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MA4248 Weeks 4-5. Topics Motion in a Central Force Field, Kepler’s Laws of Planetary Motion, Couloumb Scattering Mechanics developed to model the universe - follow the seasons, predict eclipses and comets, compute the position of the moon and planets Babylonians (2000-300BC) – arithmetical models Claudius Ptolemy (85-165AD) – geometric models based on epicycles that prevailed for 1400 years ! http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Ptolemy.html 1 PTOLEMAIC THEORY Earth Planet Earth is fixed, each planet moves in a circular epicycle whose center moves in a circle with center near the Earth. 2 REVOLUTION Nicolaus Copernicus (1473-1543) – produced a heliocentric (versus geocentric) theory of cosmology Galileo Galilei (1564-1642) – questioned authority - refuted Aristotle’s claim that heavy bodies fall faster - championed Copernican theory over Ptolemaic - sentenced to house arrest by the dreaded Inquisition Tycho Brahe (1546-1601) – observational astronomer - Danish King helped him build 37-foot quadrant - compiled, over 20 years, most accurate records - Emperor Rudolph II sponsored his move to Prague and collaboration with Kepler 3 GEOMETRY OF ELLIPSES 2a-r r 2ea a = semi-major axis (half of horizontal diameter) e = eccentricity foci angle 2 2 2 2 (2a r ) r 4e a 4ear cos 2 p / r 1 e cos , p a (1 e ) 4 ALGEBRA OF ELLIPSES ( x , y) 2a-r r 2ea ( x, y) (r cos , r sin ) 2 2 pe p 2 2 (1 e ) x y 2 2 1 e 1 e 5 KEPLER’S LAWS Johann Kepler (1571-1630) – mathematician who believed in “the simplicity and harmonious unity of the universe” (quote page 323 David Burton) I. Each planet moves around the sun in an ellipse, with the sun at one focus. II. The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time. III. The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits: T ~ a 3/2. 6 ANGULAR MOMENTUM AND TORQUE The angular momentum , torque (about any fixed point) of a body with movement, force is d r L r (m ) Torque r F dt (vector cross-products is orthogonal to both vectors and its magnitude equals the area of the parallelogram) 2 d d r d r d r L (m ) r (m 2 ) dt dt dt dt 2 d r r (m 2 ) r F Torque 7 dt CENTRAL FORCE A force acting on a body is central if its direction is along the line connecting the body to a fixed point. r p r body p fixed point F central force 8 CENTRAL FORCE For a central force Torque r F r (Fr ) 0 therefore d L Torque 0 dt Therefore the angular momentum L is constant. Therefore, since r L 0 (why?), the body moves in a plane. When does it move in a line? 9 POLAR COORDINATES Construct an orthonormal coordinate system r̂ r xx̂ yŷ ŷ θ̂ x̂ Define polar coordinates x r cos , y r sin and unit-vector valued functions (for r > 0) r̂ ( r , ) (cos , sin ), ˆ ( r , ) ( sin , cos ) ˆ dr̂ d ( sin , cos ) ˆ , r̂ 10 dt dt POLAR COORDINATES Therefore, the velocity of a particle is dr d (rr̂) dr r̂ r dr̂ r r̂ r θ θ̂ dt dt dt dt and its acceleration is 2 d r r r θ 2 r̂ r θ 2r θ θ̂ 2 dt 11 CENTRAL FORCE Therefore, if a body moves in a central force, then 2 hence and d r F F r̂ m 2 dt 2 m r r θ r̂ m r θ 2r θ θ̂ 2 m r r θ F 2 d 1 m r θ 2r θ mr θ 0. r dt Remark 2 F | F |, L | L | mr θ 12 KEPLER’S SECOND LAW Remark 1: Since L is constant and dA 1 (r)(rθ ) L dt 2 2m equals the rate at which the radius vector sweeps out Area, Kepler’s Second Law holds for any central force Remark 2: If F is conservative F dV(r) dr 2 and θ L mr , mr dVeff (r ) dr where Veff (r) L 2m r V(r) 2 2 13 INTEGRATING THE EQUATION Remark 3:We can obtain a differential equation for r dVeff ( r ) dr dVeff ( r ) dr mr dt dr dt dt 2 1 1 E 2 mr r V(r) 2 mr Veff (r) dt mdr 2( E Veff ( r )) In general, the integral on the right will not be an elementary function 14 GRAVITATIONAL & COULOMB FORCES Remark 4: If 2 F k r dV dr, V(r ) k r then the effective potential Veff (r) L 2m r 2 Veff has this graph --> for k > 0 (what if k < 0 ?) 2 k r E3 r E1 15 INTEGRATING THE EQUATION Remark 5:We can also substitute the identity dr dr L r θ d d mr 2 1 to obtain 2 2mE 1 2mk dθ r dr 2 2 2 L r Lr let u = 1/r 2 mk 2 L E u 2 1 1 cos( θ θ ) 0 2 L mk 16 KEPLER’S FIRST LAW Remark 6: This has the form of a conic section p r 1 e cos (θ θ0 ) 2 with semi-latus-rectum p L mk and eccentricity e 1 2L E mk 2 E E0 mk 2L , e 0 o Ellipse E E 0 , 0 e 1 r 0 E 0, e 1 Parabola 0 E, 1 e 1 Hyperbola 2 2 2 Circle 17 KEPLER’S THIRD LAW Remark 8: The semi-major axis is determined by the energy a k 2E Remark 9: The rate of area swept out dA L 1 ka 1 e 2 dt 2m 2 m Remark 10: The area of the ellipse (b semi-major axis) A πab o r Remark 11: The period πa 1 e 2 2 τ 2π m k a 32 18 KEPLER’S EQUATION Remark 12: The true anomaly is the angle α θ θ0 from pericenter and the eccentric anomaly is r - a -ea cos ψ Remark 13: Integrating the equations of motion for r yields Kepler’s (transcendental) equation 2π ψ e sin ψ (t t ) 0 τ o r Remark 14: For Earth’s orbit e 0.016732 o θ 0 102.85 fromVernal Equinox 19 SUPERINTEGRABILITY Remark 15: A mechanical system is integrable if you can express the state as a function of time (even an non-elementary function). This requires constants of motion (such as angular momentum) and is a very special condition. In very special cases additional constants of motion exist that ensure closed orbits. These superintegrable systems include motion in a central –k/r (k > 0) force where the Laplace-Runge-Lenz vector defined by o r K m r L mk r̂ is a constant of motion (Problem 12, page 25) 20