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Introduction to Astronautics Sissejuhatus kosmonautikasse Vladislav Pustõnski 2009 Tallinn University of Technology

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2 Orbital maneuvers Orbital maneuver is any intentional change of the orbit of a spacecraft. There is a number of so-called standard assumptions which are met in the first approximation if common chemical rocket engines are used (maneuvers performed using low thrust propulsion are the exception, we will discuss them later). Two bodies are generally considered, a central body (a planet, a natural satellite, the Sun etc.) with the mass M, and a spacecraft with the mass m s. The assumptions are the following. Standard assumptions of astrodynamics 1) The central body and the spacecraft are not influenced by other objects; 2) m s << M: the spacecraft does not attract the central body but moves in its gravity field; 3) Only gravitational attraction from the central body is present, the gravity field is central; 4) The geometric center of the central body is in the focus of the orbit, this center is taken as the origin of the inertial frame of reference. None of these assumptions is fulfilled in reality. Since the first assumption is not fulfilled, the spacecraft cannot move away from the planet to infinity, it will remain in the solar gravity field. When applying thrust, the exhaust gases also appear as individual body. The second assumption is nearly exact, there are nearly no problems where the mass of a spacecraft should be considered. The third and the forth assumptions are not exact since real celestial bodies are not exactly symmetric, non-gravitational forces are also generally present. However, we may apply these assumptions for most problems as the first approximation or

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3 during short periods of time (which is mostly relevant for maneuvers performed with chemical rocket engines). If these standard assumptions are applied, the following rules take place. 1) Kepler’s laws: orbits are circular, elliptical, parabolic, hyperbolic or linear; a line drawn from the central body to the spacecraft sweeps out equal areas in equal times; the square of the orbital period is proportional to the cube of the semi-major axis. 2)Without applying thrust, the orbit will remain unaltered. 3)If the engine is fired instantly in only one point of the orbit, the spacecraft will return to this point on the next revolution although the orbital shape and/or position will change (this is because this point belongs to the altered orbit as well). Thus, to move from one circular orbit to another, at least to firing are needed. 4)For a circular orbit, if the engine is fired in the same direction as the initial velocity, the spacecraft will move to an elliptical orbit with the perigee in the point where the engine was fired and the higher apogee 180 0 away from the firing point (or the orbit will be parabolic/hyperbolic, if the new velocity is equal to the escape velocity/exceeds it). If the direction in which the thrust is applied is the opposite, the point of firing will be the apogee and the point 180 0 away will be the lower perigee (if it lies deep in the atmosphere of below the surface of the central body, the spacecraft will land). Rules of space ballistic are not intuitive sometimes. For instance, if two spacecraft fly in the same circular orbit and wish to dock, its not enough for the trailing spacecraft to accelerate along the orbit to reach its target (if the distance is not very short). Acceleration will send it to an elliptical orbit with a higher apogee, so it will pass above its target. The general approach

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4 Delta-v in such case is a small deceleration, which will send the spacecraft to an elliptical orbit with a lower perigee (and thus, a shorter semi-major axis). The new orbital period will also be shorter (because of Kepler’s third law), so the spacecraft will move faster and the distance between the spacecraft will shorten. Then a second, accelerating, firing of the engine would send the spacecraft to a higher orbit for the desired encounter (see details later in this Lection). One of the important characteristics of a space maneuver (and a space mission) is the change of characteristic velocity needed to realize the maneuver/mission, the so-called delta-v ( V). Any rocket or spacecraft possesses its ideal velocity – the maximal change of speed it can provide to its payload using the fuel onboard. So, delta-v of any maneuver (and any mission in total) is limited by the ideal velocity of the vehicle. As it hase already been mentioned, characteristic velocity should be treated as exponential cost of the mission in the terms of mass. To provide heavier payloads and more complicated missions, it is critical to use the limited reserve of the ideal velocity as efficiently as possible, thus seeking for maneuvers with smaller delta-v. The general definition of delta-v is as follows: F – instantaneous thrust; m – instantaneous mass of the spacecraft; t – time from the start of the maneuver/mission. If the thrust is applied in short bursts, velocity changes due to other sources of acceleration

