Basic Principles Of Celestial Navigation - James Allen.pdf - Astronawigacja - matrobert2

Basic principles of celestial navigation

James A. Van Allen a)

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 of

identiﬁed stars, identiﬁed planets, the Sun, and the Moon. This subject has a multitude of

reﬁnements which, although valuable to a professional navigator, tend to obscure the basic

principles. I describe these principles, give an analytical solution of the classical two-star-sight

problem without any dependence on prior knowledge of position, and include several examples.

Association of Physics Teachers.

I. INTRODUCTION

is between 0° and 360°, although often it is

convenient to take the longitude westward of the prime me-

ridian to be between 0° and

Celestial navigation is a technique for determining one’s

geographic position by the observation of identiﬁed stars,

identiﬁed planets, the Sun, and the Moon. Its basic principles

are a combination of rudimentary astronomical knowledge

and spherical trigonometry. 1–3

Anyone who has been on a ship that is remote from any

terrestrial landmarks needs no persuasion on the value of

celestial navigation. There are modern electronic naviga-

tional aids such as Loran, ocean bottom soundings, and the

global positioning system

180°. The longitude of P also

can be speciﬁed by the plane angle in the equatorial plane

whose vertex is at O with one radial line through the point at

which the meridian through P intersects the equatorial plane

and the other radial line through the point G at which the

prime meridian intersects the equatorial plane

see Fig. 1

III. THE CELESTIAL SPHERE

which depends on receiving

precise timing data from a world-wide network of artiﬁcial

satellites, each carrying one or more atomic clocks. This pa-

per illustrates what can be learned with only a sextant, rudi-

mentary astronomical tables, and an accurate knowledge of

absolute time

GPS

The celestial sphere is an imaginary spherical surface

whose radius is very much greater than that of the Earth,

with its center at O and its polar axis coincident with the

Earth’s rotational axis. The position of each celestial object is

represented by the point on the celestial sphere at which a

line to it from O intersects this sphere. Inasmuch as the dis-

tances to all stars, planets, and the Sun are much greater than

the radius of the Earth, a terrestrial observer may be thought

of as viewing the sky from O . For the nearby Moon, how-

ever, its apparent position on the celestial sphere is, because

of parallax, different by as much as nearly one degree for

observers at different latitudes and longitudes. 8 It is impos-

sible to make an a priori correction for this effect if the

observer’s position is initially unknown.

On the celestial sphere, the declination

It is intended for students who are curious about the basic

principles of celestial navigation and makes no pretense of

serving a professional navigator or surveyor.

