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Calculating Sunrise/Sunset in Suffolk, Virginia

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 Sunday, March 7, 2021

       14 - month     14 - 3
  a = ------------ = ---------- = 0
          12             12

  y = year + 4800 - a

    = 2021 + 4800 - 0

    = 6821

  m = month + (12 * a ) - 3

    = 3 + (12 * 0) - 3

    = 0

  For a date in the Gregorian Calendar (at noon):

              (153*m)+2                 y       y       y
  JDN = day + --------- + (365 * y) + ----- - ----- + ----- - 32045
                  5                     4      100     400

      = 2459281 (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)

            = 2459281 - 2451545

            = 7736

  Mean Longitude of the Sun:
 
    L = 280.461 + (0.9856002585 * daynumber)

      = 280.461 + (0.9856002585 * 7736)

      = 7905.05

      = 345.05 (in range 0..360)

  Mean Anamoly of the Sun:

    g = 357.528 + (0.9856003 * daynumber)

      = 357.528 + (0.9856003 * 7736)

      = 7982.12

      = 62.1199999999999 (in range 0..360)

  Ecliptic Longitude (lambda):

    lambda = L + ( 1.915 * sin(g) ) + ( 0.020 * sin(2*g) ) 

           = 345.05 + ( 1.915 * sin(62.1199999999999) ) + ( 0.020 * sin(2*62.1199999999999) ) 

           = 345.05 + ( 1.915 * sin(62.1199999999999) ) + ( 0.020 * sin(124.24) )

           = 345.05 + ( 1.915 * 0.88 ) + ( 0.020 * 0.83 ) 

           = 345.05 + ( 1.69 ) + ( 0.02) 

           = 346.76

  Obliquity of the ecliptic plane:

    epsilon = 23.4393 - (.0000003563 * daynumber)

            = 23.4393 - (.0000003563 * 7736)

            = 23.4393 - ( 0.0027563307544467 );

            = 23.44

    Y = cos(epsilon) * sin(lambda)

      = cos(23.44) * sin(346.76)

      = 0.92 * -0.23;

      = -0.21

    X = cos(lambda)

      = cos(346.76)

      = 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) = 347.78 (degrees)

                                       = 23.19 (decimal hours)

                                       = 23H 11m 24s


    delta (Declination) = arcsin( sin(epsilon) * sin(lambda) )

                        = arcsin( sin(23.44) * sin(346.76) )

                        = arcsin( 0.40 * -0.23 )

                        = arcsin( -0.09 )

                        = -5.23


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

              = 7736 + (-0.21) + (0.00)

              = 7735.79;

  Then calculate Local Siderial Time at Midnight:

    LSTMid = 98.9818  +  (0.985647352 * daynumber)  +  (Universal Time * 15)  +  longitude

           = 98.9818  +  (0.985647352 * 7735.79)  +  ( (5 - (1 if Daylight Savings Time)) * 15)  +  -76.583

           = 98.9818  +  (0.985647352 * 7735.79)  +  ( (5 - (0)) * 15)  +  -76.583

           = 98.9818  +  7624.76  +  75.00  +  -76.583

           = 7722.16

           = 162.16 (range 0..360 degrees)

           = 10.81 (range 0..24 hours)

  Calculate when the sun will be at the Local Meridian:

    Meridian Time = (Right Ascension) - (Local Siderial Time at Midnight)

                  = 23.19 - 10.81
                  = 12.38 (range 0..24 hours)
                  = 12:22 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(-5.23) )
              =  -------------------------------------------
                           cos(36.733) * cos(-5.23)

                   0.015 - (0.598 * 0.394 )
              =  -----------------------------
                      0.801 * 0.996  

                   0.015 - (0.236)
              =  ------------------
                        0.798  

                   -0.221
              =  ---------
                   0.798

              =  -0.277


         HA0  = arccos( -0.277 )

              = 85.04 (degrees)
              = 5.67 (Hours)



      Sunrise = ( Sun at Local Meridian ) - ( horizon hour angle )

              = 12.38 - 5.67
              = 6.71 hours
      Sunrise = 6:42 AM

       Sunset = ( Sun at Local Meridian ) + ( horizon hour angle )

              = 12.38 + 5.67
              = 18.05 hours
       Sunset = 6:03 PM