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Suborbital Flight

Suborbital flight

A sub-orbital spaceflight (or sub-orbital flight) is a spaceflight that does not involve putting a vehicle into orbit. Manned and unmanned sub-orbital flights have been undertaken to test spacecraft and launch vehicles intended for later orbital flight, but some vehicles have been designed exclusively to reach space sub-orbitally: manned vehicles such as the X-15 and SpaceShipOne, and unmanned ones such as ICBMs and sounding rockets. The sub-orbital spaceflight should not be confused with a partial orbital spaceflight: a low Earth orbit, with deorbiting after less than one full orbit, as in the Fractional Orbital Bombardment System.

Overview

During freefall the trajectory is part of an elliptic orbit as given by the orbital equation. The perigee distance is less than the radius of the Earth, hence the ellipse intersects the Earth. The major axis is vertical, the semi-major axis is more than one half of the radius of the Earth, and almost always less than the radius. If the objective is just to reach space, sub-orbital flights are appealing because this is very much easier (it simply means going higher than the edge of space) than to achieve orbit (which requires a velocity of about 8 km/s (18,000 mph)). A dedicated sub-orbital spacecraft can therefore be built and operated much more cheaply than an orbital spacecraft. Less powerful sub-orbital craft may not reach speeds much higher than around 1.1 km/s to 1.3 km/s (2,500-3,000 mph). However, for intercontinental ballistic space flights, like that of an ICBM, or a possible future commercial spacecraft, a typical speed is / might be 7 km/s For more information on the difference between sub-orbital and orbital spaceflights, refer to the article Difference between sub-orbital and orbital spaceflights.

Flight profiles

While there are a great many possible sub-orbital flight profiles, it is expected that some will be more common than others.

Tourist flights

Sub-orbital tourist flights will initially focus on attaining the altitude required to qualify as reaching space. The flight path will probably be either vertical or very steep, with the spacecraft landing back at its take-off site. The spacecraft will probably shut off its engines well before reaching maximum altutude, and then coast up to its highest point. During a few minutes, from the point when the engines are shut off to the point where the craft begins to slow its descent for landing, the passengers will experience weightlessness. In 2004, a number of companies worked on vehicles in this class as entrants to the Ansari X Prize competition. SpaceShipOne was officially declared by Rick Searfoss to have won the competition on October_4,2004 after completing two flights within a two week period.

Scientific experiments

Another potentially large market is research payloads. Often researchers want to run experiments in microgravity or above the atmosphere. There have reportedly been several offers from researchers to launch experiments on SpaceShipOne, which have been turned down until the next version of the vehicle[http://news.bbc.co.uk/1/hi/sci/tech/3722596.stm].

Intercontinental flights

Another possibly lucrative market for sub-orbital spacecraft is intercontinental flight. Research, such as that done for the X-20 Dyna-Soar project suggests that a semi-ballistic sub-orbital flight could travel from Europe to North America in less than an hour. Due to the high cost, this is likely to be initially limited to high value cargo such as courier flights, or as the ultimate business jet.

Reaching for orbit

Commercial spacecraft operators may use sub-orbital flights to allow a constant progression towards full orbital flight. The test craft will reach higher and higher velocities until they reach low earth orbit. There is considerable debate about the validity of this approach, however, as the scale of the two problems (sub-orbital and orbital flight) are very different. Still, winged, single stage to orbit designs like Skylon do exist, so it might not be a totally unreasonable approach.

History of manned sub-orbital spaceflight


- U.S. — X-15, 1963, Joseph A. Walker — two flights above 100km altitude
- U.S. — Mercury-Redstone 3 & Mercury-Redstone 4, 1961, Alan Shepard & Virgil Grissom
- U.S.S.R. — Soyuz 18a, 1975, Vasili Lazarev & Oleg Makarov — launch emergency caused suborbital flight
- U.S. (private) — SpaceShipOne, 2004, Mike Melvill & Brian Binnie — Ansari X-Prize winner

Future of manned sub-orbital spaceflight

Privately-held companies such as Blue Origin are taking an interest in sub-orbital spaceflight, due in part to ventures like the Ansari X Prize. NASA and others are experimenting with scramjet based hypersonic aircraft which may well be used with flight profiles that qualify as sub-orbital spaceflight.

See also


- Rocket launch site
- Office of Commercial Space Transportation Category:Spaceflight



Orbit

.]] :For other meanings of the term "orbit", see orbit (disambiguation) In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity.

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Planetary orbits

Within a planetary system, planets, asteroids, comets and space debris orbit the central star in elliptical orbits. Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet. Due to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Pluto and Mercury have the most eccentric orbits. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune. As an object orbits another, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. In the elliptical orbit, the centre of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity. See also: Kepler's laws of planetary motion

Understanding orbits

There are a few common ways of understanding orbits.
- As the object moves sideways, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall away beneath it.
- A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
- As the object falls, it moves sideways fast enough (has enough tangential velocity) to miss the orbited object. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center. As an illustration of the orbit around a planet (eg Earth), the much-used cannon model may prove useful (see image below). Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and we can ignore the effects of air friction on the cannon ball. 300px If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground (A). As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above). If the cannonball is fired with sufficient velocity, the ground will curve away from the ball at the same rate as the ball falls - it is now in orbit (D). The orbit may be circular like (D) or if the firing velocity is increased even more, the orbit may become more (E) and more (F) elliptical. At a certain even faster velocity (called the escape velocity) the motion changes from an elliptical orbit to a parabola.

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron # As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms. One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :\frac - r\left( \frac \right)^2 and :\frac\frac\left( r^2\frac \right). Since the force is always radial, the transverse acceleration is zero, and it follows that: :\frac = hu^2, where h is a constant of integration and we have introduced the auxiliary variable u defined as 1/r. If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields: :\frac + u = \frac. In the case of an inverse square force law the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The equation of the orbit described by the particle is thus: :r = \frac = \frac, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

See: Orbital elements For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are: :Semimajor axis = (periapsis + apoapsis)/2 = mean of the extreme radii :Periapsis = semimajor axis × (1 - eccentricity) = least distance :Apoapsis = semimajor axis × (1 + eccentricity) = largest distance Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (mean anomaly), or over the orbital angle as observed by the primary (true anomaly), then you get a different result. See here for details.

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Earth orbits

See Earth orbit for more details.
- Low Earth orbit
- High Earth Orbit
- Intermediate circular orbit
- Geostationary orbit
- Geosynchronous orbit
- Geostationary transfer orbit
- Molniya orbit
- Polar orbit
- Polar Sun Synchronous Orbit (this is not a complete list).

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10−11 N·m2/kg2
- 6.6742 × 10−11 m3/(kg·s2)
- 6.6742 × 10−11(kg/m3)-1s-2. Thus the constant has dimension density-1 time-2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula : GT^2 \sigma = 3\pi \left( \frac \right)^3, for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.