from radio signals or a precise chronometer

II. THE EARTH AND GEOGRAPHIC POSITION

The surfaces of the land masses of the Earth and of the

oceans are approximated as the surface of a spherically sym-

metric body of unit radius, hereafter called the terrestrial

sphere. 4 The Earth as a whole is assumed to be rotating at a

constant angular velocity about an inertially oriented axis

through its geometric center O . A geographic position P is

referenced to a right-hand spherical coordinate system whose

center is at O and whose positive Z -axis is coincident with

the Earth’s rotational axis and in the same sense as the

Earth’s angular momentum vector. The plane through O that

is perpendicular to this axis is called the equatorial plane

of a celestial

of a terrestrial

observer as deﬁned in the above. The second spherical coor-

dinate of a celestial object, called right ascension

,isthe

dihedral angle analogous to eastward terrestrial longitude,

but with the fundamental difference that it is measured from

a different reference point. That reference point on the celes-

tial equator is called the vernal equinox, usually denoted by

the

, and deﬁned as follows. The equatorial plane of the Earth

is tilted to the plane of its orbit

. Any plane that contains the rotational axis in-

tersects the spherical surface in a great circle called a merid-

ian. By convention, the X axis lies in the meridian plane

about the

Sun by 23.44°. These two planes intersect along a line that

pierces the celestial sphere at two points, called nodes. The

vernal equinox is the ascending node of the ecliptic on the

equator, that is, the position of the Sun on or about 21 March

of each year as the Sun ascends from southerly to northerly

declination. The right ascension

the ecliptic plane

the

prime meridian

that passes through a point in Greenwich,

England. 1

The position P is speciﬁed by two coordinates, its latitude

. The latitude is the plane angle whose

apex is at O , with one radial line passing through P and the

other through the point at which the observer’s meridian in-

tersects the equator. Northern latitudes are taken to be posi-

tive

and longitude

of a celestial object, al-

ways taken to be positive eastward, is measured from the

vernal equinox. Astronomers usually measure right ascension

in hours, minutes, and seconds of time, but it can be con-

verted to the equivalent angular measure by taking one hour

The longitude is the dihedral angle, 3,5–7 measured eastward

from the prime meridian to the meridian through P . The

0° to 90°

and southern latitudes negative

0° to

90°

15°. Another commonly used quantity is called the hour

angle. It is the dihedral angle between any speciﬁed meridian

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DOI: 10.1119/1.1778391

longitude

object is strictly analogous to the latitude

X – Y plane

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

IV. OBSERVATIONAL CONSIDERATIONS

. The celestial

sphere rotates westward from the observer’s point of view at

an angular rate such that the vernal equinox transits

and longitude

Fig. 1. Deﬁnition of latitude F and longitude L . ~ a ! Meridian cross-section

of the Earth through the observer at P . The center of the Earth is at O and

NP and SP are the north pole and south pole, respectively. ~ b ! Equatorial

cross-section of the Earth. G represents the point at which the prime merid-

ian through Greenwich intersects the equator and P represents the point at

which the meridian through P intersects the equator. E means eastward and

W means westward. The longitude L is the dihedral angle ~ Ref. 7 ! between

the two named meridian planes, but is more easily visualized in this equa-

torial diagram.

passes

the observer’s meridian from east to west at inter-

vals of 23 hour, 56 minute, 4 second of mean solar time, also

called the sidereal rotational period of the Earth or the side-

real day.

The observer’s zenith is deﬁned by the outward extension

of the line from the center of the Earth through the observer

as in Fig. 3. The instantaneous angle between the observer’s

zenith and the line to a star is called the star’s zenith distance

and is denoted by

plane and the meridian plane through a celestial object, al-

ways taken to be positive westward and lying in the range

0–24 h or 0°–360°. The sidereal hour angle

. As the celestial sphere rotates westward

that is, clockwise as viewed from above the

north celestial pole

of a

celestial object, taken to be positive westward from the ver-

nal equinox, is related to the right ascension

SHA

decreases from 90° as the star rises

from below the observer’s eastern horizon, has a minimum

value

Fig. 2

z min as S crosses the observer’s meridian, then in-

creases to 90° as the star sets below the observer’s western

horizon. These comments are applicable to all celestial ob-

jects for which

SHA

360°

We will take

, and the SHA to be ﬁxed quantities for any

. Stars whose declinations meet

this condition transit the observer’s meridian at intervals of

exactly one sidereal day; but the intervals between the suc-

cessive transits of planets, the Sun and the Moon vary, al-

though predictably.

Stars for which

d,~

90°

2F!

beyond the solar sys-

tem; however, they vary with time for the planets, the Sun,

and the Moon. Their values are tabulated in almanacs, 9–11 of

which The Air Almanac 9

for example, stars or galaxies

is the most convenient for a navi-

gator.

constitute the observer’s cir-

cumpolar star ﬁeld which, as viewed from O or any point on

the Earth, rotates counterclockwise for a northern hemi-

sphere observer or clockwise for a southern hemisphere ob-

server as time progresses. Such stars neither rise nor set,

being always above the observer’s horizon as they move

along small circles centered on the celestial pole. Their ze-

nith distances have a maximum value of 180°

d>~

90°

2F!

2~F1d!

upper culmination as they transit the observer’s meridian

twice per sidereal day.

Usually, a practitioner of celestial navigation employs a

marine sextant or an aircraft bubble sextant to determine the

instantaneous altitude of a celestial object, the vertical angle

of the object above the local horizontal plane. After appro-

priate corrections, 1,2 this quantity is denoted by h . However,

because of its simpler geometrical signiﬁcance, I prefer the

zenith distance

~F2d!

~d2F!

Fig. 2. Right ascension ~a! and sidereal hour angle ~ SHA ! are both dihedral

angles ~ Ref. 7 ! measured from the meridian plane through the vernal equi-

nox g , with a positive eastward and SHA positive westward. In this equa-

torial diagram, S represents the intersection of the meridian plane through a

star or any other celestial object with the equator.

to represent the observed quantity, with

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James A. Van Allen

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Imagine that a terrestrial observer is located at a ﬁxed

point P of unknown latitude

through

about the Earth

object

lower culmination and a minimum value

Fig. 4. P represents the position of the observer and S the position of the

substellar point of a star at the moment that it transits the observer’s merid-

ian. O is the center of the Earth, d the declination of the star, F the latitude

of the observer, and z the zenith distance of the star. NP is the north pole and

SP is the south pole.

