3.1 Kepler's laws and orbital elements

Kepler's laws and orbital elements are the building blocks of planetary motion. They explain how planets move around the Sun, from their elliptical paths to their varying speeds. These laws connect the dots between orbit shapes, sizes, and periods.

Understanding these concepts is crucial for grasping planetary dynamics. They help us predict celestial movements, launch satellites, and explore space. Kepler's laws and orbital elements are the foundation for studying how objects behave in space.

Kepler's Laws of Planetary Motion

Elliptical Orbits and the Law of Ellipses

• Kepler's first law states that the orbit of a planet around the Sun is an ellipse with the Sun at one focus
• This law is also known as the law of ellipses
• Elliptical orbits have two foci, with the Sun located at one focus
• The shape of the ellipse is determined by its eccentricity (e), which ranges from 0 (circular orbit) to 1 (parabolic orbit)
• Planets move faster when they are closer to the Sun (perihelion) and slower when they are farther away (aphelion)

Equal Areas in Equal Times and the Law of Areas

• Kepler's second law, the law of equal areas, states that a line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time
• This means that a planet's orbital speed varies depending on its distance from the Sun
• The area swept out by the line segment connecting the planet and the Sun is proportional to the time taken to sweep out that area
• The law of equal areas is a consequence of the conservation of angular momentum in the absence of external torques

Orbital Period and the Law of Periods

• Kepler's third law, the law of periods, states that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit
• Mathematically, this can be expressed as $T^2 \propto a^3$, where T is the orbital period and a is the semi-major axis
• This law relates the size of an orbit (semi-major axis) to the time it takes to complete one orbit (orbital period)
• The proportionality constant depends on the mass of the central body (Sun for planets, planet for moons)

Orbital Elements for Characterizing Orbits

Size and Shape of the Orbit

• The semi-major axis (a) is half the longest diameter of an elliptical orbit and determines the size of the orbit
• Eccentricity (e) describes the shape of the orbit
  • e=0 for a circular orbit
  • 0<e<1 for an elliptical orbit
  • e=1 for a parabolic orbit
  • e>1 for a hyperbolic orbit

Orientation of the Orbital Plane

• Inclination (i) is the angle between the orbital plane and a reference plane
  • For Earth-orbiting objects, the reference plane is typically the Earth's equatorial plane
  • For Solar System objects, the reference plane is usually the ecliptic plane (the plane of Earth's orbit around the Sun)
• The longitude of the ascending node (Ω) is the angle from a reference direction (usually the vernal equinox) to the ascending node (where the orbit crosses the reference plane from south to north)

Orientation of the Orbit within the Orbital Plane

• The argument of periapsis (ω) is the angle from the ascending node to the periapsis (the closest approach to the central body) measured in the direction of motion
• The true anomaly (ν) is the angle between the periapsis and the current position of the orbiting object measured in the direction of motion
• These angles help to specify the orientation of the elliptical orbit within the orbital plane

Orbital Period and Velocity Calculation

Orbital Period Calculation

• The orbital period (T) can be calculated using the equation $T^2 = 4π^2a^3 / GM$, where a is the semi-major axis, G is the gravitational constant, and M is the mass of the central body
• This equation is derived from Kepler's third law and Newton's law of universal gravitation
• The orbital period depends on the size of the orbit (semi-major axis) and the mass of the central body

Orbital Velocity Calculation

• The orbital velocity (v) at any point in the orbit can be determined using the vis-viva equation: $v^2 = GM(2/r - 1/a)$, where r is the distance from the central body to the orbiting object
• For a circular orbit, the orbital velocity is constant and can be calculated using the equation $v = \sqrt{GM/r}$, where r is the radius of the orbit
• In elliptical orbits, the velocity varies depending on the distance from the central body, with the highest velocity at periapsis and the lowest at apoapsis

Kepler's Laws and Conservation Principles

Conservation of Angular Momentum

• Kepler's second law is a consequence of the conservation of angular momentum in the absence of external torques
• The constant areal velocity results from the conservation of angular momentum
• The specific angular momentum (h) of a planet is related to the semi-major axis (a) and eccentricity (e) of its orbit by the equation $h = \sqrt{GMa(1-e^2)}$, demonstrating the connection between the orbital elements and the conservation of angular momentum

Conservation of Energy

• The total energy (sum of kinetic and potential energy) of a planet in an elliptical orbit is conserved, as the gravitational force is a conservative force
• The planet's speed varies throughout its orbit, with the highest speed at perihelion and the lowest at aphelion, but the total energy remains constant
• The specific orbital energy (ε) is related to the semi-major axis by the equation $ε = -GM/2a$, showing the link between the size of the orbit and the total energy of the system
• The conservation of energy explains why planets move faster when they are closer to the Sun and slower when they are farther away