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84-451: A single-stage-to-orbit ( SSTO ) vehicle reaches orbit from the surface of a body using only propellants and fluids and without expending tanks, engines, or other major hardware. The term exclusively refers to reusable vehicles . To date, no Earth-launched SSTO launch vehicles have ever been flown; orbital launches from Earth have been performed by either fully or partially expendable multi-stage rockets . The main projected advantage of
168-475: A thrust to weight ratio in excess of 1, enabling them to lift off. Clearly, one of the main issues with nuclear propulsion would be safety, both during a launch for the passengers, but also in case of a failure during launch. As of February 2024, no current program is attempting nuclear propulsion from Earth's surface. Because they can be more energetic than the potential energy that chemical fuel allows for, some laser or microwave powered rocket concepts have
252-404: A body is proportional to the product of the masses of the two attracting bodies and decreases inversely with the square of the distance between them. To this Newtonian approximation, for a system of two-point masses or spherical bodies, only influenced by their mutual gravitation (called a two-body problem ), their trajectories can be exactly calculated. If the heavier body is much more massive than
336-427: A certain time called the period. 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: Note that while bound orbits of a point mass or a spherical body with a Newtonian gravitational field are closed ellipses , which repeat the same path exactly and indefinitely, any non-spherical or non-Newtonian effects (such as caused by
420-412: 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
504-402: A more accurate calculation and understanding of the exact mechanics of orbital motion. Historically, 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
588-420: A practical sense, both of these trajectory types mean the object is "breaking free" of the planet's gravity, and "going off into space" never to return. In most situations, relativistic effects can be neglected, and Newton's laws give a sufficiently accurate description of motion. The acceleration of a body is equal to the sum of the forces acting on it, divided by its mass, and the gravitational force acting on
672-557: A proposed SSTO. It is one of only a few prototype SSTO vehicles ever built. Several other prototypes were intended, including the DC-X2 (a half-scale prototype) and the DC-Y, a full-scale vehicle which would be capable of single stage insertion into orbit. Neither of these were built, but the project was taken over by NASA in 1995, and they built the DC-XA, an upgraded one-third scale prototype. This vehicle
756-481: A relatively small delta-v increase can be helpful, and outside assistance for a vehicle is therefore desirable. Proposed launch assists include: And on-orbit resources such as: Due to weight issues such as shielding, many nuclear propulsion systems are unable to lift their own weight, and hence are unsuitable for launching to orbit. However, some designs such as the Orion project and some nuclear thermal designs do have
840-547: A reusable vehicle must be able to reenter without damage, and land safely. While single-stage rockets were once thought to be beyond reach, advances in materials technology and construction techniques have shown them to be possible. For example, calculations show that the Titan II first stage, launched on its own, would have a 25-to-1 ratio of fuel to vehicle hardware. It has a sufficiently efficient engine to achieve orbit, but without carrying much payload. Hydrogen fuel might seem
924-515: A significantly higher degree of regular maintenance. It is considered to be marginally possible to launch a single-stage-to-orbit chemically fueled spacecraft from Earth. The principal complicating factors for SSTO from Earth are: high orbital velocity of over 7,400 metres per second (27,000 km/h; 17,000 mph); the need to overcome Earth's gravity, especially in the early stages of flight; and flight within Earth's atmosphere , which limits speed in
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#17328591530851008-410: A single point called the barycenter. The paths of all the star's satellites are elliptical orbits about that barycenter. Each satellite in that system will have its own elliptical orbit with the barycenter at one focal point of that ellipse. At any point along its orbit, any satellite will have a certain value of kinetic and potential energy with respect to the barycenter, and the sum of those two energies
1092-413: A star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a planet, moon, asteroid, or Lagrange point . Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits , with the center of mass being orbited at
1176-674: A sufficiently efficient propulsion system and discontinued development. Single-stage-to-orbit is much easier to achieve on extraterrestrial bodies that have weaker gravitational fields and lower atmospheric pressure than Earth, such as the Moon and Mars, and has been achieved from the Moon by the Apollo program 's Lunar Module , by several robotic spacecraft of the Soviet Luna program , and by China's Chang'e 5 and Chang'e 6 lunar sample return missions. Before
