No-Instrument, Field-Expedient Celestial Navigation

[Steven Carleton]

Reproduced below is an article I wrote that accompanied a talk at the winter Wilderness Medicine Society meeting in Park City, UT, in 2019. The talk was fun to prepare, and may be useful if you ever find yourself lost. It’s long (no apologies for this unless it violates Forum rules), but hey, it’s free!

Field-Expedient Celestial Navigation
Steven C. Carleton, MD, PhD

Celestial Navigation

Celestial navigation relies on the predictable relationship between celestial bodies and the surface of the Earth. At any instant of time a celestial body is directly above a spot on the Earth’s surface termed its Geographical Position, or GP. The GPs of the Sun, Moon, four navigational planets (Venus, Mars, Jupiter, Saturn), and Aries (a reference point on the celestial sphere) calculated for every hour of each year are published as The Nautical Almanac and Astronomical Ephemeris. Tables allow for interpolation to minutes and seconds of time. GPs are expressed in Greenwich Hour Angle (GHA), equivalent to celestial longitude, and declination, equivalent to celestial latitude. The hour angles of 57 navigational stars relative to Aries (Sidereal Hour Angles [SHA]) and their declinations are also included for each day. The Nautical Almanac is bulky, but relevant pages for the dates of any planned travel can be copied and carried.

Measurement of the angle between a celestial body and a natural (sea) or artificial horizon places the navigator on a circle of equal altitude centered on the GP of the celestial body. A sextant is used to measure this angle. Accuracy to within several minutes of arc can be accomplished under ideal circumstances. A small segment of a circle forms a Line of Position (LOP), or Sumner Line, that is at a right angle to the bearing line (azimuth) to the body. A single sight puts the navigator somewhere on this line. Two sights yield lines that intersect at two points on the Earth’s surface. Three sights yield three LOPs that intersect at a single point – a fix. This is the navigator’s position. A modification of this method, the Altitude-Intercept Method of Marcq St. Hilaire, permits determination of a fix with only two sights.

To reduce the sight to the navigator’s local area of travel, a navigational triangle is created and solved. Sight reduction involves formidable math (spherical trigonometry) that can be accomplished with a scientific calculator. Pre-calculated inspection tables also can be used. These are bulky, but individual pages for the latitude of travel can be copied and carried.

While a latitude line from a noon sight of the Sun can be determined without reference to time, the most powerful and flexible methods of celestial navigation require a clock accurate to within a few seconds, and keyed to Co-ordinated Universal Time (UTC) [formerly Greenwich Mean Time (GMT)].

Celestial navigation as described above was the standard for the determination of position and direction from the 19th century to the 1950s. It has been supplanted by satellite navigation, which is accurate to within several meters. Even optimally practiced, the accuracy of celestial navigation is measured in miles. This is sufficient for ocean navigation, since a landfall can generally be identified from a distance of many miles, but is insufficient for land navigation under most circumstances – one to three miles is an unacceptable margin of error when searching for a trailhead, a camp or a supply cache. The equipment required for celestial navigation can be reduced to a minimal kit weighing several pounds but is extremely unlikely to be carried by an expedition, no less by an individual trekker.

A functioning, mapping-GPS makes it virtually impossible to become lost. A functioning GPS. The ready availability of GPS has the unintended result of allowing individuals to confidently confront the wilderness without any grasp of more self-reliant methods of navigation. GPS receivers are subject to the frailties of all electronic devices; they require batteries, they may not be particularly shock- or water-resistant, and they function poorly under heavy forest canopy, in slot canyons and in other situations where access to the open sky is limited. Civilian GPS service can be shut-down by the government in times of threat, and can be interfered with by international adversaries and solar flares. Anyone traveling in the wilderness and relying solely on their cell-phone or hand-held GPS for route-finding is asking for trouble.

Navigation by map and compass is the best self-reliant back-up to satellite navigation. But successful map and compass navigation requires practice, pre-planning, and knowledge of magnetic declination in the area of planned travel. Local magnetic variation can also seriously impact compass accuracy. Unless factors that influence the magnetic compass are taken into account, serious navigation errors can result. Map and compass methods are only useful if one is actually carrying a relevant map and a compass.

