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| [[File:Hours_of_daylight_vs_latitude_vs_day_of_year_cmglee.svg|thumb|400px|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 Sunrise equation, Latitude 40° N (approximately New York City, Madrid and Beijing) is highlighted for reference]]
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| The '''sunrise equation''' as follows 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:
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| :<math>\cos \omega_\circ = -\tan \phi \times \tan \delta </math>
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| where:<br />
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| :<math>\omega_\circ</math> is the [[hour angle]] at either [[sunrise]] (when ''negative'' value is taken) or [[sunset]] (when ''positive'' value is taken);
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| :<math>\phi</math> is the [[latitude]] of the observer on the [[Earth]];
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| :<math>\delta</math> is the sun [[declination]].
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| == Theory of the Equation ==
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| The [[Earth]] rotates at an [[angular velocity]] of 15°/hour, therefore the expression <math>\omega_\circ \times \frac{\mathrm{hour}}{{15}^\circ}</math> gives the interval of time before and after local [[solar noon]] that [[sunrise]] or [[sunset]] will occur.
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| The sign convention is typically that the observer latitude <math>\phi</math> is 0 at the [[Equator]], positive for the [[Northern Hemisphere]] and negative for the [[Southern Hemisphere]], and the solar declination <math>\delta</math> 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.
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| 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 <math>-90^\circ+\delta < \phi < 90^\circ - \delta</math> during the Northern Hemisphere summer, and when <math>-90^\circ - \delta < \phi < 90^\circ + \delta</math> during the Northern Hemisphere winter. Out of these latitudes, it is either 24-hour [[day]]time or 24-hour [[nighttime]].
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| == Generalized Equation ==
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| Also note that the above equation neglects the influence of [[atmospheric refraction]] (which lifts the solar disc by approximately 0.6° when it is on the horizon) and the non-zero angle subtended by the solar disc (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
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| :<math>\cos \omega_\circ = \dfrac{\sin a - \sin \phi \times \sin \delta}{\cos \phi \times \cos \delta }</math>
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| with the [[altitude]] (a) of the center of the solar disc set to about −0.83° (or −50 arcminutes).
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| == Complete calculation on Earth ==
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| {{verify|date=March 2011}}
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| 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.
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| === Calculate current Julian Cycle ===
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| :<math>n^{\star} = J_{date} - 2451545.0009 - \dfrac{l_w}{360^\circ}</math>
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| :<math>n = \left[ n^{\star} + \frac{1}{2} \right] </math>
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| where:
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| :<math>J_{date}</math> is the [[Julian day|Julian Date]];
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| :<math>l_\omega</math> is the longitude west (west is positive, east is negative) of the observer on the Earth;
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| :<math>n</math> is the Julian cycle since Jan 1st, 2000.
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| === Approximate Solar Noon ===
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| :<math>J^{\star} = 2451545.0009 + \dfrac{l_w}{360^\circ} + n</math>
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| where:
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| :<math>J^{\star}</math> is an approximation of solar noon at <math>l_w</math>.
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| === Solar Mean Anomaly ===
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| :<math>M = [357.5291 + 0.98560028 \times (J^{\star} - 2451545)] \mod 360</math>
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| where:
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| :M is the solar [[Mean Anomaly]].
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| === Equation of Center ===
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| :<math>C = 1.9148 \sin(M) + 0.0200 \sin(2 M) + 0.0003 \sin(3 M)</math> | |
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| where:
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| :C is the [[Equation of the center]].
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| === Ecliptic Longitude ===
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| :<math>\lambda = (M + 102.9372 + C + 180) \mod 360</math>
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| where:
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| :λ is the [[ecliptic longitude]].
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| === Solar Transit ===
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| :<math>J_{transit} = J^{\star} + 0.0053 \sin M - 0.0069 \sin \left( 2 \lambda \right) </math>
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| where:
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| :J<sub>transit</sub> is the hour angle for solar transit (or [[solar noon]]).
