# Equation of time

As time equation (symbol ZG or ZGL ) the difference between the true solar time ( local apparent time WOZ ) and the mean solar time is called (mean time or mean time MOZ).

In the older literature, the inverse difference, so called between mean solar time and true solar time as the equation of time. It is still the medieval meaning of " equation " as " inflict a correction " to recognize: they had the true solar time add the difference to the mean solar time to go to the latter.

The causes of the equation of time, the slightly fluctuating velocity of the annual motion of the Earth on its elliptical orbit around the sun and the fact that the Earth's axis is not perpendicular to the orbital plane. The almost invariable direction of the Earth's axis is relatively much to the fixed stars, so that their direction relative to the sun changes with year period.

In particular cause

The equation of time is the same for all places on earth. But your shares true solar time and mean time hang because of the daily rotation of the earth on its axis from the longitude of the place. The change in the equation of time is carried out throughout the year slowly, so can each be used for the entire day in tables for 12:00 UT values given with sufficient approximation.

A simple (not corrected with the equation of time ) Sundial usually shows the true solar time at which to about 16 minutes, proceed ( about November 3 ) compared to the mean time or up to about 14 minutes follow up ( to 11 February ) can. A dial adjacent to the specified times time equation table (or equation of time chart) to help the user to calculate the mean time. In central European latitudes, the mean time of the 15th degree of longitude is sometimes displayed, performs correction to the value of the equation of time to local true solar time or to also displayed by the watch Central European Time CET.

The annual curve contained in the diagram beside the equation of time (red line ) can be used for several decades. An accuracy to about 1 minute in four- year cycle is caused by the necessary adjustment of the calendar to the solar year by intercalary. In seconds, accurate, annually calculated values are given in astronomical almanacs.

- 2.1 First Cause: ellipticity of the Earth's orbit
- 2.2 Second Cause: inclination of the Earth
- 2.3 Equation of time, the superposition of two causes

- 3.1 difference of two right ascensions ( geocentric)
- 3.2 The Earth in its elliptical orbit ( heliocentric )
- 3.3 The sun on its apparent path ( the ecliptic, geocentric)
- 3.4 Right Ascension of the True Sun
- 3.5 Right Ascension of the Sun Middle
- 3.6 Summary: The Equation of Time
- 3.7 Calculation Example

## Star and sunny and mean solar

### Star and sunny day

The period between two meridian passages of the sun is a solar day; he averages 24 hours. In contrast, the period between two meridian passages of a fixed star is called a sidereal day. This is the time for one revolution of the earth around itself and is 23 hours 56 minutes and 4 seconds ( 365.25 solar days ≈ 366.25 Star days). The difference between the length of the sidereal day and the length of the solar day results from the annual movement of the earth around the sun. From day to day the earth is in its orbit around the sun almost an arc degrees above (360 degrees in 365 days ). Since both movements have the same sense, the Earth just one degree over the full rotation is also necessary to continue to rotate as well, until the sun rises again through the meridian. This takes an average of 3 minutes and 56 seconds.

The rotation of the earth is very uniform, and therefore the duration of the star tag can be assumed to be constant. Different size are solely the small additional rotation and the associated small additional time resulting from the daily train ride the earth. This Zusatzeit can be up about 30 seconds longer or until about 20 seconds shorter than their mean value of just under 4 minutes, which can add up to up to about a quarter of an hour for months before the effect is reversed again. The depth reading on Sun true solar time goes by WOZ thus unevenly. Its deviation from the uniform time passing, for example, can be read from a mechanical clock time MOZ is called the equation of time.

### Mean Solar

Than that of the (apparent ) movement of the ( true ) sun " made" solar day has been recognized as unevenly long, the solar day but should remain a fundamental measure of time, dodged to the formal use of a fictitious so-called mean sun and the so-called mean solar time, a uniform created tempo. The artificial medium sun rotates evenly and not on the ecliptic, but on the celestial equator and "makes" doing the mean solar day.