How can you recover from being lost if you have only your wits and the contents of your pockets? The environment is full of useful navigational cues. The reliability of some of these is variable (moss on trees, the orientation of termite mounds), but celestial cues are absolutely reliable for the determination of direction with an accuracy superior to that of the magnetic compass. The fidelity of celestial methods for direction finding is evident in the historical use of these techniques for the correction of both magnetic compasses and gyrocompasses on military ships and aircraft prior to the advent of GPS.

Field-expedient celestial navigation will determine the cardinal directions with low-tech tools that can be carried in a wallet or manufactured on the fly. It is not magic: If a traveller becomes lost and has no familiarity with their general location, no familiarity with the night sky, no familiarity of the movement of the Sun with the seasons, or no familiarity with the limitations of their magnetic compass if they have one, they are in deep trouble. Some pre-planning is required for no-instrument celestial navigation. Thirty minutes on the internet should suffice to provide the information required. Online and text resources are included below. This information should not be trusted to memory – it should be written on a slip of paper and kept somewhere where it will always be available. Studying the night sky to allow easy recognition of useful stars and planets, and awareness of the changing but predictable behavior of the Sun with the seasons are important. Photocopies of one or two diagrams reduced to credit-card size, a pin, and some string added to this knowledge will allow you to perform sophisticated feats of self-contained directional navigation – and perhaps to recover when lost.

A list of the information and gear that I suggest carrying follows:

Button compass. While not a celestial technique, it will fit in your wallet and is undeniably useful…IF you know the magnetic declination of the area of travel and how to correct for it.
The magnetic declination off the area of travel (you can figure this out by celestial methods if forgotten or if you didn’t look it up in advance).
The amplitudes and declination of the Sun at the beginning, middle and end of the anticipated period of travel.
A list of any navigational planets and several navigational stars that will either be rising, setting, or near the meridian or the zenith at night during the period of travel. This should include the amplitudes and declinations of these bodies.
A photocopy of the Universal Pocket Navigator (UPN) (see Owendoff or Carleton references), a pin for use as a gnomon.
Small scraps of paper for recording information, or for constructing a UPN on the fly.
The stub of a pencil (but if you want to get Medieval, you can use the pin from #5 above and your own blood as ink for recording information).
A watch and knowledge of the time of local apparent noon (LAN) for the dates of travel.
A credit-card sized fresnel lens (great fire-starter) with a cm rule embossed on one edge, or other ruler/protractor combination for estimating angular distances.
Six to 10 feet of string.
A scrap of paper with your own hand measurements (see below) recorded on it.

This sounds like a lot, but it will all fit in your wallet (except perhaps the pencil stub).

Angle Estimation

Portable methods of estimating angles are critical to field-expedient navigation. If carrying string and a foldable cm ruler (even one photocopied on acetate), an accurate device for measuring angular distances is easily constructed. When a ruler is held at a distance of 57 cm from the plane of the face, each cm on the ruler will subtend about 1o of arc. When held half this distance from the plane of the face, each cm subtends an angle of about 2o. Prior to travel, drill or punch a small hole in the center of the ruler. Affix a string through this hole with a knot, then tie overhand knots at 28.5 cm and 57 cm from the ruler. Now, if either knot is held between the teeth or at the corner of the eye, the cm scale on the ruler can be used to measure angular distances along the horizon. This is extremely useful for exploiting the use of amplitude to establish the cardinal directions (amplitude methods will be discussed in a later section). If a ruler and string are not carried, degrees of arc can be estimated with the angles subtended by your fingers, thumb, fist and the span between various digits. There are many sources which suggest that these measurements are consistent between different people – that the length of the arm and size of the hand are in the same ratio from individual to individual. This is crazy talk. Do not use these published angles. Rather, know thyself – measure in cm the width of a few fingers, your thumb, your fist, the distance between your spread digits and the distance between your outstretched had and your eye. Apply a little trigonometry (side opposite/side adjacent = tan θ) to determine what angles your hand and fingers measure, and write it down or memorize the results. Then, as long as you have hands, you can estimate angles to minimize correctable errors in your no-instrument navigation.

