《航海学》课程参考文献（地文资料）CHAPTER 22 NAVIGATIONAL CALCULATIONS

CHAPTER 22NAVIGATIONALCALCULATIONSINTRODUCTION2200.PurposeAnd Scopethese skills do not fail when the calculator or computerdoes.This chapter discusses the use of calculators and com-In using a calculator for any navigational task, it im-puters in navigation and summarizes the formulastheportanttorememberthat theaccuracy of theresult, even ifnavigatordependson duringvoyageplanning,piloting,cecarried tomany decimal places,is only as good as the leastlestial navigation,and variousrelated tasks.Tofully utilizeaccurate entry.If a sextant observation is taken to an accu-this chapter,the navigator should be competent in basicracy ofonlya minute,thatis thebest accuracyof thefinalmathematics including algebra and trigonometry (Seesolution, regardless of a calculator's ability to solve to 12Chapter21,Navigational Mathematics),andpossess agooddecimal places.SeeChapter23,NavigationalErrors,foraworking familiarity with a basic scientific calculator.Thediscussion of the sources oferror in navigation.In addition to the 4arithmeticfunctions, a basic navi-brand ofcalculator is not important,and noeffort is madeheretorecommend orspecifyanyonetype.Thenavigatorgational calculatorshouldbeabletoperformreciprocalsshouldchooseacalculatorbasedonpersonalneeds.whichroots,logarithms,trigonometry,and haveatleastonemem-mayvarygreatlyfrompersontopersonaccordingtoindi-ory.Atthe otherend of thescaleare special navigationalvidual abilities and responsibilities.computerprograms with almanac data andtidetables,inte-grated with programs for sight reduction, great circle2201.Use Of Calculators In Navigationnavigation,DR,routeplanning,andotherfunctions.Anycommon calculatorcan be used in navigation,even2202.Calculator Keysoneproviding onlythe4basicarithmeticfunctions of addi-tion, subtraction,multiplication,and division In generalIt is not within the scope of this text to describe thehowever,themore sophisticated thecalculatorand themoresteps whichmustbetakentosolvenavigational problemsmathematical functions it can perform,thegreater will be itsusing any given calculator.There arefar too many calcula-use in navigation.Any modern hand-held calculatorlabeledtors available andfar too many ways to enter the data.Theas a“scientific"model will have all the functions necessarypurpose of this chapter is to summarize the formulas usedfor thenavigator.Programmable calculators can be presetinthe solutionof commonnavigational problems.withformulastosimplifysolutionsevenmore,and specialDespitethewidevariety ofcalculators availablefromnavigational calculators and computerprograms reducethenumerous manufacturers,a fewbasickeystrokes are com-navigator's task to merely collecting the data and entering itmontonearlyallcalculators.Most scientificcalculatorsintotheproperplacesintheprogram.Ephemeral (Almanac)havetwoormoreregisters.oractivelines.knownasthexdata is included in the more sophisticated navigational cal-register and y-register.culatorsandcomputerprogramsThe+,-,x, and + keys perform the basic arithmet-Calculators or computers can improvecelestial naviga-tion by easily solving numerous sights to refine one'sicalfunctions.The Deg Rad Grad key selects degrees, radians, orposition and by reducing mathematical and tabular errorsinherent in themanualsightreductionprocess.In othernav-grads as the method of expression of values.igational tasks,theycan improve accuracyin calculations360°=2元radians=400grads.and reduce thepossibility of errors in computation.ErrorsThe+/-keys changes the sign of thenumber in theindataentryarethemostcommonproblemincalculatorx-register.TheCorCLkeyclearstheproblemfromtheandcomputernavigation.While this is extremely helpful.the navigator mustcalculatorneverforgethowtodotheseproblemsbyusingthetablesThe CEkey clears the number just entered, but notthe problem.andothernon-automatedmeans.Soonerorlaterthecalcu-lator or computer will fail,and solutions will havetobeThe CM key clearsthememory,but not theworked outby hand andbrainpower.Theprofessionalnav-problem.TheF2ndF,Arc,or Invkey activates thesecondigatorwill regularlypracticetraditional methods to ensure339

339 CHAPTER 22 NAVIGATIONAL CALCULATIONS INTRODUCTION 2200. Purpose And Scope 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 possess a good working familiarity with a basic scientific calculator. The brand of calculator is not important, and no effort is made here to recommend or specify any one type. 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 4 basic arithmetic functions of addi￾tion, subtraction, multiplication, and division. In general, however, the more sophisticated the calculator and the more mathematical functions it can perform, the greater will be its use in navigation. Any modern hand-held calculator labeled as a “scientific” model will have all the functions necessary for the navigator. Programmable calculators can be preset with formulas to simplify solutions even more, and special navigational calculators and computer programs reduce the navigator’s task to merely collecting the data and entering it into the proper places in the program. Ephemeral (Almanac) data is included in the more sophisticated navigational cal￾culators and computer programs. Calculators or computers can improve celestial naviga￾tion by easily solving numerous sights to refine one’s position and by reducing mathematical and tabular errors inherent in the manual sight reduction process. In other nav￾igational tasks, they can improve accuracy in calculations and reduce the possibility of errors in computation. Errors in data entry are the most common problem in calculator and computer navigation. While this is extremely helpful, the navigator must never forget how to do these problems by using the tables and other non-automated means. Sooner or later the calcu￾lator or computer will fail, and solutions will have to be worked out by hand and brain power. The professional nav￾igator will regularly practice traditional methods to ensure these skills do not fail when the calculator or computer does. 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. In addition to the 4 arithmetic functions, a basic navi￾gational calculator should be able to perform reciprocals, roots, logarithms, trigonometry, and have at least one mem￾ory. At the other end of the scale are special navigational computer programs with almanac data and tide tables, inte￾grated with programs for sight reduction, great circle navigation, DR, route planning, and other functions. 2202. Calculator Keys It is not within the scope of this text to describe the steps which must be taken to solve navigational problems using any given calculator. There are far too many calcula￾tors available and far too many ways to enter the data. The purpose of this chapter is to summarize the formulas used in the solution of common navigational problems. Despite the wide variety of calculators available from numerous manufacturers, a few basic keystrokes are com￾mon to nearly all calculators. Most scientific calculators have two or more registers, or active lines, known as the x￾register and y-register. • The +,–,×, and ÷ keys perform the basic arithmet￾ical functions. • The Deg Rad Grad key selects degrees, radians, or grads as the method of expression of values. 360° = 2π radians = 400 grads. • The +/– keys changes the sign of the number in the x-register. • The C or CL key clears the problem from the calculator. • The CE key clears the number just entered, but not the problem. • The CM key clears the memory, but not the problem. • The F, 2nd F, Arc, or Inv key activates the second