1260-487: A technical sense—they are describing a portion of an elliptical path around the center of gravity—but the orbits are interrupted by striking the Earth. If the cannonball is fired with sufficient speed, the ground curves away from the ball at least as much as the ball falls—so the ball never strikes the ground. It is now in what could be called a non-interrupted or circumnavigating, orbit. For any specific combination of height above
1344-479: A vacuum bell in atmosphere would have disastrous consequences for the engine. Engines designed to fire in atmosphere therefore have to shorten the nozzle, only expanding the gasses to atmospheric pressure. The efficiency losses due to the smaller bell are usually mitigated via staging, as upper stage engines such as the Rocketdyne J-2 do not have to fire until atmospheric pressure is negligible, and can therefore use
1428-505: Is a constant value at every point along its orbit. As a result, as a planet approaches periapsis , the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis , its velocity will decrease as its potential energy increases. There are a few common ways of understanding orbits: The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes: Orbital rockets are launched vertically at first to lift
1512-523: 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 then 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. Joseph-Louis Lagrange developed a new approach to Newtonian mechanics emphasizing energy more than force, and made progress on
1596-407: Is adopted of taking the potential energy as zero at infinite separation, the bound orbits will have negative total energy, the parabolic trajectories zero total energy, and hyperbolic orbits positive total energy. An open orbit will have a parabolic shape if it has the velocity of exactly the escape velocity at that point in its trajectory, and it will have the shape of a hyperbola when its velocity
1680-464: Is also a vector. Because our basis vector r ^ {\displaystyle {\hat {\mathbf {r} }}} moves as the object orbits, we start by differentiating it. From time t {\displaystyle t} to t + δ t {\displaystyle t+\delta t} , the vector r ^ {\displaystyle {\hat {\mathbf {r} }}} keeps its beginning at
1764-487: Is an important concept in the engineering of a rocket. However, mass fraction may have little to do with the costs of a rocket, as the costs of fuel are very small when compared to the costs of the engineering program as a whole. As a result, a cheap rocket with a poor mass fraction may be able to deliver more payload to orbit with a given amount of money than a more complicated, more efficient rocket. Many vehicles are only narrowly suborbital, so practically anything that gives
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#17328591530851848-518: Is because of both the low density and the additional insulation required to minimize boiloff (a problem which does not occur with kerosene and many other fuels). The low density of hydrogen further affects the design of the rest of the vehicle: pumps and pipework need to be much larger in order to pump the fuel to the engine. The result is the thrust/weight ratio of hydrogen-fueled engines is 30–50% lower than comparable engines using denser fuels. This inefficiency indirectly affects gravity losses as well;
1932-404: Is greater than the escape velocity. When bodies with escape velocity or greater approach each other, they will briefly curve around each other at the time of their closest approach, and then separate, forever. All closed orbits have the shape of an ellipse . A circular orbit is a special case, wherein the foci of the ellipse coincide. The point where the orbiting body is closest to Earth is called
2016-581: Is located in the plane using vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis with the origin coinciding with the center of force. Let r {\displaystyle r} be the distance between the object and the center and θ {\displaystyle \theta } be the angle it has rotated. Let x ^ {\displaystyle {\hat {\mathbf {x} }}} and y ^ {\displaystyle {\hat {\mathbf {y} }}} be
2100-402: 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 significantly easier to use and sufficiently accurate. Within a planetary system , planets, dwarf planets , asteroids and other minor planets , comets , and space debris orbit
2184-588: The SSME ) + 10 x turboramjets. Around 1985 the NASP project was intended to launch a scramjet vehicle into orbit, but funding was stopped and the project cancelled. At around the same time, the HOTOL tried to use precooled jet engine technology, but failed to show significant advantages over rocket technology. The DC-X, short for Delta Clipper Experimental, was an uncrewed one-third scale vertical takeoff and landing demonstrator for
2268-461: The apoapsis is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.) In the case of planets orbiting a star, the mass of the star and all its satellites are calculated to be at
2352-461: The eccentricities of the planetary orbits vary over time. Mercury , the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch , Mars has the next largest eccentricity while the smallest orbital eccentricities are seen with Venus and Neptune . As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and
