Remote-Url: https://en.wikipedia.org/wiki/Sunrise_equation Retrieved-at: 2021-11-28 21:58:49.466006+00:00 Sunrise equation From Wikipedia, the free encyclopedia Jump to navigation Jump to search Equation to derive time of sunset and sunrise This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. [50px-Q] Unsourced material may be challenged and removed. Find sources: "Sunrise equation" ? news ? newspapers ? books ? scholar ? JSTOR (June 2018) (Learn how and when to remove this template message) [400px-Hours_of_daylight_vs_latitude_vs_day_of_year_with_] A contour plot of the hours of daylight as a function of latitude and day of the year, using the most accurate models described in this article. It can be seen that the area of constant day and constant night reach up to the polar circles (here labeled "Anta. c." and "Arct. c."), which is a consequence of the earth's inclination. File:Daylight Hours.webmPlay media A plot of hours of daylight as a function of the date for changing latitudes. This plot was created using the simple sunrise equation, approximating the sun as a single point and does not take into account effects caused by the atmosphere or the diameter of the Sun. The sunrise equation can be used to derive the time of sunrise and sunset for any solar declination and latitude in terms of local solar time when sunrise and sunset actually occur. It is: cos ? ? ? = ? tan ? ? ? tan ? ? {\displaystyle \cos \omega _{\circ }=-\tan \phi \times \tan \delta } \cos \omega _{\circ }=-\tan \phi \times \tan \ delta where: ? ? {\displaystyle \omega _{\circ }} \omega _{\circ } is the hour angle at either sunrise (when negative value is taken) or sunset (when positive value is taken); ? {\displaystyle \phi } \phi is the latitude of the observer on the Earth; ? {\displaystyle \delta } \delta is the sun declination. [ ] Contents * 1 Theory of the equation + 1.1 Hemispheric relation * 2 Generalized equation * 3 Complete calculation on Earth + 3.1 Calculate current Julian day + 3.2 Mean solar time + 3.3 Solar mean anomaly + 3.4 Equation of the center + 3.5 Ecliptic longitude + 3.6 Solar transit + 3.7 Declination of the Sun + 3.8 Hour angle + 3.9 Calculate sunrise and sunset * 4 See also * 5 References * 6 External links Theory of the equation[edit] The Earth rotates at an angular velocity of 15?/hour. Therefore, the expression ? ? / 15 ? {\displaystyle \omega _{\circ }/\mathrm {15} ^{\circ }} {\ displaystyle \omega _{\circ }/\mathrm {15} ^{\circ }}, where ? ? {\displaystyle \omega _{\circ }} \omega _{\circ } is in degree, gives the interval of time in hours from sunrise to local solar noon or from local solar noon to sunset. The sign convention is typically that the observer latitude ? {\displaystyle \ phi } \phi is 0 at the equator, positive for the Northern Hemisphere and negative for the Southern Hemisphere, and the solar declination ? {\ displaystyle \delta } \delta is 0 at the vernal and autumnal equinoxes when the sun is exactly above the equator, positive during the Northern Hemisphere summer and negative during the Northern Hemisphere winter. The expression above is always applicable for latitudes between the Arctic Circle and Antarctic Circle. North of the Arctic Circle or south of the Antarctic Circle, there is at least one day of the year with no sunrise or sunset. Formally, there is a sunrise or sunset when ? 90 ? + ? < ? < 90 ? ? ? {\displaystyle -90^{\circ }+\delta <\phi <90^{\circ }-\delta } -90^{\circ }+\ delta <\phi <90^{\circ }-\delta during the Northern Hemisphere summer, and when ? 