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5 Oberth’s effect may be neglected, and the velocity change is nearly equal to the respective delta-v. The total delta-v of the mission in such cases is the sum of the discrete delta-v (although between the bursts the velocity may change due to other forces). Since effectiveness of maneuvers depend on the way they are performed, let us study an important principle called the Oberth’s effect (Hermann Oberth, German physicist, was one of the fathers of the modern Astronautics). Work produced by a rocket engine depends on the velocity of the rocket. In a static firing, no useful work is produced, the released energy accelerates the exhaust gases only. If the rocket moves with the velocity V (its specific kinetic energy is thus V 2 /2) and during the burn the velocity increases by V, the increase of the specific energy will be when the engine was fired. Greater increase of the specific energy of the spacecraft is usually desirable, since it permits, for example, to escape from a deeper potential well. This fact have the following consequence. If a spacecraft moves along an elliptical orbit, its velocity is higher in the perigee and is lower in the apogee. So if the engine is fired in the perigee, the effect will be more pronounced than if it is fired in the apogee. For example, while a velocity increase by V in the perigee may make the velocity parabolic or hyperbolic and the spacecraft will move to infinity, the same increase in the apogee will not be enough to surpass the escape velocity limit, and the spacecraft will remain bound to the planet. Obviously the energy increase depends on the velocity which the spacecraft had

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6 Direct and angular firing velocity vector and gives a delta-v increase of V, the velocity will rise (according to the cosine rule) by This result may seem paradoxical, since the velocity change is independent of the velocity V. The intuitive explanation is that the third body participates in the process, that is the fuel that transforms to the exhaust gases. When the engine is fired in the perigee, the fuel is left closer to the planet than if the engine is fired in the apogee. That means that in the perigee the left fuel possesses lower potential energy. Thus, in the case of the perigee firing, the difference of the potential energy of the fuel (in comparison with the apogee) passes to the spacecraft. The conclusion from the Oberth’s effect is evident. To apply the impulse more effectively, one should always do it as close to the central body as possible. If the thrust is low and the burn lasts for a noticeable time, the spacecraft may pass a noticeable distance along the raising trajectory and the distance from the central body may increase during the burn. This is not desirable because of the Oberth’s effect. So it is better to have more powerful engine that would be able to burn the propellant as quickly as possible. For the same reason, one should try to apply the thrust horizontally. Vertical component of the thrust vector always decreases efficiency and thus increases losses of delta-v. It is obvious that the maximal effect will be in the cases = 0 0 or = 180 0 : thus the initial velocity will increase or decrease by V. That means that if change of the velocity direction is of no importance, it is always better to fire the engine along the velocity vector. If a spacecraft moving with the velocity V init applies the thrust under the angle with the

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7 Change of orbital plane Let a spacecraft move with a velocity V and we wish to change the direction of its flight by an angle , keeping its speed the same. In this case the target orbit will retain the size and the shape (i.e. the semi-major axis and the eccentricity) of the starting orbit but will lie in a new plane which will differ by the angle from the starting plane. From simple vector geometry it is obvious that delta-v of this maneuver is found from the relation If this maneuver is performed on the equator, it will change the orbital inclination, retaining untouched the longitude of the ascending node and the argument of periapsis. Thus, this plane change maneuver is frequently performed for inclination change, that may be needed for many purposes, like launching GSO satellites or Moon & planetary probes (since the Moon and the planets move in planes close to the ecliptic, but the launch site is often situated in higher latitudes, and as we have seen earlier, it is possible to launch a satellite to an orbit with lower inclination when the latitude of the launch site only with an inclination change maneuver). The plane change is often performed before space rendezvous to correct out-of-plane injection errors and to make the orbits of the objects coplanar. The maneuver is performed at one of the two nodes where the actual and the target planes intersect. It is seen that the plane change maneuver is quite expensive in terms of delta-v. For instance, to change an 300-km equatorial orbit to a polar orbit (V = 7730 m/s, = 90 0 ), delta-v of V of ~ 10 930 m/s is needed, i.e. the escape velocity for this orbit! Such delta-v is enough to fly to the Moon from this orbit, soft-land and return! To change inclination from the lowest

