Kepler's Laws of Planetary Motion

Introduction

• Johannes Kepler was an assistant to Tycho Brahe.
• Brahe collected extensive data on stars and planetary motions, but couldn’t complete a working model of planetary orbits.
• Kepler developed a model for planetary orbits leading to the formulation of three fundamental laws, known as Kepler's Laws.

Kepler's Three Laws

First Law: Elliptical Orbits

• Planets move in elliptical orbits with the Sun at one focus of the ellipse.
• Earlier beliefs held that orbits were circular; however, Earth's orbit is nearly circular, which led to this misconception.
• Other planets exhibit more eccentric, elliptical orbits.

Second Law: Equal Areas in Equal Times

• As a planet orbits the Sun, it sweeps out equal areas in equal time intervals.
• The planet moves faster when closer to the Sun and slower when further away.
• This is analogous to a gravitational slingshot effect.

Third Law: Harmonic Law

• The square of the orbital period (T^2) is proportional to the cube of the semi-major axis (a^3) of its orbit.
• Formulated as T^2 ∝ a^3.
• For Earth, T is one year, and semi-major axis is half of the longest diameter of the orbit.

Connection with Newton's Gravitation

• Kepler's laws were later shown to be derivable from Newton's Universal Law of Gravitation.
• Newton's law integrates the understanding of planetary motion with gravitational attraction.
• The force of gravity (F) between two masses (M1 and M2) is given by F = G(M1M2)/R², where G is the gravitational constant.

Derivation of Third Law from Newtonian Physics

• For a circular orbit, the gravitational force provides the necessary centripetal force for circular motion.
• By setting F = mv²/R and using the relationship between velocity, orbit radius (R), and period (T), one can derive that T² ∝ R³.
• The constant of proportionality is the Kepler constant (K), which is calculated as K = 4π²/GM.

Eccentricity of Orbits

• Orbits can be characterized by their eccentricity:
  • Highly elliptical (eccentric) orbits like Halley’s Comet.
  • Nearly circular orbits like Earth.
• Halley’s Comet has a period of about 76 years, as opposed to Earth's 1-year period.
• The eccentricity affects the orbital period and semi-major axis.

Observational Insights

• Halley’s Comet is observable from Earth approximately every 76 years.
• The speaker notes personal experiences observing the comet, indicating its predictability and regularity.

Conclusion

• Kepler’s Laws, especially the third law, helped solidify the understanding of planetary motions and paved the way for Newton's gravitational theory.
• Kepler’s constant applies to all bodies orbiting the Sun, reflecting universal applicability in our solar system.