## Two equation of time causes superimposed

The causes for the equation of time can be seen easily from the heliocentric view, because they follow from the movements of the Earth relative to the Quiet Sun. Sake of simplicity is sometimes still speak of " solar time " even when it is movements of the soil, not to the sun, as a function of time.

### First Cause: ellipticity of the Earth's orbit

The orbit of the Earth around the Sun is an ellipse, in one focus the sun is. Kepler's second law describes the change in the line speed during one rotation of the earth. In the vicinity of the perihelion - the point closest to the Sun - the Earth moves faster than the average and places for one day a greater distance, so they have to make a slightly larger additional rotation until the sun rises again through the meridian. It takes longer than average. The area around the aphelion - the sonnenfernsten point - it's the other way around.

In perihelion environment ( winter season), the true solar time passes more slowly because of the greater ground speed ( and thus required larger additional rotation ), in aphelion environment ( summer months ) it goes by faster than the uniform mean solar time. The variation of the true length of the day is about ± 8 seconds. The summation gives about ± 7 ½ minutes annual variation of the true solar time due to ellipticity of the Earth's orbit (sine -shaped blue line in the diagram above, the zero crossings at perihelion (currently, ie from 2000 to 2025: January 2 to 5 ) and at aphelion ).

### Second Reason: inclination of the Earth

The Earth's axis intersects its orbital plane is not perpendicular ( in the scheme shown above the image plane is not perpendicular ). The deviation from the right angle is about ε = 23.44 °. This deviation is the second cause of the equation of time, which is crucial that the observed from the sun Earth's axis changes its direction every day, makes a full tumbling motion per year. The daily train the earth is a rotation ( about 1 ° ) around the track axis (or the sun). The resulting required additional rotation of the earth takes place around its own axis. Since the two axes are not parallel, both rotations are not equal.

The two axes intersect at the day - and -night match seen from the Sun ε under the Ekliptikwinkel. The cutting angle is now the largest (about 23.44 °). The rotation of the earth on its axis has a maximum gain as a rotation around the to its orbital plane ( ecliptic ) perpendicular axis. Result is that the additional rotation, and the time required is less than in the middle. The true solar time passes faster than the mean solar time.

Both axes cover seemingly at the solstices. However, one of the two poles of the earth is closer to the Sun than the other. The daily 1 ° rotation of the earth around the track axis is formed as a sheet onto one of its two turning circles. The associated arc on the Earth's equator is larger. The required additional rotation of the Earth around its axis and the time required are larger than the average. The true solar time passes more slowly than the mean solar time.

This second cause of the equation of time alone causes a variation of the true solar day by about ± 20 seconds. The summation results in almost ± 10 minutes semi- annual variation of the true solar time due to the specific direction of the Earth's axis ( sinusoidal magenta -colored line in the diagram above, the zero crossings in the days of the solstices and the day - and -night matches, starting at the winter solstice ).

### Equation of time, the superposition of two causes

The effects of the ellipticity of the Earth's orbit and the special direction of the Earth's axis overlap and result in the equation of time (red line in the diagram above ). Since the two sine shaped curves are slightly shifted in time against each other, the extreme values in the result is less than the sum of the two individual extreme values . The equation of time has at present (2011) the following characteristics:

- 4 nodes: 13 April, 13 June, 1 September and 25 December,
- 2 main extremes: February 11 (-14 min 14 s ) and November 3 ( 16 min 26 s ),
- 2 In addition to extreme values : on 14 May ( 3 min 40 s ) and July 26 (-6 min 32 s ).

Negative values : the true solar time or the true sun goes according to the mean solar time or mean solar. positive numerical values : the true solar time or true solar runs ahead of mean solar time or mean solar.

