Physics 23 Jonathan Dowling Isaac Newton (642 727) Physics 23 Lecture 03: FRI 29 AUG CH3: Gravitation III Version: 8/28/4 Michael Faraday (79 867)
3.7: Planets and Satellites: Kepler s st Law. THE LAW OF ORBITS: All planets move in elliptical orbits, with the Sun at one focus. Laws were based on data fits! Tycho Brahe 546 60 Johannes Kepler 57 630
3.7: Planets and Satellites: Kepler s 2 nd Law 2. THE LAW OF AREAS: A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet s orbit in equal time intervals; that is, the rate da/dt at which it sweeps out area A is constant. Angular momentum, L: A t
3.7: Planets and Satellites: Kepler s 3 rd Law 3. THE LAW OF PERIODS: The square of the period of any planet is proportional to the cube of the semi-major axis of its orbit. Consider a circular orbit with radius r (the radius of a circle is equivalent to the semimajor axis of an ellipse). Applying Newton s second law to the orbiting planet yields Using the relation of the angular velocity, ω, and the period, T, one gets: T = 2π ω
3.7: Planets and Satellites: Kepler s 3 rd Law ICPP: 2 (a) The larger the orbit the longer the period: SAT-2. T = 4π 2 r 3 GM r 3/2 (b) The smaller the orbit the greater the speed: SAT-. v = ωr = 2π T = GM r r r LEO = R Earth + a LEO = 0 7 m r GEO = R Earth + a GEO = 4.22 0 7 m T LEO = 4π2 kg s 2 6.67 0 m 3 5.97 0 24 kg (0 7 m) 3 = 6307s T GEO = 4π2 kg s 2 6.67 0 m 3 5.97 0 24 kg (4.22 0 7 m) 3 = 8.62 0 4 s = 24hrs v LEO = 6.67 0 m 3 kg s 2 5.97 0 24 kg = 7.4km/s 0 7 m v GEO = 6.67 0 m 3 kg s 2 5.97 0 24 kg = 3km/s 4.22 0 7 m
3.7: Newton Derived Kepler s Laws from Inverse Square Law! http://galileo.phys.virginia.edu/classes/52.mfi.spring02/keplerslaws.htm Kepler s Second Law First: Equal Areas Proportional to Equal Time! Rate of sweeping out of area, da / dt = c Angular Momentum is proportional to the angular momentum L, and equal to L/2m = Constant = C. A t
3.7: Newton Derived Kepler s Laws from Inverse Square Law! http://galileo.phys.virginia.edu/classes/52.mfi.spring02/keplerslaws.htm Kepler s First Law: Ellipse with Sun at Focus This is equivalent to the standard (r, q ) equation of an ellipse of semi-major axis a and eccentricity e, with the origin the Sun at one focus. Note /L 2 is from inverse Square Law.
3.7: Newton Derived Kepler s Laws from Inverse Square Law! http://galileo.phys.virginia.edu/classes/52.mfi.spring02/keplerslaws.htm Kepler s 3 rd Law: For Ellipse T 2 a 3
Example, Halley s Comet ICPP: Estimate comet s speed at farthest distance? v = v = ωr ω = 2π / T v = 2πr T 6 9 00 m 76y 3 0 7 s/y 02 m 0 9 s,000m/s
3.8: Satellites: Orbits and Energy As a satellite orbits Earth in an elliptical path, the mechanical energy E of the satellite remains constant. Assume that the satellite s mass is so much smaller than Earth s mass. The potential energy of the system is given by For a satellite in a circular orbit, Thus, one gets: For an elliptical orbit (semimajor axis a),
ICPP E = GMm 2r de + r 2 dr r T = 4π 2 GM r 3 +r 3 dt +r 2 dr (a) path : As E decreases (de < 0); r decreases (dr < 0) (b) Less: As r decreases (dr < 0); T decreases (dt < 0)
NASA Gravity Recovery and Climate Experiment What Do the Two Satellites Measure? Changing g field! http://www.jpl.nasa.gov/missions/gravity-recovery-and-climate-experiment-grace/ Earth is NOT a Uniform Sphere > Gravitational Field Changes in Orbit. Rocky Mtn. High ΔM mid-atl. Low g = GM r 2 + r 2 v = GM r +r 2 dg r 3 dr As g increases (dg > 0); r decreases (dr < 0). dv r 3 2 dr As r decreases (dr < 0); v increases (dv > 0). Changing field Δg give rise to changing velocity Δv. Changing Δv gives changing satellite-to-sattellite distance. Microwave link measures changing distance between satellites. Measuring Δg allows computation of ΔM Earth s Mass Distribution.
Example, Mechanical Energy of a Bowling Ball
3.9: Einstein and Gravitation: Curvature of Space
3.9: Einstein and Gravitation: Gravity Waves Two Orbiting Black Holes LIVINGSTON LASER INTERFEROMETER GRAVITATIONAL-WAVE OBSERVATORY Disturbances in the Gravitational Field Move Outward As Waves
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