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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
picture.

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
navigation.

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
crystal-clear.

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
thing.

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
ambiguities.

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
difficult.

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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