Similarly, it is useful to be able to trace celestial bodies to their rising or setting points by projecting the co-latitude (90o – latitude) angle between the body and the horizon (to be discussed further in a later section). If you carry a protractor this is as easy as holding it up and directly measuring the angle. If not, you can estimate these angles using your hand as a protractor. As in the above discussion, many sources misrepresent the consistency of the angles between the fingers of an outstretched hand in different individuals. Do not use these published results. Rather, measure your own. The simplest method is to photocopy your spread hand, then measure the angles between fingers using the crease where the thenar and hypothenar eminences meet as the vertex. Your own individual results don’t matter – it only matters that you know what angles your hand can estimate. Write this down or memorize it.

Directions From the Daylight Sky

Direction by Amplitude

The Sun rises in the East and sets in the West, but only rises due East and sets due West on the Spring and Fall Equinoxes. The rest of the time, the rising and setting points vary from due East and West in a predictable fashion. The variation is termed amplitude, and is determined by the declination of the Sun and the latitude of the observer. Whether the amplitude is added to or subtracted from 90o or 270o depends on whether the declination of the Sun is positive or negative, and whether the observer is in the tropics or the temperate zones. This requires a little thought but is not difficult to sort out.

Anytime the Sun can be observed on the horizon or within an hour of so of rising/setting, its amplitude can be used to determine its true direction, and thus the cardinal directions. Amplitude can be calculated by formula:

Amplitude = sin-1 (sin d/cos L), where d = declination of the Sun, and L is observer’s latitude.

Or, looked up in Bowditch, Table 22 (older editions) or Table 27 (since 1980s).

Or, estimated diagrammatically (see Seidman). To use the diagram, the maximum possible amplitude for a given latitude must be know. It is provided as a table with the diagram but can also be determined from the Bowditch table or by calculation.

If the Sun is not observed on the horizon, its rising or setting point can be closely estimated by following a line from the Sun to the horizon at its rising/setting angle. The rising or setting angle is equal to the co-latitude of the observer (90o – latitude). As an example, at Park City UT (latitude about 40o), the co-latitude and rising/setting angle of the Sun = 50o.

Direction by Shadow Methods

The Sun casts long Westward shadows at rising, and long Eastward shadows at setting. At local apparent noon (LAN), when the Sun is on the meridian, it casts its shortest shadow and this shadow lies on a true North-South line. The tip of an object’s shadow cast on a flat surface will follow a shape during the course of the day that depends on the declination of the Sun on that date and the latitude of the observer. The length and bearing of the shadow can be predicted by formula or determined empirically. The Universal Pocket Navigator (UPN) is a Sun compass that has a pre-calculated series of curves and gnomon positions for various latitudes (from 0o to 60o) and dates (a surrogate for declination). If a gnomon of proper length is placed in the proper position, its shadow-tip will exactly follow the declination curve for the date listed in the appended table. When the UPN is held flat and rotated until the shadow tip touches the appropriate curve, the center line of the figure will indicate true North (in the Northern hemisphere) or true South (in the Southern hemisphere). The accuracy of the indicated direction is superior to that of a magnetic compass, and the UPN can be used to correct a compass for magnetic declination.

A sun compass can be constructed in the field from a scrap of paper, a chip of wood or a sheet of bark. The surface to be used is laid flat and a gnomon of a length that will cast shadows within the surface is set at the center one edge. After Sunrise, the tip of the gnomon shadow is marked at convenient intervals on the surface until the Sun passes the meridian. The portion of the curve for the afternoon can be traced in the same manner, or duplicated as the mirror image of the morning curve. Any two points on the curve that are equidistant from the base of the gnomon will lie on a true East-West line, and a line from the base of the gnomon that bisects this line will indicate true North and South. As long as the surface is held reasonably flat, and the gnomon position and length are consistent, the figure will indicate the cardinal directions in the manner described above. A Sun compass constructed in this manner should retain accuracy until the declination of the Sun changes significantly. Several days to a week of useful service can be expected, at which point another compass can be constructed with an up-to-date declination curve.

