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Celestial navigation

Computed altitude and azimuth – solving the navigational triangle

Every celestial position line needs the altitude the body should have from the DR position and the direction in which it lies. From DR latitude, declination and local hour angle NauticTools2 calculates the computed altitude Hc and the azimuth – the working step between the ephemeris and the position line.

• DR latitude φ
• Declination δ
• Local hour angle
• Computed altitude Hc
• Azimuth
• Navigational triangle

The navigational triangle

The celestial north pole, the observer's zenith and the body form a spherical triangle. Two sides and the angle between them are known: from pole to zenith 90° − φ, from pole to body 90° − δ, and the angle at the pole is the local hour angle. Wanted are the third side – the zenith distance 90° − Hc – and the angle at the zenith, the azimuth.

When do you need it?

• Altitude position line: Hc and azimuth, together with the observed altitude Ho, are the building blocks of the position line by the intercept method (Marcq St. Hilaire).
• Pre-computing a body: Where and how high will a star be at twilight? With Hc and azimuth you can preset it on the sextant and look in the right direction.
• Compass check: The computed azimuth of the sun is the basis for finding the compass error.

Formulas and working

1. Inputs

   DR latitude φ, declination δ (both signed, north positive) and local hour angle LHA = GHA + λ from the ephemeris calculation.

2. Computed altitude

   With the formula above, the cosine rule of the navigational triangle.

3. Azimuth

   From the auxiliary values hy = −sin LHA and hz = cos φ · tan δ − sin φ · cos LHA; their ratio and the sign of hy fix the quadrant, azimuth from 0° to 360°.

4. Intercept

   With the observed altitude Ho from the sextant corrections: intercept = Ho − Hc, in minutes of arc equal to nautical miles.

Worked examples

Things to watch out for

• Signs: South latitude and declination are negative. A forgotten sign with contrary names gives a completely wrong altitude.
• Forming the LHA correctly: add east longitude, subtract west longitude, bring it into 0° to 360°.
• Body below the horizon: If Hc is negative, NauticTools2 says so instead of showing a meaningless result – usually a sign of an input error.
• DR position instead of assumed position: NauticTools2 works directly from the DR position. Rounding to an “assumed position” as with sight reduction tables is not needed.

How it works in NauticTools2

1. Enter the values

   DR latitude, declination and local hour angle in degrees and minutes, each with N/S. At sea, “Take latitude from GPS” fills in the current latitude as DR latitude.

2. Calculate

   Hc and azimuth appear in the result, the working below.

3. Carry on

   For two or three bodies, Fix from altitudes brings all steps together to the fix. Inputs can be saved and the result printed.

Frequently asked questions

What is the computed altitude Hc?

The altitude the body should have at the time of the sight as seen from the DR position. Comparing it with the observed altitude Ho gives the intercept by which the position line is offset from the DR position.

What does a positive intercept mean?

The observed altitude is greater than the computed one – you are closer to the body's geographical position. The position line then lies intercept miles towards the body along the azimuth; a negative intercept means away from it.

Why is one minute of arc one nautical mile?

A nautical mile is defined as one minute of arc on a great circle of the Earth. An altitude difference of 1′ therefore offsets the position line by exactly 1 nm.

Related tools

• Ephemeris calculation – GHA and declination for the time of the sight
• Sextant altitude corrections – from the reading Hs to the observed altitude Ho
• Fix from 2 or 3 altitudes – position lines and the fix