See also


- Astrojax
- Circular orbit
- Clarke orbit
- Difference between sub-orbital and orbital spaceflights
- Elliptic orbit
- Escape velocity
- Gravitational slingshot
- Hohmann transfer orbit
- Hyperbolic trajectory
- Kepler's laws of planetary motion
- Orbit equation
- Orbital maneuver
- Orbital period
- Orbital spaceflight
- Orbital speed
- Parabolic trajectory
- Retrograde motion
- Specific orbital energy
- Sub-orbital spaceflight
- Trajectory

External links


- An on-line orbit plotter: http://www.bridgewater.edu/departments/physics/ISAW/PlanetOrbit.html
- [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) Category:Celestial mechanics Category:Solar System als:Umlaufbahn ja:軌道 (力学) simple:Orbit th:วงโคจร

Spacecraft

, 2004.]] A spacecraft is a vehicle that travels through space. Spacecraft include robotic or unmanned space probes as well as manned vehicles. The term is sometimes also used to describe artificial satellites, which have similar design criteria.

Overview

The term spaceship is generally applied only to spacecraft capable of transporting people. A space suit has at times been called a miniature spacecraft or spaceship, emphasizing its purpose of keeping its wearer alive while traveling in the vacuum of outer space. The spacecraft is one of the primal elements in science fiction. Numerous short stories and novels are built up around various ideas for spacecraft. Some hard science fiction books focus on the technical details of the craft, while others treat the spacecraft as a given and delve little into its actual implementation.

Examples of past or existing spacecraft

Manned
- Apollo Spacecraft
- Gemini Spacecraft
- International Space Station
- Mir
- Mercury Spacecraft
- Shuttle Buran
- Shenzhou Spacecraft
- Space Shuttle
- Soyuz Spacecraft
- SpaceShipOne
- Voskhod Spacecraft
- Vostok Spacecraft Unmanned
- Cassini-Huygens
- Cluster
- Deep Space 1
- Genesis
- Mars Exploration Rover
- Mars Global Surveyor
- Mars Pathfinder
- Pioneer 10
- Pioneer 11
- Progress
- SOHO
- Stardust
- Viking 1
- Viking 2
- Voyager 1
- Voyager 2
- WMAP

Spacecraft under development


- Crew Exploration Vehicle
- Kliper
- Automated Transfer Vehicle
- H-II Transfer Vehicle
- Ansari X Prize (incl. a list of spacecraft in various stages of completion as of 2005) The US Space Command, according to its "Long Range Plan", is currently planning to develop a weaponized spaceship, which has yet to be announced.[http://www.fas.org/spp/military/docops/usspac/]

See also


- Attitude control
- Expendable launch system
- Human spaceflight
- List of fictional spaceships
- List of spacecraft
- Spacecraft propulsion
- Space shuttle
- Starship
- Thruster
- Unidentified flying object
- Unmanned space mission

External links


- [http://science.hq.nasa.gov/missions/phase.html NASA: Space Science Spacecraft Missions]
- [http://www.skyrocket.de/space/ Gunter's Space Page - Complete information on spacecraft]
- [http://www.cinespaceships.net/ Cinespaceships - Database on spaceships in movie]
- ja:宇宙船

X-15

The North American X-15 rocket plane was perhaps the most important of the USAF/USN X-series of experimental aircraft. Although not as famous as the Bell X-1, the X-15 set numerous speed and altitude records in the early 1960s, reaching the edge of space and bringing back valuable data that was used in the design of later aircraft and spacecraft. During the X-15 program, 13 flights met the US criterion for a spaceflight by passing an altitude of 50 miles (80 km) and the pilots were accordingly awarded astronaut status by the USAF. Out of these, 2 also qualified for the international FAI definition of a spaceflight by passing the 62.1 miles (100 km) mark.

History

FAI The original Request for Proposals was issued for the airframe December 30, 1954, and for the rocket engine on February 4, 1955. North American received the airframe contract in November 1955, and Reaction Motors contracted in 1956 to build the engines. As with many of the X-aircraft, the X-15 was designed to be carried aloft under the wing of a B-52. The fuselage was long and cylindrical, with fairings towards the rear giving it a flattened look, and it had thick wedge-shaped dorsal and ventral fins. The retractable landing gear consisted of a nose wheel and two skids — to provide sufficient clearance part of the ventral fin had to be jettisoned before landing. The two XLR-11 rocket engines of the initial model X-15A delivered 36 kN (8,000 lbf) of thrust; the "real" engine that came later was a single XLR-99 that delivered 254 kN (57,000 lbf) at sea level, and 311 kN (70,000 lbf) at peak altitude. The first flight was an unpowered test made by Scott Crossfield on June 8, 1959 (making him the first man to go supersonic in a glider), who followed up with the first powered flight on September 17. The first flight with the XLR-99 was on 15 November 1960. Three X-15s were built in all, and they made a total of 199 test flights, the last one on October 24, 1968. Plans were made for a 200th X-15 flight to be launched over Smith Ranch, Nevada. It was scheduled for November 21, 1968 with William J. Knight as the pilot. Various technical and weather delays caused the planned launch to slip at least six times until late December, 1968. Finally after a cancellation on December 20, 1968 due to weather, it was decided there would not be a 200th flight. The X-15 ground crew de-mated the aircraft from the NB-52A, and prepared it for indefinite storage. X-15 #1 was sent to the National Air and Space Museum in Washington, DC. X-15 #2 is on display at the National Museum of the United States Air Force at Wright-Patterson Air Force Base near Dayton, Ohio. X-15 #3, 56-6672, was destroyed in a crash on November 15, 1967. Twelve test pilots flew the plane, including Neil Armstrong, later the first man on the Moon and Joe Engle who went on to command Space Shuttle missions. In July and August, 1963, pilot Joe Walker crossed the 100 km altitude mark twice, becoming the first person to enter space twice. Test pilot Michael J. Adams was killed on November 15, 1967 when his X-15-3 began to spin on descent and then disintegrated when the acceleration reached 15 g (147 m/s²), scattering wreckage over 50 square miles. On June 8, 2004 a memorial monument was erected at the location of cockpit [http://www.check-six.com/Crash_Sites/X-15A_crash_site.htm crash site] near Randsburg, California. Michael Adams was posthumously awarded astronaut wings for his last flight in the X-15-3, which had attained an altitude of 266,000 feet (81.1 Km). In 1991 Adams' name was added to the Astronaut Memorial at the Kennedy Space Center in Florida. The second X-15A was rebuilt after a landing accident. It was lengthened by about 0.74 m (2.4 ft), received a pair of auxiliary fuel tanks slung under the fuselage, and was given a heat-resistant surface treatment, the result being called the X-15A-2. It first flew June 28, 1964, and eventually reached a speed of 7,274 km/h (4,520 mi/h or 2,021 m/s). The altitudes attained by the X-15 remained unsurpassed by any piloted aircraft except the Space Shuttle until the 3rd spaceflight of SpaceShipOne in 2004. The speeds and altitudes have, also, frequently been exceeded by unpiloted air-launched rockets, such as the Pegasus rocket which has carried several satellites all the way into orbit. The widely reported record achieved by the diminutive X-43A scramjet testbed on November 16, 2004 of nearly Mach 10 (10,621 km/h or 2.95 km/s) at 95,000 ft (29 km) is only a record for an air-breathing jet engine.