340NAVIGATIONALCALCULATIONSfunction of another key.(Some keys have morecalculated by the formula:than one function.)3600The x2 key squares the number in the x-register.S=TThe yx key raises the number in the y-register tothe x power.where S is the speed in knots and T is the time insin, cos, and tan keys determine the trigonometricseconds.functionofangles,whichmustbeexpressed in de-grees and tenths.The distance traveled at a given speed is computedby the formula:R->PandP>Rkeysconvertfromrectangularto polar coordinates and viceversa.STD =The p key enters the value for Pi, the circumfer-60ence ofa circle divided by its diameter.where D is the distance in nautical miles,S is theTheM, STO,RCLkeys entera number intomem-speed in knots, and T is the time in minutes.ory and recall it.M+,M-keys add or subtract the number in mem-Distancetothe visiblehorizon in nautical miles can beory without displaying it.calculated usingtheformula:The Inkey calculates the natural logarithm (loge)ofa number.D= 1.17 /hg,orlog calculates the common logarithm of a number.The I/x key calculates the reciprocal of a numberD = 2.07 /h.(Very useful for finding the reciprocal of trigono-metric functions).depending upon whether the height ofeye of the ob-Some basic calculators require the conversion of de-server above sea level is in feet (h)or in meters (hm)grees, minutes and seconds (or tenths) to decimal degreesbefore solution.A good navigational calculator,however,Dip of the visible horizon in minutes of arc can be cal-should permit entry of degrees, minutes and tenths of min-culated using the formula:utes directly.Though many non-navigational computer programsD=0.97/hfohave an on-screen calculator, these are generally very sim-ple versions with only the four basic arithmetical functions.D = 1.76 /h,Theyarethustoosimpleformanynavigational problemsConversely,a good navigational computerprogram re-quires no calculator, since thedesired answer is calculateddepending upon whether the height of eye ofthe obautomaticallyfromtheentereddataserver above sea level is infeet (h)or in meters (hm)2203.Calculations Of PilotingDistance to the radar horizon innautical miles canbecalculated usingtheformula:Hull speed in knots is found byD = 1.22,/hf,orS=1.34/waterline length(infeet)D = 2.21 /hmThis is an approximate valuewhichvaries accord-ing to hull shape.depending upon whether the height of the antennaNautical and U.S.surveymiles can be interconvertedabove sea level is in feet (h)or in meters (hm).by the relationships:Dip ofthe sea short of the horizon can be calculated using1nautical mile=1.15077945U.S.surveymiles.theformula:hfdsDs = 60 tan1 U.S. survey mile =0.86897624 nautical miles.6076.1 d8268The speed of a vessel over a measured mile can bewhereDs is thedipshort of thehorizon in minutes ofarc;

340 NAVIGATIONAL CALCULATIONS function of another key. (Some keys have more than one function.) • The x2 key squares the number in the x-register. • The yx key raises the number in the y-register to the x power. • sin, cos, and tan keys determine the trigonometric function of angles, which must be expressed in de￾grees and tenths. • R—>P and P—>R keys convert from rectangular to polar coordinates and vice versa. • The p key enters the value for Pi, the circumfer￾ence of a circle divided by its diameter. • The M, STO, RCL keys enter a number into mem￾ory and recall it. • M+, M– keys add or subtract the number in mem￾ory without displaying it. • The ln key calculates the natural logarithm (log e) of a number. • log calculates the common logarithm of a number. • The 1/x key calculates the reciprocal of a number. (Very useful for finding the reciprocal of trigono￾metric functions). 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. Though many non-navigational computer programs have an on-screen calculator, these are generally very sim￾ple versions with only the four basic arithmetical functions. They are thus too simple for many navigational problems. Conversely, a good navigational computer program re￾quires no calculator, since the desired answer is calculated automatically from the entered data. 2203. Calculations Of Piloting • Hull speed in knots is found by: This is an approximate value which varies accord￾ing to 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: where S is the speed in knots and T is the time in seconds. • 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. • Distance to the visible horizon in nautical miles can be calculated using the formula: depending upon whether the height of eye of the ob￾server above sea level is in feet (hf) or in meters (hm). • Dip of the visible horizon in minutes of arc can be cal￾culated using the formula: depending upon whether the height of eye of the ob￾server above sea level is in feet (hf) or in meters (hm). • Distance to the radar horizon in nautical miles can be calculated using the formula: depending upon whether the height of the antenna above sea level is in feet (hf) or in meters (hm). • Dip of the sea short of the horizon can be calculated using the formula: where Ds is the dip short of the horizon in minutes of arc; S 1.34 waterline length(in feet) . = S 3600 T = - D ST 60 = - D 1.17 hf = , or D 2.07 hm = D 0.97' hf = , or D 1.76' hm = D 1.22 hf = , or D 2.21 hm = Ds 60 tan–1 hf 6076.1 ds - ds 8268 + -       =

