Mathematics and Society - OS X Lion Server

COMETS ANDCONIC5 ECTIQt\l5(From Kepler's discovery (see "Hep" KepZer Didn't Go Around In CircZes) we knowthat each planet's orbit is an ellipse. Ellipses belong to a special group of curvesknown as the conic sections. These curves have interested mathematicians and astronomersfor years.A conic section is the curve of intersectionbetween a hollow cone and a plane that cuts throughit. The pictures show some conic sections.i,, /,,--d~\lPERBOLA, '" , ", ,,//', , ,,,,, ,,PARABOLAWhy do the planets have e111ptica1 orbits? Newtondiscovered that ::he gravitational pull of the sun onthe planets causes their orbits to be elliptical. Hethen wondered whether comets were affected by thegravitational pull of the sun·and if so he predictedthat their orbits would be conic sections. In particularif the comets had nearly circular orbits thevwould be visi~le at regular and somewhat frequentintervals. Edmund Ha1iey supported Newton's beliefby discovering; that the comet he was observing wasthe same one that was seen 75 years earlier.Halley's comet· last appeared:f,n 1910 and is dueagain about the spring of 1986.---Cb-~'-,\/---HYPE.RBOLA~ ~~GO1980 1970TIll: ORBIT Of" WALU:VS COMET(We now know that the orbit of a comet will be anellipse, a parabola or one branch of a hyperbola asshown in the bottom figure. It is believed that mostcomets travel on elliptical orbits but that the orbitsare so stretched out that the portion we can seeappears like a parabola.ELLIPSE• SU~PARABOLAI-lYPtRlOOLAThe following two pages describe methods that canbe used to draw ellipses, parabolas and hyperbolas. (114