Ecliptic coordinate system

The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions and orbits of Solar System objects. Because most planets (except Mercury) and many small Solar System bodies have orbits with slight inclinations to the ecliptic, using it as the fundamental plane is convenient. The system's origin can be the center of either the Sun or Earth, its primary direction is towards the vernal (northward) equinox, and it has a right-hand convention. It may be implemented in spherical or rectangular coordinates.[1]

Ecliptic grid globe
Earth-centered ecliptic coordinates as seen from outside the celestial sphere. Ecliptic longitude (red) is measured along the ecliptic from the vernal equinox. Ecliptic latitude (yellow) is measured perpendicular to the ecliptic. A full globe is shown here, although high-latitude coordinates are seldom seen except for certain comets and asteroids.

Primary direction

Ecliptic vs equator small
The apparent motion of the Sun along the ecliptic (red) as seen on the inside of the celestial sphere. Ecliptic coordinates appear in (red). The celestial equator (blue) and the equatorial coordinates (blue), being inclined to the ecliptic, appear to wobble as the Sun advances.

The celestial equator and the ecliptic are slowly moving due to perturbing forces on the Earth, therefore the orientation of the primary direction, their intersection at the Northern Hemisphere vernal equinox, is not quite fixed. A slow motion of Earth's axis, precession, causes a slow, continuous turning of the coordinate system westward about the poles of the ecliptic, completing one circuit in about 26,000 years. Superimposed on this is a smaller motion of the ecliptic, and a small oscillation of the Earth's axis, nutation.[2][3]

In order to reference a coordinate system which can be considered as fixed in space, these motions require specification of the equinox of a particular date, known as an epoch, when giving a position in ecliptic coordinates. The three most commonly used are:

Mean equinox of a standard epoch
(usually the J2000.0 epoch, but may include B1950.0, B1900.0, etc.) is a fixed standard direction, allowing positions established at various dates to be compared directly.
Mean equinox of date
is the intersection of the ecliptic of "date" (that is, the ecliptic in its position at "date") with the mean equator (that is, the equator rotated by precession to its position at "date", but free from the small periodic oscillations of nutation). Commonly used in planetary orbit calculation.
True equinox of date
is the intersection of the ecliptic of "date" with the true equator (that is, the mean equator plus nutation). This is the actual intersection of the two planes at any particular moment, with all motions accounted for.

A position in the ecliptic coordinate system is thus typically specified true equinox and ecliptic of date, mean equinox and ecliptic of J2000.0, or similar. Note that there is no "mean ecliptic", as the ecliptic is not subject to small periodic oscillations.[4]

Spherical coordinates

Summary of notation for ecliptic coordinates[5]
Spherical Rectangular
Longitude Latitude Distance
Geocentric λ β Δ
Heliocentric l b r x, y, z[note 1]
  1. ^ Occasional use; x, y, z are usually reserved for equatorial coordinates.
  1. ^ Occasional use; x, y, z are usually reserved for equatorial coordinates.
Ecliptic longitude
Ecliptic longitude or celestial longitude (symbols: heliocentric l, geocentric λ) measures the angular distance of an object along the ecliptic from the primary direction. Like right ascension in the equatorial coordinate system, the primary direction (0° ecliptic longitude) points from the Earth towards the Sun at the vernal equinox of the Northern Hemisphere. Because it is a right-handed system, ecliptic longitude is measured positive eastwards in the fundamental plane (the ecliptic) from 0° to 360°. Because of axial precession, the ecliptic longitude of most "fixed stars" (referred to the equinox of date) increases by about 50.3 arcseconds per year, or 83.8 arcminutes per century, the speed of general precession.[6][7] However, for stars near the ecliptic poles, the rate of change of ecliptic longitude is dominated by the slight movement of the ecliptic (that is, of the plane of the earth's orbit), so the rate of change may be anything from minus infinity to plus infinity depending on the exact position of the star.
Ecliptic latitude
Ecliptic latitude or celestial latitude (symbols: heliocentric b, geocentric β), measures the angular distance of an object from the ecliptic towards the north (positive) or south (negative) ecliptic pole. For example, the north ecliptic pole has a celestial latitude of +90°. Ecliptic latitude for "fixed stars" is not affected by precession.
Distance
Distance is also necessary for a complete spherical position (symbols: heliocentric r, geocentric Δ). Different distance units are used for different objects. Within the Solar System, astronomical units are used, and for objects near the Earth, Earth radii or kilometers are used.