Specifications (X-15)

jet engine

General characteristics


- Crew: 1
- Length: 50.7 ft (15.45 m)
- Wingspan: 22.3 ft (6.8 m)
- Height: 13.5 ft (4.12 m)
- Wing area: 200 ft² (18.58 m²)
- Empty: 14,600 lb (6,623 kg)
- Loaded: 34,000 lb (15,422 kg)
- Maximum takeoff: 34,000 lb (15,422 kg)
- Powerplant: 1x Thiokol XLR99-RM-2 liquid-fuel rocket engine, 70,400 lbf (313 kN) thrust (at 30 km)

Performance


- Maximum speed: 4,520 mph (7,274 km/h) Mach 6.72
- Range: 280 miles (450 km)
- Service ceiling: 67 miles (108 km)
- Rate of climb: 60,000 ft/min (18,000 m/min)
- Wing loading: kg/m² ( lb/ft²)
- Thrust/weight:
- Serial Numbers: (Five main aircraft were involved in the X-15 program. The three X-15's and two B-52 carrier aircraft.)
  - X-15A-1 - 56-6670, 82 powered flights
  - X-15A-2 - 56-6671, 53 powered flights
  - X-15A-3 - 56-6672, 64 powered flights
  - NB-52A - 52-003 (retired October 1969)
  - NB-52B - 52-008 (retired November 2004)

Record flights

Highest flights

In the United States there are two definitions of how high a person must go to be referred to as an astronaut. The USAF decided to award astronaut wings to anyone who achieved a altitude of 50 miles (80 km) or more. However the FAI set the limit of space at 100 km. Thirteen X-15 flights went higher than 50 miles (80 km) and two of these reached over 100 km.

Fastest flights

X-15 Pilots


References


- Robert Godwin, ed., X-15 (The NASA Mission Reports), (Apogee Books, 2001) ISBN 1896522653
- Milton O. Thompson and Neil Armstrong, At the Edge of Space: The X-15 Flight Program (Smithsonian Institution Press, 1992) ISBN 1560981075
- Richard Tregaskis, X-15 Diary: The Story of America's First Space Ship (iUniverse.com, 2000) ISBN 0595002501
- [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20000068530_2000075022.pdf Hypersonics Before the Shuttle: A Concise History of the X-15 Research Airplane - NASA report (PDF format)]
- [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19650010561_1965010561.pdf X-15 research results with a selected bibliography - NASA report (PDF format)]
- [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920075739_1992075739.pdf Flight experience with shock impingement and interference heating on the X-15-2 research airplane 1968 - NASA (PDF format)]
- [http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19680016245_1968016245.pdf Thermal protection system X-15A-2 Design report 1968 - NASA report (PDF format)]
- [http://history.nasa.gov/monograph31.pdf American X-Vehicles: An Inventory X-1 to X-50, SP-2000-4531 - June 2003; NASA online PDF Monograph]

External links


- [http://history.nasa.gov/x15/cover.html NASA's X-15 website]
- [http://www.hq.nasa.gov/office/pao/History/SP-60/cover.html X-15 Research Results With a Selected Bibliography (NASA SP-60, 1965)]
- [http://www.hq.nasa.gov/office/pao/History/hyperrev-x15/cover.html "Transiting from Air to Space: The North American X-15" (1998)]
- [http://www.hq.nasa.gov/office/pao/History/x15conf/cover.html "Proceedings of the X-15 First Flight 30th Anniversary Celebration of June 8, 1989"]
- [http://history.nasa.gov/monograph18.pdf (PDF) Hypersonics Before the Shuttle: A Concise History of the X-15 Research Airplane (NASA SP-2000-4518, 2000)]
- [http://www.x15.com unofficial X-15 website]
- [http://www.x-15.com Another unofficial X-15 website]
- [http://www.hq.nasa.gov/office/pao/History/SP-60/cover.html X-15 Research Results (1964)]
- [http://www.dfrc.nasa.gov/gallery/photo/X-15/ X-15 photos at Dryden]
- [http://www.astronautix.com/craft/x15a.htm Encyclopedia Astronautica's X-15 chronology]
- [http://www.xb-70.com/wmaa/x15/monument/ Major Michael Adams Monument]
- [http://perso.wanadoo.fr/prototypes.com/x15/index.html X-15 site in french, all missions details]

Related content

Related development: Comparable aircraft: SpaceShipOne Designation series: X-12 - X-13 - X-14 - X-15 - X-16 - X-17 - X-18 See also:
- List of experimental aircraft
- List of rocket planes
- X-plane
- List of X-15 flights
- Dyna-Soar
- List of space disasters Category:Manned spacecraft Category:Rocket-powered aircraft X-15 Category:X-15 program Category:Parasite aircraft Category:U.S. experimental aircraft 1950-1959 Aerospace Legacy Foundation: http://www.aerospacelegacyfoundation.org

ICBM

An intercontinental ballistic missile, or ICBM, is a very-long-range (greater than 5500 km) ballistic missile typically designed for nuclear weapons delivery, i.e., delivering one or more nuclear warheads. It uses a ballistic trajectory involving a significant ascent and descent, including sub-orbital flight. ICBMs are differentiated by maximum range from other ballistic missiles: intermediate-range ballistic missiles (IRBMs), short-range ballistic missiles, and the newly named theater ballistic missiles. One particular weapon developed by the Soviet Union (FOBS) had a partial orbital trajectory, and unlike most ICBMs its target could not be deduced from its orbital flight path. It was decommissioned in compliance with arms control agreements, which address the maximum range of ICBMs and prohibit orbital or fractional-orbital weapons. The following nations currently have operational ICBM systems: Russia, the United States, France [http://www.nrdc.org/nuclear/nudb/datab16.asp], the UK, and China. India has IRBMs but is developing ICBMs, see ballistic missiles of India. (Pakistan's ballistic missiles are IRBMs.) In 2002, the United States and Russia agreed in the SORT treaty to reduce their deployed stockpiles to not more than 2,200 warheads each.

