Latitude
MERIDIAN ALTITUDE
It is surprising to us, in these advanced days of nautical science, to
read of our adventurous ancestors of a century ago navigating their
ships to all parts of the known and unknown world with nothing to guide
them but their dead reckoning and the latitude crudely obtained by the
method of meridian altitude. Many of our finest ships, as late as the
first decade of the nineteenth century, sailed to China and back with
no knowledge of their longitude save what the master guessed it to be.
Even in later days much navigating has been done in the less lucrative
trades by mariners who had no knowledge of the method of finding
longitude. It required more time and distance to navigate by latitude
and dead reckoning only, as it was not always safe to lay a course from
an indefinite position directly for the coast. It was the custom in the
old days to keep off soundings until on the latitude of the port of
destination, then steer due west, and whatever the longitude might turn
out to have been the master would sooner or later make the land in the
vicinity of his port.
The first step in obtaining the latitude by meridian altitude is the
measurement with the sextant of the sun’s altitude. This is done when
it reaches its highest point in its course across the sky; this occurs
when it bears due N. or S. true and this moment is local apparent noon.
A few minutes before this time the image of the sun should be brought
to the horizon, and by swinging the lower part of the instrument the
image will be made to swing likewise in an arc; the lowest point of its
lower edge (limb) should then be brought in contact with the horizon as
closely as the circumstances will permit. The image will keep rising
from the horizon, but by using the tangent screw it can be continually
brought back to contact. At noon it will hang, and dip below; the
reading of the sextant at this moment is the meridian altitude.
In working the problem three quantities are used and the navigator must
be familiar with them:
The first is the zenith distance (_z_), which as its name implies is
the sun’s (or stars) distance from the zenith. Zenith is 90° from the
horizon, so the true altitude of the body subtracted from 90° is _z_
the quantity desired.
The second element is the declination (_d_), which is the distance in
degrees, minutes, seconds, of the body either north or south of the
equator. This is taken from the Nautical Almanac.
The third and resulting quantity is the latitude, which is the distance
in degrees, minutes, seconds, of the ship either north or south of the
Equator.
The altitude observed taken with the sextant at noon is corrected for
semi-diameter, parallax, dip, refraction and instrument error (if any
exists). These corrections are explained in detail in _Corrections for
Observed Altitudes_.
The declination of the sun is constantly changing between 23½° N. and
23½° S. This is given in the Nautical Almanac for each two hours of
Greenwich mean time with the difference for each hour given for each
day. So it becomes necessary to ascertain the declination at the moment
of observation, namely, at local noon. This anywhere in the Atlantic
will occur subsequent to Greenwich noon, as the sun (apparently) passes
around the world from the eastward to the westward once a day--24
hours--which corresponds to 360° of longitude. The rate of travel
is therefore equivalent to 15° in an hour. Hence if the sun crosses
Greenwich meridian and five hours later crosses the meridian of the
ship, say in 75° W., the interval is 75 divided by 15, or 5 hours.
During this interval the sun has changed in declination northward or
southward and should be picked out of the Almanac for 5 hours Greenwich
mean time.
When the zenith distance and declination are at hand the latitude is
obtained by a mere algebraic addition, which is, _z_ + _d_ = latitude;
where, if the body bears south the _z_ is marked +, if north it is
marked -; if the declination is south it is marked - and if north it is
+. The result of the addition if - indicates south latitude, if + north
latitude. The meridian altitude of a star, planet or moon is found in a
similar manner. The formula of _z_ + _d_ = latitude, having regard to
signs named as above, is applicable to each. The declination and the
correction of the observed altitude are picked out of the Almanac and
Bowditch tables in a somewhat different manner peculiar to each body.
It is found by many navigators to be more convenient to observe a
body for meridian altitude by time than in waiting for the “dip.” The
altitude is taken at exactly local apparent noon in case of the sun
and the time of meridian passage in the cases of other bodies. This
expedient is especially desirable in observing stars, as the horizon is
not as distinct and the “dip” not so easily detected as with the sun.
In order to secure the local mean time (L. M. T.) of a star’s transit,
the G. M. T. of the star’s transit over the Greenwich meridian is found
in the Nautical Almanac (p. 96) for the first day of the month and
correct for the day by table on next page (N. A.). The ship’s mean time
of transit will be the same, as both sun and star hold their relative
positions as the star moves from Greenwich to the ship’s meridian
except for the small retardation of the sun’s movement over the star’s
movement. This is best found at the foot of page 2, Nautical Almanac,
where the longitude in time gives the correction to be subtracted from
(G. M. T.) of transit which will give the local mean time of transit
at ship--the time to observe the star. An observation of a planet is
similarly handled. The moon is somewhat unreliable owing to its rapid
changes in position and the large corrections necessary to correct the
altitude, and is consequently rather an unpopular body to observe.
However, there are times when she might prove valuable in giving
position when much needed.
In the case of the sun the time of transit is local apparent noon, by
applying the longitude in time gives Greenwich apparent time of local
noon, and corrected for equation of time gives Greenwich mean time of
transit.