2436-453: The perigee , and when orbiting a body other than earth it is called the periapsis (less properly, "perifocus" or "pericentron"). The point where the satellite is farthest from Earth is called the 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. Bodies following closed orbits repeat their paths with
2520-718: 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 in gravity propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide
2604-446: The three-body problem ; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian points , no method is known to solve the equations of motion for a system with four or more bodies. Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms: Differential simulations with large numbers of objects perform
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2688-460: The 1990s) which were never constructed include: Star-raker : In 1979 Rockwell International unveiled a concept for a 100-ton payload heavy-lift multicycle airbreather ramjet/ cryogenic rocket engine , horizontal takeoff/horizontal landing single-stage-to-orbit spaceplane named Star-Raker , designed to launch heavy Space-based solar power satellites into a 300 nautical mile Earth orbit. Star-raker would have had 3 x LOX/LH2 rocket engines (based on
2772-565: The DC-X, often said that he thought the first successful orbital SSTO would more likely be fueled by propane. Some SSTO concepts use the same engine for all altitudes, which is a problem for traditional engines with a bell-shaped nozzle . Depending on the atmospheric pressure, different bell shapes are required. Engines designed to operate in a vacuum have large bells, allowing the exhaust gasses to expand to near vacuum pressures, thereby raising efficiency. Due to an effect known as Flow separation , using
2856-410: The Earth at the point half an orbit beyond, and directly opposite the firing point, below the circular orbit. At a specific horizontal firing speed called escape velocity , dependent on the mass of the planet and the distance of the object from the barycenter, an open orbit (E) is achieved that has a parabolic path . At even greater speeds the object will follow a range of hyperbolic trajectories . In
2940-461: The SSTO concept is elimination of the hardware replacement inherent in expendable launch systems. However, the non-recurring costs associated with design, development, research and engineering (DDR&E) of reusable SSTO systems are much higher than expendable systems due to the substantial technical challenges of SSTO, assuming that those technical issues can in fact be solved. SSTO vehicles may also require
3024-580: The Sun are proportional to the squares of their orbital periods. Jupiter and Venus, for example, are respectively about 5.2 and 0.723 AU distant from the Sun, their orbital periods respectively about 11.86 and 0.615 years. The proportionality is seen by the fact that the ratio for Jupiter, 5.2 /11.86 , is practically equal to that for Venus, 0.723 /0.615 , 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,
3108-420: 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 previously 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
3192-403: The accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is − μ / r 2 {\displaystyle -\mu /r^{2}} and the second is zero. Equation (2) can be rearranged using integration by parts. We can multiply through by r {\displaystyle r} because it is not zero unless
3276-421: The atmosphere to reduce the take-off weight of the vehicle. Some of the issues with this approach are: Thus with for example scramjet designs (e.g. X-43 ) the mass budgets do not seem to close for orbital launch. Similar issues occur with single-stage vehicles attempting to carry conventional jet engines to orbit—the weight of the jet engines is not compensated sufficiently by the reduction in propellant. On
3360-693: The atmosphere when it is at low altitude, and then using onboard liquid oxygen after switching to the closed cycle rocket engine at high altitude, the McDonnell Douglas DC-X , the Lockheed Martin X-33 and VentureStar which was intended to replace the Space Shuttle, and the Roton SSTO , which is a helicopter that can get to orbit. However, despite showing some promise, none of them have come close to achieving orbit yet due to problems with finding
3444-468: The atmosphere, and achieving a high enough mass-ratio to carry sufficient propellant to achieve orbit, plus a meaningful payload weight. Air-breathing designs typically fly at supersonic or hypersonic speeds, and usually include a rocket engine for the final burn for orbit. Whether rocket-powered or air-breathing, a reusable vehicle must be rugged enough to survive multiple round trips into space without adding excessive weight or maintenance. In addition
Single-stage-to-orbit - Misplaced Pages Continue
3528-458: The atmosphere, in an act commonly referred to as an aerobraking maneuver. As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a ' thought experiment ', in which a cannon on top of a tall mountain is able to fire a cannonball horizontally at any chosen muzzle speed. The effects of air friction on the cannonball are ignored (or perhaps
3612-461: The calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated. The following derivation applies to such an elliptical orbit. We start only with the Newtonian law of gravitation stating that the gravitational acceleration towards the central body is related to the inverse of the square of