Flight phases

The following flight phases can be distinguished:
- boost phase - 3 to 4 minutes (for a solid rocket shorter than for a liquid-propellant rocket); altitude at the end of this phase is 150 -200 km, typical burn-out speed is 7 km/s
- midcourse phase - ca. 25 minutes - suborbital flight in an elliptic orbit, i.e. the orbit is part of an ellipse with vertical major axis; the apogee (halfway the midcourse phase) is at an altitude of typically ca. 1200 km; the semi-major axis is between one half of the radius of the Earth and the radius; the projection of the orbit on the Earth's surface is a great circle - the missile may release several independent warheads, a large number of decoys, and chaff
- reentry phase (starting at an altitude of 100 km) - 2 minutes See also Missile Defense Agency.

History

The progenitor of the ICBM was the German A9/10, which was never developed but only proposed by Wernher von Braun. The progenitor of the IRBM was the German V2 (Vergeltung, or "vengeance") rocket designed by von Braun that used liquid propellant and an inertial guidance system. It was launched from a mobile launcher in order to make it less susceptible to Allied air attacks. Following World War 2 von Braun and his lead scientist went to work directly for the US Army through Operation Paperclip developing the V2 into the Redstone IRBM and Jupiter IRBM. Due to treaty agreements the US was able to base these IRBMs in countries close to the USSR within strategic range. The USSR had no similar territory in the 1950s so under the direction of Sergei Korolev a crash programme to develop an ICBM began which at one stage consumed 5% of the entire Soviet military budget. Korolev was given access to captured V2 materials but evolved a distinct design, the R-7, that was declared 'operational' in 1957. Competition between the US armed services meant that each force developed its own ICBM programme slowing progress. The US's first ICBM was the Atlas operational in 1959. Both the R7 and Atlas required a large launch facility making them vulnerable to attack and could not be kept in a ready state. Early ICBMs formed the basis of many space launch systems. Examples include: Atlas, Redstone rocket, Titan, R-7, and Proton, derived from the earlier ICBMS, but never deployed as an ICBM. The UK built its own ICBM Black Knight but it was never made operational due to the difficulty of finding a launch site away from population centres. Under the direction of Robert McNamara the US initiated the LGM-30 Minuteman, Polaris and Skybolt solid fuel ICBMs. Modern ICBMs tend to be smaller than their ancestors (due to increased accuracy and smaller and lighter warheads) and use solid fuels, making them less useful as orbital launch vehicles. Deployment of these systems was governed by the strategic theory of Mutually Assured Destruction. In the 1970s development began of Anti-Ballistic Missile Systems by both the US and USSR but these were restricted by treaty in order to preserve the value of the existing ICBM systems. President Ronald Reagan launched the Strategic Defense Initiative as well as the MX and Midgetman ICBM programmes. This led to the agreement of a series of Strategic Arms Reduction Treaty negotiations. Countries in the early stages of developing ICBMs have all used liquid propellants for simplicity's sake.

Modern ICBMs

Modern ICBMs typically carry multiple independently targetable reentry vehicles (MIRVs), each of which carries a separate nuclear warhead, allowing a single missile to hit multiple targets. MIRV was an outgrowth of the rapidly shrinking size and weight of modern warheads and the Strategic Arms Limitation Treaties which imposed limitations on the number of launch vehicles(SALT I and SALT II). It has also proved to be an "easy answer" to proposed deployments of ABM systems – it is far less expensive to add more warheads to an existing missile system than to build an ABM system capable of shooting down the additional warheads; hence, most ABM system proposals have been judged to be impractical. The only operational ABM systems were deployed in the 1970s, the US Safeguard ABM facility was located in North Dakota and was operational from 1975-1976. The USSR deployed its Galosh ABM system around Moscow in the 1970s, which remains in service. ICBMs are based:
- in missile silos, which offer some protection from military attack (including, the designers hope, some protection from a nuclear first strike)
- on submarines: submarine-launched ballistic missiles (SLBMs); most or all SLBMs have the long range of ICBMs (as opposed to IRBMs)
- on heavy trucks; this applies to one version of the RT-2UTTH Topol M which may be deployed from a self-propelled mobile launcher, capable of moving through roadless terrain, and launching a missile from any point along its route
- mobile launchers on rails; this applies, for example, to РТ-23УТТХ "Молодец" (RT-23UTTH "Molodets" -- SS-24 "Sсаlреl") The last three kinds are mobile and therefore hard to find. During storage, one of the most important features of the missile is its serviceability. One of the key features of the first computer-controlled ICBM, the Minuteman missile was that it could quickly and easily use its computer to test itself. In flight, a booster pushes the warhead, and then falls away. Most modern boosters are solid-fueled rocket motors, which can be stored easily for long periods of time. Early missiles used liquid-fueled rocket motors. Liquid-fueled ICBMs were generally not kept fueled all the time, and therefore fueling the rocket was necessary before a launch. This annoying procedure was a source of significant operational delay, and therefore might cause the rockets to be destroyed before they could be used. It also provided opponents with intelligence because it was a definite observable event that indicated the start of an attack. Once the booster falls away, the warhead falls on an unpowered path, much like an orbit, except that it hits the earth at some point. Moving in this way is stealthy. No rocket gases or other emissions occur to indicate the missile's position to defenders. Also, it is the fastest way to get from one part of the Earth to another. This increases the element of surprise. The high speeds of a ballistic warhead (near 5 miles per second) also make it difficult to intercept. Many authorities say that missiles also release aluminized balloons, electronic noisemakers, and other items intended to confuse interception devices and radars. The high speed can cause the missile to get very hot as it reenters the atmosphere. Ballistic warheads are protected by heatshields constructed of materials such as pyrolytic graphite, and in early missiles, thick plywood. Plywood approaches the strength per weight of carbon fiber/epoxy composites, and chars slowly, protecting the missile. Accuracy is crucial, because doubling the accuracy decreases the needed warhead energy by a factor of four. Accuracy is limited by the accuracy of the navigation system, and the available geophysical information. Many authorities believe that most government-supported geophysical mapping initiatives, such as GPS, and ocean satellite altitude systems such as Seasat, probably have a covert purpose to map mass concentrations and determine local gravitic anomalies, in order to improve accuracies of ballistic missiles. Strategic missile systems are thought to use custom integrated circuits designed to calculate navigational differential equations thousands to millions of times per second in order to reduce navigational errors caused by calculation alone. These circuits are usually a network of binary addition circuits that continually recalculate the missile's position. The inputs to the navigation circuit are set by a general purpose computer according to a navigational input schedule loaded into the missile before launch. The low flying, guided cruise missile is an alternative to ballistic missiles.