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8 available from Baikonur (i 46 0 ) to equatorial, about 6000 m/s is needed for 300-km orbit, which is much more than a Moon direct land needs. So, due to the high cost of plane change maneuvers, mission planners always try to avoid them than possible, and this is the reason why they try to launch satellites directly into their target planes. This highly limits launch window choice (the launch should be performed when the launch site passes through the target plane due to the Earth’s rotation) or forces to use a parking orbit (see further). If its impossible to launch directly to the target plane, other methods are used to reduce delta-v. Since plane change delta-v is directly proportional to the spacecraft velocity V, it is obvious that it is much better to perform this maneuver in the apoapsis, since the velocity in the apoapsis has the minimum value. Thus, the inclination of a GTO is often corrected in the apogee simultaneously with the apogee kick, first, because the apogee of GTO lies in the equatorial plane and the plane change will correct the inclination, and second, because the in the apogee the delta-v of the inclination change maneuver is the smallest. The delta-v needed for such maneuver is found from the cosine rule as In some cases it requires even less delta-v to raise the apoapsis to change inclination only with a minimal cost and to lower the apoapsis again. Sometimes such operation is performed with GSO satellites, when the apogee of their GTO is intentionally chosen higher than the GSO altitude. Let us analyze the extreme case, the bi-parabolic transfer, when a spacecraft is sent to infinity where the plane change maneuver requires zero delta-v. If in the previous example we send the spacecraft from its 300-km orbit to a near infinite apogee (providing it V init, V new – initial and final velocities, – angle between the vectors of the initial and final final velocities.

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9 with the near escape velocity), this would require delta-v of 7730·(1.414-1) 3200 m/s. The inclination change in the infinity is free since the velocity tends to zero. On the return we spend another 3200 m/s to circularize the orbit again. Thus the inclination change will require only 6400 m/s instead of 10 900 m/s, which is an enormous gain. Such biparabolic transfer may be useful only for significant angular changes. In practice, the apogee cannot lie in infinity, it should be chosen at some finite distance defined from the considerations of the acceptable time of the mission, communication range etc. So the trajectories will be ellipses (bi-elliptic transfer) and the limiting angle is higher. Aerodynamic maneuvers may also be used for a direct orbital change, either through aerobraking technique either using a direct atmospheric entry for a lateral maneuver. It is sometimes possible to use gravity assists from the Moon and the planets to change inclination, that makes possible missions that otherwise would be impossible. An example is the HGS-1 satellite, the former AsiaSat 3. AsiaSat 3 should have become a GSO satellite. Due to its launch vehicle Proton malfunction, AsiaSat 3 was left on a highly inclined useless orbit, not on the planned GTO. The propellant onboard was not sufficient to put the satellite to GSO because of the need to change inclination in a high extent. The insurers transferred the satellite to Hughes Global Services, which raised the apogee by several perigee burns and finally performed a Moon close fly-by at the distance of ~ 8200 km for a gravity slingshot maneuver. This fly-by (followed by another one) nearly zeroed the inclination, and the satellite (renamed to HGS-1) was transferred to GSO. This maneuver consumed much propellant, so The limiting value of the angle may be deduced from the inequality

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10 Hohmann transfer orbit and its alternatives If transition from one circular orbit to another coplanar (lying in the same plane) circular orbit is needed, the Hohmann transfer (by the German scientist Hermann Hohmann) is usually applied as the most effective (in most cases) maneuver considering the minimum delta-v. Hohmann transfer is one half (from periapsis to apoapsis) of an elliptical orbit touching the starting and the target orbits in its periapsis and apoapsis respectively; the orbit is coplanar with the circular orbits, the angular distance between the burns is 180 0 (they lie on the opposite sides of the central body). the lifetime of the satellite decreased half of the normal. However, it was a huge gain since initially the mission was considered a total loss. The disadvantages of such a complex maneuvers are long times needed to fly far away and return, additional weight of systems, longer vehicle lifetime needed and so on. In the case of HGS-1, it was not designed for such maneuvers, and after transfer to GSO one of its solar batteries failed to deploy, which was attributed to unplanned thermal effects during the deep space trip. Another example is the Ulysses solar probe mission. This spacecraft should have studied the regions of the solar poles, so its orbit should have been perpendicular to the ecliptic. However, the Earth orbits the Sun in the plane of the ecliptic, the same do space probes sent to heliocentric orbits. Solar orbit inclination change by ~90 0 needs enormous delta-v which is very problematic to provide with the current technologies. So, Ulysses was first sent to the Jupiter to perform a swing-by maneuver, which put it to a near polar solar obit (inclination 80.2 0, aphelion ~5.4 astronomical units, perihelion ~1.3 AU).

