If 0 e 1, then the denominator of the equation of free orbits varies with the true anomaly θ, but re

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton s laws of motion and Newton s law of universal gravitation. It is a core discipline within space mission design and control. Celestial mechanics treats more broadly the orbital 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 los angeles galaxy trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, los angeles galaxy and is used by mission planners to predict the results of propulsive maneuvers. General relativity los angeles galaxy is a more exact theory than Newton s laws for calculating orbits, and is sometimes necessary for greater los angeles galaxy accuracy or in high-gravity situations (such as orbits close to the Sun).
Until the rise of space travel in the twentieth century, there was little distinction between orbital and celestial mechanics. 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. 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 1605. Isaac Newton published more general laws of celestial motion in his 1687 book, Philosophiæ Naturalis Principia Mathematica.
The following rules of thumb are useful for situations approximated by classical mechanics under the standard assumptions of astrodynamics. 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 los angeles galaxy s laws of planetary motion, which can be mathematically derived from Newton s laws, hold strictly only in describing the motion los angeles galaxy of two gravitating bodies, in the absence of non-gravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects:
o Orbits are elliptical, with the planet at one focus of the ellipse. Special cases of this are circular orbits (a circle being simply an ellipse of zero eccentricity) with the planet at the center, and parabolic orbits los angeles galaxy (which are ellipses with eccentricity of exactly 1, which is simply an infinitely long ellipse) with the planet at the focus.
los angeles galaxy * A satellite in a low orbit (or low part of an elliptical orbit) moves more quickly with respect to the surface of the planet than a satellite in a higher orbit (or a high part of an elliptical orbit), due to the stronger gravitational attraction closer to the planet.
* If a brief rocket firing is made 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 to move from one circular orbit to another, at least two brief firings are needed.
* From a circular orbit, a brief firing of a rocket in the direction which slows the satellite down, will create an elliptical orbit with a lower periapse (lowest orbital point) at 180 degrees los angeles galaxy away from the firing point, which will be the apoapse (highest orbital point). If the rocket is fired to speed the rocket, it will create an elliptical orbit with a higher apoapse 180 degrees away from the firing point (which will become the periapse).
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, 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 miss its target. One approach is to actually fire a reverse thrust to slow down, and then fire again to re-circularize los angeles galaxy the orbit at a lower altitude. Because lower orbits are faster than higher orbits, the trailing craft will begin to catch up. A third firing at the right time will put the trailing craft in an elliptical orbit which will intersect the path of the leading craft, approaching from below.
To the degree 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 Earth orbit. los angeles galaxy 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). (Celestial mechanics uses more general rules applicable to a wider variety of situations.) The differences between classical mechanics and general relativity can also become important for large objects like planets.
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. However, because the planet is much, much more massive than the spacecraft, the effect on the planet s orbit is negligible.
The Oberth effect can be employed, particularly during a gravity assist operation. This effect is that use of a propulsion system works better at high speeds, and hence course changes are best done when close to a gravitating body; this can multiply the effective delta-v.
It is now possible to use computers to search for routes using the nonlinearities in the gravity of the planets and moons of the solar system. For example, it is possible to plot an orbit from high earth orbit to Mars, passing close to one of the Earth s Trojan points. Collectively referred to as the Interplanetary Transport Network, these highly perturbative, even chaotic, orbital trajectories in principle need no fuel (in practice keeping to the trajectory requires some course corrections). The biggest los angeles galaxy problem with them is they can be exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart.
Every orbit and trajectory outside atmospheres is in principle reversible, i.e., in the space-time function the time is reversed. The velocities are reversed and the accelerations are the same, including those due to rockets bursts. Thus if a rocket burst is in the direction of the velocity, in the reversed case it is opposite to the velocity. Of course in the case of rocket bursts there is no full reversal of events, both ways the same delta-v is used and the same mass ratio applies.
Standard assumptions in astrodynamics include non-interference from outside bodies, los angeles galaxy negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate los angeles galaxy 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 los angeles galaxy 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 los angeles galaxy orbit will be different but will once again be described by Kepler s laws. The three laws are:
The formula for escape velocity is easily derived as follows. The specific energy los angeles galaxy (energy per unit mass) of any space vehicle is composed los angeles galaxy 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
does not depend on the distance, r, from the center of the central body to the space vehicle in question. Therefore, the object can reach infinite r only if this quantity is nonnegative, which implies
The escape velocity from the Earth s surface los angeles galaxy is about 11 km/s, but that is insufficient to send the body an infinite distance because of the gravitational los angeles galaxy 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 part credit for the Earth s orbital velocity for spacecraft launched los angeles galaxy from Earth, if their further acceleration los angeles galaxy (due to the propulsion system) carries them in the same direction as Earth travels in its orbit.
where μ is called the gravitational parameter, G is the gravitational constant, m1 and m2 are the masses of objects 1 and 2, and h is the angular momentum of object 2 with respect to object 1. The parameter θ is known as the true anomaly, p is the semi-latus rectum, while e is the orbital los angeles galaxy eccentricity, all obtainable from the various forms of the six independent orbital elements.
All bounded orbits where the gravity of a central body dominates los angeles galaxy 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 is
If 0 e 1, then the denominator of the equation of free orbits varies with the true anomaly θ, but remains positive, never becoming zero. Therefore, the relative position vector remains bounded, having its smallest magnitude los angeles galaxy at periapsis rp which is given by:
desbribes the geometry of the hyperbolic orbit. The system consists of two symmetric curves. the orbiting body occupies one of them. The other one is its empty mathematical los angeles galaxy image. Clearly, the denominator los angeles galaxy of the equation above goes to zero when cosθ = -1/e. we denote los angeles galaxy this value of true anomaly
since the radial distance approaches infinity as the true anamolly approaches θ∞. θ∞ is known as the true anamolly of the asymptote. Observe that θ∞ lies between 900 and 1800. From the t