Basic principles of celestial navigation - James Allen

Basic principles of celestial navigationJames A. Van Allena)

Department of Physics and Astronomy, The University of Iowa, Iowa City, Iowa 52242

~Received 16 January 2004; accepted 10 June 2004!

Celestial navigation is a technique for determining one’s geographic position by the observation ofidentified stars, identified planets, the Sun, and the Moon. This subject has a multitude ofrefinements which, although valuable to a professional navigator, tend to obscure the basicprinciples. I describe these principles, give an analytical solution of the classical two-star-sightproblem without any dependence on prior knowledge of position, and include several examples.Some approximations and simplifications are made in the interest of clarity. ©2004 American

Association of Physics Teachers.

@DOI: 10.1119/1.1778391#

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I. INTRODUCTION

Celestial navigation is a technique for determining ongeographic position by the observation of identified staidentified planets, the Sun, and the Moon. Its basic principare a combination of rudimentary astronomical knowledand spherical trigonometry.1–3

Anyone who has been on a ship that is remote fromterrestrial landmarks needs no persuasion on the valucelestial navigation. There are modern electronic navitional aids such as Loran, ocean bottom soundings, andglobal positioning system~GPS! which depends on receivinprecise timing data from a world-wide network of artificisatellites, each carrying one or more atomic clocks. Thisper illustrates what can be learned with only a sextant, rumentary astronomical tables, and an accurate knowledgabsolute time~from radio signals or a precise chronomete!.It is intended for students who are curious about the baprinciples of celestial navigation and makes no pretenseserving a professional navigator or surveyor.

II. THE EARTH AND GEOGRAPHIC POSITION

The surfaces of the land masses of the Earth and ofoceans are approximated as the surface of a sphericallymetric body of unit radius, hereafter called the terrestsphere.4 The Earth as a whole is assumed to be rotatingconstant angular velocity about an inertially oriented athrough its geometric centerO. A geographic positionP isreferenced to a right-hand spherical coordinate system whcenter is atO and whose positiveZ-axis is coincident withthe Earth’s rotational axis and in the same sense asEarth’s angular momentum vector. The plane throughO thatis perpendicular to this axis is called the equatorial plane~theX–Y plane!. Any plane that contains the rotational axis itersects the spherical surface in a great circle called a meian. By convention, theX axis lies in the meridian plane~theprime meridian! that passes through a point in GreenwicEngland.1

The positionP is specified by two coordinates, its latitudF and longitudeL. The latitude is the plane angle whosapex is atO, with one radial line passing throughP and theother through the point at which the observer’s meridiantersects the equator. Northern latitudes are taken to be ptive ~0° to 90°! and southern latitudes negative~0° to 290°!.The longitude is the dihedral angle,3,5–7 measured eastwarfrom the prime meridian to the meridian throughP. The

1418 Am. J. Phys.72 ~11!, November 2004 http://aapt.org

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longitude L is between 0° and 360°, although often itconvenient to take the longitude westward of the prime mridian to be between 0° and2180°. The longitude ofP alsocan be specified by the plane angle in the equatorial plwhose vertex is atO with one radial line through the point awhich the meridian throughP intersects the equatorial planand the other radial line through the pointG at which theprime meridian intersects the equatorial plane~see Fig. 1!.

III. THE CELESTIAL SPHERE

The celestial sphere is an imaginary spherical surfwhose radius is very much greater than that of the Eawith its center atO and its polar axis coincident with thEarth’s rotational axis. The position of each celestial objecrepresented by the point on the celestial sphere at whicline to it from O intersects this sphere. Inasmuch as the dtances to all stars, planets, and the Sun are much greaterthe radius of the Earth, a terrestrial observer may be thouof as viewing the sky fromO. For the nearby Moon, however, its apparent position on the celestial sphere is, becof parallax, different by as much as nearly one degreeobservers at different latitudes and longitudes.8 It is impos-sible to make ana priori correction for this effect if theobserver’s position is initially unknown.

