Orbit

Understanding Orbits: A Journey Through Space

Imagine a celestial dance where planets twirl around their stars in perfect harmony. This cosmic ballet is governed by the laws of physics and mathematics, specifically orbital mechanics.

The Basics of Orbital Mechanics

Orbits are the curved paths that objects follow due to gravitational forces. They can be regular or irregular, with Newtonian mechanics providing a good approximation for most cases. But when we delve into the depths of space, general relativity offers a more accurate description.

Ancient Philosophers and Modern Science

Historically, ancient philosophers used the concept of celestial spheres to explain planetary motions. Later, Copernicus shifted the center from Earth to the Sun, while Kepler formulated his laws describing elliptical orbits with varying speeds. These laws were derived from Newton’s theory of gravitation, showing that orbits are conic sections and their sizes inversely proportional to masses.

Newtonian Mechanics and Beyond

Advances in Newtonian mechanics explored variations from simple assumptions behind Kepler orbits, including perturbations due to other bodies and spheroidal rather than spherical objects. Joseph-Louis Lagrange developed a new approach emphasizing energy over force, making progress on the three-body problem and discovering Lagrangian points.

General Relativity: A New Perspective

Albert Einstein explained that gravity is due to curvature of space-time in his 1916 paper ‘The Foundation of the General Theory of Relativity.’ In relativity theory, orbits follow geodesic trajectories, which are usually approximated well by Newtonian predictions except where there are very strong gravity fields and high speeds.

Orbital Elements and Dynamics

The original vindication of general relativity is that it accounted for unexplained amounts in Mercury’s perihelion precession. However, Newton’s solution remains used for short-term purposes due to its ease of use and sufficient accuracy. Within a planetary system, bodies orbit the barycenter in elliptical orbits with mutual gravitational perturbations causing orbital eccentricities to vary over time.

Periapsis and Apoapsis: The Closest and Farthest Points

As two objects orbit each other, there is a periapsis (closest point) and apoapsis (farthest point). Planets in a star’s system have an elliptical orbit with the barycenter as one focal point. The sum of kinetic and potential energy remains constant at every point along the orbit.

Orbital Velocity Relationships

The Newton’s cannonball model illustrates how gravity affects motion in two-body problems, including orbits around planets. By firing a cannonball at different speeds, it demonstrates various orbital paths, such as circular, elliptical, and hyperbolic trajectories. Newton’s laws of motion describe the acceleration of an object as the sum of forces acting on it divided by its mass, with gravitational force proportional to the product of masses and inversely proportional to distance.

Kepler’s Laws: A Mathematical Description

The law of gravitation describes the gravitational force between two bodies, decreasing with distance. Gravitational potential energy is associated with gravitational fields and changes when objects are brought closer or farther apart. When two gravitational bodies interact, their orbits follow conic sections depending on total energy (kinetic + potential energy). Open orbits have positive total energy and are parabolic or hyperbolic shapes. Closed orbits have negative total energy and are ellipses.

Orbital Periods and Decay

Bodies following closed orbits repeat their paths with a certain time called the period. The orbit of a planet is described by Kepler’s laws, which can be formulated as follows: – The orbit lies in a plane – The point on the orbit closest to the attracting body is periapsis – The point farthest from the attracting body is apoapsis – There are specific terms for orbits about particular bodies

Orbital Decay and Perturbations

An orbit around any star has periastron and apastron. The planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area.

Limitations of Newton’s Law

The limitations of Newton’s law of gravitation include non-spherical or non-Newtonian effects causing the orbit’s shape to depart from closed ellipses. The two-body solutions were published by Newton in 1687, but there is no known method to solve the equations of motion for a system with four or more bodies.

Orbital Dynamics and Perturbations

Perturbations occur when external forces cause acceleration, changing the orbit’s parameters over time. Radial, prograde, and transverse perturbations can change the eccentricity and/or orbital period. Orbital decay occurs due to atmospheric drag, tidal forces, or gravitational waves.

Artificial Influences on Orbits

Orbital decay can occur due to tidal forces for objects below the synchronous orbit for the body they’re orbiting, which raises tidal bulges in the primary, causing a slight acceleration on one side and deceleration on the other. Artificial satellites are too small to have an appreciable tidal effect, but some moons like Phobos undergo orbital decay by this mechanism.

Orbital Mechanics in Practice

The time-averaged orbital distance is given by: r¯ = a(1 + e^2/2). An unperturbed orbit is two-dimensional, but extending it to three dimensions requires rotating the plane into space. The orbital period is simply the time taken to complete one orbit.

Orbital Dynamics in Space Missions

Perturbations occur when external forces cause acceleration, changing the orbit’s parameters over time. Radial, prograde, and transverse perturbations can change the eccentricity and/or orbital period. Orbital decay occurs due to atmospheric drag, tidal forces, or gravitational waves.

Orbital Mechanics in Space

The effects of other gravitating bodies can be significant, particularly for bodies orbiting within the heavier body’s Hill sphere. The n-body problem refers to systems with multiple gravitating bodies, which often have no closed form solution but some special cases have been formulated.

Light Radiation and Stellar Wind

Light radiation and stellar wind can cause significant perturbations to the attitude and direction of motion of smaller bodies, particularly asteroids that are rotating relative to the Sun. Strange orbits have been discovered mathematically, including non-elliptical orbits that repeat periodically.

The Future of Orbital Mechanics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to spacecraft motion. It uses Newton’s laws and gravity to calculate orbits, trajectories, and maneuvers. The discipline treats the orbital dynamics of systems under gravity, including space missions.

Conclusion: A Dance of Forces

The dance of forces that governs our universe is a complex yet beautiful symphony. From ancient philosophers to modern astrophysicists, we continue to unravel the mysteries of orbits and their intricate patterns. Understanding these dynamics not only helps us navigate through space but also deepens our appreciation for the cosmos.