Specific missiles

Land-based intercontinental ballistic missiles (ICBMs) and cruise missiles

The US Air Force currently operates just over 500 ICBMs at around 15 missile complexes located primarily in the northern Rocky Mountain states and the Dakotas. These are of the LGM-30 Minuteman III and Peacekeeper ICBM variants. Peacekeeper missiles are being phased out by 2005. All USAF Minuteman II missiles have been destroyed in accordance to START, and their launch silos sealed or sold to the public. To comply with the START II most US multiple independently targetable reentry vehicles, or MIRVs, have been eliminated and replaced with single warhead missiles. However, since the abandonment of the START II treaty, the U.S. is said to be considering retaining 800 warheads on 500 missiles.[http://www.thebulletin.org/issues/nukenotes/mj04nukenote.html] The United States Air Force awards two badges for performing duty in a nuclear missile silo. The Missile Badge is presented to commissioned officers while the Space and Missile Pin is awarded to silo ground and support personnel.

Sea-based ICBMs


- The US Navy currently has 14 Ohio-class SSBNs deployed. Each submarine is equipped with a complement of 24 Trident missiles, eight with Trident I missiles, and ten with Trident II missiles.
- The French Navy constantly maintains at least four active units, relying on two classes of nuclear-powered ballistic submarines (SSBN): the older Redoutable class, which are being progressively decommissioned, and the newer Triomphant class. These carry 16 M45 missiles with TN75 warheads, and are scheduled to be upgraded to M51 nuclear missile around 2010.
- The UK's Royal Navy has four Vanguard class submarines, each armed with 16 Trident II SLBMs.
- China's People's Liberation Army Navy has one Xia class submarine with 12 single-warhead JL-1 SLBMs. The PLAN is also developing the new Type 094 SSBN that will have up to 16 JL-2 SLBMs (possibly MIRV), which are also in development.

Current and former US ballistic missiles


- Atlas (SM-65, CGM-16) former ICBM launched from silo, now the rocket is used for other purposes
- Titan I (SM-68, HGM-25A)
- Titan II (SM-68B, LGM-25C) - former ICBM launched from silo, now the rocket is used for other purposes
- Minuteman I (SM-80, LGM-30A/B, HSM-80)
- Minuteman II (LGM-30F)
- Minuteman III (LGM-30G) - launched from silo - as of June 28, 2004, there are 517 Minuteman III missiles in active inventory
- LG-118A Peacekeeper / MX (LG-118A, MX) - silo-based; 29 missiles were on alert at the beginning of 2004; all are to be removed from service by 2005.
- Midgetman - has never been operational - launched from mobile launcher
- Polaris A1, A2, A3 - (UGM-27/A/B/C) former SLBM
- Poseidon C3 - (UGM-73) former SLBM
- Trident - (UGM-93A/B) SLBM - Trident II (D5) was first deployed in 1990 and is planned to be deployed past 2020.

Soviet/Russian

Specific types of Soviet/Russian ICBMs include:
- SS-6 SAPWOOD / R-7 / 8K71
- SS-7 SADDLER / R-16
- SS-8 SASIN / R9
- SS-9 SCARP
- SS-11 SEGO
- SS-17 SPANKER
- SS-18 SATAN / R-36M2 / Voivode
- SS-19 STILLETO
- SS-24 SCALPEL / RT-23
- SS-25 SICKLE / Topol
- SS-27 / Topol-M

People's Republic of China

Specific types of Chinese ICBMs called Dong Feng ("East Wind").
- DF-3 - cancelled. Program name transferred to a MRBM.
- DF-4 (CSS-3) - silo, 7,000km range
- DF-5 CSS-4 - silo, 12,000km range (replaced now with DF-5A 13,000km)
- DF-6 - cancelled
- DF-22 - cancelled by 1995.
- DF-31 CSS-9 - silo and road mobile, 8,000km range (DF-31A 10,000km)
- DF-41 CSS-X-10 - in development.

Ballistic missile submarines

Specific types of ballistic missile submarines include:
- George Washington class
- Ethan Allen class
- Lafayette class
- Benjamin Franklin class
- Ohio class
- Resolution class
- Vanguard class
- Typhoon class
- Delta IV class
- Redoutable class
- Triomphant class
- Xia class
- Additional Soviet/Russian ballistic missile submarines

See also


- The United States and weapons of mass destruction
- Russia and weapons of mass destruction
- China and weapons of mass destruction
- France and weapons of mass destruction
- SLBM
- Anti-ballistic missile
- Anti-Ballistic Missile Treaty
- nuclear disarmament
- nuclear navy
- nuclear warfare
- Force de frappe
- submarine
- Fractional Orbital Bombardment System
- Strategic triad
- Air Force Space Command
- ICBM address

External links


- [http://es.rice.edu/projects/Poli378/Nuclear/f04.stratg_invent.html Estimated Strategic Nuclear Weapons Inventories (September 2004)]
- [http://www.fas.org/nuke/guide/usa/icbm/index.html Intercontinental Ballistic and Cruise Missiles] Category:Intercontinental ballistic missiles ms:Peluru berpandu balistik jarak benua ja:大陸間弾道ミサイル

Sounding rocket

A sounding rocket, sometimes called an elevator research rocket, is an instrument-carrying suborbital rocket designed to take measurements and perform scientific experiments during its flight. The rockets are commonly used to take readings or carry instruments from 50 to 200 km above the surface, the region above the maximum altitude for balloons and below the minimum for satellites. The term "sounding" is taken from the maritime expression. Certain sounding rockets, such as the Black Brant X and XII, have an apogee between 1,000 and 1,500 km, well above Low Earth Orbit. A common sounding rocket consists of a solid-fuel rocket motor and a payload. The freefall part of the flight is an elliptic trajectory with vertical major axis and the average flight time is less than forty minutes. The rocket consumes its fuel on the first stage of the rising part of the flight, then separates and falls away, leaving the payload to complete the arc and return to the ground with a parachute.

