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Retrograde Motion

Retrograde motion

:This article is about the movement of the planets. For the musical term retrograde see Counterpoint, Musical set theory, Operation, Permutation, and Transformation. Prograde motion is the rotational or orbital motion of a body in a direction similar to that of other bodies within a given system, and is sometimes called direct motion. Retrograde motion is in the contrary direction. The word 'retrograde' derives from the Latin words retro, backwards, and gradus, step.

Two notations

The north orbital pole of a celestial body is defined by the right-hand rule: If you curve the fingers of your right hand along the direction of orbital motion, with your thumb extended parallel to the orbital axis, the direction your thumb points is defined to be north. Similarly, the north rotational pole of a body is defined by the direction of your thumb if you were to wrap your fingers around its equator in the direction it spins. There are two notations for retrograde motion that are mathematically equivalent: The body can be considered to orbit backwards, or it can be considered to orbit forwards, but with its orbit upside-down. For example, a moon in a retrograde orbit that is inclined from the pole of its planet by 10°, and with a 6-hour orbital period, could be said to have the orbital parameters of:
- 10° (rightside-up) and −6 h (backwards), in which case no inclination would ever exceed 90° (anything more than 90° would be upside-down), or of:
- 170° (upside-down) and +6 h (forwards), in which case no period would ever be negative. Similarly, a moon spinning backwards on an axis inclined by 10° from the axis of its orbit can instead be described as being flipped upside-down and spinning forwards. It is more common to keep the orbital or rotational period positive and let the inclination vary between 90° and 180° for retrograde motion, and between 0° and 90° for prograde motion, but when this inclination isn't listed, a negative orbital period is the only indication that an object is retrograde. (See natural satellite.)

Retrograde orbits

In the Solar system, most bodies orbit in a similar (prograde) direction to the rotation of the Sun. All planets and most smaller bodies orbit the Sun counterclockwise as seen from the [http://www.astro.uiuc.edu/~kaler/celsph.html north ecliptic pole] (which is in Draco, about 23° from the pole star, Polaris). The exceptions are mostly comets, which generally have highly disturbed orbits. Similarly, the larger and closer moons orbit their planet in the same direction as the planet's rotation, and so are also considered prograde. However, the gas giant planets have large numbers of small "irregular" moons in highly inclined or elliptical orbits, thought to be captured asteroids or Kuiper belt objects, or fragments thereof, and the majority of these are retrograde: 48 retrograde to 7 prograde for Jupiter, 18 to 8 for Saturn, and 8 to 1 for Uranus. One of the largest is the Saturnian moon Phoebe. Neptune is somewhat different: It seems to have captured its only surviving large moon, the retrograde but otherwise regular Triton, from the Kuiper Belt Object. The six irregular moons beyond Triton's orbit are evenly divided between prograde and retrograde; some of these may be original Neptunian moons whose orbits were distubed by Triton's capture, rather than being captured bodies themselves.

Retrograde rotation

Most planets, including Earth, spin in the prograde sense: That is, the north rotational pole and north orbital pole point in similar directions, more or less in the direction of the Solar north pole. The exceptions among the planets are Venus, Uranus and Pluto. Uranus rotates nearly on its side relative to its orbit. It has been described as having an axial tilt of 82° and a negative rotation of −17 hours, or, equivalently, of having an axis tilted at 98° and a positive rotation. Since current speculation is that Uranus started off with a typical prograde orientation and was knocked on its side by a large impact early in its history, it is most commonly described as having the higher axial tilt and positive rotation. (Since Uranus' moons are considered relative to Uranus itself, their description is unaffected by the choice made for the planet.) Retrograde Venus, on the other hand, has an axial tilt of less than 3°, and a very slow rotation of 243 days. Perhaps because it is easier to conceive of Venus as rotating slowly backwards than being 'upside down' relative to its near-twin Earth, but also because it is thought that an early massive impact may have resulted in Venus' current rotation while leaving its axis more or less unaffected, Venus is nearly always described as having its axis at 3° and a rotation of −243 days, rather than 187° and +243 days. When we observe the sky, we expect most objects to appear to move in a particular direction with the passing of time (diurnal motion). The apparent motion of most bodies in the sky is from east to west. However it is possible to observe a body moving west to east, such as an artificial satellite or Space Shuttle that is orbiting eastward (the preferred direction, because the rotation of the Earth assists in acquiring the required orbital speed). This orbit might be considered retrograde motion in this sense. However, as the Space Shuttle and such satellites you see going eastward would be seen orbiting the Earth counterclockwise if seen from the Pole Star, they are considered direct satellites. There are also artificial satellites which go clockwise as seen from the pole star; they are called retrograde satellites and you can see them in the sky going westward.

Retrogradation, or apparent retrograde motion

Retrograde motion should not be confused with retrogradation. The latter term is used in reference to the motion of the outer planets (Mars, Jupiter, Saturn, Neptune, Uranus, and Pluto). Though these planets appear to move from east to west on a nightly basis in response to the spin of Earth, they are most of the time drifting slowly eastward with respect to the background of stars, which can be observed by noting the position of these planets for several nights in a row. This motion is normal for these planets, so it is called direct motion (not retrograde). However, since Earth completes its orbit in a shorter period of time than these outer planets, we occasionally overtake an outer planet, like a faster car on a multiple-lane highway. When this occurs, the planet we are passing will first appear to stop its eastward drift, and it will then appear to drift back toward the west. This is retrogradation, since the planet seems to be moving in a direction opposite to that which is typical for planets. Finally as Earth swings past the planet in its orbit, it appears to resume its normal west-to-east drift on successive nights. Mars goes through retrogradation about every 25.7 months. The more distant outer planets retrograde more frequently. The period between such retrogradations is the synodic period of the planet. This retrogradation puzzled ancient astronomers, and was one reason why they named these bodies 'planets' which in Greek means 'wanderers'. In the geocentric model of the solar system, this motion was accounted for by having the planets travel in deferents and epicycles. In modern astronomy, the term retrograde motion refers to objects which are actually moving in a direction opposite that which is normal to spatial bodies within a given system, as opposed to merely observed phenomena (retrogradation) such as that described above.

Examples

Some significant examples of retrograde motion in the solar system:
- Venus rotates slowly in the retrograde direction.
- The moons Ananke, Carme, Pasiphaë and Sinope all orbit Jupiter in a retrograde direction. Many other minor moons of Jupiter orbit retrograde.
- The moon Phoebe orbits Saturn in a retrograde direction, and is thought to be a captured Kuiper belt object.
- The moon Triton orbits Neptune in a retrograde direction, and is also thought to be a captured Kuiper belt object.
- The planet Uranus has an axial tilt of 98°, which is near to 90°, and can be considered to be rotating in a retrograde direction depending on one's interpretation.

