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Another method for doing celestial navigation computations on a 
calculator is to use the Bygrave formulas instead of the Sine-Cosine 
formulas.

Just use the standard Bygrave formulas in the three step process 
following along on the form I have posted and using my terminology.

First calculate co- latitude and save in a memory on the calculator. If 
you are using a value for hour angle that is not a whole number of 
degrees you might want to make the conversion to decimal degrees and 
save it since it will we used twice. If you are using whole degrees then 
this step is not useful.

Then you calculate "W" using the formula:

tan W = tan D / cos H

and sum it to the memory where you have saved co-latitude which is then 
X and then make any adjustment necessary to convert X to Y. (If you are 
just making trials you can avoid this step by your choice of the trial 
values.) There is no reason to store W itself since it is not used 
again. You can then convert W to degree and minute format to compare 
with the Bygrave derived value.


Then you compute azimuth angle using the formula:

tan Az = (cos W / cos Y ) x tan H.

If you want you can also convert Az to degree and minute format to 
compare with the Bygrave.

The last step is to calculate altitude with the formula:

tan Hc = cos Az x tan Y.

Then convert to degree and minute format to compare.






(When entering values in the format of degrees minutes seconds, change 
decimal minutes to seconds, 6 seconds per tenth of a minute, in your 
head  before punching in the assumed latitude, declination  and hour 
angle if necessary.)

Using whole degrees for declination, assumed latitude and hour angle, 
using a TI-30 with only 3 memory locations the key strokes are:

90
-
Assumed Lat
=
STO 1  (co-latitude stored in memory 1)

---------------------------------------

Declination
tan
/
H
cos
=   
inv
tan    (computed W)
SUM 1 (X or Y now stored in memory 1)(change X to Y if necessary)
cos
/
RCL 1  (recalls Y from memory 1)
cos
x
H
tan
=
inv
tan (computed Azimuth angle)
cos
x
RCL 1 (recalls Y from memory 1)
tan
=
inv
tan  (computed altitude)


2nd
D.D - DMS (changed Hc in decimal degrees to degrees, minues and seconds)


-----------------------------------------------------------------------------------------------------


Too use the standard Sine- Cosine formulas for sight reduction you can 
use the following keystroke sequence which works with a TI-30 which has 
only three memories.



Assumed Lat
2nd DMS-D.D (changes to decimal degree format)
STO 1  (stored A. LAT in 1)

Declination
2nd DMS-D.D
STO 2   (stored DEC in 2)

GHA
2nd DMS-D.D
-
Assumed Longitude
2nd DMS - D.D
= (computed LHA)
STO 3   (LHA stored in 3)

COS
X
RECALL 2  (recalled declination)
COS
X
RECALL 1  (recalled Assumed latitude)
COS
+
RECALL 1 (recalled A. LAT)
SIN
X
RECALL 2  (recalled DEC)
SIN
=
2nd SIN  (ARCSIN, computed Hc)

2nd D.D-DMS (changes decimal degree Hc to degree-minute-second so it
can be written down)

2nd DMS - D.D  (changes it back to decimal degrees)

COS
1/x   (converts COS Hc to SEC Hc)
X
RECALL 3  (recalled LHA)
SIN
X
RECALL 2  (recalled DEC)
COS
=
2nd SIN  (ARCSIN, computed Z)

Since the azimuth angle (Z) will be in the range of 0° to 90° there are 
situations where it will not
be obvious whether the body's azimuth (Zn) is in the northern or 
southern semicircle.

In most situations there is no ambiguity as to which quarter the Zn lies 
since you know the approximate direction you are looking when you take 
the sight. The problem arises because the azimuth angle is limited to 
the range of zero to 90 degrees and when the Zn is near east or west the 
correct Zn might fall either side of the line so there is an ambiguity 
in converting from azimuth angle to Zn.

One easy rule to apply first is that if the declination is greater than 
the latitude then the azimuth can never be in the opposite semicircle. 
To generalize this rule, if the declination has the same name as the 
latitude and the declination is greater than the latitude, then you 
start with the direction of the elevated pole (the nearer pole) when 
converting from azimuth angle to azimuth (Zn.)

The second rule to apply is that if the declination is contrary name 
then the Zn must be in the opposite semicircle. To generalize this rule, 
if the declination and the latitude have contrary names then you start 
with the direction of the depressed pole (the further pole) when 
converting from azimuth angle to Zn.

These two rules take care of most of the cases, especially for 
navigators in low latitudes.

The remaining ambiguity concerns situations in which the declination is 
the same name as the latitude but is less than the latitude. In this 
situation the azimuth of the body will be both north and south of the 
east - west line during part of each day. There is no simple rule to
deal with this situation.




For you to understand the Bygrave method look at:

http://sites.google.com/site/fredienoonan/other-flight-navigation-information/modern-bygrave-slide-rule 


You may want to make a flat Bygrave for yourself in case your batteries 
go dead in your calculator.

gl



Alan wrote:
>
> Kermit:
>
> Yes, I got note 13924, and opened the link you provided.
>
> I don't understand the stuff that is there, likely beyond my limited 
> understanding of programming and computers, but thanks anyhow.
> I sent it off to my younger brother who is an EE, with years of 
> experience including programming, and such. Possibly he can clarify 
> the thing for me.
> If not, I will simply use the 11C as I've been doing, run individual 
> problems as they come up, akin to the old slide rule, a device I once 
> had some familiarity with.
>
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