On the celestial sphere, the declinationd of a celestialobject is strictly analogous to the latitudeF of a terrestrialobserver as defined in the above. The second spherical cdinate of a celestial object, called right ascensiona, is thedihedral angle analogous to eastward terrestrial longitubut with the fundamental difference that it is measured fra different reference point. That reference point on the cetial equator is called the vernal equinox, usually denotedg, and defined as follows. The equatorial plane of the Eais tilted to the plane of its orbit~the ecliptic plane! about theSun by 23.44°. These two planes intersect along a linepierces the celestial sphere at two points, called nodes.vernal equinox is the ascending node of the ecliptic onequator, that is, the position of the Sun on or about 21 Maof each year as the Sun ascends from southerly to northdeclination. The right ascensiona of a celestial object, al-ways taken to be positive eastward, is measured fromvernal equinox. Astronomers usually measure right ascenin hours, minutes, and seconds of time, but it can be cverted to the equivalent angular measure by taking one h515°. Another commonly used quantity is called the hoangle. It is the dihedral angle between any specified merid

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plane and the meridian plane through a celestial object,ways taken to be positive westward and lying in the ran0–24 h or 0°–360°. The sidereal hour angle~SHA! of acelestial object, taken to be positive westward from the vnal equinox, is related to the right ascension~Fig. 2! by

SHA5360°2a. ~1!

We will takea, d, and the SHA to be fixed quantities for anobject ~for example, stars or galaxies! beyond the solar system; however, they vary with time for the planets, the Sand the Moon. Their values are tabulated in almanacs,9–11 ofwhich The Air Almanac9 is the most convenient for a navgator.

Fig. 1. Definition of latitudeF and longitudeL. ~a! Meridian cross-sectionof the Earth through the observer atP. The center of the Earth is atO andNP and SP are the north pole and south pole, respectively.~b! Equatorialcross-section of the Earth.G represents the point at which the prime meriian through Greenwich intersects the equator andP represents the point awhich the meridian throughP intersects the equator.E means eastward andW means westward. The longitudeL is the dihedral angle~Ref. 7! betweenthe two named meridian planes, but is more easily visualized in this etorial diagram.

Fig. 2. Right ascension~a! and sidereal hour angle~SHA! are both dihedralangles~Ref. 7! measured from the meridian plane through the vernal eqnox g, with a positive eastward and SHA positive westward. In this eqtorial diagram,S represents the intersection of the meridian plane througstar or any other celestial object with the equator.

1419 Am. J. Phys., Vol. 72, No. 11, November 2004

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IV. OBSERVATIONAL CONSIDERATIONS

Imagine that a terrestrial observer is located at a fixpoint P of unknown latitudeF and longitudeL. The celestialsphere rotates westward from the observer’s point of viewan angular rate such that the vernal equinox transits~passesthrough! the observer’s meridian from east to west at intvals of 23 hour, 56 minute, 4 second of mean solar time, acalled the sidereal rotational period of the Earth or the sireal day.

The observer’s zenith is defined by the outward extensof the line from the center of the Earth through the obseras in Fig. 3. The instantaneous angle between the obserzenith and the line to a star is called the star’s zenith distaand is denoted byz. As the celestial sphere rotates westwaabout the Earth~that is, clockwise as viewed from above thnorth celestial pole!, z decreases from 90° as the star risfrom below the observer’s eastern horizon, has a minimvalue zmin as S crosses the observer’s meridian, thencreases to 90° as the star sets below the observer’s wehorizon. These comments are applicable to all celestialjects for whichd,~90°2F!. Stars whose declinations methis condition transit the observer’s meridian at intervalsexactly one sidereal day; but the intervals between the scessive transits of planets, the Sun and the Moon vary,though predictably.

Stars for whichd>~90°2F! constitute the observer’s circumpolar star field which, as viewed fromO or any point onthe Earth, rotates counterclockwise for a northern hesphere observer or clockwise for a southern hemisphereserver as time progresses. Such stars neither rise norbeing always above the observer’s horizon as they malong small circles centered on the celestial pole. Theirnith distances have a maximum value of 180°2~F1d! atlower culmination and a minimum value~F2d! or ~d2F! atupper culmination as they transit the observer’s meridtwice per sidereal day.

