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111-537: On 7 May 2009, the KEI program was canceled due primarily to financial reasons. There were five first-stage rocket motor tests planned to be carried out by Alliant Techsystems (ATK) in Promontory, Utah. The second test firing of a KEI first stage rocket motor was conducted on 14 June 2007. The static firing included a full duration burn and a demonstration of the thrust vector control nozzle. The fourth test firing of

222-404: 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

333-427: 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

444-417: A crane. This is generally not practical for larger space vehicles, which are assembled off the pad and moved into place on the launch site by various methods. NASA's Apollo / Saturn V crewed Moon landing vehicle, and Space Shuttle , were assembled vertically onto mobile launcher platforms with attached launch umbilical towers, in a Vehicle Assembly Building , and then a special crawler-transporter moved

555-510: A dragon's head with an open mouth. The British scientist and historian Joseph Needham points out that the written material and depicted illustration of this rocket come from the oldest stratum of the Huolongjing , which can be dated roughly 1300–1350 AD (from the book's part 1, chapter 3, page 23). Another example of an early multistaged rocket is the Juhwa (走火) of Korean development. It

666-408: A higher cost for deployment. Hot-staging is a type of rocket staging in which the next stage fires its engines before separation instead of after. During hot-staging, the earlier stage throttles down its engines. Hot-staging may reduce the complexity of stage separation, and gives a small extra payload capacity to the booster. It also eliminates the need for ullage motors , as the acceleration from

777-518: A launch vehicle, a useful performance metric to examine is the thrust-to-weight ratio, and is calculated by the equation: The common thrust-to-weight ratio of a launch vehicle is within the range of 1.3 to 2.0. Another performance metric to keep in mind when designing each rocket stage in a mission is the burn time, which is the amount of time the rocket engine will last before it has exhausted all of its propellant. For most non-final stages, thrust and specific impulse can be assumed constant, which allows

888-472: A multistage rocket introduces additional risk into the success of the launch mission. Reducing the number of separation events results in a reduction in complexity . Separation events occur when stages or strap-on boosters separate after use, when the payload fairing separates prior to orbital insertion, or when used, a launch escape system which separates after the early phase of a launch. Pyrotechnic fasteners , or in some cases pneumatic systems like on

999-492: 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 a focal point of the ellipse, as described by Kepler's laws of planetary motion . For most situations, orbital motion

1110-420: 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

1221-466: A rocket system will be when performing optimizations and comparing varying configurations for a mission. For initial sizing, the rocket equations can be used to derive the amount of propellant needed for the rocket based on the specific impulse of the engine and the total impulse required in N·s. The equation is: where g is the gravity constant of Earth. This also enables the volume of storage required for

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1332-410: 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

1443-399: A technical algorithm that generates an analytical solution that can be implemented by a program, or simple trial and error. For the trial and error approach, it is best to begin with the final stage, calculating the initial mass which becomes the payload for the previous stage. From there it is easy to progress all the way down to the initial stage in the same manner, sizing all the stages of

1554-491: 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

1665-431: Is a commonly used rocket system to attain Earth orbit. The spacecraft uses three distinct stages to provide propulsion consecutively in order to achieve orbital velocity. It is intermediate between a four-stage-to-orbit launcher and a two-stage-to-orbit launcher. Other designs (in fact, most modern medium- to heavy-lift designs) do not have all three stages inline on the main stack, instead having strap-on boosters for

1776-505: 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

1887-528: 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

1998-415: Is a safe and reasonable assumption to say that 91 to 94 percent of the total mass is fuel. It is also important to note there is a small percentage of "residual" propellant that will be left stuck and unusable inside the tank, and should also be taken into consideration when determining amount of fuel for the rocket. A common initial estimate for this residual propellant is five percent. With this ratio and

2109-418: 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. Historically, the apparent motions of

2220-407: 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

2331-464: 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

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2442-453: Is generally assembled at its manufacturing site and shipped to the launch site; the term vehicle assembly refers to the mating of all rocket stage(s) and the spacecraft payload into a single assembly known as a space vehicle . Single-stage vehicles ( suborbital ), and multistage vehicles on the smaller end of the size range, can usually be assembled directly on the launch pad by lifting the stage(s) and spacecraft vertically in place by means of

2553-404: 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

2664-428: Is intermediate between a five-stage-to-orbit launcher and a three-stage-to-orbit launcher, most often used with solid-propellant launch systems. Other designs do not have all four stages inline on the main stack, instead having strap-on boosters for the "stage-0" with three core stages. In these designs, the boosters and first stage fire simultaneously instead of consecutively, providing extra initial thrust to lift

