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    Re: New compact backup CELNAV system
    From: Gary LaPook
    Date: 2009 Feb 27, 10:34 -0800

    "Question: Are "plastic transparency sheets" the punched clear
    inserts you see enclosing often-used pages in a three-ring binder or
    are they something specially made for ink-jet printing?"

    They are something different, the ink won't stick to the sheets you are talking about. The kind I use are 3M "Transparency Film for Ink Jet Printers, code number "CG3480" available at Fryes, Office Depot, etc.

    Don't try using them with a laser printer they melt and gunk up the printer, don't ask.


    --- On Fri, 2/27/09, Hewitt Schlereth <hhew36@gmail.com> wrote:
    From: Hewitt Schlereth <hhew36@gmail.com>
    Subject: [NavList 7456] Re: New compact backup CELNAV system
    To: NavList@fer3.com
    Date: Friday, February 27, 2009, 9:13 AM

    Gary, this is really neat.

    I've printed the scales on 8X11.5 paper and it looks like my printer
    has the proportions correct.

    Question: Are "plastic transparency sheets" the punched clear
    inserts you see enclosing often-used pages in a three-ring binder or
    are they something specially made for ink-jet printing?

    I ask because I have nothing transparent here to hand and I really
    want to try this.

    Thanx, Hewitt

    On 2/24/09, glapook@pacbell.net <glapook@pacbell.net> wrote:
    > On Feb 24, 12:06 pm, <glap...@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
    > >
    > > [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
    > >
    > > 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
    > >
    > > [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
    > >
    > > 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
    > > 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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