## Calculation

The following quantitative, ie, computational treatment of the equation of time is essentially - namely when resulting from the elliptical orbital motion of the Earth time equation share - an application of the so-called Kepler 's equation. In particular, it is the place of the earth is determined on its elliptical orbit ( Kepler track ) at a predetermined time.

This equation is limited to the system described by Kepler two-body problem ( Kepler problem), which is used in this case earth and sun. The influence of the Moon and other solar planets on the motion of the earth and thus the equation of time is ignored.

Time equation values are calculated for any number of moments in the future, usually for a moment per calendar day. The daily values are repeated relatively accurately every four calendar years ( leap year ). Since properties of the Earth's orbit and the direction of the Earth's axis slowly change ( recorded from year to year with so-called annual constants), the bill is usually performed annually submits but new and only for the following year and published their results in astronomical almanacs.

### Difference of two right ascensions ( geocentric)

First definition:

The WOZ or MOZ - values corresponding to the respective state of the real or the fictitious mean sun in the sky. Since the time of the day in connection with the rotation of the Earth about its axis is interested only the respective right ascension ( not declination ) of the sun (s). In other words: from the perpendicular to each other were carried out in two apparent annual motion of the true Sun is interested only taking place on the celestial equator, but not periodically to mount and dismount. The equation of time is proportional to the difference between the right ascension () and the central fictive and ( ) of the actual true sun.

Second definition:

The factor 4 arises from the fact that two celestial bodies with 1 ° Rektaszensionsdifferenz the Meridian 4 min happen one after another. The order of the two Subtraktionsterme has been reversed because the directions for hour angle ( WOZ meet him and MOZ) and right ascension are defined opposite to each other.

### The Earth in its elliptical orbit ( heliocentric )

Task: At a given time the location on earth to determine its orbit.

For labeling of the current location X of the earth to Choose the sun centric angle ( the so-called true anomaly ) between it and the point nearest the Sun the Earth's orbit, the perihelion P ( see figure ). growing non-uniformly with time. This non-uniformity can be illustrated by comparison with the current location of a notional Y earth whose angle ( the so-called mean anomaly ) increases uniformly. The leading to X and Y angle thigh let as two around the sun B rotating pointer understand ( peaks at X or Y).

The uniformly increasing angles mean anomaly led Kepler as a representative of uniformly passing time. He is " normalized time ".

The mean anomaly must be corrected in the calculation of the equation of time by the angle, because the latter at the time of lying about 3 days before the perihelion passage initial years (January 1, 12:00 UT, point K), is obtained: see equation ( 3a).

Is negative and one years constant whose values differ, in particular within a leap year period significantly.

With the dependence of the path angle of the earth ( true anomaly ) of the time ( indirectly from the mean anomaly ) in the equation of time the one influence which follows from the ellipticity of the Earth's orbit, expressed. The relationship follows from the law of motion of the elliptical rail journey, the Kepler's second law. Kepler has shown the relationship with the help of the later so-called Kepler 's equation. A given point in time ( represented by the mean anomaly ) is converted into an intermediate step in the so-called Kepler eccentric anomaly. This is an elliptical centric ( ellipse center ) angle is measured from the perihelion from the auxiliary point Z on the perimeter of the ellipse. The Kepler equation is the numerical eccentricity ( see derivation of Kepler 's equation ):

It is according to dissolve, which is not possible in closed form, but for example with the numerical method succeeds, it developed Newton at this task.

The result for E as a function of M is used in the following by means of purely geometrical consideration in the ellipse and in their area (see illustration above ) derived equation:

Whereupon the desired dependence of the true anomaly V is found from the average anomaly M and of time.

Partial result: The first cause of the equation of time is recognized as a path angle of the earth as a function of time.

### The sun on its apparent path ( the ecliptic, geocentric)

Task: The path angle of the earth shall be converted to the ecliptic longitude of the sun.

