Dawn Journal: Revisiting orbital mechanics
Posted by Marc Rayman
01-03-2013 9:02 CST
Dear Impordawnt Readers,
The indefatigable Dawn spacecraft is continuing to forge through the main asteroid belt, gently thrusting with its ion propulsion system. As it gradually changes its orbit around the sun, the distance to dwarf planet Ceres slowly shrinks. The pertinacious probe will arrive there in 2015 to explore the largest body between the sun and Neptune that has not yet been glimpsed by a visitor from Earth. Meanwhile, Vesta, the fascinating alien world Dawn revealed in 2011 and 2012, grows ever more distant. The mini-planet it orbited and studied in such detail now appears only as a pinpoint of light 15 times farther from Dawn than the moon is from Earth.
Climbing through the solar system atop a column of blue-green xenon ions, Dawn has a great deal of powered flight ahead in order to match orbits with faraway Ceres. Nevertheless, it has shown quite admirably that it is up to the task. The craft has spent more time thrusting and has changed its orbit under its own power more than any other ship from Earth. While most of the next two years will be devoted to still more thrusting, the ambitious adventurer has already accomplished much more than it has left to do. And now it is passing an interesting milestone on its interplanetary trek.
With all of the thrusting Dawn has completed, it has now changed its speed by 7.74 kilometers per second (17,300 mph), and the value grows as the ion thrusting continues. For space enthusiasts from Earth, that is a special speed, known as "orbital velocity." Many satellites, including the International Space Station, travel at about that velocity in their orbits. So does this mean that Dawn has only now achieved the velocity necessary to orbit Earth? The short answer is no. The longer answer constitutes the remainder of this log.
We have discussed some of these principles before, but they are counterintuitive and questions continue to arise. Rather than send our readers on a trajectory through the history of these logs even more complicated than Dawn's flight through the asteroid belt, we will revisit a few of the ideas here. (After substantial introspection, your correspondent granted and was granted permission to reuse not only past text but also future text.)
While marking Dawn's progress in terms of its speed is a convenient description of the effectiveness of its maneuvering, it is not truly a measure of how fast it is moving. Rather, it is a measure of how fast it would be moving under very special (and unrealistic) circumstances. To understand this, we need to look at the nature of orbits in general and Dawn's interplanetary trajectory in particular.
The overwhelming majority of craft humans have sent into space have remained in the vicinity of Earth, accompanying that planet on its annual revolutions around the sun. All satellites of Earth (including the moon) remain bound to it by its gravity. (Similarly, Dawn spent much of 2011 and 2012 as a satellite of distant Vesta, locked in the massive body's gravitational grip.) As fast as satellites seem to travel compared to terrestrial residents, from the larger solar system perspective, their incessant circling of Earth means their paths through space are not very different from Earth's itself. Consider the path of a car racing around a long track. If a fly buzzes around inside the car, to the driver it may seem to be moving fast, but if someone watching the car from a distance plotted the fly's path, on average it would be pretty much like the car's.
Everything on the planet and orbiting it travels around the sun at an average of 30 kilometers per second (67,000 mph), completing one full solar orbit every year. To undertake its interplanetary journey and travel elsewhere in the solar system, Dawn needed to break free of Earth's grasp, and that was accomplished by the rocket that carried it to space more than five years ago. Dawn and its erstwhile home went their separate ways, and the sun became the natural reference for the spacecraft's position and speed on its voyage in deep space.
Despite the enormous push the Delta II rocket delivered (with affection!) to Dawn, the spacecraft still did not have nearly enough energy to escape from the powerful sun. So, being a responsible resident of the solar system, Dawn has remained faithfully in orbit around the sun, just as Earth and the rest of the planets, asteroids, comets, and other members of the star's entourage have.
Whether it is for a spacecraft or moon orbiting a planet, a planet or Dawn orbiting the sun, the sun orbiting the Milky Way galaxy, or the Milky Way galaxy orbiting the Virgo supercluster of galaxies (home to a sizeable fraction of our readership), any orbit is the perfect balance between the inward tug of gravity and the inexorable tendency of objects to travel in a straight path. If you attach a weight to a string and swing it around in a circle, the force you use to pull on the string mimics the gravitational force the sun exerts on the bodies that orbit it. The effort you expend in keeping the weight circling serves constantly to redirect its path; if you let go of the string, the weight's natural motion would carry it away in a straight line (ignoring the effect of Earth's gravity).
The force of gravity diminishes with distance, so the sun's pull on a nearby body is greater than on a more distant one. Therefore, to remain in orbit, to balance the relentless tug of gravity, the closer object must travel faster, fighting the stronger pull. The same effect applies at Earth. Satellites that orbit very close (including, for example, the International Space Station, around 400 kilometers, or 250 miles, from the surface) must streak around the planet at about 7.7 kilometers per second (17,000 mph) to keep from being pulled down. The moon, orbiting almost 1000 times farther above, needs only to travel at about 1.0 kilometers per second (less than 2300 mph) to balance Earth's weaker hold at that distance.
Notice that this means that for an astronaut to travel from the surface of Earth to the International Space Station, it would be necessary to accelerate to quite a high speed to rendezvous with the orbital facility. But then once in orbit, to journey to the much more remote moon, the astronaut's speed eventually would have to decline dramatically. Perhaps speed tells an incomplete story in describing the travels of a spacecraft, just as it does with another example of countering gravity.
