The Nautical Almanac
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The world's most complete source of free celestial navigation information

Celestial Navigation Formulas

About Calculators

The Casio fx-300ES Plus is an excellent and inexpensive calculator at about $11. It features natural input so you enter a formula just as it would be written on paper. Entering degrees, minutes and seconds is simple. The Casio has 1 memory position but there are 8 positions for variables so you have 9 memory locations. You can review many of the previous entries you make.

To clearly see this page the background is white.

‣ Determine Hc using a calculator	‣ Determine Zn (azimuth) using a calculator
‣ Determine Refraction	‣ Determine Local Apparent Noon using GHA
‣ Determine Dip	‣ Meridian by Equal Altitudes- obtain Longitude
‣ Semi-Diameter of Moon	‣ Noon sight- Determine Latitude
‣ Distance from Position to Destination	‣ Rules to Calculate Latitude

To determine Hc using a calculator

Hc = sin^-1[sin(declination) x sin(Latitude) + (cos(Latitude) x cos(declination) x cos(LHA)]
‣ sin^-1 means press the arc sin key

‣ And now as it would be entered into the Casio calculator...

Sin^-1(Sin(Latitude) x Sin(declination) + Cos(Latitude) x Cos(declination) x Cos(LHA)

NOTE- If the heavenly body's declination is Contrary name to your DR Latitude enter a negative sign before it.

‣ Why would you want to determine Hc using a calculator?

It's faster than looking up in Pub. No. 249, highly accurate, you don't need a lot of printed out pages of Latitudes from Pub. No. 249.

Pub. No. 249 doesn't cover any sight of a heavenly body with a declination greater than 29º (except Vol 1 for a few select stars) so you'd have to use Pub. No. 229 which is extremely large.

Determine Zn (azimuth)
Z is Azimuth angle

First calculate Z

Z  = cos^-1[(sin(declination) – sin(Latitude) x sin(Hc)) / (cos(Latitude) x cos(Hc))]
‣ cos^-1 means press the arc cos key
‣ The "/" means press the divide key

‣ And now as it would be entered into the Casio calculator...

Casio- Cos^-1((Sin(declination) – Sin(AP Latitude) x Sin(Hc)) / (Cos(AP Latitude) x Cos(Hc)

To obtain Zn apply the following rules for Northern and Southern latitudes;.

In Northern Latitudes

LHA greater than 180°....Zn=Z

LHA less than 180°...........Zn=360° – Z

In Southern Latitudes

LHA greater than 180°....Zn= 180° – Z

LHA less than 180°...........Zn= 180° + Z

‣ The "/" means press the Divide key

Another way (independent of Hc)

"L” is latitude and “d” is declination. When the heavenly body's declination is Contrary name to your Ap Latitude enter a negative sign before it.

The sign convention used in the calculation of this azimuth formula is as follows:
from Bowditch Chapter 22 CALCULATIONS AND CONVERSIONS, page 331

1) If latitude and declination are of contrary name, declination is treated as a negative quantity;

2) If the local hour angle is greater than 180°, it is treated as a negative quantity. If the azimuth angle as calculated is negative, add 180° to obtain the desired value.

As it would be entered into the Casio calculator...

Z = tan-1 ((sin (LHA) ÷ (cos(AP latitude) x tan(declination) – (sin(AP latitude) x cos(LHA))

To obtain Zn apply the following rules for Northern and Southern latitudes;.

In Northern Latitudes

LHA greater than 180°....Zn=Z

LHA less than 180°...........Zn=360° – Z

In Southern Latitudes

LHA greater than 180°....Zn= 180° – Z

LHA less than 180°...........Zn= 180° + Z

Determine Local Apparent Noon using GHA
...when your Longitude is known...

Form- Determine Local Apparent Noon using GHA

In Western Longitudes

((Your Longitude – Sun GHA less than your Longitude) / 15) + GHA integral hour

Example

‣ You're located at W 075º 44'. When is LAN (Local Apparent Noon) on January 21, 2016?

‣The GHA less than your Longitude on January 21, 2016 is 72° 11.7' (at 17:00 GMT).

‣Arrange the formula like this: ((75º 44' – 72° 11.7') / 15) + 17

‣ Answer= 17:14:09.2 GMT (LAN for January 21, 2016 at W 075º 44')

Another way
‣ (Your Longitude – GHA less than your Longitude)

‣ Take the answer to “TABLE 3-Conversion of Arc to Time” and convert the figure to time.

‣ Then add the result to the GHA integral hour.

In Eastern Longitudes

((360 – Your Longitude) – Sun GHA less than your Longitude) / 15 + GHA integral hour

Example

‣ You're located at E 124º 37'. When is LAN (Local Apparent Noon) on Febuary 8, 2017?
First find your longitude's angular distance West of Greenwich

        ‣Arrange the formula like this: 360°–124º 37' = 235°23.0'

‣Using The Nautical Almanac for 2017 and the date of Febuary 8, 2017 find the GHA which is less than your Longitude's angular distance West of Greenwich.
                        ‣ Answer= 221° 27.7' (at 03:00 GMT).

