Chapt-17.pdf

CHAPTER 17

AZIMUTHS AND AMPLITUDES

INTRODUCTION

1700. Checking Compass Error

man-made) is the sky. The primary compass should be

checked occasionally by comparing the observed and

calculated azimuths and amplitudes of a celestial body. The

difference between the observed and calculated values is the

compass error. This chapter discusses these procedures.

Theoretically, these procedures work with any celestial

body. However, the Sun and Polaris are used most often

when measuring azimuths, and the rising or setting Sun

when measuring amplitudes.

While errors can be computed to the nearest tenth of a

degree or so, it is seldom possible to steer a ship that

accurately, especially when a sea is running, and it is

reasonable to round calculations to the nearest half or

perhaps whole degree for most purposes.

Various hand-held calculators and computer programs

are available to relieve the tedium and errors of tabular and

mathematical methods of calculating azimuths and ampli-

tudes. Naval navigators will find the STELLA program

useful in this regard. Chapter 20 discusses this program in

greater detail.

The navigator must constantly be concerned about the

accuracy of the ship’s primary and backup compasses, and

should check them regularly. A regularly annotated compass

log book will allow the navigator to notice a developing error

before it becomes a serious problem.

As long as at least two different types of compass (e.g.

mechanical gyro and flux gate, or magnetic and ring laser

gyro) are consistent with each other, one can be reasonably

sure that there is no appreciable error in either system. Since

different types of compasses depend on different scientific

principles and are not subject to the same error sources, their

agreement indicates almost certainly that no error is present.

A navigational compass can be checked against the

heading reference of an inertial navigation system if one is

installed. One can also refer to the ship’s indicated GPS track

as long as current and leeway are not factors, so that the

ship’s COG and heading are in close agreement.

The navigator’s only completely independent

directional reference (because it is extra-terrestrial and not

AZIMUTHS

1701. Compass Error by Azimuth of the Sun

difference between the respondent azimuth angle

and the base azimuth angle and label it as the

azimuth angle difference (Z Diff.).

Mariners may use Pub 229, Sight Reduction Tables for

Marine Navigation to compute the Sun’s azimuth. They

compare the computed azimuth to the azimuth measured

with the compass to determine compass error. In computing

an azimuth, interpolate the tabular azimuth angle for the

difference between the table arguments and the actual

values of declination, latitude, and local hour angle. Do this

triple interpolation of the azimuth angle as follows:

3. Reenter the tables with the base declination and

LHA arguments, but with the latitude argument 1

greater or less than the base latitude argument,

depending upon whether the actual (usually DR)

latitude is greater or less than the base argument.

Record the Z Diff. for the increment of latitude.

1. Enter the Sight Reduction Tables with the nearest

integral values of declination, latitude, and local

hour angle. For each of these arguments, extract a

base azimuth angle.

4. Reenter the tables with the base declination and

latitude arguments, but with the LHA argument 1

greater or less than the base LHA argument,

depending upon whether the actual LHA is greater

or less than the base argument. Record the Z Diff.

for the increment of LHA.

greater or less than the base declination argument,

depending upon whether the actual declination is

greater or less than the base argument. Record the

5. Correct the base azimuth angle for each

increment.

271

2. Reenter the tables with the same latitude and LHA

arguments but with the declination argument 1

272

AZIMUTHS AND AMPLITUDES

Actual

Base

Arguments

Base

Tab*

Z Diff. Increments

Correction

(Z Diff x Inc.¸ 60)

Dec.

20˚13.8' N

20˚

97.8˚

96.4˚

–1.4˚

13.8'

–0.3˚

DR Lat.

33˚24.0' N

33˚(Same)

97.8˚

98.9˚

+1.1˚

24.0'

+0.4˚

LHA

316˚41.2'

317˚

97.8˚

97.1˚

– 0.7˚

18.8'

–0.2˚

Base Z

97.8˚

Total Corr.

–0.1˚

Corr.

(–) 0.1˚

N 97.7˚ E

*Respondent for the two base arguments and 1˚

change from third base argument, in vertical

order of Dec., DR Lat., and LHA.

097.7˚

Zn pgc

096.5 ˚

Gyro Error

1.2˚ E

Figure 1701. Azimuth by Pub. No. 229.

Example:

In DR latitude 33

. Record the tabulated azimuth for these arguments.

As the third and final step in the triple interpolation

process, decrease the value of LHA to 316

pgc. At the time of the observation, the declination of the Sun

is 20

24.0'N, the azimuth of the Sun is 096.5

13.8'N; the local hour angle of the Sun is 316

41.2'.

because the

actual LHA value was smaller than the base LHA. Enter the

Sight Reduction Tables with the following arguments: (1)

Declination = 20

Determine compass error.