Usually, a practitioner of celestial navigation employsmarine sextant or an aircraft bubble sextant to determineinstantaneous altitude of a celestial object, the vertical anof the object above the local horizontal plane. After apppriate corrections,1,2 this quantity is denoted byh. However,because of its simpler geometrical significance, I preferzenith distancez to represent the observed quantity, with

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Fig. 3. Definition of the zenith distancez of a star or other celestial object

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The quantityz is treated as intrinsically positive and is in thrange 0° to 90°.

V. DETERMINATION OF LATITUDE BY MERIDIANTRANSIT OF A STAR

As illustrated in Fig. 4,

F5d1zmin , ~3!

whereF is the observer’s latitude,d is the declination of thestar, andzmin is the observed minimum value of the zenidistance of the star~a southerly angle in this example! as ittransits the observer’s meridian.

If the star transits the observer’s meridian northward ofP,the corresponding relation is

F5d2zmin . ~4!

For a celestial object havingd5F, the value ofzmin is zero atthe moment that the object transits the observer’s meridthat is, at the observer’s zenith. Thus, in this special caseobserver’s latitude is equal to the declination of the star.

The same considerations are applicable to determinlatitude by observing the meridian transit of planets orcenter point of the disk of the Sun. But stars are conceptusimpler because their values ofa andd are almost constantwhereas these coordinates for all celestial objects of the ssystem have changing values ofa andd which must be takenfrom a daily ephemeris.9

In any case, an observer also obtains the ‘‘true north’’‘‘true south’’ direction, namely the direction of any celestiobject at the moment of its upper or lower culmination.

VI. DETERMINATION OF LATITUDE BYOBSERVING POLARIS

An especially valuable star for northern hemisphere nagators is the bright star Polaris~the North Star ora UrsaMinoris! which presently has declinationd5189.27°. Thus,the observer’s latitudeF is approximately equal to the altitude of Polaris or

F'90°2z, ~5!

Fig. 4. P represents the position of the observer andS the position of thesubstellar point of a star at the moment that it transits the observer’s mian. O is the center of the Earth,d the declination of the star,F the latitudeof the observer, andz the zenith distance of the star. NP is the north pole aSP is the south pole.


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wherez is the zenith distance of Polaris, and, at this levelapproximation, is constant and independent of the timeday. Hence, Polaris has had a special simplicity for determing latitude throughout maritime history. For example, a vsel can sail from Europe to America without an accurknowledge of the time by simply maintaining a westercourse during whichz of Polaris is approximately constant ochanges day-by-day in such a way as to assure landfaAmerica at approximately an intended latitude. Becausediurnal motion of Polaris lies along a small circle around tnorth celestial pole of angular radius 0.73°, an improvemin accuracy in determining the latitude is to observe Polaat specific moments identified from its ephemeris.10,11 Atlower culminationsF590.73°2z, at upper culminationsF589.27°2z, and at maximum eastern or western elongtions,F590°2z. Also the observation of Polaris providesconvenient method for determining true north. Unfortunatethere is no comparably bright star in the near vicinity of tsouth celestial pole.2

Note that on the terrestrial sphere, one arcminute~18! oflatitudeF corresponds to a north–south distance of one ntical mile or 6080 ft ~easily remembered as ‘‘a mileminute’’!, whereas one arcminute of longitudeL correspondsto cosF nautical mile.

VII. DETERMINATION OF LONGITUDE ANDLATITUDE BY OBSERVATION OF TWO OR MORESTARS

Suppose that the zenith distancez1 of a star has beenobserved at a particular time. It is evident from Fig. 5 ththe same value ofz would have been observed simultneously at an infinite number of other points on a smcircle, called the ‘‘circle of position,’’ on the terrestriasphere. This circle is the intersection with the sphere ocircular cone having its vertex at the center of the Earth, hanglez, and its axis through the substellar pointS, the pointon the terrestrial sphere at which the line fromO to the starpierces that sphere.