It is often necessary to report the latitude at noon very quickly to
the master. This can be accomplished by calculating the problem to a
point where the addition or subtraction of the observed altitude is
all that is necessary to give the latitude. The corrections are applied
in advance by the estimated altitude, and declination corrected by the
estimated longitude. Art. 325, Bowditch, gives the constants to be used
in four different situations.
CIRCUM- OR EX-MERIDIAN ALTITUDE
It frequently happens, especially in the higher latitudes, that an
aggravating mass of cloud drives over the sun or other objects that
you are chasing, with the tangent screw, and it is lost from view
together with all hope of a meridian altitude. But such an unfortunate
occurrence as the loss of the mid-day latitude may be averted by
employing the Circum-meridian sight or Ex-meridian, as our English
cousins call it.
The mariner accustomed to its use “shoots” the sun and notes the time
by chronometer or watch. Or on cloudy days, he would be standing by,
near apparent noon watching for a chance to catch a glimpse of the
object through a rift of cloud, and thereby forestall the loss of his
latitude.
The theory of this observation is extremely simple, being merely to add
to the observed altitude, taken before or after apparent noon when the
sun is being considered, the amount of rise or fall between the time
of sight and the time of culmination, and proceed with this amended
altitude as in an ordinary meridian altitude sight.
The use of this method of obtaining the latitude is restricted to
certain limits. Those who use Bowditch Tables will find themselves
restricted to 26 minutes from the time of transit and a declination of
63°, while Brent’s Ex-meridian Tables allow a greater scope and their
limit of 70° of declination includes many stars that would be otherwise
unavailable. A good guide is to never allow the number of degrees in
the zenith distance to be exceeded by the number of minutes from noon.
In very high altitudes circum-meridians are not to be recommended, and
the higher the altitude, the more accurate must be the time used. This
is plain when it is realized that the lower the sun’s altitude at noon,
the more nearly its diurnal path approaches the line of the horizon;
with the lessening curve of its course, comes a lessening rise near
noon, hence less accuracy is needed in the exact time of sight from
that of transit. In the tropics, however, where high altitudes of the
sun prevail, the clouds do not offer such an element of bother as they
do farther north or south, and there this problem as applied to the sun
loses its popularity.
In practice the use of the tables of Bowditch makes this problem an
exceedingly simple one, requiring but few figures. Table 27 contains
the value of rise of the body for one minute, but as this rise varies
as the square of the interval from noon, it becomes necessary to resort
to another table (26) of constants for a multiplier, in lieu of the
number of minutes from noon. That is, if we should multiply the amount
of rise or fall for 1 minute by the number of minutes from noon, we
would not be taking into account the decreasing rapidity of rise or
the increasing rapidity of fall as the body approaches or leaves the
meridian. But Table 26 provides a multiplier which reconciles this
inequality and gives the proper correction to apply to the observed
altitude.
This quantity is added in every case where the upper transit is
observed but subtracted when a sight is taken below the pole where the
conditions are reversed.
There are several pamphlets and books on the market from which the
correction to the observed altitude may be obtained. All are simple in
form and with their explanations are readily understood. Notable among
these Ex-meridian Tables are those by Capt. Armisted Rust, U. S. N.
The circum-meridian is a reliable method of finding the latitude, but
the time used should be accurate to produce satisfactory results. If,
however, the conditions be favorable, it is not necessary to discard
this observation even if the correctness of the time is somewhat in
doubt, for in Towson’s Ex-meridian Tables is found this note:
“If equal altitudes be taken before and after the meridian passage,
half the elapsed time may be employed as the hour angle for determining
the reduction. Or, when the altitudes before and after noon differ by
only a few minutes, the mean of the two may be reduced by employing
half the elapsed time as the hour angle for reducing the mean altitude.”
In practicing this suggestion it is necessary, in order to preserve
accuracy, to put the vessel on the nearest east or west course during
the run between these equal altitude observations. This is imperative
in a swift vessel.
The stars and planets offer themselves for use in this problem as in
all others, and here they possess special advantages of which the
mariner may well avail himself. Indeed, it may be said in truth that
when a horizon can be obtained the latitude is always available
through this problem.
And right here should be impressed upon the navigator the great
advantage of becoming familiar with the stars, not merely those of
greatest brilliancy, but the “lesser lights” that can be observed.
Among the latter, especial acquaintance should be sought with those
whose right ascensions place them in the gaps between the larger stars,
thus almost the entire heavens are included in the scope of operations,
making the latitude and longitude practically always available,
provided again there is a horizon.
Star charts, planispheres, and globes are for sale everywhere and no
study is more interesting than that of the ways of these celestial
travelers. They appear and pass each day, year after year, until
you consider them as old friends, and, as you come on deck for the
mid-watch, you look for Orion, for instance, the same as you look for
the members of your watch at their proper stations.
