Chapt - 22.pdf - po ang - Januszek66

CHAPTER 22

CALCULATIONS AND CONVERSIONS

INTRODUCTION

2200. Purpose and Scope

tor for basic calculations should the computer fail.

Handheld calculators are dependable enough that the navi-

gator can expect to never have to solve celestial sights,

sailings, and other problems by tables or calculations.

In using a calculator for any navigational task, it im-

portant to remember that the accuracy of the result, even if

carried to many decimal places, is only as good as the least

accurate entry. If a sextant observation is taken to an accu-

racy of only a minute, that is the best accuracy of the final

solution, regardless of a calculator’s ability to solve to 12

decimal places. See Chapter 23, Navigational Errors, for a

discussion of the sources of error in navigation.

Some basic calculators require the conversion of de-

grees, minutes and seconds (or tenths) to decimal degrees

before solution. A good navigational calculator, however,

should permit entry of degrees, minutes and tenths of min-

utes directly, and should do conversions at will. Though

many non-navigational computer programs have an on-

screen calculator, these are generally very simple versions

with only the four basic arithmetical functions. They are

thus too simple for many navigational problems. Converse-

ly, a good navigational computer program requires no

calculator per se, since the desired answer is calculated au-

tomatically from the entered data.

The following articles discuss calculations involved in

various aspects of navigation.

This chapter discusses the use of calculators and com-

puters in navigation and summarizes the formulas the

navigator depends on during voyage planning, piloting, ce-

lestial navigation, and various related tasks. To fully utilize

this chapter, the navigator should be competent in basic

mathematics including algebra and trigonometry (See

Chapter 21, Navigational Mathematics), and be familiar

with the use of a basic scientific calculator. The navigator

should choose a calculator based on personal needs, which

may vary greatly from person to person according to indi-

vidual abilities and responsibilities.

2201. Use of Calculators in Navigation

Any common calculator can be used in navigation, even

one providing only the four basic arithmetic functions of ad-

dition, subtraction, multiplication, and division. Any good

scientific calculator can be used for sight reduction, sailings,

and other tasks. However, the use of a computer program or

handheld calculator specifically designed for navigation

will greatly reduce the workload of the navigator, reduce the

possibility of errors, and increase the accuracy of results

over those obtained by hand calculation.

Calculations of position based on celestial observa-

tions are becoming increasingly obsolete as GPS takes its

place as a dependable position reference for all modes of

navigation. This is especially true since handheld, battery-

powered GPS units have become less expensive, and can

provide a worldwide backup position reference to more so-

phisticated systems with far better accuracy and reliability

than celestial.

However, for those who still use celestial techniques, a

celestial navigation calculator or computer program can im-

prove celestial positions by easily solving numerous sights,

and by reducing mathematical and tabular errors inherent in

the manual sight reduction process. They can also provide

weighted plots of the LOP’s from any number of celestial

bodies, based on the navigator’s subjective analysis of each

sight, and calculate the best fix with lat./long. readout.

On a vessel with a laptop or desktop computer conve-

nient to the bridge, a good choice would be a

comprehensive computer program to handle all navigation-

al functions such as sight reduction, sailings, tides, and

other tasks, backed up by a handheld navigational calcula-

2202. Calculations of Piloting

• Hull speed in knots is found by:

1.34 waterline length (in feet).

This is an approximate value which varies with hull

shape.

• Nautical and U.S. survey miles can be interconverted

by the relationships:

1 nautical mile = 1.15077945 U.S. survey miles.

1 U.S. survey mile = 0.86897624 nautical miles.

• The speed of a vessel over a measured mile can be

calculated by the formula:

329

330

CALCULATIONS AND CONVERSIONS

3600

level, in feet and d s is the distance to the waterline of

the object in nautical miles.

where S is the speed in knots and T is the time in

seconds.

• Distance by vertical angle between the waterline

and the top of an object is computed by solving the

right triangle formed between the observer, the top of

the object, and the waterline of the object by simple

trigonometry. This assumes that the observer is at sea

level, the Earth is flat between observer and object,

there is no refraction, and the object and its waterline

form a right angle. For most cases of practical signifi-

cance, these assumptions produce no large errors.

• The distance traveled at a given speed is computed

by the formula:

where D is the distance in nautical miles, S is the speed

in knots, and T is the time in minutes.

tan 2 a

0.0002419 2

–

0.7349

–

tan

0.0002419

• Distance to the visible horizon in nautical miles can be

calculated using the formula:

where D is the distance in nautical miles, a is the cor-

rected vertical angle, H is the height of the top of the

object above sea level, and h is the observer’s height of

eye in feet. The constants (0.0002419 and 0.7349) ac-

count for refraction.

D 1.17 h f

, or

2.07 h m

depending upon whether the height of eye of the observer

above sea level is in feet (h f ) or in meters (h m ).