From the Earth seen from the movement of the earth around the sun reflected reflected in the apparent motion of the sun in the ecliptic, the section of the Earth's orbital plane with the beaten around the earth as the center of a spherical globule ( see illustration ).

Reference point for the ecliptic longitude (and also the right ascension ) is in accordance with general custom of the vernal equinox. The ecliptic longitude Λ (t ) of the sun is obtained by the related to the perihelion of the Earth's orbit angle V (t ) is the angle (L) between perihelion P and the corresponding spring equinox place ( F) is added:

Taking into account the equation (8) instead of writing equation (7):

Partial result: The effected by the first cause of the equation of time non-uniform change in the attitude angle of the earth is in the non-uniform change in the ecliptic longitude of the sun converted (apparent motion of the Sun on the ecliptic ).

### Right Ascension of the True Sun

Task: which belongs to the non-uniform change in the length of the true solar ecliptic uneven change in its right ascension is to be calculated.

The second cause is the equation of time for the Earth's orbital plane non-perpendicular position of Earth's axis. Since only the component of motion parallel to the sun interested Äquatorebnene, the true Sun is " herunterzuprojizieren " of the ecliptic on the celestial equator ( see figure ).

The equatorial coordinate right ascension can be eg with the known transformation equations or with the following simple relationship in the corresponding right-angled spherical triangle ( see illustration ) from the ecliptic coordinate determine:

Is the angle between the ecliptic and equatorial circle ( the ecliptic ):

The value differs by only a few degrees from the value.

Partial result: With the determination of the right ascension of the true sun next to the first and the second cause of the equation of time is taken into account.

### Right Ascension of the Sun Middle

Task: The equation of motion of a uniformly running mean sun is to be determined ( = minuend in time equation (2a ) ).

The time equation (2a ) is proportional to the difference of the steady growth of a right ascension on the celestial equator evenly rotating fictitious mean sun and the uneven RA - growth ( Eq. ( 10) ) of the true sun.

The motion of this mean sun makes the uniform passage of time the same as that of the orbiting at the Earth's orbit fictional Earth ( point Y ) vividly. Its running is as close as possible to the coupling of the true sun so that it about " averages " the run. This was achieved with the following definition:

This also applies: Reminder: The values of and are negative.

Thus, the right ascension of the mean sun is at any time equal to the ecliptic longitude of the point Y ( see figures, fictional earth). By adding the ecliptic longitude of the perihelion to the mean anomaly of the fictional world created their ecliptic longitude and mean right ascension of the mean sun.

### Abstract: The equation of time

With the two partial equations (10) and (11) and with the time equation of the form

Values for any number of moments in the future calculated. In the conventional allocation of time values in equation calendar days (usually for 12h UT) of the current 0Uhr January 1 is selected as the time zero point. In the mean anomaly is to be observed according to equation ( 3a) the constant. For low accuracy requirements calculated for the period of the first year values can be used in the next few years again and again. Higher accuracy demands are met by the use of four different seasons equation of time tables for the years constant varies significantly only within a leap year cycle. After four years, the first of these four tables, etc. is used. For even higher claims the bill is made new with the now slightly modified so-called constants years for each year.

The annual constants and and accurate values for the very slowly varying values of, , and are given in astronomical almanacs.

### Calculation example

To calculate the equation of time for the April 2, 2015, 12:00 UT (t ' = (t - t K ) = 91 days).

The annual constants are:

The Rechnungenen are:

## Equation of time values for the passage excellent path points

From the calendar, and thus of the annual constant are independent equation of time values for the passage excellent points through the earth in its orbit (or by the sun on the ecliptic ). Spring, summer, autumn and winter start point called perihelion and aphelion

*) The values are for the year 2004 with L0 = -76.99 ° and Jtr = 365.2428 days.