Next, suppose that the zenith distancez2 of a second staris observed simultaneously. This observation defines aond small circle on the terrestrial sphere. The plane conting the circle of position for star 1 and the one containingcircle of position for star 2 intersect in a line throughP andP8 ~see Fig. 6!. The observer’s latitude is that of either poi

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Fig. 5. Cross-section of the terrestrial sphere in any plane containing thefrom the center of the EarthO through a substellar pointS. The circle ofposition is the intersection with the sphere of the circular cone whose axOSand whose half-angle isz, the zenith distance of the star. The dashed lirepresents an edge-on view of the plane containing the circle of posiThe normal to this plane throughO has lengthOR5p.


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P or point P8. Auxiliary information can make possible thchoice between usually widely separated pointsP andP8. Ifnot, simultaneous observation of the zenith distancez3 of athird star resolves the twofold ambiguity and providesunique choice betweenP andP8.

The resulting position is referenced to the substellar poof the stars at the moment of observation. But these pomove progressively westward as the celestial sphere roand the derived position of the observer also moves progsively westward at the known, constant latitude, thatalong a small circle on the terrestrial sphere in a plane pallel to the equator.

To determine the longitudeL of the observer, a knowledgof the simultaneous longitudes of the respective substepoints is essential. In other words, it is essential to knowabsolute time~for example, GMT, the mean solar timeGreenwich!. The historical challenge of developing an accrate chronometer for maritime use rests on this simple fac12

Truly simultaneous observations of stars may not be fsible in practice. But we make this assumption so aspresent the basic geometric principle. Suppose that theserver has made simultaneous observations of the zenithtances of two~or more! stars at a particular moment, as dscribed above, and that the GMT of the momentobservation also has been recorded.

It is usual to specify the hour angle of the ‘‘mean Surelative to the prime meridian as GMT or Universal Tim~UT!. The mean Sun is a fictitious point on the celestequator that moves eastward on the celestial sphere at astant rate so as to match the yearly average, or mean, rathe actual Sun.

The definition of a Greenwich hour angle~GHA! is thedihedral angle measured westward from the prime meridto any other specified meridian. The Greenwich SiderTime ~GSiT! is the instantaneous Greenwich hour anglethe vernal equinox, GHAg, which increases at the rate o360° in 23 hour 56 minute 4 second~the sidereal rotationaperiod of the Earth! or 15.04108° per mean solar hour. Thvalues of the GHA andd for the real Sun, the Moon, anselected planets are tabulated at 10 min intervals for eachof the year together with interpolation tables in Ref. 9.

The Greenwich Hour Angle of a celestial object, GHA*, isgiven by

GHA* 5GHAg1SHA* . ~6!

The corresponding substellar point on the terrestrial sphelocated at latituded and east longitudel, where

Fig. 6. In this sketch of the terrestrial sphere,S1 andS2 are the simulta-neous geographic positions of the respective substellar points of star 1star 2. The two circles of position correspond to the observed zenithtances of the two stars and intersect at pointsP and P8, the two possiblegeographic positions of the observer.


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The observed value ofz plus the GHA* for each observedstar complete the basic data set required for determinatiothe latitude and longitude of the observer.~The word ‘‘star’’is used for convenience in this and other sections, butanalysis is equally applicable to any celestial body,~exceptthe Moon, as noted above! given its instantaneous values od and GHA!.

The following solution of the two-star-sight problemadapted from Ref. 13. Figure 6 depicts two intersectcircles of position on the terrestrial sphere. A unique positof the observer occurs in the special case of either internaexternal tangency of the two circles. But, in general tcircles intersect at two pointsP and P8. Each circle is theintersection of a circular cone of half anglez with the terres-trial sphere of radius unity, or otherwise stated, the interstion of the sphere with a plane perpendicular to the lineOSfrom the center of the sphereO through the substellar pointSand at a normal distancep from O ~see Fig. 5!. In Fig. 5 it isnoted that a line from any point on the terrestrial spherethe identified star on the celestial sphere is parallel toOS~that is, zero parallax!.

The equation of the plane containing a circle of positiis6

x cosl 1y cosm1z cosn2p50, ~8!

wherel, m, andn are the angles to the geographicX, Y, andZ axes. respectively. of the normal to the plane throughO; p

is the length of the normalORW ~see Fig. 5!. Also

p5cosz, ~9!

and

n5902d. ~10!