2775-581: 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

2886-402: 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

2997-393: Is the initial to final mass ratio, which is the ratio between the rocket stage's full initial mass and the rocket stage's final mass once all of its fuel has been consumed. The equation for this ratio is: Where m E {\displaystyle m_{\mathrm {E} }} is the empty mass of the stage, m p {\displaystyle m_{\mathrm {p} }}

3108-402: Is the mass of the oxidizer and m f u e l {\displaystyle m_{\mathrm {fuel} }} is the mass of the fuel. This mixture ratio not only governs the size of each tank, but also the specific impulse of the rocket. Determining the ideal mixture ratio is a balance of compromises between various aspects of the rocket being designed, and can vary depending on

3219-404: Is the mass of the propellant, and m P L {\displaystyle m_{\mathrm {PL} }} is the mass of the payload. The second dimensionless performance quantity is the structural ratio, which is the ratio between the empty mass of the stage, and the combined empty mass and propellant mass as shown in this equation: The last major dimensionless performance quantity

3330-455: Is the payload ratio, which is the ratio between the payload mass and the combined mass of the empty rocket stage and the propellant: After comparing the three equations for the dimensionless quantities, it is easy to see that they are not independent of each other, and in fact, the initial to final mass ratio can be rewritten in terms of structural ratio and payload ratio: These performance ratios can also be used as references for how efficient

3441-480: The Falcon 9 Full Thrust , are typically used to separate rocket stages. A two-stage-to-orbit ( TSTO ) or two-stage rocket launch vehicle is a spacecraft in which two distinct stages provide propulsion consecutively in order to achieve orbital velocity. It is intermediate between a three-stage-to-orbit launcher and a hypothetical single-stage-to-orbit (SSTO) launcher. The three-stage-to-orbit launch system

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3552-734: The RTV-G-4 Bumper rockets tested at the White Sands Proving Ground and later at Cape Canaveral from 1948 to 1950. These consisted of a V-2 rocket and a WAC Corporal sounding rocket. The greatest altitude ever reached was 393 km, attained on February 24, 1949, at White Sands. In 1947, the Soviet rocket engineer and scientist Mikhail Tikhonravov developed a theory of parallel stages, which he called "packet rockets". In his scheme, three parallel stages were fired from liftoff , but all three engines were fueled from

3663-509: The Singijeon , or 'magical machine arrows' in the 16th century. The earliest experiments with multistage rockets in Europe were made in 1551 by Austrian Conrad Haas (1509–1576), the arsenal master of the town of Hermannstadt , Transylvania (now Sibiu/Hermannstadt, Romania). This concept was developed independently by at least five individuals: The first high-speed multistage rockets were

3774-490: The Soviet and U.S. space programs, were not passivated after mission completion. During the initial attempts to characterize the space debris problem, it became evident that a good proportion of all debris was due to the breaking up of rocket upper stages, particularly unpassivated upper-stage propulsion units. An illustration and description in the 14th century Chinese Huolongjing by Jiao Yu and Liu Bowen shows

3885-464: 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

3996-467: 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

4107-453: 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

4218-737: 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

4329-446: 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

4440-523: The "stage-0" with two core stages. In these designs, the boosters and first stage fire simultaneously instead of consecutively, providing extra initial thrust to lift the full launcher weight and overcome gravity losses and atmospheric drag. The boosters are jettisoned a few minutes into flight to reduce weight. The four-stage-to-orbit launch system is a rocket system used to attain Earth orbit. The spacecraft uses four distinct stages to provide propulsion consecutively in order to achieve orbital velocity. It

4551-410: 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

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4662-664: The FY10 defense budget. First stage (rocketry) A multistage rocket or step rocket is a launch vehicle that uses two or more rocket stages , each of which contains its own engines and propellant . A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched. By jettisoning stages when they run out of propellant,

4773-427: 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,

4884-403: 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

4995-462: 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

5106-401: The breakup of a single upper stage while in orbit. After the 1990s, spent upper stages are generally passivated after their use as a launch vehicle is complete in order to minimize risks while the stage remains derelict in orbit . Passivation means removing any sources of stored energy remaining on the vehicle, as by dumping fuel or discharging batteries. Many early upper stages, in both

5217-466: 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

5328-517: 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

5439-459: 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,

5550-440: The cost of the lower stages lifting engines which are not yet being used, as well as making the entire rocket more complex and harder to build than a single stage. In addition, each staging event is a possible point of launch failure, due to separation failure, ignition failure, or stage collision. Nevertheless, the savings are so great that every rocket ever used to deliver a payload into orbit has had staging of some sort. One of

