 # Essay On Kepler's Three Laws Of Planetary Motion

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Law of Orbits The first of Kepler’s Three Laws of Planetary Motion is the Law of Orbits, which describes the general motion of the planet in regard to its shape. As shown in Figure 6 below, it states that the planets’ orbits about the sun are elliptical, with one focal point located at the center of the sun. As a result, before commencing the proof, I hypothesized that: if planetary orbits are elliptical with one focal point located at the center of the sun, then to prove this I must reach the equation of an ellipse with the Sun as one of the focal points, r=ed1+e(cos()).

Figure 6. Orbits

Figure 7. Vectors describing the motion of the planet in its orbit

Before beginning the proof for Kepler’s First Law of Planetary Motion, however, it is essential to illustrate the scenario, as shown in Figure 7 above, and state facts based upon it. Firstly, we have r, the position vector that moves as a function of time, v, the position vector’s derivative, and a , the position vector’s second derivative. If the acceleration is always straight in towards the origin, which is the case with centripetal acceleration, the
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Therefore, the distance from the origin is a fixed proportion of the distance of the line. Based on this, it can be stated that d is the position of the directrix of the conic section considering that the eccentricity is greater than zero and less than one. Furthermore, the equation, r=ed1+e(cos(), is the exact equation for an ellipse in polar coordinates with a directrix at x=d and focal points at both the origin and the negative x-axis, further proving Kepler’s First Law of Planetary Motion: that the shape of the orbits is an ellipse with one focal point being the center of the