Equinox

The true equinoxes are the moments when the sun crosses the celestial equator during its apparent annual movement on the ecliptic .

Equinox ( plural equinoxes , from Latin aequus equal 'and nox , night') or equinox (also day and night are equally long ), the two calendar days a year called, where lights day and night are of approximately equal length.

This applies to every place ( except the poles ) on earth. And on these days the sun rises almost exactly in the east or sets in the west all over the world. The exact times of the so-called true equinoxes are the moments when the sun rises (spring equinox) or descends (autumn equinox, see figure on the right) and the declination has the value δ = 0 ° over the celestial equator . Because δ changes slightly during the length of a day, day and night are not exactly the same length at the equinoxes either.

The fact that there are only two days a year with this special property is due to the fact that the earth's axis does not intersect its orbit around the sun perpendicularly (see figure on the right), but maintains its direction in space and, viewed from the sun, relatively during the earth's orbit stumbles to the railway level.

The equinoxes fall on March 19, 20, or 21 (spring equinox), and on September 22, 23, or 24 (autumn equinox). They mark the calendar beginning of the astronomically defined seasons of spring and autumn . In 2019, the moments of were true equinoxes 22:58 CET on 20 March and 9:50 CEST on 23 September.

On other planets, too, there are equinoxes and therefore also seasons. At Saturn the axis inclination is ε = 26.7 °. With an orbital period around the sun of around 30 earth years, an equinox occurs there only about every 15 earth years.

The equinoxes have been celebrated by many peoples throughout human history. For example, the Maya built the El Castillo step pyramid in Chichén Itzá in Mexico. The architecture of the building is designed in such a way that the sunlight on the equinoxes brushes the stairs on the north side at noon, while the areas next to the stairs remain in the shade.

Actual earth orbit around the sun. Equinoxes in March and September, when the earth's axis is not tilted towards the sun or not tilted away.

Definitions

In relation to the northern hemisphere, the following facts are referred to as the equinox :

Points in time

• Equinoxes: The calendar days on which the sun crosses the celestial equator , and with which therefore spring and autumn begin.
• Primary equinox: crossing from south to north , beginning of spring.
• Secondary equinox: crossing from north to south, beginning of autumn.
• True equinoxes: The exact times at which the sun crosses the celestial equator on the equinoxes , and thus the exact start of the astronomical seasons of spring or autumn.
• Spring equinox: Exact time of the beginning of spring , around 19/20/21 March.
• Autumn equinox: Exact time of the beginning of autumn, around 22./23./24. September.

Places in the sky

• Equinox: The two points on the ecliptic where the sun is at true equinox:
• Mean equinox: The two points on the ecliptic at which the sun is located in the long-term mean at true equinox:
• Aries point
• Balance point

Equinoxes as the beginning of the year and the determining date for religious festivals

In some calendar systems , the equinox is the beginning of the year and one of the central festivals of the year, such as the Nouruz (literally “new light”) of the Iranian astronomical-solar calendar and the Baha'i calendar . Rosh Hashanah , the Jewish New Year's Day, is not identical with the autumn equinox, but depends on it (a limited time before until a limited time after). The same applies to the Jewish Passover festival and Christian Easter , which always take place within about a month after the spring equinox.

Equinox as the beginning of the season

Astronomical beginning of spring and autumn, true equinox

Equinox (2008 to 2020)
year Primary- secondary
Day Time
CET
Day Time
CEST
2008 March 20 06:48 22 Sep 17:44
2009 March 20 12:44 22 Sep 23:19
2010 March 20 18:32 23 Sep 05:09
2011 March 21 00:21 23 Sep 11:05
2012 March 20 06:14 22 Sep 16:49
2013 March 20 12:02 22 Sep 22:44
2014 March 20 17:57 23 Sep 04:29
2015 March 20 23:45 23 Sep 10:21
2016 March 20 05:30 22 Sep 16:21
2017 March 20 11:29 22 Sep 22:02
2018 March 20 17:15 23 Sep 03:54
2019 March 20 22:58 23 Sep 09:50
2020 March 20 04:50 22 Sep 15:31

The exact definition is:

The equinoxes are the points in time at which the apparent geocentric ecliptical longitude of the sun is 0 ° and 180 °, respectively.
* Apparently means: after mathematical elimination of the displacements caused by aberration and nutation .
* Geocentric means: seen from a ( hypothetical ) observer in the center of the earth .