Many sources confidently assert that a line drawn between the two shadow tips cast 15 to 30 minutes apart will indicate East and West. While such a line will point Eastward and Westward, it will only point due East-West on the Equinoxes or if the two shadow points straddle noon by equal intervals. Errors of 25o are possible in high-temperate latitudes close to the solstices (December solstice in the Northern hemisphere, June solstice in the Southern hemisphere) when using this method. Since errors of 5o in bearing are sufficient to get you totally lost, this shortcut should be regarded with caution.

Similarly, many sources assert that true North/South can be determined using a shadow cast across the face of an analog watch. In this method the watch is held flat and the hour hand is aligned with a shadow cast by a gnomon held vertically at the watch rim. A line through the center of the dial and halfway between the hour hand and 12 is purported to be a North-South line. This method is fraught with errors that increase as latitude decreases, and are greatest near the Summer Solstice. Errors of 40o are possible even at temperate latitudes (such as the Southern continental United States). At upper temperate latitudes and in the Arctic and Antarctic, particularly near the Winter Solstice, the watch method may permit a quick and dirty determination of general direction while on the move. Errors in the watch method result from the discordance between civil time and sun time (the Equation of Time, Daylight Savings Time), the longitude difference between the observer and the central meridian of the time zone to which the watch is set, and the difference between the movement of the Sun in the sky and its shadow as cast on a watch dial held out-of-plane to this movement.

Directions From the Night Sky

Direction by North Circumpolar Stars

The most familiar directional reference in the Northern Hemisphere is Polaris, the “North Star” or “Pole Star”. Polaris has a declination (89.25o) that is close enough to the North Celestial Pole (NCP, declination 90o) to act as its surrogate for directional purposes. If you can find Polaris, you can always establish the cardinal directions. Polaris is the tip of the tail of the constellation Ursa Minor, the Little Bear (or the Little Dipper). If Polaris is obscured by clouds or terrain its location can be accurately estimated through observation of other circumpolar constellations, particularly Ursa Major. The portion of Ursa Major that is useful for this purpose is more familiarly known as the Big Dipper. The two leading stars of the bowl of the Dipper, Dubhe and Merak, are known as The Pointers because a line drawn between them in the direction the opening of the bowl will cross close to the NCP at a distance five times the angular distance between the two stars. A vertical line dropped from this point to the horizon establishes North. If the Big Dipper is unfavorably placed, the constellation Cassiopeia can be used to similar purpose – visualize a line between the stars forming the top of the W-shaped asterism (Caph, the brighter, and Segin, the fainter). Another line perpendicular to this, twice its length and originating at Segin, will approximate the position of the NCP.

Direction by South Circumpolar Stars

Unfortunately, the Southern sky has no equivalent to Polaris to serve as a directional reference. However, several South circumpolar constellations can estimate the location of the South Celestial Pole (SCP) to a degree of accuracy sufficient to establish the cardinal directions. In particular, the constellation Crux (the Southern Cross) is useful in this regard. When Crux stands vertically, the upright of the cross will point very close to geographic South at the horizon. At other times, a line drawn through the stars forming the upright of the cross and extended from the foot of the cross will pass near to the SCP. Closest passage will occur at a distance five times that between the two stars forming the long, upright line of the cross. The location of the SCP can be refined by drawing a line between two brightest stars of the constellation Centaurus (Rigel Kentaurus and Agena). A line bisecting this, perpendicular to it, and extended until it intersects the line previously described will identify the SCP to a sufficient degree of accuracy. A vertical line dropped from the point of intersection to the horizon will indicate South. If the Southern Cross is obscured by clouds, the stars Achernar (the brightest star in the constellation Eridanus) and Canopus (the second brightest star in the sky, in the constellation Carina) can be used to estimate the location of the SCP. If a line drawn between these two conspicuous stars is considered as the base of an equilateral triangle, the apex of the triangle will approximate the SCP.

Direction from Scorpius

The constellation Scorpius, a prominent feature in the Summer night sky of the Northern Hemisphere and containing the conspicuous red star Antares, can be used to determine South and establish the cardinal directions When a line between the stars forming the claws of the scorpion is perpendicular to the horizon, the three stars that form the base of the tail of the asterism will point South.

