The square of the period of revolution of the planet around the sun is proportional to the cube of the semi-major axis of the ellipse. We know that a planet moving in a circular orbit around the sun i...The square of the period of revolution of the planet around the sun is proportional to the cube of the semi-major axis of the ellipse. We know that a planet moving in a circular orbit around the sun is accelerating toward the sun with the centripetal acceleration \(a \) = \(v\) 2 \(∕R\), where \(v \) is the speed of the planet’s motion in its orbit and \(R \) is the orbit’s radius.
Using analytic geometry in the limit of small Δθ , the sum of the areas of the triangles in Figure 25.9 is given by The period of revolution T of a planet about the sun is related to the semi-major ax...Using analytic geometry in the limit of small Δθ , the sum of the areas of the triangles in Figure 25.9 is given by The period of revolution T of a planet about the sun is related to the semi-major axis a of the ellipse by \(T^{2}=k a^{3}\) where k is the same for all planets. Using Equation (25.2.1) for reduced mass, the square of the period of the orbit is proportional to the semi-major axis cubed,