Is there a simple way to derive Kepler's Three Laws from Newton's Three Laws?
#113670. Asked by Bronxiteone. (Mar 26 10 5:06 PM)
The first law of Kepler states that a planet moves in an elliptical orbit around the Sun that is located at one of the two foci of the ellipse. An ellipse is one of the conic curves originally studied by Greek geometers. See also Conic section; Ellipse.
In 1687, Isaac Newton demonstrated that any body, moving in an orbit around another body that attracts it with a force that varies inversely as the square of the distance between them, must move in a conic section. This path will be an ellipse when the velocity is below a certain limit in relation to the attracting force. Thus the first law is a general law that applies to all satellites held in orbit by an inverse-square force.
Kepler needed some method by which to predict the locations of planets at given times, and this he provided with his second law. It states that the radius vector of the ellipse (the imaginary line between the planet and the Sun) sweeps out areas that are proportional to time (see illustration). A planet moves more swiftly when it is closer to the Sun and more slowly when it is farther removed.
Demonstration of Kepler's first and second laws. The planet moves along an elliptical orbit at a nonuniform rate, so that the radius vector drawn to the Sun, which is located at one focus of the ellipse, sweeps out areas that are proportional to time. Thus, the planet would take equal times (corresponding to the equal areas) to traverse the unequal distances along the ellipse that correspond to the two shaded areas. The diagram greatly exaggerates the eccentricity of any orbital ellipse in the solar system.
Again Newton demonstrated the dynamic cause behind Kepler's second law. In this case, it is not restricted to forces that vary inversely as the square of the distance; rather it is valid for all forces of attraction between the two bodies. The second law expresses the principle of the conservation of angular momentum. See also Angular momentum.
Kepler's third law defines the relations that hold within the system of planets. It states that the ratio between the square of a planet's period (the time required to complete one orbit) to the cube of the mean radius (the average distance from the Sun during one orbit) is a constant. The four satellites that Galileo had discovered around Jupiter were found to obey the third law, as did the satellites later found around Saturn. Newton demonstrated once again that the third law is valid for every system of satellites around a central body that attracts them, as the Sun attracts the planets, with a force that varies inversely as the square of the distance.
I wanted to know HOW to derive them from newton's.|
For the 3rd law, assume a planet is in a circular orbit. This is because the perimeter of an ellipse is extremely difficult to calculate, and the eccentricities of the elliptical orbits are small anyway.|
The gravitational attractive force exerted on the planet by the Sun is given by the Newtonian formula F = GMm/d² where G is the gravitational constant, M is the mass of the Sun, m is the mass of the planet and d is the Sun-to-planet distance.
The centripetal force needed to hold the planet in orbit is given by the expression F = mv²/d where v is the orbital velocity.
These two forces are the same.
Therefore GMm/d² = mv²/d
hence GM/d = v² (eliminating m/d from both sides).
Now v, the orbital velocity, is given by 2πd/p where p is the period of the planet.
Hence GM/d = 4π²d²/p²
and GM = 4π²d³/p²
Rearranging we have p²/d³ = 4π²/GM
Since 4π²/GM is a constant we have Kepler's 3rd Law - the ratio of the square of the period of a planet to the cube of its distance from the Sun is a constant.
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