Fig. 5. Cross-section of the terrestrial sphere in any plane containing the line

from the center of the Earth O through a substellar point S . The circle of

position is the intersection with the sphere of the circular cone whose axis is

OS and whose half-angle is z , the zenith distance of the star. The dashed line

represents an edge-on view of the plane containing the circle of position.

The normal to this plane through O has length OR 5 p .

is the zenith distance of Polaris, and, at this level of

approximation, is constant and independent of the time of

day. Hence, Polaris has had a special simplicity for determin-

ing latitude throughout maritime history. For example, a ves-

sel can sail from Europe to America without an accurate

knowledge of the time by simply maintaining a westerly

course during which

90°

h .

is treated as intrinsically positive and is in the

range 0° to 90°.

V. DETERMINATION OF LATITUDE BY MERIDIAN

TRANSIT OF A STAR

of Polaris is approximately constant or

changes day-by-day in such a way as to assure landfall in

America at approximately an intended latitude. Because the

diurnal motion of Polaris lies along a small circle around the

north celestial pole of angular radius 0.73°, an improvement

in accuracy in determining the latitude is to observe Polaris

at speciﬁc moments identiﬁed from its ephemeris. 10,11

As illustrated in Fig. 4,

F5d1z min ,

where

is the observer’s latitude,

is the declination of the

lower culminations

90.73°

, at upper culminations

z min is the observed minimum value of the zenith

distance of the star

89.27°

, and at maximum eastern or western elonga-

a southerly angle in this example

as it

. Also the observation of Polaris provides a

convenient method for determining true north. Unfortunately,

there is no comparably bright star in the near vicinity of the

south celestial pole. 2

Note that on the terrestrial sphere, one arcminute

90°

transits the observer’s meridian.

If the star transits the observer’s meridian northward of P ,

the corresponding relation is

F5d2z min .

8 !

corresponds to a north–south distance of one nau-

tical mile or 6080 ft

z min is zero at

the moment that the object transits the observer’s meridian,

that is, at the observer’s zenith. Thus, in this special case, the

observer’s latitude is equal to the declination of the star.

The same considerations are applicable to determining

latitude by observing the meridian transit of planets or the

center point of the disk of the Sun. But stars are conceptually

simpler because their values of

d5F

, the value of

easily remembered as ‘‘a mile a

minute’’

, whereas one arcminute of longitude

corresponds

to cos

nautical mile.

are almost constant,

whereas these coordinates for all celestial objects of the solar

system have changing values of

and

VII. DETERMINATION OF LONGITUDE AND

LATITUDE BY OBSERVATION OF TWO OR MORE

STARS

and

which must be taken

z 1 of a star has been

observed at a particular time. It is evident from Fig. 5 that

the same value of

from a daily ephemeris. 9

In any case, an observer also obtains the ‘‘true north’’ or

‘‘true south’’ direction, namely the direction of any celestial

object at the moment of its upper or lower culmination.

would have been observed simulta-

neously at an inﬁnite number of other points on a small

circle, called the ‘‘circle of position,’’ on the terrestrial

sphere. This circle is the intersection with the sphere of a

circular cone having its vertex at the center of the Earth, half

angle

VI. DETERMINATION OF LATITUDE BY

OBSERVING POLARIS

, and its axis through the substellar point S , the point

on the terrestrial sphere at which the line from O to the star

pierces that sphere.

Next, suppose that the zenith distance

An especially valuable star for northern hemisphere navi-

gators is the bright star Polaris

the North Star or

Ursa

z 2 of a second star

is observed simultaneously. This observation deﬁnes a sec-

ond small circle on the terrestrial sphere. The plane contain-

ing the circle of position for star 1 and the one containing the

circle of position for star 2 intersect in a line through P and

Minoris

which presently has declination

d51

89.27°. Thus,

the observer’s latitude

is approximately equal to the alti-

tude of Polaris or

90°

8 ~ see Fig. 6 ! . The observer’s latitude is that of either point

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James A. Van Allen

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where

The quantity

star, and

tions,

latitude

For a celestial object having

Suppose that the zenith distance

360°

GHA * .

plus the GHA* for each observed

star complete the basic data set required for determination of

the latitude and longitude of the observer.