A person throwing a ball is not that different from a rocket launching a satellite (although the former is usually somewhat less expensive and often involves fewer toxic chemicals). Both represent struggles against Earth's gravitational pull. To throw a ball higher, you have to give it a harder push, imparting more energy to make it climb away from Earth, but as soon as it leaves your hand, it begins slowing. For a harder (faster) throw, it will take longer for Earth's gravity to stop the ball and bring it back, so it will travel higher. But from the moment it leaves your hand until it reaches the top of its arc, its speed constantly dwindles as it gradually yields to Earth's tug. The astronaut's trip from the space station to the moon would be accomplished by starting with a high speed "throw" from the low starting orbit, and then slowing down until reaching the moon.
The rocket that launched Dawn threw it hard enough to escape from Earth, sending it well beyond the International Space Station and even the moon. Dawn's maximum speed relative to Earth on launch day was so high that Earth could not pull it back. As we saw in the explanation of the launch profile, Dawn was propelled to 11.46 kilometers per second (25,640 mph), well in excess of the space station's orbital speed given three paragraphs above. But it has remained under the sun's control.
Now we can think of the general problem of flying elsewhere in space as similar to climbing a hill. For terrestrial hikers, the rewards of ascent come only after doing the work of pushing against Earth's gravity to reach a higher elevation. Similarly, Dawn is climbing a solar system hill with the sun at the bottom. It started part way up the hill at Earth; and its first rewards were found at a higher elevation, where Vesta, traveling around the sun at only about two thirds of Earth's speed, revealed its fascinating secrets to the visiting ship. The ion thrusting now is propelling it still higher up the hill toward Ceres, which moves even more slowly to balance the still-weaker pull of the sun.
If Dawn had been in zero-gravity and not been obligated to obey the laws of orbital motion, the thrusting to date would have accelerated it by the 7.74 kilometers per second (17,300 mph) mentioned near the beginning. Instead of making the spacecraft go faster, however, that work was designed to climb the solar system hill. If Dawn had been targeted to a destination closer to the sun than Earth, the same amount of thrusting would have helped it speed up to descend the hill, dropping into a lower solar orbit, where it would have to zip around the gravitational master of the solar system faster than Earth.
To orbit a body that orbits the sun, a spacecraft has to match its target's solar orbit. Except in science fiction, no spacecraft in history other than Dawn has been designed to orbit two different destinations around the sun. Without its ion propulsion system, this mission would be quite impossible. Tighter orbits require greater velocity in order to counterbalance the stronger pull of gravity. Mercury and Venus orbit the sun faster than Earth. Mars moves around the sun more slowly than Earth, and all residents of the more distant main asteroid belt (including Dawn) revolve at an even more leisurely pace.
Because spacecraft wind up at different speeds relative to the sun, their final velocity is not as important in their design and operation as is the amount by which they change their velocity after being released from the rocket. Because of these complexities, rocket scientists generally put all spacecraft on a level playing field (or, in this case, a zero-gravity field free of the complications of the physics of orbits) by using the change of velocity as a measure of the spacecraft's maneuvering capability.
Dawn has slowed down tremendously since it departed Earth, but what is noteworthy is the amount by which it has propulsively changed its speed. If it had begun at a starting line with all other spacecraft on that simplified playing field, by now it would be racing along at 7.74 kilometers per second (17,300 mph), far faster than any other spacecraft. By the end of its mission, it would be flying at an extraordinary 11 kilometers per second (24,600 mph).
Most satellites in low Earth orbit hardly change their speed at all, relying instead on the momentum imparted to them by the rockets that took them into space. As you can see by comparing the numbers above, a rocket to Earth orbit delivers about the same speed that Dawn has achieved already, and the rocket that sent the probe on its interplanetary course provides roughly the same speed that Dawn will attain over the coming years. (Of course, Dawn and the rocket have different objectives. For example, our spacecraft did not have to plow through Earth's atmosphere under its own thunderous power. Rockets do. Nevertheless, the more petite Dawn is gracefully accomplishing its unique space mission without the burden of enormous propellant tanks and multiple stages.)
Having changed its speed by the same amount needed to go from the surface of Earth to Earth orbit is only a coincidence. Dawn's rocket gave it an even larger boost. But for maneuvering after launch, this spaceship is in a class by itself.
Each spacecraft is designed for a specific mission. As no other spacecraft has attempted a mission like Dawn's, no other spacecraft has needed such an exceptional capability to change its own speed. (Some others have used gravitational boosts from planets to change their speed by more than Dawn. That is not a reflection of the spacecraft's capability, however, but rather the particular trajectory it follows.) Together, all the probes humankind has dispatched on interplanetary journeys have helped provide us with new perspectives and new insights on the nature of the solar system, including its origin and evolution. And the people who are interested in them cannot help but be in awe of the daunting challenges, the remarkable engineering, the vast distances, the inspiring adventures, the thrilling sights, and the amazing new knowledge. With its extraordinary ion propulsion system, Dawn is making exciting contributions to this grand endeavor. It has already conducted a richly detailed exploration of one exotic world and, as it thrusts with its ion propulsion system to climb the solar system hill to another, it looks forward to more treasures on its ambitious expedition.
Dawn is 5.8 million kilometers (3.6 million miles) from Vesta and 56 million kilometers (35 million miles) from Ceres. It is also 2.28 AU (341 million kilometers or 212 million miles) from Earth, or 910 times as far as the moon and 2.30 times as far as the sun today. Radio signals, traveling at the universal limit of the speed of light, take 38 minutes to make the round trip.
Dr. Marc D. Rayman
6:00 p.m. PST February 28, 2013
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