Introduction to Astronautics Sissejuhatus kosmonautikasse

Tallinn University of Technology Introduction to AstronauticsSissejuhatus kosmonautikasse Vladislav Pustõnski 2009

Sphere of influence. Patch conics method Lunar & planetary missions When the mission of a spacecraft is to fly to the Moon or a planet, the spacecraft is influenced by gravity fields of different celestial bodies on its way. In the vicinity of the Earth, our planet exerts the strongest force on the probe, on the heliocentric orbit the solar attraction is the principle force, and when approaching the planet, the role of the gravity force of the planet is predominant. At different portions of the trajectory different attracting forces are the strongest; and there are certain portions on the path where gravitational attraction of two bodies (for example, the Earth and the Moon, the Earth and the Sun) is comparable. To calculate the trajectory of the space probe, one should apply analytical and numerical methods of celestial mechanics (numerical methods are unavoidable, since problems involving motion of three or more bodies cannot be solved analytically in the general case). However, for simplistic rough estimations a so-called patch conics method is frequently used. This method usually gives quite good first-order approximations, which later may be rectified with the aid of more precise methods of celestial mechanics. The general idea of this method is that in each moment the motion of the probe is assumed to be an ordinary Keplerian (i.e. a conic section) in the gravity field of only one of the celestial bodies; the presence of other attracting bodies is ignored. Such Keplerian motion takes place inside the so-called sphere of influence (SOI) of the respective body. Let us analyze this concept in detail. Consider a space probe on the line Sun-Earth between the two bodies at the distance of 149 · 106km from the Sun and 0.5 · 106km from the Earth. We shall see which of the celestial bodies mostly defines the motion of the probe.

It is simple to calculate that the Sun-indunce acceleration would be aSun5.97 · 10-3m/s2 and the Earth-induced acceleration would be aEarth1.59 · 10-3m/s2. Obviously aSun > aEarth, but it does not mean that the probe will leave the Earth and will be captured by the Sun. In fact, the Earth is also accelerated by the Sun in the same direction as the probe, the respective acceleration being aSun/Eath5.93 · 10-3m/s2. The probe experiences from the Sun an acceleration relative to the Earth which is equal to the difference aperturb/Sun = aSun - aSun/Earth 0.04 · 10-3m/s2. This perturbational acceleration is much smaller than aEarth: aperturb/Sun/aEarth· 100%  2.5%, which is quite a small value. If we regard the motion of the probe relative to the Sun, we shall consider that the Sun-induced acceleration is aSun5.97 · 10-3m/s2. The perturbation from the Earth is aperturb/Earth aEarth - aEarth/Sun = 1.59 · 10-3m/s2, where aEarth/Sun 0 – the acceleration of the Sun caused by the Earth, it is negligible due to the enormous mass of the Sun. So, the perturbation of the heliocentric motion is aSun/Earth/aSun· 100%  27%. Obviously the heliocentric motion of the probe is largely perturbed by the Earth. Thus it is jusified to regard the motion of the satellite as Earth bound with a relatively small solar perturbation perturbation. The error introduced by such choice will be several percent. If we do vice versa and regard the motion as heliocentric, we should count with a huge perturbation from the Earth amounting tens of percent, which will make the problem quite complicated. If we repeat the same estimations for a point on the same line at the the distance of 1.5·106km from the Earth in the solar direction, we will see that the Sun perturbs the geocentric motion of the space probe by more than 68% of the geocentric acceleration, but the Earth’s

perturbations of the heliocentric motion will be less than 3%. That means that at the distance of 1.5 · 106km from the Earth it much better to regard the motion as heliocentric (with some perturbations from the Earth) rather than geocentric. We may repeat the same estimations for other points in the near-Earth space (regarding accelerations as vectors). Thus we define the regions where the motion may be regarded as geocentric and the regions where it is rather heliocentric. The region where the motion is rather Earth-bound than heliocentric is called the sphere of influence (SOI) of the Earth. Analogically, such SOI may be found for other planets of the Solar System. If we analyze the motion in a vicinity of a natural satellite, we also may introduce the concept of SOI of the satellite, for instance, of the Moon. In this case a space probe which leaves the SOI of the satellite continues its flight inside the SOI of the planet. The radius of the SOI of a body with the mass m orbiting the body with the mass M at the distance of a is defined by formula (a more precise version of this formula includes an angular term taking into account that the SOI is closer to the orbiting body on the side directed towards the orbited body, but this asymmetry is quite small). Let us list the SOI of the Moon and the planets in a table. From the table it is clearly seen that outer planets have very large SOI since they are very distant from the Sun (so, the SOI of the Neptune is much larger than that of the Jupiter although the Jupiter is the largest planet). When the space probe is deep inside the SOI of a planet (or a natural satellite), the errors that arise from ignoring gravity fields of other bodies are small. However, when the

