Orbital mechanics

Orbital mechanics, also known as astrodynamics, is the study of how spacecraft move through space. It applies the principles of ballistics and celestial mechanics to rockets and satellites. This field is essential for space-mission design and control. Mission planners use these rules to predict the results of propulsive maneuvers. Without these calculations, we could not navigate between planets or maintain satellites. While celestial mechanics looks at the motion of all natural bodies, like stars and comets, astrodynamics focuses specifically on human-made trajectories.

The motion of an object in orbit is a balance between two forces. Gravity pulls the object toward the center of a massive body, like a planet. At the same time, the object has tangential velocity, which is its forward motion. In a circular orbit, the centrifugal acceleration matches the acceleration due to gravity. This balance keeps the object at a steady distance. If the object moves too slowly, gravity pulls it down. If it moves too fast, it might escape the gravity entirely.

Most orbits follow specific geometric shapes called conic sections. A circular orbit is a special case where the eccentricity is zero. Most other orbits are elliptical, which are stretched-out circles. An ellipse has two points called foci. According to Kepler's laws, a massive body like the Sun sits at one of these foci. Within an elliptical orbit, there are two extreme points. The periapsis is the point closest to the central body. The apoapsis is the point farthest away. The distance between these two points is known as the apse line.

Scientists have spent centuries perfecting the math used to predict these paths. Johannes Kepler published his laws of planetary motion in 1609. Later, Isaac Newton published his laws of motion and universal gravitation in 1687. Newton's work allowed scientists to find the orbit of a body using just three observations. Edmund Halley used these methods to establish the orbits of various comets. In 1744, Leonhard Euler formalized an analytic method for these calculations. Johann Lambert later generalized this work to include elliptical and hyperbolic orbits between 1761 and 1777.

Another major milestone occurred in 1801 when Carl Friedrich Gauss helped recover the dwarf planet Ceres. Gauss used only three observations to find the six orbital elements that describe an orbit. In the 1930s, astronomer Samuel Herrick began developing astrodynamics. He worked with rocket scientist Robert Goddard to create space navigation techniques. By the 1960s, the arrival of powerful computers allowed humans to travel to the Moon. Today, these mathematical techniques are used in everything from GPS receivers to tracking new minor planets.

Changing an orbit requires precise engine firings, and the results can be counter-intuitive. If a satellite in a circular orbit applies thrust in the direction of its motion, it creates an elliptical orbit. The highest point, or apoapsis, will be 180 degrees away from where the engine fired. The orbital period will become longer. Conversely, if thrust is applied opposite to the direction of motion, the satellite descends. It will reach its lowest point, the periapsis, 180 degrees away from the firing point. This results in a shorter orbital period.

One surprising fact involves the process of a space rendezvous. If two spacecraft are in the same circular orbit, the trailing craft cannot simply accelerate to catch the leader. If the trailing craft fires its engines to speed up, it actually moves into a higher orbit. In this higher orbit, the craft slows down relative to the leader and moves further away. To successfully dock, pilots must perform multiple precisely calculated engine firings. This process can take many hours or even days to complete.

To leave a planet forever, a spacecraft must reach a specific speed called escape velocity. This speed is derived from the object's specific potential energy and specific kinetic energy. For Earth, the escape velocity is approximately 11 km/s. However, reaching this speed is only enough to escape Earth's gravity. To escape the entire Solar System from Earth's distance from the Sun, a velocity of about 42 km/s is required. Scientists can gain "partial credit" by launching in the same direction that Earth travels in its orbit.