A free-return trajectory is a trajectory of a spacecraft traveling away from a primary body where gravity due to a secondary body causes the spacecraft to return to the primary body without propulsion. The first spacecraft to use a free-return trajectory was the Soviet Luna 3 mission in October 1959, it used the Moon's gravity to send it back towards the Earth so that the photographs it had taken of the far side of the Moon could be downloaded by radio. Symmetrical free-return trajectories were studied by Arthur Schwaniger of NASA in 1963 with reference to the Earth–Moon system, he studied cases in which the trajectory at some point crosses at a right angle the line going through the centre of the Earth and the centre of the Moon, cases in which the trajectory crosses at a right angle the plane containing that line and perpendicular to the plane of the Moon's orbit. In both scenarios we can distinguish between: A circumlunar free-return trajectory around the Moon; the spacecraft passes behind the Moon.
It moves there in a direction opposite to that of the Moon, or at least slower than the Moon in the same direction. If the craft's orbit begins in a normal direction near Earth it makes a figure 8 around the Earth and Moon when plotted in a coordinate system that rotates as the moon goes around the earth. A cislunar free-return trajectory; the spacecraft goes beyond the orbit of the Moon, returns to inside the Moon's orbit, moves in front of the Moon while being diverted by the Moon's gravity to a path away from the Earth to beyond the orbit of the Moon again, is drawn back to Earth by Earth's gravity. In both the circumlunar case and the cislunar case, the craft can be moving from west to east around the Earth, or from east to west. For trajectories in the plane of the Moon's orbit with small periselenum radius, the flight time for a cislunar free-return trajectory is longer than for the circumlunar free-return trajectory with the same periselenum radius. Flight time for a cislunar free-return trajectory decreases with increasing periselenum radius, while flight time for a circumlunar free-return trajectory increases with periselenum radius.
The speed at a perigee of 6555 km from the centre of the Earth for trajectories passing between 2000 and 20 000 km from the Moon is between 10.84 and 10.92 km/s regardless of whether the trajectory is cislunar or circumlunar or whether it is co-rotational or counter-rotational. Using the simplified model where the orbit of the Moon around the Earth is circular, Schwaniger found that there exists a free-return trajectory in the plane of the orbit of the Moon, periodic. After returning to low altitude above the Earth the spacecraft would start over on the same trajectory; this periodic trajectory is counter-rotational. It has a period of about 650 hours. Considering the trajectory in an inertial frame of reference, the perigee occurs directly under the Moon when the Moon is on one side of the Earth. Speed at perigee is about 10.91 km/s. After 3 days it reaches the Moon's orbit, but now more or less on the opposite side of the Earth from the Moon. After a few more days, the craft reaches its apogee and begins to fall back toward the Earth, but as it approaches the Moon's orbit, the Moon arrives, there is a gravitational interaction.
The craft passes on the near side of the Moon at a radius of 2150 km and is thrown back outwards, where it reaches a second apogee. It falls back toward the Earth, goes around to the other side, goes through another perigee close to where the first perigee had taken place. By this time the Moon has moved half an orbit and is again directly over the craft at perigee. Other cislunar trajectories are similar but do not end up in the same situation as at the beginning, so cannot repeat. There will of course be similar trajectories with periods of about two sidereal months, three sidereal months, so on. In each case, the two apogees will be further away from Earth; these were not considered by Schwaniger. This kind of trajectory can occur of course for similar three-body problems. While in a true free-return trajectory no propulsion is applied, in practice there may be small mid-course corrections or other maneuvers. A free-return trajectory may be the initial trajectory to allow a safe return in the event of a systems failure.
In such a case a free return to a suitable reentry situation is more useful than returning to near the Earth, but needing propulsion anyway to prevent moving away from it again. Since all went well, these Apollo missions did not have to take advantage of the free return and inserted into orbit upon arrival at the Moon; the atmospheric entry interface velocity upon return from the Moon is 36,500 ft/s whereas the more common spacecraft return velocity from low-Earth orbit is 7.8 km/s. Due to the landing-site restrictions that resulted from constraining the launch to a free return that flew by the Moon, subsequent Apollo missions, starting with Apollo 12 and including the ill-fated Apollo 13, used a hybrid trajectory that launched to a
The Lunar Atmosphere and Dust Environment Explorer was a NASA lunar exploration and technology demonstration mission. It was launched on a Minotaur V rocket from the Mid-Atlantic Regional Spaceport on September 7, 2013. During its seven-month mission, LADEE orbited around the Moon's equator, using its instruments to study the lunar exosphere and dust in the Moon's vicinity. Instruments included a dust detector, neutral mass spectrometer, ultraviolet-visible spectrometer, as well as a technology demonstration consisting of a laser communications terminal; the mission ended on April 18, 2014, when the spacecraft's controllers intentionally crashed LADEE into the far side of the Moon, determined to be near the eastern rim of Sundman V crater. LADEE was announced during the presentation of NASA's FY09 budget in February 2008, it was planned to be launched with the Gravity Recovery and Interior Laboratory satellites. Mechanical tests including acoustic and shock tests were completed prior to full-scale thermal vacuum chamber testing at NASA's Ames Research Center in April 2013.
