Solving Navigational Triangles

This article will show how to "solve" the navigational triangle to find computed altitude (Hc) and azimuth (Zn) for a celestial sight. By comparing Hc with observed altitude (Ho), the navigator finds the altitude intercept (a). With the assumed position coordinates, azimuth, and inter­cept, a line of position can be established.

The two basic methods for solving a navigational triangle or other spherical tri­angle are by sight reduction table or by mathematical solution.

A sight reduction table presents solutions of the navigational triangle for com­putation of the altitude and azimuth of a celestial body. The variable components are co-latitude, polar distance, and meridian angle "t". Their values depend on time of the observation and the observer's position. The parts of the triangle we wish to find are computed co-altitude (in order to find Hc) and azimuth angle (in order to determine Zn).

Mathematical solutions use the same formulas used to pre-compute the values found in the sight reduction tables. Final results, found through the correct use of either method, seldom differ by more than one-tenth minute of arc for intercept values, or one tenth degree for azimuth values.

SIGHT REDUCTION TABLE - PUBLICATION 229
The Sight Reductton Table for Marine Navtgatton (Pub 229) is a form of sight reduction table which is produced by the Defense Mapping Agency. They are designed to facilitate solution of navigational triangles to find computed altitude (Hc) and azimuth (Zn).

There are six volumes of Publication 229, each volume covering 15 degrees of latitude. All of the problems used in USCG license exams, which may be solved with Publication 229, are in latitudes 15 degrees to 30 degrees covered by Volume II of the tables.

The entering arguments for 229 are assumed latitude, the whole degree LHA which results when assumed longitude is applied to GHA, and declination.

The main part of the tables is divided in half. The first half of Volume II covers latitudes 15 degrees through 22 degrees, and the second half covers latitudes 23 through 30 degrees. The latitude which is used to enter the table may be either north or south. Each half covers LHA's from 0 degrees through 360 degrees.

There are two pages for each LHA value from 0 through 90 degrees, and 270 through 360 degrees. Left-hand pages are for situations where declination and latitude are both north or both south, they are said to have the "same name." Each "same name" page covers declinations from 0 through 90 degrees.

The adjacent right-hand pages cover "contrary name" situations (meaning the declination is north and latitude south or vice versa) for the LHA values at the top of the page, and "same name" situations when the LHA values are between 90 degrees and 270 degrees located at the bottom of the page. The dividing line between "contrary" and "same name" situations is the step-like line across the right-hand page.

The step-like line across the right-hand page represents the horizon. Entry values of LHA, latitude, and declination (same or contrary) should never result in crossing this line, as this would mean the body is below the horizon.

The inside front and back covers of each 229 are an interpolation table to be used for correcting the the altitude and azimuth angle obtained in the main part of the table for the declination increment. Declination increment is the minutes and tenths of minutes part of the declination of the body. For example, if the body's declination is N 43°­23.7', the declination increment is 23.7'. In other words, declination is tabulated on each page in whole degrees, and you must interpolate between those whole degrees for your actual declination value.

Declination increments 00.0 through 31.9 minutes are located inside the front cover, and declination increments 28.0 through 59.9 minutes are inside the back cover.

Detailed, step-by-step, procedures for finding computed altitude (Hc) and azimuth (Zn) will be presented later in another article in a series of example problems. I will cover all the different possible situations you might find in working with Publication 229.

Procedures For Applying Sextant Altitude Corrections For A Planet

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Prepare Your Boat For High Winds

This is the start of the hurricane season for boaters. You should be prepared well before the hurricane season begins, and know what to do in the event a storm makes landfall.
Get a plan in place early to make preparing your boat as efficient and quick as possible when a hurricane watch is issued .

Purchase items needed for a storm preparation kit.
Check your cleats for adequate strength, replace weak back plates and bolts with aluminum back plates and stainless steel bolts.
Make a checklist and keep it handy. If the time comes to implement your plan, you will be less likely to forget something important.
Take an inventory, preferably by video or camera of the interior, exterior, and the items in your boat. Keep it in a safe place in case you need to file an insurance claim.
Find a sheltered "anchorage" if you plan to anchor your boat out, and make a plan to get there quickly and safely.

