Orbital mechanics

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from laws of motion and of universal gravitation derived by Isaac Newton. Astrodynamics is a core discipline within space-mission design and control.
Celestial mechanics treats more broadly the orbit dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers.
General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations.

History

Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics. At the time of Sputnik, the field was termed 'space dynamics'. The fundamental techniques, such as those used to solve the Keplerian problem, are therefore the same in both fields. Furthermore, the history of the fields is almost entirely shared.
Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1609. Isaac Newton published more general laws of celestial motion in the first edition of Philosophiæ Naturalis Principia Mathematica, which gave a method for finding the orbit of a body following a parabolic path from three observations. This was used by Edmund Halley to establish the orbits of various comets, including that which bears his name. Newton's method of successive approximation was formalised into an analytic method by Leonhard Euler in 1744, whose work was in turn generalised to elliptical and hyperbolic orbits by Johann Lambert in 1761–1777.
Another milestone in orbit determination was Carl Friedrich Gauss's assistance in the "recovery" of the dwarf planet Ceres in 1801. Gauss's method was able to use just three observations, to find the six orbital elements that completely describe an orbit. The theory of orbit determination has subsequently been developed to the point where today it is applied in GPS receivers as well as the tracking and cataloguing of newly observed minor planets. Modern orbit determination and prediction are used to operate all types of satellites and space probes, as it is necessary to know their future positions to a high degree of accuracy.
Astrodynamics was developed by astronomer Samuel Herrick beginning in the 1930s. He consulted the rocket scientist Robert Goddard and was encouraged to continue his work on space navigation techniques, as Goddard believed they would be needed in the future. Numerical techniques of astrodynamics were coupled with new powerful computers in the 1960s, and humans were ready to travel to the Moon and return.

Practical techniques

Rules of thumb

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics outlined below. The specific example discussed is of a satellite orbiting a planet, but the rules of thumb could also apply to other situations, such as orbits of small bodies around a star such as the Sun.

Kepler's laws of planetary motion:
*Orbits are elliptical, with the more massive body at one focus of the ellipse. A special case of this is a circular orbit with the planet at the center.
*A line drawn from the planet to the satellite sweeps out equal areas in equal times no matter which portion of the orbit is measured.
*The square of a satellite's orbital period is proportional to the cube of its average distance from the planet.
Without applying force, the period and shape of the satellite's orbit will not change.
A satellite in a low orbit moves more quickly with respect to the surface of the planet than a satellite in a higher orbit, due to the stronger gravitational attraction closer to the planet.
If thrust is applied at only one point in the satellite's orbit, it will return to that same point on each subsequent orbit, though the rest of its path will change. Thus one cannot move from one circular orbit to another with only one brief application of thrust.
From a circular orbit, thrust applied in a direction opposite to the satellite's motion changes the orbit to an elliptical one; the satellite will descend and reach the lowest orbital point at 180 degrees away from the firing point; then it will ascend back. The period of the resultant orbit will be less than that of the original circular orbit. Thrust applied in the direction of the satellite's motion creates an elliptical orbit with its highest point 180 degrees away from the firing point. The period of the resultant orbit will be longer than that of the original circular orbit.

The consequences of the rules of orbital mechanics are sometimes counter-intuitive. For example, if two spacecraft are in the same circular orbit and wish to dock, the trailing craft cannot simply fire its engines to accelerate towards the leading craft. This will change the shape of its orbit, causing it to gain altitude and slow down relative to the leading craft, thus moving away from the target. The space rendezvous before docking normally takes multiple precisely calculated engine firings in multiple orbital periods, requiring hours or even days to complete.
To the extent that the standard assumptions of astrodynamics do not hold, actual trajectories will vary from those calculated. For example, simple atmospheric drag is another complicating factor for objects in low Earth orbit.
These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system. Celestial mechanics uses more general rules applicable to a wider variety of situations. Kepler's laws of planetary motion, which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating bodies in the absence of non-gravitational forces; they also describe parabolic and hyperbolic trajectories. In the close proximity of large objects like stars the differences between classical mechanics and general relativity also become important.

Laws of astrodynamics

The fundamental laws of astrodynamics are Newton's law of universal gravitation and Newton's laws of motion, while the fundamental mathematical tool is differential calculus.
In a Newtonian framework, the laws governing orbits and trajectories are in principle time-symmetric.
Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces. More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy often does not make enough of a difference in the calculation to be worthwhile.
Kepler's laws of planetary motion may be derived from Newton's laws, when it is assumed that the orbiting body is subject only to the gravitational force of the central attractor. When an engine thrust or propulsive force is present, Newton's laws still apply, but Kepler's laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws which have been set out above. The three laws are:

The orbit of every planet is an ellipse with the Sun at one of the foci.
A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.
The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axis of the orbits.
Escape velocity

The formula for an escape velocity is derived as follows. The specific energy of any space vehicle is composed of two components, the specific potential energy and the specific kinetic energy. The specific potential energy associated with a planet of mass M is given by
where G is the gravitational constant and r is the distance between the two bodies. The specific kinetic energy of an object is given by
where v is its velocity. So the total specific orbital energy is
Since energy is conserved, cannot depend on the distance,, from the center of the central body to the space vehicle in question, i.e. v must vary with r to keep the specific orbital energy constant. Therefore, the object can reach infinite only if this quantity is nonnegative, which implies
The escape velocity from the Earth's surface is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational pull of the Sun. To escape the Solar System from a location at a distance from the Sun equal to the distance Sun–Earth, but not close to the Earth, requires around 42 km/s velocity, but there will be "partial credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration carries them in the same direction as Earth travels in its orbit.

Formulae for free orbits

Orbits are conic sections, so the formula for the distance of a body for a given angle corresponds to the formula for that curve in polar coordinates, which is
where is called the gravitational parameter, and are the masses of objects 1 and 2, and is the specific angular momentum of object 2 with respect to object 1. The parameter is known as the true anomaly, is the semi-latus rectum, while is the orbital eccentricity, all obtainable from the various forms of the six independent orbital elements.

Circular orbits

All bounded orbits where the gravity of a central body dominates are elliptical in nature. A special case of this is the circular orbit, which is an ellipse of zero eccentricity. The formula for the velocity of a body in a circular orbit at distance r from the center of gravity of mass M can be derived as follows:
Centrifugal acceleration matches the acceleration due to gravity. So,
Therefore,
where m³/ is the gravitational constant.
To properly use this formula, the units must be consistent; for example, must be in kilograms, and must be in meters, then the answer will be in meters per second.
The quantity is often termed the standard gravitational parameter, which has a different value for every planet or moon in the Solar System.
Once the circular orbital velocity is known, the escape velocity is easily found by multiplying by :
To escape from gravity, the kinetic energy must at least match the negative potential energy. Therefore,