Sun path
The Sun is the apparent instantaneous position of the sun above the horizon landscape of a location. It is constantly changing as a result of Earth's rotation and can be specified by two angles on the celestial sphere: the direction and the height ( elevation angle ) of the sun.
The daily course of the sun hangs next to the location also depends on the season and is characterized by three distinctive points that are following in Central Europe: sunrise ( between Northeast and Southeast ), noonday peak ( in the south) and sunset ( between Northwest and Southwest ). Morning or in the evening we speak of the sun is low, around noon (especially in the summer months ) from the sun is high. The difference between winter and summer coined the terms low, or high solar orbit.
The seasonal course of the sun determines the latitude of the location. This calculated solar diagrams show the variation of elevation angle and horizontal angle ( azimuth) for a whole year in the horizontal coordinate system. Parameter for this is the day and the season. You are encodings of hour angle and declination of the sun in the equatorial coordinate system. With the Latitude for translating into direction and elevation angle.
In a sundial this parameter lines are used to reading the time of day ( hour lines ) and limited extent also the time of year ( day lines ).
- 3.1 Nature and Man
- 3.2 Human Culture
- 4.1 Calculation results used in year charts
- 4.2 Accurate calculation of the sun for a time
- 4.3 Ekliptikalkoordinate the sun
- 4.4 equatorial coordinates of the sun
- 4.5 Horizontal coordinates of the sun
- 4.6 Correction of the amount due to refraction
- 4.7 Calculation Example
- 4.8 Comparison of Accuracy
Hour angle and analemma
By the end of the Middle Ages the hour angle of the sun was used as a measure of the time of day. He gives the hours before / after local noon, which is why he has this name. Because the movement of the sun up to 15 minutes is inconsistent with the seasons, the so-called equation of time was introduced for correction. It specifies to be corrected as much the true solar time, to get to the uniform mean solar time. In the sun diagrams, the time scale is distorted in order to read the position of the true sun at a given mean solar time can. Because the correction every season is different, the true hour lines are not only shifted, but replaced by the typical double loops called the analemma.
Conversely, can be read off the time of day from the position of the sun. The Analemmata give the local mean time or shift to the correct longitude zone time (in Central Europe CET). The drawn on a sphere sun chart (see figure), the situation on the celestial sphere represents the realistic Skaphe, an ancient sundial, used as a projection also a spherical surface.
With the sun chart can also tanning of a building or the usable solar energy charge of a place. But while the theoretical sunshine duration of each month is only dependent on the latitude, the actual duration of sunshine is also subject to meteorological conditions (clouds, haze ) and the height of the landscape horizon.
Observation of the sun
Daily Sun ( diurnal arc )
The diurnal arc of the sun is of running over the horizon of their apparent daily circulation in the sky. The theoretical diurnal arc begins at Sunrise and ends with the astronomical astronomical downfall. The actual dawn and demise takes place about 3-4 minutes earlier or later due to the refraction of light in the atmosphere. The height of the horizon landscape (mountains, buildings) counteracts - about 6-8 minutes per degree.
The diurnal arc starts at the eastern horizon and ends in the west. The mnemonic
Applies to a mean latitude in the northern hemisphere - and also for the southern half of the earth when the South and North interchanged. The up and sunsets in Central Europe but soft from east or west from depending on the season by up to 45 °.
The moment of meridian passage of the sun ( approximately its culmination) is the True lunch which deviates by no more than about ± 15 min from the Middle noon throughout the year. From the zone time (12 clock CET) he differs from addition by a constant value ( for CET 15 ° östl.Greenwich ) results from the longitude difference from the standard meridian.
Saisoneller sun ( change of height and length of the days arc )
The diurnal arc is higher in summer and longer than in winter. For Central Europe the difference is about 16: 8 hours.
His lunch amounted to an example at ± 50 ° latitude 63.45 ° at the summer solstice and the winter solstice 16.55 °.
Statement: angle between horizon and pole ± obliquity of the ecliptic; in the example: 90 ° - 50 ° ± 23.44 ° to 63.44 ° and 16.56 ° equal.
In the tropics the sun is at noon once a year, at the zenith (90 ° elevation ), between the tropics and the equator, however, twice. Beyond the Arctic Circle in alljährlichem rhythm occurs with midnight sun and polar night to the effect that the sun for a few weeks neither - nor goes. Solar diagrams for such places extend for 24 hours or 360 ° azimuth.
