Hayabusa

The Hayabusa mission profile has been modified several times, both before and after launch. Georg Cantor makes an appearance as a character, and the hero finds a physical correlate for Cantor's Continuum Problem. Second, since it was the first-ever soft contact with the surface of an asteroid it has enormous influence on further asteroid missions. Rudy Rucker's novel White Light describes a mathematician who leaves his body and travels to a kind of afterworld that includes a mountain whose Absolute Infinite height matches that of the class of all ordinals. First it will help JAXA to further test its technologies in the fields of ion engines, autonomous and optical navigation, deep space communication, and close movement on objects with low gravity among others. Looking up into the night sky is looking into infinity -- distance is incomprehensible and therefore meaningless.". The Hayabusa mission has a very deep engineering importance for JAXA, too. Another quote from The Hitchhiker's Guide to the Galaxy states: "Infinity itself looks flat and uninteresting.

¹ Also in comparing the data from the onboard instruments of the Hayabusa with the data from the Near Shoemaker mission will put the knowledge on a wider level. The Hitchhiker's Guide to the Galaxy contains the following definition of infinity:. Accordingly, Hayabusa "will bridge the gap between ground observation data of asteroids and laboratory analysis of meteorite and cosmic dust collections," says mission scientist Hajime Yano. Brouwer, David Hilbert, Bertrand Russell, Kurt Gödel and Georg Cantor. Hayabusa will solve this problem by bringing back pristine samples from a specific, well-characterized asteroid. The footnote on p.335 of his book suggests the consideration of the following names: Abraham Robinson, Plato, Thomas Aquinas, L.E.J. Scientists' current understanding of asteroids depends greatly on meteorite samples, but it is very difficult to match up meteorite samples with the exact asteroids from which they came. Rudy Rucker, in his book Infinity and the Mind -- the science and philosophy of the mind (1982), has worked out a model list of representatives of each of the eight possible standpoints.

The United States space agency NASA had originally planned to build a miniature rover as part of the Hayabusa mission, but the project, developed by JPL and called Muses-CN, was cancelled in November, 2000, for budgeting reasons. And in between there are the various possibilities. Instead, it joins ranks with the hoppers carried on the failed Soviet Phobos missions, which also never saw use. There are scientists who hold that all three really exist and there are scientists who hold that none of the three exist. Had it been successful, MINERVA would have been the first 'space hopper' to see action. Besides the mathematical infinity and the physical infinity, there could also be a philosophical infinity. [2] [3] [4]. If the universe is indeed ever expanding as science suggests then you could never get back to your starting point even on an infinite time scale.

Instead, it escaped Itokawa's gravitational pull and tumbled into space. The universe, at least in principle, might have a similar topology; if you fly your space ship straight ahead long enough, perhaps you would eventually revisit your starting point. Early inspection of data suggests that mission control is in contact with MINERVA but that it was not successfully dropped onto the asteroid's surface. By walking/sailing/driving straight long enough, you'll return to the exact spot you started from. As a result, when the MINERVA release command arrived, MINERVA was released while the probe was ascending and at a higher altitude than intended. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. The lander release command was sent from Earth, but before the command could arrive, Hayabusa's altimeter measured its distance from Itokawa to be 44m and thus started an automatic altitude keeping sequence. Note that the question of being infinite is logically separate from the question of having boundaries.

MINERVA was deployed on November 12, 2005. An intriguing question is whether actual infinity exists in our physical universe: Are there infinitely many stars? Does the universe have infinite volume? Does space "go on forever"? This is an important open question of cosmology. This solar-powered, box-shaped vehicle was designed to take advantage of Itokawa's very low gravity by hopping great distances across the surface of the asteroid, relaying images from its cameras to Hayabusa whenever the two spacecraft were in sight of one another.[1]. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization. Unfortunately, an error during deployment resulted in the craft's failure. Physicists however require that the end result be physically meaningful. Hayabusa carried a tiny mini-spacecraft (weighing only 591 g) named MINERVA (short for MIcro/Nano Experimental Robot Vehicle for Asteroid). For convenience sake, calculations, equations, theories and approximations often use infinite series, unbounded functions, etc., and may involve infinite quantities.