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11 If the radius of the starting orbit R is smaller than the radius of the target orbit R’, the transfer begins with first firing the engine in prograde direction along the trajectory to transform the orbit to elliptical with the apogee touching the target orbit. It is simple to deduce the needed delta-v for this maneuver V V circ start is the circular velocity of the starting orbit. Such burn provides the minimal delta-v necessary to reach the target orbit. When the apogee of the ellipse is reached, the second burn is made in order to circularize the orbit again. The velocity of the spacecraft in the apogee and the circular velocity of the target orbit are correspondingly and So, delta-v of the second burn V’ is the difference of these two velocities: The total delta-v of the Hoffman transfer is the sum of delta-v’s for the two burns: V total = V + V’. The time of transfer is the half-period of the elliptical orbit. Since the semi-major axis of this orbit is a = (R + R’)/2, the time is A transfer from a higher starting orbit to a lower target orbit is realized in the same way, but in the reverse order. The first burn is retrograde with the delta-v of V’, it

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12 sends the spacecraft to the elliptical orbit touching in its periapsis the target orbit (this occurs on the angular distance of 180 0 from the first burn). In the periapsis the second retrograde burn with the delta-v of V cancels the excess over the local circular velocity. The delta-v and the transfer time remain the same. Let us examine a Hohmann transfer from LEO with the altitude of 300 km (R 6680 km) to a GSO orbit with the radius of R’ 42 000 km. The circular velocity at the starting orbit is V circ 7730 m/sec, so V 2420 m/s and V’ 1470 m/s, thus in total V total 3890 m/s. It is larger than delta-v for the escape orbit (equal to ~ 7730·(1.414-1) 3200 m/s. The time of such Hohmann transfer will be t 4 h 20 m. Transfers between orbits are often realized by the Hohmann method. However, the Hohmann transfer is an idealization, since real orbits are never exactly circular nor coplanar. Thus, in real application it is needed to take into account eccentricity and eliminate the difference in inclinations. Sometimes this difference is eliminated before the transfer and sometimes simultaneously with it during the first or the second burn. In the case of GTO it is ordinarily done with the second burn, since it costs less delta-v to change inclination at a low apogee velocity than at a high perigee velocity. If the Hohmann transfer is applied to reach the Moon, in the apogee the spacecraft will be in the circumlunar space and will be influenced by the lunar gravity. In this case the second burn is performed not to circularize the orbit around the Earth, but to put the spacecraft into a lunar orbit or brake it down to soft-land the Moon. This burn should decrease (not increase) the velocity since the spacecraft moves with a hyprerbolic velocity in respect to the Moon. One of the disadvantages of the Hohmann transfer is that it needs a lot of time to reach distant

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13 orbits. The time of a trip to the lunar orbit is about 5 d, the time of a Mars trip is about 9 months. However, in some cases time may be critical. For example, for manned missions to the Moon additional time means a need for additional resources of energy, the life support system and so on. The coast to outer planets may continue for many years and need additional energy resources and systems to increase the lifetime of the space probe. Such considerations may force the mission planners to prefer less time-consuming trajectories, although not optimized by delta-v: a choice of a more delta-v consuming trajectory may be better from the point of view of payload optimization. A less time-consuming alternative to the Hohmann transfer is a one-tangent burn. That is a transfer orbit tangential to the starting orbit (in order to keep the first burn parallel to the initial velocity vector and thus not to waste delta-v) and crossing the target obit at a non-zero angle. Such transfer orbit has a larger semi-major axis than the Hohmann transfer orbit. Infinite number of one-tangent burn transfers exist, the choice depends on the compromise between the time and the delta-v. For a trip to the Moon, only 50 m/s of additional delta-v would reduce the time of the voyage twice, from ~5 d to ~2.5 d. An orbital transfer may be performed also with a low-thrust engines based, for example, on electric propulsion. Such engines would provide continuous very small acceleration along the instant velocity vector. This kind of transfer is called spiral transfer, since the orbit of the spacecraft would be a tangential spiral winding between the starting and the target orbits. The total delta-v of the spiral transfer is approximately V’ |V init, - V new |, i.e. the difference between the circular velocities of the starting and the final orbits. In the example analyzed above, it is V 7730 – 3080 4650 m/s, i.e. by ~ 760 m/s larger than the Hohmann transfer needs. But since electric engines are much more efficient and lighter, they will consume less propellant (by mass) than a separate high-thrust engine with its own propellant tanks would