probe approaches the boundary of the SOI, these errors increase. So, the method of patched conics works good if the probe does not spend much time near the bondary of SOI. To do that, it should have sufficient speed at crossing this boundary and the total flight time should be significantly larger than the time spent near the boundary. These conditions generally keep for interplanetary missions, this is why the method give impressively good results for such missions. However, for lunar missions the situation is worse due to the short radius of the lunar SOI and smaller total times of lunar missions. This is the reason why this method is not perfect if applied to lunar missions, although it may provide rough estimations. Let us look how the patched conics method works. Transition from one conic section to another one occurs when the space probe crosses the boundary between the spheres of influence of two celestial bodies. For example, a Martian space probe which leaves the Earth first moves inside the Earth’s SOI and its motion is regarded as Earth bound. Later the probe leaves the Earth’s SOI and its orbit is considered heliocentric. Finally the probe approaches the Mars and enters its SOI, continuing its flight as a Mars bound body. The conics on boundaries of the SOI are patched together by the velocity vector of the probe. The concept of SOI should be distinguished from the Hill sphere, that is the volume inside which a satellite may stay bound to the celestial body regardless of gravitational perturbations.

The Hill sphere is larger than SOI, for the Earth it is about 1.5 · 106km. Let us make one interesting remark. In the example above we have seen that an object at the distant of 0.5 million km from the Earth is attracted by the Sun stronger than by the Earth. The same is true also for less distant points. The sphere at which the gravitational pull by the Sun is equal to the gravitational pull by the Earth has the radius of approximately 260 000 km. So, the Moon lies well out of this sphere and thus is attracted by the Sun stronger than by the Earth. If the Moon stops for a moment, it will fall to the Sun, not to the Earth. However, since the Moon moves, its orbit is rather Earth-bound than Sun-bound. Flights to the Moon Flights of automatic space probes to the Moon were one of the first problems Astronautics had to solve in the early era; this problem was followed by the the problem of lunar manned missions. Today lunar studies regained currency, so let us study this problem in detail. In any space mission, delta-v is one of the critical parameters since smaller delta-v make it possible to increase the mass of the payload. As we have seen earlier, burns should be performed horizontally to decrease losses, and the most economic transfer is the Hoffmann transfer orbit, which means the angular distance of 1800 from the target. The Moon orbits the Earth on an elliptical orbit with the perigee distance of ~ 364 000 km and the apogee distance of ~ 407 000 km. The orbital inclination (to the Earth’s equator) changes from 18018’ to 28036’ and back with the half-period of 9.3 years. That means that it is possible to perform a direct injection with the angular distance of 1800 only from launch

sites with inclinations less than about 18018’, and some years also less than about 28036’ (which rarely includes also Cape Canaveral). Such launch is possible when the launch site crosses the lunar orbital plane. However, it is not guaranteed that at the arrival of the probe to its apogee it will meet the Moon, since the Moon may occupy any position on its orbit. So, direct Hohmann transfer to the Moon in problematic and we should seek for other options, specially if the launch site has higher inclination. Another option is to apply the direct injection but to refuse the Hoffmann transfer and choose a one-tangent burn in a plane different from the plane of the lunar orbit. In this case we should seek for a compromise between the launch losses and the gain from the Earth rotation. The minimum angular distance is provided if the launch is performed in the polar direction across the pole towards the southernmost point of the lunar orbit, but this means no gain from the Earth’s rotation. In practice, some intermediate inclination is chosen, for the first Soviet lunar probes it was 650. Let us look for the conditions of the lunar direct flight for a given trajectory inclination, for instance for the polar path. It is seen that when the Moon is in the southernmost point of its trajectory, the optimal launch conditions correspond to the point A, when the angular distance is the highest (path 1). A launch from the point B would mean nearly vertical ascent trajectory and high gravity losses (path 1’). When the Moon is in the Northern hemisphere, the situation is opposite, but the optimal path 2 is worse than the optimal path 1 (the angular distance is smaller). So, there is a short period during the sidereal month when the launch conditions are the best, they correspond to the southernmost position of the Moon (of course, the position of the Moon at the probe arrival time is meant).

