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    Re: New compact backup CELNAV system
    From: Gary LaPook
    Date: 2009 Feb 24, 12:28 -0800
    I just noticed that the forms I used for the illustration used Bygrave's notation using "y" (lower case y) for the first intermediate value while my explanation uses 'W" for this value so remember this equivelence when reqding the explanation. Sorry for any confusion. (I switched to using "W" to avoid confusion, DUH) The Bygrave form included uses the "W" notation.

    gl

    --- On Tue, 2/24/09, glapook{at}PACBELL.NET <glapook{at}PACBELL.NET> wrote:
    From: glapook{at}PACBELL.NET <glapook{at}PACBELL.NET>
    Subject: [NavList 7415] New compact backup CELNAV system
    To: "NavList" <NavList@fer3.com>
    Date: Tuesday, February 24, 2009, 12:13 PM



    On Feb 24, 12:06 pm, <glap...{at}PACBELL.NET> wrote:
    > There are often posts on the Navlist regarding using celestial as a backup
    to GPS and finding a simple way to do this. I think I have found a method that
    is simple, self contained, takes up little space, needs no almanac or sight
    reduction tables and requires no batteries. This method is meant for those who
    already know celestial navigation and provides a very compact and self-contained
    system needing no books, almanacs or tables.
    >
    > The first part of the kit is simply the long term almanac from H.O. 249
    along with the Polaris tables and the Precession & Nutation  table to allow
    the use of the tabulated coordinates for the stars for a long period of time.
    Since this almanac only covers the sun and the stars no provision is made for
    handling planets or the moon but that is not that important for a
    "backup" celestial method.. However, if you have the Nautical Almanac
    then you can work with these bodies as well. I have included a form to use with
    the long term almanac which also includes  tables for correction of
    observations. The long term almanac consists of nine pages from H.O. 249 plus a
    computation form which can be printed back to back on only five sheets of paper.
    >
    > The main part of this method relies on my adaptation of the long extinct
    Bygrave slide Rule which is elegant in its simplicity but which produces
    altitudes and azimuths within in one or two minutes of arc and takes less than
    two minutes to do the computation. I have included three PDF files that can be
    printed out to create a working copy of my adaptation which consists of three
     sheets containing the modified scales which can be hole punched and kept in a
    thin three ring binder (or in an envelope) with the long term almanac and extra
    forms for recording the computation. The Cotangent scale should be printed out
    on paper and can be sealed in a plastic sheet for good durability. The Cosine
    scale and the Vernier are printed out on plastic transparency sheets in an
    inkjet printer and can also be sealed in additional plastic sheets for
    durability. The entire backup method is contained on a total of only eight
    sheets, five for the almanac and three for the Bygrave slide rule!
    >
    > When using the Bygrave slide rule the azimuth and altitude are calculated
    in three steps using the same manipulations of the slide rule for each step. I
    will first describe the use of the normal cylindrical Bygrave slide rule which
    utilizes a cursor or pointers to align the scales. I have also enclosed a copy
    of the original Bygrave instruction book. The Bygrave has also been discussed on
    the Navlist.
    >
    > We first calculate the intermediate value "y" (lower case y)
     which is found by the formula:
    >
    > tan y =  tan declination / cos H          (H = Hour angle)
    >
    > This is the formula listed in the Bygrave manual but, in fact, the slide
    rule does the calculation by modifying this formula to allow the use of the
    cotangent scale. The actual manipulation of the slide rule uses the re-arranged
    formula of:
    >
    > cotan y  =  cotan declination x cos H
    >
    > You accomplish this computation by setting one of the pointers (or the
    cursor on my copy) to zero on the cosine scale and while holding it there rotate
    the cosine scale and slide it up or down on the cotangent scale so that the
    other pointer (or cursor) is aligned with the declination on the cotangent
    scale. Now, holding the cosine scale still, rotate the pointer (cursor) to point
    at the hour angle (H) on the cosine scale and then read out "y" from
    the other pointer (cursor) where it points on the cotangent scale.
    >
    > Next you find the second intermediate value,"Y" (upper case Y)
    by adding "y" to co-latitude (if latitude and declination have the
    same name) or by subtracting "y" from co-latitude (if of opposite
    names.)
    >
    > [I kept the original nomenclature from Bygrave so that the original
    literature can be followed . I think it is confusing to use the same letter of
    the alphabet for two variables, "y" lower case y, and "Y"
    upper case Y, and I don't know why Bygrave chose this system. I prefer to
    replace "y" with "W" in the formulas and have done so on my
    forms.]
    >
    > Next we find azimuth with the formula :
    >
    > tan Az  =  (tan H x cos y ) /  cos Y
    >
    > which is re-arranged into the form:
    >
    > cot  Az = (cotan H / cos y  ) x cos Y
    >
    > Using the same manipulations as before, set one pointer to "y"
    on the cosine scale and the other pointer on H on the cotangent scale, move the
    cursor to "Y" on the cosine scale and read out azimuth from the other
    pointer on the cotangent scale.