Direction from Orion

Orion is among the most recognizable constellations in the Winter sky. This constellation can be used to establish East and rising and West at setting (discussed below), and is also useful to identify South when it is near the meridian. When the constellation stands vertically, a line between the center star of Orion’s belt (Alnilam) and the portion of the asterism forming Orion’s sword will intersect the horizon very near to true South.

Direction by Amplitudes

Any celestial body can be used to determine an amplitude as long as its declination is known. A list with pre-calculated amplitudes for prominent navigational stars or planets for the period of travel ensure that the cardinal directions can be established at any time of night when either the Eastern or Western horizon are clear. The winter constellation Orion is a particularly useful in this regard: The three stars forming the familiar belt of the figure (Alnitak, Alnilam [a navigational star, declination -1.2o], and Mintaka) have declinations so close to 0o that their rising and setting points can be considered to indicate East and West, respectively. Several other navigational stars (Menkar [declination 4.06o] and Procyon [declination 5.2o]) are sufficient for on-the-fly, rough estimates of East and West if a list of amplitudes is not carried.

As with the Sun, the rising and setting points of other celestial bodies can be estimated, within an hour or so of Sunrise/Sunset, by tracing their path to the horizon at the co-latitude angle of the area of travel.

Example: On the night of 02/25/2019 – 02/26/2019 the following prominent celestial bodies are visible in the night sky when they are rising or setting. The ability to recognize just a few of these should suffice for establishing the cardinal directions over the course of the night if the amplitudes of the bodies are known, even if either the Eastern or Western horizon are partially obscured by weather. Note: The declinations of stars can be considered to be constant over the course of several lifetimes, while those of the Moon, Planets and Sun vary on a daily basis.

Body Time Declinationo Amplitudeo at Park City

Sun sets at1830 – 9 12 South of West
Arcturus rises at 2020 + 19 25 North of East
Spica rises at 2120 – 11 14 South of East
Mars sets at 2330 + 14 19 North of West
Rigel sets at 0020 02/26 – 8 11 South of West
Aldebaran sets at 0105 02/26 + 16 21 North of West
Alnilam sets at 0105 02/26 – 1 1 South of West
Sirius sets at 0115 02/26 – 16 21 South of West
Moon rises at 0130 02/26 – 14 18 South of East
Antares rises at 0130 02/26 – 26 31 South of East
Betelgeuse sets at 0200 02/26 + 7 9 North of West
Altair rises at 0245 02/26 + 9 12 North of East
Jupiter rises at 0254 02/26 – 23 31 South of East
Procyon sets at 0340 02/26 + 5 7 North of West
Venus rises at 0512 02/26 – 20 27 South of East
Regulus sets at 0640 02/26 + 12 16 North of West
Sun rises at 0717 02/26 – 9 12 South of East

If Mars (amplitude = 19o North of West) is observed setting just before midnight, its azimuth is 289o. The cardinal directions can then be accurately established by estimating 19o of angular distance South of this setting point using the span between selected fingers on your outstretched hand, or a cm ruler held 57 cm from the plane of the face (where 1 cm = 1o, as discussed in a prior section).

Direction from Zenith Stars

Any celestial body that crosses an observer’s meridian near the zenith with follow a discernible East-West path over a span of 15 to 30 minutes. If you lie on your back and look straight up at the sky with an overhead object (branch, cliff edge, suspended length of string) as a reference, the movement of the zenith stars relative to that reference will rapidly be obvious, and the cardinal directions can be established. A body that crosses the observer’s zenith will have a declination equal to the observer’s latitude. Bodies passing within +/- 5o to 10o of the zenith will yield sufficient accuracy. You don’t need to know the identity of the stars used, but knowing which zenith stars are available is helpful in predicting when to look. At Park City on the dates of this conference, Capella (declination 46o) crosses the meridian near the time of Sunset, and might be suitable for this purpose. Other zenith navigational stars for 40o latitude include Vega (declination 39o), Deneb (declination 45o), and Alcaid (declination 49o) but these are not visible near the zenith in late February. There are several non-navigational stars that are visible at this time of year and have declinations compatible with this method. Their declinations can be found from internet or text resources (I believe the version of the Nautical Almanac produced by the government lists the declinations of 173 stars, including the 57 navigational stars).