; (2) DR Latitude = 33

; (3) LHA = 316

Solution:

See Figure 1701 Enter the actual value of declination,

DR latitude, and LHA. Round each argument to the nearest

whole degree. In this case, round the declination and the

latitude down to the nearest whole degree. Round the LHA

up to the nearest whole degree. Enter the Sight Reduction

Tables with these whole degree arguments and extract the

base azimuth value for these rounded off arguments.

Record the base azimuth value in the table.

As the first step in the triple interpolation process,

increase the value of declination by 1

Record the tabulated azimuth for these arguments.

Calculate the Z Difference by subtracting the base

azimuth from the tabulated azimuth. Be careful to carry the

correct sign.

Z Difference = Tab Z - Base Z

because the

actual declination value was greater than the base declination.

Enter the Sight Reduction Tables with the following

arguments: (1) Declination = 21

(to 21

°)

Next, determine the increment for each argument by

taking the difference between the actual values of each

argument and the base argument. Calculate the correction

for each of the three argument interpolations by

multiplying the increment by the Z difference and dividing

the resulting product by 60.

The sign of each correction is the same as the sign of the

corresponding Z difference used to calculate it. In the above

example, the total correction sums to -0.1'. Apply this value

to the base azimuth of 97.8

; (2) DR Latitude = 33

; (3)

LHA = 317

. Record the tabulated azimuth for these

because the

actual DR latitude was greater than the base latitude. Enter

the Sight Reduction Tables with the following arguments:

(1) Declination = 20

to 34

to obtain the true azimuth 97.7

Compare this to the compass reading of 096.5

pgc. The

compass error is 1.2

E, which can be rounded to 1

for

; (2) DR Latitude = 34

; (3) LHA =

steering and logging purposes.

AZIMUTH OF POLARIS

1702. Compass Error By Azimuth Of Polaris

at 02-00-00 GMT, Polaris bears 358.6

pgc. Calculate the

compass error.

N. Figure 2012 in Chapter 20 shows this table. Compare a

compass bearing of Polaris to the tabular value of Polaris to

determine compass error. The entering arguments for the

table are LHA of Aries and observer latitude.

Date

17 March 2001

Time (GMT)

02-00-00

GHA Aries

204

43.0'

Longitude

045

00.0'W

LHA Aries

159

43.0'

Example:

On March 17, 2001, at L 33

15.0' N and

045

00.0'W,

Solution:

Enter the azimuth section of the Polaris table with the

317

arguments.

As the second step in the triple interpolation process,

increase the value of latitude by 1

The Polaris tables in the Nautical Almanac list the

azimuth of Polaris for latitudes between the equator and 65

AZIMUTHS AND AMPLITUDES

273

calculated LHA of Aries. In this case, go to the column for

LHA Aries between 160

is 359.3

. Follow that column

down and extract the value for the given latitude. Since the

increment between tabulated values is so small, visual

interpolation is sufficient. In this case, the azimuth for

Polaris for the given LHA of Aries and the given latitude

and 169

Tabulated Azimuth

359.2

Compass Bearing

358.6

Error

0.6

AMPLITUDES

1703. Amplitudes

because the body’s computed altitude is zero at this instant.

The angle is prefixed E if the body is rising and W if it

is setting. This is the only angle in celestial navigation

referenced FROM East or West, i.e. from the prime

vertical. A body with northerly declination will rise and set

North of the prime vertical. Likewise, a body with southerly

declination will rise and set South of the prime vertical.

Therefore, the angle is suffixed N or S to agree with the

name of the body’s declination. A body whose declination

is zero rises and sets exactly on the prime vertical.

The Sun is on the celestial horizon when its lower limb

is approximately two thirds of a diameter above the visible

horizon. The Moon is on the celestial horizon when its

upper limb is on the visible horizon. Stars and planets are

on the celestial horizon when they are approximately one

Sun diameter above the visible horizon.

When observing a body on the visible horizon, a

correction from Table 23 must be applied. This correction

accounts for the slight change in bearing as the body moves

between the visible and celestial horizons. It reduces the

bearing on the visible horizon to the celestial horizon, from

which the table is computed.

For the Sun, stars, and planets, apply this correction to

the observed bearing in the direction away from the

elevated pole. For the moon, apply one half of the

correction toward the elevated pole. Note that the algebraic

sign of the correction does not depend upon the body’s

declination, but only on the observer’s latitude. Assuming

the body is the Sun the rule for applying the correction can

be outlined as follows:

A celestial body’s amplitude angle is the complement

of its azimuth angle. At the moment that a body rises or sets,

the amplitude angle is the arc of the horizon between the

body and the East/West point of the horizon where the

observer’s prime vertical intersects the horizon (at 90

Figure 1703. The amplitude angle (A) subtends the arc of

the horizon between the body and the point where the prime

vertical and the equator intersect the horizon. Note that it

is the compliment of the azimuth angle (Z).