However, using of the parking orbit makes it possible to launch anytime and to use any trajectory including the Hohmann transfer. A parking orbit intersects with the plane of the lunar orbit, so if the translunar injection is performed in the intersection point, the trajectory may intersect with the lunar orbit again at the distance of the Moon (it will be the apogee of the trajectory in the case of a Hohmann transfer). Of course the parking orbit and the parameters of the translunar injection maneuver should be chosen so that the probe arrives to the lunar orbit simultaneously with the Moon (which moves ~130 along its orbit). The first Soviet & US lunar probes were launched using a direct injection since parking orbit method needs advanced technology with special requirements to engines, guidance etc. Later probes were launched mostly from parking orbits, which increased the payload. The time of the trip strongly depends on the departure velocity. It is the longest for the Hohmann transfer, being about 5 days. An increase of the departure velocity by only 50 m/s enables to shorten this time twice. Such gain of time may be useful in spite of some additional propellant expenditure, since it means smaller amount of consumables (for example, batteries or, in the case of manned flights, less oxygen, water etc. and a lighter life support system). Parabolic trajectory (i.e. if the probe is provided with the escape velocity, which is ~ 100 m/s higher than that needed for the Hohmann transfer) means a 2-day trip. Substantially hyperbolic velocities enables even shorter times, for example, a 1-day trip for the excess of 0.5 km/s above the velocity the Hohmann transfer needs. But so high velocities would decrease significantly the useful payload, so they have not been applied. The eccentricity of the lunar orbit plays very small role in the energetic requirements of a lunar mission, since small increments of the departure velocity raise the apogee in a high extent. Thus, there is only ~10 m/s difference in delta-v between the mission to the perigee

and the apogee of the lunar orbit. However, the position of the Moon influences the time of the trip. If the departure velocity is close to parabolic, the difference in time between the trip to the perigee and the apogee is several hours (~ 10h, depending on the departure velocity). In precise computations of the trajectory, effects of the lunar attraction, the Earth’s oblateness and solar perturbations also should be taken into account. The effects of the lunar gravity field will be discussed further. The oblateness of the Earth decreases the time of the trip by some tens of minutes, solar perturbations have much smaller effects. Precision issues of targeting are also of great importance. Hohmann transfer trajectories are quite sensitive to errors of the departure speed. This is because an error of 1 m/s leads to a change in the apogee height of ~ 4000 km, and the time of the trip also changes significantly. So the probe will arrive to its target point much earlier or much later than the Moon. Higher departure speeds lead to decrease of sensitivity to the injection errors: the time of arrival will be less sensitive to the initial speed. For example, for the parabolic departure velocity, the probe aimed to the center of the lunar disc will hit the target even if the velocity differs from the nominal by 50 m/s. On the other hand, Hohmann transfer trajectories are less sensitive to pointing errors (deviations of the departure velocity vector from the right direction). Even a pointing error of 10 would not prevent from hitting the Moon if the Hohmann transfer is applied. Higher departure speeds require higher precision: for the parabolic departure speed only a ~ 0.50 pointing error is permitted. The maximum deviation permitted for very high hyperbolic velocities is defined by the angular size of the lunar disc viewed from the Earth, it is ~ 0.50. So to hit the Moon, one should provide the precision of ~ 0.250 when pointing to the center of the lunar disc. The time of launch should also be within the required limits, since an early or delayed launch will lead to rotation of the whole trajectory (together with the rotation of the Earth or with the probe on its parking orbit), so the targeting point would shift.

Flight inside the SOI of the Moon Lunar attraction increases the range of allowable errors. Actually, to hit the Moon one should get to the distance of the effective radius of the Moon: the trajectory of the probe (which has a hyperbolic velocity relative to the Moon) is curved towards the Moon and may touch it on the backside. The effective radius depends on the departure speed and may be calculated as RM 1740 km – radius of the Moon; M – gravitational parameter of the Moon; V – speed in the infinity. The maximum value of the effective radius (corresponding to the Hohmann transfer and so, to the minimum speed in the Moon’s infinity) is ~ 5400 km, for the parabolic departure speed it decreases to ~ 3000 km. Generally, to cancel launch errors and to provide more precise arrival to the selected point in the circumlunar space, corrections are performed during the flight to the Moon, their number may be different, generally 1-3 corrections are sufficient. Using the concept of SOI, we may analyze the motion of the probe in the circumlunar space. The radius of the lunar SOI is ~ 66 000 km, so when the probe enters the SOI, we may regard its motion as Keplerian relative to the Moon, thus ignoring the Earth’s gravity. Any probe sent to an orbit with the apogee at the lunar orbit or higher, has a hyperbolic velocity on the boundary of the lunar SOI (the corresponding escape velocity is ~0.4 km/s), so its motion relative to the Moon is hyperbolic. Thus, if the no engine burn is performed, the probe will flyby the Moon and leave it having a hyperbolic velocity also. The figure represents an example of motion of a probe which entered the SOI in the point A1.