Reference

This article originated from Jason Harris' Astroinfo which comes along with KStars, a Desktop Planetarium for Linux/KDE. See http://edu.kde.org/kstars/index.phtml See also: Hipparchus, positional astronomy, Ptolemy Category:Astrodynamics Category:Celestial mechanics

Counterpoint

:This article is about the concept of counterpoint in music. For the Star Trek: Voyager episode of the same title, see Counterpoint (Voyager episode). Counterpoint is a musical technique involving the simultaneous sounding of separate musical lines. It is especially prominent in Western music. In all eras, writing of counterpoint has been subject to rules, sometimes strict. Counterpoint written before approximately 1600 is usually known as polyphony. The term comes from the Latin punctus contra punctum ("note against note"). The adjectival form contrapuntal shows this Latin source more transparently. By definition, chords occur when multiple notes sound simultaneously; however, chordal, harmonic, "vertical" features are considered secondary and almost incidental when counterpoint is the predominant textural element. Counterpoint focuses on melodic interaction rather than harmonic effects generated when melodic strands sound together. It was elaborated extensively in the Renaissance period, but composers of the Baroque period brought counterpoint to a kind of culmination, and it may be said that, broadly speaking, harmony then took over as the predominant organising principle in musical composition. The late Baroque composer Johann Sebastian Bach wrote most of his music incorporating counterpoint, and explicitly and systematically explored the full range of contrapuntal possibilities in such works as The Art of the Fugue. Given the way terminology in music history has evolved, such music created from the Baroque period on is described as contrapuntal, while music from before Baroque times is called polyphonic. Hence, the earlier composer Josquin Des Prez is said to have written polyphonic music. Homophony, by contrast with polyphony, features music in which chords or vertical intervals work with a single melody without much consideration of the melodic character of the added accompanying elements, or of their melodic interactions with the melody they accompany. As suggested above, most popular music written today is predominantly homophonic — governed by considerations of chord and harmony. But these are only strong general tendencies, and there are many qualifications one could add. The form or compositional genre known as fugue is perhaps the most complex contrapuntal convention. Other examples include the round (familiar in folk traditions) and the canon. In musical composition, counterpoint is an essential means for the generation of musical ironies; a melodic fragment, heard alone, may make a particular impression, but when it is heard simultaneously with other melodic ideas, or combined in unexpected ways with itself, as in a canon or fugue, surprising new facets of meaning are revealed. This is a means for bringing about development of a musical idea, revealing it to the listener as conceptually more profound than a merely pleasing melody.

Species counterpoint

Species counterpoint is a type of strict counterpoint, developed as a pedagagical tool, in which a student progresses through several "species" of increasing complexity, gradually attaining the ability to write free counterpoint according to the rules at the given time. The idea is at least as old as 1532, when Giovanni Maria Lanfraco described a similar concept in his Scintille de musica. The late 16th century Venetian theorist Zarlino elaborated on the idea in his influential Le institutioni harmoniche, and it was first presented in a codified form in 1619 by Lodovico Zacconi in his Prattica di musica. Zacconi, unlike later theorists, included a few extra contrapuntal techniques as species, for example invertible counterpoint. By far the most famous pedagogue to use the term, and the one who made it famous, was Johann Fux. In 1725 he published Gradus ad Parnassum (Step by Step Up Mount Parnassus) a work intended to help teach students how to compose, using counterpoint — specifically, the contrapuntal style as practiced by Palestrina in the late 16th century — as the principal technique. Fux described five species: #Note against note; #Two notes against one; #Four notes against one; #Notes offset against each other (as suspensions); #All the first four species together, as "florid" counterpoint.

Considerations for all species

Students of species counterpoint usually practice writing counterpoint in all the modes (Ionian, Dorian, Phrygian, Lydian, Mixolydian and Aeolian). The following rules apply to melodic writing in all species: #The counterpoint must begin and end on a perfect consonance. #The final must be approached by step. If approached from below, the leading tone must be raised, except in the case of the Phrygian mode. Thus, in the Dorian mode on D, a C# is necessary at the cadence. #Permitted melodic intervals are the perfect fourth, fifth, and octave, as well as the major and minor second, major and minor third, and ascending minor sixth. When the ascending minor sixth is used it must be immediately followed by motion downwards. #If writing two skips in the same direction--something which must be done only rarely--the second must be smaller than the first, and the interval between the first and the third note may not be dissonant. #If writing a skip in one direction, it is best to proceed after the skip with motion in the other direction. #Contrary motion should predominate. #The interval of a tenth should not be exceeded between the two parts, unless necessary. #The interval of a tritone in three notes is to be avoided (for example, a melodic motion F - A - B natural), as is the interval of a seventh in three notes.

First species

In first species counterpoint, each note in an added part
- (or parts) sounds against one note in the cantus firmus. Notes in all parts are sounded simultaneously, and move against each other simultaneously. The species is said to be expanded if any of the added notes is broken up (simply repeated). A few further rules given by Fux, by study of the Palestrina style, and usually given in the works of later counterpoint pedagogues, are as follows. Some are vague, and since good judgement and taste have been regarded by contrapuntists as more important than strict observance of mechanical rules, there are many more cautions than prohibitions. #Begin and end on either the unison, octave, or fifth, unless the added part is underneath, in which case begin and end only on unison or octave. #Use no unisons except at the beginning or end. #Avoid hidden or parallel fifths or octaves. #Attempt to keep the two parts within a tenth of each other, unless an exceptionally pleasing line can be written outside of that range. #Avoid moving in parallel thirds or sixths for too long. #Avoid having both parts move in the same direction by skip. #Attempt to have as much contrary motion as possible. In the following examples, all in two voices, the cantus firmus — the given part — is in the lower voice. The same cantus firmus is used for each, and each is in the Dorian mode. Dorian mode

Second species

In second species counterpoint, two notes in the added part (or parts) work against each longer note in the given part. The species is said to be expanded if one of the two shorter notes differs in length from the other. Additional considerations in second species counterpoint are as follows, and are in addition to the considerations for first species: #It is permissible to begin on an upbeat, leaving a half-rest in the added voice. #The accented beat must have only consonance (perfect or imperfect). The unaccented beat may have dissonance, but only as a passing tone, i.e. it must be approached and left by step in the same direction. #Avoid the interval of the unison except at the beginning or end of the example, except that it may occur on the unaccented portion of the bar. #Use caution with successive accented perfect fifths or octaves. They must not be used as part of a sequential pattern. Dorian mode

Third species

In third species counterpoint, four (or three) notes move against each longer note in the given part. As with second species, it is expanded if the shorter notes vary in length among themselves. Dorian mode

Fourth species

In fourth species counterpoint, a note is sustained or suspended in an added part while notes move against it in the given part, creating a dissonance, followed by the suspended note then changing (and "catching up") to create a subsequent consonance with the note in the given part as it continues to sound. Fourth species counterpoint is said to be expanded when the added-part notes vary in length from each other. The technique requires chains of notes sustained across the boundaries determined by beat, and so creates syncopation. syncopation

Florid counterpoint

In fifth species counterpoint, sometimes called florid counterpoint, the other four species of counterpoint are combined within the added part (or added parts). In the example, the first and second bars are second species, the third bar is third species, and the fourth and fifth bars are third and embellished fourth species. syncopation

General notes

It is a common and pedantic misconception that counterpoint is defined by these five species, and therefore anything that does not follow the strict rules of the five species is not counterpoint. This is not true; although much contrapuntal music of the common practice period indeed adheres to the rules, there are exceptions. Fux's book and its concept of "species" was purely a method of teaching counterpoint, not a definitive or rigidly prescriptive set of rules for it. He arrived at his method of teaching (or so he believed, at least) by examining the works of Giovanni Pierluigi da Palestrina, an important late 16th century composer and one who in Fux's time was held in the highest esteem as a contrapuntist. Works in the contrapuntal style of the 16th century—the "prima pratica" or "stile antico," it was called by modernist composers then—were often said by Fux's contemporaries to be in "Palestrina style." Indeed, Fux's treatise is a rather accurate compendeum of Palestrina's techniques.
- (Note: in counterpoint, the parts or individual melodic strands are often called voices, even if the music is thought of as instrumental.)

Contrapuntal derivations

Since the Renaissance period in European music, much music which is considered contrapuntal has been written in imitative counterpoint. In imitative counterpoint, two or more voices enter at different times, and (especially when entering) each voice repeats some version of the same melodic element. The fantasia, the ricercar, and later, the fugue (the contrapuntal form par excellence) all feature imitative counterpoint, which also frequently appears in choral works such as motets and madrigals. Imitative counterpoint has spawned a number of devices that composers have turned to in order to give their works both mathematical rigor and expressive range. Some of these devices include:
- Inversion: The inverse of a given fragment of melody is the fragment turned upside down – so if the original fragment has a rising major third (see interval), the inverted fragment has a falling major (or perhaps minor) third. (Compare, in twelve tone technique, the inversion of the tone row, which is the so-called prime series turned upside down.) In a completely separate sense, a contrapuntal inversion of melodies being simultaneously sounded by voices is the subsequent switching of the melodies between voices, so that for example an upper-voice melody is now sounded in some lower voice, and vice versa.
- Retrograde refers to the contrapuntal device whereby notes in an imitative voice sound backwards in relation to their order in the original.
- Retrograde inversion is where the imitative voice sounds notes both backwards and upside down.
- Augmentation is when in one of the parts in imitative counterpoint the notes are extended in duration compared to the rate at which they were sounded when introduced.
- Diminution is when in one of the parts in imitative counterpoint the notes are reduced in duration compared to the rate at which they were sounded when introduced.

