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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?

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