During August 2013, LADEE underwent final balancing and mounting on the launcher, all pre-launch activities were complete by August 31, ready for the launch window which opened on September 6. NASA Ames was responsible for the day-to-day functions of LADEE while the Goddard Space Flight Center operated the sensor suite and technology demonstration payloads as well as managing launch operations; the LADEE mission cost $280 million, which included spacecraft development and science instruments, launch services, mission operations, science processing and relay support. The Moon may have a tenuous atmosphere of moving particles leaping up from and falling back to the Moon's surface, giving rise to a "dust atmosphere" that looks static but is composed of dust particles in constant motion. According to models proposed starting from 1956, on the daylit side of the Moon, solar ultraviolet and X-ray radiation is energetic enough to knock electrons out of atoms and molecules in the lunar soil. Positive charges build up until the tiniest particles of lunar dust are repelled from the surface and lofted anywhere from metres to kilometres high, with the smallest particles reaching the highest altitudes.
They fall back toward the surface where the process is repeated. On the night side, the dust is negatively charged by electrons in the solar wind. Indeed, the "fountain model" suggests that the night side would charge up to higher voltages than the day side launching dust particles to higher velocities and altitudes; this effect could be further enhanced during the portion of the Moon's orbit where it passes through Earth's magnetotail. On the terminator there could be significant horizontal electric fields forming between the day and night areas, resulting in horizontal dust transport; the Moon has been shown to have a "sodium tail" too faint to be detected by the human eye. It is hundreds of thousands of miles long, was discovered in 1998 as a result of Boston University scientists observing the Leonid meteor storm; the Moon is releasing atomic sodium gas from its surface, solar radiation pressure accelerates the sodium atoms in the anti-sunward direction, forming an elongated tail which points away from the Sun.
As of April 2013, it had not yet been determined whether ionized sodium gas atoms or charged dust are the cause of the reported Moon glows. China's Chang'e 3 spacecraft, launched on December 1, 2013, entered lunar orbit on December 6, was expected to contaminate the tenuous lunar exosphere with both propellant from engine firings and lunar dust from the vehicle's landing. While concern was expressed that this could disrupt LADEE's mission, such as its baseline readings of the Moon's exosphere, it instead provided additional science value since both the quantity and composition of the spacecraft's propulsion system exhaust were known. Data from LADEE was used to track the distribution and eventual dissipation of the exhaust and dust in the Moon's exosphere, it was possible to observe the migration of water, one component of the exhaust, giving insight on how it is transported and becomes trapped around the lunar poles. The LADEE mission was designed to address three major science goals: Determine the global density and time variability of the tenuous lunar exosphere before it is perturbed by further human activity.
LADEE was launched on September 7, 2013, at 03:27 UTC, from the Wallops Flight Facility at the Mid-Atlantic Regional Spaceport on a Minotaur V carrier rocket. This was the first lunar mission to be launched from that facility; the launch had the potential for visibility along much of the U. S. eastern seaboard, from Maine to South Carolina. As the Minotaur V is a solid-propellant rocket, spacecraft attitude control on this mission operated a bit differently from a typical liquid-fueled rocket with more continuous closed-loop feedback; the first three Minotaur stages "fly a pre-programmed attitude profile" to gain velocity and deliver the vehicle to its preliminary trajectory, while the fourth stage is used to modify the flight profil
Apollo 16 was the tenth manned mission in the United States Apollo space program, the fifth and penultimate to land on the Moon, the second to land in the lunar highlands. The second of the so-called "J missions," it was crewed by Commander John Young, Lunar Module Pilot Charles Duke and Command Module Pilot Ken Mattingly. Launched from the Kennedy Space Center in Florida at 12:54 PM EST on April 16, 1972, the mission lasted 11 days, 1 hour, 51 minutes, concluded at 2:45 PM EST on April 27. Young and Duke spent 71 hours—just under three days—on the lunar surface, during which they conducted three extra-vehicular activities or moonwalks, totaling 20 hours and 14 minutes; the pair drove the Lunar Roving Vehicle, the second produced and used on the Moon, for 26.7 kilometers. On the surface and Duke collected 95.8 kilograms of lunar samples for return to Earth, while Command Module Pilot Ken Mattingly orbited in the command and service module above to perform observations. Mattingly spent 64 revolutions in lunar orbit.