Dissonant counterpoint

Dissonant counterpoint was first theorized by Charles Seeger as "at first purely a school-room discipline," consisting of species counterpoint but with all the traditional rules reversed. First species counterpoint is required to be all dissonances, establishing "dissonance, rather than consonance, as the rule," and consonances are "resolved" through a skip, not step. He wrote that "the effect of this discipline" was "one of purification." Other aspects of composition, such as rhythm, could be "dissonated" by applying the same principle (Charles Seeger, "On Dissonant Counterpoint," Modern Music 7, no. 4 (June-July 1930): 25-26). Seeger was not the first to employ dissonant counterpoint, but was the first to theorize and promote it. Other composers who have used dissonant counterpoint, if not in the exact manner prescribed by Charles Seeger, include Ruth Crawford-Seeger, Carl Ruggles, Dane Rudhyar, and Arnold Schoenberg.

External links

- [http://www.ntoll.org/interests/music/species/ A guide to species counterpoint]
- [http://www.musique.umontreal.ca/personnel/Belkin/bk.C/index.html Principles of Counterpoint]
- [http://www.o-art.org/history/early/Seeger.html On Dissonant Counterpoint by David Nicholls]
- [http://www.findarticles.com/cf_dls/m2298/2_17/61551810/p6/article.jhtml?term= Dane Rudhyar's Vision of American Dissonance by Carol J. Oja]
- [http://www.music.vt.edu/musicdictionary/textd/Dissonantcounterpoint.html Dissonant counterpoint examples and definition]
- [http://www.music.columbia.edu/~chris/ctrpnt.html De-Mystifying Tonal Counterpoint or How to Overcome Your Fear of Composing Counterpoint Exercises] by Christopher Dylan Bailey, composer at Columbia
- [http://www.greenwych.ca/musicmid.htm New Tonal Music composed with emphasis on counterpoint]
- [http://www.greenwych.ca/drone.htm Role of the drone in the evolution of counterpoint and harmony] Category:Counterpoint ko:대위법 ja:対位法

Musical set theory

Musical set theory is an atonal or post-tonal method of musical analysis and composition which is based on explaining and proving musical phenomena, taken as "sets" and subsets, using mathematical rules and notation and using that information to gain insight to compositions or their creation.

Mathematical set theory and musical set theory

Although musical set theory is often assumed to be the application of mathematical set theory to music, there is little coincidence between the terminology and even less between the methods of the two. In fact musical set theory is better characterized as the application of group theory and combinatorics to certain aspects of music theory. In musical set theory what is called a set is often in fact a tuple, an ordered collection of things (such as the term set form for tone row). Musical set theory also uses the terms linear and nonlinear for ordered and unordered sets, which has nothing to do with what these terms mean in mathematics. Allen Forte's book, The Structure of Atonal Music (ISBN 0300021208), one of the primary developments in musical set theory, is sometimes criticised for its supposedly faulty calculations and terminology. Musical set theory is best regarded however as an unrelated field from mathematical set theory, with its own vocabulary, whose only connection to mathematical set theory is in sometimes using the language of naive set theory to talk about finite sets.

Assumptions of atonal theory

In addition to octave and enharmonic equivalency assumed in twelve tone theory and equal tempered tonal theory, set theory also makes use of inversional and transpositional equivalency, though the degree of equivalency varies among theorists. Set theory does not, however, use diatonic functionality that is assumed in tonal theory, and this is the reason for the use of integer notation and modulo 12. Since the structures of tonal theory may then be constructed rather than assumed, tonal theory can be regarded as a specific area of atonal theory.

The set and set types

The fundamental concept of musical set theory is the set. A set is a collection of any musical materials or qualities, ordered or unordered, although most often sets of pitch classes are considered. Sets may be simultaneities or successions. A set is indicated by being enclosed in brackets: , an ordered set is indicated by <>, and an unordered set by (). Thus the set of pitch classes 0, 1, and 2 is , the ordered set <0,1,2>, and the unordered set (0,1,2). The domain of all pitch class sets may be partitioned into types or equivalence classes based on cardinality or number of pitch classes, or other criteria. There are thirteen cardinalities from 0-12: the null set, monad, dyad, trichord, tetrachord, pentachord, hexachord, septachord, octachord, nonachord, decachord, undecachord, and aggregate or dodecachord.

Basic operations

The basic operations that may be performed on a set are transposition and inversion and multiplication. Order operations include retrograde and rotation. Compound operations, the result of two basic operations, may be performed and the product of operations X and Y on z is written "Y(X(z))" with X performed on z, and then Y performed on that result. These operations may also be called transformations, mappings, morphisms, or permutations; and in music theory, but not in mathematics, derivations. Taking all combinations of a certain number of basic operations (for example taking all combinations of transposition, inversion, and multiplication by 7) produces permutation groups. Transposition is moving a set up or down in pitch by a constant interval. If x is the original pitch transposed n semitones, T_n=x+n (mod12). Inversion is turning a set upside-down reversing the order of the intervals between pitch classes. More specifically the compound operation transpositional inversion is T_nI(x)=-x+n (mod12). Multiplication is multiplying the pitch class numbers of a set, the most useful multipliers are 1, 5, 7, 11, as multiplication by 1 is the same, multiplication by 11 is inversion, multiplication of the chromatic scale by 5 produces the circle of fourths and multiplication by 7 produces the circle of fifths. Retrograde is reversing the order of the set so the first member is last and the last is first. Rotation is placing the last member of the set first.

Normal form

Another useful concept used in musical set theory is that of normal form. Since sets may be listed in any order without changing their identity, normal form is used as a way to compare sets (sometimes called normal order). Normal order is that which is stacked to the left, rises from left to right, within one octave and fits within the smallest interval. In the event of any ties for what produces the smallest outside interval, one compares the next most outside interval until the tie is broken, or the ordering that starts on the smallest pitch class integer is chosen. Normal order can be used to quickly compare if two sets may be transposed onto each other. For example, it is harder to compare and as quickly as and .

Transpositional and inversional types

Each of the cardinality types listed above may be further partitioned into transpositional type (T_n type) and/or inversional type (T_n/T_nI type). A list of all sets which are in the same transpositional type as a given set may be found by transposing the original set by all intervals. Thus the trichord , in normal form, is in the same transpositional type as +1=, +2=, , , , , , , , , and . All of the above are in the transpositional type _{T_n}, as the representative set is that which is in the most normal form. is equivalent under transposition and/or inversion with twenty four rather than twelve sets, the twelve above and their inversions. It happens to be the representative set for its class: _{T_n/T_nI}, as it is the most normal ordered form between the most normal ordered form uninverted, , and the most normal ordered transposition of its inversion, (T₇I=). Thus, to find the type of a set:
- List the set in normal form.
- Transpose the set so that the first pitch class is zero. This is the representative form of the T_n type.
- Perform T_nI and repeat the steps above. This is the representative form of the inversions T_n type.
- Compare the T_n type representative forms. The most normal form of the two representative types above is the representative form of the set's T_nI type. Given any set of numbers from zero to eleven, there is a corresponding indexing integer ranging from 0 to 4095, defined as the sum of the numbers 2ⁱ for each number i in the set. Transpositions, or transpositions with inversion, are examples of permutation groups. Given any such group on the numbers from 0 to 11, we can find a corresponding representative form by finding the smallest index in the orbit of the set under the transformations of the group.

