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Introduction Corrections to Sextant Time Finding GHA and Declination Assumed Position and Local Hour Angle Computed Altitude and Azimuth Interpolation Altitude Intercept Using Position Plotting Sheets Plotting Lines of Position
FINDING COMPUTED ALTITUDE AND AZIMUTH
YOU HAD A BRIEF GLANCE AT PART OF A PAGE OF 229 IN THE LAST LESSON. NOW LET'S TAKE A CLOSER LOOK.
ALL THE PAGES OF 229 THAT YOU WILL NEED TO WORK THE PROBLEMS IN THIS COURSE ARE PROVIDED IN A SEPARATE BOOK. YOU WILL USE THESE PAGES TO FIND THE COMPUTED ALTITUDE (HC) AND AZIMUTH ZN) OF THE CELESTIAL BODIES YOU ARE WORKING WITH.
COMPUTED ALTITUDE (HC) COULD BE DEFINED AS WHAT HO WOULD BE IF IT WERE TAKEN FROM A KNOWN POSITION. IT IS THE ANGLE UPWARD FROM THE CELESTIAL HORIZON TO THE BODY. HO IS FOUND BY ACTUALLY MEASURING THE ANGLE WITH A SEXTANT AND APPLYING THE CORRECTIONS. HC IS CALCULATED OR "COMPUTED" STARTING WITH A KNOWN ASSUMED POSITION AND KNOWN G.P. BY USING SPHERICAL TRIGONOMETRY.
AZIMUTH (ZN) IS THE TRUE BEARING FROM THE ASSUMED POSITION TO THE BODY AND ITS G.P. IT CAN ALSO BE CALCULATED BY TRIGONOMETRY BY WORKING WITH KNOWN COORDINATES OF A.P. AND G.P.
THOSE PIECES OF INFORMATION FROM 229, HC AND ZN, WILL ENABLE US TO PLOT A LINE OF POSITION ON A CHART WHICH WILL EVENTUALLY LEAD TO A FIX.
THE SIGHT REDUCTION TABLES, 229, IS JUST A LISTING OF SOLUTIONS FOR HC AND ZN WORKED OUT FOR MILLIONS OF COMBINATIONS OF A.P.'S AND G.P.’S. ZN'S AREN'T LISTED IN THEIR FINAL FORM. INSTEAD, YOU WILL FIND AZIMUTH ANGLE (Z) IN THE TABLES. Z IS CONVERTED TO THE ZN YOU'LL USE BY APPLYING A SIMPLE RULE YOU DON'T EVEN HAVE TO REMEMBER. IT APPEARS ON EVERY PAGE OF THE TABLES.
IN THIS CHAPTER YOU'LL GET ACQUAINTED WITH THE 229 TABLES AND LEARN THE SIMPLEST METHOD OF INTERPOLATION WHICH WILL GIVE YOU HC AND ZN PRECISELY ENOUGH TO SATISFY THE EXAMINERS. THE SECTION ON INTERPOLATION HAS BONUS VALUE. THE SAME METHOD CAN BE USED FOR ALL THE TABLES YOU WILL NEED TO USE.
IF YOU HAPPEN TO FIND ACTUAL COPIES OF 229, YOU WILL FIND THAT EACH VOLUME HAS INTERPOLATION TABLES ON THE INSIDE FRONT AND BACK COVERS. IGNORE THEM. THE METHOD WE'LL SHOW YOU IS QUICKER AND EASIER THAN USING THOSE TABLES. ALSO, WE WILL. EVENTUALLY GET INTO TABLES 27 AND 28 IN BOWDITCH. THEY HAVE NO INTERPOLATION TABLES. YOU'LL HAVE TO LEARN TO INTERPOLATE FROM SCRATCH ANYWAY, SO IT MAKES SENSE TO LEARN ONE METHOD WHICH WILL SERVE ALL PURPOSES.
INTERPOLATION TABLES WERE USEFUL IN THE DAYS BEFORE EVERYBODY COULD AFFORD CALCULATORS, BUT THOSE DAYS ARE GONE AND NOT REGRETTED. DO IT OUR WAY. BEFORE YOU CAN INTERPOLATE YOU HAVE TO FIND YOUR STARTING POINT IN THE TABLES. YOU GET INTO THE BOOK BY USING SOME OF THE INFORMATION YOU'VE ALREADY FOUND: ASSUMED LATITUDE, DECLINATION, AND LHA.
