(455) Kepler's 3 laws of orbital motion [IB Physics SL/HL]

Context

• Timeframe: Late 1500s to early 1600s; Newton's universal law of gravitation established later.

• Knowledge was based on empirical observations of planetary orbits.

Kepler's Laws of Motion

Kepler's First Law

• Elliptical Orbits: Planets move in elliptical orbits with the Sun at one focus.

  • Comparison to circular orbits; real orbits are elliptical.

  • Example shown: Earth's orbit with varying distances from the Sun.

Kepler's Second Law

• Equal Area in Equal Time: A planet sweeps out equal areas in equal time, leading to variable speeds.

  • Closer to the Sun: Planet moves faster

  • Farther from the Sun: Planet moves slower

  • Visual demonstration of areas A1 and A2 being equal in given time intervals.

Kepler's Third Law

• Orbital Period Squared Proportional to Semi-Major Axis Cubed: T² is proportional to R³.

  • In circular orbits, simplifies to T² = k * R³, where k is a constant.

  • Significance of Newton's contributions to understanding this relationship.

Derivation of Kepler's Third Law

Force Equations

• Gravitational Force (Fg): Fg = GMm/R²

• Centripetal Force (Fc): Fc = mv²/R

• Set Fg = Fc, leading to GM = v²R.

Velocity Determination

• Orbital velocity (v) calculated as v = 2πR/T.

• Substituting v into the gravitational equation leads to: GM/R = (2πR/T)².

Final Derivation

• Expanded equation results in T² = (4π²/GM) * R³.

• The constant of proportionality identified as 4π²/GM.

Conclusion

• Kepler's principles defined early understanding of planetary motion, later clarified and quantified by Newton.