The definition is therefore independent of the location of a real observer; the equinoxes occur worldwide at the same point in time, which, however, corresponds to different times in different time zones .

Except for a few seconds, these times coincide with the times at which the center of the solar disk crosses the celestial equator, i.e. at which the sun moves from the southern to the northern half of the sky (ecliptical longitude 0 °) or from the northern to the southern half of the sky (ecliptical longitude 180 °) transferred. The time difference results from the fact that it is actually the center of gravity of the earth-moon system that moves around the sun in the middle orbit plane , while the earth itself orbits this center of gravity ("true earth orbit ") and thus - because the orbital plane of the earth-moon system is slightly inclined compared to the earth's orbital plane - usually a little above or below this plane. From the geocentric observer's point of view, the sun is therefore not exactly on the ecliptic (and has an ecliptical latitude not equal to zero). It also does not pass exactly the spring or autumn point and crosses the equator before or after it has reached the ecliptical longitude of these points. This time difference is a few seconds.

Because the average duration of an orbit of the earth around the sun in relation to the spring equinox ( tropical year ) is about 365.2422 days, almost six hours longer than the duration of the calendar common year with exactly 365 days, the calendar date of the equinoxes shifts from one common year to the next to a time about six hours later. With the insertion of February 29th in a leap year , the equinox is about 18 hours earlier than the previous year. The following equinoxes are rounded to the nearest minute (for the beginning of the four seasons, see also the table in the Seasons article ).

Equinoxes

The equinoxes are the times of the astronomical beginning of spring or autumn, which are shortened to the calendar day. In the northern hemisphere, spring begins in March and autumn in September. In the southern hemisphere it is the other way round.

The sun crosses the celestial equator at the equinox , so on this day it is perpendicular to the earth's equator at the time of the equinox . Day and night are then roughly the same length everywhere on earth, since half of the daily solar path lies above ( day arc ), the other half below the horizon. Everywhere on earth on this day the sun rises almost exactly in the east and sets in the west (see point of rise ).

In spherical astronomy , celestial objects are treated in a simplified manner and the extent of the solar disk is initially not taken into account, as is atmospheric influences . Due to the atmospheric refraction of sunlight and the reference to the first and last rays of the sun, the periods of day and night do not actually have the same duration at the time of an "equinox" , but the night is a few minutes shorter (see Equilux below).

The solstices lie between the equinoxes, i.e. the days on which the sun reaches its greatest distance from the celestial equator and is perpendicular to one of the tropics of the earth. The two equinoxes and the two solstices in a year represent the beginning of the astronomical seasons .

Equilux

“Equilux” is a calendar day on which the exposure time on the earth's surface with an ideal (mathematical) horizon, measured between the first sunbeam in the morning and the last sunbeam in the evening, would be exactly twelve hours; so this definition relates to the edge of the solar disk, not its center. The date of the Equilux therefore does not fall on the date of an equinox ("Equinox"), but takes place a few days before the primary or after the secondary equinox during the course of the year. In contrast to the equinoxes which are related to the center of the earth and are therefore the same worldwide, the Equilux date also depends on the latitude of the location. For the 40th parallel it is around March 17th or September 26th, for the 5th parallel it is around February 25th or October 15th.

The sun rises when its upper edge becomes visible above the horizon line , before its center appears. Sunset occurs after the center of the sun disk appears to have sunk below the horizon, when the last ray of the sun at the top of the sun goes out. Compared to a point-like view of the center of the sun, half a diameter arc of the sun (about 0.25 ° or 16 ') is added. In addition, the refraction of light by the earth's atmosphere causes the solar disk to appear to rise (by around 0.6 ° or 34 ′). This extension of the clear day at the expense of the night by just under 7 minutes (1.7 degrees × 4 minutes / degrees) at the equator (in Central Europe by just under 11 minutes) is taken into account when determining the Equilux.

