Angle of the Sun by Time of Day 1155 Reading Blvd, Wyomissing, Pa

Calculating the Dominicus's location in the sky at a given time and place

The position of the Sun in the heaven is a part of both the time and the geographic location of observation on Earth'south surface. As Globe orbits the Sun over the grade of a year, the Lord's day appears to move with respect to the fixed stars on the celestial sphere, forth a circular path chosen the ecliptic.

Earth'south rotation near its axis causes diurnal motility, so that the Lord's day appears to move across the heaven in a Sunday path that depends on the observer'south geographic latitude. The time when the Lord's day transits the observer'due south tiptop depends on the geographic longitude.

To find the Sunday'southward position for a given location at a given time, one may therefore go on in 3 steps as follows:^{[1] [2]}

1. summate the Sunday's position in the ecliptic coordinate system,
2. convert to the equatorial coordinate organisation, and
3. convert to the horizontal coordinate system, for the observer's local time and location. This is the coordinate system unremarkably used to calculate the position of the Sun in terms of solar zenith bending and solar azimuth bending, and the ii parameters can be used to draw the Sun path.^[three]

This calculation is useful in astronomy, navigation, surveying, meteorology, climatology, solar energy, and sundial design.

Estimate position [edit]

Ecliptic coordinates [edit]

These equations, from the Astronomical Almanac,^{[4] [v]} tin can be used to calculate the credible coordinates of the Dominicus, mean equinox and ecliptic of date, to a precision of about 0°.01 (36″), for dates between 1950 and 2050. These equations are coded into a Fortran ninety routine in Ref.^[3] and are used to calculate the solar zenith angle and solar azimuth angle as observed from the surface of the Earth.

Commencement by calculating due north, the number of days (positive or negative, including fractional days) since Greenwich apex, Terrestrial Time, on i January 2000 (J2000.0). If the Julian date for the desired time is known, then

${\displaystyle due north=\mathrm {JD} -2451545.0}$

The hateful longitude of the Dominicus, corrected for the aberration of light, is:

${\displaystyle Fifty=280.460^{\circ }+0.9856474^{\circ }n}$

The hateful anomaly of the Sun (actually, of the World in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is:

${\displaystyle m=357.528^{\circ }+0.9856003^{\circ }northward}$

Put ${\displaystyle L}$ and ${\displaystyle g}$ in the range 0° to 360° by adding or subtracting multiples of 360° every bit needed.

Finally, the ecliptic longitude of the Lord's day is:

${\displaystyle \lambda =L+one.915^{\circ }\sin k+0.020^{\circ }\sin 2g}$

The ecliptic latitude of the Sun is nearly:

${\displaystyle \beta =0}$ ,

as the ecliptic breadth of the Sun never exceeds 0.00033°,^[six]

and the distance of the Dominicus from the Globe, in astronomical units, is:

${\displaystyle R=1.00014-0.01671\cos yard-0.00014\cos 2g}$ .

Obliquity of the ecliptic [edit]

Where the obliquity of the ecliptic is non obtained elsewhere, it can exist approximated:

${\displaystyle \epsilon =23.439^{\circ }-0.0000004^{\circ }north}$

Equatorial coordinates [edit]

${\displaystyle \lambda }$ , ${\displaystyle \beta }$ and ${\displaystyle R}$ grade a consummate position of the Sunday in the ecliptic coordinate system. This can be converted to the equatorial coordinate organisation by calculating the obliquity of the ecliptic, ${\displaystyle \epsilon }$ , and standing:

Right ascension,

${\displaystyle \blastoff =\arctan(\cos \epsilon \tan \lambda )}$ , where ${\displaystyle \blastoff }$ is in the aforementioned quadrant every bit ${\displaystyle \lambda }$ ,

To get RA at the correct quadrant on computer programs utilize double statement Arctan function such equally ATAN2(y,x)

${\displaystyle \blastoff =\arctan 2(\cos \epsilon \sin \lambda ,\cos \lambda )}$

and declination,

${\displaystyle \delta =\arcsin(\sin \epsilon \sin \lambda )}$ .

Rectangular equatorial coordinates [edit]

Right-handed rectangular equatorial coordinates in astronomical units are:

${\displaystyle X=R\cos \lambda }$

${\displaystyle Y=R\cos \epsilon \sin \lambda }$

${\displaystyle Z=R\sin \epsilon \sin \lambda }$

Where ${\displaystyle X}$ centrality is in the direction of the March equinox, the ${\displaystyle Y}$ axis towards June Solstice, and the ${\displaystyle Z}$ centrality towards the North celestial pole.^[vii]

Horizontal coordinates [edit]

Declination of the Sun as seen from World [edit]

The path of the Dominicus over the celestial sphere through the course of the day for an observer at 56°N breadth. The Sun's path changes with its declination during the year. The intersections of the curves with the horizontal axis prove azimuths in degrees from North where the Sun rises and sets.