2203. Tide Calculations

• The rise and fall of a diurnal tide can be roughly cal-

culated from the following table, which shows the

fraction of the total range the tide rises or falls during

flood or ebb.

• Dip of the visible horizon in minutes of arc can be cal-

culated using the formula:

D 0.97' h f

, or

Hour

Amount of flood/ebb

1.76' h m

1/12

2/12

depending upon whether the height of eye of the observer

above sea level is in feet (h f ) or in meters (h m )

3/12

2/12

1/12

• Distance to the radar horizon in nautical miles can

be calculated using the formula:

2204. Calculations of Celestial Navigation

D 1.22 h f

, or

Unlike sight reduction by tables, sight reduction by

calculator permits the use of nonintegral values of latitude

of the observer, and LHA and declination of the celestial

body. Interpolation is not needed, and the sights can be

readily reduced from any assumed position. Simultaneous,

or nearly simultaneous, observations can be reduced using

a single assumed position. Using the observer’s DR or MPP

for the assumed longitude usually provides a better repre-

sentation of the circle of equal altitude, particularly at high

observed altitudes.

2.21 h m

depending upon whether the height of the antenna

above sea level is in feet (h f ) or in meters (h m ).

• Dip of the sea short of the horizon can be calculated

using the formula:

60 tan 1

–

h f

6076.1 d s

d s

8268

• The dip correction is computed in the Nautical Alma-

nac using the formula:

where Ds is the dip short of the horizon in minutes

of arc; h f is the height of eye of the observer above sea

0.97 h

------------

-------

----------------------------

----------------

-------------------------

----------------------

------------

CALCULATIONS AND CONVERSIONS

331

where dip is in minutes of arc and h is height of eye in

feet. This correction includes a factor for refraction.

The Air Almanac uses a different formula intended for

air navigation. The differences are of no significance in

practical navigation.

tical and the body, L is the latitude at the point of

observation, and d is the declination of the celestial body.

2205. Calculations of the Sailings

• The computed altitude (Hc) is calculated using the ba-

sic formula for solution of the undivided navigational

triangle:

• Plane sailing is based on the assumption that the me-

ridian through the point of departure, the parallel

through the destination, and the course line form a

plane right triangle, as shown in Figure 2205 .

sin

L d

sin

cos

L d LHA,

cos

From this: cos C= 1

---- , sin C= p

---- , and tan C

--- .

in which h is the altitude to be computed (Hc), L is the

latitude of the assumed position, d is the declination of

the celestial body, and LHA is the local hour angle of the

body. Meridian angle (t) can be substituted for LHA in

the basic formula.

Restated in terms of the inverse trigonometric function:

From this: 1=D cos C, D=1 sec C, and p=D sin C .

From this, given course and distance (C and D), the dif-

ference of latitude (l) and departure (p) can be found,

and given the latter, the former can be found, using

simple trigonometry. See Chapter 24.

sin

–

[

(

sin

)

(

cos

LHA

)]

• Traverse sailing combines plane sailings with two or

more courses, computing course and distance along a

series of rhumb lines. See Chapter 24.

When latitude and declination are of contrary name,

declination is treated as a negative quantity. No special

sign convention is required for the local hour angle, as in

the following azimuth angle calculations.

• The azimuth angle (Z) can be calculated using the al-

titude azimuth formula if the altitude is known. The

formula stated in terms of the inverse trigonometric

function is:

cos

– sin d

-----------------------------------------------------

–

cos L cos Hc

(

sin

L sin Hc

)

(

)

If the altitude is unknown or a solution independent of

altitude is required, the azimuth angle can be calculated

using the time azimuth formula:

tan

–

-----------------------------------------------------------------------------

sin LHA

cos L tan d

(

)

–

(

sin L cos LHA

)

The sign conventions used in the calculations of both

azimuth formulas are as follows: (1) if latitude and dec-

lination are of contrary name, declination is treated as a

negative quantity; (2) if the local hour angle is greater

than 180

Figure 2205. The plane sailing triangle.

to obtain the desired value.

• Parallel sailing consists of interconverting departure

and difference of longitude. Refer to Fi gure 2205.

• Amplitudes can be computed using the formula:

A sin

–

(

sin d sec L

)

DLo

p sec L, and p DLo cos L

this can be stated as

• Mid-latitude sailing combines plane and parallel sail-

ing, with certain assumptions. The mean latitude (Lm)

is half of the arithmetical sum of the latitudes of two

places on the same side of the equator. For places on

A sin

– sin d

cos L

(

-------------

)

where A is the arc of the horizon between the prime ver-

, it is treated as a negative quantity.

If the azimuth angle as calculated is negative, add 180

332

CALCULATIONS AND CONVERSIONS

opposite sides of the equator, the N and S portions are

solved separately.

of departure (DLo v ) can be calculated from the formula:

DLo v

sin

–

----------------

ö .