3696-517: The center of gravity and mass of the planet, there is one specific firing speed (unaffected by the mass of the ball, which is assumed to be very small relative to the Earth's mass) that produces a circular orbit , as shown in (C). As the firing speed is increased beyond this, non-interrupted elliptic orbits are produced; one is shown in (D). If the initial firing is above the surface of the Earth as shown, there will also be non-interrupted elliptical orbits at slower firing speed; these will come closest to
3780-459: The coordinate system at the center of the mass of the system. Energy is associated with gravitational fields . A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational potential energy . Since work is required to separate two bodies against the pull of gravity, their gravitational potential energy increases as they are separated, and decreases as they approach one another. For point masses,
3864-683: The distance r {\displaystyle r} of the orbiting object from the center as a function of its angle θ {\displaystyle \theta } . However, it is easier to introduce the auxiliary variable u = 1 / r {\displaystyle u=1/r} and to express u {\displaystyle u} as a function of θ {\displaystyle \theta } . Derivatives of r {\displaystyle r} with respect to time may be rewritten as derivatives of u {\displaystyle u} with respect to angle. Plugging these into (1) gives So for
3948-434: The distance between them, namely where F 2 is the force acting on the mass m 2 caused by the gravitational attraction mass m 1 has for m 2 , G is the universal gravitational constant, and r is the distance between the two masses centers. From Newton's Second Law, the summation of the forces acting on m 2 related to that body's acceleration: where A 2 is the acceleration of m 2 caused by
4032-516: The early stages of flight due to drag, and influences engine performance. Advances in rocketry in the 21st century have resulted in a substantial fall in the cost to launch a kilogram of payload to either low Earth orbit or the International Space Station , reducing the main projected advantage of the SSTO concept. Notable single stage to orbit concepts include Skylon , which used the hybrid-cycle SABRE engine that can use oxygen from
4116-428: The entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations x = A cos ( t ) {\displaystyle x=A\cos(t)} and y = B sin ( t ) {\displaystyle y=B\sin(t)} of the ellipse. The location of the orbiting object at the current time t {\displaystyle t}
4200-421: The exhaust gases down to near vacuum pressures. As a result, these engine bells are counterproductive due to their excess weight. Some SSTO concepts use very high pressure engines which permit high ratios to be used from ground level. This gives good performance, negating the need for more complex solutions. Some designs for SSTO attempt to use airbreathing jet engines that collect oxidizer and reaction mass from
4284-408: The force of gravitational attraction F 2 of m 1 acting on m 2 . Combining Eq. 1 and 2: Solving for the acceleration, A 2 : where μ {\displaystyle \mu \,} is the standard gravitational parameter , in this case G m 1 {\displaystyle Gm_{1}} . It is understood that the system being described is m 2 , hence
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#17328591530854368-417: The gravitational energy decreases to zero as they approach zero separation. It is convenient and conventional to assign the potential energy as having zero value when they are an infinite distance apart, and hence it has a negative value (since it decreases from zero) for smaller finite distances. When only two gravitational bodies interact, their orbits follow a conic section . The orbit can be open (implying
4452-490: The highest accuracy in understanding orbits. In relativity theory , orbits follow geodesic trajectories which are usually approximated very well by the Newtonian predictions (except where there are very strong gravity fields and very high speeds) but the differences are measurable. Essentially 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
4536-481: The larger bell. One possible solution would be to use an aerospike engine , which can be effective in a wide range of ambient pressures. In fact, a linear aerospike engine was to be used in the X-33 design. Other solutions involve using multiple engines and other altitude adapting designs such as double-mu bells or extensible bell sections . Still, at very high altitudes, the extremely large engine bells tend to expand
4620-410: The last drop of specific impulse, and shaving off the last pound, costs money and/or reduces reliability. The Tsiolkovsky rocket equation expresses the maximum change in velocity any single rocket stage can achieve: where: Orbit In celestial mechanics , an orbit (also known as orbital revolution ) is the curved trajectory of an object such as the trajectory of a planet around
4704-536: The main challenge is achieving a high enough mass-ratio to carry sufficient propellant to achieve orbit , plus a meaningful payload weight. One possibility is to give the rocket an initial speed with a space gun , as planned in the Quicklaunch project. For air-breathing SSTO, the main challenge is system complexity and associated research and development costs, material science , and construction techniques necessary for surviving sustained high-speed flight within
