DERIVING KEPLER’SLAWS OF PLANETARYMOTIONBy: Emily Davis
WHO IS JOHANNES KEPLER? German mathematician, physicist,and astronomer Worked under Tycho Brahe Observation alone Founder of celestial mechanics
WHAT ABOUT ISAAC NEWTON? “If I have seen further it is bystanding on the shoulders ofGiants.”Laws of Motion Universal Gravitation Explained Kepler’s laws The laws could be explainedmathematically if his laws ofmotion and universal gravitationwere true.Developed calculus
KEPLER’S LAWS OF PLANETARYMOTION1.Planets move around the Sun in ellipses,with the Sun at one focus.2.The line connecting the Sun to a planetsweeps equal areas in equal times.3.The square of the orbital period of aplanet is proportional to the cube of thesemimajor axis of the ellipse.
INITIAL VALUES AND EQUATIONSUnit vectors of polar coordinates(1)
INITIAL VALUES AND EQUATIONSFrom (1),(2)Differentiate with respect to time t(3)
INITIAL VALUES AND EQUATIONSCONTINUED Vectors follow the right-hand rule(8)
INITIAL VALUES AND EQUATIONSCONTINUED Force between the sun and a planet(9)Newton’s 2nd law of motion: F ma(10)F-forceG-universal gravitationalconstantM-mass of sunm-mass of planetr-radius from sun to planet
INITIAL VALUES AND EQUATIONSCONTINUED Planets accelerate toward the sun, and a is a scalar multiple of r.(11)
INITIAL VALUES AND EQUATIONSCONTINUED Derivative of(12)(11) and (12) together(13)
INITIAL VALUES AND EQUATIONSCONTINUED Integrates to a constant(14)
INITIAL VALUES AND EQUATIONSCONTINUED When t 0,1.2.3.4.5.
KEPLER’S LAWS OF PLANETARYMOTION1.Planets move around the Sun in ellipses,with the Sun at one focus.2.The line connecting the Sun to a planetsweeps equal areas in equal times.3.The square of the orbital period of aplanet is proportional to the cube of thesemimajor axis of the ellipse.
KEPLER’S LAWS OF PLANETARYMOTION1.Planets move around the Sun in ellipses,with the Sun at one focus.2.The line connecting the Sun to a planetsweeps equal areas in equal times.3.The square of the orbital period of aplanet is proportional to the cube of thesemimajor axis of the ellipse.
WHAT DOES LAW 2 MEAN?The line connecting the Sun to a planetsweeps equal areas in equal times.
PROOF OF LAW 2 CONTINUED Insert (4) into (14)(17)(17) is simplified when t 0(18)
PROOF OF LAW 2 CONTINUED C is always constant so (17) (18)(19)Differential area equation for polar coordinatesDifferential area equation is half of the constant function.(20)
PROOF OF LAW 2 CONTINUED LAW 2 isPROVED!!!!
KEPLER’S LAWS OF PLANETARYMOTION1.Planets move around the Sun in ellipses,with the Sun at one focus.2.The line connecting the Sun to a planetsweeps equal areas in equal times.3.The square of the orbital period of aplanet is proportional to the cube of thesemimajor axis of the ellipse.
TERMINOLOGY OF LAW 3Semi-MajorAxisaSemi-MinorAxisbMajor AxisMinor Axisa semi-major axisb semi-minor axisT period
WHAT DOES LAW 3 MEAN?The square of the orbital period of a planet isproportional to the cube of the semimajor axis ofthe ellipse.
PROOF OF LAW 3There is a concept from the proof of Law 1 that isneeded EccentricityTwo formulas (31) and (32)(32)(31)
PROOF OF LAW 3 CONTINUED Kepler’s law says(33)Formula 1 for areaFormula 2 uses (20)
PROOF OF LAW 3 CONTINUED Combining Formula 1 and 2 and solving for T(34)rmax is equal to (31) when θ π
PROOF OF LAW 3 CONTINUED a is found using the maximum position vector(35)
PROOF OF LAW 3 CONTINUED a can be found from (35)
PROOF OF LAW 3 CONTINUED Square both sides of (34) and use (32) and (35)
PROOF OF LAW 3 CONTINUED Solve T2
PROOF OF LAW 3 CONTINUED LAW 3 isPROVED!!!!
REFERENCES"Johannes Kepler: His Life, His Laws and Times." Kepler Mission. 4Aug. 2009. Web. http://kepler.nasa.gov/johannes/."Johannes Kepler: The Laws of Planetary Motion." Astronomy 161The Solar System. Department of Physics & AstronomyUniversity of Tennessee, 10 Aug. 2000. Web. pler.htmal ."Newton's Three Laws of Motion." Astronomy 161 The Solar System.Department of Physics & Astronomy University of Tennessee, 10Aug. 2000. Web. wton3laws.html .Stern, David P. "Kepler's Three Laws of Planetary Motion." NASAPolar, Wind, and Geotail Projects. 7 Feb. 2008. Web. http://wwwistp.gsfc.nasa.gov/stargaze/Kep3laws.htm .Weir, Maurice D. "Planetary Motion and Satellites." Thomas'calculus. 11th ed. Boston: Pearson Addison Wesley, 2005. 950-56.Print.
THANKS TO Dr. Lunsford All the Math Professors Seniors
WHAT ABOUT ISAAC NEWTON? “If I have seen further it is by standing on the shoulders of Giants.” Laws of Motion Universal Gravitation Explained Kepler’s laws The laws could be explained mathematically if his laws of motion and uni
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