Equinox as coordinate zero point

True equinox and middle equinox

The true equinoxes are the actual intersections of the celestial equator with the ecliptic:

• The passage of the sun through the vernal equinox defines the astronomical beginning of spring.
• The passage of the sun through the autumn point defines the astronomical beginning of autumn.

The mean equinox, however, are fictitious. They should only reflect the long-period orbital movement, which is why no short-term disturbances ( e.g. nutation and aberration ) are taken into account when determining them. Therefore, the mean equinox points can differ from the real ones by several hours.

The usual symbol for the Aries point, which has a prominent meaning in celestial mechanics , is or ( U + 2648 ). It is the coordinate zero point for ecliptical coordinates and equatorial coordinates and several other basic astronomical quantities. Its English name is first point of aries.${\ displaystyle {\ mathcal {W}}}$

Spring equinox and autumn equinox

The spring and autumn points themselves, i.e. those points on which the sun is at the time of an equinox in the above sense in front of the fixed star background, are called equinoxes. In more clearly differentiating language usage, they are also referred to as "equinox points".

The vernal equinox is the point on the imaginary celestial sphere at which the sun, on its orbit projected onto this sphere , the ecliptic, cuts through the celestial equator on the way from south to north ( right ascension  = 0 h).

Accordingly, the autumn point is the point on the imaginary celestial sphere at which the sun, on its orbit projected onto this sphere, cuts through the celestial equator on its way from north to south (right ascension = 12 h).

The summer point (right ascension = 6 h) and winter point (right ascension = 18 h), in which the sun is at the solstices , lie at an angle of 90 ° to the spring point and autumn point .

Equinox line

The line connecting the two positions of the earth at the time of an equinox is called the equinox. This line goes through the middle of the sun, its extension outside the earth's orbit through the equinox. It is perpendicular to the solitary line .

Spring point as the coordinate zero point

In connection with astronomical coordinate systems , the term "equinox" always refers to the spring equinox, never the autumn equinox. The spring point serves as the zero point for both the equatorial and the ecliptical coordinate system, from which the right ascension or ecliptical longitude is counted (positive to the east). The vernal equinox is not a directly observable and measurable point, but its position can always be calculated from suitable observations.

Migration of the equinoxes

The gravitation of the sun, moon and planets causes tidal forces . They pull the equatorial bulge of the rotating earth, the axis of which is inclined by about 23.5 ° to the plane of the ecliptic , into this plane - and thus tend to straighten the axis of rotation with respect to the plane of the ecliptic.

However, the earth's axis does not straighten up. Rather, it maintains the angle of inclination and slowly changes its orientation, similar to a top , so that the direction in which it is inclined passes through a full 360 ° for about 25,800 years. The equatorial plane defined perpendicular to the earth's axis also performs this movement, so that the equinox points, as the intersection points of the equatorial plane and the ecliptic plane, pass through the ecliptic once every 25,800 years. This movement of the earth's axis or the equinox points is called precession ( Latin for “going ahead”).

The spring equinox moves along the ecliptic in six millennia by almost a quarter of the celestial sphere

The equinox shifts per year by about 50 arc seconds in a westerly direction along the ecliptic. This effect is so great that it is noticeable over an observation period of several decades and was already known in antiquity.

Additional periodic fluctuations are superimposed on the precession movement . They are due to the inclination of the moon's orbit , which is inclined by 5 ° 9 ′ to the ecliptic, and the continuously shifting nodal line of the lunar orbit as well as slight periodic shifts in the earth's axis of rotation. These various periodic movements, which the earth's axis performs in addition to precession, are summarized in astronomy under the term nutation . The displacement of the equinox points along the ecliptic does not take place completely evenly, but with periodically slightly fluctuating speed.