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14 weight. Thus, there may be no sense to provide a separate high-thrust engine for a delta-v efficient Hohmann transfer. However, low-thrust spiral transfer may also be applied if there are no electric propulsion engines onboard, but the main engine of the satellite or the upper stage of the launch vehicle has failed. Steering engines burning ordinary chemical fuel may be used for the transfer. Since their low thrust is insufficient to provide high thrust, spiral transfer (or a series of Hohmann transfers to intermediate orbits) is unavoidable. Such transfer requires extra delta-v and thus extra propellant, so the lifetime of the satellite will reduce. If we need to pass from one circular orbit to another coplanar circular orbit the radius of which is significantly larger, a more economic alternative to the Hohmann transfer is the bi- parabolic transfer described earlier. It means that we may first send the spacecraft to the infinity providing it with the escape velocity, and than with an infinitely small impulse return it back along a parabolic path tangential to the target orbit. The third burn (retrograde) will kill the parabolic excess circularizing the orbit. It may be proved that such a transfer needs less delta-v than the Hohmann transfer if the ratio of the starting and the target orbital radii is R new / R init > 11.9. In practice, since infinite distances are idealization, the apogee of the second burn should lie at some finite distance, and the transfer trajectories will be ellipses (bi-elliptic transfer). So the delta-v of the second burn is non-zero, and the practical limiting ratio is higher. If the starting orbit has a 300-km height, the height of the targeting orbit in the ideal case is >73 000 km. Such orbital heights are not frequent, so be-elliptical transfers are not of frequent use as well. Their largest disadvantage is that they need much time to perform (more than twice of the Hohmann transfer), and the gain is quite modest even if the second burn is very distant. For orbits inside the lunar orbit be-elliptic transfer would give only few tens of m/s of delta-v.

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15 Multiple burn transfers. Parking orbit The Hohmann transfer described above is an example of so-called multiple burn transfers. These are transfers produced by multiple short burns of the engine instead of one long burn. Such maneuvers are often useful and/or only possible. For instance, it may be impossible for the upper stage of a launch vehicle to send a satellite to a high-altitude circular orbit. The first reason is that such long burn involves high gravity losses discussed further (due to non- horizontal thrust and burning propellant far from the planet – the Oberth’s effect). The second reason is technical: the engine may be unable to realize a long low-thrust burn, since rocket engines usually cannot be throttled down significantly, and the normal thrust of the upper stage should be large to avoid growth of launch losses. However, even if the altitude of the target orbit is not very large (for instance, 500 km) and this orbit may be reached by a single burn of the upper stage, it is often better first to deliver the payload to a lower orbit (for instance, 180 km) and then realize a Hohmann transfer to the target orbit. Such a multiple burn transfer enables to increase the useful payload since it is associated with smaller losses and thus requires a smaller total delta-v. Thus, it is often useful – or only possible – to send the spacecraft to an intermediate low orbit which serves as a starting orbit for the further mission. This orbit is called parking orbit. Usually the height of this orbit is chosen following the considerations of propellant saving and communication with the spacecraft during its coast along the orbit. The alternative to the parking orbit is direct injection when a spacecraft is delivered to the target orbit by continuous firing of the launch vehicle motors until the separation of the payload. If a parking

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16 orbit is used, the spacecraft performs multiple burns before it reaches the target orbit. There are certain cases when a launch to a parking orbit is usually applied. Launch to an orbit with a high apogee above the launch site. Such satellites like Molniya should be placed on a high-apogee orbit, and the apogee is not far from the launch site. Such injection cannot be performed by a single burn due to a high position of the target point. The satellite has the highest velocity in the perigee situated in the opposite point of its orbit, i.e. at the angular distance of about 180 0 from the launch site. In such case the satellite is placed to a circular low parking orbit and coasts along it for a half of revolution. Than the second burn is performed and increases the velocity up to the perigee velocity of the target orbit. Launch to an orbit with a high perigee. To deliver a satellite to a large-radius orbit or an orbit with a high perigee, an elliptical parking orbit is necessary. Actually the parking orbit for such launch is a Hohmann transfer orbit with its apogee coinciding with the perigee of the target orbit. When the apogee of the parking orbit is reached, an apogee kick burn along the velocity vector rises the perigee of the parking orbit transforming it into the apogee of the target orbit. Launch to GSO is a particular case of such transfer. Launch to the geostationary orbit. GSO lies in the equatorial plane, so if Hohmann transfer should be applied, the starting point for this transfer should be in the angular distance of 180 0 from the targeting point on GSO, thus the starting point also should be in the equatorial plane. If the launch site is not on the equator, it is suitable first to deliver the upper stage of the launch vehicle with its payload to the intermediate parking orbit, let it coast to the equator and than initiate the Hohmann transfer (which is the second intermediate orbit in this kind of launch). Arian 5G launch vehicle does not use parking orbit since its launch site is close to the equator (latitude 5 0 10’), and a direct inject enables to avoid restart capabilities of the upper stage engine.

