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    Calculating accurate apparent-angles between stars
    From: George Huxtable
    Date: 2003 Jan 6, 15:37 +0000

    Fred Hebard wishes to compute accurate apparent-angles between stars, to
    check his sextant. Bruce Stark says that there are problems in using his
    lunar tables for that purpose.
    Fred has another choice, of doing the job by a pocket-calculator which has
    trig capability, rather than using tables. One problem with using tables is
    that it involves using logs, and so the standard trig formulae need to be
    bent and twisted to avoid negative numbers appearing, because the log of a
    negative number is meaningless. That's how haversines came into the
    Nowadays a pocket calculator (or a computer) provides all the trig
    functions, multiplications and divisions in a moment at the touch of a
    button to far more accuracy than a navigator will ever need. This makes
    logs and tables unnecessary except for traditional reasons of "doing the
    job the old way" (which I respect, and don't wish to belittle). If pocket
    calculators had been invented first, logs would never have been needed for
    So I suggest Fred gets hold of a pocket-calculator with trig functions, for
    a few dollars.. Make sure it's set to handle decimal degrees, not radians
    or grads. If it's programmable, so much the better. It should calculate to
    at least six significant decimal places: I think all do much better than
    that. (A computer which can run a program such as a version of Basic, or
    even a spreadsheet, will do the same job, but in some cases it will then be
    necessary to express angles in radians rather than degrees.)
    Make a distance observation, of the sextant angle between two stars, at a
    known GMT from a known lat and long.
    Then take from an almanac Dec and GHA for star1, and similarly for star2,
    at the moment of that observation.
    All these quantities need to be converted from degrees-and-minutes to
    decimal degrees, a simple matter.
    It's necessary to get the signs right. Hour Angles are, as ever, measured
    positive Westward. The formulae below expect declination and latitude to be
    positive for North, negative for South, and longitudes as positive
    Westward, to match Hour Angles. (Note that some authorities use the
    opposite convention for longitudes, unfortunately). I think of lat and long
    in terms of Northitude and Westitude, which makes this convention
    Easterly longitudes, such as 1 degree East, can be expressed as -1 degree
    of Westitude, or 359 degrees of Westitude, which mean exactly the same
    Start with star 1.
    Procedure A: converting dec and GHA to alt and az.
    Subtract the observer's Westitude (long.) from the star's GHA to obtain the
    Local Hour Angle (LHA) of the star, which is its Westerly displacement from
    the observer.
    LHA = GHA - long
    Then its altitude is found, using -
    alt = arcsin (sin lat sin dec + cos lat cos dec cos LHA)
    This is the calculated altitude of star1, which we will name alt1. If it's
    negative, the star is below the true horizon.
    The azimuth of star 1 is its direction from the observer, measured
    clockwise from North, in degrees. Note that some astronomical texts,
    notably Meeus, define azimuths from the South, but we navigators differ
    from that.
    az  =  arctan (sin LHA / (cos LHA sin lat -cos lat tan dec))
    The arctan function will give an answer in the region between -90 deg and
    +90 deg, so the following rules must be applied, in order, to resolve the
    1. If the resulting az is negative, add 180 deg. to az.
    2. Also, if LHA is in the range 0 to 180 deg., add 180 deg. to az.
    This will provide an azimuth measured clockwise from true North, 0 to 360 deg.
    In the expression above for az, if the denominator (which is a subtraction)
    gives a result of zero, then an infinity arises which the calculator will
    reject. This occurs for azimuths that are due East and West (90 or 270
    degrees respectively). If that denominator turns out to be exactly zero,
    just replace the calculation of az by choosing an az of 270 when LHA is in
    the range 0 to 180 deg., and an az of 90 otherwise.
    Some calculators and computer programs offer a function named POL, or
    perhaps ATAN2, meant for converting rectangular to polar coordinates, and
    this can provide an easy way of obtaining azimuths. If anyone would like to
    know how to implement this, I will explain on request, as there's a trick
    involved in adapting it to our definition of az.
    If intermediate values need to be written down in the course of any
    calculations above, then I recommend that 6 significant figures should be
    recorded. If a calculator can be programmed, nothing needs to be written
    down until the final result appears.
    Astute readers may recognise in the procedure above a method for obtaining
    calculated altitude and azimuth of any Almanac body from its dec and GHA.
    This can be used for ordinary astronavigation, for defining a position line
    after comparing an observed and corrected sextant altitude. It provides an
    alternative to altitude/azimuth lookup tables, and to a better accuracy.
    Although I have spelled it all out in some detail, it's quite quick and
    easy to implement. on a calculator, computer, or spreadsheet. You can try
    it out and compare the result with your altitude/azimuth tables.
    End of procedure A.
    Taking star 2, repeat procedure A. We now have computed values from the
    almanac data, for alt1 and az1 for star 1, and alt2 and az2 for star 2.
    We need to find the angle between the directions that the two stars appear
    to have in the sky to calibrate a sextant. These are slightly different
    from the computed direction because of the effect of refraction. We have
    had to convert from dec and GHA to alt and az, for each star, because
    refraction, acting in a vertical direction, affects only alt and not az, so
    the refraction correction becomes very simple.
    Usually an altitude observed by a sextant is corrected by subtracting a
    quantity for the refraction, which is taken from a table. Here, however, we
    are converting a calculated altitude to deduce what an observed altitude
    would be: the other way around. So the refraction correction is a small
    angle to be ADDED to the calculated alt.
    The refraction correction in minutes is provided in a table which has as
    its argument the apparent altitude. We can use that same table for
    corrections the other way, entering the same table with our calculated
    altitude. Because the refraction correctons are so small at altitudes
    above, say, 10 degrees, any errors in doing this are quite insignificant.
    Procedure B
    So the next step is to look up alt1 for star 1 in a refraction table,
    convert from minutes to degrees, and ADD it to alt1 to give an apparent
    altitude app1 for star 1.
    End of procedure B
    Do the same for star 2.
    Finally we need to obtain the angle (the angular "distance" in degrees)
    between the apparent positions of the two stars.
    Procedure C
    dist = arc cos (sin app1 sin app2 + cos app1 cos app2 cos(az2 - az1))
    The decimal part of this result can be converted to arc-minutes.
    End of procedure C
    This result can be compared directly with the distance-angle between the
    two stars, as observed by the sextant, as required by Fred Hebard.
    The same set of procedures can be used to check an observer's skill at
    measuring lunar distances, from a known position at a known time, if
    (reversed) parallax corrections for the two bodies are included in
    procedure B, something that's rather easy to do. This set of three
    procedures, A, B, and C, then include the necessary "clearing" of the lunar
    distance (which is not otherwise necessary), providing a result which can
    be compared directly with an observed lunar distance between the centres of
    the bodies, by sextant. Unfortunately, doing the reverse calculation, to
    discover the time and the longitude from a lunar distance, is somewhat more
    Instead of using a lookup table for the refraction correction, it can
    instead be rather simply computed: details on request. Same applies to
    parallax corrections, in the case of a lunar distance.
    The method described here is suitable for electronic calculation only,
    because of the ease with which high-accuracy result can be obtained. It's
    not suitable for any form of hand-calculation.
    George Huxtable.

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