90 ? ? ? < ? < 90 ? + ? {\displaystyle -90^{\circ }-\delta <\phi <90^{\circ } +\delta } -90^{\circ }-\delta <\phi <90^{\circ }+\delta during the Northern Hemisphere winter. For locations outside these latitudes, it is either 24-hour daytime or 24-hour nighttime. Hemispheric relation[edit] Suppose ? N {\displaystyle \phi _{N}} {\displaystyle \phi _{N}} is a given latitude in Northern Hemisphere, and ? ? N {\displaystyle \omega _{\circ N}} {\ displaystyle \omega _{\circ N}} is the corresponding sunrise hour angle that has a negative value, and similarly, ? S {\displaystyle \phi _{S}} \phi _{S} is the same latitude but in Southern Hemisphere, which means ? S = ? ? N {\ displaystyle \phi _{S}=-\phi _{N}} {\displaystyle \phi _{S}=-\phi _{N}}, and ? ? S {\displaystyle \omega _{\circ S}} {\displaystyle \omega _{\circ S}} is the corresponding sunrise hour angle, then it is apparent that cos ? ? ? S = ? cos ? ? ? N = cos ? ( ? 180 ? ? ? ? N ) {\displaystyle \cos \omega _{\circ S}=-\cos \omega _{\circ N}=\cos(-180^{\circ }-\omega _{\circ N})} {\displaystyle \cos \omega _{\circ S}=-\cos \omega _{\circ N}=\cos (-180^{\circ }-\omega _{\circ N})}, which means ? ? N + ? ? S = ? 180 ? {\displaystyle \omega _{\circ N}+\omega _{\circ S}= -180^{\circ }} {\displaystyle \omega _{\circ N}+\omega _{\circ S}=-180^{\ circ }}. The above relation implies that on the same day, the lengths of daytime from sunrise to sunset at ? N {\displaystyle \phi _{N}} {\displaystyle \phi _{N}} and ? S {\displaystyle \phi _{S}} \phi _{S} sum to 24 hours if ? S = ? ? N {\ displaystyle \phi _{S}=-\phi _{N}} {\displaystyle \phi _{S}=-\phi _{N}}, and this also applies to regions where polar days and polar nights occur. This further suggests that the global average of length of daytime on any given day is 12 hours without considering the effect of atmospheric refraction. Generalized equation[edit] [220px-Corrections_for_Sextant_] Sextant sight reduction procedure showing solar altitude corrections for refraction and elevation. The equation above neglects the influence of atmospheric refraction (which lifts the solar disc ? i.e. makes the solar disc appear higher in the sky ? by approximately 0.6? when it is on the horizon) and the non-zero angle subtended by the solar disc ? i.e. the apparent diameter of the sun ? (about 0.5?). The times of the rising and the setting of the upper solar limb as given in astronomical almanacs correct for this by using the more general equation cos ? ? ? = sin ? a ? sin ? ? ? sin ? ? cos ? ? ? cos ? ? {\displaystyle \ cos \omega _{\circ }={\dfrac {\sin a-\sin \phi \times \sin \delta }{\cos \ phi \times \cos \delta }}} {\displaystyle \cos \omega _{\circ }={\dfrac {\ sin a-\sin \phi \times \sin \delta }{\cos \phi \times \cos \delta }}} with the altitude angle (a) of the center of the solar disc set to about ?0.83? (or ?50 arcminutes). The above general equation can be also used for any other solar altitude. The NOAA provides additional approximate expressions for refraction corrections at these other altitudes.^[1] There are also alternative formulations, such as a non-piecewise expression by G.G. Bennett used in the U.S. Naval Observatory's "Vector Astronomy Software".^[2] Complete calculation on Earth[edit] The generalized equation relies on a number of other variables which need to be calculated before it can itself be calculated. These equations have the solar-earth constants substituted with angular constants expressed in degrees. Calculate current Julian day[edit] n = ? J d a t e ? 