5661-489: The delta-v into fractions. As each lower stage drops off and the succeeding stage fires, the rest of the rocket is still traveling near the burnout speed. Each lower stage's dry mass includes the propellant in the upper stages, and each succeeding upper stage has reduced its dry mass by discarding the useless dry mass of the spent lower stages. A further advantage is that each stage can use a different type of rocket engine, each tuned for its particular operating conditions. Thus

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5772-419: The different stages of the rocket should be clearly defined. Continuing with the previous example, the end of the first stage which is sometimes referred to as 'stage 0', can be defined as when the side boosters separate from the main rocket. From there, the final mass of stage one can be considered the sum of the empty mass of stage one, the mass of stage two (the main rocket and the remaining unburned fuel) and

5883-683: 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

5994-434: 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

6105-404: The drawbacks of a less efficient specific impulse rating. But suppose the defining constraint for the launch system is volume, and a low density fuel is required such as hydrogen. This example would be solved by using an oxidizer-rich mixture ratio, reducing efficiency and specific impulse rating, but will meet a smaller tank volume requirement. The ultimate goal of optimal staging is to maximize

6216-428: 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}

6327-598: The entire vehicle stack to the launch pad in an upright position. In contrast, vehicles such as the Russian Soyuz rocket and the SpaceX Falcon 9 are assembled horizontally in a processing hangar, transported horizontally, and then brought upright at the pad. Spent upper stages of launch vehicles are a significant source of space debris remaining in orbit in a non-operational state for many years after use, and occasionally, large debris fields created from

6438-444: The equation for burn time to be written as: Where m 0 {\displaystyle m_{\mathrm {0} }} and m f {\displaystyle m_{\mathrm {f} }} are the initial and final masses of the rocket stage respectively. In conjunction with the burnout time, the burnout height and velocity are obtained using the same values, and are found by these two equations: When dealing with

6549-411: The equations for determining the burnout velocities, burnout times, burnout altitudes, and mass of each stage. This would make for a better approach to a conceptual design in a situation where a basic understanding of the system behavior is preferential to a detailed, accurate design. One important concept to understand when undergoing restricted rocket staging, is how the burnout velocity is affected by

6660-486: The first stage of the American Atlas I and Atlas II launch vehicles, arranged in a row, used parallel staging in a similar way: the outer pair of booster engines existed as a jettisonable pair which would, after they shut down, drop away with the lowermost outer skirt structure, leaving the central sustainer engine to complete the first stage's engine burn towards apogee or orbit. Separation of each portion of

6771-639: The first-stage rocket motor was completed on November 13, 2008. The test demonstrated a successful operation of the first-stage rocket motor in its final flight configuration that was to be used during a Summer 2009 flight test. Due to the cancellation of KEI in May 2009, this test did not occur. On 7 May 2009, Missile Defense Agency executive director David Altwegg announced that KEI would be canceled due to technical and financial reasons. The announcement had previously been overlooked during Defense Secretary Robert Gates ' April 6 press conference on changing priorities in

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6882-408: 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

6993-467: The fuel to be calculated if the density of the fuel is known, which is almost always the case when designing the rocket stage. The volume is yielded when dividing the mass of the propellant by its density. Asides from the fuel required, the mass of the rocket structure itself must also be determined, which requires taking into account the mass of the required thrusters, electronics, instruments, power equipment, etc. These are known quantities for typical off

7104-414: The fueled-to-dry mass ratio and on the effective exhaust velocity of the engine. This relation is given by the classical rocket equation : where: The delta v required to reach low Earth orbit (or the required velocity of a sufficiently heavy suborbital payload) requires a wet to dry mass ratio larger than has been achieved in a single rocket stage. The multistage rocket overcomes this limit by splitting

7215-414: The full launcher weight and overcome gravity losses and atmospheric drag. The boosters are jettisoned a few minutes into flight to reduce weight. Orbit This is an accepted version of this page 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 a star, or of a natural satellite around

7326-417: 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

7437-490: 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

7548-451: The initial rocket stages usually have a lower specific impulse rating, trading efficiency for superior thrust in order to quickly push the rocket into higher altitudes. Later stages of the rocket usually have a higher specific impulse rating because the vehicle is further outside the atmosphere and the exhaust gas does not need to expand against as much atmospheric pressure. When selecting the ideal rocket engine to use as an initial stage for

