**Fixes from Sightings of Stars and Planets. ** There are 59 navigational stars and 4 navigational planets which we can use to achieve position fixes. However, there are only two short periods during the day in which we can do this because we need it to be dark enough to see the bodies in the sky yet light enough to see the horizon. In other words, we are restricted to taking star and planet sights during the times of morning and evening nautical twilight.

Because nautical twilight gives us only a short period of time to make observations, advanced planning is essential. We need to establish what our estimated position at nautical twilight will be in advance and select the stars and planets that we intend to use for the fix. To be sure of taking three reliable sights for a three point fix, we should select at least four or five bodies for our observations.

Having selected our stars and/or planets, the next step is to calculate what the altitude and azimuth of each of them will be at the estimated position at the time of the planned observation. In this way, we will know what the approximate altitude and azimuth of each of the bodies will be when we come to make the sightings from the true position and this will cut the time we take to locate them in the sky.

There is also the added advantage that by making the calculations for the estimated position in advance, most of the work required to calculate the intercept will have been completed by the time the observation is made. This will of course, speed up to process of calculating the position lines and the ultimate fix.

**Sight Reduction. **This is the process of reducing the data gathered from observations of celestial bodies down to the information needed to establish an astronomical position line. The two essential items of data that we need to begin the process of sight reduction are the azimuth and the altitude of the celestial body in question. Methods of sight reduction usually fall under two categories, tabular and formula. Tabular methods such as Rapid Sight Reduction involve interpolating large tables of data by entering latitude, declination and LHA to extract altitude and azimuth. Formula methods involve mathematically calculating the altitude and azimuth from the same input data.

**Using Spherical Trigonometry For Sight Reduction. **With spherical trigonometry, we have the tools to quickly calculate the altitude and azimuth of selected celestial bodies at the estimated position without being encumbered by large tables of data.

**Why Spherical Trigonometry?** At first sight, the term ‘spherical trigonometry’ might seem quite daunting but with the knowledge of just two simple formulas and with a little practice of the methods demonstrated below, it will be found to be quick and easy to apply. The method is comprehensively taught in my book ‘Celestial Navigation – Theory and Practice’.

**Why Calculate Azimuth? ** The true azimuth and the azimuth angle provide exactly the same directional information albeit in different formats. This begs the question: “why go to the trouble of calculating the azimuth angle and then converting it to the true azimuth when it is easier just to measure the true azimuth directly with a compass?” However, we calculate the azimuth angle by finding the angle PZX from the values of the sides PZ, PX and ZX in the spherical triangle ZPX and these values are derived from data relating to the DR position. If we measure the azimuth by compass, we can only do so from the true position. At the time of taking the altitude, we would not know where the true position is so our aim must be to find the direction of the true position from the DR position and we can only do this by calculating the azimuth angle at the DR position. There is also the point that, unless you are in the fortunate position of having a gyro compass, you must take magnetic compass readings and these have to be corrected for variation and deviation; so you might just as well calculate the azimuth in the first place.

There is another important reason to be able to use spherical trigonometry for this task as the following statement in the ‘International Maritime Organization Regulations’ makes clear. “The provision of trigonometric tables onboard and regular practice with them by all officers and navigation related staff is compulsory”. In the event of GPS and other electronic navigation systems failure, it would be irresponsible of a ship’s master if his ship were to go dangerously off course simply because trigonometric calculations could not be made”.

**Example. Using Spherical Trigonometry To Calculate the altitude and azimuth of the star Alioth at the estimated position at the planned time of observation using the data provided in the scenario. **

**Scenario. **

Estimated Position: Lat. 50^{o}N Long. 45^{o}W

Data from Nautical Almanac re. Alioth: SHA = 166. Declination = 56^{o}N GHA Aries = 300

** ****Step 1.** Estimated position at planned time of observation: Lat. 50^{o}N Long. 45^{o}W

**Step 2. ** From the Nautical Almanac, extract data for planned time of observation as follows: **Alioth.** SHA = 166. Declination = 56^{o}N GHA Aries = 300

**Step 3. Calculate LHA.**

SHA Alioth 166

GHA Aries __300__

. 466

Long __ -45__ (subtract if long is west, add if long is east)