    >
    > The third step calculates altitude, Hc. using the formula:
    >
    > tan Hc = cos Az x tan Y
    >
    > with the formula re-arranged into the form:
    >
    > cot  Hc  = cot Y / cos Az
    >
    > set one pointer to Az on the cosine scale with the other pointer to
    "Y" on the cotangent scale. move the pointer to zero on the cosine
    scale and read out Hc from the other pointer on the cotangent scale.
    >
    > I have developed an even simpler implementation of the Bygrave, one that
    is very easy to make since it doesn't require concentric tubes. I made this
    by printing out the cotangent scale twice on a piece of paper. I then printed
    the cosine scale in red on a transparent sheet so that the cosine scale can be
    placed directly on top of the cotangent scale and aligned much like a normal
    slide rule. We follow the same steps as already described but it is even simpler
    since no cursor needs to be used. I will illustrate how easy it is to use with
    an example. I have include a form and pictures of the scales showing this
    computation.
    >
    > Since the Bygrave doesn't require that the latitude or the LHA to be
    whole degrees  you can compute Hc for your D.R. position but for convenience in
    this example we will use 34º N for latitude, for declination, 14º N and the
    LHA is 346º.
    >
    > Look at illustration number 1 (i.jpg) which shows the form to be used with
    this simplified model of the Bygrave. The top of the form is used to compute
    hour angle, "H", in the range of 0º through 90º. You enter the LHA
    in the proper column and make the computation. You can see in our example we
    have placed the LHA, 346º, in the column for LHAs in the range of 270º to
    360º. The form shows that in this case we subtract LHA from 360º to find
    "H", hour angle, in this example, 14º.  We have carried this 14º
    down to the "H" blank on the form and we have entered declination and
    latitude in the appropriate blanks.(2.jpg)
    >
    > [The scales on the original Bygrave ran from 20' up to 89º 40'
    and then back in the reverse direction from 90º20' to 179º40'. My
    simplified version eliminates the second set of numbers keeping the scale less
    cluttered. The original Bygrave allowed hour angles of 0º to 180º east and
    west (which had been normal celestial practice prior to the introduction of the
    concept of LHA) but because my version eliminated the second set of numbering on
    the scales it is necessary to get hour angle, "H", into the range of
    0º and 90º which is accomplished on the top part of the form. Other changes
    were also necessary because of my simplification of the scales and they will be
    pointed out later.]
    >
    > (To avoid confusion I have switched to using "W" to replace
    Captain Bygrave's lower case "y.")
    >
    > Next we subtract the latitude from 90º to form co-latitude. If we were
    using our D.R. latitude we would subtract it from 89º 60' since this
    notation makes it easy to subtract degrees and minutes. In our example the
    co-latitude is 56º. We use the top of the form to determine if we will be
    adding or subtracting the intermediate value of "W" to co-latitude.
    This is determined by the column of the LHA and by the names of the latitude and
    declination. In our example we can see that we will be adding "W"
    since LHA is in the last column and the names of the latitude and the
    declination are the same. Just circle the "+" mark at the bottom of
    this column and also place a "+" mark on the "W" line under
    the co-latitude line.(3,4 and 5.jpg)
    >
    > [With the original Bygrave you always added if latitude and declination
    had the same name and subtracted if of opposite names. Because of my simplified
    scales you must subtract "W" if same name and LHA between 90º and
    270º, and the form accomplishes this. This puts you at exactly the same place
    on the scales as with the original Bygrave. You still always subtract if of
    opposite names.]
    >
    > Next we will follow the zig-zag diagram along the right edge of the form
    to find the value of "W". The left side of the zig-zag indicates the
    cosine scale and the right side indicates the cotangent scale. The same process
    is used three times and values are always taken off from the cotangent scale,
    never from the cosine scale. The zig-zag tells us to line up 0 (zero) on the red
    cosine scale with "D" (declination) on the black cotangent scale. We
    then look at "H" (hour angle) on the red scale and take out
    "W" from the adjacent cotangent scale. (6.jpg)
    >
    > On both scales the numbering is above and to the right of the marks on the
    scales. I have aligned the red cosine scale slightly below the black cotangent
    scale for clarity. We can see that the red zero is lined up directly below the
    black tick mark for 20º (the declination). You usually use visual interpolation
    for the minutes of declination. Now, without allowing the scales to shift, we
    locate "H" on the red scale and read out "W" from the
    adjacent black scale, in this case 20º 33', and write this value on the
    "W" line below the co-latitude. (7.jpg)
    >
    > Next we find he second intermediate value, "Y", by first
    determining "X" by adding "W" to co-latitude, in this
    example "X" = 76º 33'. Then, following the rule on the form,
    since "X" &lt; 90 then "Y" = "X" and we carry
    it down to the "Y" line. If "X" > 90 we would subtract
    "X" from 180º (or from 179-60) to form "Y".