Overview [edit]

The Dominicus appears to move northward during the northern spring, crossing the celestial equator on the March equinox. Its declination reaches a maximum equal to the angle of Earth's centric tilt (23.44°)^{[eight] [9]} on the June solstice, then decreases until reaching its minimum (−23.44°) on the December solstice, when its value is the negative of the axial tilt. This variation produces the seasons.

A line graph of the Sun's declination during a year resembles a sine wave with an aamplitude of 23.44°, but one lobe of the wave is several days longer than the other, amongst other differences.

The following phenomena would occur if Earth were a perfect sphere, in a circular orbit around the Sun, and if its axis is tilted 90°, and so that the axis itself is on the orbital plane (similar to Uranus). At one appointment in the twelvemonth, the Lord's day would be direct overhead at the Due north Pole, so its declination would exist +ninety°. For the next few months, the subsolar point would move toward the South Pole at abiding speed, crossing the circles of latitude at a constant rate, so that the solar declination would decrease linearly with time. Eventually, the Sunday would be directly above the South Pole, with a declination of −ninety°; then it would start to move due north at a constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a triangle moving ridge rather than a sine wave, zigzagging betwixt plus and minus ninety°, with linear segments between the maxima and minima.

If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less astute, being curved to resemble the maxima and minima of a sine wave. Withal, even when the centric tilt equals that of the bodily Earth, the maxima and minima remain more acute than those of a sine wave.

In reality, Earth's orbit is elliptical. Earth moves more rapidly around the Sun near perihelion, in early on Jan, than most aphelion, in early July. This makes processes like the variation of the solar declination happen faster in Jan than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of alter before and afterward are non quite equal.

The graph of apparent solar declination is therefore different in several means from a sine wave. Computing information technology accurately involves some complexity, as shown below.

Calculations [edit]

The declination of the Sun, δ_☉, is the bending between the rays of the Sunday and the plane of the Globe'south equator. The World'south centric tilt (called the obliquity of the ecliptic by astronomers) is the angle between the Earth'southward axis and a line perpendicular to the Globe's orbit. The Globe'south axial tilt changes slowly over thousands of years only its current value of virtually ε = 23°26' is about constant, so the change in solar declination during one yr is nearly the aforementioned equally during the side by side year.

At the solstices, the angle between the rays of the Sun and the airplane of the World'southward equator reaches its maximum value of 23°26'. Therefore, δ_☉ = +23°26' at the northern summertime solstice and δ_☉ = −23°26' at the southern summer solstice.

At the moment of each equinox, the center of the Sun appears to pass through the celestial equator, and δ_☉ is 0°.

The Sun's declination at any given moment is calculated by:

${\displaystyle \delta _{\odot }=\arcsin \left[\sin \left(-23.44^{\circ }\right)\cdot \sin \left(EL\right)\right]}$

where EL is the ecliptic longitude (substantially, the World'due south position in its orbit). Since the Globe's orbital eccentricity is modest, its orbit can exist approximated as a circle which causes up to 1° of fault. The circle approximation means the EL would be ninety° ahead of the solstices in World'south orbit (at the equinoxes), so that sin(EL) tin be written as sin(ninety+NDS)=cos(NDS) where NDS is the number of days after the Dec solstice. Past as well using the approximation that arcsin[sin(d)·cos(NDS)] is close to d·cos(NDS), the following often used formula is obtained:

${\displaystyle \delta _{\odot }=-23.44^{\circ }\cdot \cos \left[{\frac {360^{\circ }}{365}}\cdot \left(Due north+10\right)\correct]}$

where N is the day of the year commencement with North=0 at midnight Universal Time (UT) as Jan 1 begins (i.due east. the days office of the ordinal date −1). The number 10, in (North+10), is the approximate number of days afterward the December solstice to January i. This equation overestimates the declination near the September equinox by up to +1.5°. The sine office approximation by itself leads to an error of up to 0.26° and has been discouraged for use in solar energy applications.^[ii] The 1971 Spencer formula^[10] (based on a Fourier series) is also discouraged for having an error of upwards to 0.28°.^[eleven] An boosted error of up to 0.5° can occur in all equations around the equinoxes if not using a decimal place when selecting Due north to suit for the time later UT midnight for the beginning of that day. So the above equation can have up to 2.0° of error, about four times the Dominicus's angular width, depending on how it is used.