After Young and Duke rejoined Mattingly in lunar orbit, the crew released a subsatellite from the service module. During the return trip to Earth, Mattingly performed a one-hour spacewalk to retrieve several film cassettes from the exterior of the service module. Apollo 16's landing spot in the highlands was chosen to allow the astronauts to gather geologically older lunar material than the samples obtained in three of the first four Moon landings, which were in or near lunar maria. Samples from the Descartes Formation and the Cayley Formation disproved a hypothesis that the formations were volcanic in origin. Mattingly had been assigned to the prime crew of Apollo 13, but was exposed to rubella through Duke, at that time on the back-up crew for Apollo 13, who had caught it from one of his children, he never contracted the illness, but was removed from the crew and replaced by his backup, Jack Swigert, three days before the launch. Young, a captain in the United States Navy, had flown on three spaceflights prior to Apollo 16: Gemini 3, Gemini 10 and Apollo 10, which orbited the Moon.
One of 19 astronauts selected by NASA in April 1966, Duke had never flown in space before Apollo 16. He served on the support crew of Apollo 10 and was a capsule communicator for Apollo 11. Although not announced, the original backup crew consisted of Fred W. Haise, William R. Pogue and Gerald P. Carr, who were targeted for the prime crew assignment on Apollo 19. However, after the cancellations of Apollos 18 and 19 were finalized in September 1970 this crew would not rotate to a lunar mission as planned. Subsequently and Mitchell were recycled to serve as members of the backup crew after returning from Apollo 14, while Pogue and Carr were reassigned to the Skylab program where they flew on Skylab 4. Anthony W. England Karl G. Henize Henry W. Hartsfield Jr. Robert F. Overmyer Donald H. Peterson The insignia of Apollo 16 is dominated by a rendering of an American eagle and a red and blue shield, representing the people of the United States, over a gray background representing the lunar surface.
Overlaying the shield is a gold NASA vector, orbiting the Moon. On its gold-outlined blue border, there are 16 stars, representing the mission number, the names of the crew members: Young, Duke; the insignia was designed from ideas submitted by the crew of the mission. Apollo 16 was the second of the Apollo type J missions, featuring the use of the Lunar Roving Vehicle, increased scientific capability, lunar surface stays of three days; as Apollo 16 was the penultimate mission in the Apollo program and there was no new hardware or procedures to test on the lunar surface, the last two missions presented opportunities for astronauts to clear up some uncertainties in understanding the Moon's properties. Although previous Apollo expeditions, including Apollo 14 and Apollo 15, obtained samples of pre-mare lunar material, before lava began to upwell from the Moon's interior and flood the low areas and basins, none had visited the lunar highlands. Apollo 14 had visited and sampled a ridge of material, ejected by the impact that created the Mare Imbrium impact basin.
Apollo 15 had sampled material in the region of Imbrium, visiting the basin's edge. There remained the possibility, because the Apollo 14 and Apollo 15 landing sites were associated with the Imbrium basin, that different geologic processes were prevalent in areas of the lunar highlands far from Mare Imbrium. Several members of the scientific community remarked that the central lunar highlands resembled regions on Earth that were created by volcanic processes and hypothesized the same might be true on the Moon, they had hoped. Two locations on the Moon were given primary consideration for exploration by the Apollo 16 expedition: the Descartes Highlands region west of Mare Nectaris and the crater Alphonsus. At Descartes, the Cayley and Descartes formations were the primary areas of interest in that scientists suspected, based on telescopic and orbital imagery, that the terrain found there was formed by magma more viscous than that which formed the lunar maria; the Cayley Formation's age was approximated to be about the same as Mare Imbrium based on the local frequency of impact craters.
