German astronomer Johannes Kepler is credited with demonstrating that abandoning an Earth-centered, or Ptolemaic, view of planetary motion for a Sun-centered, or Copernican, model implied that the motion of the planets had to be clearly elliptical. He reached this conclusion by analyzing the careful observations of his mentor, Tycho Brahe, on the motions of the heavenly bodies as well as his own observations of the orbit of Mars, which showed that planetary motion did not follow the path of a circle. From this insight, Kepler generated his three empirical laws of planetary motion. The first, the law of elliptical orbits, states that the orbits of the planets are ellipses with the Sun at one focus. The second, the law of equal areas, states that a line from the planet to the Sun sweeps over equal areas in equal intervals of time, and the third, the law of harmonies, states that the square of the period of revolution of a planet about the Sun is proportional to the cube of the semi-major axis of the planet's elliptical orbit. Kepler's laws imply a nonuniform speed of revolution for planets as they orbit around the sun; they also imply that their velocity also changes throughout the planet's year, being faster when the planet is close to the Sun and slower when the planet is far away. The theoretical framework of Kepler's laws was finally derived some 30 years after his death when Isaac Newton formulated his law of universal gravitation.
Kepler's second law is often taken to be equivalent to the statement of conservation of angular momentum, which states that the total angular momentum of a closed system remains constant. The law of equal areas describes the constantly varying speed at which any given planet moves while orbiting the Sun under the influence of gravity. This can be visualized by drawing an ellipse to describe the orbit of the planet around the Sun. If we draw a line from the center of the planet to the Sun, and another one after it has moved along the elliptical orbit for a given period, we obtain a triangle with a given surface area. Similarly, another such triangle can be drawn after the planet has moved an equal amount of time along the orbit. But the triangles will have different shapes, because the orbit is not circular, thus, the distance from the planet to the Sun will vary and so will the length of the sides of the triangles. With this approach, the areas of the triangles formed when Earth is close to the Sun yield short but wide triangles and the areas formed when Earth is far from the Sun are represented by long, narrow triangles. If the areas of all such possible triangles are calculated, Kepler's second law states that they are all the same size. Since the bases of these triangles are shorter when Earth is furthest from the Sun, Earth would have to be moving more slowly in order for this imaginary area to be the same size as when Earth is closest to the Sun.
Kepler's second law is related to the conservation of angular momentum as follows: If we describe a planet as experiencing rotational motion under the influence of gravity, we can use a vector quantity called torque (t), which is produced by any force applied to a body that causes the body to rotate. Mathematically, this can be expressed as: t = r x F, which is the cross product of r, the vector starting from the axis of rotation to where the force is applied and of the force applied, in this case, gravity. In this equation, F is given by Newton's universal law of gravitation, which states that whenever there are two objects that have a mass, they will exert a gravitational force on each other proportional to the product of their masses, and inversely proportional to the square of the distance between the two centers of mass. This is expressed as:
F = -(Gmm/r[sup2 ])r
in which F is the gravitational force, m and m are the respective masses of the two bodies separated by a distance r, G is the gravitational constant and r is a unit vector directed from the first mass to the second. Substituting in the expression for torque yields:
t = (rr) x F = (rr) x -(Gmm/r[sup2 ])r
Rearranging, t = (r x r) (Gmm/r). Since the cross product of a vector with itself is zero, the torque expression also reduces to zero, which is equivalent to stating that the sun does not exert a torque on an orbiting planet. The torque of a planet can also be defined as the instantaneous rate of change of its angular momentum (l) or: t = dl/dt. If the Sun does not exert a torque on a planet, it follows that dl/dt must also be a zero vector, which means there is no change in angular momentum; angular momentum remains constant, i.e., it is conserved.
Kepler derived these three laws strictly from observation as calculus had not yet been invented in his time. It is also interesting to note he was able to draw his conclusions because he happened to be observing Mars, the only planet visible to the naked eye with an eccentricity large enough so as to allow one to distinguish its elliptical orbit from a circular one with the rudimentary observation instruments available at the time.
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