Their calculation is simpler than that for any given time period, because the Kepler equation (4) does not need to be solved. Λ of the ecliptic preset length of one of the excellent points is easy to maintain (Eq. (8)) and on to the eccentric anomaly ( Eq. ( 5) ) to find. From the latter is followed by the mean anomaly (Eq. (4) ), ie the point on the path of the imaginary central earth. The ecliptic longitude of the perihelion to the latter added ( Gl.11 ) is the desired mean right ascension? M ( minuend in time equation (2a ) ). The true right ascension α ( subtrahend ) is to winter identical to the ecliptic length Λ at points spring. Only at points perihelion and aphelion, the coordinate transformation ( Eq. ( 10) ) results in small values differences.

In the procedure to start the calculation with a given ecliptic length or a given true anomaly, is obtained in addition to the equation of time and the time elapsed since the perihelion passage of time on earth. It is the time, which represents the mean anomaly, and is calculated from the intermediate result of the average anomaly by means of the switch in accordance with alternate ends of the following equation:

This procedure is sometimes recommended for the general work to determine the equation of time tables. It saves the time-consuming while solving the Kepler equation, takes on values for desired time points but only by trial and error or with sufficient earnings density by interpolating.

## Sunrise and sunset at winter solstice

The day of the winter solstice, usually December 21, is the shortest light of day. In contrast, the earliest sunset ( SU) will be held as early as December 10, the latest sunrise ( SA ), but only to the 5th of January. SA and SU are symmetrical in the midday sun peaking (12 clock WOZ ). Are you SA and SU in Standard Time ( CET), there is an asymmetry. Since moving WOZ and CET during the year because of the equation of time against each other, the equation of time also affects SA and SU. This is especially noticeable in January. The already increasing day length leads to an earlier and later SA SU. At the same time WOZ CET and move against each other so that both SA and SU shift to later times (given in CET). Both effects overlap and cause little change in the morning in January, during the day in the evening already noticeably growing.

## Analemma

To set the function of the equation of time from the declination of the sun as the diagram shows, creates a loop figure is referred to as Analemma. This loop shows the true position of the sun at 12 noon local mean time clock for different seasons as height above the celestial equator and the lateral distance from the meridian. This match in the lateral direction four times a degree in angular minutes. The figures shown on the left are north of the Tropic of Cancer with the line of sight to the south. They are stretched towards the sky figure in the horizontal direction by a factor of about five to six. On the right is marked on a vertical wall in a historic lunch Weiser the associated shadow curve.

The slight asymmetry between right and left stems from the fact that perihelion and winter solstice does not fall on the same day. The latter was the case recently in 1246 ( day of the winter solstice about as today). The inside interface was approximately for 16 April and 29 August (Gregorian calendar backward applied ).

In the year 6433 the perihelion is (today, ie from 2000 to 2025: January 2 to 5 ) have reached the day of the spring equinox. The equation of time will be a symmetric to this point figure on the days of the equinoxes zero and the analemma.

Most often, the analemma can be seen as an hourly loop on sundials, which are designed to display the mean solar time. Often it is, however, in two parts (one part like an S, and the other similar to a question mark) is divided, in order to prevent confusion in reading ( for declination there are two points around the figure). Each of the two parts, about one half a year. Such watches have two interchangeable dials. The dials are easy to interpret the valid zone at the installation time, so show the " normal time".

The every day at the same average time photographed sun is shining out of a standing analemma in the sky.

## Historical

The equation of time was known to the ancient astronomers. Geminus they mentioned. In the Almagest of Ptolemy it was pretty addressed accurately and concisely. The quantitative description used today is the equation of time John Flamsteed has already in 1672, so soon after the announcement of Kepler's laws, made.

In older yearbooks we find the equation of time with the opposite sign. The word part equation meant in the Middle Ages inflict a correction. They had to add the true solar time, the values of the equation of time to get at the mean time.

Today we read from the mean time of the ( always running smoothly ) Watches and added her the values of the equation of time to get the true solar time, which must indicate, for example a simple sundial. In the French yearbooks the old convention is still common.