Historical use

From antiquity through the 18th century, ecliptic longitude was commonly measured using twelve zodiacal signs, each of 30° longitude, a practice that continues in modern astrology. The signs approximately corresponded to the constellations crossed by the ecliptic. Longitudes were specified in signs, degrees, minutes, and seconds. For example, a longitude of ♌ 19° 55′ 58″ is 19.933° east of the start of the sign Leo. Since Leo begins 120° from the vernal equinox, the longitude in modern form is 139° 55′ 58″.[8]

In China, ecliptic longitude is measured using 24 Solar terms, each of 15° longitude, and are used by Chinese lunisolar calendars to stay synchronized with the seasons, which is crucial for agrarian societies.

Rectangular coordinates

Heliocentric rectangular ecliptic
Heliocentric ecliptic coordinates. The origin is the Sun's center, the plane of reference is the ecliptic plane, and the primary direction (the x-axis) is the vernal equinox. A right-handed rule specifies a y-axis 90° to the east on the fundamental plane. The z-axis points toward the north ecliptic pole. The reference frame is relatively stationary, aligned with the vernal equinox.

A rectangular variant of ecliptic coordinates is often used in orbital calculations and simulations. It has its origin at the center of the Sun (or at the barycenter of the Solar System), its fundamental plane on the ecliptic plane, and the x-axis toward the vernal equinox. The coordinates have a right-handed convention, that is, if one extends their right thumb upward, it simulates the z-axis, their extended index finger the x-axis, and the curl of the other fingers points generally in the direction of the y-axis.[9]

These rectangular coordinates are related to the corresponding spherical coordinates by

Conversion between celestial coordinate systems

Converting Cartesian vectors

Conversion from ecliptic coordinates to equatorial coordinates

[10]

Conversion from equatorial coordinates to ecliptic coordinates

where ε is the obliquity of the ecliptic.

See also

Notes and references

  1. ^ Nautical Almanac Office, U.S. Naval Observatory; H.M. Nautical Almanac Office, Royal Greenwich Observatory (1961). Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac. H.M. Stationery Office, London (reprint 1974). pp. 24–27.
  2. ^ Explanatory Supplement (1961), pp. 20, 28
  3. ^ U.S. Naval Observatory, Nautical Almanac Office (1992). P. Kenneth Seidelmann, ed. Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley, CA (reprint 2005). pp. 11–13. ISBN 1-891389-45-9.
  4. ^ Meeus, Jean (1991). Astronomical Algorithms. Willmann-Bell, Inc., Richmond, VA. p. 137. ISBN 0-943396-35-2.
  5. ^ Explanatory Supplement (1961), sec. 1G
  6. ^ N. Capitaine; P.T. Wallace; J. Chapront (2003). "Expressions for IAU 2000 precession quantities" (PDF). Astronomy & Astrophysics: 581. Bibcode:2003A&A...412..567C. doi:10.1051/0004-6361:20031539.
  7. ^ J.H. Lieske et al. (1977), "Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants". Astronomy & Astrophysics 58, pp. 1-16
  8. ^ Leadbetter, Charles (1742). A Compleat System of Astronomy. J. Wilcox, London. p. 94.; numerous examples of this notation appear throughout the book.
  9. ^ Explanatory Supplement (1961), pp. 20, 27
  10. ^ Explanatory Supplement (1992), pp. 555-558

External links

39 Laetitia

Laetitia (; minor planet designation: 39 Laetitia) is a large main-belt asteroid that was discovered by French astronomer Jean Chacornac on February 8, 1856, and named after “Laetitia”, one of the epithets of Ceres, Roman goddess of fertility and abundance.

