8.2 Orbital Elements and Kepler's Laws

Orbital elements and Kepler's laws are fundamental to understanding celestial mechanics. These concepts describe how objects move in space, from planets orbiting the sun to satellites circling Earth. They provide a framework for predicting and analyzing orbital paths.

Kepler's laws explain the elliptical nature of orbits, how orbital speed varies, and the relationship between orbital period and distance from the central body. These principles, combined with orbital elements, allow us to precisely calculate and predict the motion of objects in space.

Orbital Elements and Kepler's Laws

Elements of orbital description

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• Semi-major axis ($a$)
  • Determines the size and energy of the orbit
  • Calculated as half the distance between the closest (periapsis) and farthest (apoapsis) points of the orbit (Earth's semi-major axis is 1 AU)
• Eccentricity ($e$)
  • Measures the deviation of the orbit from a perfect circle
  • Ranges from 0 for circular orbits to 1 for parabolic escape trajectories and greater than 1 for hyperbolic orbits (Earth's eccentricity is 0.0167)
• Inclination ($i$)
  • Measures the tilt of the orbital plane relative to a reference plane, usually the equatorial plane of the primary body
  • Ranges from 0° for orbits in the reference plane to 90° for polar orbits perpendicular to the reference plane (Earth's inclination to the ecliptic is 0°)
• Right Ascension of the Ascending Node (RAAN or $\Omega$)
  • Defines the orientation of the orbit in the reference plane, measured as the angle from the vernal equinox to the ascending node (point where the orbit crosses the reference plane from south to north)
  • Ranges from 0° to 360° (Earth's RAAN is undefined as it orbits in the ecliptic plane)
• Argument of Periapsis ($\omega$)
  • Specifies the orientation of the orbit within its plane, measured as the angle from the ascending node to the periapsis (closest point to the primary body)
  • Ranges from 0° to 360° (Earth's argument of perihelion is 114.20°)
• True Anomaly ($\nu$)
  • Indicates the current position of the orbiting body along the orbit, measured as the angle from the periapsis to the body's current location
  • Ranges from 0° at periapsis to 180° at apoapsis and back to 360° at periapsis (Earth's true anomaly varies throughout its orbit)

Kepler's laws of planetary motion

• First Law: The Law of Ellipses
  • States that planets orbit the Sun in elliptical paths, with the Sun located at one focus of the ellipse
  • Implies that the distance between a planet and the Sun varies throughout its orbit (Earth is closest to the Sun in January and farthest in July)
• Second Law: The Law of Equal Areas
  • States that a line segment connecting a planet to the Sun sweeps out equal areas in equal time intervals
  • Implies that planets move faster when they are closer to the Sun and slower when they are farther away (Earth's orbital velocity is about 30 km/s at perihelion and 29 km/s at aphelion)
• Third Law: The Law of Periods
  • States that the square of a planet's orbital period is directly proportional to the cube of its orbit's semi-major axis, expressed mathematically as $P^2 = a^3$ (in units where the period is in years and semi-major axis in AU)
  • Implies that planets farther from the Sun have longer orbital periods (Earth's period is 1 year, while Jupiter's is about 12 years)

Calculations with Kepler's laws

• Orbital period ($P$) calculation
  1. Use Kepler's Third Law: $P^2 = a^3$
  2. Solve for $P$: $P = \sqrt{a^3}$, where $P$ is in years and $a$ is in AU (For Earth, $a = 1$ AU, so $P = 1$ year)
• Orbital velocity ($v$) calculation
  • Use the vis-viva equation: $v^2 = \mu (\frac{2}{r} - \frac{1}{a})$, where $\mu$ is the standard gravitational parameter, $r$ is the current orbital radius, and $a$ is the semi-major axis (For Earth, $\mu = 1.327 \times 10^{20}$ m³/s², $r$ varies, and $a = 1.496 \times 10^{11}$ m)
• Orbital position calculation
  1. Determine the true anomaly ($\nu$) and the orbital elements
  2. Calculate the radius at the given true anomaly: $r = \frac{a(1-e^2)}{1+e\cos\nu}$
  3. Express the position vector in the orbital plane: $\vec{r} = r \cos\nu \hat{i} + r \sin\nu \hat{j}$, where $\hat{i}$ and $\hat{j}$ are unit vectors in the orbital plane

Types of orbital paths

• Elliptical orbits
  • Eccentricity between 0 and 1 ($0 \leq e < 1$)
  • Closed orbits with a finite orbital period (Earth, Mars, and most satellites have elliptical orbits)
  • Characterized by negative specific orbital energy, indicating bound orbits
• Parabolic orbits
  • Eccentricity equal to 1 ($e = 1$)
  • Open orbits with an infinite orbital period
  • Represent escape trajectories with zero specific orbital energy (Spacecraft often use parabolic trajectories to leave Earth's sphere of influence)
• Hyperbolic orbits
  • Eccentricity greater than 1 ($e > 1$)
  • Open orbits with no defined orbital period
  • Characterized by positive specific orbital energy, indicating unbound orbits (Interstellar objects passing through the solar system follow hyperbolic trajectories)