Because a Geographical Position of a star is normally
thousands of miles from our position, the circle of position
is very large and the small piece that interests us - the one
near our position - may be considered a straight line, orthogonal
to the Azimuth of the star. This line is called the Line
of Position or LOP (fig. 9).
fig. 9 - Line of Position
We managed, from the measured altitude of a star at a certain time
and our assumed position, to draw a line of position. We know that
our actual position is somewhere along this line. To determine this
point we can draw another line, for another star. The point were they
intercept each other is our position - or our Astronomical Position.
fig. 10 - Triangle formed by the intersection
of three lines of Altitude
Normally, the navigator should repeat this procedure
for yet another star, just to be sure. Since measurements are
affected by minor imprecisions, the three lines will probably
not intercept in a single point, resulting in a small triangle.
Our position is probably in some point of this triangle (fig.
10). The smaller the triangle, the better. We usually assume
that our Astronomical Position is in the center of the triangle.
In figure 10 above, we can see how three circles of position determine
3 Lines of Position r1, r2 and r3.
In traditional celestial navigation the determination of
a Line of Position involves the computation of the GP of the
star (GHA and declination) using the Nautical Almanac and
the solution of the Position Triangle PXZ, formed by
the terrestrial pole (P), the GP of the star (X) and the assumed
position of the navigator (Z) (see fig.11).
This solution, using tables, yields the Calculated Altitude
and the Azimuth of the star. The difference, in minutes of
degree, between the calculated altitude and the altitude of
the star measured with the sextant is the distance between
the line of position and our assumed position - the error
Delta of our estimate. This can be away or towards
the star.
fig. 11 - Triangle of Position
PXZ
Using Navigator software, the GP of a star and the triangle
of Position are solved by the computer using formulas. All you will
have to do is enter the sextant reading (date, time and altitude),
name of the star and the assumed position (latitude and Longitude).
Determination of the Astronomical Position
It's not necessary to draw the lines of position when using Navigator
software. But let's see how this is done using pencil and
paper:
Plot your assumed position.
Using a parallel ruler, draw a line passing on the assumed position,
in the direction of the Azimuth of the star.
Over this line, measure the error Delta of the estimate - in
the direction of the star or contrary to it - according to the
sign of the Delta.
Draw the line of position, orthogonal to the Azimuth, at this
point.
Detailed Nautical Charts are usually only available for places
near the shore. When in high seas, we normally don't have charts
with the adequate scale to plot our position. Special plotting paper
is used instead.
When navigating using Navigator software, the computer determines
the altitude lines interceptions and calculates the astronomical
position. A simplified map is drawn, showing the parallels, meridians,
lines of altitude and the astronomical position.
The sextant
The sextant is an instrument that measures angles.
Fig 12 shows a schematic sextant. The eyepiece is aligned
to the small mirror, which is fixed in the frame of the
instrument. This mirror is half transparent. By the transparent
part, the navigator can see the horizon directly. The small
mirror also partially reflects the image from the big mirror,
where you see the star. The big mirror is mobile and
turns with the arm of the sextant. Doing that, we change
the angle between the two mirrors. The altitude of the star
is measured in the scale. There is a drum to make the
fine adjustments. Whole degrees are read in the scale and the
minutes in the drum.
fig. 12 - The Sextant
sextant working model (requires Flash
5.0 plug-in)
The sextant has two sets of filters (or shades) to eliminate
the excess of light, especially when observing the Sun. The use
of two or more filters in front of the big mirror is necessary
when observing the Sun. Serious eye injuries
will result from observing the Sun without filters, even for a
brief period.
fig. 13 - Image of the Sun in the sextant
When looking through the eyepiece and adjusting the sextant,
you will see something like figure 13, to the left. Sextant
readings must be made with the sextant in the vertical position.
Inclining (rocking about the axis of the eyepiece) the adjusted
instrument slightly, the image of the celestial body describes
a small arc that touches the horizon in a point near the center
of the mirror. In this situation, the angle is ready to be
read in the instrument scale.
Altitude corrections
But before we can use this apparent reading in our calculations,
some corrections must be made, in order to obtain the true observed
altitude. These corrections are: 1) the height of the eye, 2) semi
diameter of the body (only for Sun and Moon), 3) instrumental error,
4) atmospheric refraction and 5) parallax (only for the Moon).
Since most of these corrections depend only on the selected celestial
object and altitude, they are performed automatically by Navigator
software. The only information you will have to provide
to the program are the height of the eye (a.k.a. Dip)
and the instrumental error. The application of these
corrections to the instrumental altitude gives the corrected
altitude, the one used in calculations.
An observer located in a high place will see a
star with an altitude bigger than other at sea level, in the
same location. This error is called height of the eye (Dip).
fig. 14 - Error due to the height of the
eye (Dip)
fig. 15 - eyepiece image of the sextant
index error
The sextant index error (IE) is due to a small
misalignment of the scale of the sextant (the "zero"
of the instrument). To read the index error, adjust the scale
to 0°00.0' and point towards the horizon. In fig. 15 left we
can see this error. Turn the drum until the horizon forms a
single line (fig. 15 right). Then you can read the index error.
fig. 16 - Sign of the instrumental error
The index error can be positive or negative, as
shown in fig. 16. The index correction has opposite signal (i.e.
must be subtracted from altitude if positive and vice-versa)
Parallax error is illustrated in fig.17. Since the
navigator is not in the Earth's center, but in its surface,
the apparent object position is below the true geocentric
position.
Parallax is only meaningful for the Moon. Other objects are
so far, their parallax is very small.
fig. 17 - Parallax in altitude error
Nautical Almanac data is tabulated for the centers
of the celestial objects. For the Sun and Moon, however, it's
easier to measure the altitude of the lower part of the body,
as illustrated in fig. 18. This is known as the lower limb.
Of course a correction must be applied in order to obtain the
altitude of the center of the body. This correction is called
semi diameter. Sometimes, the upper limb is also used.
The first edition of this book was published
in 1802. It has been said to be one of the few things a
sailor absolutely needs before going to the sea, the other
things being a "Bible and the mother's blessing".
Overtime, some of the original Bowditch's
celestial navigation text was replaced by more modern subjects,
like radar and radio communications. Unfortunately, the
Lunar calculation section is one thing that was removed,
apparently in 1914. If you have a copy of this text, I would
like to read it !.
>> "Memento Vagnon
de la Navegacion Astronomique"
by François Meyrier
Good celestial navigation course
in French, with step-by-step approach.
>> "Navegação
astronômica" por Geraldo
Luiz Miranda de Barros
Edições Marítimas - 250 páginas