Symmetry

The number of times which a set may be mapped onto itself through different operations is its degree of symmetry. Every set has at least one degree of symmetry, as it maps onto itself under the identity operation T₀. Transpositional symmetry is the property of set which maps onto itself for T_n where n does not equal 0. Inversional symmetry is the property of a set which maps into itself under T_nI. For any given T_n/T_nI type all sets will have the same degree of symmetry. The number of distinct sets in a type is 24 (the total number of operations, transposition and inversion, for n = 0 through 11) divided by the degree of symmetry of T_n/T_nI type. Inversionally symmetrical sets have a canonical ordering. The canonical ordering is the ordering such that the interval series of the set is its own retrograde or rather is retrograde symmetrical. For each of these canonical orderings a set will map onto itself under T_xI, x being the inversional index which is the sum of the first and last members of each canonical ordering. The first and last members, and each pair of members farther in, of sets of even cardinality will all equal the inverional index. For odd cardinality sets the middle number is a 1/2 index and the center of inversional symmetry. Transpositionally symmetrical sets in normal form may be partitioned into segments which under transposition map onto each other cyclically, so that the last segment maps onto the first.

Sums

Sums are also used in musical set theory. George Perle provides the following example: :"C-E, D-F♯, E♭-G, are different instances of the same interval… the other kind of identity… has to do with axes of symmetry. C-E belongs to a family of symmetrically related dyads as follows:" ::Axis pitches italicized, the axis is pitch class determined. Thus in addition to being part of the interval-4 family, C-E is also a part of the sum-2 family (with G♯ equal to 0). The tone row to Alban Berg's Lyric Suite,

\

, is a series of six dyads, all sum 11. If the row is rotated and retrograded, so it runs

\

, the dyads are all sum 6. ::Axis pitches italicized, the axis is dyad (interval 1) determined

Theorists and books

- John Rahn: Basic Atonal Theory (ISBN 0028731603)
- Allen Forte: Structure of Atonal Music (ISBN 0300021208)
- David Lewin: Musical Form and Transformation: 4 Analytic Essays (ISBN 0300056869), Generalized Musical Intervals and Transformations (ISBN 0300034938)
- Joseph N. Straus: Introduction to Post-Tonal Theory (ISBN 0130143316)
- George Perle: Twelve Tone Tonality (ISBN 0520033876)
- Important terms: pitch (music), Z-relation, interval vector, permutation, identity (music), identity function

External links

- [http://www.mta.ca/faculty/arts-letters/music/pc-set_project/pc-set_new/pages/introduction/toc.html A Brief Introduction to Pitch-Class Set Analysis]
- [http://www.sonic.mdx.ac.uk/research/nickpitch.html Nick Collins : Uniqueness of pitch class spaces, minimal bases and Z partners]
- [http://www.lsu.edu/faculty/jperry/virtual_textbook/20th_c_pitch_theory.htm Twentieth Century Pitch Theory: Some Useful Terms and Techniques]
- [http://music.theory.home.att.net/setheory.htm Introduction to Set Theory by Larry Solomon]
- [http://www.robertkelleyphd.com/atnltrms.htm Introduction to Post-Functional Music Analysis: Post-Functional Theory Terminology by Robert T. Kelley]
- [http://www.robertkelleyphd.com/12-tone.htm Introduction to Post-Functional Music Analysis: Set Theory, The Matrix, and the Twelve-Tone Method by Robert T. Kelley]
- [http://www.flexatone.net/athenaSCv.html SetClass View (SCv)] An athenaCL netTool for on-line, web-based pitch class analysis and reference.
-

Permutation (music)

In music and the terminology of the twelve tone technique a permutation is one of the many forms a tone row or twelve tone series. That is, prime form and any transposition, inversion, retrograde or retrograde-inversion, a total of 48 permutations. However, not all prime series will yield so many variations because tranposed and/or inverse transformations may be identical to each other, this being known as invariance. More generally permutation is any reordering of the original or prime form of an ordered set of pitch classes (DeLone et. al. (Eds.), 1975, chap. 6). In other words, the concept of permutation in music is the same as that of permutation in combinatorial mathematics except that the latter is applied not only to music but to other things as well. Reordering may be called an operation or transformation.

- Permutation
- Musical set theory

Reference

- DeLone et. al. (Eds.) (1975). Aspects of Twentieth-Century Music. Englewood Cliffs, New Jersey: Prentice-Hall. ISBN 0130493465, Ch. 6.

Rotation

:This article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation). rotation (disambiguation) rotation (disambiguation) Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. For a three-dimensional body, the rotation is around an axis — it amounts to rotation in each plane perpendicular to the axis around the intersection of the plane and the axis. An example of rotation of a planar figure around a point is the movement of the propeller of an aircraft. A door attached to the wall by two or more hinges rotates around the axis going through the hinges. hinge If the axis of rotation is within the body, the body is said to rotate upon itself, or spin. Among such rotations, the simplest case is that of constant angular frequency (see also below).

Allowed rotations

An object may allow rotation with respect to an attached other object by means of one or more hinges (e.g. a door, scissors, a hinge joint). Gear couplings and universal joints connect two driveshafts at an angle (rotating about their own axes) to transmit torque, by two pairs of hinges allowing rotation about two other axes. A ball-and-socket joint, e.g. in the shoulder, allows rotations about all three axes. See also: rolling, axle, wheel, rolling, gyration, gyroscope, rotations in anatomy

Mathematics

Mathematically, a rotation is a rigid body movement which keeps a point fixed; unlike a translation. This definition is applicable both for rotations in a plane (two dimensions) and in space (three dimensions). It turns out that a rotation in the three-dimensional space keeps fixed not just a single point, but rather an entire line; that is to say, any rotation in the three dimensional space is a rotation around an axis. This is a consequence of Euler's rotation theorem. Any rigid body movement is in fact either a rotation, or a translation, or a combination of the two. If one does a rotation around a point (axis), followed by another rotation around the same point (axis), the total result is yet another rotation. The reverse (inverse) of a rotation is also a rotation. It follows that the rotations around a point or axis form a group. If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may not be a rotation anymore. group Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as pitch, roll and yaw. See also: curl, cyclic permutation, Euler angles, rigid body, rotation around a fixed axis, rotation group, rotation matrix

Astronomy

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all rotate around their axes, while planets also rotate about a star such as the Sun, and moons also rotate about a planet. The motion of the components of galaxies is complex, but it usually includes a rotation component. One consequence of the rotation of a planet is the phenomenon of precession. Precession has the overall effect of introducing a long-term "wobble" in the movement of the axis of a planet. For example, the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is currently 66.5 degrees, but this angle has slowly changed over time due to the action of precession. See also: orbital period, oblate, orbital revolution

Physics

The speed of rotation is given by the angular frequency (rad/s) or frequency (turns/s, turns/min), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. The angular velocity vector also describes the direction of the axis of rotation. Similarly the torque is a vector. According to the right-hand rule, moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw. See also: rotational energy, angular momentum, angular velocity, centrifugal force, centripetal force, circular motion, circular orbit, Coriolis effect, spin, rigid body angular momentum

Amusement rides

Many amusement rides provide rotation. A Ferris wheel and observation wheel have a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centrifugal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where the centrifugal force keeps people in their seats.

External links

- [http://www.cut-the-knot.org/Curriculum/Geometry/RotationTransform.shtml Product of Rotations] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/Connes.shtml When a Triangle is Equilateral] at cut-the-knot Category:Euclidean geometry Category:Celestial mechanics ja:自転

Orbit

.]] :For other meanings of the term "orbit", see orbit (disambiguation) In physics, an orbit is the path that an object makes around another object while under the influence of a source of centripetal force, such as gravity.

History

Orbits were first analysed mathematically by Johannes Kepler who formulated his results in his laws of planetary motion. He found that the orbits of the planets in our solar system are elliptical, not circular (or epicyclic), as had previously been believed. Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies responding to the force of gravity were conic sections. Newton showed that a pair of bodies follow orbits of dimensions that are in inverse proportion to their masses about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.