The Strasbourg cathedral clock contains a device with which the precession is displayed (but not the nutation).

The equinox of astronomical coordinates

Ecliptical and equatorial coordinate systems have the equinoxes as a common fixed point

The migration of the equinoxes has the particular consequence that the zero points of the above-mentioned astronomical coordinate systems are not fixed in space, but rather slowly move along the ecliptic with the vernal equinox. For example, the ecliptical length of a star without proper motion increases by 50 arc seconds in one year, i.e. H. by 1.4 ° in 100 years. The coordinates of a celestial object change without this corresponding to any actual movement of the object. When specifying it, the point in time, i.e. the position of the vernal equinox, must therefore always be given to which the coordinates refer. This point in time (not to be confused with one of the equinoxes) is also called the equinox and is given as a year, possibly with a fraction. The coordinates for the equinox of the time of observation (for example 2005, 432), the so-called equinox of the date, are important for observations .

Converting coordinates between different equinoxes is a common task.

Equinox and epoch

The term epoch should not be confused with the equinox . The epoch describes the actual point in time of an observation or a process: the equinox of the coordinate system in which measurements are made.

Standard equinoxes

Catalogs of celestial objects are usually related to so-called standard equinoxes. These are coordinate systems that are related to specific points in time and are also called standard epochs. The times are set at the turn of every 25th year. In the past the time difference between two standard epochs was 25  Bessel years (approx. 9131.055 days), today it is 25  Julian years (9131.25 days). These standard equinoxes are labeled with a year and a B or J in front of it. These are:

Standard
epoch / equinox
date annotation
Julian Gregorian
B1850 2396758.203 Dec. 31, 1849, 4:52 p.m. UT Insignificant these days
B1875 2405889.258 December 31, 1874, 6:12 p.m. UT After the equinox of this epoch, the exact boundaries of the constellation were defined as lines of constant right ascension or constant declination.
B1900 2415020,313 Dec. 31, 1899, 7:31 pm UT
B1925 2424151,368 December 31, 1924, 20:50 UT Insignificant these days
B1950 2433282,423 December 31, 1949, 10:09 pm UT The star positions in the fourth fundamental catalog are given with this equinox.
B1975 2442413,478 Dec. 31, 1974, 23:28 UT Last standard equinox, referring to a Bessel era (used very rarely).
J2000 2451545,000 0Jan. 1, 2000, 12:00 UT Was determined exactly in such a way, regardless of the previous points in time, in order to obtain smooth points in time. This equinox is in use today.

Example: The star Arcturus has the following equatorial coordinates right ascension and declination related to different equinoxes at different epochs:

epoch Equinox
J2000.0 of the date J2050.0
0Jan. 1, 2000 213.9153 ° / 19.1824 ° 213.9153 ° / 19.1824 ° 214.5019 ° / 18.9522 °
Aug 12, 2028 213.9061 ° / 19.1665 ° 214.2418 ° / 19.0346 ° 214.4928 ° / 18.9363 °
0Jan. 1, 2050 213.8992 ° / 19.1546 ° 214.4860 ° / 18.9244 ° 214.4860 ° / 18.9244 °

The change in coordinates for different epochs, but the same fixed equinox (J2000.0 or J2050.0) reflects the star's proper motion. The difference in the coordinates for the same epoch but different equinoxes is due to the precession. The coordinates given in the equinox of the date include the influence of both proper motion and precession.

For calculations, it is often advantageous to ignore the periodic influence of the nutation on the movement of the equinox and to refer to a fictitious evenly moved equinox (the nutation must of course be added to the results afterwards). It is then the middle equinox, while the true equinox contains the influence of nutation.

