Orbital mechanics

[1] The fundamental techniques, such as those used to solve the Keplerian problem (determining position as a function of time), are therefore the same in both fields.

Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy, publishing his laws in 1605.

Isaac Newton published more general laws of celestial motion in the first edition of Philosophiæ Naturalis Principia Mathematica (1687), which gave a method for finding the orbit of a body following a parabolic path from three observations.

Another milestone in orbit determination was Carl Friedrich Gauss's assistance in the "recovery" of the dwarf planet Ceres in 1801.

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.

Numerical techniques of astrodynamics were coupled with new powerful computers in the 1960s, and humans were ready to travel to the Moon and return.

The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics outlined below.

For example, if two spacecrafts are in the same circular orbit and wish to dock, unless they are very close, the trailing craft cannot simply fire its engines to go faster.

This will change the shape of its orbit, causing it to gain altitude and actually slow down relative to the leading craft, missing the target.

These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system (see n-body problem).

In the close proximity of large objects like stars the differences between classical mechanics and general relativity also become important.

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.

, 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.

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.

is often termed the standard gravitational parameter, which has a different value for every planet or moon in the Solar System.

, and substituting the result in the conic section curve formula above, we get: Under standard assumptions the orbital period (

from periapsis is broken into two steps: Finding the eccentric anomaly at a given time (the inverse problem) is more difficult.

The main difficulty with this approach is that it can take prohibitively long to converge for the extreme elliptical orbits.

For simple procedures, such as computing the delta-v for coplanar transfer ellipses, traditional approaches[clarification needed] are fairly effective.

The Hohmann transfer orbit alone is a poor approximation for interplanetary trajectories because it neglects the planets' own gravity.

Planetary gravity dominates the behavior of the spacecraft in the vicinity of a planet and in most cases Hohmann severely overestimates delta-v, and produces highly inaccurate prescriptions for burn timings.

To address computational shortcomings of traditional approaches for solving the 2-body problem, the universal variable formulation was developed.

It works equally well for the circular, elliptical, parabolic, and hyperbolic cases, the differential equations converging well when integrated for any orbit.

In a two-body simulation, these elements are sufficient to compute the satellite's position and velocity at any time in the future, using the universal variable formulation.

The following are some effects which make real orbits differ from the simple models based on a spherical Earth.

Most of them can be handled on short timescales (perhaps less than a few thousand orbits) by perturbation theory because they are small relative to the corresponding two-body effects.

This works because the cosine of a small angle is very nearly one, resulting in the small plane change being effectively "free" despite the high velocity of the spacecraft near periapse, as the Oberth Effect due to the increased, slightly angled thrust exceeds the cost of the thrust in the orbit-normal axis.

This maneuver can be approximated by an elastic collision at large distances, though the flyby does not involve any physical contact.

Due to Newton's Third Law (equal and opposite reaction), any momentum gained by a spacecraft must be lost by the planet, or vice versa.