Orbital mechanics los angeles times crossword or astrodynamics is the application los angeles times crossword of 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 los angeles times crossword 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 los angeles times crossword , including both spacecraft and natural astronomical bodies such as star systems, planets , moons , and comets . Orbital mechanics focuses on spacecraft trajectories , including orbital maneuvers , orbit 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 is sometimes necessary for greater accuracy or in high-gravity situations (such as orbits close to the Sun).
Rules of thumb 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's laws of planetary motion , which can be mathematically derived from Newton's laws, hold strictly only in describing the motion of two gravitating los angeles times crossword bodies, in the absence of non-gravitational forces, or approximately when the gravity of a single massive body like the Sun dominates other effects: Orbits are either circular, with the planet at the center of the circle, or elliptical , with the planet at one focus of the ellipse.
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.
los angeles times crossword 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 los angeles times crossword are needed.
los angeles times crossword 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 perigee (lowest los angeles times crossword orbital point) at 180 degrees away from the firing point, which will be the apogee (highest orbital point). If the rocket is fired to speed the rocket, it will create an elliptical orbit with a higher apogee 180 degrees away from the firing point (which will become the perigee).
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 los angeles times crossword 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 the orbit at a lower altitude. Because lower orbits are faster than higher orbits, los angeles times crossword 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 . These rules of thumb are decidedly inaccurate when describing two or more bodies of similar mass, such as a binary star system los angeles times crossword . ( 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. History 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, los angeles times crossword 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 .
Practical techniques Further information: List of orbits Transfer orbits Transfer los angeles times crossword orbits allow spacecraft to move from one orbit to another. Usually they require a burn at the start, a burn at the end, and sometimes one or more burns in the middle. The Hohmann transfer orbit typically requires the least delta-v , but any orbit that intersects both the origin orbit and destination orbit may be used. Gravity assist and the Oberth effect In a gravity assist , a spacecraft swings by a planet and leaves in a different direction, at a different velocity. This is useful to speed or slow a spacecraft instead of carrying more fuel. 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 los angeles times crossword on the planet's orbit is negligible. los angeles times crossword 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 . Interplanetary Transport Network and fuzzy orbits Main article: Interplanetary Transport Network See also: Low energy transfers 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 problem with them is they are usually exceedingly slow, taking many years to arrive. In addition launch windows can be very far apart. They have, however, been employed on projects such as Genesis los angeles times crossword . This spacecraft visited Earth's lagrange L1 point and returned using very little propellant. 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 his differential calculus . Standard assumptions in astrodynamics include non-interference from outside bodies, negligible mass for one of the bodies, and negligible other forces (such as from the solar wind, atmospheric drag, etc.). More accurate calculations can be made without these simplifying assumptions, but they are more complicated. The increased accuracy los angeles times crossword often does not make enough of a difference in the calculation to be worthwhile. Kepler's los angeles times crossword laws of planetary motion may be derived los angeles times crossword from Newton's los angeles times crossword 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 los angeles times crossword laws are invalidated. When the thrust stops, the resulting orbit will be different but will once again be described by Kepler's laws. The three laws are: The orbit of every planet is an ellipse with the sun at one of the foci .
Escape los angeles times crossword velocity The formula for escape los angeles times crossword velocity is easily derived as follows. The specific energy (energy per unit mass ) 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
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 los angeles times crossword 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 the vicinity of the Earth requires around 42 km/s velocity, but there will be "part credit" for the Earth's orbital velocity for spacecraft launched from Earth, if their further acceleration (due to the propulsion system) carries them in the same direction as Earth travels in its orbit. Formulae for free orbits Orbits are conic sections , so, naturally, the formula for the distance of a body for a given angle corresponds to the formula los angeles times crossword for that curve in polar coordinates , which is: . The parameters are given by the orbital elements los angeles times crossword . Circular los angeles times crossword orbits Although most orbits are elliptical in nature, a special case 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 where G is the gravitational constant los angeles times crossword , equal to 6.672 598 10 11 m 3 /(kg s 2 ) To properly use this formula, the units must be consistent; for example, M must be in kilograms, and r must be in meters. The answer will be in meters per second. The quantity GM 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 fou
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