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| === Declination of the Sun ===
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| :<math>\sin \delta = \sin \lambda \times \sin 23.45^\circ </math>
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| where:
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| :<math>\delta </math> is the declination of the sun.
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| === Hour Angle ===
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| ''This is the equation from above with corrections for astronomical refraction and solar disc diameter.''
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| :<math>\cos \omega_\circ = \dfrac{\sin(-0.83^\circ) - \sin \phi \times \sin \delta}{\cos \phi \times \cos \delta}</math>
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| where:
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| :ω<sub>o</sub> is the hour angle;
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| :<math>\phi</math> is the north latitude of the observer (north is positive, south is negative) on the Earth.
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| This is the main equation from above with the solar disc correction.
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| For observations on a sea horizon an elevation-of-observer correction, add <math>-1.15^\circ\sqrt{\text{elevation in feet}}/60^\circ</math>, or <math>-2.076^\circ\sqrt{\text{elevation in metres}}/60^\circ</math> 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°.
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| === Calculate Sunrise and Sunset ===
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| :<math>J_{set} = 2451545.0009 + \dfrac{\omega_\circ + l_w}{360^\circ} + n + 0.0053 \sin M - 0.0069 \sin 2 \lambda</math>
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| :<math>J_{rise} = J_{transit} - \left( J_{set} - J_{transit} \right)</math>
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| where:
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| :J<sub>set</sub> is the actual Julian Date of sunset;
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| :J<sub>rise</sub> is the actual Julian Date of sunrise.
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| <!--
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| ==Sun Rise Time : Nepali Method==
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| [[Sun rise]] at the morning is based on [[longitude]] and [[latitude]] of the place and [[standard time]] of the place. Finding the sun-rise (and sun-set) time we need three parameters; viz. [[latitude]], [[longitude]], date on which sun-rise is to be computed. Let take an example of [[Kathmandu]] (27° 42′ 0″ N, 85° 20′ 0″ E) for September 5.
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| Step 1. Find date difference since March 21 (September 4- March 21=167); then assume A=167*0.985626=164.60°.
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| Step 2. Find B, where B= Sin-1(0.398749 * Sin 164.60)= 6.07846 (this is in degree, where the sun is over in north, 6.07846° north from equator).
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| Step 3. Find C in minutes, where C=4*Sin-1(TanB*TanL) =4*Sin-1(Tan 6.07846o*Tan 27° 42′ 0″) =4*3.20496 = 12.81=13 minutes
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| Sunrise in all places of earth having 27° 42′ 0″ N is 13 minute earlier (because of +ve 13) than 6:00 i.e. 5:47 at [[local time]].
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| Sunset for that time shall be 12- sunrise time (outcome shall be in time)
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| Day-length for that time shall be 2* sunset time (outcome shall be in hours)
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| Step 4. Find the time difference with standard time (or [[Greenwich Mean Time|Greenwich Time]]) to the [[local time]]. That shall be the sun-rise (watch) time.
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| For September 5, [[Kathmandu]](85° 20′ 0″ E, 27° 42′ 0″ N) Sun-rise at 5 hour 41 minutes (23:54 Monday).
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| Time difference of [[Greenwich]] and [[Nepal]] standard time is 5 hours 45 minutes. Standard time Nepal and local time of Kathmandu is 4 minutes slow (west).
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| -->
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| ==See also==
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| *[[Sunrise]]
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| *[[Sunset]]
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| *[[Day length]]
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| ==External links==
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| *[http://www.esrl.noaa.gov/gmd/grad/solcalc/ Sunrise, sunset, or sun position for any location]
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| *[http://aa.usno.navy.mil/faq/ Astronomical Information Center]
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| ==References==
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| [[Category:Equations]]
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| [[Category:Time in astronomy]]
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| [[Category:Dynamics of the Solar System]]
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