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    Re: Fw: Chichester's Calculations
    From: Gary LaPook
    Date: 2008 Dec 09, 01:54 -0800

    Now we will look at Chichester's illustration of the old method on page 
    234 to compare it with the short method already discussed.
    
    What he is doing with this example is using the traditional Haversine- 
    Cosine method of calculating Hc and azimuth. The formulas used for this 
    were derived from the standard Sine - Cosine formulas and, in fact, uses 
    the same method and formula for calculating azimuth.
    
    The formula for calculating Hc is:
    
    hav ZD = hav LHA cos Lat cos Dec + hav (Lat ~ Dec)
    
    (Lat ~ Dec means the difference between latitude and declination, 
    subtracting the smaller from the larger if of the same name and adding 
    if of different names)
    
    (ZD is zenith distance)
    
    so Hc  = 90º - ZD
    
    For calculating azimuth we use
    
    sin Z = (sin LHA cos dec ) / cos Hc
    
    usually rearranged into the more convenient form of
    
    sin Z = sin LHA cos dec sec Hc
    
    Since csec ZD is the same as sec Hc
    
    we can rearrange this formula to
    
     sin Z = sin LHA cos dec csec ZD
    
    
    Chichester used these formulas and solved them using logarithms by using 
    this format:
    
                                                    
    Hc                                                      Az
    
    LHA ___________     log hav LHA  ___________        log sin LHA  
    ______________          
    Lat ___________        log cos Lat   ____________ 
    Dec  __________        log cos Dec + ___________       log cos Dec  
    ______________                   
                                                          
    ______________               
    
                                        inv hav    _______________  
    (L ~ D) ___________ hav (L~D)  + ____________ 
    
                 
    89-60                                                                          
                                                 
    (89-60 is a convenient way to write 90º when you will be subtracting)
    
    ZD -  ___________<<<<  inv hav  _____________>>> log csec ZD  
    +___________      
                                                   
     
    Hc ____________           
                       ..
    Ho-_____________
    A______________                     Z ______________<<<...inv log sin 
    _______________
    
    
    In contrast to the previous example using H.O 249, using the haversine - 
    cosine allows the use of the DR position and does not require the 
    selection of an AP that produces whole degrees of latitude and whole 
    degrees of LHA.
    
    I have attached a marked up version of page 234. "A" shows the 
    computation of Hc and "B" the computation of azimuth. Chichester uses 
    the DR latitude and determines LHA from the DR longitude. He transforms 
    the usual LHA into a value less than 180º by subtracting from 360º. This 
    has also been called hour angle, H.A., and angle "t".
    
    Using Chichester's numbers:
    
    
                                                    
    Hc                                                      Az
    
    LHA _35-48.5_E___   log hav LHA _  8.97548____          log sin LHA    
    9.76716_______          
    Lat ___37-08.5_N_    log cos Lat   ___9.90154____ 
    Dec  __08-02_N____log cos Dec + __9.99572___            log cos Dec  
    _9.99572_______                   
                                                          
    =__8.87274___               
    
                                                    hav    __.07459____  
    (L ~ D) 29-06.5_____ hav (L~D)  + __.06315_____
    
              89-60       
     
    ZD -  43-34______<<<<  inv hav  =__.13774___>>>  log csec ZD  
    +_10.16166____      
                                                   
     
    Hc __46-26_____           
                     
    Ho-__46-23___________
    
    A____3 away___                     Z ___57_________<<<...inv log sin 
    ___9.92454____________
    
    
    This method required referring to nine different pages in the log tables 
    including interpolating three times (I used Nories Air Tables). It 
    required subtracting Dec from Lat;  adding three five digit numbers 
    twice; adding two five digit numbers one time; and finally subtracting 
    ZD from 90º.
    
    This compares to two pages of H.O. 249 and one addition of the two digit 
    minutes correction.
    
    So take your choice.
    
    
    I have attached these nine pages from Norie's so you can follow along 
    with the computation. I have also attached a text document of this 
    explanation since the column format of emails usually gets distorted.
    
    
    
    gl
    
    
    