It will land via parachute near Woomera, Australia. This point of view does not mean that infinity cannot be used in physics. The capsule will experience peak deceleration of about 25 G and heating rates approximately 30 times those experienced by the Apollo spacecraft. Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system. This is currently planned for June 2010. There exists the concept of infinite entities (such as an infinite plane wave) but there are no means to generate such things. Once it is on its return trajectory, the re-entry capsule will be detached from the main spacecraft at a distance of about 300,000 to 400,000 km from the Earth, and the capsule will coast on a ballistic trajectory, re-entering the Earth's atmosphere. It is for example presumed impossible for any body to have infinite mass or infinite energy.

This maneuver was delayed due to problems with attitude control and the thrusters of the craft. It is therefore assumed by physicists that no measurable quantity could have an infinite value, for instance by taking an infinite value in an extended real number system (see also: hyperreal number), or by requiring the counting of an infinite number of events. After a few months in close proximity to the asteroid, the spacecraft was scheduled to fire its engines to begin its cruise back to Earth. counting). However, it is currently uncertain whether the metal projectiles were fired during contact. In physics, approximations of real numbers are used for continuous measurements and natural numbers are used for discrete measurements (i.e. Any samples that were collected are now held inside a separate re-entry capsule. The number Infinity plus 1 is also used sometimes in common speech.

At the second Hayabusa touchdown with its deployable collection horn, the spacecraft was programmed to fire tiny projectiles at the surface and then collect the resulting spray. These terms describe things that are only potential infinities; it is impossible to play a video game for an infinite period of time or keep a computer running for an infinite period of time. Autonomous optical navigation was employed extensively during this period because the long communication delay prohibits Earth-based real-time commanding. See halting problem. Afterwards, the spacecraft moved closer to the surface ("home position"), and then approached the asteroid for a series of soft landings and collection of samples at the safest site. In practice however, some programming loops considered as infinite will halt by exceeding the (finite) number range of one of its variables. Hayabusa surveyed the asteroid surface from a distance of about 20 km, the "gate position". In theory, as long as there is no external interaction, the loop will continue to run for all time.

As it arrived, the spacecraft did not go into orbit around the asteroid, but remained in a station-keeping heliocentric orbit close by. An infinite loop in computer programming is a conditional loop construction whose condition always evaluates to true. The spacecraft's xenon ion engines (two separate units, each with two exhausts) have been operating near-continuously for the last two years, slowly moving Hayabusa toward a September 2005 rendezvous with Itokawa. In video games, infinite lives and infinite ammo refer to a never-ending supply of lives and ammunition. Following launch, the spacecraft's name was changed from the original MUSES-C to Hayabusa, the Japanese word for falcon). For example, "The movie was infinitely boring, but we had to wait forever to get tickets.". The Hayabusa spacecraft was launched on 9 May 2003 at 04:29:25 UTC on an M-5 rocket from the Uchinoura Space Center (still called Kagoshima Space Center at that time). In common parlance, infinity is often used in a hyperbolic sense.

. Leopold Kronecker rejected the notion of infinity and began a school of thought, in the philosophy of mathematics called finitism, which led to the philosophical and mathematical school of mathematical constructivism. Despite its designer's intention, Hayabusa did land and sit on the asteroid surface for about 30 minutes (see the November 19 entry in the recent events section below). One example of this is Hilbert's paradox of the Grand Hotel. However, it is the first craft designed from the onset to make contact with the surface of an asteroid. Our intuition gained from finite sets breaks down when dealing with infinite sets. Technically, Hayabusa is not designed to 'land': it simply touches the surface with its sample capturing device and then moves away. Certain extended number systems, such as the hyperreal numbers, incorporate the ordinary (finite) numbers and infinite numbers of different sizes.