Catalog equinox and dynamic equinox

The exact position of the equinox, like the position of the equator and the ecliptic, must be determined by observation. Occasionally suitable observation material is evaluated particularly carefully for this purpose. The result is, for example, a star catalog, the coordinates of which indicate the position of the stars with respect to the equinox sought as precisely as possible. These coordinates embody the coordinate system for practical use and represent a fundamental system to which other position measurements can refer. For example, if the coordinates of a star are determined by measuring its distance from suitable fundamental stars, then the coordinates found automatically relate to the equinox of the fundamental system. The equinox, which is derived from catalog items (as the intersection of the hour circle of right ascension 0 with the equator), is the catalog equinox. The equinox embodied by the fundamental system never exactly coincides with the actual equinox due to inevitable measurement inaccuracies. In the case of high accuracy requirements, the catalog to whose catalog equinox the measurements are based must therefore be given. If the equinox is derived exclusively from planetary observations (the angular momentum vector of the earth's motion, for example, is perpendicular to the ecliptic plane and allows it to be determined), a dynamic equinox is obtained.

Conversion from one equinox to another

The following conversions transform equatorial coordinates from one equinox to another. The proper movement of astronomical objects is not taken into account. Method:

1. Converting the equatorial coordinates of the previous equinox into Cartesian coordinates (a unit sphere).
2. Transformation of the Cartesian coordinates into Cartesian coordinates of the target equinox with the help of a rotation matrix .
3. Conversion of the transformed Cartesian coordinates into equatorial coordinates.

Conversion of equatorial coordinates into Cartesian coordinates

With right ascension α and declination δ applies to ${\ displaystyle P = {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}}:}$

 ${\ displaystyle x = \ cos (\ alpha) \, \ cos (\ delta)}$ ${\ displaystyle y = \ sin (\ alpha) \, \ cos (\ delta)}$ ${\ displaystyle z = \ sin (\ delta)}$ ${\ displaystyle P = {\ begin {pmatrix} \ cos (\ alpha) \, \ cos (\ delta) \\\ sin (\ alpha) \, \ cos (\ delta) \\\ sin (\ delta) \ end {pmatrix}}}$
Rotations using a rotation matrix
${\ displaystyle P \ cdot M = P '}$
${\ displaystyle {\ begin {pmatrix} x \\ y \\ z \ end {pmatrix}} \ cdot {\ begin {pmatrix} m_ {11} & m_ {12} & m_ {13} \\ m_ {21} & m_ { 22} & m_ {23} \\ m_ {31} & m_ {32} & m_ {33} \ end {pmatrix}} = {\ begin {pmatrix} x '\\ y' \\ z '\ end {pmatrix}}}$

The rotation matrix results from the superposition of three rotations around the angles determined with polynomials and : ${\ displaystyle \ zeta, z}$${\ displaystyle \ Theta}$

${\ displaystyle {\ begin {pmatrix} \ cos (\ zeta) \ cdot \ cos (\ Theta) \ cdot \ cos (z) - \ sin (\ zeta) \ cdot \ sin (z) & - \ sin (\ zeta) \ cdot \ cos (\ Theta) \ cdot \ cos (z) - \ cos (\ zeta) \ cdot \ sin (z) & - \ sin (\ Theta) \ cdot \ cos (z) \\\ cos (\ zeta) \ cdot \ cos (\ Theta) \ cdot \ sin (z) + \ sin (\ zeta) \ cdot \ cos (z) & - \ sin (\ zeta) \ cdot \ cos (\ Theta) \ cdot \ sin (z) + \ cos (\ zeta) \ cdot \ cos (z) & - \ sin (\ Theta) \ cdot \ sin (z) \\\ cos (\ zeta) \ cdot \ sin (\ Theta ) & - \ sin (\ zeta) \ cdot \ sin (\ Theta) & \ cos (\ Theta) \ end {pmatrix}}}$

This is what matrix multiplication means:

${\ displaystyle x '= x \ cdot m_ {11} + y \ cdot m_ {12} + z \ cdot m_ {13}}$
${\ displaystyle y '= x \ cdot m_ {21} + y \ cdot m_ {22} + z \ cdot m_ {23}}$
${\ displaystyle z '= x \ cdot m_ {31} + y \ cdot m_ {32} + z \ cdot m_ {33}}$
Calculate the equatorial coordinates of the target equinox
${\ displaystyle \ alpha '= \ operatorname {sgn} (y') \ cdot \ arccos {\ frac {x '} {\ sqrt {x' ^ {2} + y '^ {2}}}}}$
${\ displaystyle \ delta '= \ arcsin (z')}$