The azimuth α for the location of the sunrise or sunset times vary throughout the year relative to the East or West Point, for example, in 50 ° latitude by ± 38.25 ° to the north or to the south. The hour angle for the moment of sunrise and sunset vary in places that width of ± 31.13 ° at λ = -90 ° ( rising ) or at λ = 90 ° ( sinking ) (longest / shortest day: 16 h 9 min / 7 h 51 min = 12 h ± 31.13 ° · 4 min / °).
Effects of the sun
Nature and Man
From the Sun and its variability depend on a number of important variables, especially
- The intensity of the solar radiation. From it also revealed
- Climate zone (along with the moisture and clouds conditions) and the types of vegetation
- The need for heating or cooling of
- The emergence of local winds ( see, eg, updraft ) and cloud formation, but also
- Regional winds (such as monsoon ) and many ocean currents
- The development of settlement structures, especially in the mountains.
Human Culture
The measurement of the sun by sundials allows people for thousands of years to determine the time of day. The division into seasons corresponding to the days arc height of the sun. The first determination of the Earth's diameter by Eratosthenes was performed by simultaneous measurement of the sun at two different points on the Earth's surface. The measurement of the sun with the help of simple measuring instruments was also an early method of navigation.
The Daily " path of the sun across the sky " at various mythologies plays a major role, such as Helios ' " chariot " of ancient Greece and in the interpretation of sunrise and sunset. Inhabitants of the northern hemisphere are often when they travel to the southern hemisphere, amazed at the " reversal " of the daily apparent motion of the sun " to the left".
Calculation of the sun
Calculation results used in year charts
Simple solar diagrams are parameterized with the real local time. Further simplification is the assumption that do not change the daily paths of the Sun from year to year.
Even in modern practical handling of the sun year - solar diagrams are used. Because of the parameterization with medium spatial - or time zone and the consideration of a very slow change in the apparent path of the sun ( shift of the vernal equinox ) they are the result of relatively accurate calculations, but the accuracy achieved because of the form of representation can not exploit. Also in -drawn diagrams with higher resolution at best, based on the four-year switching rhythm small differences can be seen, so you can use the currently valid diagram for many years repeated.
Exact calculation of the sun for a time
The above-mentioned relatively accurate calculation is described in the following. Its most important feature is the reference to a uniform passage of time, which is not the true solar time. The uneven annual course of the sun, and the equation of time lies at the bottom, is taken into account. Of the additional long-term influences (eg according to the planetary theory VSOP87 ) is taken into account only the change in the course of the sun in the form of displacement of the vernal equinox to the perigee of the Earth's orbit ellipse in contrast to the usual astronomical calculations.
The following statement is not fundamentally different from that for the equation of time. It can be found in the corresponding variant to determine the equation of time also. Also, in the equation of time with higher accuracy eliminates the approximation to the frequency with the year. Each position of the sun for a point is obtained in an arbitrarily long time axis.
Ekliptikalkoordinate the sun
As time variable, the number of days since the Standardäquinoktium J2000.0 (1 January 2000, 12 clock 12 clock TT ≈ UT) used ( if necessary including the fraction of the day in UT).
If the Julian day number of the desired point in time, it shall
.
The position of the Sun on the ecliptic will initially be determined without consideration caused by the Erdbahnelliptizität velocity fluctuations. It is a mean velocity of the sun at (360 ° in approximately 365.2422 days ) and obtains the mean ecliptic longitude of the Sun:
.
Subsequently to take into account the influence of the Bahnelliptizität and to maintain the length ecliptic, is to add this as a correction, the so-called midpoint equation. This correction depends on the angle between the sun and perihelion, the so-called anomaly. The center expects the equation (fictitious ) uniformly increasing mean anomaly as input. This grows by 360 ° in an anomalistic year to about 365.2596 days:
.
The midpoint equation is a periodic function of the average anomaly and thus can be broken down into a Fourier series. For small Bahnexzentrizitäten can be canceled the series after a few terms. Taking into account the ( numerical ) eccentricity only linear and quadratic terms, this is the center point equation
.
With and conversion, this reduces the ecliptic longitude of the Sun:
.
Note: The invoice will be clearer if you and brought by adding or subtracting appropriate multiples of 360 ° in the range between 0 ° and 360 °.