Photometric observations of this asteroid gathered between 1968–74 were used to build a light curve that provided shape and rotation information. It has the general shape of an elongated triaxial ellipsoid with ratios between the lengths of the axes equal to 15:9:5. Major surface features are on a scale of 10 km and the surface color does not vary significantly across the surface. In the ecliptic coordinate system, the pole of rotation is estimated to be oriented to the coordinates (λ0, β0) = (121° ± 10°, +37° ± 10°).In 1988 a search for satellites or dust orbiting this asteroid was performed using the UH88 telescope at the Mauna Kea Observatories, but the effort came up empty. Photometric observations collected during 2006–08 were used to measure time variations of the asteroid light curve. This data suggests that the asteroid may have a complex shape or it could be a binary asteroid system. Observations of an occultation on March 21, 1998, produced several chords indicating an ellipsoidal cross-section of 219 × 142 km.

Astrological sign

In Western astrology, astrological signs are the twelve 30° sectors of the ecliptic, starting at the vernal equinox (one of the intersections of the ecliptic with the celestial equator), also known as the First Point of Aries. The order of the astrological signs is Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces.

The concept of the zodiac originated in Babylonian astrology, and was later influenced by Hellenistic culture. According to astrology, celestial phenomena relate to human activity on the principle of "as above, so below", so that the signs are held to represent characteristic modes of expression. Modern discoveries about the true nature of celestial objects have undermined the theoretical basis for assigning meaning to astrological signs, and empirical scientific investigation has shown that predictions and recommendations based on these systems are not accurate.The twelve sector division of the ecliptic constitutes astrology's primary frame of reference when considering the positions of celestial bodies, from a geocentric point of view, so that we may find, for instance, the Sun in 23° Aries (23° longitude), the Moon in 7° Scorpio (217° longitude), or Jupiter in 29° Pisces (359° longitude). Beyond the celestial bodies, other astrological points that are dependent on geographical location and time (namely, the Ascendant, the Midheaven, the Vertex and the houses' cusps) are also referenced within this ecliptic coordinate system.Various approaches to measuring and dividing the sky are currently used by differing systems of astrology, although the tradition of the Zodiac's names and symbols remain consistent. Western astrology measures from Equinox and Solstice points (points relating to equal, longest and shortest days of the tropical year), while Jyotiṣa or Vedic astrology measures along the equatorial plane (sidereal year). Precession results in Western astrology's zodiacal divisions not corresponding in the current era to the constellations that carry similar names, while Jyotiṣa measurements still correspond with the background constellations.In Western and Indian astrology, the emphasis is on space, and the movement of the Sun, Moon and planets in the sky through each of the zodiac signs. In Chinese astrology, by contrast, the emphasis is on time, with the zodiac operating on cycles of years, months, and hours of the day.A common feature of all three traditions however, is the significance of the Ascendant – the zodiac sign that is rising (due to the rotation of the earth) on the eastern horizon at the moment of a person's birth.

Axes conventions

In ballistics and flight dynamics, axes conventions are standardized ways of establishing the location and orientation of coordinate axes for use as a frame of reference. Mobile objects are normally tracked from an external frame considered fixed. Other frames can be defined on those mobile objects to deal with relative positions for other objects. Finally, attitudes or orientations can be described by a relationship between the external frame and the one defined over the mobile object.

The orientation of a vehicle is normally referred to as attitude. It is described normally by the orientation of a frame fixed in the body relative to a fixed reference frame. The attitude is described by attitude coordinates, and consists of at least three coordinates.While from a geometrical point of view the different methods to describe orientations are defined using only some reference frames, in engineering applications it is important also to describe how these frames are attached to the lab and the body in motion.

Due to the special importance of international conventions in air vehicles, several organizations have published standards to be followed. For example, German DIN has published the DIN 9300 norm for aircraft (adopted by ISO as ISO 1151–2:1985).

Behenian fixed star

The Behenian fixed stars are a selection of fifteen stars considered especially useful for magical applications in the medieval astrology of Europe and the Arab world. Their name derives from Arabic bahman, "root," as each was considered a source of astrological power for one or more planets. Each is also connected with a gemstone and plant that would be used in rituals meant to draw the star's influence (e.g., into a talisman). When a planet was within six degrees of an associated star, this influence was thought to be particularly strong.

Heinrich Cornelius Agrippa discussed them in his Three Books of Occult Philosophy (Book II, chapters 47 & 52) as the Behenii (singular Behenius), describing their magical workings and kabbalistic symbols. He attributed these to Hermes Trismegistus, as was common with occult traditions in the Middle Ages. Their true origin remains unknown, though Sir Wallis Budge suspects a possible Sumerian source.