FIRST, USE ASSUMED LATITUDE TO CHOOSE THE VOLUME YOU NEED. IN THIS COURSE, AND ALSO IN THE EXAM ROOM, YOU'LL ALWAYS USE VOLUME TWO OF 229, WHICH COVERS LATITUDES 15°30°, BOTH NORTH AND SOUTH. IF YOU HAPPENED TO PICK UP THE WRONG VOLUME, YOU'D DISCOVER YOUR MISTAKE BEFORE IT GOT YOU IN TROUBLE.
THE ASSUMED LATITUDE THEN LEADS YOU TO THE FRONT OR BACK PART OF THE BOOK. INFORMATION FOR LATITUDES 15° THROUGH 22° IS IN THE FRONT HALF OF VOLUME TWO: VIEW IMAGE 36
INFORMATION FOR LATITUDES 23°-30° IS IN THE BACK HALF OF THE BOOK. VIEW IMAGE 37
ONCE YOU HAVE FOUND THE PROPER VOLUME AND PART OF 229 (FRONT OR BACK HALF) BY USING THE ASSUMED LATITUDE, YOU MUST NOW LOCATE THE CORRECT PAGE. TO DO THAT YOU USE THE LHA OF THE BODY. THAT YOU FOUND BY APPLYING ASSUMED LONGITUDE TO THE GHA.
EACH PAGE OF 229 HAS LHA NUMBERS ON THE UPPER AND LOWER OUTSIDE CORNERS. THE PAGE SHOWN IS AS IT APPEARS IN 229, EXCEPT THAT IT HAS BEEN REDUCED IN SIZE NOTICE THAT THE LEFT HAND PAGE CONTAINS INFORMATION FOR LHA’S OF 87° AND 273°, AS INDICATED BY THE NUMBERS IN UPPER AND LOWER LEFT HAND CORNERS. DON’T WORRY ABOUT TRYING TO FIGURE OUT WHICH PART OF THE PAGE IS FOR 87° AND WHICH IS FOR 273°. THE INFORMATION IS IDENTICAL FOR BOTH LHA’S. VIEW IMAGE 38
THE RIGHT HAND PAGE, IMAGE 39, ALSO HAS INFORMATION FOR LHA’S OF 87° AND 273° (INDICATED BY THE NUMBERS IN THE UPPER RIGHT HAND CORNER), AND IT HAS INFORMATION FOR LHA'S OF 93° AND 267°, AS THE LOWER RIGHT HAND NUMBERS SHOW. THE DIVISION BETWEEN THESE TWO PAGES PRINTED ON ONE SHEET IS MARKED BY THE HORIZONTAL LINES, WHICH RUN ACROSS THE COLUMNS FROM 11° TO 7° OF DECLINATION OUTLINED IN RED.
LET'S SUPPOSE YOU WERE WORKING WITH A STAR THAT HAD AN LHA OF 87° , HOW DO YOU TELL WHETHER TO USE THE LEFT OR RIGHT HAND PAGE?
NOTICE THAT THE LEFT HAND PAGE IS TITLED, "LATITUDE SAME NAME AS DECLINATION." IF OUR DR POSITION WAS IN NORTH LATITUDE AND THE DECLINATION THAT WE OBTAINED FROM THE ALMANAC WAS NORTH 18° 45', WE WOULD USE THE LEFT HAND PAGE BECAUSE LATITUDE AND DECLINATION HAD THE SAME NAME. IF BOTH THE DR POSITION AND DECLINATION WERE SOUTH, WE WOULD ALSO USE THE LEFT HAND PAGE BECAUSE THEY ARE THE SAME NAME. VIEW IMAGE 39
LET'S SUPPOSE YOU WERE WORKING WITH A STAR THAT HAD AN LHA OF 87° , HOW DO YOU TELL WHETHER TO USE THE LEFT OR RIGHT HAND PAGE?
NOTICE THAT THE LEFT HAND PAGE IS TITLED, "LATITUDE SAME NAME AS DECLINATION." IF OUR DR POSITION WAS IN NORTH LATITUDE AND THE DECLINATION THAT WE OBTAINED FROM THE ALMANAC WAS NORTH 18° 45', WE WOULD USE THE LEFT HAND PAGE BECAUSE LATITUDE AND DECLINATION HAD THE SAME NAME. IF BOTH THE DR POSITION AND DECLINATION WERE SOUTH, WE WOULD ALSO USE THE LEFT HAND PAGE BECAUSE THEY ARE THE SAME NAME. VIEW IMAGE 40
THE RIGHT HAND PAGE FOR LHA'S 87° TO 273° IS USED WHEN YOUR DR LATITUDE AND DECLINATION HAVE DIFFERENT CONTRARY NAMES. IF YOUR DR POSITION WAS 20° SOUTH AND THE DECLINATION WAS 7° NORTH, YOU WOULD USE THIS CONTRARY NAME PAGE.