341NAVIGATIONALCALCULATIONShris the height ofeyeoftheobserver above sea level,infeetwheredip is inminutes ofarc and h isheight ofeye infeet.and dis the distance tothe waterline oftheobject in nauti-This correction includes a factor for refraction. The Aircal miles.Almanacusesa different formula intendedforair naviga-tion, The differences are of no significance in practicalDistance by vertical angle between the waterlinenavigationand thetop of an object is computed by solving theright triangle formed between the observer, the top ofThe computed altitude (Hc)is calculated using the bathe object, and the waterline of theobject by simplesic formula for solution of the undivided navigationaltrigonometry.This assumes that the observer is at seatrianglelevel, the earth is flat between observer and objectthere is no refraction, and the object and its waterlinesinh= sinLsind+ cosLcosdcosLHA,form a right angle.For most cases of practical significance,these assumptions produce no large errorsin which h is the altitude to be computed (Hc), L is thelatitudeof the assumed position,d isthedeclination oftan? aH-htan athe celestial body,and LHA is thelocalhourangleoftheD0.73490.00024190.00024192body.Meridian angle (t)can be substituted forLHA inthebasicformulaRestated in terms of the inverse trigonometric functionwhere D is the distance in nautical miles, a is the cor-rected vertical angle,H is the height of the top of theHc = sin-'[(sinL sind)+(cosL cosd cosLHA)],objectabove sea level,and h is the observer's height ofeye infeet.Theconstants(.0002419and.7349)ac-When latitude and declination areof contrary namecountforrefraction.declination is treated as a negative quantity.No specialsign convention is required for the local hour angle, as in2204.Tide Calculationsthefollowingazimuth angle calculationsThe rise and fall of a diurnal tide can be roughly cal-The azimuth angle (Z) can be calculated using the al-culatedfrom the followingtable,which shows thetitude azimuth formula if the altitude is known. Thefraction of thetotal range thetiderises or falls duringformula stated in terms of the inverse trigonometricflood or ebb.function is:HourAmountofflood/ebb(sin d-(sin L sin Hc)Z = cos(cos L cos Hc)1234561/122/12If the altitude is unknown or a solution independent of3/123/12altitude is required, the azimuth angle can be calculated2/12using thetimeazimuthformula:1/12sin LHA2205.CalculationsOfCelestial NavigationZ = tan(cos L tan d)-(sin L cos LHA)Unlike sightreductionbytables,sight reduction by calculatorpermits theuse of nonintegral values of latitude ofThe sign conventions used in the calculations of boththe observer, and LHA and declination of the celestialazimuth formulas are as follows: ()if latitude and dec-body.Interpolation is not needed, and the sights can belinationareofcontraryname,declination istreatedasareadilyreducedfromanyassumedposition.Simultaneous,negative quantity;(2) if the local hour angle is greateror nearlysimultaneous,observations can be reducedusingthan 180°, it is treated as a negative quantitya single assumed position.Using the observer's DRor MPPIf the azimuth angle as calculated is negative,add 180°for theassumed longitude usuallyprovides abetter repre-to obtain the desired value.sentation ofthe circle of equal altitude,particularlyat highobserved altititudes.Amplitudes can becomputed usingtheformulaThe dip correction is computed in theNautical AlmanacA = sin-l(sin d sec L)using the formula:D=0.97/hthis can be stated as

NAVIGATIONAL CALCULATIONS 341 hf is the height of eye of the observer above sea level, in feet and ds is the distance to the waterline of the object in nauti￾cal miles. • 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. 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 (.0002419 and .7349) ac￾count for refraction. 2204. 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. 2205. Calculations Of Celestial Navigation Unlike sight reduction by tables, sight reduction by cal￾culator 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 altititudes. • The dip correction is computed in the Nautical Almanac using the formula: 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 naviga￾tion. The differences are of no significance in practical navigation. • The computed altitude (Hc) is calculated using the ba￾sic formula for solution of the undivided navigational triangle: 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: 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: If the altitude is unknown or a solution independent of altitude is required, the azimuth angle can be calculated using the time azimuth formula: 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°, it is treated as a negative quantity. If the azimuth angle as calculated is negative, add 180° to obtain the desired value. • Amplitudes can be computed using the formula: this can be stated as Hour Amount of flood/ebb 1 1/12 2 2/12 3 3/12 4 3/12 5 2/12 6 1/12 D tan2 a 0.00024192 - H h – 0.7349 + - tan a 0.0002419 = – - D 0.97 h = sinh L d L d LHA = sin sin + cos cos cos , Hc sin–1 = [(sinL d sin ) + ( cosL d LHA cos cos ) ]. Z cos–1 sin d L sin Hc – ( ) sin ( ) cos L cos Hc -    = Z tan–1 sin LHA ( ) cos L tan d – ( ) sin L cos LHA -    = A sin–1 = ( ) sin d sec L

342NAVIGATIONALCALCULATIONSParallel sailing consists of interconverting departure,sindA = sin-anddifferenceof longitude.RefertoFigure2206cosLDLo = p sec L, and p= DLo cos Lwhere A is the arc of the horizon between the prime ver-tical and the body, L is the latitude at the point ofMid-latitude sailing combines plane and parallel sail-observation,and d isthedeclinationofthe celestial bodying,withcertainassumptions.Themeanlatitude(Lm)ishalfofthearithmeticalsumofthelatitudesoftwo2206.Calculations Of The Sailingsplaces onthesamesideof theequator.Forplacesonopposite sides of the equator, theN and S portions arePlane sailing is based on the assumption that the me-solved separatelyridian through the point of departure, the parallelthrough the destination, and the course line form aIn mid-latitude sailingplane right triangle, as shown in Figure 2206.DLo = p sec Lm, and p= DLo cos LmFrom this cos C, sin C-D, and tan C= !Mercator Sailing problems are solved graphically onFromthis:I=DcosC,D=1 secC,andp=DsinCa Mercatorchart.Formathematical Mercator solutionstheformulasare:From this, given course and distance(C and D),the dif-ference of latitude (l) and departure (p) can be foundDLotan C =or DLo= m tan C.and given the latter, the former can be found, usingmsimple trigonometry. See Chapter 24.where m is the meridional part from Table 6.Following solution of the course angle by MercatorTraversesailingcombinesplanesalingswithtwoorsailing,the distance is by the plane sailingformula:more courses, computing course and distance along aseriesof rhumb lines.SeeChapter24D =LsecC.Great-circle solutions for distance and initial courseangle can be calculated fromtheformulasDep. (p)P2D=cos-1[(sin L, sinL2 + cos L,cos L,cos DLo)](L2, 入2)sin DLoC = tan-sin L,cos DLocostanLwhere D is the great-circle distance, C is the initial(2)107great-circlecourse angle,L, is the latitude ofthepointof departure,L, is the latitudeof the destination,andO0DLo is the difference of longitude of the points of de-partureand destination.Ifthenameof thelatitudeofthedestination is contraryto that of thepointof depar-ture, it is treated as a negative quantity.Thelatitudeofthevertex,Ly,isalwaysnumericallyequalto or greater the L, orL2.Ifthe initial course angleC is lessthan90o, thevertex is toward L,but if C is greater than90,the nearer vertex is in the opposite direction.The ver-P, (L,入,)tex nearerL,has the samename asL.Figure 2206. The plane sailing triangleThe latitude of the vertex can be calculated from the