Conversions between standard equinoxes

The following matrices apply to the standard equinoxes B1875, B1900, B1950, B1975 and J2000:

Matrix B1875 → B1900

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981452 & -0 {,} 005585025 & -0 {,} 002429486 \\ + 0 {,} 005585008 & + 0 {,} 999984404 & -0 {,} 000006789 \\ + 0 {,} 002429496 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

Matrix B1875 → B1950

${\ displaystyle {\ begin {pmatrix} +0 {,} 999833020 & -0 {,} 016757867 & -0 {,} 007288174 \\ + 0 {,} 016757422 & + 0 {,} 999859575 & -0 {,} 000061084 \\ + 0 {,} 007288430 & -0 {,} 000061087 & + 0 {,} 999973439 \ end {pmatrix}}}$

Matrix B1875 → B1975

${\ displaystyle {\ begin {pmatrix} +0 {,} 999703104 & -0 {,} 022343591 & -0 {,} 009717233 \\ + 0 {,} 022344281 & + 0 {,} 999750345 & -0 {,} 000108598 \\ + 0 {,} 009717840 & -0 {,} 000108210 & + 0 {,} 999952781 \ end {pmatrix}}}$

Matrix B1875 → J2000

${\ displaystyle {\ begin {pmatrix} +0 {,} 999535925 & -0 {,} 027933851 & -0 {,} 012144267 \\ + 0 {,} 027935279 & + 0 {,} 999609760 & -0 {,} 000169365 \\ + 0 {,} 012147193 & -0 {,} 000169297 & + 0 {,} 999926241 \ end {pmatrix}}}$

Matrix B1900 → B1875

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981452 & + 0 {,} 005585025 & + 0 {,} 002429486 \\ - 0 {,} 005585008 & + 0 {,} 999984404 & -0 {,} 000006784 \\ - 0 {,} 002429496 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

Matrix B1900 → B1950

${\ displaystyle {\ begin {pmatrix} +0 {,} 999925772 & -0 {,} 011173365 & -0 {,} 004858902 \\ + 0 {,} 011173233 & + 0 {,} 999937576 & -0 {,} 000027154 \\ + 0 {,} 004858978 & -0 {,} 000027925 & + 0 {,} 999988195 \ end {pmatrix}}}$

Matrix B1900 → B1975

${\ displaystyle {\ begin {pmatrix} +0 {,} 999832961 & -0 {,} 016761357 & -0 {,} 007288174 \\ + 0 {,} 016760912 & + 0 {,} 999859517 & -0 {,} 000061097 \\ + 0 {,} 007288430 & -0 {,} 000061087 & + 0 {,} 999973439 \ end {pmatrix}}}$

Matrix B1900 → J2000

${\ displaystyle {\ begin {pmatrix} +0 {,} 999702892 & -0 {,} 022350570 & -0 {,} 009717237 \\ + 0 {,} 022353004 & + 0 {,} 999750189 & -0 {,} 000108276 \\ + 0 {,} 009719586 & -0 {,} 000108210 & + 0 {,} 999952781 \ end {pmatrix}}}$

Matrix B1950 → B1875

${\ displaystyle {\ begin {pmatrix} +0 {,} 999833020 & + 0 {,} 016757867 & + 0 {,} 007288175 \\ - 0 {,} 016757422 & + 0 {,} 999859575 & -0 {,} 000061059 \\ - 0 {,} 007288430 & -0 {,} 000061087 & + 0 {,} 999973439 \ end {pmatrix}}}$