4788-494: The mass ratio to delta-v curve is very steep to reach orbit in a single stage, and this makes a 10% difference to the mass ratio on top of the tankage and pump savings. The overall effect is that there is surprisingly little difference in overall performance between SSTOs that use hydrogen and those that use denser fuels, except that hydrogen vehicles may be rather more expensive to develop and buy. Careful studies have shown that some dense fuels (for example liquid propane ) exceed
4872-504: The mountain is high enough that the cannon is above the Earth's atmosphere, which is the same thing). If the cannon fires its ball with a low initial speed, the trajectory of the ball curves downward and hits the ground (A). As the firing speed is increased, the cannonball hits the ground farther (B) away from the cannon, because while the ball is still falling towards the ground, the ground is increasingly curving away from it (see first point, above). All these motions are actually "orbits" in
4956-410: The object never returns) or closed (returning). Which it is depends on the total energy ( 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, the speed is always less than the escape velocity. Since the kinetic energy is never negative if the common convention
5040-473: The obvious fuel for SSTO vehicles. When burned with oxygen , hydrogen gives the highest specific impulse of any commonly used fuel: around 450 seconds, compared with up to 350 seconds for kerosene . Hydrogen has the following advantages: However, hydrogen also has these disadvantages: These issues can be dealt with, but at extra cost. While kerosene tanks can be 1% of the weight of their contents, hydrogen tanks often must weigh 10% of their contents. This
5124-498: The orbiting object crashes. Then having the derivative be zero gives that the function is a constant. which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h , is the angular momentum per unit mass . In order to get an equation for the orbit from equation (1), we need to eliminate time. (See also Binet equation .) In polar coordinates, this would express
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#17328591530855208-411: The orbits of bodies subject to gravity were conic sections (this assumes that the force of gravity propagates instantaneously). Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses , and that those bodies orbit their common center of mass . Where one body is much more massive than the other (as is the case of an artificial satellite orbiting a planet), it
5292-421: The origin and rotates from angle θ {\displaystyle \theta } to θ + θ ˙ δ t {\displaystyle \theta +{\dot {\theta }}\ \delta t} which moves its head a distance θ ˙ δ t {\displaystyle {\dot {\theta }}\ \delta t} in
5376-477: The other hand, LACE-like precooled airbreathing designs such as the Skylon spaceplane (and ATREX ) which transition to rocket thrust at rather lower speeds (Mach 5.5) do seem to give, on paper at least, an improved orbital mass fraction over pure rockets (even multistage rockets) sufficiently to hold out the possibility of full reusability with better payload fraction. It is important to note that mass fraction
5460-493: The performance of hydrogen fuel when used in an SSTO launch vehicle by 10% for the same dry weight. In the 1960s Philip Bono investigated single-stage, VTVL tripropellant rockets , and showed that it could improve payload size by around 30%. Operational experience with the DC-X experimental rocket has caused a number of SSTO advocates to reconsider hydrogen as a satisfactory fuel. The late Max Hunter, while employing hydrogen fuel in
5544-627: The perpendicular direction θ ^ {\displaystyle {\hat {\boldsymbol {\theta }}}} giving a derivative of θ ˙ θ ^ {\displaystyle {\dot {\theta }}{\hat {\boldsymbol {\theta }}}} . We can now find the velocity and acceleration of our orbiting object. The coefficients of r ^ {\displaystyle {\hat {\mathbf {r} }}} and θ ^ {\displaystyle {\hat {\boldsymbol {\theta }}}} give
5628-422: The potential to launch vehicles into orbit, single stage. In practice, this area is not possible with current technology. The design space constraints of SSTO vehicles were described by rocket design engineer Robert Truax : Using similar technologies (i.e., the same propellants and structural fraction), a two-stage-to-orbit vehicle will always have a better payload-to-weight ratio than a single stage designed for
5712-548: The radial and transverse polar basis with the first being the unit vector pointing from the central body to the current location of the orbiting object and the second being the orthogonal unit vector pointing in the direction that the orbiting object would travel if orbiting in a counter clockwise circle. Then the vector to the orbiting object is We use r ˙ {\displaystyle {\dot {r}}} and θ ˙ {\displaystyle {\dot {\theta }}} to denote
5796-408: The rocket above the atmosphere (which causes frictional drag), and then slowly pitch over and finish firing the rocket engine parallel to the atmosphere to achieve orbit speed. Once in orbit, their speed keeps them in orbit above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall). Occasionally a space craft will intentionally intercept
