Kepler's Laws with Derivations (Laws of ellipses, equal area).pptx

Kepler’s Laws of Planetary Motion.
Law of Ellipses
Law of Equal Areas
Law of Harmonies
Importance of Kepler’s Laws

1. Kepler’s Laws of Planetary Motion With Derivations and Visuals

2. Background • Johannes Kepler (1571–1630) • Used Tycho Brahe’s precise astronomical data • Published Astronomia Nova (1609), Harmonices Mundi (1619) • His three laws describe planetary motion empirically • Provided foundation for Newton’s Law of Gravitation

3. First Law: Law of Ellipses Final Result: Planets move in ellipses with the Sun at one focus. Equation of ellipse: (x²/a²) + (y²/b²) = 1 Derivation: 1. Ellipse definition: r1 + r2 = 2a 2. Relation: b² = a²(1-e²) 3. Polar form: r(θ) = a(1-e²)/(1+e cosθ) 4. Orbit distances: rmin=a(1-e), rmax=a(1+e)

4. Second Law: Equal Areas Final Result: A line from Sun to planet sweeps equal areas in equal times. Equation: dA/dt = constant [Insert real orbit with areas image here] Derivation: 1. Area element: dA = ½ r² dθ 2. Rate: dA/dt = ½ r² dθ/dt 3. Angular momentum: L = m r² dθ/dt 4. So dA/dt = L/2m = constant

5. Third Law: Harmonies Final Result: T² ∝ a³ Equation: T² = (4π²/GM) a³ [Insert solar system image here] Derivation: 1. Centripetal: v²/r = GM/r² 2. v = 2πr/T → (2πr/T)² = GM/r 3. Rearr: T² = 4π²/GM × r³ 4. Replace r with a for ellipse: T² ∝ a³

6. Importance & Applications • Foundation for Newton’s Law of Gravitation • Explains planetary motion with precision • Used in satellite design and space missions • Helps detect exoplanets and binary stars • Basis for orbital mechanics and astrophysics