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17 Launch to the Moon & the planets. Orbits of the Moon and the planets lie close to the ecliptic. But some launch sites (as Baikonur) are far from this plane (which is inclined to the equator by the angle about 23 0 26’. So it is impossible to deliver the spacecraft into this plane by a direct injection. But if the spacecraft is put to the parking orbit, it can coast to the lower latitudes to perform the deep space injection. However, even for launch sites on lower latitudes a parking orbit may be of great use to increase the launch window. For a direct injection, the launch window may be limited by several minutes or even less since the Earth’s rotation carries the launch site away from the right plane. If something goes wrong, the launch should be shifted to the next day, but the ballistic conditions may worsen significantly in a day. But if a parking orbit is used, the interplanetary injection may be performed in each orbit. One of the advantages of the parking orbit, in addition to delta-v savings, is that it enables to carry out additional checkout of the systems before the injection to a distant target and, if necessary, make reparations and/or corrections. It is specially useful for manned missions (for example, to the Moon): the crew have time to check the systems of the spacecraft and to decide if no serious faults are present and if the mission may be performed. During the Apollo missions to the Moon, the upper stage with the spacecraft made about 1.5-2 revolutions on the 180-km parking orbit before the translunar injections. However, the parking orbit is associated with several technical complications. 1)Multiple restart of engines. Some “tricks” are needed to make a restartable engine. This includes multiple chemical sets for ignition (if the propellant is not hypergolic) etc. 2)Engine restarts in weightlessness. In weightlessness, the propellant flies freely inside the tanks not covering the pump inlets. To settle the propellant to the bottom, special ullage motors are needed, or other ullage mechanisms, like diaphragms inside the tanks.

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18 3)Additional thermal protection. Cryogenic tanks need better insulation to prevent the propellant to boil off, other systems might also need special thermal protection since they should stay on the orbit at conditions different from that of the working orbit. 4)Batteries and other resources. Batteries should have longer lifetime and other consumables (like ullage gas) should be provided for the period of the coast. 5)Inertial guidance system. Inertial guidance system should be able to maintain precision pointing for the whole period of the coast. 6)Reaction control system. The upper stage (or the spacecraft) should be able to orientate itself before each burn, so reaction control is indispensable (sometimes the engines of the reaction control systems are used also as ullage motors). 7)Communication with the mission control. It is highly desirable that the spacecraft maintains communication with the mission control during the whole period of the coast, so that the mission control were able to take steps if something goes wrong during the coast. This presuppose an extensive net of ground stations. Several launch vehicles are provided with a special upper stage for multiple burns. This is the case of the 4-stage Russian Proton with the upper stage Blok-D/DM, Molniya (4-stage version of the Soyuz with the upper stage), Inertial Upper Stage (IUS) and Payload Assist Module (PAD-D/D2) of the Shuttle. Sometimes this stage is not even regarded as a part of the launch vehicle but rather a part of the orbital payload, since this stage does not participate in the orbital injection. However, often the upper stage of the launch vehicle provide the final impulse for the parking orbital injection and then is used for orbital transfers (like the upper stage S-IVB of the Saturn-V). In the case of s direct injection, this stage burns continuously.

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19 Atmospheric maneuvers If the planet has atmosphere, it may be used for several maneuvers related to orbital transfer. One of the possible applications is aerocapture – a procedure when a spacecraft approaching to the planet by a hyperbolic trajectory touches the upper atmospheric layers and decelerates due to atmospheric drag forces. The drag force extinguishes the velocity excess, and the spacecraft enters an elliptical orbit which may be circularized using rocket thrust or additional atmospheric maneuvers without propellant usage. Aerocapture may save hundreds of m/s of delta-v. A disadvantage of this method is that a heatshield is needed, which means additional weight of the spacecraft (the heatshield may be discarded after the final escape from the atmosphere). The spacecraft should also be capable to withstand high g-force and thermal loads, capability of precise aerodynamic maneuvers is critical as well. Zond 6, 7 used Quite often the same upper stage is used in different launch vehicles. Blok-D was built for the Soviet moon rocket N1, it is still in use (in different modifications) as the 4 th stage of the Proton launch vehicle, and its modification Blok-DM-SL is used on the Zenit-3SL in the Sea Launch. Briz is also used as the 4 th stage of Proton and as the 3 rd stage of light the launch vehicle Rokot. Agena was used as the 2 nd stage for Thor, the 2 nd (3 rd ) stage for Atlas and the 3 rd stage for Titan. S-IVB was tested as the 2 nd stage for Saturn 1B and became the 3 rd stage for Saturn V. Centaur in very different versions (including 1-motor and 2-motor) has been used in many Titan, Delta and Altas launch vehicles, the Space Shuttles “Atlantis” & “Challenger” were also adopted to carry Centaur for sending heavier payloads to high orbits, but these plans were dropped after the Challenger disaster.