5880-408: The same mission, in most cases, a very much better [payload-to-weight ratio]. Only when the structural factor approaches zero [very little vehicle structure weight] does the payload/weight ratio of a single-stage rocket approach that of a two-stage. A slight miscalculation and the single-stage rocket winds up with no payload. To get any at all, technology needs to be stretched to the limit. Squeezing out
5964-429: The second half of the twentieth century, very little research was conducted into space travel. During the 1960s, some of the first concept designs for this kind of craft began to emerge. One of the earliest SSTO concepts was the expendable One stage Orbital Space Truck (OOST) proposed by Philip Bono , an engineer for Douglas Aircraft Company . A reusable version named ROOST was also proposed. Another early SSTO concept
6048-453: The slight oblateness of the Earth , or by relativistic effects , thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion . The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves
6132-440: The smaller, as in the case of a satellite or small moon orbiting a planet or for the Earth orbiting the Sun, it is accurate enough and convenient to describe the motion in terms of a coordinate system that is centered on the heavier body, and we say that the lighter body is in orbit around the heavier. For the case where the masses of two bodies are comparable, an exact Newtonian solution is still sufficient and can be had by placing
6216-449: The spheres and was developed without any understanding of gravity. After the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although the model was capable of reasonably accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became increasingly unwieldy. Originally geocentric , it
6300-730: The standard Euclidean bases and let r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ {\displaystyle {\hat {\mathbf {r} }}=\cos(\theta ){\hat {\mathbf {x} }}+\sin(\theta ){\hat {\mathbf {y} }}} and θ ^ = − sin ( θ ) x ^ + cos ( θ ) y ^ {\displaystyle {\hat {\boldsymbol {\theta }}}=-\sin(\theta ){\hat {\mathbf {x} }}+\cos(\theta ){\hat {\mathbf {y} }}} be
6384-412: The standard derivatives of how this distance and angle change over time. We take the derivative of a vector to see how it changes over time by subtracting its location at time t {\displaystyle t} from that at time t + δ t {\displaystyle t+\delta t} and dividing by δ t {\displaystyle \delta t} . The result
6468-443: The subscripts can be dropped. We assume that the central body is massive enough that it can be considered to be stationary and we ignore the more subtle effects of general relativity . When a pendulum or an object attached to a spring swings in an ellipse, the inward acceleration/force is proportional to the distance A = F / m = − k r . {\displaystyle A=F/m=-kr.} Due to
6552-463: 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 that 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 ,
6636-436: The vehicle has to hold itself up on rocket power until it reaches orbit. The lower excess thrust of the hydrogen engines due to the lower thrust/weight ratio means that the vehicle must ascend more steeply, and so less thrust acts horizontally. Less horizontal thrust results in taking longer to reach orbit, and gravity losses are increased by at least 300 metres per second (1,100 km/h; 670 mph). While not appearing large,
6720-498: The way vectors add, the component of the force in the x ^ {\displaystyle {\hat {\mathbf {x} }}} or in the y ^ {\displaystyle {\hat {\mathbf {y} }}} directions are also proportionate to the respective components of the distances, r x ″ = A x = − k r x {\displaystyle r''_{x}=A_{x}=-kr_{x}} . Hence,
6804-403: Was a reusable launch vehicle named NEXUS which was proposed by Krafft Arnold Ehricke in the early 1960s. It was one of the largest spacecraft ever conceptualized with a diameter of over 50 metres and the capability to lift up to 2000 short tons into Earth orbit, intended for missions to further out locations in the Solar System such as Mars . The North American Air Augmented VTOVL from 1963
6888-510: Was a similarly large craft which would have used ramjets to decrease the liftoff mass of the vehicle by removing the need for large amounts of liquid oxygen while traveling through the atmosphere. From 1965, Robert Salkeld investigated various single stage to orbit winged spaceplane concepts. He proposed a vehicle which would burn hydrocarbon fuel while in the atmosphere and then switch to hydrogen fuel for increasing efficiency once in space. Further examples of Bono's early concepts (prior to
6972-803: Was lost when it landed with only three of its four landing pads deployed, which caused it to tip over on its side and explode. The project has not been continued since. From 1999 to 2001 Rotary Rocket attempted to build a SSTO vehicle called the Roton. It received a large amount of media attention and a working sub-scale prototype was completed, but the design was largely impractical. There have been various approaches to SSTO, including pure rockets that are launched and land vertically, air-breathing scramjet -powered vehicles that are launched and land horizontally, nuclear-powered vehicles, and even jet-engine -powered vehicles that can fly into orbit and return landing like an airliner, completely intact. For rocket-powered SSTO,
7056-516: 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 (or epicyclic ), as had previously been believed, and that
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