Matrix B1950 → B1900

${\ displaystyle {\ begin {pmatrix} +0 {,} 999925772 & + 0 {,} 011173365 & + 0 {,} 004858902 \\ - 0 {,} 011173233 & + 0 {,} 999937576 & -0 {,} 000027138 \\ - 0 {,} 004858978 & -0 {,} 000027925 & + 0 {,} 999988195 \ end {pmatrix}}}$

Matrix B1950 → B1975

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981433 & -0 {,} 005588515 & -0 {,} 002429486 \\ + 0 {,} 005588499 & + 0 {,} 999984384 & -0 {,} 000006797 \\ + 0 {,} 002429496 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

Matrix B1950 → J2000

${\ displaystyle {\ begin {pmatrix} +0 {,} 999925666 & -0 {,} 011178601 & -0 {,} 004858904 \\ + 0 {,} 011181959 & + 0 {,} 999937517 & -0 {,} 000026798 \\ + 0 {,} 004860723 & -0 {,} 000027925 & + 0 {,} 999988195 \ end {pmatrix}}}$

Matrix B1975 → B1875

${\ displaystyle {\ begin {pmatrix} +0 {,} 999703104 & + 0 {,} 022343591 & + 0 {,} 009717234 \\ - 0 {,} 022344281 & + 0 {,} 999750345 & -0 {,} 000108547 \\ - 0 {,} 009717840 & -0 {,} 000108210 & + 0 {,} 999952781 \ end {pmatrix}}}$

Matrix B1975 → B1900

${\ displaystyle {\ begin {pmatrix} +0 {,} 999832961 & + 0 {,} 016759612 & + 0 {,} 007288175 \\ - 0 {,} 016760912 & + 0 {,} 999859546 & -0 {,} 000061059 \\ - 0 {,} 007288430 & -0 {,} 000061087 & + 0 {,} 999973439 \ end {pmatrix}}}$

Matrix B1975 → B1950

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981433 & + 0 {,} 005586770 & + 0 {,} 002429486 \\ - 0 {,} 005588499 & + 0 {,} 999984394 & -0 {,} 000006780 \\ - 0 {,} 002429496 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

Matrix B1975 → J2000

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981399 & -0 {,} 005592006 & -0 {,} 002429487 \\ + 0 {,} 005593735 & + 0 {,} 999984365 & -0 {,} 000006411 \\ + 0 {,} 002431241 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

MatrixJ2000 → B1875

${\ displaystyle {\ begin {pmatrix} +0 {,} 999535925 & + 0 {,} 027933851 & + 0 {,} 012146003 \\ - 0 {,} 027935279 & + 0 {,} 999609760 & -0 {,} 000170004 \\ - 0 {,} 012147193 & -0 {,} 000169297 & + 0 {,} 999926220 \ end {pmatrix}}}$

Matrix J2000 → B1900

${\ displaystyle {\ begin {pmatrix} +0 {,} 999702948 & + 0 {,} 022350570 & + 0 {,} 009717230 \\ - 0 {,} 022351260 & + 0 {,} 999750189 & -0 {,} 000108954 \\ - 0 {,} 009717840 & -0 {,} 000108210 & + 0 {,} 999952781 \ end {pmatrix}}}$

Matrix J2000 → B1950

${\ displaystyle {\ begin {pmatrix} +0 {,} 999925685 & + 0 {,} 011178601 & + 0 {,} 004858900 \\ - 0 {,} 011180214 & + 0 {,} 999937517 & -0 {,} 000027528 \\ - 0 {,} 004860723 & -0 {,} 000026180 & + 0 {,} 999988195 \ end {pmatrix}}}$

Matrix J2000 → B1975

${\ displaystyle {\ begin {pmatrix} +0 {,} 999981399 & + 0 {,} 005592006 & + 0 {,} 002429485 \\ - 0 {,} 005593735 & + 0 {,} 999984365 & -0 {,} 000007162 \\ - 0 {,} 002431241 & -0 {,} 000006981 & + 0 {,} 999997049 \ end {pmatrix}}}$

example

For the conversion from B1950 to J2000, the values , and , from which the matrix is ​​made ${\ displaystyle \ zeta = 1152 {,} 4075 ''}$${\ displaystyle z = 1152 {,} 750 ''}$${\ displaystyle \ Theta = 1002 {,} 2442 ''}$