Before hurricane season begins, make a plan that includes purchasing items that will help you prep your boat quickly and efficiently. Here are a few basics suggestions for your boat’s preparation kit:

1. Duct tape
2. Plastic
3. Rags
4. Plywood
5. Trash bags to easily remove items from your boat

Add these items to the items for all boats if your boat is moored:
Extra mooring lines a 1/4” to double the size of your normal line. Purchase new line and store outside of the marine environment to ensure the integrity of the line. Use this line for storm preparation only.
Chafing gear made of garden or fire hose, leather, or other reinforced plastic.
Extra fenders or old tires
Extra line, tie straps, or bungy cords to lash loose items down

Add these items to the items for all boats if your boat is trailered:
Blocks for your boat trailer
Tie straps or heavy rope to tie your boat to the trailer
Grounds hooks or steel rebar and chains or tie straps to anchor your trailer to the ground

Pre-deparure Check List for Small Boats

Personal Flotation Devices (PFDs)
Have at least one Coast Guard-approved device per passenger and a minimum of two on board.
An additional throwable device is required if the vessel is more than 16 feet long.
Explain the location and use of all PFDs to passengers and crew that may be new to the vessel.

Sound Producing Devices
Have a horn capable of producing a four-second blast audible for at least 1/2 mile on board.
if you use portable air horn, have a spare can of air or an alternate device.
Attach a whistle to each PFD.

Lights and Shapes
Have all navigation lights as required for your boat.
Make sure all instrument lights are working.
If you intend to engage in a recreational boating activity that requires a day-shape, have the required shapes.
Have aboard a flashlight and spare batteries.

Distress Signals
Make flares, day signals, etc., accessible and ensure they are stored in a dry location.
Carry signals at all times even if not required by the Coast Guard.
Inform the crew and passengers of their location and safety rules for proper usage.

Tools and Spares
Carry a basic toolbox with tools appropriate for your boat.
Carry a box of spares including fuel filter, light bulbs, head parts, through-hull plugs, etc.

Fuel and Oil
Top off your fuel tanks.
If you can't, have enough fuel to provide a reasonable margin of safety for your return.
Check the engine oil and coolant levels.

Fire Extinguishers
Carry at least one fire extinguisher and make sure it is accessible. Make sure you have at least the number required by Coast Guard rules.
Check to be sure mounts are secure and functional before departure.
Take the time to point out locations to passengers and crew.

Ventilation
On any powered vessel or auxiliary powered sailboat, or vessels using LPG for cooking or heat, check that all interior spaces are well ventilated before departure.
If fuel smells are detected before ventilating, check after running the blowers for several minutes before starting.
If odor persists, shut down the engine and look for the source of the leak.

Bilges
Check to be sure bilges are reasonably dry and that pumps are not running excessively.
Clean up any spilled oil or waste in bilges to prevent overboard discharge.

Weather Forecast
Always check the weather forecast before boating.
Have a radio on board to receive weather updates.

Battery Care
If you have a dual charging system, make sure the selector switch in the proper position.
Make sure the power is on to the entire vessel.
Have aboard spare batteries for accessories such as your handheld radio, flashlight, portable navigational aid, etc.
If the batteries are rechargeable, make sure they're charged.

Docking and Anchoring Tips
Have at least one anchor set up and bent-on to your anchor line.
Carry two or three extra dock lines in case you encounter unusual conditions dockside.
Visually inspect the lines you use for chafe or wear.
Carry at least two fenders on-board for docking or towing if required.

Rules & Documentation
Have the ship's papers, radio license, fishing permit, etc. on board.
Have the chart or charts for the area you intend to cruise in, regardless of your level of local knowledge.

Emergency Plugs for Thru-hulls (Small Boats)

Every boat that has thru-hulls or seacocks should have safety plugs on board. In the event of a seacock or hose failure, the plug is driven into the opening, keeping the water out.
Make the plugs of softwood, such as cedar or pine, oak, mahogany or ash are too hard and may crack the seacock housing. Easy to make, you'll need a hand plane or sharp knife, saw of some kind, rasp, 80-grit sandpaper and oil sealer.

Two sizes of plugs fit all thru-hulls on board. For large plugs (1 1/2"ID), use 1 1/2" stock cut into 5" lengths. Taper plugs from 1 1/2" to 3/4", cutting into an octagon shape with the help of an electric or hand saw. Place in a vise and continue tapering with a jigsaw and finish with a rasp. Alternatively, try whittling with a sharp knife or block plane. Finish off with sandpaper and round the ends to prevent splintering. For smaller seacocks (3/4"), use 1" cedar or pine, cut 4" long and taper to 3/4". Drill a small hole in the larger end for the lanyard. Dip each plug in oil, hang to dry, attach a lanyard of 1/8" polyester cord, and tie around the base of the thru-hull.

Make a few spares for every size of seacock and stow them in your tool kit. If you ever need need them (hopefully never), just tap the plug into place. The lanyard wraps securely around the seacock base to keep water pressure from forcing the plug out. Every year, inspect the plugs for rot and cracking and oil when necessary.