    Gary J. LaPook wrote:
    > I am attaching a marked a up version of page 235, page 29 from Volume 3 
    > of H.O. 249 and the correction table from that volume. To make it easier 
    > to follow this explanation I have marked up page 235 with red boxes 
    > labeled A through G.
    >
    > "A" show his computation of GMT or Zulu time for entering the Almanac. 
    > 12:11:22 is his watch time and the watch is obviously set to GMT. 02:30 
    > + 1/2 is the correction for his watch error which is running that many 
    > minutes _slow_ on GMT, Adding these two numbers produces the GMT of the 
    > observation of 12:13:52 (he dropped the extra half second.)
    >
    > "B" shows the computation of LHA _for _entry into H.O. 249. Taking the 
    > entry of the sun's GHA (Greenwich Hour Angle) for 12:00:00 GMT he takes 
    > out the GHA of 000º 53.1'. since the sight was taken 13 minutes and 52 
    > seconds after 12:00 o'clock you look in the increments table in the 
    > Nautical Almanac or in the Air Almanac and take out 3º 28' additional 
    > for that extra time which you add to the GHA at 12:00: to find the GHA 
    > of the Sun at the time of the observation to be 004º 21'. Chichester 
    > automatically added 360º to this and wrote down 364º21' to make it 
    > easier for the next step of subtraction his assumed longitude. Since the 
    > DR position is 43N -  25W ("C") he choses an assumed longitude of 25º 
    > 21' so when he subtracts this from the GHA he ends up with a whole 
    > number of degrees of LHA which is 339º.
    >
    > "C" shows the DR position.
    >
    > "F" shows the declination of the sun taken out of the Almanac at the 
    > same time that the value of GHA was obtained.
    >
    > "D" show the computation of Hc using H.O. 249. Looking on page 29 of 
    > volume 3 for latitude 43 and declinations 15-29 same name, we go down 
    > the column for 20º declination and come across from 339º LHA and we take 
    > out the tabulated Hc of 61º 02', the "d" correction value of +50 and the 
    > azimuth of 136. The tabulated Hc is for 20º declination exactly. Since 
    > the declination of the sun was actually 6 minutes more at the time of 
    > observation we must correct this tabulated Hc for this difference. We go 
    > to the correction table and under the 50 column (the "d" value) we read 
    > down to the 6 minutes of extra declination line and extract 5' which we 
    > add to the tabulated Hc of 61º 02' to determine the actual Hc of 61º 07'.
    >
    > "E" shows the computation of Ho. Starting with the HS (sextant altitude) 
    > of 60º 50' we add the correction  for semi diameter (this is obviously a 
    > lower limb shot), subtract refraction and subtract dip which Chichester 
    > has combined into one correction of + 13'. (The SD alone is + 16 and 
    > refraction for this Hs is - 1' making a +15'. Since Chichester uses +13' 
    > he is including a dip correction of -2'.) On the next line he applied 
    > the Index correction (IC) of + 1 to arrive at the Ho of 61º 04'.
    >
    > "G" shows him subtracting the Ho from Hc to arrive at the intercept of 3 
    > away since Ho was less than Hc.
    >
    >
    > The other examples he gives are done the same way although it is 
    > interesting that in three of the examples he also combines the IC with 
    > the other corrections to Hs writing down +14 total correction.
    >
    > I will get to page 234 tomorrow.
    >
    > gl
    >
    >
    >
    >
    >
    >
    >
    > Beverley Maxwell wrote:
    >   
    >> Gary,
    >> Thank you.  I am sending page 235, which Chichester refers to as the 
    >> short method, and page 234 he calls the long old method.
    >>  
    >> Frank M.
    >>
    >> ------------------------------------------------------------------------
    >> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
    >> 
    .
    >>
    >> ------------------------------------------------------------------------
    >> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
    >> 
    .
    >> ------------------------------------------------------------------------
    >>
    >>
    >> ------------------------------------------------------------------------
    >>
    >>     
    >
    >
    >
    > >
    >   
    
    
    --~--~---------~--~----~------------~-------~--~----~
    Navigation List archive: www.fer3.com/arc
    To post, email NavList@fer3.com
    To unsubscribe, email NavList-unsubscribe@fer3.com
    -~----------~----~----~----~------~----~------~--~---
    
    Now we will look at Chichester's illustration of the old method on page 234 to 
    compare it with the short method already discussed.
    
    What he is doing with this example is using the traditional Haversine- Cosine 
    method of calculating Hc and azimuth. The formulas used for this were derived 
    from the standard Sine - Cosine formulas and, in fact, uses the same method 
    and formula for calculating azimuth.
    
    The formula for calculating Hc is:
    
    hav ZD = hav LHA cos Lat cos Dec + hav (Lat ~ Dec)
    
    (Lat ~ Dec means the difference between latitude and declination, subtracting the 
    smaller from the larger if of the same name and adding if of different names)
    
    (ZD is zenith distance)
    
    so Hc  = 90� - ZD
    
    For calculating azimuth we use
    
    sin Z = (sin LHA cos dec ) / cos Hc
    
    usually rearranged into the more convenient form of
    
    sin Z = sin LHA cos dec sec Hc
    
    Since csec ZD is the same as sec Hc
    
    we can rearrange this formula to
    
     sin Z = sin LHA cos dec csec ZD
    
    
    Chichester used these formulas and solved them using logarithms by using this format:
    
                                Hc                                      Az
    
    LHA ___________        log hav LHA  ___________        log sin LHA  ______________ 
             
    Lat ___________        log cos Lat   ____________ 
    Dec  __________        log cos Dec + ___________       log cos Dec  ______________ 
                      