    > [I introduced this new intermediate variable "X" so that the
    resulting "Y" will be less than 90º which allowed me to keep the
    scales less cluttered since the reverse numbering, 90º20' to 179º 40',
    found on the original Bygrave is not needed.](8 and 9.jpg)
    >
    > We now follow the second zig-zag to find Az . (Note we start with the
    value computed at the last step, "W.") (10.jpg) Using  the same
    manipulation as the first time, we line up "W" on the red scale with
    "H" on the black scale . We than look at "Y" on the red
    scale and take out Az from the adjacent black scale. Looking at illustration 11
    (11.jpg), we have lined up red 20-33 with black 14 by aligning the 20-30 mark
    slightly to the left of the "14" on the cotangent scale. (Remember at
    this point on the red scale the short tick marks show 1/2 degree or 30'.)
    Next we located "Y" on the red scale (76-33) and take out the Az from
    the black scale. (12.jpg) (The red 76-30 mark is lightly to the left of the
    black 45 tick mark and the red 76-40 is aligned with the black 45-20 so by
    visual interpolation the red 76-33 will align with the black 45-09, the Az. (At
    this point on the scales each tick mark is 10'.)
    >
    > Next enter the Az on the form  and compute Zn by applying the rules on
    the attached form, in this case the celestial body is to the south east so we
    subtract Az from 180º (179-60) to determine Zn, 134º 51' (or
    134.9º).(13.jpg)
    >
    > [Because of my simplified scales the determination of Zn is different than
    with the original Bygrave since you can only read out Az  in the range of 0º
    to 90º with my version while on the original Bygrave the Az was taken out in
    the range of 0º to 180º. The determination of Zn is usually not a problem in
    real life since you know the approximate direction of the body when you take the
    sight. I have include rules to resolve any ambiguity.]
    >
    > We now follow the last zig-zag to calculate Hc. ( Note we again start with
    the value computed at the last step, Az). (14.jpg) Using the same manipulation
    as before, we line up Az on the red scale with "Y" on the black scale.
     Next we locate zero on the red scale and take out the Hc from the adjacent
    black scale. Looking at no. 15 (15.jpg) we see that we have aligned he red 45
    tick mark slightly to the right of the black 76-30 (each tick mark is 10' on
    both scales) and we visually interpolate red 45-09 with black 76-33. Looking now
    at no. 16 16.jpg), we find the red zero and interpolate on the black scale to
    take out the Hc of 71º 17'and enter it on the form. (17.jpg)
    >
    > Comparing our result with H.O. 249 (see no. 18, 18.jpg) we find the Hc in
    H.O 249 of 71º 17' and Zn of 135º, the same as with the Bygrave.
    >
    > The original Bygrave had a cotangent scale marked every one minute of arc
    while the cotangent scale of my reproduction is marked every ten minutes and in
    some cases every five minutes. The original cosine scale was marked at varying
    spacing and my recreation is also, but not as frequently as on the original. At
    most places on these scales it is sufficiently accurate to visually interpolate.
    However, at places where the scale markings are far apart, visual interpolation
    is not accurate enough. To improve the accuracy of my reproduction I have added
    a scale that assists in interpolation. This scale consists of a small diagram
    with converging numbered lines and should be used on the cosine scale above 80º
    and on the cotangent scale above 80º and below 10º. To use it slide it up
    until the outside lines, labeled zero and ten, fit between two marks on the main
    scales and then use the intermediate lines for the interpolation. In most cases
    it is not necessary to use this vernier. If the scales were more finely divided
    this Vernier could be dispensed with completely.
    >
    > There are some unusual cases that require slightly different procedures
    and all of these special cases are described on the form. If "H" is
    less than 1º or greater than 89º (actually 89º 15' on my version) simply
    assume a longitude to bring "H" within the range of the scales. The
    intercept will be longer but perfectly usable for practical navigation.
    >
    > If the computed azimuth is greater than 85º the computed altitude will
    lose accuracy even though the Az is accurate. For azimuths in this range even
    rounding the azimuth up or down one half minute can change the Hc by ten
    minutes. So you use the azimuth but you compute altitude by interchanging
    declination and latitude and then doing the normal computation. You discard the
    azimuth derived during the computation of altitude and use the original azimuth.