Planetary orbits

Within a planetary system, planets, asteroids, comets and space debris orbit the central star in elliptical orbits. Any comet in a parabolic or hyperbolic orbit about the central star is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. To date, no comet has been observed in our solar system with a distinctly hyperbolic orbit. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about that planet. Due to mutual gravitational perturbations, the eccentricities of the orbits of the planets in our solar system vary over time. Pluto and Mercury have the most eccentric orbits. At the present epoch, Mars has the next largest eccentricity while the smallest eccentricities are those of the orbits of Venus and Neptune. As an object orbits another, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest from each other. In the elliptical orbit, the centre of mass of the orbiting-orbited system will sit at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in velocity. As a planet approaches apoapsis, the planet will decrease in velocity. See also: Kepler's laws of planetary motion

Understanding orbits

There are a few common ways of understanding orbits.
- As the object moves sideways, it falls toward the orbited object. However it moves so quickly that the curvature of the orbited object will fall away beneath it.
- A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
- As the object falls, it moves sideways fast enough (has enough tangential velocity) to miss the orbited object. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center. As an illustration of the orbit around a planet (eg Earth), the much-used cannon model may prove useful (see image below). Imagine a cannon sitting on top of a (very) tall mountain, which fires a cannonball horizontally. The mountain needs to be very tall, so that the cannon will be above the Earth's atmosphere and we can ignore the effects of air friction on the cannon ball. 300px If the cannon fires its ball with a low initial velocity, the trajectory of the ball will curve downwards and hit the ground (A). As the firing velocity is increased, the cannonball will hit the ground further (B) and further (C) away from the cannon, because while the ball is still falling towards the ground, the ground is curving away from it (see first point, above). If the cannonball is fired with sufficient velocity, the ground will curve away from the ball at the same rate as the ball falls - it is now in orbit (D). The orbit may be circular like (D) or if the firing velocity is increased even more, the orbit may become more (E) and more (F) elliptical. At a certain even faster velocity (called the escape velocity) the motion changes from an elliptical orbit to a parabola.

Newton's laws of motion

For a system of only two bodies that are only influenced by their mutual gravity, their orbits can be exactly calculated by Newton's laws of motion and gravity. Briefly, the sum of the forces will equal the mass times its acceleration. Gravity is proportional to mass, and falls off proportionally to the square of distance. To calculate, it is convenient to describe the motion in a coordinate system that is centered on the heavier body, and we can say that the lighter body is in orbit around the heavier body. An unmoving body that's far from a large object has more energy than one that's close. This is because it can fall farther. This is called "potential energy" because it is not yet actual. With two bodies, an orbit is a flat curve. The orbit can be open (so the object never returns) or closed (returning), depending on the total kinetic + potential energy of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity for that position, in the case of a closed orbit, always less. The path of a free-falling (orbiting) body is always a conic section. An open orbit has the shape of a hyperbola (or in the limiting case, a parabola); the bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This is often the case with comets that occasionally approach the Sun. A closed orbit has the shape of an ellipse (or in the limiting case, a circle). The point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part. Orbiting bodies in closed orbits repeat their path after a constant period of time. This motion is described by the empirical laws of Kepler, which can be mathematically derived from Newton's laws. These can be formulated as follows: # The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of the ellipse. Therefore the orbit lies in a plane, called the orbital plane. The point on the orbit closest to the attracting body is the periapsis. The point farthest from the attracting body is called the apoapsis. There are also specific terms for orbits around particular bodies; things orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or, synonymously, periselene and aposelene). An orbit around any star, not just the Sun, has a periastron and an apastron # As the planet moves around its orbit during a fixed amount of time, the line from Sun to planet sweeps a constant area of the orbital plane, regardless of which part of its orbit the planet traces during that period of time. This means that the planet moves faster near its perihelion than near its aphelion, because at the smaller distance it needs to trace a greater arc to cover the same area. This law is usually stated as "equal areas in equal time." # For each planet, the ratio of the 3rd power of its semi-major axis to the 2nd power of its period is the same constant value for all planets. Except for special cases like Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies. The 2-body solutions were published by Newton in Principia in 1687. In 1912, K. F. Sundman developed a converging infinite series that solves the 3-body problem, however it converges too slowly to be of much use. Instead, orbits can be approximated with arbitrarily high accuracy. These approximations take two forms. One form takes the pure elliptic motion as a basis, and adds perturbation terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moon, planets and other bodies are known with great accuracy, and are used to generate tables for celestial navigation. The differential equation form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces will equal the mass times its acceleration (F = ma). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial ones corresponds to solving an initial value problem. Numerical methods calculate the positions and velocities of the objects a tiny time in the future, then repeat this. However, tiny arithmetic errors from the limited accuracy of a computer's math accumulate, limiting the accuracy of this approach. Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large objects have been simulated.

Analysis of orbital motion

(see also orbit equation and Kepler's first law) To analyse the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the centre of force. In such coordinates the radial and transverse components of the acceleration are, respectively: :

\frac - r\left( \frac \right)^2

and :

\frac\frac\left( r^2\frac \right)

. Since the force is always radial, the transverse acceleration is zero, and it follows that: :

\frac = hu^2

, where h is a constant of integration and we have introduced the auxiliary variable u defined as 1/r. If magnitude of the radial force is f(r) per unit mass of the orbiting body, then the elimination of the time variable from the radial component of the equation of motion yields: :

\frac + u = \frac

. In the case of an inverse square force law the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The equation of the orbit described by the particle is thus: :

r = \frac = \frac

, where φ and e are constants of integration and L is the Semi-latus rectum. This can be recognised as the equation of a conic section in polar coordinates.

Orbital parameters

See: Orbital elements For a general elliptic orbit, the relations between the axis, eccentricity, and least and largest distance are: :Semimajor axis = (periapsis + apoapsis)/2 = mean of the extreme radii :Periapsis = semimajor axis × (1 - eccentricity) = least distance :Apoapsis = semimajor axis × (1 + eccentricity) = largest distance Note that there are alternative definitions for a "mean radius" or "average distance": if you average the radius over time for one orbit (mean anomaly), or over the orbital angle as observed by the primary (true anomaly), then you get a different result. See here for details.

Orbital period

See: orbital period

Orbital decay

If some part of a body's orbit enters an atmosphere, its orbit can decay because of drag. At each periapsis, the object scrapes the air, losing energy. Each time, the orbit grows less eccentric (more circular) because the object loses kinetic energy precisely when that energy is at its maximum. Eventually, the orbit circularises and then the object spirals into the atmosphere. The bounds of an atmosphere vary wildly. During solar maxima, the Earth's atmosphere causes drag up to a hundred kilometres higher than during solar minimums. Some satellites with long conductive tethers can also decay because of electromagnetic drag from the Earth's magnetic field. Basically, the wire cuts the magnetic field, and acts as a generator. The wire moves electrons from the near vacuum on one end to the near-vacuum on the other end. The orbital energy is converted to heat in the wire. Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails. These forms of propulsion require no propellant or energy input, and so can be used indefinitely. See statite for one such proposed use. Orbital decay can also occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges in the primary, and since below the synchronous orbit the orbiting object is moving faster than the body's surface the bulges lag a short angle behind it. The gravity of the bulges is slightly off of the primary-satellite axis and thus has a component along the satellite's motion. The near bulge slows the object more than the far bulge speeds it up, and as a result the orbit decays. Conversely, the gravity of the satellite on the bulges applies torque on the primary and speeds up its rotation. Artificial satellites are too small to have an appreciable tidal effect on the planets they orbit, but several moons in the solar system are undergoing orbital decay by this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years. Finally, orbits can decay via the emission of gravitational waves. This mechanism is extremely weak for most stellar objects, only becoming significant in cases where there is a combination of extreme mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.

