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    Lunars using Bennett
    From: Dave Walden
    Date: 2008 Apr 3, 14:20 -0700
    LUNARS USING BENNETT
    While browsing the archives, (much of what is said sinks in slowly in my case and I learn/appreciate new things with each rereading) I ran across the following paragraph included in a longer message from Alex:
     
     
    " [NavList 3357] Re: sight reduction tables
    From: eremenko---purdue.edu
    Date: 5 Oct 2007 19:24
    ...
    Of the tables I like very much the Complete on Board,
    because it contains much more than sight reduction
    tables and almanac, in a very small, convenient package.
    The main disadvantage from my point of view is that
    you cannot do the Lunars with Complete on Board.
    ...
    Alex."
     
    Being a great fan of "Complete on Board ..." by Bennett, and having outlined a method a while ago that seems to let one calculate azimuth using the included "Sight Reduction Tables" to the same accuracy as altitude, I wondered what would be possible for lunars.
     
    It would seem so far that the answer is that lunars can be done using only Bennett (ok, plus sextant and watch) for separations of up to 90 degrees. ("Complete on Board ..." is what it is. Most of us understand what that is and appreciate it. I make no claim that the method below is a "good" way to clear a lunar distance, only that it is a possible one. The effort was undertaken as an intellectual challenge. I hope it might prove interesting to others. If not, hit DELTETE.)
     
    In two sentences: Use the altitude calculation procedure with different variables to calculate the true lunar distance. Run the appropriate variables "backwards" then frontwards through the altitude calculation to produce the cleared lunar distance.
    I leave the derivation as an exercise to the reader. (If it doesn't work for you, I may need to do more derivation work.) Below is an example to demonstrate the method. I stop with the cleared lunar distance. There are a number of ways to use the true and cleared distances to calculate time. I leave that to the reader.
     
    (Errors, of course, may still remain. I invite comment.)
     
    Problem: At 1500 UT on 2 April 2008, from 39N, 77W, LD apparent center to center is 46:30. The true and cleared lunar distances from the moon to the sun are desired. Assume the altitudes are not observed. Use from calculation.
     
    Obtain GHA and DEC of sun and moon from Bennett:
     
    sun GHA=15 hr * 15 deg/hr + 179:6 + 0:3 - 360:0 (v=4) (instructions page 4, data page 43, correction page 66)
    =44:9
    sun DEC=N 4:59 + 0:14 (d=23)
    =N 5:13
    moon GHA=225:0 + 223:51 - 0:87 - 360:0 (v=-173) (data page 75)
    =87:24
    moon DEC=S 13:20 + 0:43 (d=85)
    =S 12:37
     
     
    Use steps 13 - 18 from page 6 to calculate sun and moon altitudes:
     
    SUN
    LHA=GHA-Long
    13 LHA   327 9 -> LHA 14296
    14 DR Lat 39 0 -> LAT  1427
    15 Dec    5 13 -> DEC    23
    16 (theta)28 48<- SUM 15746 -> RES 12369
    17 Lat~Dec33 47             -> L~D 16885
    18 Comp Alt 45 2            <- ALT 29254
    Moon
    13 LHA   10 24 -> LHA 27173
    14 DR Lat 39 0 -> LAT 1427
    15 Dec   12 37 -> DEC 138
    16 (theta)9 3 <- SUM 28738 -> RES 1245
    17 Lat~Dec 51 37          -> L~D 37908
    18 Comp Alt 37 29         <- ALT 39153
     
    APPARENT ALTITUDES
    sun ctr =45: 2 + 0: 1=45: 3 bennett refrac page 66, no dip,HP=58 from page 75)
    moon UL =37:29 - 0:29=37: 0 bennett moon corrections pg 101-102
    moon LL =37:29 - 0:61=36:28
    moon ctr=avg =36:44
     
     
    Use steps 13-18 to calculate true lunar distance:
     
    GHA difference between sun and moon replaces LHA.
    DEC sun replaces DR Lat.
    DEC moon replaces Dec.
    DEC~DEC replaces Lat~Dec.
    90-LD true replaces Comp Alt.
     
    13 del GHA  43 15 -> LHA 11298
    14 dec Sun   5 13 -> LAT 23
    15 dec Moon 12 37 -> DEC 138
    16 (theta)  42 36 <- SUM 11459 -> RES 26400
    17 dec~dec 17 50                -> L~D 4805
    18 90-LDtr 43 28               <- ALT 31205
     
    LD true = 90:0 - 43:28 = 46:32
     
     
    Use alt calculation "backwards" then frontwards. First "backwards":
     
    90-LD apparent replaces Comp Alt.
    moon apparent~sun apparent replaces Lat~Dec.
    sun apparent replaces Dec.
    moon apparent replaces DR Lat.
    intermediate variable PHI replaces LHA.
     
    18 90-LDap 43 30              -> ALT 31164
    17 app~app  8 19              -> L~D  1052
    16 (theta) 45 40 <- SUM 10713 <- RES 30112
    15 sun app 45 3  -> DEC 1966
    14 moon ap 36 44 -> LAT 1253
    13 PHI     62 6  <- LHA 7494
     
    Now frontwards:
    PHI replaces LHA.
    sun true replaces DR Lat.
    moon true replaces Dec.
    moon true~sun true replaces Lat~Dec.
    90-LD cleared replaces Comp Alt.
     
    13 PHI     62 6 -> LHA 7493
    14 sun tr  45 2 -> LAT 1964
    15 moon tr 7 29 -> DEC 1309
    16 (theta) 5 26 <- SUM 10766 -> RES 29836
    17 tr~tr 7 33                -> L~D   867
    18 90-LDcl 43 52             <- ALT 30693
     
    LD clear = 90:0 - 43:52 = 46:8
     
    FINAL Cleared Lunar Distance 46 Degrees 8 Minutes.
     
     
     
    From Frank's calculator:
     
    INPUT:
     
    Lunars Calculator (ver. 4)
    SETUP:
    DR Lat: 39N
    DR Lon: 77W
    I.C.:
    Temperature: 50 °F
    Pressure (SL):29.80inches Hg
    Height of Eye: feet
    Body: Sun
    ALTITUDES:
    (Leave Blank to Calculate)
    Body:
    Moon:
    LUNAR:
    Greenwich Date: April 2 , 2008
    Time: 15: :
    Distance: 45:58 NEAR (n.b. 46:30 center to center less the 2 SD's)
    Options:
    Ignore Oblateness
    Ignore Flattening
    -FER, Centennia Software, June 2004.
     
     
    OUTPUT:
     
    GHA Dec HP
    Moon: 87° 24.7' -12° 38.0' 57.68
    Sun: 44° 08.5' 5° 13.7' 0.15
    True LD: 46° 33.4'
    Moon Apparent Altitude: 36° 42.6'
    Moon Altitude correction: -0° 45.0'
    Sun Apparent Altitude: 45° 02.5'
    Sun Altitude correction: 0° 00.9'
    No flattening correction.
    No oblateness correction.
    Cleared LD: 46° 07.4'
    Error in Lunar: -26'
    Error in Longitude: -13° 01.2'
    -FER, Centennia Software, June 2004.
     
     
    Surprisingly, final watch errors may be within 3 minutes or so.
    Looks too good to be true.  Is it?


    You rock. That's why Blockbuster's offering you one month of Blockbuster Total Access, No Cost
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