External links


- ESA [http://www.spaceflight.esa.int/users/file.cfm?filename=facsrockets article on sounding rockets]
- [http://www.esa.int/esapub/bulletin/bullet88/peder88.htm 30 years of sounding rocket launches] at Esrange in Kiruna, Sweden
- [http://www.wff.nasa.gov/~code810/ NASA Sounding Rockets Program Office]
- [http://www.nsroc.com/front/mainmenu/mmframe.html NASA Sounding Rocket Operations Contract]
- [http://history.nasa.gov/SP-4401/sp4401.htm NASA Sounding Rockets, 1958-1968: A Historical Summary (NASA SP-4401, 1971)]
-

Fractional Orbital Bombardment System

Fractional Orbital Bombardment System (FOBS) was a Soviet ICBM in the 1960s that after launch would go into a low Earth orbit and would then de-orbit for an attack. It had no range limit and the orbital flight path would not reveal the target location. This would allow a path to North America over the South Pole, hitting targets from the south, which is the opposite direction from which NORAD early warning systems are oriented. The Outer Space Treaty banned nuclear weapons or weapons of mass destruction in earth orbit. However, it did not ban systems that were capable of placing weapons in orbit, and the Soviet Union avoided violating the treaty by conducting tests of its FOBS system without live warheads. The orbital missile 8K69 was initially deployed in 1968, and the first regiment with the R-36 orbital missiles was put on alert in 1969. The U.S. Defense Support Program early warning satellites enabled the US to detect a FOBS launch. The SALT II treaty (1979) prohibited the deployment of FOBS systems: : Each Party undertakes not to develop, test, or deploy: :(...) :(c) systems for placing into Earth orbit nuclear weapons or any other kind of weapons of mass destruction, including fractional orbital missiles; The missile was phased out in January 1983 in compliance with this treaty.

External links


- http://www.globalsecurity.org/wmd/world/russia/r-36o.htm Category:Intercontinental ballistic missiles Category:Space weapons ja:部分軌道爆撃システム

Freefall

Free-fall or free fall in the strict sense is the condition of acceleration which is due only to gravity. In other words, the objects undergoing free fall experience only one force: their own weight. Examples include:
- a spacecraft with the rockets off
- the Moon's trajectory around the Earth, or the Earth's orbit around the Sun.
- on Earth, falling through a vacuum tube or shaft, e.g.:
  - for a physics demonstration
  - at NASA's Zero-G Research Facility as opposed to the cases where other forces are acting, including:
- standing on the ground, sitting in a chair on the ground, etc. (gravity is cancelled by the reaction force of the ground)
- flying in a plane (gravity is cancelled by the lift the wings provide) - see below for special trajectories which form an exception
- atmospheric reentry, landing on a parachute: gravity is opposed by atmospheric drag
- during an orbital maneuver in a spacecraft: the rocket provides thrust thrust More generally, free fall is the condition of acceleration which is due only to gravity and air friction: in parachuting, free fall (skydiving) refers to the act of falling and delaying the opening of a parachute. Freeflying is skydiving in other body positions than the more standard belly flying. With air friction acting upon an object that has been dropped the object will eventually reach terminal velocity (around 120 miles/hour for a human body flying in the belly-down arched position; terminal velocity depends on many factors including mass, drag coefficient, and relative surface area) if the fall is from sufficient altitude (2,000 ft) and also otherwise uninterrupted.

People surviving free fall

At least three airmen have survived free falls of around 20,000 ft (6,000 m) without a parachute in the Second World War; Lt. I.M. Chisov was a Russian bomber, Sgt. Alan Magee an American gunner on a B-17, and Sgt. Nicholas Alkemade a British gunner on a Lancaster bomber. It is estimated that a person free falling horizontally, reaches a terminal velocity of around 120 mph (200 km/h) after a fall of just 2,000 ft (600 m), so the additional 18,000 ft (5,500 m) doesn't make these falls that much more dangerous, apart from the lack of oxygen at high altitude. All three men lost consciousness during their falls, and two of them landed on terrain covered in deep snow, which was probably a significant factor in the survivability of the falls. Vesna Vulović, a flight attendant from Yugoslavia, survived a fall from 10,160 m (33,330 ft) when the DC-9 airplane she was traveling in blew up over Srbská Kamenice, Czechoslovakia, on January 26, 1972. She remained strapped into her flight attendant's seat in the tail section of the plane, which remained attached to the washrooms. The assembly struck the snow-covered flank of a mountain. A terrorist bomb was thought to be the cause. Vulović broke both legs and was temporarily paralyzed from the waist down. No other passengers survived. [http://www.guinnessworldrecords.com/content_pages/record.asp?recordid=43941] Stories about desperate Russians deploying paratroopers without parachutes (unsuccessfully) during World War II are most likely fabricated. [http://www.greenharbor.com/fffolder/questions.html#anchor1234559] It is reported that two of the victims of the Lockerbie bombing survived for a brief period after hitting the ground.

Record free fall

Lockerbie bombing As part of Project Excelsior on 16 August 1960, Joseph Kittinger achieved the record for the longest free fall jump and the fastest maximum speed of 614 mph (982 km/h), before opening his parachute at around 18,000 feet (5,500 m). Kittinger started the jump from a specially constructed balloon at an altitude of 102,800 feet (31,300 m), which also qualified him for the highest balloon ascent and highest parachute jump. Some people claim that Kittinger's jump wasn't true free-fall as he used a drogue chute for stability. According to the Guinness book of records, Eugene Andreev (USSR) holds the official FAI record for the longest free-fall parachute jump after falling for 80,380 ft (24,500 m) from an altitude of 83,523 ft (24,458 m) near the city of Saratov, Russsia on November 1, 1962. Andreev did not use a drogue chute during his jump.

References


- [http://www.greenharbor.com/fffolder/ffresearch.html Free fall accidents, mathematics of free fall - detailed research on the topic]
- [http://members.aol.com/MercStG/FFAccPage1.html Free fall accidents] and [http://www.parachutehistory.com/other/bonusday.html parachute history] Category:Gravity Category:Parachuting Category:Introductory physics ko:자유 낙하

Elliptic orbit

In astrodynamics or celestial mechanics a elliptic orbit is an orbit with the eccentricity greater than 0 and less than 1. Specific energy of an elliptical orbit is negative. An orbit with a eccentricity of 0 is a circular orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit and tundra orbit.

Velocity

Under standard assumptions the orbital velocity (v\,) of a body traveling along elliptic orbit can be computed as: :v=\sqrt where:
- \mu\, is standard gravitational parameter,
- r\, is radial distance of orbiting body from central body,
- a\,\! is length of semi-major axis. Conclusion:
- Velocity does not depend on eccentricity but is determined by length of semi-major axis (a\,\!),
- Velocity equation is similar to that for hyperbolic trajectory with the difference that for the latter, is positive.

Orbital period

Under standard assumptions the orbital period (T\,\!) of a body traveling along elliptic orbit can be computed as: :T=a^ where:
- \mu\, is standard gravitational parameter,
- a\,\! is length of semi-major axis. Conclusions:
- The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis (a\,\!),
- The orbital period does not depend on the eccentricity (See also: Kepler's third law).

Energy

Under standard assumptions, specific orbital energy (\epsilon\,) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form: :-=-=\epsilon<0 where:
- v\, is orbital velocity of orbiting body,
- r\, is radial distance of orbiting body from central body,
- a\, is length of semi-major axis,
- \mu\, is standard gravitational parameter. Conclusions:
- Specific energy for elliptic orbits is independent of eccentricity and is determined only by semi-major axis of the ellipse. Using the virial theorem we find:
- the time-average of the specific potential energy is equal to 2ε
  - the time-average of r-1 is a-1
- the time-average of the specific kinetic energy is equal to -ε

Flight path angle

Equation of motion

See orbit equation.

