Formulas to determine Sunrise or Sunset

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Here are a few simple formulas to determine Sunrise or Sunset time.

"Cos-1" means use the arc cosine key.
"Integral hour" means the single hour figure without minutes.  Example- "12"  not 12:14.

To make things easier first get the GHA of Sunrise using the following formula.  You can also simplify the process of getting the GHA by using HO-249, (with your AP. Latitude and the Sun's declination) finding the LHA for when Hc is closest to 0° and add your Ap. Longitude to it. 

For the declination figure just use the average declination for the day you want to calculate Sunrise.

360º – Cos-1( -Tan(Declination) x Tan(AP Latitude)) + Ap. Longitude = GHA of Sunrise

Look up the GHA in The Nautical Almanac and find the GHA integral hour figure lower than the calculated GHA figure found above.  Also obtain the GHA (in degrees) of the integral hour and the declination for the integral hour.

Use the figures in the following formula to get the time of sunrise-

GHA Integral Hour + (360 – Cos-1( -Tan(Declination) x Tan(Ap. Latitude)) + Ap. Longitude – GHA of Integral Hour) / 15

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Now, to calculate Sunset do the following;

First get the GHA of Sunset using this formula.  You can also simplify the process of getting the GHA by using HO-249, (with your AP. Latitude and the Sun's declination) finding the LHA for when Hc is closest to 0° and add your Ap. Longitude to it.


Cos-1( -Tan(Declination) x Tan(AP Latitude)) + Ap. Longitude = GHA of Sunset

Look up the GHA in The Nautical Almanac and find the GHA integral hour figure lower than the calculated GHA figure found above.  Also obtain the GHA (in degrees) of the integral hour and the declination for the integral hour..  Use the figures in the following formula to get the time of Sunset-

GHA Integral Hour + Cos-1( -Tan(Declination) x Tan(Ap. Latitude)) + Ap. Longitude – GHA of Integral Hour) / 15

Pretty simple.

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You can use The Nautical Almanac and find the date you'd like to know the approximate sunrise and sunset times for using the latitude figure which is closest to you.

Here's an example;

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You can also use HO-249 to get a fairly close estimate of Sunrise or Sunset.

All of the documents mentioned here are available on our website.

1- Enter HO-249 with the Latitude you'd like to find the Sunrise time for.  You'll also need the integral declination figure for the day such as, 22°

2- Look down the Declination column of HO-249 until you see when Hc is closest to 0°.

3- Move horizontally from the approximate Hc 0° figure over to the LHA column and find the LHA.

4- Add your Longitude to the LHA figure- the result is the GHA for that day and time.

5- Using The Nautical Almanac find the GHA which is lower than the answer you got in step 4.  Note the integral hour.

6- Subtract the GHA figure you got in step 4 from the GHA you found in step 5.

7- With the "Yellow Pages" (Increments & Corrections) find the time (in minutes and seconds) figure for the degrees and minutes of arc you got in step 6.  Note- you may find it easier to use TABLE 3. — CONVERSION OF ARC TO TIME to convert the resulting figure to time.

8- Add the minutes & seconds of time you found in step 7 to the GHA integral hour figure.  The result is the time of Sunrise (or Sunset)

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Here's an example using HO-249 and today's date- May 30, 2016.

All of the documents mentioned here are available on our website.

When is Sunrise at a position of N 40°,   W 065° 30'   ?

Sun's declination- N 22°   (yes, the integral value is 21° but it's closer to 22°)

1- Using HO-249 with a Latitude of N 40° and a declination of 22° find in the LHA column- 250°

2- Add the Longitude of W 065° 30' to the LHA of 250° to get a GHA of 315° 30'

3- Using the The Nautical Almanac for May 30, 2016 find the GHA which is lower than 315° 30' which is- 300° 36.5' at UT 8 hours.

4- Subtract 300° 36.5' from 315° 30' to get 14° 53.5'

5-  Use the "Yellow Pages" (Increments & Corrections) to find the time (in minutes and seconds)   for 14° 53.5' found in step 4.  Note- you may find it easier to use TABLE 3. — CONVERSION OF ARC TO TIME to convert the resulting figure of 14° 53.5' to time. 

6- 14° 53.5' converted to time is- 59 minutes 34 seconds.

7- Combine the figures of UT 8 hours to 59 minutes 34 seconds to get a Sunrise figure of UT 8:59:34 at a position of N 40°,   W 065° 30'

Keep in mind when calculating Sunrise or Sunset "close enough....is close enough".  The result you obtain with these calculations assumes no refraction.  So if you're looking at the ocean's horizon and the Sun's upper limb doesn't appear exactly when you calculated it to appear don't be surprised.  There will most likely be an apparent delay due to refraction.

Determining Sunset is a similar process except you use the LHA of 110°  (360° minus 250°).

Note- in Eastern Longitudes your position Longitude is subtracted from the LHA to get the GHA of Sunrise or Sunset.

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