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

Historical Context

• Late 1500s to early 1600s, before Newton's universal law of gravitation was proposed.

• Kepler's laws arose from empirical observations of planetary orbits.

Kepler's First Law: Elliptical Orbits

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

  • Ellipses have two focal points, with the Sun located at one.

• Implication: Orbits are not circular; planets are closer to the Sun at one point in their orbit and farther at another.

  • Applicability: This concept applies not only to planets but also to satellites and other orbiting bodies.

• Visualization:

  • Animation shows Earth's elliptical orbit around the Sun, illustrating varying distances throughout the orbit.

Kepler's Second Law: Equal Area in Equal Time

• Definition: A planet sweeps out equal areas in equal times, regardless of its position in the orbit.

  • Example: If you take a specific time period, the area covered when the planet is close to the Sun is the same as when it is farther away.

• Speed Variation:

  • Planets move faster when closer to the Sun and slower when farther away.

• Visualization:

  • Demonstrates the speed of Earth as it orbits, faster when closer and slower when farther from the Sun.

• Joke:

  • Reference to an XKCD comic related to Kepler's consistent area-sweeping.

Kepler's Third Law: Relationship of Period to Distance

• Definition: The planet's orbital period squared (T²) is proportional to the semi-major axis cubed (R³) of its orbit.

  • Simplified relationship: T² ∝ R³ (for circular orbits, R represents radius).

• Insight: Kepler figured out T² = constant * R³; Newton later derived the constant of proportionality.

Derivation Using Newton's Law

1. Starting Point: Assume circular orbits to simplify calculations.

2. Gravitational Force: Use Newton's law of gravitation:

   • FG = G(m1 * m2) / R².

3. Centripetal Force:

   • FC = m(v² / R).

4. Equating Forces: Set gravitational and centripetal forces equal to each other:

   • G(M*m) / R² = m(v² / R).

5. Simplification: Cancel m, leading to GM = v²R.

6. Expressing Velocity: Orbital speed (v) = distance/time.

   • Distance for a full orbit: circumference = 2πR; thus, v = 2πR / T.

7. Substituting in Equation:

   • GM/R = (2πR/T)².

8. Final Steps:

   • Rearranging leads to T² = (4π² / GM)R³.

9. Conclusion:

   • Derived relationship verifies Kepler's formula and identifies the constant of proportionality.