. 421 (subtract 360 if LHA is greater than 360)

. -360

LHA Alioth __ 61 ^{o }__(West since LHA less than 180

^{o})

**Step 4. Calculate PZ, PX and ZPX**

**PZ** = 90^{o} – Lat = 90^{o} – 50^{o} = 40^{o}. **PX** = 90^{o} – Dec = 90^{o} – 56^{o} = 34^{o }. **ZPX** = LHA = 61^{o}

**Step 5. Calculate Zenith Distance (ZX).**

**Formula to Calculate Azimuth (ZX): **

Cos (ZX) = [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

**Enter values of PZ, PX and ZPX in the formula: **(decimals to 3 places).

Cos (ZX) = [Cos(PZ) . Cos(PX)] + [Sin(PZ) . Sin(PX) . Cos(ZPX)]

= [Cos(40^{o}) . Cos(34^{o})] + [Sin(40^{o}) . Sin(34^{o}) . Cos(61^{o})]

= [0.766 x 0.829] + [0.643 x 0.559 x 0.485]

= 0.635 + 0.174

Cos (ZX) = 0.809

∴ ZX = Cos^{-1} (0.809) = 36^{o}

** ****Step 6. Calculate Altitude.**

Altitude = 90^{o} – ZX = 90^{o} – 36^{o} = 54^{o}.

**Step 7. Calculate Azimuth.**

Data previously calculated: PZ = 40^{o } PX = 34^{o} ZX = 36^{o}

**Formula to Calculate Azimuth (PZX): **

Cos PZX = [Cos(PX) -(Cos(ZX) x Cos(PZ))] / [Sin(ZX) . Sin(PZ)]

**Enter values of PZ, PX AND ZX into the formula: **(decimals to 3 places).

Cos PZX = [Cos(34) – (Cos(36) x Cos(40))] / [Sin(36) . Sin(40)]

= [0.829 – (0.809 x 0.766)] / [0.588 x 0.643]

= [0.829 – 0.620] / 0.378

= 0.21 / 0.378

∴ Cos(PZX) = 0.557

∴ PZX = Cos^{-1}(0.557) = 56.15 ≈ 56^{o}

∴ Azimuth angle = N56^{o}W (True Azimuth: 304^{o}) (Lat. North and LHA less than 180^{o})

**Summary: ** The altitude and azimuth of the star Alioth at the estimated position (EP or DR) at time of planned observation have been calculated in advance to be:

Altitude: 54^{o}. Azimuth: 304^{o}

This data serves two purposes:

Firstly it helps the navigator to quickly locate the position of the star at the planned observation time by providing its approximate altitude and azimuth. Secondly, by calculating the altitude and azimuth from the EP or DR in advance, the navigator will have all the data necessary to quickly calculate the intercept.

**Calculating the Intercept. **Ho is observed altitude. Hc is calculated altitude.

To calculate intercept (p): p = Ho – Hc

If p is positive, intercept is from DR position towards azimuth.

If p is negative, intercept is from DR position away from azimuth (i.e. towards reciprocal).

**Calculating the Intercept in the Alioth Example. **Suppose the actual measurements of altitude and azimuth of Alioth at the true position at the time of observation were as follows: Altitude: 53.785^{o} Azimuth: 304^{o}

We would calculate the intercept in the following way:

Calculated altitude (Hc) = 54^{o } Observed altitude (Ho) = 53.785^{o}

p = Ho – Hc = 53.785^{o} **– **54^{o }** = –**0.125^{o} **= –**7.5′ = -7.5 nautical miles

**Therefore intercept = **7.5 n.m from 304^{o } i.e. 7.5 n.m. towards 124^{o} (reciprocal).

** **Note. 1 minute of arc at the Earth’s centre will subtend a distance of 1 nautical mile at the Earth’s surface. (See chapter 6,).

The Rapid Sight Reduction method is taught in my book ‘Astro Navigation Demystified’.

The trigonometric method of sight reduction is taught in my book ‘Celestial Navigation – Theory and Practice’.

**Books of the Astro Navigation Demystified Series:**

Applying Mathematics to Astro Navigation

Astronomy for Astro Navigation

Celestial Navigation. Theory and Practice

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