The declination can exist more than accurately calculated by non making the two approximations, using the parameters of the World's orbit to more accurately estimate EL:^[12]

${\displaystyle \delta _{\odot }=\arcsin \left[\sin \left(-23.44^{\circ }\correct)\cdot \cos \left({\frac {360^{\circ }}{365.24}}\left(Northward+10\right)+{\frac {360^{\circ }}{\pi }}\cdot 0.0167\sin \left({\frac {360^{\circ }}{365.24}}\left(Due north-2\correct)\right)\right)\correct]}$

which can exist simplified by evaluating constants to:

${\displaystyle \delta _{\odot }=-\arcsin \left[0.39779\cos \left(0.98565^{\circ }\left(North+x\right)+1.914^{\circ }\sin \left(0.98565^{\circ }\left(N-2\right)\right)\right)\right]}$

North is the number of days since midnight UT as January 1 begins (i.e. the days part of the ordinal appointment −i) and can include decimals to adjust for local times later or earlier in the day. The number 2, in (Northward-2), is the approximate number of days subsequently January 1 to the Earth's perihelion. The number 0.0167 is the current value of the eccentricity of the Earth'south orbit. The eccentricity varies very slowly over time, only for dates fairly close to the nowadays, it can exist considered to exist constant. The largest errors in this equation are less than ± 0.two°, but are less than ± 0.03° for a given twelvemonth if the number x is adjusted up or downward in fractional days equally determined by how far the previous year'due south Dec solstice occurred before or after apex on December 22. These accuracies are compared to NOAA'southward avant-garde calculations^{[thirteen] [14]} which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°.^[15]

(The in a higher place formula is related to a reasonably simple and accurate calculation of the Equation of Time, which is described hither.)

More than complicated algorithms^{[16] [17]} correct for changes to the ecliptic longitude by using terms in improver to the 1st-order eccentricity correction above. They also correct the 23.44° obliquity which changes very slightly with time. Corrections may besides include the furnishings of the moon in offsetting the Earth's position from the heart of the pair's orbit effectually the Dominicus. After obtaining the declination relative to the center of the Earth, a further correction for parallax is practical, which depends on the observer's distance away from the center of the World. This correction is less than 0.0025°. The error in calculating the position of the center of the Dominicus tin can be less than 0.00015°. For comparison, the Sun'south width is about 0.5°.

Atmospheric refraction [edit]

The declination calculations described above do non include the effects of the refraction of light in the atmosphere, which causes the credible angle of elevation of the Sunday as seen by an observer to be higher than the bodily angle of acme, particularly at low Sun elevations.^[2] For example, when the Lord's day is at an top of 10°, it appears to be at 10.1°. The Sun's declination can be used, along with its correct rising, to calculate its azimuth and also its true tiptop, which can then exist corrected for refraction to give its apparent position.^{[2] [fourteen] [eighteen]}

Equation of time [edit]

The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and beneath the axis a sundial volition announced slow.

In improver to the annual north–south oscillation of the Sunday's apparent position, corresponding to the variation of its declination described above, at that place is also a smaller but more complex oscillation in the east–westward direction. This is caused past the tilt of the World's axis, and too by changes in the speed of its orbital motion around the Sun produced by the elliptical shape of the orbit. The main effects of this east–west oscillation are variations in the timing of events such equally sunrise and sunset, and in the reading of a sundial compared with a clock showing local mean time. As the graph shows, a sundial tin be up to about 16 minutes fast or tedious, compared with a clock. Since the Earth rotates at a mean speed of one caste every four minutes, relative to the Sun, this 16-minute displacement corresponds to a shift e or westward of almost four degrees in the apparent position of the Sun, compared with its hateful position. A westward shift causes the sundial to exist ahead of the clock.

Since the main effect of this oscillation concerns time, it is chosen the equation of time, using the discussion "equation" in a somewhat archaic sense significant "correction". The oscillation is measured in units of time, minutes and seconds, respective to the amount that a sundial would exist ahead of a clock. The equation of time can be positive or negative.

Analemma [edit]

An analemma is a diagram that shows the annual variation of the Dominicus's position on the celestial sphere, relative to its mean position, equally seen from a stock-still location on Earth. (The discussion analemma is also occasionally, but rarely, used in other contexts.) It can exist considered as an image of the Sun'south apparent motion during a year, which resembles a effigy-eight. An analemma tin be pictured by superimposing photographs taken at the aforementioned time of mean solar day, a few days apart for a yr.

An analemma can likewise be considered as a graph of the Sun's declination, usually plotted vertically, against the equation of fourth dimension, plotted horizontally. Ordinarily, the scales are chosen then that equal distances on the diagram stand for equal angles in both directions on the celestial sphere. Thus iv minutes (more than precisely iii minutes, 56 seconds), in the equation of time, are represented past the same distance as i° in the declination, since Earth rotates at a hateful speed of ane° every iv minutes, relative to the Sun.

