Kepler's laws of planetary motion describe the orbits of planets around the sun and reveal the relationship between a planet's orbital velocity and its distance from the sun. The key insights can be summarized as follows:
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Kepler's Third Law: This law states that the square of the orbital period \(T\) of a planet is directly proportional to the cube of the semi-major axis \(a\) of its orbit (which is essentially the average distance from the sun). Mathematically, this is expressed as: \[ T^2 \propto a^3 \] This implies that planets further from the sun take longer to complete one orbit.
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Orbital velocity relationship: While Kepler's Third Law doesn't directly give the relationship between velocity and distance, it leads to the understanding that the orbital velocity \(v\) of a planet decreases as its distance \(r\) from the sun increases. In general terms, the gravitational force acting on the planets, and thus their orbital speed, decreases with distance. For circular orbits, the orbital velocity can be expressed as: \[ v = \sqrt{\frac{GM}{r}} \] where \(G\) is the gravitational constant, \(M\) is the mass of the sun, and \(r\) is the distance from the sun. This equation shows that as \(r\) increases, \(v\) decreases.
In summary, based on Kepler's observations, there is an inverse relationship between a planet's orbital velocity and its distance from the sun: as the distance from the sun increases, the orbital velocity of the planet decreases.
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