THE BOTTOM PART OF THIS RIGHT HAND PAGE, IMAGE 41, IS USED FOR LHA'S OF 93° TO 267° ONLY WHEN LATITUDE AND DECLINATION ARE THE SAME NAME. THERE IS NO CORRESPONDING CONTRARY PAGE FOR THESE LHA'S. VIEW IMAGE 41
WHEN YOU HAVE FOUND THE CORRECT PAGE, YOU USE THE ASSUMED LATITUDE ONCE MORE TO PICK THE COLUMN. LET'S SUPPOSE OUR ASSUMED LATITUDE IS 21° N, AND DECLINATION IS N 13°, LHA 69°. COME DOWN THE COLUMN (OR UP, IF LATITUDE AND DECLINATION ARE SAME NAME AND YOU FIND THE LHA AT THE BOTTOM .OF A RIGHT HAND PAGE) UNTIL YOU ARE STRAIGHT ACROSS FROM THE WHOLE DEGREE OF DECLINATION. VIEW IMAGE 42
A LOT OF MISTAKES ARE MADE AT THIS POINT IN THE PROCEDURE. YOUR EYE OR EVEN YOUR FINGER CAN GET OFF LINE AS IT MOVES ACROSS FROM THE DECLINATION COLUMN. IT'S BEST TO USE A STRAIGHTEDGE. USING A STRAIGHTEDGE TO KEEP FROM LOSING YOUR PLACE IS A GOOD IDEA WHEN USING ANY KIND OF TABLE, BUT ESPECIALLY SO IN 229 BECAUSE THE PAGES ARE WIDER THAN MOST. IT IS BETTER NOT TO MAKE PENCIL MARKS IN THE TABLES. THE PENCIL MARK HELPS WITH THE PROBLEM YOU'RE WORKING AT THE MOMENT, BUT LATER ANOTHER PROBLEM COULD BRING YOU BACK TO THE SAME AREA ON THAT PAGE AND YOUR EYE WOULD JUMP RIGHT TO THE PENCIL MARK!
THE HC ON THE LEFT AND THE Z ON THE RIGHT IN EACH LATITUDE COLUMN HAVE ALREADY BEEN DEFINED. THE ONE IN THE MIDDLE, UNDER "d" IS THE DIFFERENCE BETWEEN THE HC IMMEDIATELY TO THE LEFT OF IT AND THE HC IMMEDIATELY BELOW THAT ONE IN THE COLUMN. THESE DIFFERENCES WILL BE USED WHEN YOU INTERPOLATE. WHEN YOU TAKE OUT THE DIFFERENCE, YOU MUST PUT DOWN ITS SIGN, PLUS OR MINUS. THE SIGNS ARE NOT REPEATED ON EVERY LINE. THEY ARE PRINTED ONLY NEXT TO THE TOP DIFFERENCE IN EACH GROUP OF FIVE. IF NO SIGN APPEARS NEXT TO THE DIFFERENCE YOU WANT, LOOK UPWARD IN THE COLUMN UNTIL YOU COME TO A PLUS OR MINUS. IN THIS EXAMPLE, THE DIFFERENCE IS + 17.8.
YOU MAY OCCASIONALLY ENCOUNTER A DIFFERENCE PRINTED IN ITALIC (SLANTED) TYPE WITH A TINY DOT TO THE RIGHT OF IT. FOR THIS COURSE AND FOR THE TEST, YOU SHOULD TREAT THESE THE SAME AS ANY OTHERS. HOWEVER, YOU'RE BOUND TO WONDER ABOUT IT AND SOMEBODY WILL PROBABLY GET A COMPLETE COPY OF A 229 AND LOOK IT UP, SO HERE IS WHY YOU SHOULD IGNORE THE DOT.
FOR CERTAIN VERY HIGH ALTITUDES, AN EXTRA INTERPOLATION CORRECTION KNOWN AS THE DOUBLE SECOND DIFFERENCE CORRECTION CAN BE USED IF EXTREMELY ACCURATE RESULTS ARE REQUIRED. THE DOT INDICATES THE HC'S TO WHICH THAT CORRECTION SHOULD BE APPLIED. MOST PROBLEMS ON COAST GUARD TESTS WERE MADE UP BEFORE 229 WAS IN PRINT AND THAT EXTRA CORRECTION WAS AVAILABLE. IF YOU USE IT, YOU'LL CHANGE YOUR ANSWER JUST A LITTLE, PERHAPS TWO OR THREE TENTHS OF A MINUTE, AND YOU WANT AN ANSWER AS CLOSE AS POSSIBLE TO THE ONE THE TEST WRITER GOT. YOU CAN SAFELY IGNORE THE DOT AND THE EXTRA CORRECTION TO HC IN REAL LIFE, TOO. IT'S HARD TO GET AN ACCURATE SEXTANT ALTITUDE OF BODIES AT VERY HIGH ALTITUDE, SO THE FANCY EXTRA CORRECTION DOESN'T MEAN MUCH.