Corrections To Apparent Altitude

Index correction and dip correction depend on the instrument and the height of the observer. The remaining corrections depend on which body was observed and the altitude obtained. Listed below are the various corrections.

1. Refraction:
A correction made to the altitudes of all bodies is the correction for refraction. Refraction occurs because the light rays from the body encounter a denser medium, the earths atmosphere, and are bent downward.
Refraction causes bodies to appear to be at greater altitudes than they are actually at. For this reason the correction for refraction is always negative.

Refraction correction is maximum for bodies at altitudes near the horizon. (zero) and decreases as altitude increases. Bodies at the zenith (90°) do not require a refraction correction. The main part (left half) of the "STARS AND PLANETS" correction table inside the front cover of the Almanac is a table of refraction corrections for various apparent altitudes from 9° 56.0 and upwards. Like the dip correction table, the main refraction table is a "critical table" and no interpolation is required.

Refraction values for altitudes less than 10 degrees for the sun, stars, and planets are located on the next page (A3) of the Almanac. This is not a crit­ical table, meaning that interpolation is required as necessary.

Due to their great distance from earth, no corrections other than IC, dip. and refraction are required for correcting sextant altitudes (Hs) of stars, Refraction is "built in" to the altitude correction tables for the sun and moon, in that it is mathematically combined with other factors in deriving the values which appear in the tables. This is the case with the sun's val­ues in the low altitude table.

Refraction values in the main tables are for a temperature of 50 degrees Fahrenheit (10°C) and atmospheric pressure of 29.83 inches (1010 mil­libars). If temperature and/or pressure at the time of a sight vary a great deal from these fairly standard values, especially when temperature is below 10°F., an additional correction for non standard conditions should be applied. These corrections are derived from the table on page A4 of the Almanac.

2. Semidiameter:
When using a sextant to measure the altitude of a body, the center of the body is used for measurement. The semidiameter correction is incorpo­rated into the altitude correction tables for the sun and moon because it is impossible to determine the altitude of the center of these bodies with a conventional sextant. For this reason you measure the altitude to the lower limb or upper limb of the body. The choice of which limb used depends on which is visible.
The semidiameter correction is added to lower limb altitudes and subtracted from upper limb observations. In practice, the semidiame­ter correction is included as a component in the correction found in the tables for the respective bodies.

3. Augmentation:
The augmentation effect is the increase in apparent size of the body (which increases semidiameter) as a result of an increase in altitude. This occurs because of the nearness of the sun and moon to the earth, and because the observer is on the surface rather than the center of the earth. Augmentation effect is zero for a body on the horizon, and maximum at the zenith.
The augmentation effect is a component of the main correction in the moon's altitude correction tables. It is not included in the tables for the sun because the effect, though measurable by astronomers, is so small as to be considered negligible in navigation.

4. Phase:
Phase is another small correction which must be applied to altitudes of Venus and Mars. Because the light we see from these planets is reflected from the sun, the center of the planet which we see in the sextant optics may not be the true center. Like phases of the moon, only part of the body may be illuminated depending on where the body is in it's orbit. The phase correction corrects the sextant to an equivalent altitude of the center of the body. This correction is a component of the additional corrections for Venus and Mars found inside the front cover (page A2) of the Nautical Almanac.

5. Parallax:
Last, you must consider the correction that must be applied to observations of bodies within the solar system due to the observer being on earth's sur­face rather than at the center. The difference between the altitudes as mea­sured from the surface and from the center is called parallax. The value of the correction is greatest when the body is on the horizon, called horizon­tal parallax, and zero when the body is at the zenith.

The parallax correction is greatest for observations of the moon, and varies from hour to hour. Parallax corrections for the sun are next in magnitude and are a component of the sun's altitude correction tables in the Almanac. Parallax correction for Venus and Mars is included in the addi­tional correction tables for Venus and Mars.
Note: These additional corrections for Venus and Mars are only applied to altitudes from night-time observations.

The Marine Sextant (Correctable and Non-Correctable Errors)

Devices for measuring the altitude of celestial bodies above the horizon along a vertical circle have existed for centuries. The earliest devices were cross staffs which consisted of two pieces of wood. Later came quadrants, octants, and sextants which incorporated arcs of one fourth, one eighth, and one sixth of a circle, respectively.

The instruments used by early explorers were made of wood, but by the time of the American Revolution instruments of cast brass or bronze were most common. The modern "micrometer drum" sextant's accuracy can be as close as one tenth mile when used by a skilled navigator in good weather conditions.

