New! Enter your latitude and longitude.Today's sunrise/sunset calculation for 39.040759 °N, 77.04876 °W.
Find today's Julian date (days since Jan 1, 2000 + 2451545): Julian date: 2459420
Now, calculate J
n
_{*} = (J_{date} - 2451545 - 0.0009) - (l_{w}/360)n = round(n _{*})n _{*} = (2459420 - 2451545 - 0.0009) - (77.04876/360) = 7874.7850756667n = round(7874.7850756667) = 7875
Now J
J
_{*} = 2451545 + 0.0009 + (l_{w}/360) + nJ _{*} = 2451545 + 0.0009 + (77.04876/360) + 7875 = 2459420.2149243
Using J
M = [357.5291 + 0.98560028 * (J
_{*} - 2451545)] mod 360M = [357.5291 + 0.98560028 * (2459420.2149243 - 2451545)] mod 360 = 8119.3431344829 mod 360 = 199.34313448293 We need to calculate the equation of center, C:
C = (1.9148 * sin(M)) + (0.0200 * sin(2 * M)) + (0.0003 * sin(3 * M))
C = 1.9148 * sin(199.34313448293) + 0.0200 * sin(2 * 199.34313448293) + 0.0003 * sin(3 * 199.34313448293) = -0.62198268624105 We need λ which is the ecliptical longitude of the sun:
λ = (M + 102.9372 + C + 180) mod 360
λ = (199.34313448293 + 102.9372 + -0.62198268624105 + 180) mod 360 = 481.65835179669 mod 360 = 121.65835179669
Finally, calculate J
J
_{transit} = J_{*} + (0.0053 * sin(M)) - (0.0069 * sin(2 * λ))J _{transit} = 2459420.2149243 + (0.0053 * sin(199.34313448293)) - (0.0069 * sin(2 * 121.65835179669)) = 2459420.219334
Now, to get an even more accurate number, recursively recalculate M using J
I1: M = 199.34748065987, C = -0.62211740065992, λ = 121.66256325921, J
_{transit} = 2459420.2193341I2: M = 199.34748073468, C = -0.6221174029787, λ = 121.6625633317, J _{transit} = 2459420.2193341I3: M = 199.34748073468, C = -0.6221174029787, λ = 121.6625633317, J _{transit} = 2459420.2193341Ok, translate this into something we understand. i.e. When is Solar Noon?
J
_{transit} =
2459420.2193341 = 07/24/2021 at 13:15:50 -0500
Alrighty, now calculate how long the sun is in the sky at latitude 39.040759: Now we need to calculate δ which is the declination of the sun:
δ = arcsin( sin(λ) * sin(23.45) )
δ = arcsin(sin(121.6625633317) * sin(23.45)) = 19.798645405573 Now we can go about calculating H (Hour angle):
H = arccos( [sin(-0.83) - sin(l
_{n}) * sin(δ)] / [cos(l_{n}) * cos(δ)] )H = arccos((sin(-0.83) - sin(39.040759) * sin(19.798645405573))/(cos(39.040759) * cos(19.798645405573))) = 108.16562617699
Just as above, calculate J
J
_{**} = 2451545 + 0.0009 + ((H + l_{w})/360) + nJ _{**} = 2451545 + 0.0009 + ((108.16562617699 + 77.04876)/360) + 7875 = 2459420.5153844
We can use M from above because it really doesn't change that much over the course of a day, calculate J
J
_{set} = J_{**} + (0.0053 * sin(M)) - (0.0069 * sin(2 * λ))J _{set} = 2459420.5153844 + (0.0053 * sin(199.34748073468)) - (0.0069 * sin(2 * 121.6625633317)) = 2459420.5197942
Now I'm going to cheat and calculate J
J
_{rise} = J_{transit} - (J_{set} - J_{transit})J _{rise} = 2459420.2193341 - (2459420.5197942 - 2459420.2193341) = 2459419.918874Using the same idea, figure out when sunrise and sunset are:
J
_{rise} =
2459419.918874 = 07/24/2021 at 06:03:10 -0500
J _{set} =
2459420.5197942 = 07/24/2021 at 20:28:30 -0500
A couple anomalies occur. At high latitudes, you will sometimes get H = 0. This means that either the sun does not rise (in the winter) or the sun does not set (in the summer) on that day. |

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