Chapter 6 More complicated orbits and spacecraft trajectories

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1 There s three major functions under navigation Trajectory design, which is designing the trajectory that ll get you to your destination, or meet those science requirements. For Juno, we needed to get to Jupiter with a certain amount of fuel, maximum amount of fuel the spacecraft could carry. And then on top of that, orbit the planet in a way that will meet the spacecraft science instrument requirements. Orbit determination: using tracking data to determine exactly where the spacecraft is so you can reconstruct the trajectory which is useful for all the science instruments so that they can process their data and know exactly where each data point was taken from. Secondly, you can predict in the future where the spacecraft is going so that you can provide that to the science instrument leads so that they know where the spacecraft will be so they can plan their pointing of their instruments or whatever or use those predictions to determine how you need to alter the trajectory to get back to whatever the planned trajectory was because we don t model things exactly right a spacecraft is always deviating from the exact planned trajectory so we need to get it back to where we planned it to be using firing thrusters. Maneuver design: a maneuver analyst will take the predicted trajectory that orbit determination delivers that and figure out the best way to get back to the planned trajectory using the limitations of the spacecraft thrusters whatever they may be point the spacecraft in this direction and fire the thrusters by x amount. John Bordi, Navigation Team Chief for Juno Chapter 6 More complicated orbits and spacecraft trajectories In the last chapter, you saw the connection between the force of gravity and the concept of inertia that allows any object to orbit a planet. Moreover, you now know that the further the orbiting object is away from the center of the thing it s orbiting, the slower it can go and still remain in orbit. In this chapter, you ll understand why satellite trajectory designers developed so many different configuration orbits. I mean, since when was a circle not good enough? The orbital elements mathematically describe the formal orbit As noted earlier, space scientists and engineers demand numbers. Not content to describe an orbit as circular or sort of ovally, they want a mathematical description of an orbit. It turns out that any orbit of an object in a central force field is described uniquely by six numbers. Since these numbers were based on Kepler s Laws of Planetary Motion, they are called the Keplerian orbital elements. The most basic parameters of an ellipse its size and shape are covered by two of the orbital elements. The size of the orbit is described by the semi-major axis (covered in the previous chapter) and is assigned the symbol a. The semi-major axis represents a distance, so much have units of distance, like km or AU. The shape of an orbit is described by its eccentricity (also covered in the previous chapter) and is assigned the symbol e. Because it is a parameter that describes how oval the ellipse is, it has no units. Instead, it behaves like a scale, where an eccentricity of zero describes a circular 47

2 orbit, whereas a parabolic orbit (in which the object never returns) has an eccentricity of one. All elliptical orbits have an eccentricity of less than one. So far so good; why are four more numbers needed to describe the orbit? 48

3 For one thing, all astronomical objects (e.g., planets, stars) rotate around their rotational axis. The plane that is perpendicular to that axis which cuts through the full circumference of the object (e.g., the object s equator) is called the plane of reference. Not all spacecraft will orbit on an object s plane of reference. Thus, you need another number, the inclination (symbolized i ), to measure how far off the plane of reference the orbit is. When two planar shapes meet, like the plane of reference and the plane of the orbit, it is easiest to describe the inclination as the measure of the angle formed at the intersection, and thus the inclination is measured in angular units like degrees or radians. As an example, the Moon s orbit is inclined about 5 from Earth s equatorial plane (plane of reference). Similarly, the orbiting object could be above or below the plane of reference. Where the orbiting object crosses the reference plane from below to above the plane, this point is called the ascending node. If you look at the orbit from high above the orbited body s north pole, the ascending node of the orbiting object could be in any direction. But orbit after orbit, you would notice that the ascending node for this object never changes. If only there were a reference line that you could measure the angle to the ascending node, you could have another number that describes the orbit. Of course there is such a reference line; for different objects there are different reference lines with respect to the ascending node. For orbits around the Earth or the Sun, the reference line is called the first point in Aries, and the angle between that reference line and the ascending node is called the longitude of the ascending node (or the right ascension of the ascending node). Since it is an angle, it is measured in degrees or radians. The symbol for this angle is Ω (capital omega). Are we done? No, that s only four orbital elements. What else needs to be specified? For one thing, if you go back to looking down at the orbit from the orbited body s north pole, even if you know the inclination and longitude of the ascending node of the orbit, the closest the orbit gets to the orbited body (called periapsis ) can occur anywhere along the orbit. That is, the periapsis point on the orbit could be coincident with the ascending node, but it doesn t have to be. So you have to specify that number, which is the angle between the line from the orbited body to the periapsis point, and the ascending node. This angle is called the argument of periapsis (for Earth orbits, it is called the argument of perigee), and its units are degrees or radians. The symbol for this angle is ω ( little omega). Finally, have you noticed that the discussion has centered on placing the orbit in its correct orientation in space that we ve forgotten a key point: there is an actual object, like a spacecraft, orbiting the body. In other words, there s a spacecraft occupying a particular point on the orbit at a particular time. Fortunately, this can be described by a number as well. The true anomaly is the angle between the line drawn from the orbited body to the spacecraft, and the line from the orbited body to the periapsis point at a specified time. The true anomaly is measured in degrees or radians. Sometimes you will see the term 49