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20 aerocapture after return from the circumlunar flights to slow down before the final re-entry and to set the final re-entry point above the territory of the USSR. The similar method may be used for circularization of elliptical orbits. It is the so-called aerobraking. Passages through atmospheric layers in the perigee reduce the apoapsis. The height of the periapsis may also be reduced. If time is not critical, the loads at each passage may be significantly reduced by increasing the number of passages and passing through less dense layers. In this case no special heatshield is needed, braking may be performed with the body of the spacecraft or with its solar panels (however, increased thermal and g-force loads should be taken into account in the design process). Aerobraking technique was used on the Mars Global Surveyor Mars orbiter in 1997-1999. It was inserted into an orbit with the initial height of ~ 54 000 km. Later its periapsis was set at the height of 110 km and the braking with the solar panels began. Later the periapsis was raised to 120 km since g-force turned to be excessive. By March 1999, the apoapsis decreased to 450 km and the periapsis was raised to ~170 km and the orbit stabilized. The same techniques was used later on the Mars Odyssey (saving about 200 kg of the propellant) and the Mars Reconnaissance Orbiter. Aerobraking may also be applied in bi-elliptical transfers to brake down the spacecraft on its return from a distant point at plane change maneuvers. This would additionally save fuel. Other method is a direct entry into the upper atmospheric layers to perform a lateral maneuver using aerodynamic forces. After such a maneuver, it will be necessary to replenish the speed loss caused by the drag forces. An extensive shielding is crucial, as well as capability to maneuver precisely in the atmosphere. Multiple entries may help to decrease the loads.

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21 Orbital rendezvous & docking Rendezvous is a maneuver when to spacecraft arrive at the same orbit in the vicinity of each other and equalize their velocities. An orbital rendezvous may be followed by a docking – a maneuver when spacecraft join together mechanically. To perform a rendezvous, spacecraft should arrive at the same point at the same time. In a rendezvous usually take part two spacecraft, one of which plays an active role of interceptor and another one is a passive target. The operation begins when one of the spacecraft is launched to space. Often the target is launched the first and is already placed on the rendezvous orbit by the moment when the interceptor is launched (that is the case of a rendezvous with a space station). However, the spacecraft launched the first may also adjust its orbit after the launch of the second spacecraft. Since inclination change is very expensive, the second spacecraft should be launched directly into the orbital plane of the first spacecraft; this is possible two times per day, when the launch site crosses the corresponding orbital plane due to the Earth’s rotation. Launch windows have certain length depending of the characteristic velocity that the spacecraft possess: the farther is launch site from the orbital plane during the launch, the larger will be the delta-v cost of the plane correction maneuver. Smaller orbital inclinations favor longer launch windows since the launch site moves off slower from the orbital plane. If the amount of the propellant is limited, launch windows may become quite short. For some Gemini spacecraft launch windows were from tens of seconds to several minutes; launch windows for the Space Shuttles on their missions to the International Space Station are about 5 minutes.

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22 In rare occasions the interception occurs on the very first orbit, usually it is performed after several orbits. The reason is the launch window: in the case of an immediate interception, the interceptor should be placed in a close vicinity of the target, and each second of delay increases the distance between the objects by ~8 km. Gemini 11 manned spacecraft had to intercept its Agena target in the first orbit, so the launch window was only 2 seconds! In the general case, the interceptor is placed to a coplanar parking orbit (its height is usually lower than the orbit of the target since such choice requires smaller total delta-v). Since the orbital periods on this orbit and the target orbit are different, the spacecraft configuration varies with time, and the interceptor may wait for a suitable moment for starting the transfer to the target orbit. The corresponding maneuver of adjusting positions of the spacecraft is called orbit phasing and the corresponding parking orbit is called phasing orbit. If the interceptor moves initially along the same orbit as the target, it should enter the phasing orbit with a different period in order to reach the target after several revolutions. In the case of a trailing interceptor, the phasing orbit should have a smaller semi-major axis, which means higher mean speed to reach the target. In the case of a leading interceptor, the phasing orbit should have a longer semi-major axis, thus the speed of the interceptor will be smaller and it will lag behind. For larger differences between the semi-major axes of the target orbit and the phasing orbit less revolutions are needed to complete the phasing maneuver and thus the rendezvous may be performed quicker. However, this difference is limited by the characteristic velocity reserve available, since larger difference of the semi-major axes needs more delta-v for the orbital transfers. If the orbits are not coplanar, the plane corrections should also be introduced in these maneuvers as well (they are performed in the intersection points of the orbital planes).