Earth orbits

See Earth orbit for more details.
- Low Earth orbit
- High Earth Orbit
- Intermediate circular orbit
- Geostationary orbit
- Geosynchronous orbit
- Geostationary transfer orbit
- Molniya orbit
- Polar orbit
- Polar Sun Synchronous Orbit (this is not a complete list).

Scaling in gravity

The gravitational constant G is defined to be:
- 6.6742 × 10⁻¹¹ N·m²/kg²
- 6.6742 × 10⁻¹¹ m³/(kg·s²)
- 6.6742 × 10⁻¹¹(kg/m³)^-1s^-2. Thus the constant has dimension density^-1 time^-2. This corresponds to the following properties. Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar orbits without scaling the time: if for example distances are halved, masses are divided by 8, gravitational forces by 16 and gravitational accelerations by 2. Hence orbital periods remain the same. Similarly, when an object is dropped from a tower, the time it takes to fall to the ground remains the same with a scale model of the tower on a scale model of the earth. When all densities are multiplied by four, orbits are the same, but with orbital velocities doubled. When all densities are multiplied by four, and all sizes are halved, orbits are similar, with the same orbital velocities. These properties are illustrated in the formula :

GT^2 \sigma = 3\pi \left( \frac \right)^3,

for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, where T is the orbital period.

Role in the evolution of atomic theory

When atomic structure was first probed experimentally early in the twentieth century, an early picture of the atom portrayed it as a miniature solar system bound by the coulomb force rather than by gravity. This was inconsistent with electrodynamics and the model was progressively refined as quantum theory evolved, but there is a legacy of the picture in the term orbital for the wave function of an energetically bound electron state.

External links

- An on-line orbit plotter: http://www.bridgewater.edu/departments/physics/ISAW/PlanetOrbit.html
- [http://www.braeunig.us/space/orbmech.htm Orbital Mechanics] (Rocket and Space Technology) Category:Celestial mechanics Category:Solar System als:Umlaufbahn ja:軌道 (力学) simple:Orbit th:วงโคจร

Right-hand rule

:The right hand rule is also an algorithm used to solve Mazes In mathematics and physics, the right-hand rule is a convention for determining relative directions of certain vectors. In fact, there are two closely related right-hand rules.

Direction associated with an ordered pair of directions

vector The first of these occurs in situations in which a non-commutative operation must be performed on two directions a and b (in a three-dimensional space) that constructs a direction c perpendicular to both a and b. There are in fact two such directions. The right-hand rule imposes the following procedure for choosing one of the two directions. First, the hand is held flat and positioned so that the fingers are aligned with a. Then, the hand is rotated about the forearm so that the fingers curl inward toward b. The thumb indicates c. (There is also an alternative technique. First, the forefinger of the right hand is pointed directly forward, and the entire hand positioned so that the forefinger is aligned with a. Then, the middle finger is turned inward (toward the palm), and the hand is turned about the axis defined by a so that the middle finger aligns with b. The thumb indicates c.)

Direction associated with a rotation

The other form of the right-hand rule occurs in situations where a direction c must be determined based on a rotational direction, or vice versa. In this case, the fingers of the right hand are curled in the rotational direction, and the thumb indicates c. Correspondingly:
- moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw
- leftward direction is associated with the rotation of the wheels of a vehicle moving forward The relation with the previous section is established by associating with directions a and b the rotation over the shortest angle from a to b.

Applications of the right-hand rule

Perhaps the most fundamental application of the right-hand rule is the Cartesian coordinate system, where the first form is used to position the z-axis once the x- and y-axes have been determined. The first form of the rule is also used to determine the direction of the cross product of two vectors. This leads to widespread use in physics, wherever the cross product occurs. A list of physical quantities whose directions are related by the right-hand rule is given below. (Some of these are related to cross products only indirectly, and use the second form.)
- The angular velocity of a rotating object and the rotational velocity of any point on the object
- A torque, the force that causes it, and the position of the point of application of the force
- A magnetic field, the position of the point where it is determined, and the electric current (or change in electric flux) that causes it
- A magnetic field in a coil of wire and the electric current in the wire
- The force of a magnetic field on an object, the magnetic field itself, and the velocity of the object

Left-hand rule

Left-handed materials are metamaterials which have a negative refractive index. The term "left-handed material" was coined by a prediction of Russian theorist V. G. Veselago in 1968.

External links

[http://physics.syr.edu/courses/video/RightHandRule/index2.html demonstration] Category:Mechanics

Equator

The equator is an imaginary circle drawn around a planet (or other astronomical object) at a distance halfway between the poles. The equator divides the planet into a Northern Hemisphere and the Southern Hemisphere. The latitude of the equator is, by definition, 0°. The length of Earth's equator is about 40,075.0 km, or 24,901.5 miles. The equator is one of the five main circles of latitude based on the relationship of the Earth's rotation and plane of orbit around the sun. Additionally, the equator is the only line of latitude which is also a great circle The Sun, in its seasonal movement through the sky, passes directly over the equator twice each year on the Vernal and Autumnal Equinoxes, which occur in March and September (respectively). At the equator, the rays of the sun are perpendicular to the surface of the earth on these dates. Places near the equator experience the quickest rates of sunrise and sunset in the world, taking minutes. Such places also have a relatively constant amount of day/night time on every day throughout the year compared with more northerly or southerly places.

Equatorial climate

In many tropical regions people identify two seasons, wet and dry, but most places very close to the equator are wet throughout the year, although seasons can vary depending on a variety of factors including elevation and proximity to an ocean. ocean The surface of the Earth at the equator is mainly ocean. The highest point on the Equator is 4,690 m, at 77° 59' 31" W on the south slopes of Volcán Cayambe (summit 5,790 m) in Ecuador. This is a short distance above the snow line, and is the only point on the Equator where snow lies on the ground (Google Earth satellite data and photos).

Equatorial countries

The equator traverses the land and/or water of 13 countries in total:
- São Tomé and Príncipe - passing through Ilhéu das Rolas, an islet in this archipelago
- Gabon
- Republic of the Congo
- Democratic Republic of Congo
- Uganda
- Kenya
- Somalia
- Maldives - misses every island, passing between Gaafu Dhaalu Atoll and Gnaviyani Atoll
- Indonesia
- Sumatra - also small islands Tanah Masa to the West and Lingga to the East
- Borneo - Kalimantan
- Sulawesi
- Halmahera - also small islands Kayoa to the West and Gebe to the East
- Kawe, a small island near Waigeo - and other islets throughout Indonesia
- Kiribati - misses every island
- Gilbert Islands - passing between Aranuka and Nonouti Atolls
- Line Islands - passing between Kiritimati Island and Malden Island, though neither is very close to the equator
- Ecuador
- Galapagos Islands - passing through Isabela Island.
- Mainland Ecuador
- Colombia
- Brazil

Solar system

The solar system comprises our Sun and the retinue of celestial objects gravitationally bound to it. Traditionally, this is said to consist of the Sun, nine planets and their 158 currently known moons; however, a large number of other objects, including asteroids, meteoroids, planetoids, comets, and interplanetary dust, orbit the Sun as well. Although the term "solar system" is frequently applied to other star systems and the planetary systems which may comprise them, it should strictly refer to our system specifically: the word "solar" is derived from the Sun's Latin name, Sol (and the term sometimes appears as Solar System). When talking about another stellar system (or planetary system), including the star(s) and bodies associated with them through gravity, it is usual to shorten it to "the system" (e.g. "the Alpha Centauri system" or "the 51 Pegasi system").

