Question

State kelpers laws and relate them to the newtons universal gravitational law.

Answer 1

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1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semi major axis of its orbit.

Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.
Johannes Kepler proposed three laws of planetary motion. His Law of Harmonies suggested that the ratio of the period of orbit squared (T2) to the mean radius of orbit cubed (R3) is the same value k for all the planets that orbit the sun. Known data for the orbiting planets suggested the following average ratio:

k = 2.97 x 10-19 s2/m3 = (T2)/(R3)
Newton was able to combine the law of universal gravitation with circular motion principles to show that if the force of gravity provides the centripetal force for the planets' nearly circular orbits, then a value of 2.97 x 10-19 s2/m3 could be predicted for the T2/R3 ratio. Here is the reasoning employed by Newton:

Consider a planet with mass Mplanet to orbit in nearly circular motion about the sun of mass MSun. The net centripetal force acting upon this orbiting planet is given by the relationship

Fnet = (Mplanet * v2) / R
This net centripetal force is the result of the gravitational force that attracts the planet towards the sun, and can be represented as

Fgrav = (G* Mplanet * MSun ) / R2
Since Fgrav = Fnet, the above expressions for centripetal force and gravitational force are equal. Thus,

(Mplanet * v2) / R = (G* Mplanet * MSun ) / R2
Since the velocity of an object in nearly circular orbit can be approximated as v = (2*pi*R) / T,

v2 = (4 * pi2 * R2) / T2
Substitution of the expression for v2 into the equation above yields,

(Mplanet * 4 * pi2 * R2) / (R • T2) = (G* Mplanet * MSun ) / R2
By cross-multiplication and simplification, the equation can be transformed into

T2 / R3 = (Mplanet * 4 * pi2) / (G* Mplanet * MSun )
The mass of the planet can then be canceled from the numerator and the denominator of the equation's right-side, yielding

T2 / R3 = (4 * pi2) / (G * MSun )
The right side of the above equation will be the same value for every planet regardless of the planet's mass. Subsequently, it is reasonable that the T2/R3 ratio would be the same value for all planets if the force that holds the planets in their orbits is the force of gravity. Newton's universal law of gravitation predicts results that were consistent with known planetary data and provided a theoretical explanation for Kepler's Law of Harmonies.