Its velocity is vin, and since the Moon moves along its trajectory with the velocity vM1, the probe has the velocity vin’, relative to the Moon, which is calculated as the vector difference If the Moon were a motionless body, the path of the probe would be the hyperbola A1B (B is the point where the probe leaves the SOI). However, when the probe follows this hyperbola, the Moon moves along its own path. At the moment when the probe leaves the SOI the Moon, it reaches the point M2, thus the probe leaves the SOI at the point A2. In the point A2 the geocentric velocity vout of the probe is the sum of the selenocentric velocity vout’(equal to the velocity vin’ but rotated in some angle) and the transport velocity of the Moon vM2: It is seen from the picture that vout is larger than vin. So the velocity of the probe increases, it even may become hyperbolic in the geocentric frame of reference and the probe will leave the Earth. The probe moves along a loop-like path in the geocentric frame of reference. The angle  at which the trajectory changes its direction at the flyby (i.e. the angle between the vectors vin’ and vout’) depends on the flyby distance and the velocity module. Since the boundary of SOI is quite far from the center of the Moon, we may regard this distance infinite and thus the velocity vinis nearly equal to the hyperbolic excess. So the directions of the velocity vectors are nearly the same as the directions of the asymptotes of the hyperbola. The maximum value of  corresponds to the minimum vin’ and the flyby at the distance equal to the lunar radius. As we have seen in one of the previous lectures, the minimum velocity which has a probe entering the lunar SOI is about 0.8 km/s, and max 1200. So in the selenocentric frame of reference the attraction of the Moon cannot change the direction of the initial velocity more than by 1200.

The following path of the probe will bring it back to the Earth’s vicinity. If the probe enters the SOI at the point A’, it will fly around the Moon in the counterclockwise direction and will leave the SOI in the point B’. It is simple to make sure that in this case the geocentric velocity vout will be hyperbolic, and the probe will leave the Earth. This is an example of the gravity slingshot performed in the lunar gravity field (see further). Not any point of the SOI is accessible for the probe. For a Hohmann transfer, the probe moves with the velocity of ~ 0.2 km/s in the circumlunar space and cannot reach the trailing side of the SOI (moving with the velocity of ~1 km/s), so the probe may enter the SOI only in its leading side. Only for near hyperbolic velocities the lower part of the trailing side of the SOI is accessible, but the upper trailing part cannot be accessed. A lot of various trajectories are possible with the use of correcting maneuvers. Flyby trajectories with return to the Earth are of specific interest. The first such flyby was done by the Luna-3 in October 1959 which performed a flyby at the distance of ~ 6000 km above the backside (entering from the southern hemisphere) and made photos. The photos were transmitted to the ground station when the probe returned to the vicinity of the Earth (the height of the perigee ~ 40 000 km). The first manned Apollo spacecraft followed free-return trajectories which would bring them back to the Earth after a lunar flyby if the main engine had failed. Corrections with the steering engines would be enough to put the spacecraft on the correct re-entry path. Further expeditions did not use this kind of trajectories since they needed additional delta-v expense, and the confidence in the main engine had raised: this engine became necessary to return the spacecraft to the Earth. To put a flyby probe to a circular orbit around the Moon, in it is necessary to brake it down using the engine onboard. The optimal braking maneuver should be performed in the periselene of the hyperbolic flyby orbit because of the Oberth’s effect. As we have seen in

Return from the Moon one of the previous lectures, for a Hohmann transfer, the speed in the periselene will be ~2500 m/s, and since the circular velocity of a low Moon orbit is ~ 1700 m/s, the delta-v of ~0.8 m/s is required. The first natural satellite of the Moon was the Luna-10 which entered the the elliptical lunar orbit in April 1966. The initial periselene was ~ 350 km and the aposelene was ~ 1000 km. The mass of the probe was ~ 250 kg. The Apollo 8 became the first manned lunar satellite in December 1968, with the crew of 3 astronauts onboard. Orbits of lunar satellites are generally not stable, each satellite will either hit the surface or leave the Moon in months or years due to solar & Earth’s gravitational perturbations and also perturbations from the lunar gravity field (so-called mascons, local mass concentrations). To return to the Earth from the Moon, the probe should obtain a hyperbolic velocity relative to the Moon. This velocity should be provided by a horizontal burn, and the burn should occur at the angular distance of ~1800 from the target, i.e. over the backside. The probe which has a velocity of ~ 2.5 km/s near the surface of the Moon will come to the boundary of the lunar SOI with the speed of ~ 1 km/s. In the figure the corresponding vector diagram is depicted. It is obvious that the probe should not accelerate in the direction opposite to that depicted in the figure, i.e. it should not leave the SOI from the leading hemisphere. This is because if it leaves the SOI from the front, the geocentric velocity will be higher than the velocity of the Moon on its orbit, and the probe will get an Earth escape velocity, thus leaving the Earth. So, it is possible to return from the Moon only from the trailing hemisphere. A vertical launch from the surface is also possible, but only from the eastern part of the visible lunar hemisphere. The the Soviet automatic soil return probes Luna-16/-20/-24 had to land in this region since their return rockets ascended vertically for a direct flight towards the Earth.