WHEN YOU'RE STUDYING NAVIGATION ON YOUR OWN THERE IS A GOOD CHANCE THAT SOME ONE WHO REALLY LIKES NAVIGATION WILL WANT TO HELP YOU. IT'S EASY ENOUGH TO IGNORE THE DOT, BUT TRY IGNORING A KEEN NAVIGATOR WHO WANTS TO SHOW OFF. ASK HIM TO SHOW YOU THE DOUBLE SECOND DIFFERENCE TRICK AFTER YOU'VE PASSED YOUR TEST.
AS YOU WORK WITH THE 229-IN THIS PART OF THE SIGHT REDUCTION PROCESS, YOU WILL ALSO NEED THE DIFFERENCE BETWEEN THE Z ON YOUR LINE AND THE Z IMMEDIATELY BELOW IT. THESE HAVEN'T BEEN WORKED OUT AND PRINTED FOR YOU. YOU HAVE TO DO THOSE YOURSELF. YOU WILL ALSO HAVE TO NOTE WHETHER THE DIFFERENCE BETWEEN Z'S IS (+) OR (-). YOU DETERMINE THE SIGN OF THE DIFFERENCE BY NOTICING WHETHER THE Z IS INCREASING OR DECREASING AS YOU WORK DOWN THE COLUMN. IF Z IS GETTING BIGGER, THE DIFFERENCE IS PLUS. IF IT'S GETTING SMALLER, DIFFERENCE GETS A MINUS SIGN.
PLEASE NOTICE THAT BOTH HC AND Z SOMETIMES INCREASE AND SOMETIMES DECREASE GOING DOWNWARD IN THE COLUMN AND THAT ONE CAN BE INCREASING WHILE THE OTHER IS DECREASING.
ARE YOU READY FOR YOUR FIRST PLUNGE INTO 229? WE'LL MAKE IT EASY BY ASKING YOU TO FIND HC AND Z FOR WHOLE DEGREES OF LATITUDE DECLINATION. WHEN YOU HAVE GAINED SOME FAMILIARITY WITH THE BOOK, WE WILL DISCUSS INTERPOLATION.
USE THE 229 TO FIND HC AND Z FOR THE FOLLOWING: VIEW IMAGE 43
IT IS IMPORTANT TO POINT OUT AT THIS TIME THAT THE z (AZIMUTH ANGLE) THAT YOU OBTAIN FROM 229 IS NOT THE BEARING THAT YOU WILL NEED WHEN YOU PLOT YOUR LINE OF POSITION ON A CHART. THAT BEARING (CALLED THE AZIMUTH) IS FOUND BY USING THE FORMULAS THAT APPEAR ON EVERY PAGE OF THE TABLE. WE'LL DISCUSS THIS CONVERSION OF Z TO ZN AT THE END OF THIS CHAPTER.
YOU WILL PROBABLY NEVER USE AN HC OR A Z AS YOU FIND IT PRINTED IN THE TABLE UNDER A WHOLE DEGREE OF LHA AND LATITUDE AND ACROSS FROM A WHOLE DEGREE OF DECLINATION. FOR SIGHT REDUCTION YOU'LL ENTER THE TABLES WITH WHOLE DEGREES OF LATITUDE AND LHA, OF COURSE. THAT'S WHY YOU CHOOSE AN ASSUMED POSITION. BUT THE ODDS AGAINST THE DECLINATION TURNING OUT TO BE AN EXACT DEGREE WITH NO MINUTES AND TENTHS ARE ASTRONOMICAL. IT WILL BE NECESSARY TO INTERPOLATE FOR HC AND Z TO ACCOUNT FOR THE MINUTES AND TENTHS OF DECLINATION.
BEFORE YOU CAN PROCEED WITH PRACTICE PROBLEMS IN FINDING HC AND ZN, YOU MUST KNOW HOW TO DO A STRICT, FORMAL INTERPOLATION. EVEN IF YOU KNOW HOW TO INTERPOLATE, PAY CLOSE ATTENTION TO THE NEXT LESSON. YOU MAY FIND THAT THE METHOD YOU KNOW ISN'T PRECISE ENOUGH FOR TEST PURPOSES. YOU MAY ALSO FIND THAT THE METHOD SHOWN HERE IS EASIER THAN THE ONE YOU KNOW.