342 NAVIGATIONAL CALCULATIONS where A is the arc of the horizon between the prime ver￾tical and the body, L is the latitude at the point of observation, and d is the declination of the celestial body. 2206. Calculations Of The Sailings • 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 2206. 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. • Traverse sailing combines plane salings with two or more courses, computing course and distance along a series of rhumb lines. See Chapter 24. • Parallel sailing consists of interconverting departure and difference of longitude. Refer to Figure 2206. • 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 opposite sides of the equator, the N and S portions are solved separately. In mid-latitude sailing: • Mercator Sailing problems are solved graphically on a Mercator chart. For mathematical Mercator solutions the formulas are: where m is the meridional part from Table 6. Following solution of the course angle by Mercator sailing, the distance is by the plane sailing formula: • Great-circle solutions for distance and initial course angle can be calculated from the formulas: where D is the great-circle distance, C is the initial great-circle course angle, L1 is the latitude of the point of departure, L2 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. • The latitude of the vertex, Lv, is always numerically equal to or greater the L1 or L2. If the initial course angle C is less than 90°, the vertex is toward L2, but if C is greater than 90°, the nearer vertex is in the opposite direction. The ver￾tex nearer L1 has the same name as L1. Figure 2206. The plane sailing triangle. The latitude of the vertex can be calculated from the A sin–1 sin d cos L = ( ) - From this: cos C= 1 D - , sin C= p D - , and tan C p 1 = - . From this: 1=D cos C, D=1 sec C, and p=D sin C . DLo p sec L, and p DLo cos L = = DLo p sec Lm, and p DLo cos Lm = = tan C DLo m = - or DLo m tan C = . D = L sec C . D= cos–1 (sin L1 [ L2 sin cos L1 cos L2 + cos DLo )] C sin DLo cos L1 tan L2 sin L1 – cos DLo -     –1 = tan

343NAVIGATIONALCALCULATIONSformula:sinDVXDLovsinLy=cos(cosL,sinC)cOSThe difference of longitude of the vertex and the pointAcalculatorwhichconverts rectangular to polar coor-ofdeparture(DLo,)canbecalculatedfromtheformuladinates provides easy solutions to plane sailings.However,the usermustknow whetherthedifferenceDLo, = sin-'(cos C)of latitudecorrespondstothecalculator'sX-coordinate(sinL)ortothe Y-coordinateThedistancefrom the point of departureto the vertex2207.Calculations Of Meteorology AndOceanography(Dy)can becalculatedfrom theformula:Converting thermometer scales between centigrade,Dy = sin-'(cos L, sin DLoy)Fahrenheit, and Kelvin scales can be done using thefollowing formulas:Co = 5(F°- 329)The latitudes of points on the great-circle track can9bedeterminedfor equal DLo intervals each sideofthevertex (DLovx) using the formula:Co+32°FoLx= tan-(cos D Loyxtan Ly)The DLoy and Dy of the nearer vertex are never greaterK°=C+273.15°than 90°. However, when L and L2 are of contraryname,theother vertex, 180°away,maybethebetterMaximum length of sea waves can be found by theone to use in the solution for points on the great-circleformula:track if it is nearerthemid point of thetrackW=1.5./fetch in nautical milesThe method of selecting the longitude (or DLovx), anddetermining the latitude at which the great-circle cross-es the selected meridian, provides shorter legs in higherWave height=0.026 2 where S is the wind speed inlatitudes and longer legs in lower latitudes.Points atknots.desired distances or desired equal intervals of distanceon thegreat-circlefrom the vertex can be calculated usWave speed in knots:ing the formulas:= 1.34 /wavelength (in feet), or= 3.03 × wave period (in seconds),[sin Lycos Dyx]Lx = sinUNITCONVERSIONUse the conversion tables that appear on the following pages to convert between different systems of units.Conversions followedby an asterisk are exact relationships

NAVIGATIONAL CALCULATIONS 343 formula: The difference of longitude of the vertex and the point of departure (DLov) can be calculated from the formula: The distance from the point of departure to the vertex (Dv) can be calculated from the formula: • The latitudes of points on the great-circle track can be determined for equal DLo intervals each side of the vertex (DLovx) using the formula: The DLov and Dv of the nearer vertex are never greater than 90°. However, when L1 and L2 are of contrary name, the other vertex, 180° 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. The method of selecting the longitude (or DLovx), 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 can be calculated us￾ing the formulas: 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. 2207. Calculations Of Meteorology And Oceanography • Converting thermometer scales between centigrade, Fahrenheit, and Kelvin scales can be done using the following formulas: • Maximum length of sea waves can be found by the formula: • Wave height = 0.026 S2 where S is the wind speed in knots. • Wave speed in knots: 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. Lv cos–1 cos L1 = ( ) sin C DLov sin–1 cos C Lv sin = ( ) - Dv sin–1 cos L1 sin DLov = ( ) Lx tan–1 cos D Lovx tan Lv = ( ) Lx sin–1 sin Lv cos Dvx = [ ] DLovx sin–1 sin Dvx cos Lx -       = C° 5 F( ) ° - 32° 9 = - F° 9 5 = -C° + 32° K° = C 273.15 + ° W 1.5 fetch in nautical miles . = = 1.34 wavelength (in feet), or 3.03 wave period (in seconds) × · =