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23 The same phasing technique is applied when a launch vehicle should put multiple payloads to different orbital positions. The payloads are released one by one, and phasing is used to reach the next orbital position. When the target and the interceptor are close enough of each other, the final approach (distances of kilometers and less) may be performed using two general methods. The most simple is direct pointing, when the interceptor follows the target keeping constant the angle between its velocity vector and the line of sight of the target. All deviations caused by gravity are corrected with the thrusters. Due to the simplicity of the method, no computer is needed at all. That is why this method was applied on the first Soyuz spacecraft which lacked an onboard computer and used an analog device. A more propellant saving technique is the free trajectories method, when exact trajectories in the gravity field are computed and necessary corrections are performed. This method needs computers onboard and was applied on Gemini as well. For docking, the spacecraft should equalize their speeds in the immediate proximity, so that chocking loads were minimal. Use of manipulator for docking is also possible. The exact time of docking is usually chosen considering the illumination conditions (so that astronauts onboard could see the docking spacecraft and devices), and also ground relay conditions (so that the mission control could see what happens and provide assistance if necessary). First space rendezvous Dec 1965 Gemini VI / VII First space docking Mar 1966 Gemini VIII / Agena target First automated docking Nov 1967 Cosmos 186 / 187 First docking out from the Earth and on the lunar orbit May 1969 Apollo 10 CSM/LM First rendezvous on the lunar surface Nov 1969 Apollo 12 / Surveyor 3 First international rendezvous and docking Jul 1975 Apollo-Soyuz Test Project

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24 End of the Lecture 6

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Orbital inclination change Orbital inclination change performed by a plane change maneuver on the equator

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The second fly-by and final GSO injection AsiaSat 3/HGS-1 lunar gravity assist maneuver HGS-1 trip to the Moon and the first fly-by

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Hohmann transfer orbit Hohmann transfer orbit between two coplanar circular orbits

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One-tangent burn One-tangent burn is a less time-consumable alternative to the Hohmann transfer (By source)source

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Bi-elliptic transfer O 0, O F – the starting and the targeting orbits; O 1, O 2 – the first and the second transfer orbits; R 0, R F – radii of the starting and the target orbits correspondigly; R – distance to the second burn; V 0 – the first prograde burn; V – the second prograde burn; V f – the third retrograde burn. (By source)source

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Launch to the Moon. (By source)source Parking orbit Venus Express launch (By source)source

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Launch to an orbit with a high perigee 1) The spacecraft is in the perigee of its future parking orbit. 2) The first burn sends the spacecraft to the elliptical parking orbit. 3) The second burn in the apogee of the parking orbit raises the perigee. 4) The spacecraft have reached the target orbit (By source)source

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Astra 1KR transfer to GSO. Atlas 5/Centaur launch vehicle. (By source)source Parking orbit for GSO launch

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Hyperbolic access velocity with the aid of atmospheric drag (By source)source Aerocapture

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Actual timeline was different from data in this poster since braking intensity had to be reduced. Mars Global Surveyor aerobrake

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Phasing orbit Trailing interceptor enters the phasing orbit with a shorter period at the moment t 0 and after several revolutions catches it up at the moment t 1 (By D.A.Vallado, D.McClain, Fundamentals of Astrodynamics adnd applications)

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Gemini VI / VII rendezvous Gemini VI / VII orbital rendezvous on Dec 15, 1965

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Apollo 10 CSM / LM docking View from the Lunar Module View from the Command Service Module. The Earth is on the background

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Apollo-Soyuz Test Project Two dockings were performed in Jul 1975

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MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary Mechanical and Aerospace Engineering Department Florida Institute.

MAE 4262: ROCKETS AND MISSION ANALYSIS Orbital Mechanics and Hohmann Transfer Orbit Summary Mechanical and Aerospace Engineering Department Florida Institute.

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