Orbital parameters

Solar system

In the solar system planets, asteroids, comets and space debris have elliptical orbits around the Sun. Moons have an elliptic orbit around their planet. Many artificial satellites have various elliptic orbits around the Earth.

See also


- Characteristic energy
- Circular orbit
- Hyperbolic trajectory
- Orbit
- Orbital equation
- Parabolic trajectory Category:Celestial mechanics Category:Astrodynamics ja:楕円軌道

Orbital equation

In astrodynamics an orbit equation defines the path of orbiting body m_2\,\! around central body m_1\,\! relative to m_1\,\!, without specifying position as a function of time. Under standard assumptions, a body moving under the influence of a force, directed to a central body, with a magnitude inversely proportional to the square of the distance (such as gravity), has an orbit that is a conic section (i.e. circular orbit, elliptic orbit, parabolic trajectory, hyperbolic trajectory) with the central body in one the two foci, or the focus (Kepler's first law). If the conic section intersects the central body, then the actual trajectory can only be the part above the surface, but for that part the orbit equation and many related formulas still apply, as long as it is a freefall (situation of weightlessness). In polar coordinates the orbit equation is: :r= where we have the polar coordinates:
- r\, is the distance between the orbiting body and the central body
- \theta\, is the direction of the orbiting body, called the true anomaly and the parameters:
- h\,\! is the specific relative angular momentum of the orbiting body
- e\,\! is the eccentricity of the orbit
- \mu\,\! is the constant which, divided by the distance squared, gives the magnitude of the acceleration; in the case of gravity this is the standard gravitational parameter Note that is the semi-latus rectum of the conic section. This ratio, together with e\,\!, fully determines the geometry of the orbit. For a given orbit, the larger \mu\,\!, the faster the orbiting body moves in it: twice as fast if the attraction is four times as strong. The minimum value of r in the equation is :r= while, if e<1, the maximum value is :r= If the maximum is less than the radius of the central body, then the conic section is an ellipse which is fully inside the central body and no part of it is a possible trajectory. If the maximum is more, but the minimum is less than the radius, part of the trajectory is possible:
- if the energy is non-negative (parabolic or hyperbolic orbit): the motion is either away from the central body, or towards it.
- if the energy is negative: the motion can be first away from the central body, up to :r= :after which the object falls back. If r becomes such that the orbiting body enters an atmosphere, then the standard assumptions no longer apply, as in atmospheric reentry.

Low-energy trajectories

If the central body is the Earth, and the energy is only slightly larger than the potential energy at the surface of the Earth, than the orbit is elliptic with eccentricity close to 1 and one end of the ellipse just beyond the center of the Earth, and the other end just above the surface. Only a small part of the ellipse is applicable. If the horizontal speed is v\,\!, then the periapsis distance is . The energy at the surface of the Earth corresponds to that of an elliptic orbit with a=R/2\,\! (with R\,\! the radius of the Earth), which can not actually exist because it is an ellipse fully below the surface. The energy increase with increase of a is at a rate 2g\,\!. The maximum height above the surface of the orbit is the length of the ellipse, minus R\,\!, minus the part "below" the center of the Earth, hence twice the increase of a\,\! minus the periapsis distance. At the top the potential energy is g times this height, and the kinetic energy is . This adds up to the energy increase just mentioned. The width of the ellipse is 19 minutes times v\,\!. The part of the ellipse above the surface can be approximated by a part of a parabola, which is obtained in a model where gravity is assumed constant. This should be distinguished from the parabolic orbit in the sense of astrodynamics, where the velocity is the escape velocity. See also trajectory.

Categorization of orbits

Consider orbits which are at one point horizontal, near the surface of the Earth. For increasing speeds at this point the orbits are subsequently:
- part of an ellipse with vertical major axis, with the center of the Earth as the far focus (throwing a stone, sub-orbital spaceflight, ballistic missile)
- a circle just above the surface of the Earth (Low Earth orbit)
- an ellipse with vertical major axis, with the center of the Earth as the near focus
- a parabola
- a hyperbola Note that in the sequence above, h, \epsilon and a increase monotonically, but e first decreases from 1 to 0, then increases from 0 to infinity. The reversal is when the center of the Earth changes from apoapsis to periapsis (the other focus starts near the surface and passes the center of the Earth). We have :e=\left | \frac-1\right | Extending this to orbits which are horizontal at another height, and orbits of which the extrapolation is horizontal below the surface of the Earth, we get a categorization of all orbits, except the radial trajectories, for which, by the way, the orbit equation can not be used. In this categorization ellipses are considered twice, so for ellipses with both sides above the surface one can restrict oneself to taking the side which is lower as the reference side, while for ellipses of which only one side is above the surface, taking that side.

See also


- Kepler's first law
- Circular orbit
- Elliptic orbit
- Parabolic trajectory
- Hyperbolic trajectory
- Rocket equation Category:Astrodynamics Category:Celestial mechanics

Perigee

: This article is about several astronomical terms (apogee & perigee, aphelion & perihelion, generic equivalents based on apsis, and related but rarer terms). In architecture, apsis is a synonym for apse; Apogee is also the name of a video game publisher. video game publisher In astronomy, an apsis (plural apsides "ap-si-deez") is the point of greatest or least distance of the elliptical orbit of a celestial body from its center of attraction (the center of mass of the system). The point of closest approach is called the periapsis or pericentre and the point of farthest approach is the apoapsis (Greek απο, from), apocentre or apapsis (the latter term, although etymologically more correct, is much less used). A straight line drawn through the periapsis and apoapsis is the line of apsides. This is the major axis of the ellipse, the line through the longest part of the ellipse. Related terms are used to identify the body being orbited. The most common are perigee and apogee, referring to Earth orbits, and perihelion and aphelion, referring to orbits around the Sun (Greek ἡλιος).

Formulae

We have:
- Periapsis: maximum speed v_\mathrm = \sqrt \,  at minimum distance r_\mathrm=(1-e)a\!\, (periapsis distance)
- Apoapsis: minimum speed v_\mathrm = \sqrt \,  at maximum distance r_\mathrm=(1+e)a\!\, (apoapsis distance) where one easily verifies :h = \sqrt :\epsilon=-\frac (each the same for both points, like they are for the whole orbit, in accordance with Kepler's laws of planetary motion (conservation of angular momentum) and the conservation of energy) where:
- a\!\, is the semi-major axis
- e\!\, is the eccentricity
- h\!\, is the specific relative angular momentum
- \epsilon\!\, is the specific orbital energy
- \mu\!\, is the standard gravitational parameter Properties: :e=\frac=1-\frac=\frac-1 Note that for conversion from heights above the surface to distances, the radius of the central body has to be added, and conversely. The arithmetic mean of the two distances is the semi-major axis a\!\,. The geometric mean of the two distances is the semi-minor axis b\!\,. The geometric mean of the two speeds is \sqrt, the speed corresponding to a kinetic energy which, at any position of the orbit, added to the existing kinetic energy, would allow the orbiting body to escape (the square root of the sum of the squares of the two speeds is the local escape velocity).