The considerable distance between the Descartes site and previous Apollo landing sites would be beneficial for the network of geophysical instruments, portions of which were deployed on each Apollo expedition beginning with Apollo 12. At the Alphonsus, three scientific objectives were determined to be of primary int
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy; this includes the radial elliptic orbit, with eccentricity equal to 1. In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter; the relative position of one body with respect to the other follows an elliptic orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, tundra orbit. Under standard assumptions the orbital speed of a body traveling along an elliptic orbit can be computed from the vis-viva equation as: v = μ where: μ is the standard gravitational parameter, r is the distance between the orbiting bodies. A is the length of the semi-major axis; the velocity equation for a hyperbolic trajectory has either + 1 a, or it is the same with the convention that in that case a is negative.
Under standard assumptions the orbital period of a body traveling along an elliptic orbit can be computed as: T = 2 π a 3 μ where: μ is the standard gravitational parameter, a is the length of the semi-major axis. Conclusions: The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis, For a given semi-major axis the orbital period does not depend on the eccentricity. Under standard assumptions, specific orbital energy of elliptic orbit is negative and the orbital energy conservation equation for this orbit can take the form: v 2 2 − μ r = − μ 2 a = ϵ < 0 where: v is the orbital speed of the orbiting body, r is the distance of the orbiting body from the central body, a is the length of the semi-major axis, μ is the standard gravitational parameter. Conclusions: For a given semi-major axis the specific orbital energy is independent of the eccentricity. 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 It can be helpful to know the energy in terms of the semi major axis.
The total energy of the orbit is given by E = − G M m 2 a. Since gravity is a central force, the angular momentum is constant: L ˙ = r × F = r × F r ^ = 0 At the closest and furthest approaches, the angular momentum is perpendicular to the distance from the mass orbited, therefore: L = r p = r m v; the total energy of the orbit is given by E =. We may obtain E = 1 2 L 2 m r 2 − G M m r; this is true for r being the closest / furthest distance so we get two simultaneous equations which we solve for E: E = − G M m r 1 + r 2 Since r 1 = a + a ϵ and r 2 = a − a ϵ, where epsilon is the eccentricity of the orbit, we have the stated result. The flight path angle is the angle between the orbiting body's velocity vector and the local horizontal. Under standard assumptions of the conservation of a
The Soviet Union the Union of Soviet Socialist Republics, was a socialist state in Eurasia that existed from 1922 to 1991. Nominally a union of multiple national Soviet republics, its government and economy were centralized; the country was a one-party state, governed by the Communist Party with Moscow as its capital in its largest republic, the Russian Soviet Federative Socialist Republic. Other major urban centres were Leningrad, Minsk, Alma-Ata, Novosibirsk, it spanned over 10,000 kilometres east to west across 11 time zones, over 7,200 kilometres north to south. It had five climate zones: tundra, steppes and mountains; the Soviet Union had its roots in the 1917 October Revolution, when the Bolsheviks, led by Vladimir Lenin, overthrew the Russian Provisional Government which had replaced Tsar Nicholas II during World War I. In 1922, the Soviet Union was formed by a treaty which legalized the unification of the Russian, Transcaucasian and Byelorussian republics that had occurred from 1918. Following Lenin's death in 1924 and a brief power struggle, Joseph Stalin came to power in the mid-1920s.
Stalin committed the state's ideology to Marxism–Leninism and constructed a command economy which led to a period of rapid industrialization and collectivization. During his rule, political paranoia fermented and the Great Purge removed Stalin's opponents within and outside of the party via arbitrary arrests and persecutions of many people, resulting in at least 600,000 deaths. In 1933, a major famine struck the country. Before the start of World War II in 1939, the Soviets signed the Molotov–Ribbentrop Pact, agreeing to non-aggression with Nazi Germany, after which the USSR invaded Poland on 17 September 1939. In June 1941, Germany broke the pact and invaded the Soviet Union, opening the largest and bloodiest theatre of war in history. Soviet war casualties accounted for the highest proportion of the conflict in the effort of acquiring the upper hand over Axis forces at intense battles such as Stalingrad and Kursk; the territories overtaken by the Red Army became satellite states of the Soviet Union.
