Consider a Planet Moving Around a Star — Orbital Mechanics and Kepler's Laws

1. Kepler's First Law (Law of Orbits): All planets move in elliptical orbits with the star at one focus.

2. Kepler's Second Law (Law of Areas): A line joining the planet to the star sweeps out equal areas in equal times.

   • Planet moves faster when closer to the star (perihelion).
   • Planet moves slower when farther from the star (aphelion).
   • This is a consequence of conservation of angular momentum.

3. Kepler's Third Law (Law of Periods): T² ∝ R³ (for circular orbits, R is the radius; for elliptical, R is the semi-major axis) T²/R³ = 4π²/(GM) = constant for all planets orbiting the same star

Where:

• T = orbital period
• R = mean orbital radius
• G = 6.674 × 10⁻¹¹ N·m²/kg²
• M = mass of the star

For a circular orbit:

Gravitational force = Centripetal force GMm/R² = mv²/R

Solving for orbital speed: v = √(GM/R)

Note: Orbital speed is independent of the planet's mass!

Orbital time period: T = 2πR/v = 2πR/√(GM/R) = 2π√(R³/GM)

Squaring: T² = 4π²R³/(GM) This confirms T² ∝ R³ (Kepler's Third Law)

For Earth orbiting the Sun:

• G = 6.674 × 10⁻¹¹ N·m²/kg²
• M_Sun = 2 × 10³⁰ kg
• R_Earth = 1.5 × 10¹¹ m
• T_Earth = 365.25 days ≈ 3.156 × 10⁷ s

For a planet of mass m in circular orbit of radius R around a star of mass M:

Kinetic Energy: KE = ½mv² = GMm/(2R) Potential Energy: PE = −GMm/R (negative, bound system) Total Mechanical Energy: E = KE + PE = GMm/(2R) − GMm/R = −GMm/(2R)

Key observations:

1. Total energy is NEGATIVE → planet is gravitationally bound to the star.
2. |Total energy| = KE (magnitude of total energy equals kinetic energy).
3. As orbit radius R increases, total energy increases (becomes less negative).
4. To move a planet to a larger orbit, energy must be added.

Escape velocity: v_esc = √(2GM/R) = √2 × v_orbital