Useful Resources

http://www.nauticalalmanac.net

245 page online almanac for the current year with daily pages for the Sun, Moon, navigational stars, 4 navigational planets (Venus, Mars, Jupiter, Saturn) and Aries. Downloadable as a .pdf. Contains the same information that is provided by the Nautical Almanac and Astronomical Ephemeris jointly produced by the U.S. Navy Hydrographic Office and its U.K equivalent, and by commercial almanacs such as Reed’s.

http://www.thenauticalalmanac.com

Another free, online nautical almanac from 2011 to 2025. The current year comes up automatically in a highlighted tab, and can be downloaded as a .pdf. The site ncludes a long-term Sun-only almanac, downloadable copies of Bowditch, sight reduction tables and a wealth of other useful navigational information.

http://www.tecepe.com.br/nav

An online almanac that provides the above astronomical data for any chosen three-day period. Also displays and prints a polar chart of the sky for any time and location so that available navigational stars and planets can be identified.

http://www.robertoior.it

Downloadable online table of increments and corrections for almanac data to allow celestial sights to be calculated to the second of time.

sunearthtools.com

An online calculator that generates altitude and azimuth tables and planispheric plots of the sun on any given date at increments chosen by the user.

aa.usno.navy.mil

U.S. Naval Observatory website with numerous astronomical calculators including Sun/Moon altitude and azimuth tables for any chosen latitude, date and increment.

mytopo.com

Free mapping site that provides USGS-derived topographical maps for any location and at any scale chosen by the user.

Stellarium (stellarium.org)

Astonishingly powerful (and free!) planetarium software that duplicates and animates the appearance of the sky for any date, time and location. Available for Mac, iOS, Linux, Windows and Android platforms. An hour or so spent exploring the sky with Stellarium will familiarize the user with the appearance and movement of celestial bodies, the changing appearance of the night sky with the seasons, and provide astronomical data that can be used for the determination of position and direction.

Star Walk

A powerful, inexpensive iOS planetarium app with many of the same functions as Stellarium.

Celestial

An inexpensive navigation app available for mobile platforms and PCs that provides almanac data, sight logging, sight reduction and plotting for formal celestial navigation. Accurate and easy to use. Substitutes for 20 lbs. of books and tables.

msi.nga.mil

The National Geospacial-Intelligence Agency, online site for a free download of Bowditch (The American Practical Navigator) in .pdf format. Back in the day, in the Western world, the only two books that were felt to be essential on a sailing vessel were The Bible and Bowditch! Bowditch contains an encyclopedic explanation of all aspects of navigation – celestial, coastal, and electronic. It also has essential tabular data covering any aspect of navigation that can be imagined (including a very useful table of amplitudes). Over 1000 pages long in hardback, and a bargain if interested in navigation history or practice. Literally everything you need to know.

Owendoff, RS. Better Ways of Pathfinding. Stackpole Publishing, Harrisburg, PA, 1964.

Now out of print, but available in libraries and from used booksellers, this book has excellent, practical advice for field-expedient celestial navigation and includes the original Universal Pocket Navigator, an unparalleled aid for direction-finding based on Sun shadows.

Carleton, SC: Wilderness navigation and techniques. In: Auerbach’s Wilderness Medicine, 7th Ed., Elsevier, Inc., Philadelphia, 2017.

Discusses satellite navigation, celestial navigation, map and compass navigation, and methods for field-expedient navigation when instruments are lacking. Includes a copy of the Universal Pocket Navigator.

Burch, D. Emergency Navigation, 2nd Ed., McGraw-Hill, Camden, MN, 2008.

Comprehensive coverage of emergency navigation techniques at sea with excellent illustrations. Many of the techniques described are also applicable to land navigation.

Seidman, D. The Essential Wilderness Navigator, Ragged Mountain Press, Camden, MN, 1995.

Clear, concise treatment of map reading and compass use, and a small section on celestial techniques. Includes a very useful figure and table for determining the amplitude of the sun graphically…this diagram could even be drawn in the dirt to determine true direction from the rising or setting Sun. Now available in a 2nd Edition published in 2001.

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