Structure and layout of the solar system

The Sun (astronomical symbol ☉) is a main sequence G2 star that contains 99.86% of the system's known mass. Its two largest orbiting bodies, Jupiter and Saturn, account for 91% of the remainder (The Oort Cloud might hold a substantial percentage, but as yet its existence is unconfirmed). In broad terms, the charted regions of our solar system consist of the Sun and its planetary system: the eight bodies in relatively unique orbits (commonly called planets or major planets) and two belts of smaller objects (which can be called minor planets, planetoids, meteoroids, planetesimals or, in the case of Pluto, planets). Objects in orbit round the Sun all lie within the same shallow plane, called the ecliptic, and all orbit in the same direction. Many are in turn orbited by moons, and the largest are encircled by planetary rings of dust and other particles. The major planets are, in order, Mercury (☿), Venus (♀), Earth (♁), Mars (♂), Jupiter (♃), Saturn (♄), Uranus (♅/10px), Neptune (♆), and Pluto (♇), though Pluto's status has been thrown into question by the discovery of (see below). Eight of the nine planets are named after or derived from gods and goddesses from Greco-Roman mythology; Earth, a Germanic word, is known in many Romance languages as Terra, the Roman goddess of the Earth. Distances within the solar system are measured most often in astronomical units, or AU. 1 AU is the distance between the Earth and the Sun, or 149 598 000 kilometers. Pluto is roughly 38 AU from the Sun, while Jupiter lies at roughly 5.2 AU. For very large distances within the solar system, such as regions beyond Pluto or the orbital circumferences of planets, the terameter (Tm, one milliard kilometers) is sometimes used. Despite the fact that many diagrams (like the image at the top of this article), for practicality's sake, represent the solar system as having each orbit the same distance apart, in actuality the orbits are largely arranged geometrically, that is, each is roughly double the distance from the Sun as the one before it. Venus’s distance from the Sun is roughly double that of Mercury, Earth’s distance is roughly double that of Venus, Mars’s double that of Earth and so on. This relationship is roughly expressed in the Titius-Bode law, a mathematical formula for predicting the semi-major axes of planets in AU. In its simplest form, it is written :

a= 0.4 + 0.3\times k

where k=0,1,2,4,8,16,32,64,128. By this formulation, we would expect Mercury's orbit (k=0) to be 0.4 AU, and Mars's orbit (k=4) to be at 1.6 AU. In fact their orbits are 0.38 and 1.52 AU.Ceres, the largest asteroid, lies at k=8. This law is only a rough guide, and doesn't fit all of the planets (Neptune is far closer than predicted, though Pluto lies at Neptune's predicted orbit). As of now, there is no scientific explanation for why this law "works," and many claim it is merely a coincidence. Pluto

Origin and evolution of the solar system

The current hypothesis of solar system formation is the nebular hypothesis, first proposed in 1755 by Immanuel Kant. It states the solar system was formed from a gaseous cloud called the solar nebula. It had a diameter of 100 AU and was 2-3 times the mass of the Sun. Over time, the nebula began to collapse, possiby due to disturbance by a nearby supernova. This explosion sent shock waves into space, which squeezed the nebula, pushing more and more matter inward until gravitational forces overcame its internal gas pressure and it also began to collapse. As the nebula collapsed, it decreased in size, which in turn caused it to spin faster to conserve angular momentum. And as the competing forces associated with gravity, gas pressure, magnetic fields, and rotation acted on it, the contracting nebula began to flatten into a spinning pancake shape with a bulge at the center. When the nebula further condensed, a protostar was formed in the middle. This system was heated by the friction of the rocks colliding into each other. Lighter elements such as hydrogen and helium evaporated out of the centre and migrated to the edges of the disc, thus concentrating the heavier elements to form dust and rocks in the centre. These heavier elements clumped together to form planetesimals and protoplanets. In the outer regions of this solar nebula, ice and volatile gases were able to survive, and as a result, the inner planets are rocky and the outer planets were massive enough to capture large amounts of lighter gases, such as hydrogen and helium. After 100 million years, the pressures and densities of hydrogen in the centre of the collapsed nebula became great enough for the protosun to sustain thermonuclear fusion reactions. As a result of this, hydrogen was converted to helium, and a great amount of heat was released. 4×¹H → ⁴He + neutrinos + photons During that time, the protostar turned into the Sun and the protoplanets and planetesimals were transformed into planets. All of the planets formed in a relatively short time of a few million years.

Regions of the solar system

protostar's rotating magnetic field on the plasma in the interplanetary medium (Solar Wind) [http://quake.stanford.edu/~wso/gifs/HCS.html]. (click to enlarge) ]] According to their location, the objects in the solar system are divided into three zones: Zone I or the inner solar system, including terrestrial planets and the Main belt of asteroids; Zone II, including the giant planets, their satellites and the centaurs, and Zone III, or the outer solar system, comprising the area of the Trans-Neptunian objects including the Kuiper Belt, the Oort cloud, and the vast region in between.

Interplanetary medium

The environment in which the solar system resides is called the interplanetary medium. The Sun radiates a continuous stream of charged particles, a plasma known as solar wind, which forms a very tenuous "atmosphere" (the heliosphere), permeating the interplanetary medium in all directions for at least ten billion (10) miles (16 Tm or 16 km) into space. Small quantities of dust are also present in the interplanetary medium and are responsible for the phenomenon of zodiacal light. Some of the dust is likely interstellar dust from outside the solar system. The influence of the Sun's rotating magnetic field on the interplanetary medium creates the largest structure in the Solar System, the heliospheric current sheet.

The inner planets

The four inner or terrestrial planets are characterised by their dense, rocky makeup. They formed in the hotter regions close to the Sun, where lighter and more volatile materials evaporated, leaving only those with high melting points, such as silicates, which form the planets' solid crusts and semi-liquid mantles, and iron, which forms their cores. All have impact craters and many possess tectonic surface features, such as rift valleys and volcanoes. The four inner planets are: volcanoes
- Mercury (0.39 AU from the Sun): The closest planet to the Sun is also the smallest and most atypical of the inner planets, having no atmosphere and, to date, no observed geological activity save that produced by impacts. Its relatively large iron core suggests that it was once a much larger world whose outer mantle was sheared off in early formation by the Sun’s gravity.
- Venus (0.72 AU): The first truly terrestrial planet, Venus, like the Earth, possesses a thick silicate mantle around an iron core, as well as a substantial atmosphere and evidence of one-time internal geological activity, such as volcanoes. It is much drier than Earth, and its atmosphere is 90 times as dense as Earth’s, however, and composed overwhelmingly of carbon dioxide with traces of sulfuric acid.
- Earth/Moon (1 AU): The largest of the inner planets, Earth is also the only one to demonstrate unequivocal evidence of ongoing geological activity. Its liquid hydrosphere, unique among the terrestrials, is probably the reason why Earth is also the only planet where multi-plate tectonics has been observed, since water acts as a lubricant for subduction. Its atmosphere is radically different from the other terrestrials, having been altered by the presence of life to contain 21 percent free oxygen. Its satellite, the Moon, is sometimes considered a terrestrial planet in a co-orbit with its partner, since its orbit around the Sun never actually loops back on itself when observed from above. The Moon possesses many of the features in common with other terrestrial planets, though it lacks an iron core.
- Mars (1.5 AU): Smaller than the Earth or Venus, Mars possesses a tenuous atmosphere of carbon dioxide. Its surface, peppered with vast volcanoes and rift valleys such as Valles Marineris, shows that it was once geologically active and [http://www.universetoday.com/am/publish/mars_volcanoes_active.html recent evidence] suggests it may have continued to be so until very recently. Mars possesses two tiny moons thought to be captured asteroids.

The asteroid belt

Asteroids are objects smaller than planets that mostly occupy the orbit between Mars and Jupiter, between 2.3 and 3.3 AU from the Sun, and are composed in significant part of non-volatile minerals. The main belt contains tens of thousands (possibly millions) over 1 km across, though they can be as small as dust. Despite their large numbers, the total mass of the main asteroid belt is unlikely to be more than a thousandth that of the Earth. Asteroids with a diameter of less than 50 m are called meteoroids. The largest asteroid, Ceres, has a diameter of roughly 1000 km; large enough to be spherical, which would make it a planet by some definitions of the word. The asteroids are thought to be the remnants of a small terrestrial planet that failed to coalesce due to the gravitational interference of Jupiter. They are subdivided into asteroid groups and families based on their specific orbital characteristics. Asteroid moons are asteroids that orbit larger asteroids. They are not as clearly distinguished as planetary moons, sometimes being almost as large as their partners. Trojan asteroids are located in either of Jupiter's L₄ or L₅ points, though the term is also sometimes used for asteroids in any other planetary Lagrange point as well. The inner solar system is dusted with rogue asteroids, many of which cross the orbits of the inner planets.