7659-443: The largest rocket ever to do so, as well as the first reusable vehicle to utilize hot staging. A rocket system that implements tandem staging means that each individual stage runs in order one after the other. The rocket breaks free from the previous stage, then begins burning through the next stage in straight succession. On the other hand, a rocket that implements parallel staging has two or more different stages that are active at

7770-402: The largest, the second stage and subsequent upper stages are above it, usually decreasing in size. In parallel staging schemes solid or liquid rocket boosters are used to assist with launch. These are sometimes referred to as "stage 0". In the typical case, the first-stage and booster engines fire to propel the entire rocket upwards. When the boosters run out of fuel, they are detached from

7881-420: The lower-stage engines are designed for use at atmospheric pressure, while the upper stages can use engines suited to near vacuum conditions. Lower stages tend to require more structure than upper as they need to bear their own weight plus that of the stages above them. Optimizing the structure of each stage decreases the weight of the total vehicle and provides further advantage. The advantage of staging comes at

7992-403: The mass of the payload. High-altitude and space-bound upper stages are designed to operate with little or no atmospheric pressure. This allows the use of lower pressure combustion chambers and engine nozzles with optimal vacuum expansion ratios . Some upper stages, especially those using hypergolic propellants like Delta-K or Ariane 5 ES second stage, are pressure fed , which eliminates

8103-464: The mass of the propellant calculated, the mass of the empty rocket weight can be determined. Sizing rockets using a liquid bipropellant requires a slightly more involved approach because there are two separate tanks that are required: one for the fuel, and one for the oxidizer. The ratio of these two quantities is known as the mixture ratio, and is defined by the equation: Where m o x {\displaystyle m_{\mathrm {ox} }}

8214-408: The mass of the remaining rocket is decreased. Each successive stage can also be optimized for its specific operating conditions, such as decreased atmospheric pressure at higher altitudes. This staging allows the thrust of the remaining stages to more easily accelerate the rocket to its final velocity and height. In serial or tandem staging schemes, the first stage is at the bottom and is usually

8325-427: 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 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

8436-423: The most common measures of rocket efficiency is its specific impulse, which is defined as the thrust per flow rate (per second) of propellant consumption: When rearranging the equation such that thrust is calculated as a result of the other factors, we have: These equations show that a higher specific impulse means a more efficient rocket engine, capable of burning for longer periods of time. In terms of staging,

8547-504: 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

8658-566: The nearly spent stage keeps the propellants settled at the bottom of the tanks. Hot-staging is used on Soviet-era Russian rockets such as Soyuz and Proton-M . The N1 rocket was designed to use hot staging, however none of the test flights lasted long enough for this to occur. Starting with the Titan II, the Titan family of rockets used hot staging. SpaceX retrofitted their Starship rocket to use hot staging after its first flight , making it

8769-708: The need for complex turbopumps . Other upper stages, such as the Centaur or DCSS , use liquid hydrogen expander cycle engines, or gas generator cycle engines like the Ariane 5 ECA's HM7B or the S-IVB 's J-2 . These stages are usually tasked with completing orbital injection and accelerating payloads into higher energy orbits such as GTO or to escape velocity . Upper stages, such as Fregat , used primarily to bring payloads from low Earth orbit to GTO or beyond are sometimes referred to as space tugs . Each individual stage

8880-419: The number of stages that split up the rocket system. Increasing the number of stages for a rocket while keeping the specific impulse, payload ratios and structural ratios constant will always yield a higher burnout velocity than the same systems that use fewer stages. However, the law of diminishing returns is evident in that each increment in number of stages gives less of an improvement in burnout velocity than

8991-410: 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

9102-454: The oldest known multistage rocket; this was the " fire-dragon issuing from the water " (火龙出水, huǒ lóng chū shuǐ), which was used mostly by the Chinese navy. It was a two-stage rocket that had booster rockets that would eventually burn out, yet, before they did so, automatically ignited a number of smaller rocket arrows that were shot out of the front end of the missile, which was shaped like

9213-471: 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 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

9324-498: 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

9435-411: 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

9546-421: 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

9657-525: The outer two stages, until they are empty and could be ejected. This is more efficient than sequential staging, because the second-stage engine is never just dead weight. In 1951, Soviet engineer and scientist Dmitry Okhotsimsky carried out a pioneering engineering study of general sequential and parallel staging, with and without the pumping of fuel between stages. The design of the R-7 Semyorka emerged from that study. The trio of rocket engines used in