Interplanetary flights Interplanetary space probes are one of the most effective ways to study distant objects of the Solar system. They give the unique opportunity to get a close look to the planets, their satellites, to perform measurements in situ and experiments on the surface of the objects. They may bring back to the Earth samples for more detailed investigations, and may safely substitute a human being scientist in places where he cannot get personally. Let us analyze interplanetary flights using the concept of SOI, which gives very good qualitative results and a reasonable first quantitative approximation. A flight to the planet may be divided into three parts: 1) geocentric flight inside the SOI of the Earth; 2) heliocentric flight between the SOI of the Earth and the SOI of the planet; 3) planetocentric flight inside the SOI of the planet. Planning of the mission begins from the analysis of the arrival conditions and from the search for the SOI entering velocity which enables the desired mission. The second step is to choose a suitable heliocentric transfer orbit. The choice is predefined by the available characteristic velocity, the time of the transfer and the ballistic conditions. Finally, the launch parameters are defined to deliver the probe to the boundary of the Earth’s SOI with the needed velocity. However, we shall follow the analysis in the sequence the flight occurs.

Flight inside the SOI of the Earth The probe during its flight inside the Earth’s SOI also participates in the transport motion of the Earth relative to the Sun. Its heliocentric velocity after leaving the SOI depends on the direction where the probe was launched. If it is launched towards the leading SOI hemisphere, i.e. in the direction of the Earth’s orbital motion, the heliocentric velocity of the probe on the boundary of the SOI will be higher than the local heliocentric circular velocity, so the apohelion of the heliocentric orbit of the probe will lie higher than the orbit of the Earth. Such probe may fly to the outer planets: the Mars, the Jupiter etc. Vice versa, if the probe is launched towards the trailing hemisphere, is perihelion will be lower and the probe may visit the Venus or the Mercury. Launched nearly perpendicularly to the Earth’s orbit, the probe will have the same heliocentric velocity as the Earth, and its orbit will remain close to the Earth. Such probe will follow a more eccentric path than the Earth or (if the launch was performed at an angle with the plane of the ecliptic) will spend some time above and below the ecliptic, not getting far away from the Earth. The velocity of the interplanetary probe should be hyperbolic so that the probe could leave our planet. It is not difficult to prove that the velocity Vbound the probe will have on the boundary of the SOI is v0 – the velocity of the probe after the engine cut-off; Vesc – the escape velocity at the engine cut-off distance; R0 – the distance from the center at which the engine cut-off occured; RSOI 0.93·106 km – the radius of the SOI.

If the probe should leave the SOI with a predefined velocity pointed in a predefined direction, its trajectory inside the SOI may be any hyperbola, the perigee distance of which is defined by the interplanetary injection (for example, by the height of the parking orbit) and the hyperbolic excess is defined by the mission needs. The family of these hyperbolas form the axisymmetric hyperboloid with its axis pointed in the predefined direction and passing through the center of the Earth. Near the boundary of the SOI the cavity of this hyperboloid is nearly cylindrical since due to the high radius of the SOI the hyperbolas are close to their asymptotes. The intersection of the hyperboloid and the boundary of the SOI is circular. To enter one of the interplanetary hyperbolas at a certain height (lower is better due to the Oberth’s effect), the probe should be provided with the velocity v0 at one of the points the locus of which is the circle of orbital launches NB. Its angular width  (angle NOP) may is vout – the velocity of the probe at the boundary of the SOI. This circle may be achieved by a direct injection if the launch site is situated on the circle of direct launchesKM. In the general case this circle is available for several launch sites two times per day due to the rotation of the Earth. However, if the latitude of the launch site does not lay inside the sector cut of by KM, direct injection is impossible (without high losses; an ineffective non-horizontal ascent is possible anyway). So, parking orbit is a good solution; it should be perpendicular to the circle BN to avoid nonhorizontal burns. The specific parking orbit and the time of launch should be chosen considering the optimal launch conditions.

Heliocentric flight After leaving the SOI of the Earth with the velocity vout, the probe has a heliocentric velocity Vout which may be found as the vector sum VEarth  30 km/s being the heliocentric velocity of the Earth directed tangentially to the nearly circular Earth’s heliocentric orbit. To ensure effectiveness, in most cases the velocity Vout should be directed tangentially to the terrestrial orbit, so the probe should be launched in the prograde or the retrograde direction. Let us first neglect inclinations of the planetary orbits to the ecliptic and regard the interplanetary trajectory as lying in the plane of the ecliptic. The Hohmann transfer is known to be the most efficient in terms of minimal delta-v. The corresponding velocity Vout may be found as Rplanet – the radius of the orbit of the planet in units of the radius of the orbit of the Earth, ~ 150 ·106 km. The time of the flight may be found from the formula for the Hohmann transfer time, it may be expressed in days as The launch should be performed with anticipation, since the planet moves along its orbit and during the time of the transfer passes the angular distance of  = 3600·THoh/ Pplanet, Pplanet being the orbital period of the planet. Thus, since the angular distance for the Hohmann transfer is 1800, the initial configuration angle (angle Earth – Sun – planet) should be