In addition, Hayabusa is the first spacecraft designed to deliberately land on an asteroid and then take off again (NEAR Shoemaker made a controlled descent to the surface of 433 Eros in 2000, but it was not designed as a lander and was eventually deactivated after it arrived). Cantor's views prevailed and modern mathematics accepts actual infinity. Other spacecraft, notably Galileo and NEAR Shoemaker, have visited asteroids before, but the Hayabusa mission, if successful, will mark the first time that an asteroid sample is returned to Earth for analysis. If a set is too large to be put in one to one correspondence with the positive integers, it is called uncountable. . The smallest ordinal infinity is that of the positive integers, and any set which has the cardinality of the integers is countably infinite. The spacecraft also carried a detachable mini-lander but it failed to reach the surface (see Minerva mini-lander below). Cardinal numbers define the size of sets, meaning how many members they contain, and can be standardized by choosing the first ordinal number of a certain size to represent the cardinal number of that size.

In November 2005, it successfully landed on the asteroid to collect samples, and after technical difficulties with the spacecraft, it is slated to return those samples to Earth by June 2010. Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences. Having arrived at Itokawa, Hayabusa is studying the asteroid's shape, spin, topography, color, composition, density, and history. Ordinal numbers may be identified with well-ordered sets, or counting carried on to any stopping point, including points after an infinite number have already been counted. The Hayabusa spacecraft, formerly known as MUSES-C (ミューゼスＣ), was launched on 9 May 2003 and rendezvoused with Itokawa in mid-September 2005. Cantor defined two kinds of infinite numbers, the ordinal numbers and the cardinal numbers. Hayabusa (はやぶさ - peregrine falcon) is an unmanned space mission led by the Japan Aerospace Exploration Agency to collect a sample of material from a small near-Earth asteroid named 25143 Itokawa (dimensions 540 meters by 270 meters by 210 meters) and return the sample to Earth for analysis. An infinite set can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.

June 2010: Hayabusa will release its samples to return to Earth in a re-entry capsule. Dedekind's approach was essentially to adopt the idea of one-to-one correspondence as a standard for comparing the size of sets, and to reject the view of Galileo (which derived from Euclid) that the whole cannot be the same size as the part. February 2007: Hayabusa will begin its return to Earth. This modern mathematical conception of the quantitative infinite developed in the late nineteenth century from work by Cantor, Gottlob Frege, Richard Dedekind and others, using the idea of collections, or sets. If it stabilizes to the opposite direction, we'll have to wait longer so that the probe goes around and the sun is then on the other side. Georg Cantor developed a system of transfinite numbers, in which the first transfinite cardinal is aleph-null (), the cardinality of the set of natural numbers. If it stabilizes so that the solar cell and the antenna are directed to an appropriate direction, the probe will be powered up and communication will be recovered. A different type of "infinity" are the ordinal and cardinal infinities of set theory.

January 2006: Precession movement stabilizes to pure rotation around Z-axis. This is because zero times infinity is undefined. [25] [26]. Notice that . The probability of this is 60% by December 2006, 70% by spring 2007. Infinity is not a real number but may be considered part of the extended real number line, in which arithmetic operations involving infinity may be performed. After the stabilization, the rotation axis must be directed toward the Sun and Earth within specific angular range.
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It will take a month or two to wait for stabilization by conversion of precession to pure rotation. One important example of such functions is the group of Möbius transformations. It is estimated that the turbulence was caused by evaporation of 8 or 10cc of leaked fuel. The domain of a complex-valued function may be extended to include the point at infinity as well. On December 8, sudden altitude change was observed, and the communication with Hayabusa was lost. In this context is often useful to consider meromorphic functions as maps into the Riemann sphere taking the value of at the poles. Due to the power outage, the telemetry log data is faulty, and still needs investigation. When this is done, the resulting space is still a one-dimensional complex manifold and called the extended complex plane or the Riemann sphere.

As the result of telemetry analysis, it was found that there is a strong possibility of the projectile of sampler not penetrated when it landed on November 25. A point labeled can be added to the complex plane as a topological space giving the one-point compactification of the complex plane. Telemetry was obtained and analyzed. means that the magnitude | x | of x grows beyond any assigned value. On December 5, attitude was corrected enough to regain communication through the medium gain antenna. As in real analysis, in complex analysis the symbol , called "infinity", denotes an unbounded limit. Attitude control was commanded using the xenon gas. Infinity is often used not only to define a limit but as if it were a value in the extended real numbers in real analysis; if f(t) ≥ 0 then.