An analemma is drawn as it would be seen in the sky past an observer looking up. If n is shown at the top, then due west is to the right. This is usually done even when the analemma is marked on a geographical globe, on which the continents, etc., are shown with westward to the left.

Some analemmas are marked to show the position of the Sun on the graph on diverse dates, a few days apart, throughout the year. This enables the analemma to be used to make simple analog computations of quantities such as the times and azimuths of sunrise and sunset. Analemmas without appointment markings are used to right the time indicated by sundials.^[xix]

Meet also [edit]

• Ecliptic
• Issue of Sun angle on climate
• Newcomb'southward Tables of the Sunday
• Solar azimuth bending
• Solar elevation bending
• Solar irradiance
• Solar fourth dimension
• Sun path
• Sunrise equation

References [edit]

1. ^ Meeus, Jean (1991). "Chapter 12: Transformation of Coordinates". Astronomical Algorithms. Richmond, VA: Willmann Bell, Inc. ISBN0-943396-35-2.
2. ^ ^{a b c d} Jenkins, Alejandro (2013). "The Lord's day's position in the heaven". European Journal of Physics. 34 (3): 633. arXiv:1208.1043. Bibcode:2013EJPh...34..633J. doi:10.1088/0143-0807/34/3/633.
3. ^ ^{a b} Zhang, T., Stackhouse, P.West., Macpherson, B., and Mikovitz, J.C., 2021. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 part. Renewable Energy, 172, 1333-1340. DOI: https://doi.org/10.1016/j.renene.2021.03.047
4. ^ U.S. Naval Observatory; U.K. Hydrographic Office, H.M. Nautical Annual Office (2008). The Astronomical Almanac for the Year 2010. U.Due south. Govt. Printing Part. p. C5. ISBN978-0-7077-4082-nine.
5. ^ Much the same prepare of equations, roofing the years 1800 to 2200, can exist found at Approximate Solar Coordinates, at the U.S. Naval Observatory website Archived 2016-01-31 at the Wayback Machine. Graphs of the error of these equations, compared to an authentic ephemeris, tin as well be viewed.
6. ^ Meeus (1991), p. 152
7. ^ U.S. Naval Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.). Explanatory Supplement to the Astronomical Almanac. University Science Books, Mill Valley, CA. p. 12. ISBN0-935702-68-7.
8. ^ "Selected Astronomical Constants, 2022 (PDF)" (PDF). United states Naval Observatory. 2014. p. K6–K7.
9. ^ "Selected Astronomical Constants, 2022 (TXT)". Us Naval Observatory. 2014. p. K6–K7.
10. ^ J. W. Spencer (1971). "Fourier serial representation of the position of the sun".
11. ^ Sproul, Alistair B. (2007). "Derivation of the solar geometric relationships using vector analysis". Renewable Energy. 32: 1187–1205. doi:10.1016/j.renene.2006.05.001.
12. ^ "SunAlign". Archived from the original on 9 March 2012. Retrieved 28 February 2012.
13. ^ "NOAA Solar Calculator". Earth Organisation Research Laboratory. Retrieved 28 February 2012.
14. ^ ^{a b} "Solar Calculation Details". Globe Organisation Research Laboratory. Retrieved 28 Feb 2012.
15. ^ "Astronomical Algorithms". Retrieved 28 February 2012.
16. ^ Blanco-Muriel, Manuel; Alarcón-Padilla, Diego C; López-Moratalla, Teodoro; Lara-Coira, Martín (2001). "Computing the Solar Vector" (PDF). Solar Free energy. 70 (5): 431–441. Bibcode:2001SoEn...lxx..431B. doi:10.1016/s0038-092x(00)00156-0.
17. ^ Ibrahim Reda & Afshin Andreas. "Solar Position Algorithm for Solar Radiations Applications" (PDF) . Retrieved 28 February 2012.
18. ^ "Atmospheric Refraction Approximation". National Oceanic and Atmospheric Administration. Retrieved 28 Feb 2012.
19. ^ Sundial#Noon marks

External links [edit]

• Solar Position Algorithm, at National Renewable Energy Laboratory's Renewable Resources Information Centre website.
• Sun Position Calculator, at pveducation.org. An interactive calculator showing the Sun's path in the sky.
• NOAA Solar Calculator, at the NOAA Earth Organization Research Laboratory'south Global Monitoring Division website.
• NOAA's declination and sun position calculator
• HORIZONS Organisation, at the JPL website. Very authentic positions of Solar System objects based on the JPL DE serial ephemerides.
• General ephemerides of the solar arrangement bodies, at the IMCCE website. Positions of Solar System objects based on the INPOP series ephemerides.
• Solar position in R. Insol package.

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Source: https://en.wikipedia.org/wiki/Position_of_the_Sun

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