2451545.0 + 0.0008 ? {\displaystyle n=\lceil J_{date} -2451545.0+0.0008\rceil } {\displaystyle n=\lceil J_{date}-2451545.0+0.0008 \rceil } where: n {\displaystyle n} n is the number of days since Jan 1st, 2000 12:00. J d a t e {\displaystyle J_{date}} J_{{date}} is the Julian date; 2451545.0 is the equivalent Julian year of Julian days for Jan-01-2000, 12:00:00. 0.0008 is the fractional Julian Day for leap seconds and terrestrial time. TT was set to 32.184 sec lagging TAI on 1 January 1958. By 1972, when the leap second was introduced, 10 sec were added. By 1 January 2017, 27 more seconds were added coming to a total of 69.184 sec. 0.0008=69.184 / 86400 without DUT1. The ? ? ? {\displaystyle \lceil \cdot \rceil } \lceil \cdot \rceil operation rounds up to the next integer day number n. Mean solar time[edit] J ? = n ? l w 360 ? {\displaystyle J^{\star }=n-{\dfrac {l_{w}}{360^{\circ }}}} {\displaystyle J^{\star }=n-{\dfrac {l_{w}}{360^{\circ }}}} where: J ? {\displaystyle J^{\star }} J^{{\star }} is an approximation of mean solar time at n {\displaystyle n} n expressed as a Julian day with the day fraction. l ? {\displaystyle l_{\omega }} l_{\omega } is the longitude (west is negative, east is positive) of the observer on the Earth; Solar mean anomaly[edit] M = ( 357.5291 + 0.98560028 ? J ? ) mod 3 60 {\displaystyle M= (357.5291+0.98560028\times J^{\star }){\bmod {3}}60} {\displaystyle M= (357.5291+0.98560028\times J^{\star }){\bmod {3}}60} where: M is the solar mean anomaly used in the next three equations. Equation of the center[edit] C = 1.9148 sin ? ( M ) + 0.0200 sin ? ( 2 M ) + 0.0003 sin ? ( 3 M ) {\ displaystyle C=1.9148\sin(M)+0.0200\sin(2M)+0.0003\sin(3M)} {\displaystyle C=1.9148\sin(M)+0.0200\sin(2M)+0.0003\sin(3M)} where: C is the Equation of the center value needed to calculate lambda (see next equation). 1.9148 is the coefficient of the Equation of the Center for the planet the observer is on (in this case, Earth) Ecliptic longitude[edit] ? = ( M + C + 180 + 102.9372 ) mod 3 60 {\displaystyle \lambda = (M+C+180+102.9372){\bmod {3}}60} {\displaystyle \lambda =(M+C+180+102.9372) {\bmod {3}}60} where: ? is the ecliptic longitude. 102.9372 is a value for the argument of perihelion. Solar transit[edit] J t r a n s i t = 2451545.0 + J ? + 0.0053 sin ? M ? 0.0069 sin ? ( 2 ? ) {\displaystyle J_{transit}=2451545.0+J^{\star }+0.0053\sin M-0.0069\sin \ left(2\lambda \right)} {\displaystyle J_{transit}=2451545.0+J^{\star } +0.0053\sin M-0.0069\sin \left(2\lambda \right)} where: J[transit] is the Julian date for the local true solar transit (or solar noon). 2451545.0 is noon of the equivalent Julian year reference. 0.0053 sin ? M ? 0.0069 sin ? ( 2 ? ) {\displaystyle 0.0053\sin M-0.0069\ sin \left(2\lambda \right)} {\displaystyle 0.0053\sin M-0.0069\sin \left(2\ lambda \right)} is a simplified version of the equation of time. The coefficients are fractional days. Declination of the Sun[edit] sin ? ? = sin ? ? ? sin ? 23.44 ? {\displaystyle \sin \delta =\sin \lambda \times \sin 23.44^{\circ }} {\displaystyle \sin \delta =\sin \lambda \times \sin 23.44^{\circ }} where: ? {\displaystyle \delta } \delta is the declination of the sun. 23.44? is Earth's maximum axial tilt toward the sun ^[3] Hour angle[edit] This is the equation from above with corrections for atmospherical refraction and solar disc diameter. cos ? ? ? = sin ? ( ? 0.83 ? ) ? sin ? ? ? sin ? ? cos ? ? ? cos ? ? {\ displaystyle \cos \omega _{\circ }={\dfrac {\sin(-0.83^{\circ })-\sin \phi \times \sin \delta }{\cos \phi \times \cos \delta }}} \cos \omega _{\circ } ={\dfrac {\sin(-0.83^{\circ })-\sin \phi \times \sin \delta }{\cos \phi \ times \cos \delta }} where: ?