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We used the following web pages as references for these calcuations: Astronomy FAQ Wikipedia: Calculating Julian Date Calculating Position of the Sun Home Planet: An Amazing (and Free) Astronomy Program Where is Suffolk, Virginia?: Latitude: 36 deg. 44 min. N = 36 + (44/60) = 36.733 Longitute: 76 deg. 35 min. W = 1 * ( 76 + (35/60) = 76.583 What is the Julian Date?: Astronomical calculations often use a number called the Julian Date. This is the number of days that have passed since 12 noon, at the Prime Meridian (12 noon, Greenwich Mean Time) on November 24, 4714 BC. From Wikipedia: "The Julian day system was intended to provide astronomers with a single system of dates that could be used when working with different calendars and to unify different historical chronologies." Calculating the Julian Date: Today is Monday, September 9, 2024 14  month 14  9 a =  =  = 0 12 12 y = year + 4800  a = 2024 + 4800  0 = 6824 m = month + (12 * a )  3 = 9 + (12 * 0)  3 = 6 For a date in the Gregorian Calendar (at noon): (153*m)+2 y y y JDN = day +  + (365 * y) +    +   32045 5 4 100 400 = 2460563 (at noon in Greenwich, England today) To calculate the sunrise and sunset, we need to know the position of the sun on the Celestial Sphere. Calculate 'daynumber': daynumber = (Today's Julian Date)  (Julian Date at 00:00, Jan 1, 2000) = 2460563  2451545 = 9018 Mean Longitude of the Sun: L = 280.461 + (0.9856002585 * daynumber) = 280.461 + (0.9856002585 * 9018) = 9168.59 = 168.59 (in range 0..360) Mean Anamoly of the Sun: g = 357.528 + (0.9856003 * daynumber) = 357.528 + (0.9856003 * 9018) = 9245.66 = 245.66 (in range 0..360) Ecliptic Longitude (lambda): lambda = L + ( 1.915 * sin(g) ) + ( 0.020 * sin(2*g) ) = 168.59 + ( 1.915 * sin(245.66) ) + ( 0.020 * sin(2*245.66) ) = 168.59 + ( 1.915 * sin(245.66) ) + ( 0.020 * sin(491.32) ) = 168.59 + ( 1.915 * 0.91 ) + ( 0.020 * 0.75 ) = 168.59 + ( 1.74 ) + ( 0.01) = 166.86 Obliquity of the ecliptic plane: epsilon = 23.4393  (.0000003563 * daynumber) = 23.4393  (.0000003563 * 9018) = 23.4393  ( 0.0032131075730105 ); = 23.44 Y = cos(epsilon) * sin(lambda) = cos(23.44) * sin(166.86) = 0.92 * 0.23; = 0.21 X = cos(lambda) = cos(166.86) = 0.97 a = arctan(Y/X) = arctan(0.21/0.97) = arctan(0.22) = 12.22 If X < 0 then alpha = a + 180 If Y < 0 and X > 0 then alpha = a + 360 else alpha = a alpha (Right Ascension of the Sun) = 167.78 (degrees) = 11.19 (decimal hours) = 11H 11m 24s delta (Declination) = arcsin( sin(epsilon) * sin(lambda) ) = arcsin( sin(23.44) * sin(166.86) ) = arcsin( 0.40 * 0.23 ) = arcsin( 0.09 ) = 5.19 The Local Meridian is the imaginary line passing from due North, directly overhead at your location, to due South. Local Siderial Time is the projection of your Local Meridian onto the Celestial Sphere. To calculate Rise/Set time, you need to know when an object will pass through your Local Meridian. Calculate Local Siderial Time for Midnight at your location: First calculate the 'daynumber' for Midnight at your location: daynumber = daynumber for Noon, Greenwich + (5/24) + (1/24 if Daylight Savings time)  > Correction for Suffolk Time Zone, relative to Greenwich = 9018 + (0.21) + (0.04) = 9017.83; Then calculate Local Siderial Time at Midnight: LSTMid = 98.9818 + (0.985647352 * daynumber) + (Universal Time * 15) + longitude = 98.9818 + (0.985647352 * 9017.83) + ( (5  (1 if Daylight Savings Time)) * 15) + 76.583 = 98.9818 + (0.985647352 * 9017.83) + ( (5  (1)) * 15) + 76.583 = 98.9818 + 8888.40 + 60.00 + 76.583 = 8970.80 = 330.80 (range 0..360 degrees) = 22.05 (range 0..24 hours) Calculate when the sun will be at the Local Meridian: Meridian Time = (Right Ascension)  (Local Siderial Time at Midnight) = 11.19  22.05 = 13.14 (range 0..24 hours) = 1:08 PM Use some spherical geometry magic to calculate the Hour Angle (HA) for the horizon. Label this value HA0: sin(h0)  ( sin(latitude) * sin(Declination) ) cos(HA0) =  cos(lat) * cos(Dec) Set h0 to 50/60 when calculating movement of upper limb of sun across the horizon sin(50/60)  ( sin(36.733) * sin(5.19) ) =  cos(36.733) * cos(5.19) 0.015  (0.598 * 0.194 ) =  0.801 * 0.996 0.015  (0.116) =  0.798 0.101 =  0.798 = 0.127 HA0 = arccos( 0.127 ) = 92.84 (degrees) = 6.19 (Hours) Sunrise = ( Sun at Local Meridian )  ( horizon hour angle ) = 13.14  6.19 = 6.95 hours Sunrise = 6:57 AM Sunset = ( Sun at Local Meridian ) + ( horizon hour angle ) = 13.14 + 6.19 = 19.33 hours Sunset = 7:19 PM