                                        ______________               
    
                           inv hav    _______________  
    (L ~ D) ___________    hav (L~D)  + ____________ 
    
            89-60
    
    ZD -  ___________<<<<  inv hav  _____________>>> >>>>>>log csec ZD  +___________  
        
                                                   
     
    Hc ____________           
                       
    Ho-_____________
    A______________                     Z ______________<<<>>>>>>>> log csec ZD 
    +_10.16166____      
                                                   
     
    Hc __46-26_____           
                     
    Ho-__46-23___________
    
    A____3 away___                     Z ___57_________<<<<< I am attaching a marked a up version of page 235, page 29 from Volume 3 
    > of H.O. 249 and the correction table from that volume. To make it easier 
    > to follow this explanation I have marked up page 235 with red boxes 
    > labeled A through G.
    >
    > "A" show his computation of GMT or Zulu time for entering the Almanac. 
    > 12:11:22 is his watch time and the watch is obviously set to GMT. 02:30 
    > + 1/2 is the correction for his watch error which is running that many 
    > minutes slow on GMT, Adding these two numbers produces the GMT of the 
    > observation of 12:13:52 (he dropped the extra half second.)
    >
    > "B" shows the computation of LHA for entry into H.O. 249. Taking the 
    > entry of the sun's GHA (Greenwich Hour Angle) for 12:00:00 GMT he takes 
    > out the GHA of 000� 53.1'. since the sight was taken 13 minutes and 52 
    > seconds after 12:00 o'clock you look in the increments table in the 
    > Nautical Almanac or in the Air Almanac and take out 3� 28' additional 
    > for that extra time which you add to the GHA at 12:00: to find the GHA 
    > of the Sun at the time of the observation to be 004� 21'. Chichester 
    > automatically added 360� to this and wrote down 364�21' to make it 
    > easier for the next step of subtraction his assumed longitude. Since the 
    > DR position is 43N -  25W ("C") he choses an assumed longitude of 25� 
    > 21' so when he subtracts this from the GHA he ends up with a whole 
    > number of degrees of LHA which is 339�.
    >
    > "C" shows the DR position.
    >
    > "F" shows the declination of the sun taken out of the Almanac at the 
    > same time that the value of GHA was obtained.
    >
    > "D" show the computation of Hc using H.O. 249. Looking on page 29 of 
    > volume 3 for latitude 43 and declinations 15-29 same name, we go down 
    > the column for 20� declination and come across from 339� LHA and we take 
    > out the tabulated Hc of 61� 02', the "d" correction value of +50 and the 
    > azimuth of 136. The tabulated Hc is for 20� declination exactly. Since 
    > the declination of the sun was actually 6 minutes more at the time of 
    > observation we must correct this tabulated Hc for this difference. We go 
    > to the correction table and under the 50 column (the "d" value) we read 
    > down to the 6 minutes of extra declination line and extract 5' which we 
    > add to the tabulated Hc of 61� 02' to determine the actual Hc of 61� 07'.
    >
    > "E" shows the computation of Ho. Starting with the HS (sextant altitude) 
    > of 60� 50' we add the correction  for semi diameter (this is obviously a 
    > lower limb shot), subtract refraction and subtract dip which Chichester 
    > has combined into one correction of + 13'. (The SD alone is + 16 and 
    > refraction for this Hs is - 1' making a +15'. Since Chichester uses +13' 
    > he is including a dip correction of -2'.) On the next line he applied 
    > the Index correction (IC) of + 1 to arrive at the Ho of 61� 04'.
    >
    > "G" shows him subtracting the Ho from Hc to arrive at the intercept of 3 
    > away since Ho was less than Hc.
    >
    >
    > The other examples he gives are done the same way although it is 
    > interesting that in three of the examples he also combines the IC with 
    > the other corrections to Hs writing down +14 total correction.
    >
    > I will get to page 234 tomorrow.
    >
    > gl
    >
    >
    >
    >
    >
    >
    >
    > Beverley Maxwell wrote:
    >>
    >> Gary,
    >> Thank you.  I am sending page 235, which Chichester refers to as the 
    >> short method, and page 234 he calls the long old method.
    >>  
    >> Frank M.
    >>
    >> ------------------------------------------------------------------------
    >> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
    >> .
    >>
    >> ------------------------------------------------------------------------
    >> Start your day with Yahoo!7 and win a Sony Bravia TV. Enter now 
    >> .
    >> ------------------------------------------------------------------------
    >>
    >>
    >> ------------------------------------------------------------------------
    >>
    >
    >
    >
    > --~--~---------~--~----~------------~-------~--~----~
    > Navigation List archive: www.fer3.com/arc
    > To post, email NavList@fer3.com
    > To unsubscribe, email NavList-unsubscribe@fer3.com
    > -~----------~----~----~----~------~----~------~--~---
    >
    
    

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