The following table uses symbols from a 1531 quarto edition of Agrippa, but other forms exist. Where the name used in old texts differs from the one in use today, the modern form is given first.

Celestial coordinate system

In astronomy, a celestial coordinate system (or celestial reference system) is a system for specifying positions of celestial objects: satellites, planets, stars, galaxies, and so on. Coordinate systems can specify an object's position in three-dimensional space or plot merely its direction on a celestial sphere, if the object's distance is unknown or trivial.

The coordinate systems are implemented in either spherical or rectangular coordinates. Spherical coordinates, projected on the celestial sphere, are analogous to the geographic coordinate system used on the surface of Earth. These differ in their choice of fundamental plane, which divides the celestial sphere into two equal hemispheres along a great circle. Rectangular coordinates, in appropriate units, are simply the cartesian equivalent of the spherical coordinates, with the same fundamental (x, y) plane and primary (x-axis) direction. Each coordinate system is named after its choice of fundamental plane.

Celestial sphere

In astronomy and navigation, the celestial sphere is an abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphere, which may be centered on Earth or the observer. If centered on the observer, half of the sphere would resemble a hemispherical screen over the observing location.

The celestial sphere is a practical tool for spherical astronomy, allowing astronomers to specify the apparent positions of objects in the sky if their distances are unknown or irrelevant. In the equatorial coordinate system, the celestial equator divides the celestial sphere into two halves: the northern and southern celestial hemispheres.

Chinese constellations

Traditional Chinese astronomy has a system of dividing the celestial sphere into asterisms or constellations, known as "officials" (Chinese 星官 xīng guān).The Chinese asterisms are generally smaller than the constellations of Hellenistic tradition.

The Song dynasty (13th-century) Suzhou planisphere shows a total of 283 asterisms, comprising a total of 1,565 individual stars.

The asterisms are divided into four groups, the Twenty-Eight Mansions (二十八宿, Èrshíbā Xiù) along the ecliptic, and the Three Enclosures of the northern sky.

The southern sky was added as a fifth group in the late Ming Dynasty based on European star charts, comprising an additional 23 asterisms.

The Three Enclosures (三垣, Sān Yuán) are centered on the North Celestial Pole and include those stars which could be seen year-round.The Twenty-Eight Mansions form an ecliptic coordinate system used for those stars visible (from China) but not during the whole year, based on the movement of the moon over a lunar month.

Ecliptic

The ecliptic is the mean plane of the apparent path in the sky that the Sun follows over the course of one year; it is the basis of the ecliptic coordinate system. This plane of reference is coplanar with Earth's orbit around the Sun (and hence the Sun's apparent path around Earth). The ecliptic is not normally noticeable from Earth's surface because the planet's rotation carries the observer through the daily cycles of sunrise and sunset, which obscure the Sun's apparent motion against the background of stars during the year.

Equinox

An equinox is commonly regarded as the instant of time when the plane (extended indefinitely in all directions) of Earth's equator passes through the center of the Sun. This occurs twice each year: around 21 March and 22–23 September. In other words, it is the moment at which the center of the visible Sun is directly above the Equator. In the northern hemisphere, the equinox in March is called the Vernal or Spring Equinox; the September equinox is called the Autumnal or Fall Equinox.

However, because the Moon (and to a lesser extent the other planets) cause the motion of the Earth to vary from a perfect ellipse, the equinox is now officially defined by the Sun's more regular ecliptic longitude rather than by its declination. The instants of the equinoxes are currently defined to be when the longitude of the Sun is 0° and 180°. There are tiny (up to 1¼ arcsecond) variations in the Sun's latitude (discussed below), which means the Sun's center is rarely precisely over the equator under the official definition. The two understandings of the equinox can lead to discrepancies of up to 69 seconds.

On the day of an equinox, daytime and nighttime are of approximately equal duration all over the planet. They are not exactly equal, however, due to the angular size of the Sun, atmospheric refraction, and the rapidly changing duration of the length of day that occurs at most latitudes around the equinoxes. The word is derived from the Latin aequinoctium, from aequus (equal) and nox (genitive noctis) (night).

