**Kepler’s 2nd Law**An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.**Kepler’s 2nd Law**• Geometrically, in a time dt, the radius vector r sweeps out the area dA = half the area of the parallelogram • The displacement is dr = v dt • Mathematically, this means • That is: the radius vector from the Sun to any planet sweeps out equal areas in equal times**If the orbit is circular & of radius r, this follows from**Newton’s Universal Gravitation. This gravitational force supplies a centripetal force for user in Newton’s 2nd Law Ks is a constant Kepler’s 3rd Law The square of the orbital period T of any planet is proportional to the cube of the semimajor axis a of the elliptical orbit Ksis a constant, which is the same for all planets.

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