For the outer planets this angle is positive, for the inner planets it is negative since they move quicker than the probe on a Hohmann transfer trajectory. Each configuration, including the initial configuration, repeats with the period depending on the angular velocities of the planet and the Earth (the so-called synodic month) and is related to the orbital period by the formula The table to the right lists some critical parameters for Hohmann transfers to the planets. v0 is given for the height of 200 km, Vin is the heliocentric velocity of entry into the SOI of the planet. It is seen that the biggest disadvantage of the Hohmann transfer orbits is that for distant planets the transfer time may be very large: a travel to the Neptune would require tens of years. To reduce the travel time to reasonable limits, higher transfer velocities are needed. They may be provided either by higher departure velocities v0 either by gravity assists from other planets. For example, if the heliocentric escape velocity were imparted to

the probe, the Neptune may be reached in ~ 13 years and the Mars in ~ 70 days. For shorter transfer times, configuration angles are different from that of the Hohmann transfer. The planetary orbit may be inclined to the ecliptic. For example, this angle for the Mars is ~1.850. It may seem a small value, but the heliocentric orbital plane change maneuver to that extent needs the delta-v of ~ 1 km/s. In such case the interplanetary injection maneuver should include plane change. The mission is cheaper if the Earth and the planet are close to the points where their orbital planes intersect, however, it is unreal that both planets were situated in these points at the launch time and at the arrival. The compromise is needed, and it includes search for a balance between the extra delta-v losses for the inclination change and the quicker of slower transfer trajectory: quicker transfer may need less delta-v for the inclination change. Flights to the planets are always associated with the need to perform trajectory corrections because of launch errors and perturbations by other celestial bodies. An error of ~ 1 m/s at the boundary of the Earth’s SOI may shift the trajectory of the probe by many thousands of kilometers at the end of the journey. The problem of such corrections is very complicated and is actually the optimization problem. What is better? To provide a more precise (but heavier) guidance system minimizing the weight of the propellant for corrections or to choose simpler and more reliable guidance system and take more fuel? To perform the correction soon after the error had been discovered to minimize the deviation of the path or to spend more time on the analysis and perform the correction later but more precisely? If a correction is performed when the heliocentric velocity is higher, mostly the speed may be changed but not the direction, so the arrival time is mostly influenced. The smaller is the heliocentric velocity, the simpler is to change the direction of the flight. The general practice is that the number of corrections is planned before the launch, and they are aimed to keep the probe inside the permitted corridor of paths leading to the target.

Flight inside the SOI of the planet In the latest time, the idea of interplanetary flights using electric propulsion (with thrusters giving very small accelerations, but possessing very high effectiveness in terms of the specific impulse) has been implemented. Although small accelerations do not enable delta-v effective transfers and need much time (the probe moves toward the target along an unbending spiral), electric propulsion is very propellant effective and may provide high velocity changes with a small amount of propellant. The Japanese probe Hayabusa used electric propulsion to perform a close encounter and study of the asteroid Itokawa. The Dawn spacecraft uses electric propulsion to reach the asteroids Ceres & Vesta. After the probe entered the SOI of the planet, we regard its motion as planetocentric and ignore the gravitational influence of the Sun. The important issue is the planetocentric velocity of the probe at the entry. If the target planet is one of the outer planets, the heliocentric velocity of the planet is higher than that of the probe, so the planet runs after the probe, and the entry occurs at the leading hemisphere of the SOI. If the target planet is the Mercury or the Venus, the probes moves quicker and enters the trailing hemisphere. The planetocentric velocity of the probe is the vector difference for the outer planets (Vin – planetocentric velocity at the entry, Vin – heliocentric velocity at the entry, Vplanet – heliocentric velocity of the planet). For the inner planets the planetocentric velocity is The planetocentric velocity at the entry is always larger than the local hyperbolic velocity (for the Mars and the Venus the difference is ~ 3 times, for other planets it is even higher), so the probe moves with a hyperbolic velocity relative to the planet, and its planetocentric path is a