4 mean anomaly used as a Keplerian orbital element rather than the true anomaly. The symbol for true anomaly is υ ( little nu). With the six Keplerian orbit elements, you can calculate and predict the position of any orbiting object at any time. In fact, a table of predicted positions of an orbiting object is called an ephemeris (plural: ephemerides). Satellite orbits come in all shapes and sizes The shape and size of any particular satellite s orbit depends entirely on its function. For instance, the geosynchronous satellites mentioned in the last chapter must be positioned over the same point on the equator for its entire life, which means that the satellite s orbit must be circular (e = 0) and maintain a distance of about 42,164 km from the center of the Earth. Moreover, these satellites must be positioned over the equator (i = 0 ). This is because the users of these satellites, be they communications firms or weather bureaus, require a predictable line-of-sight with the satellite all day and night. It does take a fair amount of fuel to achieve that type of high orbit; also, if the satellite is taking images, the resolution of the Earth s surface is low because of the distance. For these reasons, another circular, lowinclination orbit is the equatorial low Earth orbit (LEO). This is the orbit favored by spy satellites, low-power radio communications satellites, remote-sensing satellites and any other satellite that needs good resolution of the Earth s surface or cannot send a powerful enough radio signal back to Earth. 50

5 However, some satellites require passage over higher latitudes of the Earth. For instance, the International Space Station (ISS) must be able to meet its astronauts coming up from Baikonur Cosmodrome in Kazakhstan. This means that the ISS must orbit at a much higher inclination with respect to the equator, which is why people on the surface far from the equator have a chance to see the ISS passing overhead. The ISS is in a highinclination LEO. If a satellite passes over the poles of the Earth, it is said to be in a polar orbit. The disadvantage of all LEO orbits is that the satellite cannot be over the same spot on Earth at all times. This is remedied by orbiting a constellation of satellites so that as one disappears from view of an observer on Earth, another one is appearing. The Iridium communications satellites did exactly this: the company has 66 satellites and 6 spare satellites in LEO. The US, Canada, and especially Russia have much of their landmass in the higher latitudes, so each country has a significant number of polar-orbiting satellites. It may be surprising that those countries do not place all their satellites in a polar orbit. The difficulty with a polar orbit is getting there in the first place; when satellites are launched, it is practically impossible to enter a polar orbit directly. A satellite launching from a high latitude enters, by default, a high-inclination orbit. The satellite s trajectory may then be adjusted into a polar orbit, but that requires significant expenditures in fuel. Thus, it is more economical to keep the satellite in a high-inclination orbit, and lose a little of its coverage of high latitudes. In fact, through clever adjustment of the orbital elements, high-inclination orbits with some of the advantages of a geosynchronous orbit have been found. The most famous of these orbits is the Molniya orbit (right), which allows a satellite to remain more-or-less stationary over a point in the high latitudes for several hours. In fact, the orbit is named after the first series of Soviet communications satellites to use this orbit in the 1960s. Since one satellite cannot remain over the same high-latitude spot all day and night, a constellation of satellites must be used. Further, because of the high altitude apogee, the satellite must pass through the Van Allen radiation belts four times per day, leading to an increased chance of damaging the electronics. NASA has a good resource for finding the orbit of nearly every satellite orbiting the Earth. A Java applet, it is at 51