The post-war division of Europe into capitalist and communist halves would lead to increased tensions with the United States-led Western Bloc, known as the Cold War. Stalin died in 1953 and was succeeded by Nikita Khrushchev, who in 1956 denounced Stalin and began the de-Stalinization; the Cuban Missile Crisis occurred during Khrushchev's rule, among the many factors that led to his downfall in 1964. In the early 1970s, there was a brief détente of relations with the United States, but tensions resumed with the Soviet–Afghan War in 1979. In 1985, the last Soviet premier, Mikhail Gorbachev, sought to reform and liberalize the economy through his policies of glasnost and perestroika, which caused political instability. In 1989, Soviet satellite states in Eastern Europe overthrew their respective communist governments; as part of an attempt to prevent the country's dissolution due to rising nationalist and separatist movements, a referendum was held in March 1991, boycotted by some republics, that resulted in a majority of participating citizens voting in favor of preserving the union as a renewed federation.
Gorbachev's power was diminished after Russian President Boris Yeltsin's high-profile role in facing down a coup d'état attempted by Communist Party hardliners. In late 1991, Gorbachev resigned and the Supreme Soviet of the Soviet Union met and formally dissolved the Soviet Union; the remaining 12 constituent republics emerged as independent post-Soviet states, with the Russian Federation—formerly the Russian SFSR—assuming the Soviet Union's rights and obligations and being recognized as the successor state. The Soviet Union was a powerhouse of many significant technological achievements and innovations of the 20th century, including the world's first human-made satellite, the first humans in space and the first probe to land on another planet, Venus; the country had the largest standing military in the world. The Soviet Union was recognized as one of the five nuclear weapons states and possessed the largest stockpile of weapons of mass destruction, it was a founding permanent member of the United Nations Security Council as well as a member of the Organization for Security and Co-operation in Europe, the World Federation of Trade Unions and the leading member of the Council for Mutual Economic Assistance and the Warsaw Pact.
The word "Soviet" is derived from a Russian word сове́т meaning council, advice, harmony and all deriving from the proto-Slavic verbal stem of vět-iti, related to Slavic věst, English "wise", the root in "ad-vis-or", or the Dutch weten. The word sovietnik means "councillor". A number of organizations in Russian history were called "council". For example, in the Russian Empire the State Council, which functioned from 1810 to 1917, was referred to as a Council of Ministers after the revolt of 1905. During the Georgian Affair, Vladimir Lenin envisioned an expression of Great Russian ethnic chauvinism by Joseph Stalin and his supporters, calling for these nation-states to join Russia as semi-independent parts of a greater union, which he named as the Union of Soviet Republics of Europe and Asia. Stalin resisted the proposal, but accepted it, although with Lenin's agreement changed the name of the newly proposed sta
In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Orbit refers to a repeating trajectory, although it may refer to a non-repeating trajectory. To a close approximation and satellites follow elliptic orbits, with the central mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion. For most situations, orbital motion is adequately approximated by Newtonian mechanics, which explains gravity as a force obeying an inverse-square law. However, Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of spacetime, with orbits following geodesics, provides a more accurate calculation and understanding of the exact mechanics of orbital motion; the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres. This model posited the existence of perfect moving spheres or rings to which the stars and planets were attached.
It assumed the heavens were fixed apart from the motion of the spheres, was developed without any understanding of gravity. After the planets' motions were more measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably predicting the planets' positions in the sky and more epicycles were required as the measurements became more accurate, hence the model became unwieldy. Geocentric it was modified by Copernicus to place the Sun at the centre to help simplify the model; the model was further challenged during the 16th century, as comets were observed traversing the spheres. The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular, as had been believed, that the Sun is not located at the center of the orbits, but rather at one focus. Second, he found that the orbital speed of each planet is not constant, as had been thought, but rather that the speed depends on the planet's distance from the Sun.
Third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the Sun. For the planets, the cubes of their distances from the Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are about 5.2 and 0.723 AU distant from the Sun, their orbital periods about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.23/11.862, is equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, that those bodies orbit 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.
Advances in Newtonian mechanics were used to explore variations from the simple assumptions behind Kepler orbits, such as the perturbations due to other bodies, or the impact of spheroidal rather than spherical bodies. Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus. Albert Einstein in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously; this led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory, orbits follow geodesic trajectories which are approximated well by the Newtonian predictions but the differences are measurable.
All the experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy. The original vindication of general relativity is that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion first noted by Le Verrier. However, Newton's solution is still used for most short term purposes since it is easier to use and sufficiently accurate. Within a planetary system, dwarf planets and other minor planets and space debris orbit the system's barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near or within that planet. Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time.
Mercury, the smallest planet in the Solar System, has the most eccentric orbit