There is no simple formula for the position at time t of a planet P in its orbit (an ellipse) around the sun. Introduce the auxiliary circle and angle θ in Figure 10 (note that P determines θ because it is the central angle of point B on the circle). Let a = OA and e = OS /OA (the eccentricity of the orbit).

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(a) Show that sector BSA has area (a²/2)(0 - e sin 0). (b) By Kepler's Second Law, the area of sector BSA is proportional to the time t elapsed since the planet passed point A, and because the circle has area nа², BSA has area (лa²)(t/T), where T is the period of the orbit. Deduce Kepler's Equation: 2лt T = 0 - e sin 0 (c) The eccentricity of Mercury's orbit is approximately e = 0.2. Use Newton's Method to find after a quarter of Mercury's year has elapsed (t = T/4). Convert 0 to degrees. Has Mercury covered more than a quarter of its orbit at t = T/4? Auxiliary circle B Р a Elliptical orbit S Sun