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Source: https://edwilliams.org/sunrise_sunset_example.htm
Source:
Almanac for Computers, 1990
published by Nautical Almanac Office
United States Naval Observatory
Washington, DC 20392
Inputs:
day, month, year: date of sunrise/sunset
latitude, longitude: location for sunrise/sunset
zenith: Sun's zenith for sunrise/sunset
offical = 90 degrees 50'
civil = 96 degrees
nautical = 102 degrees
astronomical = 108 degrees
NOTE: longitude is positive for East and negative for West
Worked example (from book):
June 25, 1990: 25, 6, 1990
Wayne, NJ: 40.9, -74.3
Office zenith: 90 50' cos(zenith) = -0.01454
1. first calculate the day of the year
N1 = floor(275 * month / 9)
N2 = floor((month + 9) / 12)
N3 = (1 + floor((year - 4 * floor(year / 4) + 2) / 3))
N = N1 - (N2 * N3) + day - 30
Example:
N1 = 183
N2 = 1
N3 = 1 + floor((1990 - 4 * 497 + 2) / 3)
= 1 + floor((1990 - 1988 + 2) / 3)
= 1 + floor((1990 - 1988 + 2) / 3)
= 1 + floor(4 / 3)
= 2
N = 183 - 2 + 25 - 30 = 176
2. convert the longitude to hour value and calculate an approximate time
lngHour = longitude / 15
if rising time is desired:
t = N + ((6 - lngHour) / 24)
if setting time is desired:
t = N + ((18 - lngHour) / 24)
Example:
lngHour = -74.3 / 15 = -4.953
t = 176 + ((6 - -4.953) / 24)
= 176.456
3. calculate the Sun's mean anomaly
M = (0.9856 * t) - 3.289
Example:
M = (0.9856 * 176.456) - 3.289
= 170.626
4. calculate the Sun's true longitude
[Note throughout the arguments of the trig functions
(sin, tan) are in degrees. It will likely be necessary to
convert to radians. eg sin(170.626 deg) =sin(170.626*pi/180
radians)=0.16287]
L = M + (1.916 * sin(M)) + (0.020 * sin(2 * M)) + 282.634
NOTE: L potentially needs to be adjusted into the range [0,360) by adding/subtracting 360
Example:
L = 170.626 + (1.916 * sin(170.626)) + (0.020 * sin(2 * 170.626)) + 282.634
= 170.626 + (1.916 * 0.16287) + (0.020 * -0.32141) + 282.634
= 170.626 + 0.31206 + -0.0064282 + 282.634
= 453.566 - 360
= 93.566
5a. calculate the Sun's right ascension
RA = atan(0.91764 * tan(L))
NOTE: RA potentially needs to be adjusted into the range [0,360) by adding/subtracting 360
Example:
RA = atan(0.91764 * -16.046)
= atan(0.91764 * -16.046)
= atan(-14.722)
= -86.11412
5b. right ascension value needs to be in the same quadrant as L
Lquadrant = (floor( L/90)) * 90
RAquadrant = (floor(RA/90)) * 90
RA = RA + (Lquadrant - RAquadrant)
Example:
Lquadrant = (floor(93.566/90)) * 90
= 90
RAquadrant = (floor(-86.11412/90)) * 90
= -90
RA = -86.11412 + (90 - -90)
= -86.11412 + 180
= 93.886
5c. right ascension value needs to be converted into hours
RA = RA / 15
Example:
RA = 93.886 / 15
= 6.259
6. calculate the Sun's declination
sinDec = 0.39782 * sin(L)
cosDec = cos(asin(sinDec))
Example:
sinDec = 0.39782 * sin(93.566)
= 0.39782 * 0.99806
= 0.39705
cosDec = cos(asin(0.39705))
= cos(asin(0.39705))
= cos(23.394)
= 0.91780
7a. calculate the Sun's local hour angle
cosH = (cos(zenith) - (sinDec * sin(latitude))) / (cosDec * cos(latitude))
if (cosH > 1)
the sun never rises on this location (on the specified date)
if (cosH < -1)
the sun never sets on this location (on the specified date)
Example:
cosH = (-0.01454 - (0.39705 * sin(40.9))) / (0.91780 * cos(40.9))
= (-0.01454 - (0.39705 * 0.65474)) / (0.91780 * 0.75585)
= (-0.01454 - 0.25996) / 0.69372
= -0.2745 / 0.69372
= -0.39570
7b. finish calculating H and convert into hours
if if rising time is desired:
H = 360 - acos(cosH)
if setting time is desired:
H = acos(cosH)
H = H / 15
