Kepler's Laws of Planetary Motion

Johannes Kepler (1571-1630)

• As a child, displayed a knack for math and was intrigued by the cosmos.
• Two astronomical events sparked his interest: the Great Comet of 1577 and a lunar eclipse in 1580.
• Kepler, a German astronomer and mathematician, accurately described planetary motion without knowing about gravity.
• His work paved the way for Isaac Newton's theory of gravity.

Kepler's First Law of Planetary Motion: "The Law of Ellipses"

• A planet's orbit is an ellipse, with the Sun located at one of the two foci.

Kepler's Second Law of Planetary Motion: "The Law of Equal Areas"

• A line segment connecting a planet and the Sun sweeps out equal areas during equal time intervals.

Kepler's Third Law of Planetary Motion: "The Law of Harmonies"

• The square of the orbital period (P) of a planet is proportional to the cube of the semi-major axis (a) of its orbit: P² ∝ a³

Orbits

• An orbit refers to the curved path an object in space follows around another due to gravity; examples include stars, planets, moons, asteroids, or spacecraft.

Low Earth Orbit (LEO)

• An orbit that is relatively close to Earth's surface, ranging from 180 km to 2000 km.
• LEO satellites can have their orbital planes tilted at various angles.
• Uses for LEO include satellite imaging, communication, and the International Space Station (ISS).

Geostationary Orbit (GEO)

• Satellites fly above Earth's equator at 35,786 km, moving west to east, matching Earth's rotation.
• GEO's location makes the satellites appear stationary as viewed from Earth.
• GEO is ideal for satellites needing to stay fixed at a specific location, like telecommunication and weather satellites.

Polar Orbit (PO)

• A type of low Earth orbit at an altitude of 200 to 1000 km, with a speed of 7.5 km per second.
• Satellites travel around Earth from one pole to the other, instead of from west to east; deviations of 10-30 degrees are still considered polar orbits.
• PO is useful for global Earth coverage, reconnaissance, and Earth observation.

Sun-Synchronous Orbit (SSO)

• A specific polar orbit synchronized with the Sun.
• Satellites always appear in the same position relative to the Sun, passing over the same spot on Earth at the same local time daily.
• At an altitude of 600-800 km and a speed of 7.5 km/s at 800 km, SSO allows images to be more comparable in terms of light and shadows.
• SSO is useful for monitoring weather patterns, emergencies (forest fires, flooding), and accumulating data on long-term problems (rising sea levels).

Medium Earth Orbit (MEO)

• It covers a range of altitudes between LEO.
• Satellites do not need to follow specific paths, and the orbit is used by various satellites for different purposes.
• Navigation satellites, like the European Galileo system, commonly use MEO to provide navigation across the world.

Transfer Orbits and Geostationary Transfer Orbit (GTO)

• Transfer orbits move satellites from one orbit to another.
• Satellites launched from Earth by rockets aren't always directly in their final orbit.
• Satellites are often placed into a transfer orbit, using onboard motors to move from one orbit to another.
• Satellites use engines to change orbit eccentricity and achieve higher/lower altitudes, also modifying orbital inclination and the semi-major axis.

Highly Eccentric Orbit (HEO)

• Spacecraft distance varies greatly, coming close at high speeds and then moving far into space and slowing down.
• Comets have highly eccentric orbits formed at the edges of the Solar System with orbital eccentricity close to one.
• Halley's Comet has an eccentricity of 0.967 and takes millions of years to orbit the Sun fully.
• HEO is useful for missions observing Earth or focusing on space from high altitudes for extended periods.

Isaac Newton (1642-1727)

• Realized the physical laws that operate on Earth also operate in the heavens.
• In 1666, Newton developed his laws of motion.
• In 1687, he discussed these laws in his work "Principia Mathematica Philosophiae Naturalis", explaining how outside forces affect movement.

Newton's First Law of Motion: "Law of Inertia"

• An object maintains a constant velocity unless a net force acts upon it, altering its speed or direction.
• An object at rest will remain at rest.

Newton's Second Law of Motion: "Law of Acceleration"

• When a net force is present on an object, it causes a net acceleration of that object.
• F = mass x acceleration

Newton's Third Law of Motion: "Law of Action and Reaction"

• For every force, there is an equal and opposite reaction force.

The Universal Law of Gravitation

• Every particle attracts every other particle in the universe with a force directly proportional to the product of the masses and inversely proportional to the square of the distance.
• Equation: Fg = G(M₁M₂ / d²)
• G = 6.67 x 10^-11 N⋅m²/kg², also known as the gravitational constant.
• M = mass of the bodies
• d = distance between the bodies.

Conservation of Angular Momentum

• Angular momentum of an object remains constant unless an external force acts on it.

Total Orbital Energy

• Total orbital energy stays constant if there is no external force.
• Total orbital energy = gravitational potential energy + kinetic energy

Geometry of an Ellipse

• Key aspects include co-vertex, vertex, semi-minor axis, focus, center, linear eccentricity, and semi-major axis.

Circular Orbits

• Velocity is constant.
• Formula to calculate the orbital velocity: V = √(GM/r)

Elliptical Orbits

• The velocity is not constant.
• The distance between the two bodies orbiting each other changes.
• µ = G(M1 + M2) or GM
• v (orbital speed of body in an elliptical orbit) = √µ(2/r - 1/a)
  • µ = standard gravitational parameter
  • r = distance between orbiting bodies
  • a = length of the semi major axis

Eccentricity of an Orbit

• Can be obtained through apoapsis and periapsis using the formula: e = (ra - rp) / (ra + rp)

Orbital Maneuver

• A planned action that changes a spacecraft's orbit in space by firing thrusters or engines.
• Adjusting velocity/direction allows moving to a different altitude, inclination, or orbital period to meet mission objectives; it's a controlled change in spacecraft trajectory.

Impulsive Maneuver

• A quick burst of thrust changes a spacecraft's velocity to change its orbit.

Phasing Maneuver

• Changes the position of a spacecraft in its orbit.

Chase Maneuver

• One spacecraft actively adjusts trajectory to intercept another in a specific timeframe.

Hohmann Maneuver

• Transfers a spacecraft between two orbits of different altitudes around a central body.
• It achieves transfer by placing the spacecraft in an elliptical transfer orbit that is tangential to both initial and target orbits.
• The maneuver uses two impulsive engine burns, first to establish the transfer orbit and second to adjust it to match the target.
• It uses the lowest possible impulse, resulting in longer travel times than higher-impulse transfers.