The outer planets

The four outer planets, or gas giants, (sometimes called Jovian planets) are so large they collectively make up 99 percent of the mass known to orbit the Sun. Their large sizes and distance from the Sun meant they could hold on to much of the hydrogen and helium too light for the smaller and hotter terrestrial planets to retain.
- Jupiter (5.2 AU), at 318 Earth masses, is 2.5 times the mass of all the other planets put together. Its composition of largely hydrogen and helium is not very different from that of the Sun. Three of its 63 satellites, Ganymede, Io and Europa, share elements in common with the terrestrial planets, such as volcanism and internal heating. Jupiter has a faint, smoky ring.
- Saturn (9.5 AU), famous for its extensive ring system, shares many qualities in common with Jupiter, including its atmospheric composition, though it is far less massive, being only 95 Earth masses. Two of its 49 moons, Titan and Enceladus, show signs of geological activity, though they are largely made of ice. Titan is the only satellite in the solar system with a substantial atmosphere.
- Uranus (19.6 AU) and Neptune (30 AU), while having many characteristics in common with the other gas giants, are nonetheless more similar to each other than they are to Jupiter or Saturn. They are both substantially smaller, being only 14 and 17 Earth masses, respectively. Their atmospheres contain a smaller percentage of hydrogen and helium, and a higher percentage of “ices”, such as water, ammonia and methane. For this reason some astronomers suggested that they belong in their own category, “Uranian planets,” or “ice giants.” Both planets possess dark, insubstantial ring systems. Neptune’s largest moon Triton is geologically active. Centaurs are icy comet-like bodies that have less-eccentric orbits so that they remain in the region between Jupiter and Neptune. The first centaur to be discovered, 2060 Chiron, has been called a comet since it has been shown to develop a tail, or coma, just as comets do when they approach the sun.

The trans-Neptunian region

The area beyond Neptune, often referred to as the outer solar system or simply the "trans-Neptunian region", is still largely unexplored.

The Kuiper belt

This region's first formation, which actually begins inside the orbit of Neptune, is the Kuiper belt, a great ring of debris, similar to the asteroid belt but composed mainly of ice and far greater in extent, which lies between 30 to 50 AU from the Sun. This region is thought to be the place of origin for short-period comets, such as Halley's comet. Though there are estimated to be over 70,000 Kuiper belt objects with a diameter greater than 100 km, the total mass of the Kuiper belt is relatively low, perhaps equalling or just exceeding the mass of the Earth. Many Kuiper belt objects have orbits that take them outside the plane of the ecliptic.
- Pluto, the solar system's smallest planet, is considered to be part of the Kuiper Belt population. Like others in the belt, it has a relatively eccentric orbit inclined 17 degrees to the ecliptic and ranging from 29.7 AU from the Sun at perihelion to 49.5 AU at aphelion. It has a large moon (the largest in the solar system relative to its own size), called Charon, and, new observations suggest, two other, much smaller moons. Like the Earth/Moon, Pluto and Charon are often considered a double planet. A member of the traditional nine planets, Pluto's tiny mass (less than 1% of Earth's) and diameter have called this status into question. Kuiper belt objects with Pluto-like orbits are called Plutinos. Other Kuiper belt objects have resonant orbits and are grouped accordingly. The remaining Kuiper belt objects, in more "classical" orbits, are classified as Cubewanos. The Kuiper Belt has a very sharply defined edge. At around 49 AU, a sharp dropoff occurs in the number of objects observed. This dropoff is known as the "Kuiper Cliff", and as yet its cause is unknown. Some speculate that something must exist beyond the belt large enough to sweep up the remaining debris, perhaps as large as Earth or Mars. This view is still controversial, however.

The scattered disc

Overlapping the Kuiper belt but extending much further outwards is the scattered disc. Scattered disc objects are believed to have been originally native to the Kuiper belt, but were ejected into erratic orbits in the outer fringes. One particular scattered disc object, originally found in 2003 but confirmed two years later by Mike Brown, has renewed the old debate about what constitutes a planet since, though its size is not yet known, it is almost certainly larger than Pluto. It currently has no name, but has been given the provisional designation , and has been nicknamed "Xena" by its discoverers, after the television character. It has many similarities with Pluto: its orbit is highly eccentric, with a perihelion of 38.2 AU (roughly Pluto's distance from the Sun) and an aphelion of 97.6 AU, and is steeply inclined to the ecliptic plane, indeed, at 44 degrees, more so than any known object in the solar system. Like Pluto, it is believed to consist largely of rock and ice, and has a [http://www.gps.caltech.edu/%7Embrown/planetlila/moon/index.html moon]. Whether it and the largest Kuiper belt objects should be considered planets or whether instead Pluto should be reclassified as a minor planet has not yet been resolved.

A new region?

Sedna, the newly discovered Pluto-like object with a gigantic, highly elliptical 10,500-year orbit that takes it from about 76 to 928 AU, has too distant a perihelion to be a scattered member of the Kuiper Belt and could be the first in an entirely new population. is also believed to be a member of this population.

Comets

Comets are composed largely of volatile ices and have highly eccentric orbits, generally having a perihelion within the orbit of the inner planets and an aphelion far beyond Pluto. Short-period comets exist with apoapses closer than this, however, and old comets that have had most of their volatiles driven out by solar warming are often categorized as asteroids. Long period comets have orbits lasting thousands of years. Some comets with hyperbolic orbits may originate outside the solar system.

And beyond

The point at which the solar system ends and interstellar space begins is not precisely defined, since its outer boundaries are delineated by two separate forces: the solar wind and the Sun's gravity. gravity The heliosphere expands outward in a great bubble to about 95 AU, or three times the orbit of Pluto. The edge of this bubble is known as the termination shock; the point at which the solar wind collides with the opposing winds of the interstellar medium. Here the wind slows, condenses and becomes more turbulent, forming a great oval structure known as the heliosheath that looks and behaves very much like a comet's tail; extending outward for a further 40 AU at its stellar-windward side, but tailing many times that distance in the opposite direction. The outer boundary of the sheath, the heliopause, is the point at which the solar

Retrograde motion

Two notations

Retrograde orbits

Retrograde rotation

Retrogradation, or apparent retrograde motion

Examples

Reference

Counterpoint

Species counterpoint

Considerations for all species

First species

Second species

Third species

Fourth species

Florid counterpoint

General notes

Contrapuntal derivations

Dissonant counterpoint

External links

Musical set theory

Mathematical set theory and musical set theory

Assumptions of atonal theory

The set and set types

Basic operations

Normal form

Transpositional and inversional types

Symmetry

Sums

Theorists and books

External links

Permutation (music)

Related article

Reference

Rotation

Allowed rotations

Mathematics

Astronomy

Physics

Amusement rides

External links

Orbit

History

Planetary orbits

Understanding orbits

Newton's laws of motion

Analysis of orbital motion

Orbital parameters

Orbital period

Orbital decay

Earth orbits

Scaling in gravity

Role in the evolution of atomic theory

See also

External links

Right-hand rule

Direction associated with an ordered pair of directions

Direction associated with a rotation

Applications of the right-hand rule

Left-hand rule

See also

External links

Equator

Equatorial climate

Equatorial countries

See also

Solar system

Structure and layout of the solar system

Origin and evolution of the solar system

Regions of the solar system

Interplanetary medium

The inner planets

The asteroid belt

The outer planets

The trans-Neptunian region

The Kuiper belt

The scattered disc

A new region?

Comets

And beyond