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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

ASSUMED POSITION AND LOCAL HOUR ANGLE

  NOW WE'VE REACHED THE POINT IN THE PROBLEM WHERE YOU CAN PUT THE ALMANAC AWAY AND GO THE NEXT BOOK, THE SIGHT REDUCTION TABLES. WE'LL BE USING "SIGHT REDUCTION TABLFES FOR MARINE NAVIGATION," WHICH IS THE DEFENSE MAPPING AGENCY'S PUBLICATION NUMBER "229." WE'LL JUST CALL IT WHAT EVERYONE ELSE CALLS IT, "229."

  HERE IS A PART OF A PAGE FROM 229. YOU WILL NOTICE THAT IT CONTAINS A MULTITUDE OF COMPUTED ALTITUDES (HC) AND AZIMUTH ANGLES (Z) AT VARIOUS LATITUDES (16° TO 22° ON THIS EXCERPT) AND DECLINATIONS. VIEW IMAGE

  THE LITTLE "d" BETWEEN THE HC AND Z COLUMNS IS SIMPLY THE DIFFERENCE BETWEEN DESCENDING HC'S IN A COLUMN. IT CAN BE USED FOR INTERPOLATION.

  IN THE UPPER RIGHT HAND CORNER YOU WILL FIND A NEW TERM, "LHA," WHICH STANDS FOR LOCAL HOUR ANGLE.

  THE LOCAL HOUR ANGLE (LHA) IS THE DISTANCE THE BODY (AND ITS G.P.} IS WEST OF YOUR POSITION, MEASURED IN DEGREES. YOU MUST KNOW THE LHA OF THE BODY IN ORDER TO FIND THE PROPER PAGE OF 229 TO GET HC AND Z.

  LET'S SUPPOSE YOUR LHA WAS EXACTLY 64°, YOUR LATITUDE WAS EXACTLY 21° , AND THE DECLINATION WAS EXACTLY 5°. ENTERING THE TABLE WITH THESE VALUES, HC WOULD BE 22° -06.9’ AND Z WOULD BE 104°.9.

  UNFORTUNATELY, LIFE IS NEVER THAT EASY. YOUR LATITUDE, LONGITUDE, AND DECLINATION WILL NOT BE "ROUND" NUMBERS. DOES THAT MEAN YOU WILL HAVE TO DO A LENGTHY INTERPOLATION TO FIND THE CORRECT HC AND Z?

  NO, BECAUSE WE CAN AVOID INTERPOLATION FOR LATITUDE AND LHA. INSTEAD OF WORKING WITH DR POSITION, WE CHOOSE AN "ASSUMED LATITUDE" IN WHOLE DEGREES, AND AN "ASSUMED LONGITUDE" WHICH WILL FORCE THE LHA TO COME OUT IN WHOLE DEGREES.

  THESE TWO MAKE THE ASSUMED POSITION (A.P.): THE WHOLE DEGREE OF LATITUDE CLOSEST TO THE DR LATITUDE, AND A LONGITUDE CHOSEN TO MAKE THE LHA COME OUT IN WHOLE DEGREES.

  CHOOSING THE ASSUMED LATITUDE IS EASY. JUST PICK THE CLOSEST WHOLE DEGREE TO THE DR LATITUDE:

VIEW IMAGE 21

  IF THE DR LATITUDE SHOULD END IN 30.0', EXACTLY HALFWAY BETWEEN WHOLE DEGREES, PICK THE EVEN WHOLE DEGREE. FOR EXAMPLE:

33°30.0' N BECOMES 34° N

32°30.0' N BECOMES 32° N

  CHOOSING THE ASSUMED LONGITUDE REQUIRES A LITTLE MORE EFFORT BECAUSE IT MUST PRODUCE AN LHA THAT IS WHOLE DEGREES.