Sextants are used to measure the angle between two points. Its main use is to measure the altitude (angle) of celestial bodies above the sea horizon. The sextant is also used to measure the angle (or difference in bearing) between two terrestrial objects by turning it on its side.

The Frame: Constructed of either brass or aluminum. It is the part of the sextant to which all the other parts are attached.

The limb: The bottom part of the frame which is cut with gear teeth.

The Arc: The arc is graduated with a scale representing altitude readings from a few degrees negative to over 120°.

The Index Arm: A movable bar pivoted about the center of curvature of the limb. The index mirror and micrometer drum/tangent screw assembly are fixed to it.

The Tangent Screw: It is the worm gear and shaft which engages the limb.

The Release Levers: These spring actuated clamps hold the tangent screw against the limb and keeps the worm gear engaged. Pinching them togeth­er releases the tangent screw so that the navigator may make large changes to the position of the arm.

The Micrometer Drum: Sixty minutes of arc are graduated around its cir­cumference. It is rigidly mounted on the shaft of the tangent screw and one revolution of the drum changes the altitude measurement one degree.

The Vernier Scale: Adjacent to the micrometer drum and fixed to the arm, it allows reading to the nearest tenth of a minute of arc.

The Index Mirror: Mounted on the index arm perpendicular to the plane of the frame of the sextant. It is directly adjacent with the pivot point of the arm.

The Horizon Glass: One half mirror and one half clear glass. It is mount­ed perpendicular to the frame of the sextant directly in the line of sight of the telescope. The silvered half of the glass is closest to the frame of the sextant.

The Shade Glasses: Those for the index mirror allow observation of the sun or bright moon and the horizon glass shades reduce glare on the hori­zon for low altitude sights.

The Telescope: Amplifies both direct and reflected images observed.

The Handle: Made of plastic or wood and is designed for right-hand use. Often, the batteries for the light which illuminates the arc for night read­ings are in the handle.

One of the interesting things about the sextant is that it can measure an angle of over 120 degrees, even though its arc is actually only sixty degrees. This is possible because of the optical principle of the sextant. The angle between the body and the horizon is about twice the angle between the index mirror and horizon glass. In fact, it is exactly twice the angle, and that is why a sextant with an arc of 60° can read angles of up to 120°.

Reading a marine sextant is not difficult, but must be done in a certain order. First, read the number of degrees indicated by the index arm. Next, note where the zero mark on the vernier is pointing to read the minutes of arc. Next, note where the zero mark on the vernier is pointing to read the min­utes of arc. Finally, to read tenths of minutes, check to see which two marks are most closely aligned between the vernier and the drum.

CORRICTABLE AND NON-CORRECTABLE ERRORS OF SEXTANTS

There are four correctable and three non-correctable errors which may be found in marine sextants. The correctable errors are:

1. Error of perpendicularity - This error results from the index mirror not being perpendicular to the sextant frame.

2. Side Error - This error results from the horizon glass not being perpen­dicular to the frame of the sextant.

3. Index Error - This error results from the horizon glass not being paral­lel to the index mirror when the sextant is set on zero.

4. Collimation Error - An error which results from the telescope not being parallel to the frame.

These errors may be corrected by the navigator using various adjustment points on the sextant.
The non-correctable errors on the sextant are:

1. Graduation Error - These small errors are caused by imperfections in machining the arc, cutting the limb gears, or marking the scale of the arc or micrometer drum.

2. Prismatic Error - This error is caused by the planes of a mirror not being exactly parallel.

3. Centering Error - This error results when the index arm is not pivoted at the exact center of curvature of the arc.

The sum of these errors, called instrument error, may be found on the manufacturer's certificate located inside the lid of the sextant case. The amount of error may change with the altitude being measured. To correct a sextant reading for instrument error, the sign of the error is simply reversed to become the instrument correction, which is applied to the sextant altitude observation.

Of all these errors index error is the most important because it is an error which directly effects altitude measurements. Because sextant readings can be effected by changing temperatures, which tend to expand or contract the metal parts of a sextant, the navigator should determine the amount of index error at least once each day.

Index error is determined by setting the sextant near zero, pointing it at the horizon and then turning the micrometer drum slowly until the actual and reflected images of the horizon are aligned. If the sextant reads zero when the horizons are aligned, there is no error. If not, note the sextant reading when the horizons are aligned. If the reading is a positive angle (greater than 0°-00.0') the error is said to be "on the arc." If the reading is less than zero, or a negative angle, it is aid to be "off the arc." If the error determined is on the arc it must be taken off (the index cor­rection (IC) is negative). If off the arc, put it on (the IC is positive). As a memory aid remember, "If it's ON it's OFF, and if it's OFF it's ON."