In mid-latitude sailing:

sin

L v

DLo

p sec Lm, and p DLo cos Lm

The distance from the point of departure to the vertex

(D v ) can be calculated from the formula:

• Mercator Sailing problems are solved graphically on

a Mercator chart. For mathematical Mercator solutions

the formulas are:

D v

sin

–

(

cos L 1

sin DLo v

tan C

----------- or DLo m tan C

• The latitudes of points on the great-circle track can

be determined for equal DLo intervals each side of the

vertex (DLo vx ) using the formula:

where m is the meridional part from Table 6 in the Ta-

bles Part of this volume. Following solution of the

course angle by Mercator sailing, the distance is by the

plane sailing formula:

L x

tan

–

(

cos D Lo vx tan L v

)

l sec C.

The DLo v and D v of the nearer vertex are never greater

than 90

. However, when L 1 and L 2 are of contrary

name, the other vertex, 180

• Great-circle solutions for distance and initial course

angle can be calculated from the formulas:

away, may be the better

one to use in the solution for points on the great-circle

track if it is nearer the mid point of the track.

–

D = cos

[

(

sin L

sin

L 2

cos L 1 cos L 2 cos DLo )] ,

The method of selecting the longitude (or DLo vx ), and

determining the latitude at which the great-circle cross-

es the selected meridian, provides shorter legs in

higher latitudes and longer legs in lower latitudes.

Points at desired distances or desired equal intervals of

distance on the great-circle from the vertex (D vx ) can

be calculated using the formulas:

and

–

sin DLo

tan

-------------------------------------------------------------------------------------------

(

cos L 1

tan L 2

)

–

(

sin L 1

cos DLo

)

where D is the great-circle distance, C is the initial

great-circle course angle, L 1 is the latitude of the point

of departure, L 2 is the latitude of the destination, and

DLo is the difference of longitude of the points of de-

parture and destination. If the name of the latitude of

the destination is contrary to that of the point of depar-

ture, it is treated as a negative quantity.

L x

sin

–

[

sin L v

cos D vx

]

and

–

sin D vx

cos L x

DLo vx

sin

------------------

• The latitude of the vertex ,L v , is always numerically equal

to or greater than L 1 or L 2 . If the initial course angle C is

less than 90

, the vertex is toward L 2 , but if C is greater than

A calculator which converts rectangular to polar coor-

dinates provides easy solutions to plane sailings.

However, the user must know whether the difference

of latitude corresponds to the calculator’s X-coordinate

or to the Y-coordinate.

, the nearer vertex is in the opposite direction. The ver-

tex nearer L 1 has the same name as L 1 .

The latitude of the vertex can be calculated from the

formula:

2206. Calculations Of Meteorology And Oceanography

–

L v

cos

(

cos L 1

sin C

)

• Converting thermometer scales between centigrade,

Fahrenheit, and Kelvin scales can be done using the

The difference of longitude of the vertex and the point

cos

DLo

CALCULATIONS AND CONVERSIONS

333

following formulas:

formula:

---------------------------- ,

(

- 32

)

1.5 fetch in nautical miles .

--- C

, and

• Wave height = 0.026 S 2 where S is the wind speed in

knots.

• Wave speed in knots

273.15

1.34 wavelength in feet, or

• Maximum length of sea waves can be found by the

3.03 wave period in seconds

UNIT CONVERSION

Use the conversion tables that appear on the following pages to convert between different systems of units.

Conversions followed by an asterisk are exact relationships.

MISCELLANEOUS DATA

Are 1 square inch ______________= 6.4516 square centimeters*

1 square foot ______________= 144 square inches*

= 0.09290304 square meter*

= 0.000022957 acre

1 square yard ______________= 9square feet*

= 0.83612736 square meter

1 square (statute) mile ___________= 27,878,400 square feet*

= 640 acres*

= 2.589988110336 square kilometers*

1 square centimeter ____________= 0.1550003 square inch

= 0.00107639 square foot

1 square meter ______________= 10.76391 square feet

= 1.19599005 square yards

1 square kilometer ____________= 247.1053815 acres

= 0.38610216 square statute mile

= 0.29155335 square nautical mile

Astronomy

1 mean solar unit _____________= 1.00273791 sidereal units

1 sidereal unit ______________= 0.99726957 mean solar units

1 microsecond ______________= 0.000001 second*

1 second ________________= 1,000,000 microseconds*

= 0.01666667 minute

= 0.00027778 hour

= 0.00001157 day

1 minute ________________= 60 seconds*

= 0.01666667 hour

= 0.00069444 day

1 hour _________________= 3,600 seconds*

= 60 minutes*

= 0.04166667 day

1 mean solar day _____________= 24 h 03 m 56 s .55536 of mean sidereal time

= 1 rotation of Earth with respect to Sun (mean)*

= 1.00273791 rotations of Earth

with respect to vernal equinox (mean)

= 1.0027378118868 rotations of Earth

with respect to stars (mean)

Chapt - 22.pdf

Plik z chomika:

Inne pliki z tego folderu:

Inne foldery tego chomika: