Copyright © 2001 H.L. Cohen Ver. 081501AST 1002 Slide Handouts: Topic B: Part 4 Page 1Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815011Topic BTopic B“The New Astronomy”Part 4. Newton(Web Version: 08-15-01)Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815012Issac Newton (1643–1727)§ English• Mathematician• Scientist• Head of Mint§ Born• 1 year after Galileo’s death• 13 years after Kepler’s deathSee Study Guide Fig. 3Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815013Contributions• Invented differential calculusIssac Newton (1643–1727)d(x2) 2xdx=dy dy• Formulated laws of mechanics & gravitation(Philosophiae Naturalis Principia Mathematica, 1687)• Theories of light & color (Opticks, 1704)• Invented “Newtonian” reflecting telescopeDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815014Issac Newton (1643–1727)§ Defensive of discoveries§ Sometimes reluctant to credit others(Robert Hooke may have provided some of his ideas)§ Reluctant to publish(Edmund Halley funded publication of Principia)§ Spent most of life onAlchemy, theology & biblical chronologyDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815015Kepler’s ApproachTycho’s ObservationsYesDataTrial &ErrorFit Data?PredictionsDoneNo“Laws”(Empirical Method)Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815016Kepler’s Successü Gradual adoption during2nd half of 17th centuryü Why?• Predictions much better• Tests/finds valid for other planets besides Mars• Newton provides “cause” (allows basis for mechanics)Copyright © 2001 H.L. Cohen Ver. 081501AST 1002 Slide Handouts: Topic B: Part 4 Page 2Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815017Newton’s ApproachYesDefinitions(Force, etc.)DoneNoAssumptions(Law of Grav.)K’sLaws?SolveEquationsInvents new math tools to aid solution — result is development of the calculus(Theoretical Method)+Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815018Newton’s 2nd Law of MotionDefines relation between Force, Mass & AccelerationF = mass x acceleration• The Effect — Acceleration• The Cause — Force (F )• The Mass — Property of matter that resists acceleration(Thus forces required to accelerate masses)Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 0815019Postulates a specific type of forceacting between massesm — First MassM — Second Massd — Distance(between centers of 2 masses)FGravity— Gravitational ForceNewton’s Law of GravitationdMmm x MFGravity∝∝d 2NoteDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150110#1Inverse Square Law#24F GravitymMdF GravitymM#32F Gravitym1/2 M1/2d1/2dDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150111Results§ Newton derives Kepler’s Three Laws§ Derivations show each law . . . More “general” or “elaborate” than Kepler realized§ Briefly here is how each differs . . .Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150112§ Possible orbits . . .Any conic section if force varies inversely as distance squared§ Forces need not be gravitational§ Conic sections are . . .• circle• ellipse• parabola• hyperbolaLaw One: Newton’s VersionCopyright © 2001 H.L. Cohen Ver. 081501AST 1002 Slide Handouts: Topic B: Part 4 Page 3Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150113The Conic Sections§ Why so named?§ Answer . . . Right circular cone cut by a plane forms these curves• circle (c)• ellipse (e)• parabola (p)• hyperbola (h)chepDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150114Examples . . .§ Some comets observed leaving solar system on parabolic or hyperbolicorbits++Sun§ Two charged particles(same sign) repel each along hyperbolic orbitsDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150115Law Two: Newton’s VersionLaw of Areas obeyed for any“central force”(Central forces always pointtoward same point)So, an “inverse square law” force unnecessaryDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150116§ For Gravitational Forcem & M = two masses,d = distance§ Harmonic Law wasP 2= a 3where P = orbit period & a = mean distanceLaw Three: Newton’s VersiondMmaMmm x MF ∝∝d 2Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150117§ For Gravitational Forcem & M = two masses,d = distance§ Harmonic Law becomes( M + m ) P 2= a 3Law Three: Newton’s VersiondMmaMmNew factor containing sum of the two massesm x MF ∝∝d 2Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150118§ For Sun and planet• M = Sun’s mass• m = planet’s massWhy Kepler “Had No Mass”Mmm very small!§ Assume m small compared to Mä Result . . . can omit m if approximatingäSo (regardless of planet)M P 2= a 3M much bigger!(M + m) P 2= a 3Copyright © 2001 H.L. Cohen Ver. 081501AST 1002 Slide Handouts: Topic B: Part 4 Page 4Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150119Why?§ BecauseM + m = Big No. + Small No.= Big No. (approximately)§ Example1.000 + 0.001 = 1.001= 1.000 (approximately!)Planet’s MassSun’s MassDept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150120Planet Mass* Period Mean DistanceMercury 0.24 yr 0.39 AUVenus 0.62 yr 0.72 AUEarth 0.000003 1.00 yr 1.00 AUMars 1.88 yr 1.52 AUJupiter 0.000954 11.9 yr 5.20 AUSaturn 29.4 yr 9.56 AUExample: Harmonic LawCompare Earth to Jupiter using (M+m)P2 ∝ ∝ a3(1+0.000954)x11.92(1+0.000003)x1.0025.2031.003*Mass compared to Sun’s mass (i.e., mass Sun = 1 solar mass)Dept. Astronomy – University of Florida – Copyright © H.L. Cohen 2001, Ver. 08150121Planet Mass* Period Mean DistanceMercury 0.24 yr 0.39 AUVenus 0.62 yr 0.72 AUEarth 0.000003 1.00 yr 1.00 AUMars 1.88 yr 1.52 AUJupiter 0.000954 11.9 yr 5.20 AUSaturn 29.4 yr 9.56 AUExample: Harmonic LawCompare Earth to Jupiter using (M+m)P2 ∝ ∝ a3(1.000954)x141(1.000003)x 1 1411Multiplying