344NAVIGATIONALCALCULATIONSMISCELLANEOUSDATAArea1 square inch=6.4516 square centimeters*1 squarefoot= 144 square inches*=0.09290304squaremeter*=0.000022957acre1 square yard=9 square feet*=0.83612736 square meterI square (statute)mile=27,878,400 square feet*=640 acres*=2.589988110336squarekilometers*1 square centimeter=0.1550003 square inch=0.00107639 squarefoot1 square meter= 10.76391 square feet=1.19599005 square yards1 square kilometer=247.1053815acres=0.38610216 square statute mile=0.29155335squarenauticalmileAstronomyI mean solar unit=1.00273791 sidereal units1 sidereal unit=0.99726957mean solar units1 microsecond=0.000001second*1 second=1,000,000 microseconds*=0.01666667minute=0.00027778hour=0.00001157 day1minute=60 seconds*= 0.01666667 hour=0.00069444 day1hour_=3,600 seconds*=60 minutes*= 0.04166667 day1 mean solar day=24h03m56s.55536ofmean sidereal time= I rotation of earth with respect to sun (mean)*=1.00273791rotationsofearthwithrespecttovernalequinox(mean)=1.0027378118868rotationsofearthwith respect to stars (mean)I mean sidereal day=23h56m04$09054ofmeansolartime=27.321661 days1 sidereal month=27do7h43m11s.51 synodical month=29.530588 days=29d12h44m02s.8=31,556,925.975 secondsI tropical (ordinary) year=525,948.766minutes= 8,765.8128 hours=365d.242198790d.0000000614(t-1900),where t= the year (date)= 365dosh48m46s () 0s.0053t=365d.25636042+0.0000000011(t1900),1 sidereal yearwhere 1= the year (date)=365d06ho9m09s.5 (+) 0s.0001t1 calendaryear (common)=31,536,000seconds*=525,600 minutes*=8,760 hours*= 365 days*1 calendar year (leap)=31,622,400seconds*=527,040 minutes*=8,784 hours*= 366 days*

344 NAVIGATIONAL CALCULATIONS MISCELLANEOUS DATA Area 1 square inch _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 6.4516 square centimeters* 1 square foot _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 144 square inches* = 0.09290304 square meter* = 0.000022957 acre 1 square yard _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 9 square 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 _ _ _ _ _ _ _ _ _ _ _ _ = 24h03m56s.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) 1 mean sidereal day _ _ _ _ _ _ _ _ _ _ _ = 23h56m04s09054 of mean solar time 1 sidereal month _ _ _ _ _ _ _ _ _ _ _ _ = 27.321661 days = 27d07h43m11s.5 1 synodical month _ _ _ _ _ _ _ _ _ _ _ _ = 29.530588 days = 29d12h44m02s.8 1 tropical (ordinary) year _ _ _ _ _ _ _ _ _ = 31,556,925.975 seconds = 525,948.766 minutes = 8,765.8128 hours = 365d.24219879 – 0d.0000000614 (t–1900), where t = the year (date) = 365d05h48m46s (–) 0s.0053t 1 sidereal year _ _ _ _ _ _ _ _ _ _ _ _ _ = 365d.25636042 + 0.0000000011 (t–1900), where t = the year (date) = 365d06h09m09s.5 (+) 0s.0001t 1 calendar year (common)_ _ _ _ _ _ _ _ _ = 31,536,000 seconds* = 525,600 minutes* = 8,760 hours* = 365 days* 1 calendar year (leap) _ _ _ _ _ _ _ _ _ _ = 31,622,400 seconds* = 527,040 minutes* = 8,784 hours* = 366 days*

345NAVIGATIONALCALCULATIONS1 light-year=9,460,000,000,000kilometers=5,880,000,000,000 statute miles=5,110.000,000,000nauticalmiles=63,240astronomical units=0.3066parsecs1 parsec=30,860,000,000,000kilometers=19,170,000,000,000statutemiles=16,660,000,000,000nauticalmiles=206,300 astronomical units= 3.262 light yearsIastronomical unit=149,600,000 kilometers=92,960,000 statute miles=80,780,000nautical miles=499s.012 light-time= mean distance, earth to sunMean distance,earthtomoon=384,400 kilometers=238,855 statute miles=207,559nautical milesMean distance,earth to sun=149,600,000 kilometers=92,957,000 statutemiles=80,780,000 nautical miles=1 astronomical unitSun's diameter= 1,392,000 kilometers=865,000 statute miles=752,000nautical milesSun'smass=1,987,000,000,000,000,000,000,000,000,000,000grams=2,200,000,000,000,000,000,000,000,000shorttons=2,000,000,000,000,000,000,000,000,000 longtonsSpeed of sun relative to neighboring stars=19.4 kilometers per second= 12.1 statute miles per second= 10.5nautical miles per secondOrbital speed of earth= 29.8 kilometers per second= 18.5 statute miles per second= 16.1 nautical miles per secondObliquity of the ecliptic=23°27'08"26-0".4684(-1900),where 1= the year (date)= 50".2564 +0".000222 (t1900), per year,General precession of the equinoxeswhere t= the year (date)Precession ofthe equinoxes in right ascension=46".0850+0".000279 (t-1900),per year,where = the year (date)Precession of the equinoxes in declination=20".0468-0".000085(t-1900),per year,where = the year (date)= 2.512Magnitude ratio5/100*=ChartsNautical miles per inch=reciprocal of natural scale+72,913.39Statute miles per inch=reciprocal ofnatural scale+63,360*Inches per nautical mile=72,913.39 × natural scaleInches per statute mile=63,360× natural scale*Natural scale= 1:72,913.39 × nautical miles per inch=1:63,360× statute milesper inch*EarthAcceleration due to gravity (standard)=980.665centimeterspersecondpersecond=32.1740 feetper second per secondMass-ratio—Sun/Earth=332,958Mass-ratioSun/(Earth & Moon)=328,912= 81.30Mass-ratioEarth/Moon= 5.517 grams per cubic centimeterMean densityVelocity of escape= 6.94 statute miles per secondCurvature of surface=0.8 foot per nautical mile