On December 4, as an emergency measure, xenon propellant from the ion engines was blown to correct the spin, and it was confirmed successful. Projective geometry also introduces a line at infinity in plane geometry, and so forth for higher dimensions. On December 3, the probe's Z-axis was found to be 20 to 30 degrees from the sun direction and increasing. We can also treat and as the same, leading to the one-point compactification of the real numbers, which is the real projective line. On December 2, an attitude correction was tried, but the thruster did not generate enough force. Adding algebraic properties to this gives us the extended real numbers. [23] [24] On November 27, the probe experienced a power outage when trying attitude correction, probably due to a fuel leakage. Points labeled and can be added to the real numbers as a topological space, producing the two-point compactification of the real numbers.

JAXA held a press conference about the situation so far. means that x grows beyond any assigned value, and means x is eventually less than any assigned value. On December 6, Hayabusa was 550km from Itokawa. In real analysis, the symbol , called "infinity", denotes an unbounded limit. [22]. The infinity symbol is represented in Unicode by the character ∞ (∞). Mission control is working to resolve the problem before the craft's upcoming launch window for return to Earth. Another conjecture is that he derived it from the Greek letter ω (omega), the last letter in the Greek alphabet.

On November 30, JAXA announced that control and communication with Hayabusa had been restored, but a problem remains with the craft's reaction control system, perhaps involving a frozen pipe. One conjecture about why he chose this symbol is that he derived it from a Roman numeral for 1000 that was in turn derived from the Etruscan numeral for 1000, which looked somewhat like CIƆ and was sometimes used to mean "many". [21]. John Wallis is usually credited with introducing ∞ as a symbol for infinity in 1655 in his De sectionibus conicus. However, due to a leak in the thruster system, the probe was put in a safe hold mode. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before August Ferdinand Möbius and Johann Benedict Listing discovered the Möbius strip in 1858. This time, the sampling device was activated [20]. Again, one can imagine walking along its surface forever.

On November 25, a second touchdown attempt was performed. A popular explanation is that the infinity symbol is derived from the shape of a Möbius strip. This mode did not permit a sample to be taken, but there is a high probability that some dust may have whirled up into the sampling horn when it touched the asteroid, so the sample canister currently attached to the sampling horn was sealed. One can imagine walking forever along a simple loop formed from a ribbon. [19] Unfortunately, the sampling sequence was not triggered since a sensor detected an obstacle during descent; the probe tried to abort the landing, but since its attitude was not appropriate for ascent, it chose instead a safe descent mode. One possibility is suggested by the name it is sometimes called — the lemniscate, from the Latin lemniscus, meaning "ribbon". The probe had entered into a safe mode, slowly spinning to stabilize altitude.[17] [18] However, after regaining control and communication with the probe, the data from the landing attempt were downloaded and analyzed, and on November 23, JAXA announced that the probe had indeed landed on the asteroid surface. The precise origins of the infinity symbol ∞ are unclear.

Ground control sent a command to abort and ascend, and by the time the communication was regained, the probe had moved 100 km away from the asteroid. Unlike the traditional empiricists, he thought that the infinite was in some way given to sense experience. It was initially reported that Hayabusa had stopped at approximately 10 meters from the surface, hovering for 30 minutes for unknown reasons. An exception was Wittgenstein, who made an impassioned attack upon axiomatic set theory, and upon the idea of the actual infinite, during his "middle period".². There was considerable confusion during and after the manoeuvre about precisely what had happened, due to the fact that the high-gain antenna of the probe cannot be used during final phase of touch-down, as well as the blackout during handover of ground station antenna from DSN to Usuda station. Modern discussion of the infinite is now regarded as part of set theory and mathematics, and generally avoided by philosophers. On November 19, Hayabusa landed on the asteroid. A potential infinity is allowed by letting an infinitely-large quantity be cancelled out by an infinitely-small quantity.