hyperbola. So, if no maneuver is performed inside the SOI or the probe does not meet the planet, it will leave the SOI with the same speed of vin but pointed in other direction. To hit the planet, the impact parameter (the distance from the center of the planet that the probe would have if it traveled in a straight line, without gravity pull) should not exceed the effective radius of the planet corresponding to the speed vin. To put the probe into a planetocentric orbit, it should be slowed down. This maneuver is better to perform in the periapsis of the hyperbola and as close to the planet as possible. So, the point of entry into the SOI should be chosen so that the periapsis were not be distant from the planet. The required orbit may be achieved by a series of multiple burn maneuvers, which may include atmospheric maneuvers like aerobraking or even aerocapture. The precise targeting inside the SOI of the planet is gained by corrections. Corrections are most effective near the boundary of the SOI, where the planetocentric velocity is the smallest and a fixed delta-v burn introduces the largest relative change. Planetary flyby and gravity assist maneuver Flyby trajectories in the SOI of the planets are quite diverse. We may introduce some simplifications to our analysis. It is possible to neglect the time of the flyby compared to the total time of the mission since the time of the transfer is much longer. The probe first accelerates when moves towards the planet and then decelerates when moves away, if the planet is one of the outer planets; in the case on an inner planet all occurs in the reverse order. Thus the total time of the mission remains nearly unaltered.

The velocity vector of the planet may be regarded as constant. Thus, the influence of the planet to the motion of the probe is like an instantaneous hit which the probe experiences. This hit changes the speed and the direction of the velocity of the probe. The figure illustrates planetocentric motion of the probe in its hyperbolic orbit and also the vector diagram of its planetocentric and heliocentric velocities. The probe enters the SOI with the heliocentric velocity Vin, which defines its planetocentric velocity vin. If the impact parameter of the entry is b, the path of the probe inside the SOI will be a hyperbola with the deflection angle  , which may be found as Vcirc being the circular velocity at the distance equal to the impact parameter b. The smaller is the impact parameter, the larger is the deflection angle and higher is the change of initial direction. However, the maximum value of the deflection angle is limited by the radius of the planet or the height of its atmosphere, so it is impossible, for example, to reverse the initial velocity vector. If the escape velocity for the largest value of the impact parameter is Vesc, the maximum deflection angle is found as The probe leaves the SOI with the same planetocentric speed as it entered, thus vout = vin. The orientation of the vector vout on the vector diagram may be found with the use of the deflection angle. The heliocentric speed Vout of the probe at leaving the SOI differs from its entry heliocentric speed Vin due to the gravitational “hit” from the planet. The velocity change V is found as The difference arises from the fact that the probe exchanges its energy and momentum with the planet, but that exchange is not noticeable for the planet because of its very high mass.

It is obvious that the maximum increase of the heliocentric velocity takes place for the maximum deflection angle, so Actually, the vector vout may not lie in the same plane as the vector vin, so the vector diagram should be treated as 3-dimensional: the final heliocentric velocity vector Vout may lie in the plane different from the plane of the planetary orbit. Increase of vin, on the one hand, tends to increase V, but on the other hand, it makes the deflection angles smaller (the hyperbola unbends to a straight line), and this favors a decrease of V. It may be demonstrated that for any vin the largest V takes place if the deflection angle is  = 600, the largest value is Vmax = Vcirc – the circular velocity at at the distance equal to the impact parameter b. Of course, the final value and direction of the heliocentric velocity is predefined by the mission requirements, so it is generally impossible to obtain maximum velocity increase. The table to the left gives the maximal value of the velocity increase which the planets and the Moon may impart to the probe. To find the heliocentric velocity Vout after the flyby, it is necessary to calculate the vector sum Vout = Vin+ V. Depending on the geometry of the flyby, this velocity may be larger or smaller than the initial velocity Vin. Increase of Voutenables to speed up travels to outer planets, decrease of Voutpermits to travel to Mercury and circumsolar space with smaller expense of delta-v. This kind of space maneuvers is called gravity assist, or gravitational slingshot (gravitational swing-by). This maneuver easies very much journeys to distant planets and/or

enables to significantly increase the payload without additional propellant expenses. It is also used for traveling between satellites of the giant planets (the Galileo mission to the Jupiter and the Cassini-Huygens mission to the Saturn) using gravity fields of their large satellites (the Galilean satellites of the Jupiter and the Titan satellite of the Saturn) to change velocity vector of the probe and to pass from one orbit to another, thus visiting various satellites and zones, with the minimal expenditure of the propellant. To change the velocity of the probe, the mission planners may vary the module and the direction of the heliocentric velocity at the entry to the SOI of the planet, as well as the place of the entry. However, the speed and the direction of the entry are mostly predefined by the transfer trajectory (although may be altered somehow by varying this trajectory in the limits set by the energy considerations of the transfer). So altering the impact parameter and the entry point (by corrections near the boundary of the SOI) are the main control parameters of a slingshot maneuver. We have analyzed the passive slingshot, when the probe does not perform any engine burn during the flyby. Active slingshot is another type of such maneuver, its idea is to realize an engine burn in the periapsis of the trajectory to increase the velocity. Because of the Oberth’s effect, such burn will lead to significant increase of the final planetocentric velocity vin on the boundary of the SOI, this increase will be much higher than if the burn were performed far from the planet. Aerogravity assist is a proposed type of the gravity assist maneuver where a probe uses aerodynamic lift in the atmosphere of the planet to curve its trajectory for a larger deflection angle than the radius of the planet permits at a passive slingshot. Increasing the deflection angle favors greater velocity changes at the flyby. However, such maneuver would require intensive thermal protection, shielding and very precise guiding, so it is much more difficult to perform. So far aerogravity assists have not

