| Kepler's
Problem can be described as follows. In this diagram, AMP is an ellipse,
ABP a circumscribed circle, S and F are foci, A and P the aphelion and
perihelion, and M the planetary position. For any planetary position (M)
on an ellipse, the time taken to move from A to M is to the planet's period
as sector ASM is to the area of the ellipse. In order to simplify calculating
area ASM, Kepler constructed BMD perpendicular to AP so that sector ASM
is to the area of the ellipse as sector ASB is to the area of the circle,
thus, SM and SB sweep our equal areas in equal times. As Kepler lamented,
however, there is no directly calculable relationship between sector ASM
and angle ASM or between sector ASB and angle ASB. As a result, Kepler
was forced to rely on trial and error methods of approximation. In identifying
the scope and difficulty of the problem, Kepler urged geometers to find
a direct solution. As a result, it was thereafter known as Kepler's Problem,
and was taken up by several mathematicians, especially after the middle
of the century when Kepler's works became better known. |