The word ‘‘star’’

is used for convenience in this and other sections, but the

analysis is equally applicable to any celestial body,

except

the Moon, as noted above

given its instantaneous values of

The following solution of the two-star-sight problem is

adapted from Ref. 13. Figure 6 depicts two intersecting

circles of position on the terrestrial sphere. A unique position

of the observer occurs in the special case of either internal or

external tangency of the two circles. But, in general the

circles intersect at two points P and P

and GHA

Fig. 6. In this sketch of the terrestrial sphere, S 1and S 2 are the simulta-

neous geographic positions of the respective substellar points of star 1 and

star 2. The two circles of position correspond to the observed zenith dis-

tances of the two stars and intersect at points P and P

, the two possible

geographic positions of the observer.

. Each circle is the

intersection of a circular cone of half angle

with the terres-

trial sphere of radius unity, or otherwise stated, the inter sec-

tion of the sphere with a plane perpendicular to the line OS

from the center of the sphere O through the substellar point S

and at a normal distance p from O

. Auxiliary information can make possible the

choice between usually widely separated points P and P

.If

.InFig.5itis

noted that a line from any point on the terrestrial spher e to

the identiﬁed star on the celestial sphere is parallel to OS

see Fig. 5

z 3 of a

third star resolves the twofold ambiguity and provides a

unique choice between P and P

The resulting position is referenced to the substellar points

of the stars at the moment of observation. But these points

move progressively westward as the celestial sphere rotates

and the derived position of the observer also moves progres-

sively westward at the known, constant latitude, that is,

along a small circle on the terrestrial sphere in a plane par-

allel to the equator.

To determine the longitude

that is, zero parallax

The equation of the plane containing a circle of position

is 6

where l , m , and n are the angles to the geographic X , Y , and

Z axes. respectively. of the normal to the plane through O ; p

is the length of the normal O W ~

x cos l

y cos m

z cos n

of the observer, a knowledge

of the simultaneous longitudes of the respective substellar

points is essential. In other words, it is essential to know the

absolute time

see Fig. 5

. Also

p 5 cos z ,

~ 9 !

for example, GMT, the mean solar time at

and

. The historical challenge of developing an accu-

rate chronometer for maritime use rests on this simple fact. 12

Truly simultaneous observations of stars may not be fea-

sible in practice. But we make this assumption so as to

present the basic geometric principle. Suppose that the ob-

server has made simultaneous observations of the zenith dis-

tances of two

n 5 90 2d .

~ 10 !

Let

a [ cos l 5 cos l cos d

11a

[

cos m

sin

cos

11b

stars at a particular moment, as de-

scribed above, and that the GMT of the moment of

observation also has been recorded.

It is usual to specify the hour angle of the ‘‘mean Sun’’

relative to the prime meridian as GMT or Universal Time

or more

[

cos n

sin

11c

cos

11d

In terms of the known quantities

, and

, the equation of

. The mean Sun is a ﬁctitious point on the celestial

equator that moves eastward on the celestial sphere at a con-

stant rate so as to match the yearly average, or mean, rate of

the actual Sun.

The deﬁnition of a Greenwich hour angle

each plane is

a i x

b i y

c i z

p i 5

with i

1 for one of the planes and i

2 for the other. The

is the

dihedral angle measured westward from the prime meridian

to any other speciﬁed meridian. The Greenwich Sidereal

Time

GHA

two planes intersect along a line PP

, given by the simulta-

neous solution of Eq.

for i

1, 2. The result is

is the instantaneous Greenwich hour angle of

the vernal equinox, GHA

GSiT

, which increases at the rate of

360° in 23 hour 56 minute 4 second

the sidereal rotational

where

or 15.04108° per mean solar hour. The

values of the GHA and

for the real Sun, the Moon, and

selected planets are tabulated at 10 min intervals for each day

of the year together with interpolation tables in Ref. 9.

The Greenwich Hour Angle of a celestial object, GHA*,is

given by

GHA* 5 GHA g1 SHA*.

a 1 b 2 2

a 2 b 1 !

14a

[ ~

b 2 c 1 2

b 1 c 2 !

14b

[ ~

b 2 p 1 2

b 1 p 2 !

14c

[ ~

a 1 c 2 2

a 2 c 1 !

14d

The corresponding substellar point on the terrestrial sphere is

located at latitude

[ ~

b 1 c 2 2

b 2 c 1 !

14e

and east longitude

, where

[ ~

c 2 p 1 2

c 1 p 2 !

