13.5 Kepler's Laws of Planetary Motion

Kepler's laws of planetary motion revolutionized our understanding of the cosmos. These three laws describe how planets orbit the Sun in elliptical paths, move faster when closer to the Sun, and have orbital periods related to their distance from it.

These laws form the foundation of celestial mechanics and orbital dynamics. They explain the motion of planets, moons, and artificial satellites, allowing us to predict their positions and plan space missions with incredible accuracy.

Kepler's Laws of Planetary Motion

Kepler's laws of planetary motion

Kepler's First Law: The Law of Ellipses

• Planets orbit the Sun following elliptical paths with the Sun located at one focus of the ellipse (Earth, Mars)
• Elliptical orbits characterized by their eccentricity ($e$) which describes the deviation from a perfect circle
  • $e = 0$ represents a circular orbit
  • $0 < e < 1$ represents an elliptical orbit (most planets in our solar system)
  • $e = 1$ represents a parabolic orbit
  • $e > 1$ represents a hyperbolic orbit (some comets)
• These orbits are examples of conic sections in orbital mechanics

Kepler's Second Law: The Law of Equal Areas

• A line segment connecting a planet to the Sun sweeps out equal areas in equal time intervals
• Planets move faster when closer to the Sun and slower when farther away (Mercury at perihelion vs aphelion)
• Consequence of the conservation of angular momentum in the absence of external torques

Kepler's Third Law: The Law of Periods

• The square of a planet's orbital period ($T$) directly proportional to the cube of the semi-major axis ($a$) of its orbit
• Expressed mathematically as $T^2 = k a^3$ where $k$ is a constant dependent on the mass of the central body
• Relates the orbital period to the size of the orbit allowing comparisons between different planets

Mathematical applications of Kepler's laws

Kepler's Third Law used to calculate orbital periods and distances

• $T^2 = \frac{4\pi^2}{G(M+m)} a^3$ where $G$ is gravitational constant, $M$ is mass of central body, $m$ is mass of orbiting body
• For most cases, $m \ll M$, so equation simplifies to $T^2 = \frac{4\pi^2}{GM} a^3$

Vis-viva equation relates orbital speed ($v$), semi-major axis ($a$), and distance from central body ($r$)

• $v^2 = GM (\frac{2}{r} - \frac{1}{a})$
• At perihelion (closest approach), $r = a(1-e)$
• At aphelion (farthest point), $r = a(1+e)$

Orbital energy ($E$) given by $E = -\frac{GMm}{2a}$, negative for bound orbits (elliptical, circular)

Conservation principles in elliptical orbits

Conservation of angular momentum ($L$)

• Angular momentum conserved in absence of external torques
• $L = mvr\sin\theta$ where $\theta$ is angle between velocity and radial vectors
• Explains why planets move faster at perihelion and slower at aphelion

Conservation of energy

• Total energy ($E$) is sum of kinetic energy ($KE$) and potential energy ($PE$)
• $E = KE + PE = \frac{1}{2}mv^2 - \frac{GMm}{r}$
• Total energy determines shape of orbit
  1. Circular: $E < 0$ and $e = 0$
  2. Elliptical: $E < 0$ and $0 < e < 1$
  3. Parabolic: $E = 0$ and $e = 1$
  4. Hyperbolic: $E > 0$ and $e > 1$

Interplay between angular momentum and energy conservation

• As planet moves from aphelion to perihelion:
  1. Potential energy decreases
  2. Kinetic energy increases
  3. Total energy remains constant
• Increase in kinetic energy results in increased orbital speed
• Conservation of angular momentum causes planet to move closer to Sun

Celestial mechanics and orbital dynamics

• Celestial mechanics applies Kepler's laws and Newtonian physics to study the motion of celestial bodies
• Orbital mechanics focuses on spacecraft trajectories and maneuvers in space
• Both fields rely on the principles of angular momentum and conservation of energy to analyze and predict orbital behavior