2.1 Kepler's laws and orbital dynamics

Kepler's laws revolutionized our understanding of planetary motion. They describe elliptical orbits, varying orbital speeds, and the relationship between orbital period and distance from the Sun. These laws form the foundation for modern celestial mechanics.

Orbital dynamics builds on Kepler's work, providing mathematical tools to analyze orbits in detail. We can calculate an object's position, velocity, and energy at any point in its orbit. This knowledge is crucial for space exploration and satellite operations.

Kepler's Laws of Planetary Motion

Kepler's laws of planetary motion

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• First Law: The Law of Ellipses describes planetary orbits as elliptical paths with the Sun positioned at one focus enabling accurate predictions of planetary positions
• Second Law: The Law of Equal Areas states a line connecting a planet to the Sun sweeps out equal areas in equal time intervals explaining variations in orbital velocity (faster near perihelion, slower near aphelion)
• Third Law: The Law of Periods establishes the relationship between orbital period and semi-major axis applies universally to all objects orbiting the same central body (planets, moons, artificial satellites)

Orbital period and semi-major axis

• Mathematical form: $T^2 = \frac{4\pi^2}{GM}a^3$ relates orbital period (T) to semi-major axis (a) and central body mass (M)
• Derivation steps:
  1. Equate gravitational force to centripetal force
  2. Solve for velocity in terms of radius
  3. Express period using circumference and velocity
  4. Substitute and simplify to obtain final equation
• Applications include calculating orbital periods of exoplanets and determining satellite orbits for communication systems

Orbital Dynamics

Velocity and position in elliptical orbits

• Orbital elements: semi-major axis (a), eccentricity (e), true anomaly (θ) define the orbit's size, shape, and object's position
• Position in polar coordinates:
  • Radius: $r = \frac{a(1-e^2)}{1 + e\cos\theta}$ gives distance from focus
  • Angle: θ (true anomaly) measures angular position from periapsis
• Velocity components:
  • Radial: $v_r = \sqrt{\frac{GM}{a(1-e^2)}}e\sin\theta$ (towards or away from focus)
  • Tangential: $v_\theta = \sqrt{\frac{GM}{a(1-e^2)}}(1 + e\cos\theta)$ (perpendicular to radial)
• Total velocity: $v = \sqrt{v_r^2 + v_\theta^2}$ varies throughout orbit (fastest at periapsis, slowest at apoapsis)
• Kepler's equation $M = E - e\sin E$ relates mean anomaly (M) to eccentric anomaly (E) for time-dependent calculations

Effects of orbital eccentricity

• Eccentricity (e) measures orbit deviation from circular shape ranges from 0 (circular) to nearly 1 (highly elliptical)
• Orbit shapes: circular (e = 0), elliptical (0 < e < 1), parabolic (e = 1, escape velocity), hyperbolic (e > 1, interstellar trajectories)
• Total energy $E = -\frac{GMm}{2a}$ remains constant throughout orbit more eccentric orbits have higher energy for given semi-major axis
• Periapsis $r_p = a(1-e)$ and apoapsis $r_a = a(1+e)$ distances vary with eccentricity affecting orbital velocity extremes
• Angular momentum conservation explains velocity changes throughout orbit (faster at periapsis, slower at apoapsis)