9768-632: The overall payload ratio of the entire system. It is important to note that when computing payload ratio for individual stages, the payload includes the mass of all the stages after the current one. The overall payload ratio is: Where n is the number of stages the rocket system comprises. Similar stages yielding the same payload ratio simplify this equation, however that is seldom the ideal solution for maximizing payload ratio, and ΔV requirements may have to be partitioned unevenly as suggested in guideline tips 1 and 2 from above. Two common methods of determining this perfect ΔV partition between stages are either

9879-573: The payload ratio (see ratios under performance), meaning the largest amount of payload is carried up to the required burnout velocity using the least amount of non-payload mass, which comprises everything else. This goal assumes that the cost of a rocket launch is proportional to the total liftoff mass of the rocket, which is a rule of thumb in rocket engineering. Here are a few quick rules and guidelines to follow in order to reach optimal staging: The payload ratio can be calculated for each individual stage, and when multiplied together in sequence, will yield

9990-627: 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

10101-477: 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 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

10212-402: The previous increment. The burnout velocity gradually converges towards an asymptotic value as the number of stages increases towards a very high number. In addition to diminishing returns in burnout velocity improvement, the main reason why real world rockets seldom use more than three stages is because of increase of weight and complexity in the system for each added stage, ultimately yielding

10323-441: The problem of calculating the total burnout velocity or time for the entire rocket system, the general procedure for doing so is as follows: The burnout time does not define the end of the rocket stage's motion, as the vehicle will still have a velocity that will allow it to coast upward for a brief amount of time until the acceleration of the planet's gravity gradually changes it to a downward direction. The velocity and altitude of

10434-548: 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

10545-430: The rest of the rocket (usually with some kind of small explosive charge or explosive bolts ) and fall away. The first stage then burns to completion and falls off. This leaves a smaller rocket, with the second stage on the bottom, which then fires. Known in rocketry circles as staging , this process is repeated until the desired final velocity is achieved. In some cases with serial staging, the upper stage ignites before

10656-408: 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

10767-430: The rocket after burnout can be easily modeled using the basic physics equations of motion. When comparing one rocket with another, it is impractical to directly compare the rocket's certain trait with the same trait of another because their individual attributes are often not independent of one another. For this reason, dimensionless ratios have been designed to enable a more meaningful comparison between rockets. The first

10878-427: The rocket system. Restricted rocket staging is based on the simplified assumption that each of the stages of the rocket system have the same specific impulse, structural ratio, and payload ratio, the only difference being the total mass of each increasing stage is less than that of the previous stage. Although this assumption may not be the ideal approach to yielding an efficient or optimal system, it greatly simplifies

10989-518: The same time. For example, the Space Shuttle has two Solid Rocket Boosters that burn simultaneously. Upon launch, the boosters ignite, and at the end of the stage, the two boosters are discarded while the external fuel tank is kept for another stage. Most quantitative approaches to the design of the rocket system's performance are focused on tandem staging, but the approach can be easily modified to include parallel staging. To begin with,

11100-419: The separation—the interstage ring is designed with this in mind, and the thrust is used to help positively separate the two vehicles. Only multistage rockets have reached orbital speed . Single-stage-to-orbit designs are sought, but have not yet been demonstrated. Multi-stage rockets overcome a limitation imposed by the laws of physics on the velocity change achievable by a rocket stage. The limit depends on

11211-518: The shelf hardware that should be considered in the mid to late stages of the design, but for preliminary and conceptual design, a simpler approach can be taken. Assuming one engine for a rocket stage provides all of the total impulse for that particular segment, a mass fraction can be used to determine the mass of the system. The mass of the stage transfer hardware such as initiators and safe-and-arm devices are very small by comparison and can be considered negligible. For modern day solid rocket motors, it

11322-509: 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

11433-440: 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

11544-437: 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 the Sun is not located at the center of the orbits, but rather at one focus . Second, he found that

11655-730: 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

11766-412: 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

11877-443: 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

11988-463: 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 ,

12099-406: The type of fuel and oxidizer combination being used. For example, a mixture ratio of a bipropellant could be adjusted such that it may not have the optimal specific impulse, but will result in fuel tanks of equal size. This would yield simpler and cheaper manufacturing, packing, configuring, and integrating of the fuel systems with the rest of the rocket, and can become a benefit that could outweigh

12210-498: 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,

12321-515: Was proposed by medieval Korean engineer, scientist and inventor Ch'oe Mu-sŏn and developed by the Firearms Bureau (火㷁道監) during the 14th century. The rocket had the length of 15 cm and 13 cm; the diameter was 2.2 cm. It was attached to an arrow 110 cm long; experimental records show that the first results were around 200m in range. There are records that show Korea kept developing this technology until it came to produce

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