  AN LHA IS THE DISTANCE THAT THE BODY IS WEST OF YOUR POSITION MEASURED IN DEGREES. IF YOU WERE AT 20° W LONGITUDE, AND A BODY HAD A GHA OF 60°, HOW MANY DEGREES WEST OF YOUR POSITION IS IT? IT IS 40 DEGREES BEYOND (WEST) OF YOUR POSITION. YOU OBTAINED THAT ANSWER BY SUBTRACTING YOUR 20° W LONGITUDE FROM THE GHA OF 60°. VIEW IMAGE 22

  IF YOUR POSITION IS 135° W AND THE GHA OF A CELESTIAL BODY IS 292° , WHAT IS ITS LHA? AGAIN, THE SOLUTION IS FOUND BY SUBTRACTING YOUR LONGITUDE FROM THE GHA OF THE BODY TO GET AN LHA OF 157° . DO YOU SEE A PATTERN DEVELOPING?

  IF YOUR DR IS IN WEST LONGITUDE, YOU CAN FIND LHA BY SUBTRACTING THAT LONGITUDE FROM GHA:

GHA- WEST LONGITUDE = LHA

292°- 135° = 157°

 

  HERE IS ONE THAT USES THE SAME FORMULA BUT IS A BIT TOUGHER. YOUR LONGITUDE IS 95°W AND THE BODY HAS A GHA OF 25°. WHAT IS THE LHA? WHEN YOU SET UP YOUR FORMULA YOU GET. : 

25°- 95° = LHA

   TO UNDERSTAND WHAT'S HAPPENING, VIEW IMAGE 23

   LHA IS THE DISTANCE A BODY IS WEST OF YOUR POSITION. GOING WEST FROM A POSITION AT 95° W, THE LHA IS 290°. IN ORDER TO DO THE MATH WHEN THE GHA IS SMALLER THAN THE LONGITUDE, YOU SIMPLY ADD 360° TO THE GHA TO GET: 385°- 95°= 290° LHA

  TO REVIEW, IF YOUR POSITION IS IN WEST LONGITUDE, YOU FIND LHA BY SUBTRACTING YOUR LONGITUDE FROM THE GHA.

  NOW LET'S SEE WHAT HAPPENS IF YOU ARE IN EAST LONGITUDE. SUPPOSE YOU WERE AT LONGITUDE 20°E AND A BODY HAD A GHA OF 50° . HOW MANY DEGREES IS IT WEST OF YOUR POSITION? VIEW IMAGE 24. THE LHA IS FOUND BY ADDING THE LONGITUDE TO THE GHA.

TO FIND LHA WHEN YOUR LONGITUDE IS EAST: GHA + EAST LONGITUDE = LHA  50° + 20° = 70° LHA.

IF YOUR LONGITUDE IS 65° E AND THE GHA IS 218° , WHAT IS THE LHA? FOLLOWING OUR FORMULA, THE ANSWER IS 218°+ 65° = 283. LHA.

IF YOUR LONGITUDE IS 175°E AND THE GHA 350° , WHAT IS THE LHA? USING THE FORMULA, WE GET 350° + 175° = 525°LHA.

  AN LHA CANNOT BE MORE THAN 360°, HOWEVER! IF THE ANSWER IS MORE THAN 360° , YOU SIMPLY SUBTRACT 360° TO GET AN LHA OF 165°.

  WE NOW KNOW THE RULES FOR FINDING AN LHA, WHICH WILL BE USED TO GET THE RIGHT PAGE IN 229:

  THE FORMULAS ARE SIMPLE ENOUGH, BUT WE STILL NEED TO FIGURE A WAY TO MAKE A LONGITUDE LIKE 117°-16.3'W PRODUCE A "WHOLE" NUMBER WHEN WE COMPUTE AN LHA. TO DO THIS, WE MUST FIND AN "ASSUMED LONGITUDE" THAT WILL FORCE THE LHA TO COME OUT AS A WHOLE NUMBER.

THE WORKFORM FOR FINDING AN LHA TO USE IN THE 229 FOR SIGHT REDUCTION LOOKS LIKE THIS: VIEW IMAGE 25

FOLLOW THIS PROBLEM THROUGH.