Terminology

Various related terms are used for other celestial objects. The '-gee', '-helion' and '-astron' and '-galacticon' forms are frequently used in the astronomical literature, while the other listed forms are occasionally used, although '-saturnium' has very rarely been used in the last 50 years. The '-gee' form is commonly (although incorrectly) used as a generic 'closest approach to planet' term instead of specifically applying to the Earth. The term peri/apomelasma was used by Geoffrey A. Landis in 1998 before peri/aponigricon appeared in the scientific literature in 2002.
(1) Properly pronounced 'affelion', although 'ap-helion' is commonly heard.
Since "peri" and "apo" are Greek, it is considered by purists more correct to use the Greek form for the body, giving forms such as '-zene' for Jupiter and '-krone' for Saturn. For Venus, the alternate form '-krition' (from Kritias, an older name for Aphrodite) has also been suggested. In the Moon's case, in practice all three forms are used, albeit very infrequently. The '-cynthion' form is, according to some, reserved for artificial bodies, whilst others reserve '-lune' for an object launched from the Moon and '-cynthion' for an object launched from elsewhere. For Jupiter, the '-jove' form is occasionally used by astronomers whilst the '-zene' form is never used, like the other pure Greek forms ('-cytherion' (Venus), '-areion' (Mars), '-hermion' (Mercury), '-krone' (Saturn), '-uranion' (Uranus), '-poseidion' (Neptune) and '-hadion' (Pluto)). The daunting prospect of having to maintain a different word for every orbitable body in the solar system (and beyond) is the main reason why the generic '-apsis' has become the almost universal norm.

See also


- Eccentric anomaly
- Elliptic orbit Category:Celestial mechanics Category:Astrodynamics ja:近地点・遠地点

Ellipse

Elliptical redirects here, for the exercise machine, see Elliptical trainer. Elliptical trainer In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An ellipse is a type of conic section: if a cone is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres. Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form :A x^2 + B xy + C y^2 + D x + E y + F = 0 such that B^2 < 4 AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. An ellipse can be drawn with two pins, a loop of string, and a pencil. The pins are placed at the foci and the pins and pencil are enclosed inside the string. The pencil is placed on the paper inside the string, so the string is taut. The string will form a triangle. If the pencil is moved around so that the string stays taut, the sum of the distances from the pencil to the pins will remain constant, satisfying the definition of an ellipse. triangle The line segment which passes through the foci and terminates on the ellipse is called the major axis. The major axis is along the longest segment that passes through the ellipse. The line which passes through the center (halfway between the foci), at right angles to the major axis, is called the minor axis. A semimajor axis is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. Likewise, the semiminor axis is one half the minor axis. If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. An ellipse centred at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix A = PDP^T, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. Then the axes of the ellipse will lie along the eigenvectors of A, and the squares of the lengths of the axes are the inverses of the eigenvalues.

Parametrisation

The size of an ellipse is determined by two constants, conventionally denoted a and b. The constant a equals the length of the semimajor axis; the constant b equals the length of the semiminor axis. length An ellipse centered at the origin of an x-y coordinate system with its major axis along the x-axis is defined by the equation :\frac + \frac = 1 The derivation of this formula is quite instructive and not overly difficult. The following diagram shows an ellipse demonstrating the Pythagoras equation a² = b² + c² as a special case of the non-parametric equation above (x=0, y=b). center The same ellipse is also represented by the parametric equations: :x = a\,\cos t :y = b\,\sin t :0 \leq t < 2\pi which use the trigonometric functions sine and cosine. If an ellipse is not centered at the origin of an x-y coordinate system, but again has its major axis along the x-axis, it may be specified by the equation :\frac + \frac = 1 where (h,k) is the center. A Gauss-mapped form: :\left(\frac,\frac\right) has normal (\cos\phi,\sin\phi).

Eccentricity

The shape of an ellipse is usually expressed by a number called the eccentricity of the ellipse, conventionally denoted e (not to be confused with the mathematical constant e). The eccentricity is related to a and b by the statement :e = \sqrt or where c (the linear eccentricity of the ellipse) equals the distance from the center to either focus :e = \frac The eccentricity is a positive number less than 1, or 0 in the case of a circle. The greater the eccentricity is, the larger the ratio of a to b is, and therefore the more elongated the ellipse is. The ellipse shown in the image below has an eccentricity of approximately 0.8733. The distance between the foci is 2ae.

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted l\,\! (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to a\,\! and b\,\! (the ellipse's semi-axes) by the formula al=b^2\,\! or, if using the eccentricity, l=a(1-e^2)\,\!. perpendicular In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation : r (1 + e \cos \theta) = l \,\! An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°.

Area

The area enclosed by an ellipse is \pi ab\,\!, where \pi is Archimedes' constant.

Circumference

The circumference of an ellipse is 4aE(e), where the function E is the complete elliptic integral of the second kind. The exact infinite series is: :c = 2\pi a \left[\right]\!\, A good approximation is Ramanujan's: :c \approx \pi \left[3(a+b) - \sqrt\right]\!\, which can also be written as: :c \approx \pi a \left[ 3 (1+\sqrt) - \sqrt \right] \!\, More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by an incomplete elliptic integral. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.

Stretching and Projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a point). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse.

Ellipses in physics

Indian astronomer Aryabhata discovered that the orbits of the planets around the sun are ellipses in 499, which he described in his book, the Aryabhatiya [http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata_I.html]. In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses, which is Kepler's first law. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are similar ellipses with the common barycenter being one of the foci of each ellipse. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus. The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. Albert Einstein also used the ellipse to prove his theory of relativity by using an elliptical shaped mass. Einstein's contributions to modern physics may not have been discovered if it were not for ellipses.

Ellipses in computer graphics

Drawing an ellipse is a common graphics primitive in standard display libraries, such as the QuickDraw and GDI interfaces on the Macintosh and Windows systems. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. Jack Bresenham at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984). A more challenging task is to perform these drawing operations with antialiasing, to create a smooth-looking curve. The curve drawing algorithms of Xiaolin Wu (SIGGRAPH 91) are an example.

See also


- Ellipsoid, a higher dimensional analog of an ellipse
- Spheroid, the ellipsoids obtained by rotating an ellipse a