NAVIGATIONAL CALCULATIONS 345 1 light-year _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 9,460,000,000,000 kilometers = 5,880,000,000,000 statute miles = 5,110,000,000,000 nautical miles = 63,240 astronomical units = 0.3066 parsecs 1 parsec _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 30,860,000,000,000 kilometers = 19,170,000,000,000 statute miles = 16,660,000,000,000 nautical miles = 206,300 astronomical units = 3.262 light years 1 astronomical unit _ _ _ _ _ _ _ _ _ _ _ _ = 149,600,000 kilometers = 92,960,000 statute miles = 80,780,000 nautical miles = 499s.012 light-time = mean distance, earth to sun Mean distance, earth to moon _ _ _ _ _ _ _ _ = 384,400 kilometers = 238,855 statute miles = 207,559 nautical miles Mean distance, earth to sun_ _ _ _ _ _ _ _ _ = 149,600,000 kilometers = 92,957,000 statute miles = 80,780,000 nautical miles = 1 astronomical unit Sun’s diameter _ _ _ _ _ _ _ _ _ _ _ _ _ = 1,392,000 kilometers = 865,000 statute miles = 752,000 nautical miles Sun’s mass _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1,987,000,000,000,000,000,000,000,000,000,000 grams = 2,200,000,000,000,000,000,000,000,000 short tons = 2,000,000,000,000,000,000,000,000,000 long tons Speed of sun relative to neighboring stars _ _ _ = 19.4 kilometers per second = 12.1 statute miles per second = 10.5 nautical miles per second Orbital speed of earth _ _ _ _ _ _ _ _ _ _ _ = 29.8 kilometers per second = 18.5 statute miles per second = 16.1 nautical miles per second Obliquity of the ecliptic _ _ _ _ _ _ _ _ _ _ = 23°27′08″.26 – 0″.4684 (t–1900), where t = the year (date) General precession of the equinoxes _ _ _ _ _ = 50″.2564 + 0″.000222 (t–1900), per year, where t = the year (date) Precession of the equinoxes in right ascension _ = 46″.0850 + 0″.000279 (t–1900), per year, where t = the year (date) Precession of the equinoxes in declination_ _ _ = 20″.0468 – 0″.000085 (t–1900), per year, where t = the year (date) Magnitude ratio _ _ _ _ _ _ _ _ _ _ _ _ _ = 2.512 Charts Nautical miles per inch _ _ _ _ _ _ _ _ _ _ = reciprocal of natural scale ÷ 72,913.39 Statute miles per inch _ _ _ _ _ _ _ _ _ _ _ = reciprocal of natural scale ÷ 63,360* Inches per nautical mile _ _ _ _ _ _ _ _ _ _ = 72,913.39 × natural scale Inches per statute mile _ _ _ _ _ _ _ _ _ _ = 63,360 × natural scale* Natural scale _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1:72,913.39 × nautical miles per inch = 1:63,360 × statute miles per inch* Earth Acceleration due to gravity (standard) _ _ _ _ = 980.665 centimeters per second per second = 32.1740 feet per second per second Mass-ratio—Sun/Earth _ _ _ _ _ _ _ _ _ _ = 332,958 Mass-ratio—Sun/(Earth & Moon) _ _ _ _ _ _ = 328,912 Mass-ratio—Earth/Moon _ _ _ _ _ _ _ _ _ = 81.30 Mean density _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 5.517 grams per cubic centimeter Velocity of escape _ _ _ _ _ _ _ _ _ _ _ _ = 6.94 statute miles per second Curvature of surface _ _ _ _ _ _ _ _ _ _ _ = 0.8 foot per nautical mile 100 5 = *

346NAVIGATIONALCALCULATIONSWorld GeodeticSystem(WGS)Ellipsoid of 1984Equatorial radius (a)=6,378,137meters=3,443.918 nautical milesPolar radius (b)=6,356,752.314 meters=3432.372 natical milesMean radius (2a + b)/3=6,371,008.770meters=3440.069nautical milesFlattening or ellipticity (f=1-b/a)=1/298.257223563=0.003352811Eccentricity (e=(2f-f2)1/2)=0.081819191=0.006694380Eccentricity squared (e2)Length1 inch= 25.4 millimeters*=2.54 centimeters*= 12 inches*1 foot (U.S.)= 1 British foot= l/g yard*= 0.3048 meter*= '/g fathom*I foot (U.S. Survey)=0.30480061meter1 yard= 36 inches*= 3 feet*=0.9144meter*=6 feet*1fathom= 2 yards*=1.8288 meters*1 cable=720 feet*= 240 yards*=219.4560 meters*I cable (British)= 0.1 nautical mile1 statute mile=5,280feet*= 1,760 yards*=1,609.344meters*=1.609344 kilometers*=0.86897624 nautical mile=6,076.11548556 feetI nautical mile=2.025.37182852yards=1,852meters*=1.852 kilometers*=1.150779448statutemiles1 meter=100 centimeters*=39.370079 inches=3.28083990 feet=1.09361330 yards=0.54680665fathom=0.00062137statute mile=0.00053996nautical mile1 kilometer=3,280.83990 feet= 1,093.61330 yards=1,000 meters*=0.62137119 statute mile=0.53995680nautical mileMass= 437.5 grains*lounce=28.349523125grams*=0.0625pound*=0.028349523125kilogram*