Let

a[cosl 5cosl cosd, ~11a!

b[cosm5sinl cosd, ~11b!

c[cosn5sind, ~11c!

p5cosz. ~11d!

In terms of the known quantitiesz, d, andl, the equation ofeach plane is

aix1biy1ciz2pi50, ~12!

with i 51 for one of the planes andi 52 for the other. Thetwo planes intersect along a linePP8, given by the simulta-neous solution of Eq.~12! for i 51, 2. The result is

x52Bz1C

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D, ~13!

where

A[~a1b22a2b1!, ~14a!

B[~b2c12b1c2!, ~14b!

C[~b2p12b1p2!, ~14c!

D[~a1c22a2c1!, ~14d!

E[~b1c22b2c1!, ~14e!

F[~c2p12c1p2!. ~14f!

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The two possible positions of the observerP and P8 arethe intersections of the above line with the unit sphere; tis, they are the two points for which

x21y21z251. ~16!

The substitution of Eq.~15! into Eq. ~16! yields a quadraticequation forx with two real rootsxp and xp8 . The corre-sponding values ofyp andzp andyp8 andzp8 are calculatedby Eq.~15! to define the two possible positionsP (xp ,yp ,zp)andP8 (xp8 ,yp8 ,zp8).

Finally, the latitudeF and longitudeL, of P and P8 arecalculated from the Cartesian coordinatesx, y, andz by therelations

tanF5z

Ax21y2, ~17a!

tanL5y

x, ~17b!

with attention to resolving quadrant ambiguity, to yield tdesired results: P (Fp ,Lp) andP8 (Fp8 ,Lp8).

Recall that the input data for the above analytical solutare as follows.~a! From the observer:z1 andz2 , the simul-taneously observed zenith distances of the two stars andobserver’s Greenwich Mean Time of the stellar observatio~b! Derived from Ref. 9: the values of the longitudel1 andlatituded1 for the substellar point of star 1 and the longitul2 and the latituded2 for the substellar point of star 2, botat the GMT of the observations.

The above analytical solution is too tedious to be pformed by hand by a practicing navigator, but it can be eaprogrammed for a computer of modest capability. The timindependent values of the latitudesd1 and d2 of the twosubstellar points are taken from standard tables9,10 and thetime-dependent values of their longitudesl1 and l2 aregiven by

l5360°2GHA* ~18a!

or

l5360°2~SHA* 1GHAg!. ~18b!

In Eq. ~18b! SHA* and GHAg are found~with interpolationif necessary! from Ref. 9 for the observer’s recorded GM~Fig. 7!.

The observer then enters the values ofz1 , d1 , andl1 andz2 , d2 , andl2 , into the computer, which yieldsLp , Fp ,Lp8 , andFp8 . If it is not obvious whetherP or P8 is thedesired solution, the process is repeated for another pastars, say, 1 and 3. Note that the method is the same forpractical combination of stars, planets, and the Sun.

The traditional practice1,2 of celestial navigation uses observations to refine the observer’s position relative to an


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VIII. NOTES ON PRACTICALITY

The use of a marine sextant requires simultaneous visity of the celestial object and the local horizon, that is,least partially clear skies and twilight conditions for stars aplanets or daylight conditions for the Sun. Under favoraconditions at sea a skilled observer can determinez to anaccuracy of about one arcminute or better. A bubble aircsextant provides an artificial horizon and greatly expandsopportunities for observation, although usually with lessaccuracy than that provided by a marine sextant. Obsetion, when practical, of a given celestial object at intervof, say, a few hours is equivalent to the two-star-sight tenique, although movement of the observer between obsetions must be taken into account.

It should be mentioned that the simultaneous observaof the zenith distance and azimuth of a single celestial obat a given GMT, as is possible with a pier-mounted theolite at a fixed point, yields bothL andF. The solution of thecorresponding geometric problem is straightforward, bunot within the scope of this paper. Note that an error in GMof four seconds of time is equivalent to an error of one aminute of longitude.