Fundamental plane (spherical coordinates)

The fundamental plane in a spherical coordinate system is a plane of reference that divides the sphere into two hemispheres. The latitude of a point is then the angle between the fundamental plane and the line joining the point to the centre of the sphere.For a geographic coordinate system of the Earth, the fundamental plane is the Equator. Celestial coordinate systems have varying fundamental planes:

The horizontal coordinate system uses the observer's horizon.

The Besselian coordinate system uses Earth's terminator (day/night boundary). This is a Cartesian coordinate system (x, y, z).

The equatorial coordinate system uses the celestial equator.

The ecliptic coordinate system uses the ecliptic.

The galactic coordinate system uses the Milky Way's galactic equator.

Jamshīd al-Kāshī

Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (Persian: غیاث الدین جمشید کاشانی‎ Ghiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician.Much of al-Kāshī's work was not brought to Europe, and much, even the extant work, remains unpublished in any form.

Midheaven

The midheaven (MC) is a point of definition in the ecliptic coordinate system. It aims to find the part of the ecliptic that corresponds to the highest point in a celestial object's apparent daily traverse of the visible sky, midway between its ascension on the eastern horizon and descension on the western horizon. The midheaven does not represent the point immediately overhead (our local zenith) or even the point of the ecliptic closest to it, but the point at which the local meridian intersects with the ecliptic.

In the northern hemisphere, the more northerly the latitude of the observer, the lower down on the horizon the midheaven point is likely to be, but it will always be the part of the zodiac that is due south at any time, indicating the point where the planets reach their highest declination in their arc between the ascendant and descendant. The reverse is true in the southern hemisphere where the planets culminate on the midheaven in alignment with due north.

Minute and second of arc

A minute of arc, arcminute (arcmin), arc minute, or minute arc is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn (or complete rotation), one minute of arc is 1/21600 of a turn – it is for this reason that the Earth's circumference is almost exactly 21,600 nautical miles. A minute of arc is π/10800 of a radian.

A second of arc, arcsecond (arcsec), or arc second is 1/60 of an arcminute, 1/3600 of a degree, 1/1296000 of a turn, and π/648000 (about 1/206265) of a radian. These units originated in Babylonian astronomy as sexagesimal subdivisions of the degree; they are used in fields that involve very small angles, such as astronomy, optometry, ophthalmology, optics, navigation, land surveying, and marksmanship.

To express even smaller angles, standard SI prefixes can be employed; the milliarcsecond (mas) and microarcsecond (μas), for instance, are commonly used in astronomy.

The number of square arcminutes in a complete sphere is 148510660 square arcminutes (the surface area of a unit sphere in square units divided by the solid angle area subtended by a square arcminute, also in square units - so that the final result is a dimensionless number).

Position of the Sun

The position of the Sun in the sky is a function of both the time and the geographic location of observation on Earth's surface. As Earth orbits the Sun over the course of a year, the Sun appears to move with respect to the fixed stars on the celestial sphere, along a circular path called the ecliptic.

Earth's rotation about its axis causes the fixed stars to apparently move across the sky in a way that depends on the observer's geographic latitude. The time when a given fixed star transits the observer's meridian depends on the geographic longitude.

To find the Sun's position for a given location at a given time, one may therefore proceed in three steps as follows:

calculate the Sun's position in the ecliptic coordinate system,

convert to the equatorial coordinate system, and

convert to the horizontal coordinate system, for the observer's local time and location.This calculation is useful in astronomy, navigation, surveying, meteorology, climatology, solar energy, and sundial design.

Triplicity

In astrology, a triplicity is a group of three signs belonging to the same element.

Zodiac

The zodiac is an area of the sky that extends approximately 8° north or south (as measured in celestial latitude) of the ecliptic, the apparent path of the Sun across the celestial sphere over the course of the year. The paths of the Moon and visible planets are also within the belt of the zodiac.In Western astrology, and formerly astronomy, the zodiac is divided into twelve signs, each occupying 30° of celestial longitude and roughly corresponding to the constellations Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces.The twelve astrological signs form a celestial coordinate system, or more specifically an ecliptic coordinate system, which takes the ecliptic as the origin of latitude and the Sun's position at vernal equinox as the origin of longitude.

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