Observer’s Lat. Rising/Setting Observed bearing

North

Rising

Add to

North

Setting

Subtract from

South

Rising

Subtract from

In practical navigation, a bearing (psc or pgc) of a body

can be observed when it is on either the celestial or the

visible horizon. To determine compass error, simply

convert the computed amplitude angle to true degrees and

compare it with the observed compass bearing.

The angle is computed by the formula:

South

Setting

Add to

The following two articles demonstrate the procedure

for obtaining the amplitude of the Sun on both the celestial

and visible horizons.

1704. Amplitude of the Sun on the Celestial Horizon

sin A = sin Dec / cos Lat.

This formula gives the angle at the instant the body is

on the celestial horizon. It does not contain an altitude term

24.6' N. The navigator

observes the setting Sun on the celestial horizon. Its decli-

which is also the point where the plane of the equator

intersects the horizon (at an angle numerically equal to the

observer’s co-latitude). See Figure 1703.

Example:

The DR latitude of a ship is 51

274

AZIMUTHS AND AMPLITUDES

Actual

Base

Base Amp. Tab. Amp.

Diff.

Inc.

Correction

L=51.4

N51

32.0

32.8

+0.8

0.4

+0.3

dec=19.67

19.5

32.0

32.9

+0.9

0.3

+0.3

Total

+0.6

Figure 1704. Interpolation in Table 22 for Amplitude.

nation is N 19

40.4'. Its observed bearing is 303

pgc.

1705. Amplitude of the Sun on the Visible Horizon

Required:

Gyro error.

In higher latitudes, amplitude observations should be

made when the body is on the visible horizon because the

value of the correction is large enough to cause significant

error if the observer misjudges the exact position of the

celestial horizon. The observation will yield precise results

whenever the visible horizon is clearly defined.

Solution:

Interpolate in Table 22 for the Sun’s calculated

amplitude as follows. See Figure 1704 . The actual values

for latitude and declination are L = 51.4

N and dec. = N

Example:

Observer’s DR latitude is 59

. Find the tabulated values of latitude and

declination closest to these actual values. In this case, these

tabulated values are L = 51

47’N, Sun’s declination

11.3’S. At sunrise the Sun is observed on the visible

horizon bearing 098.5

. Record the

amplitude corresponding to these base values, 32.0

and dec. = 19.5

pgc.

,asthe

base amplitude.

Next, holding the base declination value constant at

Required:

Compass error.

, increase the value of latitude to the next tabulated

value: N 52

Solution:

Given this particular latitude and declination, the

amplitude angle is E100.4

. Note that this value of latitude was increased

because the actual latitude value was greater than the base

value of latitude. Record the tabulated amplitude for L =

S, so that the Sun’s true bearing

and dec. = 19.5

: 32.8

. Then, holding the base latitude

at the moment it is on the celestial horizon, that is,

when its Hc is precisely 0

. Applying the Table 23

correction to the observed bearing using the rules given in

Article 1703, the Sun would have been bearing 099.7

, increase the declination value to the

next tabulated value: 20

. Record the tabulated amplitude

pgc

had the observation been made when the Sun was on the

celestial horizon. Therefore, the gyro error is 0.7

for L = 51

The latitude’s actual value (51.4

and dec. = 20

: 32.9

) is 0.4 of the way

) and the value used to

determine the tabulated amplitude (52

). The declination’s

1706. Amplitude by Calculation

actual value (19.67

) is 0.3 of the way between the base

) and the value used to determine the tabulated

amplitude (20.0

As an alternative to using Table 22 and Table 23, a

visible horizon amplitude observation can be solved by the

“altitude azimuth” formula, because azimuth and amplitude

angles are complimentary, and the co-functions of compli-

mentary angles are equal; i.e., cosine Z = sine A.

). To determine the total correction to base

amplitude, multiply these increments (0.4 and 0.3) by the

respective difference between the base and tabulated values

(+0.8 and +0.9, respectively) and sum the products. The

total correction is +0.6

. Add the total correction (+0.6

)

Sine A = [SinD - (sin L sin H)] / (cos L cos H)

to the base amplitude (32.0

) to determine the final

) which will be converted to a true bearing.

Because of its northerly declination (in this case), the

Sun was 32.6

For shipboard observations, the Sun’s (computed)

altitude is negative 0.7

north of west when it was on the celestial

horizon. Therefore its true bearing was 302.6

when it is on the visible horizon.

Using the same entities as in Article 1705 , the amplitude

angle is computed as follows:

(270

) at this moment. Comparing this with the gyro

bearing of 303

gives an error of 0.4

W, which can be

Sin A = [sin 5.2

- (sin 59.8

X sin -0.7

)] / (cos 59.8

rounded to 1/2

X cos 0.7

°)

19.67

is 5

19.5

is 100.4

value constant at 51

between the base value (51

value (19.5

amplitude (32.6

32.6

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