Example:
H = 360 - acos(-0.39570)
= 360 - 113.310 [ note result of acos converted to degrees]
= 246.690
H = 246.690 / 15
= 16.446
8. calculate local mean time of rising/setting
T = H + RA - (0.06571 * t) - 6.622
Example:
T = 16.446 + 6.259 - (0.06571 * 176.456) - 6.622
= 16.446 + 6.259 - 11.595 - 6.622
= 4.488
9. adjust back to UTC
UT = T - lngHour
NOTE: UT potentially needs to be adjusted into the range [0,24) by adding/subtracting 24
Example:
UT = 4.488 - -4.953
= 9.441
= 9h 26m
10. convert UT value to local time zone of latitude/longitude
localT = UT + localOffset
Example:
localT = 9h 26m + -4
= 5h 26m
= 5:26 am EDT
Source: https://edwilliams.org/sunrise_sunset_algorithm.htm
Archive: https://web.archive.org/web/20161022214335/http://williams.best.vwh.net/sunrise_sunset_algorithm.htm
Sunrise/Sunset Algorithm
Source:
Almanac for Computers, 1990
published by Nautical Almanac Office
United States Naval Observatory
Washington, DC 20392
Inputs:
day, month, year: date of sunrise/sunset
latitude, longitude: location for sunrise/sunset
zenith: Sun's zenith for sunrise/sunset
offical = 90 degrees 50'
civil = 96 degrees
nautical = 102 degrees
astronomical = 108 degrees
NOTE: longitude is positive for East and negative for West
NOTE: the algorithm assumes the use of a calculator with the
trig functions in "degree" (rather than "radian") mode. Most
programming languages assume radian arguments, requiring back
and forth convertions. The factor is 180/pi. So, for instance,
the equation RA = atan(0.91764 * tan(L)) would be coded as RA
= (180/pi)*atan(0.91764 * tan((pi/180)*L)) to give a degree
answer with a degree input for L.
1. first calculate the day of the year
N1 = floor(275 * month / 9)
N2 = floor((month + 9) / 12)
N3 = (1 + floor((year - 4 * floor(year / 4) + 2) / 3))
N = N1 - (N2 * N3) + day - 30
2. convert the longitude to hour value and calculate an approximate time
lngHour = longitude / 15
if rising time is desired:
t = N + ((6 - lngHour) / 24)
if setting time is desired:
t = N + ((18 - lngHour) / 24)
3. calculate the Sun's mean anomaly
M = (0.9856 * t) - 3.289
4. calculate the Sun's true longitude
L = M + (1.916 * sin(M)) + (0.020 * sin(2 * M)) + 282.634
NOTE: L potentially needs to be adjusted into the range [0,360) by adding/subtracting 360
5a. calculate the Sun's right ascension
RA = atan(0.91764 * tan(L))
NOTE: RA potentially needs to be adjusted into the range [0,360) by adding/subtracting 360
5b. right ascension value needs to be in the same quadrant as L
Lquadrant = (floor( L/90)) * 90
RAquadrant = (floor(RA/90)) * 90
RA = RA + (Lquadrant - RAquadrant)
5c. right ascension value needs to be converted into hours
RA = RA / 15
6. calculate the Sun's declination
sinDec = 0.39782 * sin(L)
cosDec = cos(asin(sinDec))
7a. calculate the Sun's local hour angle
cosH = (cos(zenith) - (sinDec * sin(latitude))) / (cosDec * cos(latitude))
if (cosH > 1)
the sun never rises on this location (on the specified date)
if (cosH < -1)
the sun never sets on this location (on the specified date)
7b. finish calculating H and convert into hours
if if rising time is desired:
H = 360 - acos(cosH)
if setting time is desired:
H = acos(cosH)
H = H / 15
8. calculate local mean time of rising/setting
T = H + RA - (0.06571 * t) - 6.622
9. adjust back to UTC
UT = T - lngHour
NOTE: UT potentially needs to be adjusted into the range [0,24) by adding/subtracting 24
10. convert UT value to local time zone of latitude/longitude
localT = UT + localOffset
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