IX. SOME EXAMPLES

A. F by meridian transit

~1! In a sequence of twilight observations, an obserfinds that the minimum value of the northerly zenith distan~that is, at meridian transit! of the bright star Vega,zmin

525.51°. From tables, the declination of Vegad5138.38°.Hence, the observer’s latitude is given by Eq.~4!, namely,

F538.38°225.51°5112.87°. ~19!

~2! In a sequence of twilight observations, an obserfinds the minimum zenith distance of Polaris~at upper cul-mination! to be zmin542.15°. The declination of Polaris id589.27°. Hence,

Fig. 7. GHAg, SHA*, GHA*, a and l are all dihedral angles~Ref. 7! asdefined in Secs. II and III. Their relationships are most simply represenby this diagram of the equatorial plane in whichg is the vernal equinox,Gis the intersection of the Greenwich meridian with the equator, andS is theintersection of the meridian through the star with the equator.


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F589.27°242.15°5147.12°. ~20!

~3! The minimum zenith distance of the center of tSun’s disk~that is, at local noon! is observed on 21 June tbe 38.35° to the north of the zenith. On that date the Sudeclination is 123.44°. Hence, F523.44°238.35°5214.91°~that is, south latitude!.

B. Latitude from simultaneous observation of the zenithdistancesz1 and z2 of two stars

The essential data for determining an observer’s latitudFare the observed valuesz1 andz2 and the time-independenvalues ofd and SHA for the two stars~see Sec. VII!.

C. Determination of both latitude F and longitude L

During evening twilight on 19 February 2004, the navigtor on an imaginary ship in the central Atlantic Ocean oserved simultaneously the zenith distances of four bristars—Sirius, Procyon, Aldebaran, and Pollux—all at exac20:00 GMT. A copy of the Air Almanac9 for 2004 is avail-able. The job of the navigator is to report the ship’s latituand longitude to the captain of the ship.

In lieu of an actual set of zenith distances, I have adopa position of the ship~unknown to the navigator! and havecalculated the four relevant zenith distances. These vaare then treated as ‘‘observed’’ values. The navigator tfills out Table I. The right-hand column of Table I lists threspective values of zenith distancez, calculated by the author, but regarded as ‘‘observed’’ values. The valuesGHAg, SHA*, and d are taken directly from Ref. 9. Thvalues of GHA* are found by Eq.~6! and the values ofl byEq. ~7!. See also Fig. 7. Recall thatd andl are, respectively,the constant latitude and the time-dependent longitude ofsubstellar point of each star at the moment of observatio

The navigator then uses the suggested computer progto calculate the latitudeF and longitudeL of the two pos-sible positions of the ship for each of the six pairs of staFor each pair there are six numbers to enter into the cputer, namelyz1 , d1 , andl1 for star 1 andz2 , d2 , andl2for star 2, etc. The output of the computer program is givin Table II.

In every two-star case, there is the twofold ambiguitydiscussed in Sec. VII. But it is clear that only a gross knoedge of the ship’s position~for example, in the North Atlan-tic Ocean or on a mountain in Tibet! would make it possibleto select the appropriate one of the two mathematically psible solutions. The uniqueness of such a choice becooverwhelmingly clear as the other five cases are examin

Table I. Calculated values of zenith distances for an adopted positionGMT. Greenwich Date: 19 February 2004. Greenwich Mean Time: 20GHAg589.11°.@All numerical data are in degrees~°!.#

SHA d GHA* l z

Sirius 258.67 216.72 347.78 12.22 70.45Procyon 245.12 15.22 334.23 25.77 61.50Aldebaran 290.95 116.52 20.06 339.94 26.87Pollux 243.60 128.02 332.71 27.29 48.02


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X. CONCLUDING REMARKS

The analytical solution of the two-star sight problem rquires only ~corrected! observed values of the zenith distances at recorded GMT and basic astronomical data fthe annualAir Almanac.9 And, in practice, it requires a computer of modest capability. A single program covers all casHowever, it does not require an assumed position, massight reduction tables, or massive tables of computed altitand azimuth.