MINERVA was released but due to an error failed to reach the surface (see above). Potentiality lies in the definitions of this operation, as well-defined and interconsistent mathematical axioms. On November 12, Hayabusa closed in to 55m from the asteroid's surface. Such seeming paradoxes are resolved by taking any finite figure and stretching its content infinitely in one direction; the magnitude of its content is unchanged as its divisions drop off geometrically but the magnitude of its bounds increases to infinity by necessity. [16]. Not reported, this motivation of Hobbes came too late as curves having infinite length yet bounding finite areas were known much before. From analysis of the closeup images, the Woomera Desert site (Point B) was found to be too rocky to be suitable for landing so that a landing will now be performed only at the Muses Sea site (Point A) even if the landing is to be performed twice. Famously, the ultra-empiricist Hobbes tried to defend the idea of a potential infinity in the light of the discovery by Evangelista Torricelli, of a figure (Gabriel's horn) whose surface area is infinite, but whose volume is finite.

After that, Hayabusa backed off to a higher position, then descended again to 500m and released one of the target markers into space to test the craft's ability to track it (this was confirmed). Our idea of infinity is merely negative or privative. On November 9, Hayabusa performed a descent to 70m to test the landing navigation and the laser altimeter. They believed all our ideas were derived from sense data or "impressions", and since all sensory impressions are inherently finite, so too are our thoughts and ideas. On November 7, Hayabusa was 7.5 km from Itokawa. Locke, in common with most of the empiricist philosophers, also believed that we can have no proper idea of the infinite. [14] [15]. The idea that size can be measured by one-to-one correspondence is today known as Hume's principle, although Hume, like Galileo, believed the principle could not be applied to infinite sets.

It will take a few days to evaluate the situation and set new schedule. He thought this was one of the difficulties which arise when we try, "with our finite minds", to comprehend the infinite. [13] The project manager, Jun-ichiro Kawaguchi, explained that the optical navigation system was not tracking the asteroid very well, probably caused by the complex shape of Itokawa. It appeared, by this reasoning, as though a set which is naturally smaller than the set of which it is a part (since it does not contain all the members of that set) is in some sense the same size. However, at 1:50 am UTC (10:50 am JST) on November 4, it was announced that due to a detection of an anomalous signal at the Go/NoGo decision, the descent, including release of Minerva and the target marker had been canceled. For example, we can match up the "set" of even numbers {2, 4, 6, 8 ...} with the natural numbers {1, 2, 3, 4 ...} as follows:. The descent went well initially, and navigation images with wide-angle cameras were been obtained. Galileo (during his long house arrest in Siena after his condemnation by the Inquisition) was the first to notice that we can place an infinite set into one-to-one correspondence with one of its proper subsets (any part of the set, that is not the whole).

On November 3, Hayabusa began its descent which was to have included delivery of a target marker, and release of the Minerva mini-lander. Aquinas also argued against the idea that infinity could be in any sense complete, or a totality. On November 3, Hayabusa was 3.0 km from Itokawa. However, on this view, no infinite magnitude can have a number, for whatever number we can imagine, there is always a larger one: "There are not so many (in number) that there are no more". [12]. The parts are actually there, in some sense. On November 2, JAXA held a press conference about scientific and engineering achievements during proximity operations around Itokawa. The second view is found in a clearer form by medieval writers such as William of Ockham:.

However, it was also announced that the spacecraft's second reaction wheel, governing the Y-axis, had failed, and that it is now being pointed by the craft's rotation thrusters.[11]. For example, ∀n∈Z(∃m∈Z[m>n∧P(m)]), which reads, "for any integer n, there exists an integer m > n such that P(m)". Closeup pictures were released. The other is that we may quantify over infinite sets without restriction. On October 4, JAXA announced that the spacecraft had successfully moved to its 'Home Position' 7 km from Itokawa. One is that it is always possible to find a number of things that surpasses any given number, even if there are not actually such things. On September 15, a 'colour' image of the asteroid was released (which is, however, grey in colouring) [10]. This is often called potential infinity; however there are two ideas mixed up with this.