${\ displaystyle {\ begin {pmatrix} +0 {,} 9999257352 & -0 {,} 0111761178 & -0 {,} 0048598834 \\ + 0 {,} 0111761178 & + 0 {,} 9999375449 & -0 {,} 0000271607 \\ + 0 {,} 0048598834 & -0 {,} 0000271560 & + 0 {,} 9999881903 \ end {pmatrix}}}$

results. This means for matrix multiplication:

${\ displaystyle x '= x \ cdot m_ {11} + y \ cdot m_ {12} + z \ cdot m_ {13} = 0 {,} 9999257352x-0 {,} 0111761178y-0 {,} 0048598834z}$
${\ displaystyle y '= x \ cdot m_ {21} + y \ cdot m_ {22} + z \ cdot m_ {23} = 0 {,} 0111761178x + 0 {,} 9999375449y-0 {,} 0000271607z}$
${\ displaystyle z '= x \ cdot m_ {31} + y \ cdot m_ {32} + z \ cdot m_ {33} = 0 {,} 0048598834x-0 {,} 0000271560y + 0 {,} 9999881903z}$

For example, for the celestial pole of the equinox B1950:

${\ displaystyle \ alpha = 0 ^ {\ circ}}$ (freely selectable)
${\ displaystyle \ delta = + 90 ^ {\ circ}}$
${\ displaystyle x = \ cos (\ alpha) \ cdot \ cos (\ delta) = 0}$
${\ displaystyle y = \ sin (\ alpha) \ cdot \ cos (\ delta) = 0}$
${\ displaystyle z = \ sin (\ delta) = 1}$
${\ displaystyle {\ begin {pmatrix} 0 \\ 0 \\ 1 \ end {pmatrix}} \ cdot {\ begin {pmatrix} +0 {,} 9999257352 & -0 {,} 0111761178 & -0 {,} 0048598834 \\ +0 {,} 0111761178 & + 0 {,} 9999375449 & -0 {,} 0000271607 \\ + 0 {,} 0048598834 & -0 {,} 0000271560 & + 0 {,} 9999881903 \ end {pmatrix}} = {\ begin {pmatrix } -0 {,} 0048619076 \\ - 0 {,} 0000271833 \\ + 0 {,} 9999881805 \ end {pmatrix}}}$

and it

${\ displaystyle \ alpha '= 180 {,} 320341641 ^ {\ circ}}$
${\ displaystyle \ delta '\, = \; \; 89 {,} 721427765 ^ {\ circ}}$

literature

Wiktionary: Equinox  - explanations of meanings, word origins, synonyms, translations
Commons : Equinox  - collection of images, videos, and audio files

Individual evidence

1. There are only places with exactly the same moment of sunrise and other places with exactly the same moment of sunset. They are each on the same longitude. It is one and the same moment, namely the time of the (astronomical) equinox. Rising and setting moments relate to the center of the sun in the mathematical horizon , and the deflection of light in the earth's atmosphere is neglected.
2. Christoph Neumüller : The beginning of spring from 1900 to 2100. At: dasinternet.net.
3. Christoph Neumüller: beginning of autumn from 1900 to 2100. At: dasinternet.net.
4. ^ Institut de Mecanique Celeste et de Calcul des Ephemerides (IMCCE). Retrieved March 19, 2019 .
5. The equinox is here! But what is it really? At: National Geographic.
6. ^ A b Jean Meeus: Astronomical Algorithms. Willmann-Bell, Richmond 2000. ISBN 0-943396-61-1 .
7. Season table of the USNO. ( Memento of October 8, 2015 in the Internet Archive ). At: usno.navy.mil. Retrieved September 23, 2014.
8. Equinoxes. ( Memento from September 25, 2015 in the Internet Archive ). At: usno.navy.mil. Retrieved January 28, 2014.