346 NAVIGATIONAL CALCULATIONS World Geodetic System (WGS) Ellipsoid of 1984 Equatorial radius (a) _ _ _ _ _ _ _ _ _ _ _ = 6,378,137 meters = 3,443.918 nautical miles Polar radius (b) _ _ _ _ _ _ _ _ _ _ _ _ _ = 6,356,752.314 meters = 3432.372 natical miles Mean radius (2a + b)/3 _ _ _ _ _ _ _ _ _ _ = 6,371,008.770 meters = 3440.069 nautical miles Flattening or ellipticity (f = 1 – b/a) _ _ _ _ _ = 1/298.257223563 = 0.003352811 Eccentricity (e = (2f – f2)1/2) _ _ _ _ _ _ _ = 0.081819191 Eccentricity squared (e2) _ _ _ _ _ _ _ _ _ = 0.006694380 Length 1 inch _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 25.4 millimeters* = 2.54 centimeters* 1 foot (U.S.) _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 12 inches* = 1 British foot = 1/3 yard* = 0.3048 meter* = 1 /6 fathom* 1 foot (U.S. Survey) _ _ _ _ _ _ _ _ _ _ _ = 0.30480061 meter 1 yard_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 36 inches* = 3 feet* = 0.9144 meter* 1 fathom_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 6 feet* = 2 yards* = 1.8288 meters* 1 cable _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 720 feet* = 240 yards* = 219.4560 meters* 1 cable (British)_ _ _ _ _ _ _ _ _ _ _ _ _ = 0.1 nautical mile 1 statute mile _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 5,280 feet* = 1,760 yards* = 1,609.344 meters* = 1.609344 kilometers* = 0.86897624 nautical mile 1 nautical mile _ _ _ _ _ _ _ _ _ _ _ _ _ = 6,076.11548556 feet = 2,025.37182852 yards = 1,852 meters* = 1.852 kilometers* = 1.150779448 statute miles 1 meter _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 100 centimeters* = 39.370079 inches = 3.28083990 feet = 1.09361330 yards = 0.54680665 fathom = 0.00062137 statute mile = 0.00053996 nautical mile 1 kilometer_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 3,280.83990 feet = 1,093.61330 yards = 1,000 meters* = 0.62137119 statute mile = 0.53995680 nautical mile Mass 1 ounce _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 437.5 grains* = 28.349523125 grams* = 0.0625 pound* = 0.028349523125 kilogram*

347NAVIGATIONALCALCULATIONS1 pound=7,000 grains*=16 ounces*=0.45359237kilogram*1 short ton= 2,000 pounds*=907.18474kilograms*=0.90718474metric ton*=0.8928571 longton1 long ton= 2,240 pounds*=1,016.0469088 kilograms*= 1.12 short tons*=1.0160469088metric tons*1kilogram=2.204623 pounds= 0.00110231 short ton=0.0009842065longton1 metric ton,= 2,204.623 pounds=1,000 kilograms*=1.102311 short tons=0.9842065 longtonMathematics元=3.1415926535897932384626433832795028841971元2=9.8696044011元=1.7724538509=2.718281828459Base of Naperian logarithms (e)Modulus of common logarithms (log1oe)=0.4342944819032518=206,264."80625Iradian=3,437.7467707849=572957795131=57°17'44".80625Icircle=1,296,000"*=21,600**=3600*=2元 radians*180°= 元 radians*10=3600**=60**=0.0174532925199432957666radianI'=60"*=0.000290888208665721596radian1"=0.000004848136811095359933radianSineofi=0.00029088820456342460Sine ofI"=0.00000484813681107637MeteorologyAtmosphere (dry air)Nitrogen=78.08%99.99%=20.95%Oxygen= 0.93%Argon=0.03%CarbondioxideNeon=0.0018%Helium=0.000524%=0.0001%KryptonHydrogen=0.00005%Xenon=0.0000087%Ozone=0to0.000007% (increasingwithaltitude)Radon=0.000000000000000006%(decreasingwith altitude)Standard atmospheric pressure at sea level=1,013.250 dynes per square centimeter=1,033.227grams persquarecentimeter=1,033.227centimeters ofwater=1,013.250 millibars*=760millimeters ofmercury=76centimeters of mercury=33.8985 feet of water=29.92126 inches ofmercury=14.6960 pounds per square inch=1.033227kilogramsper squarecentimeter=1.013250 bars*

NAVIGATIONAL CALCULATIONS 347 1 pound _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 7,000 grains* = 16 ounces* = 0.45359237 kilogram* 1 short ton _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 2,000 pounds* = 907.18474 kilograms* = 0.90718474 metric ton* = 0.8928571 long ton 1 long ton _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 2,240 pounds* = 1,016.0469088 kilograms* = 1.12 short tons* = 1.0160469088 metric tons* 1 kilogram _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 2.204623 pounds = 0.00110231 short ton = 0.0009842065 long ton 1 metric ton _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 2,204.623 pounds = 1,000 kilograms* = 1.102311 short tons = 0.9842065 long ton Mathematics π _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 3.1415926535897932384626433832795028841971 π2 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 9.8696044011 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1.7724538509 Base of Naperian logarithms (e) _ _ _ _ _ _ _ = 2.718281828459 Modulus of common logarithms (log10e) _ _ _ = 0.4342944819032518 1 radian _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 206,264.″80625 = 3,437′.7467707849 = 57°.2957795131 = 57°17′44″.80625 1 circle_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1,296,000″* = 21,600′* = 360°* = 2π radians* 180°_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = π radians* 1°_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 3600″* = 60′* = 0.0174532925199432957666 radian 1′ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 60″* = 0.000290888208665721596 radian 1″_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.000004848136811095359933 radian Sine of 1′ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.00029088820456342460 Sine of 1″ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.00000484813681107637 Meteorology Atmosphere (dry air) Nitrogen _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 78.08% Oxygen _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 20.95% Argon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.93% Carbon dioxide _ _ _ _ _ _ _ _ _ _ _ = 0.03% Neon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.0018% Helium _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.000524% Krypton _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.0001% Hydrogen _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.00005% Xenon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.0000087% Ozone _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0 to 0.000007% (increasing with altitude) Radon _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 0.000000000000000006% (decreasing with altitude) Standard atmospheric pressure at sea level_ _ _ = 1,013.250 dynes per square centimeter = 1,033.227 grams per square centimeter = 1,033.227 centimeters of water = 1,013.250 millibars* = 760 millimeters of mercury = 76 centimeters of mercury = 33.8985 feet of water = 29.92126 inches of mercury = 14.6960 pounds per square inch = 1.033227 kilograms per square centimeter = 1.013250 bars* π      99.99%

