Kepler's Third Law: Square of Planet's Orbital Period is Proportional to the Cube of its Major Axis

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In this video, I derive Kepler's 3rd Law of planetary motion and obtain a formula for a planet's period of orbit around the Sun in terms of the length of its major axis. Using the results of Laws 1 and 2, and using the formula for a horizontally shifted ellipse in Cartesian coordinates, as well as its area, I show that the square of the period is proportional to the cube of the major axis. This proportionality is independent of the planet as it is just in terms of the gravitational constant and mass of the Sun.

#math #vectors #calculus #ellipse #keplerslaws

Timestamps

• Applied Project Problem 2: Kepler's 3rd Law – 0:00
• Solution to (a): Obtaining the formula for the area swept by the orbit using the equation from the 2nd Law – 1:37
  • Setting area equal to the area of an ellipse to solve for the period – 5:47
• Solution to (b): Obtaining the formula for eccentricity e and directrix d in terms of h, G, M using the equation from the 1st Law – 8:30
  • Recap on the horizontal-shifted ellipse in polar and Cartesian coordinates – 11:12
  • Combining the equations of the ellipse's major and minor axes to obtain e·d – 14:00
• Solution to (c): Combining parts (a) and (b) to relate the period with the major axis – 18:32
  • The square of the period is proportional to the length of the major axis – 22:10

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