What is the mathematical proof for Kepler's first law?

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SUMMARY

Kepler's first law states that planets move in elliptical orbits with the Sun at one focus, a conclusion derived from the mathematical properties of gravitational forces and orbital mechanics. The discussion highlights the misconception that a planet's orbit must be circular if the velocity is perpendicular to the force; in reality, the absence of a force to maintain a circular path leads to elliptical orbits. The eccentricity of an orbit quantifies its deviation from circularity, ranging from 0 (circular) to 1 (parabolic). Understanding these principles is essential for grasping the dynamics of planetary motion.

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  • Understanding of gravitational forces and their effects on motion
  • Familiarity with the concept of orbital eccentricity
  • Basic knowledge of Newtonian mechanics
  • Mathematical proficiency in geometry and algebra
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  • Study the mathematical derivation of Kepler's laws of planetary motion
  • Learn about gravitational force equations and their implications on orbits
  • Explore the concept of orbital eccentricity in detail
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babbi.mamgain
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I wanted to ask how come kepler say that the orbits on which planet rotates is elliptical ... just by obvious saying, what we have learned is that when the velocity is Perpendicular to force the path traced is circular...even if we say in reality the actual angle B/W velocity and force is not 90 ...but i question is what makes a planet to go that far from the focci ? try to give an answer easy to understand. or please try to explain the mathematical proof in detail.
 
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That's a very strange question!
just by obvious saying, what we have learned is that when the velocity is Perpendicular to force the path traced is circular... even if we say in reality the actual angle B/W velocity and force is not 90
I'm not sure what you mean by "by obvious saying". You appear to be saying that even if something is NOT true, we can say it is! That is certainly not true.

It is not a matter of something "forcing" a planet to move in a non-circular orbit. It is a matter of there not being anything to force the planet to go in a circular orbit! A circular orbit has "eccentricity" exactly equal to 0. An elliptic orbit can have eccentricity anywhere from 0 to 1. It would be very surprising if a parameter were exactly 0 rather than some range of values.
 

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