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Celestial Navigation Theory

The Navigational Triangle

A flat plane triangle can be solved using trigonometry. Knowing 2 sides and an angle, SAS, the third side and a second angle can be determined.

The Celestial Triangle, which is laid out on the face of the planet, is a spherical triangle.

In this triangle we also know 2 sides and an angle, Latitude, Declination and LHA.

Pub 229 is a set of tables which solves the triangle using spherical trigonometry.

Enter the table with:
1. Latitude
2. Declination
3. LHA

We extract :
1. Hc (height computed)
2. Z (azimuth)

Hc is compared to Ho (height observed) along with Z converted to Zn and then plotted from your Assumed Position to give a line of position.

The Celestial Triangle’s three points are defined by the Pole, your DR Position (latitude and longitude) and the Objects Geographical Position (GHA and Declination)

The Component Parts

Ho - Height Observed: This is the Sextant observation (Hs) corrected for height of the eye above sea level and for refraction. (Dip, Altitude Correction)Triangle

Lat - Latitude: This is your DR Latitude rounded to the nearest whole latitude.

Dec - Declination: This is exact Declination for the body observed at the time of the sighting.

LHA - Local Hour Angle: This is Meridian Angle (t). The Angle between your Longitude and the objects GHA (Greenwich Hour Angle). It is expressed as a whole number.

LHA = GHA - West Long or + East Long

Navigational Triangle

AP - Assumed Position
DR - Position
GP - Geographical Position, GHA and Dec of the Body

PUB 229 Entering Argument:

LHA Local Hour Angle LHA= GHA - W Long
         GHA + E Long
Note: Long is changed to make LHA a whole number, Lat and new Long become AP.

LAT DR - Lat rounded to nearest whole Lat.
DEC - Exact Declination of body.Sphere triangle
HS - Height Shot
HA - Apparent Altitude     Corrected for Index Error and Dip Angle
HO - Height Observed    All Corrections Applied
The angle that would be formed at the center of the earth between the Observer’s celestial horizon and the line of sight to center of the body.
HC - Height Computed for DR position
Index Correction:
Corrections for inaccuracy in reading the sextant. Add or subtract.
Dip Angle:
Height of Eye. Difference between the Horizontal reference plane and the visible horizon. Always subtracted.
Altitude Corrections:
Entered with HA (apparent altitude). A correction that take the following factors in account: Always added for lower limb.
Correction for bending of light rays coming from the body shot.
Correction for 1/2 of the body’s diameter.
Increase in apparent size of the sun and moon as a result of the increase in apparent altitude.
The difference in apparent altitude as viewed from the surface of the Earth and in the center of the Earth.
Additional Corrections:
Correction for non-standard conditions. Barometric pressure and temperature.
Altitude Correction is done in two parts using apparent altitude and H.P. (horizontal parallax). Follow instruction on correction table. Always added.

See Celestial Navigation DiagramCelistial Navigation
Click image for more detail      

Download Sunline Worksheet
Download Local Apparent Noon Worksheet
Download Latitude by Polaris Worksheet

Great Circle Calculator

L1 = Departure Latitude
L2 = Destination Latitude
DLO = Difference in Longitude
D = Distance
1. If L1 and L2 are Contrary in name, Treat L2 as a negative.
2. If course is negative, add 180 degrees.
3. If DLO is greater than 180, enter as negative.
4. Distance in miles = = CosD x 60
5. Subtract when crossing the equator

Cos D = (Cos L1 x Cos L2 x Cos DLO) + -(Sin L1 x Sin L2)

Initial Course Angle:

Cos Angle = Sin L2 - (Cos Dist x Sin L1)
(Sin Dist x Cos L1)
Note: Distance in degrees

Points along the route.
Determine Lat for known Long.

L1 = Departure Latitude
L2 = Destination Latitude
DLO1 = Diff of Dep Long to known Long
DLO2 = Diff of Arrival Long to known Long
DLO12 = Diff between Dep and Arr Long.

Tan Lat = Tan L2 x Sin DLO1 = Tan L1 x Sin SLO2
Sin DLO12

Cosine – Haversine

Sight Reduction
Hc = Height Computed
Z = Azimuth Angle
L = Assumed Latitude
D = Exact Declination
LHA = Local Hour Angle
Note: Angles must be in degrees and tenths

= Arcsin (sin L sin D + cos L cos D dos LHA)

Z = Arccos (sin D - (sin L sin Hc)
cos L cos Hc
  • If Latitude and Declination are contrary or LHA is greater than 180 Degrees, Declination is negative.
  • If Z as Calculated is less than zero, add 180 degrees.

Great Circle
Publication 229 Solution:
Enter 229 with:
Lat = Latitude of point of departure
Dec = Latitude of destination
LHA = Difference in longitude

1. If HC and Z are of body above the horizon
Then D = 90 - HC (D = Distance)
(Z = Initial Course)
2. If HC and Z are of body below the horizon
Then D = 90 + HC and C = 180 - Z

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