6 and shows a three-dimensional view of the Earth and the satellites that orbit it. Not only is the model scaled properly, you can literally see where each satellite is at the current time. By clicking on the individual satellites, you will highlight its orbit and get details about the satellite itself. But what if you want your satellite or spacecraft to leave the Earth entirely? There are a couple of such orbits. Recall from the previous chapter the idea of escape velocity; the trajectories these spacecraft take are also orbits, albeit open-ended orbits. A spacecraft with exactly escape velocity upon launch will achieve a parabolic orbit; a spacecraft with more velocity than that will be in a hyperbolic orbit. These orbits, being open-ended, will never bring the spacecraft back to Earth. Lagrange points are stable orbits around two massive bodies The popular definition of escape velocity is the minimum velocity a spacecraft needs in order to escape the Earth s gravity. However, from Newton s Law of Universal Gravitation, any mass is attracted to any other mass regardless of distance. In other words, there are protons in furthest quasar that are attracted to you, albeit the force is incredibly small. But incredibly small is not zero, so, by extension, a spacecraft will always be attracted to the Earth, no matter how great its distance from it. So how does a spacecraft ever leave the influence of the Earth s gravity? In an ideal system composed only of the spacecraft and the Earth, escape velocity is the minimum velocity needed such that the apogee of the spacecraft s orbit is at infinity. In other words, the spacecraft, once launched with this velocity, will continue to move away from the Earth slower and slower but never turning around and coming back. In the actual universe, which contains nearly countless masses of different sizes and shapes, not only is the spacecraft attracted to Earth, but it is also attracted to every other mass in the universe. Nearly all of these attractions are trivial, but there are a few masses, such as the Moon or the Sun, that will exert a similar-strength attraction to the spacecraft as the Earth does. The points where the force of attraction to the Sun and the force of attraction to the Earth are equal are called Lagrange points, 52

7 named after the Prussian mathematician Joseph-Louis Lagrange, who worked out the mathematical details of these calculations in There are five Lagrange points, some in places that aren t necessarily intuitively obvious. They are numbered; for instance, L1 for the Earth-Sun system is the Lagrange point located between the Sun and the Earth, whereas L2 is the Lagrange point located past the Earth on the Earth-Sun line. Translating these into distances is a complex calculation, but the result is that L1 is about 1.5 million km towards the Sun from the Earth, and L2 is about 1.5 million kilometers away from the Sun from the Earth. The interesting thing about these Lagrange points is that spacecraft can enter semi-stable orbits, called halo orbits, around these points with a minimal expenditure of fuel. Thus, the huge James Webb Space Telescope will enter a halo orbit in the Earth-Sun L2 point. Interplanetary trajectories orbit the Sun But suppose you don t want your spacecraft to achieve mere Earth orbit; your sights are set much higher (or further): you literally want to reach for the stars. Interstellar travel, at best, is a dicey proposition. Even with our best technology, getting to the nearest star, Proxima Centauri, would be a several tens of thousands of years endeavor. Instead, you can visit our nearest astronomical neighbors, like the Moon or the planets. Clearly, the same physics that got you into Earth orbit in the first place will get you to your further destinations. As an example, your target is the Moon: you want your spacecraft to start off in LEO and end up in orbit around the Moon. Very nice! Here, though, you encounter the first couple of difficulties. From LEO, you will need a significantly higher speed to leave LEO and gain altitude. That higher speed though will be too fast to enter into a stable orbit around the Moon (in other words, you are going much faster than the Moon s escape velocity). To avoid these problems, you have a couple of choices: the Hohmann transfer orbit or the ballistic capture trajectory. The first has the advantages of being well-understood and therefore quite predictable (as seen last chapter) and typically quicker than the second, but the second has the advantage of using less fuel. Hohmann proposed a simple solution to the problem of speeding up and slowing down: burn the rocket motors once in LEO to attain the speed needed to enter into an orbit around the Sun. For planets further from the Sun than the Earth, this new orbit s perihelion (closest point to the Sun) should be at the Earth s orbit and its aphelion (furthest point from the Sun) should be at the planet s orbit. For planets closer to the Sun than the Earth, aphelion should be the Earth s orbit and perihelion should be the planet s orbit. For the Moon, you simply need to make the spacecraft s perigee (closest to Earth) distance to be the altitude of LEO and the spacecraft s apogee (furthest from Earth) 53