[o] is the hour angle from the observer's meridian; ? {\displaystyle \phi } \phi is the north latitude of the observer (north is positive, south is negative) on the Earth. For observations on a sea horizon needing an elevation-of-observer correction, add ? 1.15 ? elevation in feet / 60 {\displaystyle -1.15^{\circ }{\sqrt {\text {elevation in feet}}}/60} {\displaystyle -1.15^{\circ }{\sqrt {\text{elevation in feet}}}/60}, or ? 2.076 ? elevation in metres / 60 {\displaystyle -2.076^{\ circ }{\sqrt {\text{elevation in metres}}}/60} {\displaystyle -2.076^{\circ }{\ sqrt {\text{elevation in metres}}}/60} to the ?0.83? in the numerator's sine term. This corrects for both apparent dip and terrestrial refraction. For example, for an observer at 10,000 feet, add (?115?/60) or about ?1.92? to ?0.83?.^[4] Calculate sunrise and sunset[edit] J r i s e = J t r a n s i t ? ? ? 360 ? {\displaystyle J_{rise}=J_{transit} -{\dfrac {\omega _{\circ }}{360^{\circ }}}} J_{{rise}}=J_{{transit}}-{\ dfrac {\omega _{\circ }}{360^{\circ }}} J s e t = J t r a n s i t + ? ? 360 ? {\displaystyle J_{set}=J_{transit}+{\ dfrac {\omega _{\circ }}{360^{\circ }}}} J_{{set}}=J_{{transit}}+{\dfrac {\ omega _{\circ }}{360^{\circ }}} where: J[rise] is the actual Julian date of sunrise; J[set] is the actual Julian date of sunset. See also[edit] * Day length * Equation of time References[edit] 1. ^ NOAA (U.S. Department of Commerce). "Solar Calculation Details". ESRL Global Monitoring Laboratory - Global Radiation and Aerosols. 2. ^ "Correction Tables for Sextant Altitude". www.siranah.de. 3. ^ https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html 4. ^ The exact source f these numbers are hard to track down, but Notes on the Dip of the Horizon provides a description yielding one less significant figure, with another page in the series providing -2.075. External links[edit] * Sunrise, sunset, or sun position for any location ? U.S. only * Sunrise, sunset and day length for any location ? Worldwide * Rise/Set/Transit/Twilight Data ? U.S. only * Astronomical Information Center * Converting Between Julian Dates and Gregorian Calendar Dates * Approximate Solar Coordinates * Algorithms for Computing Astronomical Phenomena * Astronomy Answers: Position of the Sun * A Simple Expression for the Equation of Time * The Equation of Time * Equation of Time * Long-Term Almanac for Sun, Moon, and Polaris V1.11 * Evaluating the Effectiveness of Current Atmospheric Refraction Models in Predicting Sunrise and Sunset Times * Retrieved from "https://en.wikipedia.org/w/index.php?title=Sunrise_equation& oldid=1053984191" Categories: * Equations * Time in astronomy * Dynamics of the Solar System Hidden categories: * Articles with short description * Short description matches Wikidata * Articles needing additional references from June 2018 * All articles needing additional references Navigation menu Personal tools * Not logged in * Talk * Contributions * Create account * Log in Namespaces * Article * Talk [ ] Variants expanded collapsed Views * Read * Edit * View history [ ] More expanded collapsed Search [ ] [Search] [Go] Navigation * Main page * Contents * Current events * Random article * About Wikipedia * Contact us * Donate Contribute * Help * Learn to edit * Community portal * Recent changes * Upload file Tools * What links here * Related changes * Upload file * Special pages * Permanent link * Page information * Cite this page * Wikidata item Print/export * Download as PDF * Printable version In other projects * Wikimedia Commons Languages * ????? * ??????? * Sloven??ina * ???? 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