PROBLEM: YOUR DR POSITION IS LAT 21°43.2' N, LONG. 117°16.3' W. THE GHA OF THE SUN AT THE MOMENT OF THE OBSERVATION WAS 150° 54.7'. WHATIS THE  ASSUMED LATITUDE, ASSUMED LONGITUDE, AND LHA?

SOLUTION: THE ASSUMED LATITUDE IS EASY. ALL YOU DO IS ROUND TO THE NEAREST WHOLE DEGREE: 22° N. FOR THE ASSUMED LONGITUDE AND LHA, SET UP THE WORKFORM AND FILL IN WHAT YOU KNOW: VIEW IMAGE 26

  BECAUSE YOUR LONGITUDE IS WEST, YOU WILL SUBTRACT TO FIND THE LHA. BECAUSE YOU WANT LHA TO COME OUT AS A WHOLE NUMBER, THE ASSUMED LONGITUDE MUST HAVE THE SAME MINUTES AND TENTHS AS THE GHA. YOU CAN THEREFORE CONTINUE TO FILL OUT THE FORM AS SHOWN BELOW:  VIEW IMAGE 27

  WE NOW KNOW THAT THE MINUTES AND TENTH OF LEA WILL BE 54.7° WHAT ABOUT THE WHOLE DEGREES OF THE ASSUMED LONGITUDE? YOU WANT THE ASSUMED LONGITUDE TO BE AS CLOSE AS POSSIBLE TO YOUR ACTUAL DR LONGITUDE, IN THE SAME WAY THAT WE CHOSE ASSUMED LATITUDE THAT WAS WITHIN 30' OF THE ACTUAL LATITUDE. CAN YOU TELL WHICH OF THESE POSSIBLE ASSUMED LONGITUDES IS CLOSEST TO YOUR DR LONGITUDE?

  1. ACTUAL DR LONGITUDE: 117° -16 .3’W

  2. POSSIBLE ASSUMED LONGITUDES: 

  • 116°-54.7’, 

  • 117°-54.7’, 

  • and 118°-54.7’.

  YOU COULD CHECK BY COMPARING THE POSSIBLE LONGITUDES WITH THE DR LONGITUDE BY ARITHMETIC TO SEE WHICH WAS WITHIN 30’. MOST TIMES, HOWEVER, IT'S EASIER TO CHECK IT WITH A LITTLE SKETCH.  VIEW IMAGE 28

  DRAW THREE LINES TO REPRESENT MERIDIANS OF LONGITUDE. LABEL THE ONE IN THE MIDDLE FOR THE WHOLE DEGREE OF YOUR DR LONGITUDE AND THE OTHER TWO ACCORDINGLY. MARK YOUR DR LONGITUDE. IN THIS PROBLEM, ASSUMED LONGITUDE HAS TO HAVE 54.7' MARK 116°54.7'W, AND ALSO 117° 54.7'W. IT'S PRETTY OBVIOUS THAT 116° 54.7’W IS THE ONE CLOSER TO THE DR.

  YOU CAN SKETCH THIS QUITE ACCURATELY IF YOU HAVE LINED NOTEBOOK PAPER AND USE A SCALE OF FOUR SPACES PER DEGREE OF LONGITUDE. IF IT'S A REALLY CLOSE CALL, YOU COULD DO IT ON THE EDGE OF A CHART OR PLOTTING SHEET, OR USE ARITHMETIC.

NOW WE CAN COMPLETE THE SOLUTION: VIEW IMAGE 29

HERE'S ANOTHER ONE:

PROBLEM: AT DR POSITION LAT. 15°12.5' SF LONG. 75°43.81 E YOU OBSERVE THE SUN AND FIND THE GHA TO BE 315°47.2’. FIND THE ASSUMED LATITUDE, ASSUMED LONGITUDE AND THE LHA IN WHOLE DEGREES TO USE IN 229.