8 distance to be the Moon s orbital distance. Hohmann transfer orbits for the Moon are also known as trans-lunar injections. The bi-elliptic transfer is a variation of the Hohmann transfer; instead of immediately defining the new orbit by the first burn of the rocket motor (and the second burn slows down the spacecraft to orbit the target body), there is an additional burn approximately midway, so the first burn does not have to be as long. The bi-elliptic transfer saves fuel in some cases, such as boosting a satellite from LEO to a higher Earth orbit. The ballistic capture trajectory utilizes the target body s gravitational field to capture a spacecraft that had just enough speed to get to the point where the target body s gravitation exceeded the gravitation of any other body. Thus, there would be no need to have rockets (and their heavy fuel) to slow down the spacecraft at the target body, and, indeed, the spacecraft would not be sped up as much (i.e., not use as much fuel) as on a Hohmann transfer orbit. For this reason, this trajectory is also known as the low-energy trajectory. A spacecraft on a ballistic capture trajectory would take significantly longer to reach its target than one on a Hohmann transfer orbit. Moreover, some fuel would be needed once the spacecraft was captured by the target body to maneuver into a suitable orbit around the target body (without this maneuver, the spacecraft would be orbiting the target at a great distance, too far away for most scientific instruments). Perhaps the greatest drawback of the ballistic capture trajectory is that humans have not performed it very much. In fact, the first instance of it being used was as a makeshift solution for a lunar probe that was originally slated to fly past the Moon and not return, but had to, in fact, be parked in lunar orbit. This was in 1991, and very few missions since then have intentionally used this trajectory. Ballistic capture is a special case of a general class of trajectories involving gravity assist. Gravity assist allows a spacecraft to gain speed by making a close open-ended trajectory (parabolic or hyperbolic) around a planet. This extra speed is the result of transferring some of the energy of the orbital motion of the planet to the spacecraft. An example of the gravity assist orbit is shown on the next page with the Juno mission. After its launch in 2011, Juno s initial orbit s aphelion was just beyond the orbit of Mars. A couple of small burns later, it was falling back toward the inner solar system and flew by the Earth in 2013, picking up some of the Earth s orbital speed and attaining enough velocity to arrive at Jupiter. In fact, all of the outer solar system missions, such as 54

9 Voyager, Pioneer and Cassini, have used gravity assist as a method cutting down on the amount of fuel needed. The Oberth effect is a bonus for fastmoving spacecraft A cool consequence of moving very fast is that the amount of energy you can gain from burning your fuel increases. This seems to violate the first law of thermodynamics (the energy can neither be created nor destroyed one). This will take a little explanation. The total energy of a system is the sum of all the chemical, kinetic, electrical, etc., energies of all of its components. In the case of a rocket launching, as the fuel is burned and directed out the nozzle, the chemical energy in the fuel itself is converted to kinetic energy of the exhaust. Similarly, since the fuel was stationary to begin with, the exhaust it creates when it burns is accelerating out the nozzle, directed downward to the launch pad. This, by Newton s Second Law (the F = ma one), means that there is a force directed downward on the exhaust itself. By Newton s Third Law, if there s a force on the exhaust downwards, there must be an equal and opposite (i.e., upwards) force on the rocket itself, and the rocket accelerates upwards. We have a launch! The launchpad did not move or provide any pushback for the rocket exhaust; the motion of the rocket was all a reaction to the force on the escaping exhaust. If the launchpad were truly necessary, consider the problem of firing a rocket in orbit it has nothing to push back on. Imagine a rocket using gravity assist around a planet. In accordance with Kepler s Second Law (the equal areas/equal times one), as the rocket approaches periapsis, it increases in speed. If you were to fire the rocket at periapsis, where the rocket itself reached a maximum speed, the change in energy of the rocket would be the greatest there than at any other point in the rocket s trajectory. This is called the Oberth effect. How is this possible? Suppose the rocket exhaust comes out the back of the nozzle with a high velocity, measured from the point of view of the rocket. But step back a bit, so that you are viewing the rocket from some distance. Because the rocket is moving so quickly at periapsis, the individual rocket exhaust particles before they fly through the nozzle are 55