14f

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

James A. Van Allen

1421

The observed value of

P or point P

not, simultaneous observation of the zenith distance

Greenwich

period of the Earth

, x may be regarded as the parameter that desig-

nates a general point on the line ¯ P

and y and z are the

other two coordinates of the point. For any x , we have

F 2 Dx

15a

15b

are

the intersections of the above line with the unit sphere; that

is, they are the two points for which

x 2

The two possible positions of the observer P and P

y 2

z 2

yields a quadratic

equation for x with two real roots x p and x p

into Eq.

Fig. 7. GHA g , SHA * , GHA * , a and l are all dihedral angles ~ Ref. 7 ! as

deﬁned in Secs. II and III. Their relationships are most simply represented

by this diagram of the equatorial plane in which g is the vernal equinox, G

is the intersection of the Greenwich meridian with the equator, and S is the

intersection of the meridian through the star with the equator.

. The corre-

sponding values of y p and z p and y p

and z p

are calculated

by Eq.

to deﬁne the two possible positions P ( x p , y p , z p )

and P

Finally, the latitude

( x p

, y p

, z p

are

calculated from the Cartesian coordinates x , y , and z by the

relations

and longitude

,of P and P

sumed position, based on dead reckoning or other approxi-

mate information. The distinctive feature of the present

solution is that it obviates the need for such prior knowledge.

tan

y 2 ,

17a

x 2

VIII. NOTES ON PRACTICALITY

tan

x ,

17b

The use of a marine sextant requires simultaneous visibil-

ity of the celestial object and the local horizon, that is, at

least partially clear skies and twilight conditions for stars and

planets or daylight conditions for the Sun. Under favorable

conditions at sea a skilled observer can determine

Recall that the input data for the above analytical solution

are as follows.

F p ,

L p ) and P

(

F p

L p

to an

accuracy of about one arcminute or better. A bubble aircraft

sextant provides an artiﬁcial horizon and greatly expands the

opportunities for observation, although usually with lesser

accuracy than that provided by a marine sextant. Observa-

tion, when practical, of a given celestial object at intervals

of, say, a few hours is equivalent to the two-star-sight tech-

nique, although movement of the observer between observa-

tions must be taken into account.

It should be mentioned that the simultaneous observation

of the zenith distance and azimuth of a single celestial object

at a given GMT, as is possible with a pier-mounted theodo-

lite at a ﬁxed point, yields both

z 2 , the simul-

taneously observed zenith distances of the two stars and the

observer’s Greenwich Mean Time of the stellar observations.

From the observer:

z 1 and

Derived from Ref. 9: the values of the longitude

l 1 and

d 1 for the substellar point of star 1 and the longitude

l 2 and the latitude

d 2 for the substellar point of star 2, both

at the GMT of the observations.

The above analytical solution is too tedious to be per-

formed by hand by a practicing navigator, but it can be easily

programmed for a computer of modest capability. The time-

independent values of the latitudes

d 2 of the two

substellar points are taken from standard tables 9,10

d 1 and

. The solution of the

corresponding geometric problem is straightforward, but is

not within the scope of this paper. Note that an error in GMT

of four seconds of time is equivalent to an error of one arc-

minute of longitude.

and

and the

time-dependent values of their longitudes

l 1 and

l 2 are

given by

360°

GHA *

18a

IX. SOME EXAMPLES

360°

SHA* 1 GHA g ! .

18b

by meridian transit

In Eq.

18b

SHA* and GHA g are found ~ with interpolation

In a sequence of twilight observations, an observer

ﬁnds that the minimum value of the northerly zenith distance

if necessary

from Ref. 9 for the observer’s recorded GMT

Fig. 7

The observer then enters the values of

that is, at meridian transit

of the bright star Vega,

z min

z 1 ,

d 1 , and

l 1 and

25.51°. From tables, the declination of Vega

d51

38.38°.

z 2 ,

d 2 , and

l 2 , into the computer, which yields

L p ,

F p ,

Hence, the observer’s latitude is given by Eq.

, namely,

is the

desired solution, the process is repeated for another pair of

stars, say, 1 and 3. Note that the method is the same for any

practical combination of stars, planets, and the Sun.

The traditional practice 1,2 of celestial navigation uses ob-

servations to reﬁne the observer’s position relative to an as-

, and

F p

. If it is not obvious whether P or P

38.38°

25.51°

12.87°.

In a sequence of twilight observations, an observer

ﬁnds the minimum zenith distance of Polaris

at upper cul-

mination

to be

z min 5

42.15°. The declination of Polaris is

89.27°. Hence,

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

James A. Van Allen

1422

In Eq.

The substitution of Eq.

with attention to resolving quadrant ambiguity, to yield the

desired results: P (

latitude

L p

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