348NAVIGATIONALCALCULATIONS= (-)273.16℃CAbsolutezero=(-)459.69°FPressure=0.001 millibar*I dyne per square centimeter=0.000001 bar*I grampersquare centimeter=1 centimeter ofwater=0.980665millibar*=0.07355592centimeterofmercury=0.0289590 inchofmercury=0.0142233 pound per square inch=0.001 kilogram per square centimeter*=0.000967841atmosphere1 millibar= 1,000 dynes per square centimeter*=1.01971621grams per square centimeter=0.7500617millimeterofmercury=0.03345526footofwater=0.02952998 inchofmercury=0.01450377poundpersquare inch= 0.001 bar*=0.00098692atmosphere1 millimeterof mercury=1.35951grams per square centimeter=1.3332237millibars=0.1 centimeter of mercury*=0.04460334footofwater=0.039370079 inchofmercury=0.01933677 pound per square inch=0.001315790atmosphere1 centimeter of mercury= 10 millimeters of mercury*1 inch of mercury=34.53155grams per square centimeter=33.86389 millibars=25.4millimeters ofmercury*=1.132925feet of water=0.4911541 pound per square inch=0.03342106atmosphere1 centimeterofwater=1 gram per square centimeter=0.001kilogram per square centimeter1footofwater=30.48000grams persquarecentimeter=29.89067millibars=2.241985centimeters ofmercury=0.882671inchofmercury=0.4335275poundper squareinch=0.02949980atmosphereI pound per square inch=68,947.57dynespersquarecentimeter=70.30696gramspersquarecentimeter=70.30696centimeters of water=68.94757millibars=51.71493millimetersofmercury=5.171493centimeters ofmercury=2.306659feet ofwater=2.036021 inches of mercury=0.07030696kilogrampersquarecentimeter=0.06894757bar=0.06804596atmosphereI kilogram per square centimeter=1,000 grams per square centimeter*=1,000 centimeters ofwater1bar=1,000,000 dynes per square centimeter*=1,000 millibars*SpeedI foot per minute=0.01666667footper second=0.00508 meter per second*1 yard per minute=3 feet per minute*= 0.05 foot per second*=0.03409091 statute mileper hour=0.02962419knot=0.01524meter per second*

348 NAVIGATIONAL CALCULATIONS Absolute zero_ _ _ _ _ _ _ _ _ _ _ _ _ _ = (–)273.16°C = (–)459.69°F Pressure 1 dyne per square centimeter_ _ _ _ _ _ _ _ = 0.001 millibar* = 0.000001 bar* 1 gram per square centimeter _ _ _ _ _ _ _ = 1 centimeter of water = 0.980665 millibar* = 0.07355592 centimeter of mercury = 0.0289590 inch of mercury = 0.0142233 pound per square inch = 0.001 kilogram per square centimeter* = 0.000967841 atmosphere 1 millibar _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1,000 dynes per square centimeter* = 1.01971621 grams per square centimeter = 0.7500617 millimeter of mercury = 0.03345526 foot of water = 0.02952998 inch of mercury = 0.01450377 pound per square inch = 0.001 bar* = 0.00098692 atmosphere 1 millimeter of mercury_ _ _ _ _ _ _ _ _ _ = 1.35951 grams per square centimeter = 1.3332237 millibars = 0.1 centimeter of mercury* = 0.04460334 foot of water = 0.039370079 inch of mercury = 0.01933677 pound per square inch = 0.001315790 atmosphere 1 centimeter of mercury _ _ _ _ _ _ _ _ _ = 10 millimeters of mercury* 1 inch of mercury _ _ _ _ _ _ _ _ _ _ _ _ = 34.53155 grams per square centimeter = 33.86389 millibars = 25.4 millimeters of mercury* = 1.132925 feet of water = 0.4911541 pound per square inch = 0.03342106 atmosphere 1 centimeter of water_ _ _ _ _ _ _ _ _ _ _ = 1 gram per square centimeter = 0.001 kilogram per square centimeter 1 foot of water _ _ _ _ _ _ _ _ _ _ _ _ _ = 30.48000 grams per square centimeter = 29.89067 millibars = 2.241985 centimeters of mercury = 0.882671 inch of mercury = 0.4335275 pound per square inch = 0.02949980 atmosphere 1 pound per square inch_ _ _ _ _ _ _ _ _ _ = 68,947.57 dynes per square centimeter = 70.30696 grams per square centimeter = 70.30696 centimeters of water = 68.94757 millibars = 51.71493 millimeters of mercury = 5.171493 centimeters of mercury = 2.306659 feet of water = 2.036021 inches of mercury = 0.07030696 kilogram per square centimeter = 0.06894757 bar = 0.06804596 atmosphere 1 kilogram per square centimeter _ _ _ _ _ _ = 1,000 grams per square centimeter* = 1,000 centimeters of water 1 bar _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ = 1,000,000 dynes per square centimeter* = 1,000 millibars* Speed 1 foot per minute _ _ _ _ _ _ _ _ _ _ _ _ = 0.01666667 foot per second = 0.00508 meter per second* 1 yard per minute _ _ _ _ _ _ _ _ _ _ _ _ = 3 feet per minute* = 0.05 foot per second* = 0.03409091 statute mile per hour = 0.02962419 knot = 0.01524 meter per second*