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 Latitude Longitude °N °W
Today's sunrise/sunset calculation for 39.040759 °N, 77.04876 °W.

Note: I am not using the appropriate number of significant figures to calculate sunrise, sunset, and solar noon. While it may appear that my calculations are accurate to one second, they actually have a precision of closer to 3 minutes (mostly due to approximation). Also, this calculation does not take into account the effect of air temperature, altitude, etc. Together, these may affect the time by 5 minutes or more.

Find today's Julian date (days since Jan 1, 2000 + 2451545):

Julian date: 2459116

n* = (Jdate - 2451545 - 0.0009) - (lw/360)
n = round(n*)

n* = (2459116 - 2451545 - 0.0009) - (77.04876/360) = 7570.7850756667
n = round(7570.7850756667) = 7571

Now J*:

J* = 2451545 + 0.0009 + (lw/360) + n

J* = 2451545 + 0.0009 + (77.04876/360) + 7571 = 2459116.2149243

Using J*, calculate M (mean anomaly) and then use that to calculate C and λ:

M = [357.5291 + 0.98560028 * (J* - 2451545)] mod 360

M = [357.5291 + 0.98560028 * (2459116.2149243 - 2451545)] mod 360 = 7819.7206493629 mod 360 = 259.72064936293

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(259.72064936293) + 0.0200 * sin(2 * 259.72064936293) + 0.0003 * sin(3 * 259.72064936293) = -1.8767854155646

We need λ which is the ecliptical longitude of the sun:

λ = (M + 102.9372 + C + 180) mod 360

λ = (259.72064936293 + 102.9372 + -1.8767854155646 + 180) mod 360 = 540.78106394737 mod 360 = 180.78106394737

Finally, calculate Jtransit:

Jtransit = J* + (0.0053 * sin(M)) - (0.0069 * sin(2 * λ))

Jtransit = 2459116.2149243 + (0.0053 * sin(259.72064936293)) - (0.0069 * sin(2 * 180.78106394737)) = 2459116.2095213

Now, to get an even more accurate number, recursively recalculate M using Jtransit until it stops changing. Notice how close the approximation was.

I1: M = 259.71532413302, C = -1.8767502117342, λ = 180.77577392129, Jtransit = 2459116.2095227
I2: M = 259.71532547501, C = -1.8767502206078, λ = 180.7757752544, Jtransit = 2459116.2095227
I3: M = 259.71532547455, C = -1.8767502206048, λ = 180.77577525394, Jtransit = 2459116.2095227
I4: M = 259.71532547455, C = -1.8767502206048, λ = 180.77577525394, Jtransit = 2459116.2095227

Ok, translate this into something we understand. i.e. When is Solar Noon?

Jtransit = 2459116.2095227 = 09/23/2020 at 13:01:42 -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(180.77577525394) * sin(23.45)) = -0.30871076152915

Now we can go about calculating H (Hour angle):

H = arccos( [sin(-0.83) - sin(ln) * sin(δ)] / [cos(ln) * cos(δ)] )

H = arccos((sin(-0.83) - sin(39.040759) * sin(-0.30871076152915))/(cos(39.040759) * cos(-0.30871076152915))) = 90.818276961155

Just as above, calculate J*, but this time using hour-angle:

J** = 2451545 + 0.0009 + ((H + lw)/360) + n

J** = 2451545 + 0.0009 + ((90.818276961155 + 77.04876)/360) + 7571 = 2459116.4671973

We can use M from above because it really doesn't change that much over the course of a day, calculate Jset in the same way:

Jset = J** + (0.0053 * sin(M)) - (0.0069 * sin(2 * λ))

Jset = 2459116.4671973 + (0.0053 * sin(259.71532547455)) - (0.0069 * sin(2 * 180.77577525394)) = 2459116.4617957

Now I'm going to cheat and calculate Jrise:

Jrise = Jtransit - (Jset - Jtransit)

Jrise = 2459116.2095227 - (2459116.4617957 - 2459116.2095227) = 2459115.9572497

Using the same idea, figure out when sunrise and sunset are:

Jrise = 2459115.9572497 = 09/23/2020 at 06:58:26 -0500
Jset = 2459116.4617957 = 09/23/2020 at 19:04:59 -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.