10 already moving quickly in the same direction as the rocket. As the particles accelerate in the opposite direction as the rocket s motion, from your point of view far away, it seems like the particles are still moving in the same direction as the rocket, only not as fast. In an extreme case, the particles may even appear to slow down until they are stationary. Let s calculate the total kinetic energy of the rocket and fuel particle system. To avoid any worries about burning fuel, let s make the propulsion system an ion drive, which sends charged particles out of its fuel tank and through the nozzle without altering the particle. Before the particle is ejected out the nozzle, it was in the fuel tank, and therefore moving at the same speed as the rocket. The rocket has its own kinetic energy, of course, since it is moving. Thus, the total kinetic energy of this system is the kinetic energy of the rocket plus the kinetic energy of the particle. The principle of conservation of energy (the first law of thermodynamics) states that the total energy of a system cannot change. Of course, energy can change forms, but since we aren t burning anything, we can assume that the total kinetic energy is conserved. So, even after the particle is ejected through the nozzle, the total kinetic energy must be the same as the total kinetic energy before the particle was ejected. A slow or motionless particle certainly has less kinetic energy than a fast particle of the same mass. Thus, to keep the total kinetic energy of the system the same, the kinetic energy of the rocket must increase. The only way to accomplish this is to speed the rocket up, so that its kinetic energy increases to compensate for the loss of kinetic energy of the particle. Yes, this sounds weird, especially considering that the particle was accelerated out of the nozzle. The ejected particles slow down more (from the observer point of view) when the rocket is moving faster, so the gain in speed of the rocket is greatest when the rocket is moving the fastest. Thus, one additional boost you can give a spacecraft is to have a carefullytimed burn of the rocket motor just as the rocket arrives at periapsis during a gravity assist trajectory. This particular maneuver has been performed by numerous interplanetary spacecraft, including both Pioneer missions, both Voyager missions and the Messenger mission. Smaller changes in trajectory are necessary in adjusting spacecraft orbits Of course, not all rocket motor burns are related to putting the spacecraft into interplanetary trajectories. In fact, many of the motors on a spacecraft are capable of very little force, do not carry much fuel and are not necessarily aligned to move the spacecraft forward. These are called thruster motors, and they are often grouped into a reaction control system. For instance, there are 12 thrusters in Juno s reaction control system. The smallest of this type of motors are called Vernier motors, and are capable of fine adjustments, such as the docking of one spacecraft with another. In the photo to the left, the Vernier motors are the small flames seen on the side of the spacecraft (the main engine, of course, is firing downwards). 56

11 The International Space Station, as an example, has to fire its thruster rockets (and thus replenish their fuel supplies) every so often to maintain a certain altitude orbit around the Earth, and not be slowed down (and thus losing altitude) by the drag of the Earth s atmosphere. Another use of a minor thruster burn is orbit phasing. When the Apollo-Soyuz docking mission occurred in 1973, the Soviet Soyuz module was already in a nearly circular orbit around the Earth. The American Apollo spacecraft entered a similar altitude orbit. Because the two spacecraft were in nearly the same orbit, Apollo could not catch up to Soyuz and the docking mission would have failed. However, by using a thruster to lift Apollo into a slightly higher and more eccentric orbit, Apollo slowed down enough to allow Soyuz to catch up. Of course, another thruster burn was needed later to speed up Apollo to enter into the same orbit as Soyuz. Finally, a last thruster burn was needed to close the gap between spacecraft and accomplish the docking. Similarly, there are thruster burns that allow changes to the orbit inclination of a spacecraft as well as other orbital parameters. But how do all of these rocket motors work? And how can they accommodate such a range in thrusts needed for different actions on a spacecraft? That will be the topic of the next chapter. 57