            ‣Finally, arrange the formula like this: (235°23.0' – 221° 27.7') / 15 + 3

‣ Answer= 03:55:41.2 GMT (LAN for Febuary 8, 2017 at E 124º 37')

Another way
‣ (360° – Your Longitude)= your longitude's angular distance West of Greenwich ‣Then (Your longitude's angular distance West of Greenwich – GHA less than your Longitude)

‣ Take the answer to “TABLE 3-Conversion of Arc to Time” and convert the figure to time.

‣ Then add the result to the GHA integral hour.

Meridian by Equal Altitudes

obtain Longitude

A Noon sight will help determine your Latitude fairly accurately and Longitude approximately.

Form- Meridian by Equal Altitudes

‣Take and record 3 Sun sights one-half hour before an estimated LAN- Local Apparent Noon. Space the shots about 5 minutes apart. Record the date and Greenwich time of each shot. Label the shots in the order they were taken- 1A, 2A and 3A.

‣Wait for the Sun to MP (Meridian Passage) and drop back down to the last Height of sextant (Hs) Sun sight you took (which was sight number 3A) and record the time of the sight in Greenwich time. Label it 3P.

‣Set your Sextant to the Height of sextant (Hs) of the 2nd shot you took before LAN and wait for the Sun to drop back down to that Hs and record the time of that sight in Greenwich time. Label it 2P.

‣Set your Sextant to the Height of sextant (Hs) of the 1rst shot you took before LAN and wait for the Sun to drop back down to that Hs and record the time of that sight in Greenwich time. Label it 1P.

‣Add the time of sights 1A & 1P together then divide them by 2. Write the result down.

‣Add the time of sights 2A & 2P together then divide them by 2. Write the result down.

‣Add the time of sights 3A & 3P together then divide them by 2. Write the result down.

You should now have are 3 averaged time figures that are fairly close together.

‣Add the 3 averaged times together then divide them by 3. Write the result down.

You now have, hopefully, a fairly accurate time when LAN occured. Label this "LAN Averaged"

‣Get The Nautical Almanac and find the GHA of the Sun for the day and hour on which you took the sights.

‣Record the whole hour GHA in degrees.

‣Using the Increments & Corrections Table find the "LAN Averaged" minutes and seconds (of time) in degrees for the time of the sight.

‣Add the GHA degrees and GHA increments together- this is the approximate Longitude of your location. But.....if you were sailing along in a boat that Longitude is behind you now by the distance you travelled since the determined Local Apparent Noon IF you're travelling due East or due West only. So, depending upon your speed add or subtract a few estimated miles to your position depending upon the direction you're heading. If heading West add a few estimated miles. If heading East subtract a few estimated miles. It shouldn't be much if you're sailing.

Noon sight- Determine Latitude

‣ Click here for the complete method sheet- Method to determine Latitude

‣Get a general idea of when Local Apparent Noon is for your approximate Longitude.
For example-If you're near W 060º then LAN will be at Greenwich time PLUS 4 hours or 16:00 GMT.

‣Using your sextant "follow the Sun up" until it just hangs in the sky neither rising nor falling.

‣Record the sextant height and the date and time.

‣ Correct the sextant for Dip, Index Error and use the Altitude Corrections Table for the main correction to get the Ho (Height observed).

‣ Use The Nautical Almanac to find the declination of the Sun for date and time of the measurement.

‣ Latitude= (90º – Ho) + declination of Sun- see Rules below.

Example

(90º – Ho (53º 5.3') + declination (N04º 47.6')= 41º 42.3'

‣ Make sure to enter the correct declination sign!

‣ If the declination is of Contrary Name to your Latitude then enter a - (minus) sign before it

Rules to Calculate Latitude

1- Latitude and declination Same name but latitude is greater than declination:

‣ Latitude= (90º – Ho) + declination

2- Latitude and declination Same name but declination greater than latitude:

‣ Latitude= declination – (90º – Ho)

3- Latitude and declination Contrary name:

‣ Latitude= (90º – Ho) – declination

Determine Refraction        0.96 / Tan(Ha) Gives good results down to about 8° from the horizon but not less.

            Refraction (good overall formula from 90° to below 8° from the horizon)

‣ And now as it would be entered into the Casio calculator...
                                                    1 ÷ Tan((Ha + (7.31 ÷ (Ha + 4.4))
Both refraction formulas use the standard pressure and temperature of;
               1010 mb        10° C
                29.83 in        53° F

Determine Dip

Determine Dip using feet- 0.97 x (Square Root of He (Height of Eye) in feet)
Determine Dip using meters- 1.76 x (Square Root of He (Height of eye) in meters )

Semi-Diameter of Moon .2724 x HP

Distance from Position to Destination

Distance = 90 - Sin^-1(sin(Destination Latitude) x sin(Your present Latitude) + 
cos(Destination Latitude) x cos(Your present Latitude) x 
cos(Difference in Longitude between Your Location and Destination Location)

Multiply answer by 60 to get Nautical miles.

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