SOLUTION: THE ASSUMED LATITUDE IS 15° S. NOW SET UP THE FORM TO DO THE ASSUMED LONGITUDE AND LHA: VIEW IMAGE 30

  BECAUSE OUR DR LONGITUDE IS EAST, WE MUST ADD IT TO GHA. WHAT NUMBER OF MINUTES MUST BE ADDED TO THE GHA TO RESULT IN 60.0’? THE ANSWER IS OBTAINED BY SUBTRACTING THE GHA MINUTES FROM 60.0’ (60.0’- 47.2’ = 12.8’) .NOW WE CAN CONTINUE FILLING IN THE FORM, BUT BE SURE TO CARRY THE 60.0’OVER TO THE WHOLE DEGREE COLUMN: VIEW IMAGE 31

  THAT TAKES CARE OF THE MINUTES OF ASSUMED LONGITUDE. YOUR SKETCH AND DECIDE ON THE WHOLE DEGREES. NOW MAKE YOUR SKETCH AND DECIDE ON THE WHOLE DEGREE:  VIEW IMAGE 32

 IT ISN'T SO OBVIOUS FROM THE SKETCH WHICH OF THE A.P.'S IS CLOSER TO THE DR. IF WE HAD A PLOTTING SHEET AND A PAIR OF DIVIDERS HANDY, WE COULD PLOT IT MORE ACCURATELY AND COMPARE THAT WAY. WITH NO TOOLS HANDY, LET'S TRY THE ARITHMETIC:

   DR LONG.   75° 43.8E ?                ? ASS LONG. 76° 12.8E

? ASS LONG. 75° 12.8E                        DR LONG. 75° 43.8E

                       DIFF 31.0'                                        DIFF 29.0' THIS ONE IS CLOSER

NOW YOU CAN COMPLETE THE SOLUTION: VIEW IMAGE 33

REMEMBER, AN LHA CANNOT BE MORE THAN 360° . IN THIS CASE, WE WE MUST SUBTRACT 360° TO GET A FINAL ANSWER OF 32°.

 

HERE ARE SOME PRACTICE PROBLEMS WITH THE DR POSITION AND GHA GIVEN. CHOOSE THE A.P. AND FIND THE LHA IN WHOLE DEGREES. VIEW IMAGE 34

ANSWERS: VIEW IMAGE 35

  WITH THE ASSUMED POSITION (A.P.) YOU HAVE CHOSEN TO CREATE A NICE, CLEAN LHA WITH NO MINUTES TO COMPLICATE THINGS, YOU ARE READY TO ENTER THE WONDERFUL WORLD OF PUBLICATION H.O. 229. AND WHAT A WORLD IT IS! IN H.O. 229 YOU WILL FIND COUNTLESS "COMPUTED ALTITUDES" AND "AZIMUTHS" ALL WORKED OUT FOR THE ASSUMED POSITION YOU CHOSE.

  WHAT'S AN ASSUMED POSITION, YOU ASK? IMAGINE THAT THERE IS AN INVISIBLE NAVIGATOR AT THE ASSUMED POSITION YOU CHOSE. HE TAKES A SEXTANT READING OF THE SAME CELESTIAL BODY YOU ARE WORKING WITH AND CONVERTS IT TO AN HO. THAT CORRECTED SEXTANT READING TAKEN FROM THE ASSUMED POSITION IS A COMPUTED ALTITUDE. NATURALLY, THE COMPUTED ALTITUDE WILL BE DIFFERENT FROM THE HO YOU ACTUALLY GOT TAKING THE SIGHT FROM YOUR BOAT, BECAUSE THE A.P. IS IN A SLIGHTLY DIFFERENT LOCATION THAN YOUR BOAT'S DR POSITION. THAT'S FINE, BECAUSE THE DIFFERENCE BETWEEN THE TWO READINGS (HO AND COMPUTED ALTITUDE) WILL GET US ONE STEP CLOSER TO FIXING OUR POSITION. IN ADDITION TO THIS COMPUTED ALTITUDE, H.O. 229 WILL ALSO GIVE YOU A DIRECTION, WHICH IS CALLED AN "AZIMUTH." THIS AZIMUTH IS THE DIRECTION FROM THE A.P. TO THE CELESTIAL BODY.

COMPUTED ALTITUDE AND AZIMUTH