Alternatively to the use of the center point equation can be calculated from the mean length of the ecliptic length with the aid of the Kepler equations, which, however, requires an iterative solution method.
Equatorial coordinates of the sun
For the thus calculated, counted along the ecliptic, ecliptic longitude now the associated counted along the celestial equator right ascension must be determined. With the obliquity of the ecliptic
Gives the right ascension as
.
Note: If the denominator in the argument of the arctan has a value less than zero, 180 ° must be added to the result to get the angle in the correct quadrant (must lie in the same quadrant as ). For a more detailed explanation of the quadrant determination, see the article in position angle.
As an alternative to exact formula used here can be used for the calculation of a series expansion, which is also possible in the time equation.
The perpendicular to the celestial equator counted declination is obtained as
.
Horizontal coordinates of the Sun
The purpose of calculating the sun's position for a given time are azimuth ( direction ) and height of the sun. First is to determine the right ascension of the hour angle of the sun.
To this end, we determine the Julian day number for 0h UT of the considered date, compute
And thus the mean sidereal time at Greenwich for this time ( universal time UT in hours):
The sidereal time is the hour angle of the vernal equinox, expressed in the time scale (). Are integer multiples of 24 can optionally be removed from the result. Multiplying by the conversion factor of 15 ° / h returns the Greenwich hour angle of the vernal equinox in degrees:
For a place on the longitude (east positive counted ) is the hour angle of the vernal equinox
And subtracting the right ascension of the sun supplies the hour angle of the sun for that place:
Azimuth and elevation angles can be calculated with the latitude to
Or to
Note: If the denominator in the argument of the arctan has a value less than zero, 180 ° must be added to the result to get the angle in the correct quadrant.
The calculated azimuth is counted from south. Should it be counted from the north, 180 ° to add to earnings.
Correction of the height due to refraction
Finally, if required, is still the refraction to be considered ( light refraction in the atmosphere), which allows the solar disk appear slightly higher than it actually is. The mean refraction ( in minutes of arc ) for an object that is located on the height h ( in degrees) can be calculated approximately by
.
The refraktionsbehaftete height in degrees is then
.
It is noted that the refraction is dependent on the detailed state of the atmosphere. The formula given assumes an atmospheric pressure of 1010 mbar and a temperature of 10 ° C. Deviating conditions can be taken into account by appropriate corrections, but even then the formula describes only a mean refraction, while the actual values , especially in the immediate vicinity horizon depending on the current temperature stratification may differ markedly from that agent.
Calculation example
It is the position of the sun for the August 6, 2006 at 8 clock CEST ( = 6 clock UT ) in Munich ( = 48.1 ° N, 11.6 ° = O) to be determined. This results in
Note: The accounts shall be kept with a sufficient number of digits ( for example, double precision, with eight-digit calculators caution ); sufficient, particularly for many locations need to be considered. It should be noted that some computer programs and programming languages expect angles in radians and not in degrees; the angles are then converted accordingly.
Comparison of Accuracy
As the following chart shows, reaching the calculated values here for the Sun for the period 1950-2050, an accuracy of about 0.01 °. Most striking is the difference in ecliptic longitude with a regular period of 18.6 years and an amplitude of 0.0047 °; is in the bill is not taken into account nutation in length. Towards the edges of the graph towards the variation in the residual error grows significantly. This is caused by the change did not take account of the eccentricity of Earth's orbit, which had been recognized in the calculation of the coefficients of the midpoint equation to be constant with the value for the year 2000. This error has the anomalistic year as a period; its amplitude grows in 100 years to 0.0048 °. Furthermore, neglecting those perturbations, which directly affect the ecliptic longitude; especially the interference by Jupiter ( terms with amplitudes 0.0019 °, 0.0014 °, ...), Moon ( terms with amplitudes 0.0017 °, ...), Mars ( terms with amplitudes 0.0014 °, 0.0011 °, ...) and Venus ( terms with amplitudes 0.0014 °, 0.0011 °, ...). That the ecliptic latitude was tacitly constant set to zero produces no appreciable error. The calculated coordinates and the comparative data are for a geocentric observer; for a real observer on the Earth's surface, the observed position of the sun by up to 0.0024 ° deviate ( the solar parallax ) of it.
If more detailed data is required, this can be calculated with more complex procedures or purchased from one of the numerous Ephemeridenserver on the web (see links).