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Potting Orbits

The equation

describes the polar coordinates of an orbit measured from one of the orbit’s two fer cal points. For objects in orbit .around the sun, the sun is at one of the focal poin Thus r is the distance of the object from the sun. The parameters p and E determine the size of the orbit and its eccentricity, respectively. A circular orbit has an eccentrici • of 0; if 0 < E < I, the orbit is elliptical; and if E > I, ‘the orbit is hyperbolic. “0 tain the polar plot that represents an orbit having E = 0.5 and p = 2 AU (AU stan for “astronomical unit”; 1 AU is. the mean distance from the sun to Earth). How f away does the orbiting object get from the sun? How close does it approach Earth’
orbit?

Solution
Figure 5.3-7·shows the polar plot of the orbit. The plot was generated by the following session

»theta = [O:pi/90:2*pi);
»r = 2./(1-O.5*cos(theta));
»po~ar(checa,r),ciCle(‘Orbital Eccentricity 0.5

A polar plot showing an orbit having an
eccentricity of 0.5.

The sun is at the origin, and the plot’s concentric circular grid enables us to determine that the closest and farthest distances the object is from the sun are approximately J.3.and 4 AU. Earth’s orbit, which is nearly circular, is represented by the innermost circle. Thus the closest the object gets to Earth’s orbit is approximately 0.3 AU. The radial grid lines allow us to determine that when 8 = 900 and 2700 , the object is 2 AU from the sun.