    >
    > When declination is less than 55' on my version (less than 20' on
    the original) you can't compute "W" because you start the process
    with declination on the cotangent scale. In this case, Bygrave says to use the
    same process as when the azimuth exceeds 85º, you simply interchange
    declination and latitude and compute altitude. But Bygrave didn't tell us
    how to calculate azimuth in this case. In my testing I have found a method that
     produces quite accurate azimuths. You simply skip the computation of
    "W" and simply set "W" equal to declination. The  worst
    case I have found is that the azimuth is within 0.9º of the true azimuth but
    most are much closer. If the declination is less than one degree and the
    latitude is also less than one degree, follow this procedure and also assume a
    latitude equal to one degree. After you have computed the Az you then follow the
    same procedure discussed above for azimuths exceeding 85º by interchanging the
    latitude and declination and then computing  Hc.
    >
    > Another rare possibility is that "Y" will exceed 89º 15'
    after adding "W" to co-declination so it won't fit on the scale.
    The simple way to handle this situation is to assume a latitude so the
    "Y" does fit on the scale even though the resulting intercept is
    longer but still usable.
    >
    > An extremely unlikely case (I only mention it to be complete) is that
    "W" exceeds the range of the cotangent scale, 89º15', so cannot
    be computed in the first step of the process.  This can only happen when
    shooting one star, Kochab, which has a declination of 74º13' north and then
    only if "H" exceeds 87º 20', an extremely unlikely event.
    >
    > I am attaching a revised form to use with the Bygrave slide rule. This
    form steps you through the computation and contains
    >  the rules for the special cases. The special cases are likely to come up
    only very rarely in practice.
    >
    >  The first rule for H &lt; 1º or H > 89º only involves LHAs
    covering 4 degrees out of 360º (LHA in the ranges of 0 -1, 89-91, 269-271, and
    359-360) so only occurs by chance very rarely and these can be avoided if sights
    are preplanned as is the normal procedure for flight navigation. Worst case, you
    have to change the time of the observation by four minutes.
    >
    > Rule 3 covers the case when Y exceeds 89º which covers a range of two
    degrees out of a possible 180º so is also very rare. Co-lat is in the range of
    0-90 and W is also in the same range so X comes in the range of 0 -180. If X is
    less than 89 then Y is also less than 89. If X is greater than 91 then Y is less
    than 89 also. Only in the case of X between 89 and 91 will Y exceed 89. This
    situation can't be avoided in advance because you can't predict what the
    value of W will be but just assuming a latitude that differs by one degree
    solves the problem which will result in a longer intercept but one that is still
    usable.
    >
    > The fourth rule deals with cases of bodies bearing almost directly east or
    west and this situation can be avoided by choosing a different body to shoot or,
    if only the sun is available,by waiting a few minutes to allow the azimuth to
    change out of this range.
    >
    > The remaining situation covered by rule two (declinations less than one
    degree) concerns only bodies in the solar system since none of the navigational
    stars have declinations less than one degree. Obviously the most important body
    is the sun and its declination is between 1º north and 1º south for five days
    in March and again in September so this situation can't be avoided and this
    is the most important special case. The special rule handles it nicely and the
    Hc is completely accurate. The computed azimuth is an approximation but is never
    more than one degree different than the actual azimuth and is usually much
    closer. Since you can use your D.R. for the A.P. the intercepts are short and
    this slight inaccuracy in the azimuth will not make a noticeable difference in
    the LOP.
    >
    > So give it a try and let me know what you think.
    >
    > Gary J. LaPook
    >
    > Linked File:http://www.fer3.com/arc/imgx/1.JPG
    > Linked File:http://www.fer3.com/arc/imgx/2.JPG
    > Linked File:http://www.fer3.com/arc/imgx/3.JPG
    > Linked File:http://www.fer3.com/arc/imgx/4.JPG
    > Linked File:http://www.fer3.com/arc/imgx/5.JPG
    > Linked File:http://www.fer3.com/arc/imgx/6.JPG
    > Linked File:http://www.fer3.com/arc/imgx/7.JPG
    > Linked File:http://www.fer3.com/arc/imgx/8.JPG
    > Linked File:http://www.fer3.com/arc/imgx/9.JPG
    > Linked File:http://www.fer3.com/arc/imgx/10.JPG
    > Linked File:http://www.fer3.com/arc/imgx/11.JPG
    > Linked File:http://www.fer3.com/arc/imgx/12.JPG
    > Linked File:http://www.fer3.com/arc/imgx/13.JPG
    > Linked File:http://www.fer3.com/arc/imgx/14.JPG
    > Linked File:http://www.fer3.com/arc/imgx/15.JPG
    > Linked File:http://www.fer3.com/arc/imgx/16.JPG
    > Linked File:http://www.fer3.com/arc/imgx/17.JPG
    > Linked File:http://www.fer3.com/arc/imgx/18.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Almanac-form.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Backup-navigation-tables.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Bygrave-form-.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Vernier.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Cosine-Scale.pdf
    > Linked File:http://www.fer3.com/arc/imgx/Cotangent-.pdf



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