been realized in practice. The Luna-3 may be considered the first use of the gravity assist, since the lunar gravity field was applied to return the probe in the vicinity of the Earth in order to transmit back the images. The first time the gravity assist was used for an interplanetary transfer was the probe Mariner 10, which used a Venus slingshot in February 1974 for the following Mercury flyby in March 1974 (actually Mariner 10 passed to a heliocentric resonance orbit, which enabled to send data in 3 successive flybys, until the propellant for the steering engines depleted). The most known use of the gravity assist is probably the Voyager 2 journey to the giant planets (launched in 1977): in the years 1979-1989 it performed the so-called Grand Tour. Taking advantage of the unique configuration of the giant planets, Voyager 2 successively used gravity assists from the Jupiter, the Saturn and the Uranus and spent 12 years in the journey that would have otherwise taken tens of year and with less scientific payload onboard. Nowadays many spacecraft are launched to their targets with the use of multiple gravity assists (even by the inner planets including the Earth, although they cannot change the velocity so much as the giant planets). This is due to that multiple gravity assists, although significantly increase the time of the journey, enable to take more scientific payload. In the case of a direct fly (without gravity assists) the weight of the probe would be noticeably smaller since the launch vehicle should provide the whole characteristic velocity. Gravity assists help to get free a significant fraction of this velocity. For instance, the Galileo Jupiter orbiter (launched October 1989) performed one Venus flyby and 2 Earth flybys before the it arrived to the Jupiter in December 1995. In the Jovian system multiple gravity assists were undertaken to alter the orbit and to send the probe to different targets. The trajectory of the Messenger Mercury orbiter (launched August 2004) includes one Earth flyby, 2 Venus flybys and 3 Mercury flybys before the final orbital insertion that should occur in March 2011.

End of the Lecture 7

(By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.) Direct launch to the Moon Orbital plane for a lunar trip from the Nothern hemisphere. i – inclination; – inclination of the lunar orbit;  – latitude of the launch site.

Translunar trajectories Ln – Moon in the Nothern hemisphere; Ls – Moon in the Nothern hemisphere; – inclination of the lunar orbit; – latitude of the launch site. 1, 2 – optimal trajectories; 1’, 2’ – maximal losses (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

Effective radius of the Moon reff – effective radius of the Moon; V– velocity of the probe (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

Moon flyby vin – initial velocity vector; vM1 – Moon velocity vector at the point M1; vin’ – initial velocity vector relative to the Moon; vout – final velocity vector; vM2 – Moon velocity vector at the point M2; vout’ – initial velocity vector relative to the Earth Numbers on the path designate the time after the launch in days (By V.V.Beletsky, Essays on the Motion of Celestial Bodies, and V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

Return from the Moon vout – velocity vector relative to the Earth; vM – Moon velocity vector; vout’ –velocity vector relative to the Moon (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

vout – velocity at the boundary of the SOI; vo – velocity after engine cun-off; O – the center of the Earth; BN – the circle at which the engine cut-off occurs when placing the probe to one of the hyperbolas; KM – the circle on the Earth there the launch site should be situated for a direct injection to one of the hyperbolas (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.) Escape from the Earth’s SOI Geometry of the interplanetary orbits inside the SOI of the Earth

Interplanetary Hohmann transfers E – position of the Earth at the launch; O – position of the outer planet at the launch; I – position of the inner planete at the launch; E-Sun-O – configuration angle for the outer planet; E-Sun-I – configuration angle for the inner planet (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

Motion inside a planetary SOI Vin – initial heliocentric velocity; Vout – finar heliocentric velocity; Vplanet – planetary heliocentric velocity; vin – initial planetocentric velocity; vout – final planetocentric velocity; V – velocity change b – impact parameter  – deflection angle (By V.I.Levantovski, Mekhanika kosmicheskogo polyota v elementrarnom izlozhenii.)

Voyager 2 launch (By source) Voyager 2 trajectory with gravity assists Voyager 1 & 2 heliocentric trajectories (By source) Voyager 2 velocity change history (By source)

Galileo trajectory in the Jovian system Galileo’s trajectory in the Jovian system during the prime mission. (By source) Galileo deploy from the Shuttle Atlantis. (By source)

Introduction to Astronautics Sissejuhatus kosmonautikasse