[9]. In Europe, the traditional view derives from Aristotle:. On September 12, Hayabusa was 20 km from Itokawa and JAXA scientists announced that Hayabusa had officially 'arrived'. [1] [2] The concept of different orders of infinity would remain unknown in Europe until the late 19th century. [8]. It recognises five different types of infinity: infinite in one and two directions, infinite in area, infinite everywhere, and infinite perpetually. From September 11, individual hills could be discerned on the asteroid. 400 BC) classifies all numbers into three sets: enumerable, innumerable and infinite.

[7]. The Indian Jaina mathematical text Surya Prajinapti (ca. From September 4, Hayabusa's cameras were able to confirm Itokawa's elongated shape. The earliest known documented knowledge of infinity is presented in the Veda- Yajur Veda which states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity". On August 28, the orbit maneuver of Hayabusa was handed over from the ion engines to the bi-propellant thrusters. . [6]. For a discussion about infinity and the physical universe, see Universe.

[5] Other images were taken from August 22 to August 24. In popular culture, we have Buzz Lightyear's rallying cry, "To infinity — and beyond!", which may also be viewed as the rallying cry of set theorists considering large cardinals.¹. The picture was taken by the star tracker and shows a point of light, believed to be the asteroid, moving across the starfield. By some, infinity is considered to be not a number but a concept of increase beyond bounds. On August 14, Hayabusa 's first image of Itokawa was released. In mathematics, infinity is relevant to, or the subject matter of, articles such as mathematical limits, aleph numbers, classes in set theory, Dedekind-infinite sets, large cardinals, Russell's paradox, hyperreal numbers, projective geometry, extended real numbers and the Absolute Infinite. On July 31, 2005, the X-axis reaction wheel failed. In both theology and philosophy, infinity is explored in articles such as the Ultimate, the Absolute, God, and Zeno's paradoxes.

The November 12 release of the MINERVA mini-probe ended in failure. In philosophy, infinity can be attributed to space and time, as for instance in Kant's first antinomy. The original decision to sample two different sites on the asteroid was changed when one of the sites, Woomera Desert, was found to be too rocky for a safe landing. In theology, for example in the work of theologians such as Duns Scotus, the infinite nature of God invokes a sense of being without constraint, rather than a sense of being unlimited in quantity. The November 4 'rehearsal' landing on Itokawa failed, and was rescheduled. The word infinity comes from the Latin infinitas, "unboundedness". JAXA claims that since global mapping of Itokawa has been completed, this is not a major problem, but the mission plan was altered. Popular or colloquial usage of the term often does not accord with its more technical meanings.

After the latter failure, the spacecraft is now being turned on the X and Y axes with its thrusters. Infinity refers to several distinct concepts which arise in theology, philosophy, mathematics and everyday life. Two reaction wheels that govern the attitude movement of Hayabusa failed in 2005; the X-axis wheel failed on July 31, and the Y-axis on October 2. and . Since orbital mechanics dictate that the spacecraft must still leave the asteroid by November, the amount of the time it was able to spend at Itokawa was greatly reduced and the number of landings on the asteroid were reduced from three to two. and . This reduction in electrical power reduced the efficiency of the ion engines, thus delaying the arrival at Itokawa from June to September 2005. If then and .

In 2003, while Hayabusa was en-route to Itokawa, a large solar flare damaged the solar cells aboard the spacecraft. If then and . Hayabusa was to deploy a small rover supplied by NASA, called Muses-CN, onto the surface of the asteroid, but the rover was cancelled by NASA in November 2000 due to budget constraints. and . As a result, the target asteroid was changed from Nereus to Itokawa. and . However, a July 2000 failure of Japan's M-5 rocket forced a delay in the launch, putting Nereus out of reach. means that the area under f(t) approaches 1.

The spacecraft was originally intended to launch in July 2002 to the asteroid 4660 Nereus. means that the area under f(t) is not finite. means that f(t) does not bound a finite area from 0 to 1.