ACKNOWLEDGMENTS

The author’s interest in celestial navigation is based onexperience as a naval officer in the South Pacific Fleet duWorld War II. Numerous versions of the manuscript for thpaper were prepared by Christine Stevens and the figureJoyce Chrisinger. The author is especially indebted tocolleague Robert L. Mutel for checking the exposition aTables I and II. In addition, Mutel wrote an analytical soltion of the three-star-sight problem, which yields a uniqgeographic position.

APPENDIX: SUGGESTED EXERCISES FOR THESTUDENT

~1! Verify the astronomical data in the first four columnsTable I. Then use the author’s adopted positiF5142.00°, L5239.00° to calculate the four zenitdistances in the fifth column.

~2! Write a computer program for solving the two-star-sigproblem. Then use the data in Table I and verify tauthor’s calculated values ofF andL in Table II.

~3! Show, for a given date of the year, that the time interbetween sunrise and sunset yields a crude measureobserver’s latitudeF. Estimate the accuracy by youown observations.

a!Electronic mail: [email protected] Practical Navigator, H.O. Publ. No. 9, U.S. Navy HydrographicOffice ~USGPO, Washington, DC, 1966!.

2F. W. Wright, Celestial Navigation~Cornell Maritime Press, CambridgeMD, 1976!.

3R. M. Green,Spherical Astronomy~Cambridge U.P., Cambridge, 1985!.4This approximation ignores the oblateness of the actual Earth as wethe heights of mountains, etc., and thereby eliminates the distinctamong astronomical latitude, geocentric latitude, and geodetic latitud

nd.Table II. The geographic coordinates of the observer as derived fromdata of Table I.@All numerical data are in degrees~°!.#

First solution Second solution

Sirius—Procyon L 230.00 185.16F 142.00 211.99

Sirius—Aldebaran L 230.00 247.99F 142.00 121.84

Sirius—Pollux L 230.00 176.65F 142.00 113.53

Procyon—Aldebaran L 234.79 230.00F 26.06 142.00

Procyon—Pollux L 230.00 183.96F 142.00 136.23

Aldebaran—Pollux L 26.82 230.00F 26.94 142.00


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5D. T. Sigley and W. T. Stratton,Solid Geometry~Dryden, New York,1956!, revised edition.

6H. G. Ayre and R. Stephens,A First Course in Analytic Geometry~VanNostrand, Princeton, NJ, 1956!.

7A dihedral angle is the ‘‘hinge’’ angle between two intersecting planesis measured by the plane angle between two lines that pass throucommon point on the line of intersection of the planes, one line lyingeach plane and perpendicular to the line of intersection of the two pla

8A primitive technique for determining longitude without the use ofaccurate time piece exploits this parallactic effect~‘‘linear distance’’ tech-nique!.


ta

s.

9The Air Almanac 2004, The National Almanac Office, United States NavObservatory~USGPO, Philadelphia, PA, 2003!.

10The Astronomical Almanac for the Year 2004, The Nautical AlmanacOffice, United States Naval Observatory~USGPO, Washington, DC,2002!.

11Observer’s Handbook 2004~The Royal Astonomical Society of CanadaToronto, 2003!.

12D. Sobel,Longitude~Walker, New York, 1995!.13J. A. Van Allen, ‘‘An analytical solution of the two star sight problem o

celestial navigation,’’ Navig.: J. Inst. Navig.28~1!, 40–43~1981!.

1850. The

body t.

Horizontal Electromagnet. This horizontal helix on a stand was probably made by Daniel Davis or one of his successors at some time after ca.hollow coil produces a magnetic field that is enhanced when an iron bar is placed inside. The 1842 edition of Daniel Davis’sManual of Magnetismnotes that‘‘the magnetizing power will be greatly increased if the wire be coiled in the manner of a cork-screw, so as to form a hollow cylinder into which theobe magnetized can be inserted. Such a coil is denominated a Helix.’’ The helix is in the Greenslade Collection.~Photograph and notes by Thomas BGreenslade, Jr., Kenyon College!


Basic principles of celestial navigation - James Allen

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