But we are off our course. The increasing popularity of the
circum-meridian and its undoubted accuracy when used with time obtained
from a carefully rated chronometer, is breaking the hold of the
time-honored meridian altitude. There is no waiting with cold fingers,
perhaps, for the body to dip for this sight, just shoot the star, note
the time and duck for the chart room to work it up.
The most favorable position of a body for a circum-meridian altitude
is one in which the rise and fall near the meridian are slow. In the
case of the sun, it was explained that a low altitude proved the
best, but, in the case of the stars, we find another condition; those
near the pole, or in other words, of large declinations, describe
such small diurnal circles that here also the change in altitude is
correspondingly small, thus fulfilling a desired condition for the
successful working of this problem. To illustrate this point the reader
is referred to Polaris. Now this star has an extremely small diurnal
circle and it will be remembered that the altitude is for all practical
purposes the same for a half hour either side of the meridian, showing
the extreme slowness of its movement of revolution.
The stars are used in the same way as the sun except, of course, that
the distance from the meridian becomes the star’s hour angle instead of
local apparent time. This is readily obtained as follows: Adding to the
Greenwich mean time the sidereal time of the preceding Greenwich mean
noon (Nautical Almanac), together with the acceleration of Greenwich
mean time (Table 9 Bowditch), gives the right ascension of the
meridian. Taking the difference between this latter quantity and the
right ascension of the star, we have the star’s hour angle, west, if
the right ascension of the meridian is greater than that of the star,
and east, if contrary conditions exist.
The circum-meridian, as well as the straight meridian altitude, is
available for use of stars near the meridian below the pole, and, as
one proceeds into higher latitudes, the pole becomes more and more
elevated, offering thereby more opportunities for practicing this phase
of the problem. The only feature to be remembered in this case is
that the body is higher at the time of a circum-meridian than when it
transits, so the correction to be applied to the observed altitude must
be subtracted (-) in order to obtain the meridian altitude.
The planets, too, are used by the ex-meridian altitude method, but
being wanderers in the heavens their right ascensions and declinations
must be determined for the Greenwich date from the Nautical Almanac.
The amended altitude of any body is assumed to be the meridian altitude
and is used in the familiar formula _z_ + _d_ = latitude (see Latitude
by meridian altitude); but it must be borne in mind that the result is
not the latitude at noon but at the time of sight. If the observation
was made say 9 minutes before noon and the latitude considered to
be the position at local apparent noon as in an ordinary meridian
altitude, there would be an error of 3 miles from the correct position
for a 20-knot steamer.
Another point to be guarded against is that when taking several
altitudes and their corresponding times their mean cannot be obtained
in the ordinary way, but each altitude must be separately reduced and
the mean taken of the results.
It is again necessary to diverge from the subject to impress on the
mariner an urgent warning against anything but the most untiring
vigilance in the care of his chronometer, and the keeping of accurate
time. If this element cannot be depended upon there will be many
hours of anxiety coming to him and probably sooner or later downright
disaster. The almost universal establishment of time signals in all
good-sized sea ports of the world together with radio time signals sent
broadcast allows but little excuse for not obtaining a good rate by the
time a vessel is ready for sea. Every well-known work on navigation
deals with the subject of rating chronometers and so no space will
here be given to it. After reading this talk on one of the most
important and up-to-date observations where so much depends upon the
accuracy of the time, the reader cannot fail to appreciate this earnest
admonishment.
POLARIS
The process of finding the latitude by means of Polaris is valuable,
comparatively short and the result, if the conditions are favorable, is
accurate. We will consider it first in a general way.
The imaginary line representing the earth’s axis, if extended
indefinitely, is presumed to pierce the celestial sphere at the
celestial pole, therefore for an observer standing at our north pole
this imaginary point would be exactly in the zenith and hence 90° from
the horizon just as the pole is 90° from the equator, these amounts
evidently bear a relation to one another. Should the person at the
pole leave his frigid surroundings and proceed toward the equator, he
would note that the pole had dropped lower and lower in the heavens,
precisely in proportion to his progress southward, until at length,
when the equator (latitude 0°) was reached, the pole would be observed
to be exactly in the horizon (altitude 0°). From this it is easy to
deduce the statement that the altitude of the celestial pole is equal
to the latitude of the place of observation.
The object of this problem then is to obtain the altitude of the
celestial pole. This point, unfortunately, is marked by no star of
which a direct altitude may be observed to aid the navigator in
reaching this desired result. There is, however, a star of the 2d
magnitude, called Polaris (because of its proximity to the pole) with
a polar distance of only 1¼°. As all fixed stars are apparently
revolving in circles around the celestial pole, this star joins the
grand procession with its little radius of 1¼°.
It is plain that at no time can this star be more than the amount of
this radius (1¼°) from the pole, and when on the meridian either above
or below the pole the full amount of the radius is subtracted from or
added to the corrected altitude of the star to obtain the true altitude
of the pole. When the star is on a line passing through the pole
and parallel to the horizon at its elongations as it is called, the
altitude is then equal to the latitude, for its elevation is the same
as that of the pole.
It requires 24 hours for this star to complete the small circle of
revolution, the same time required by every star; its movement is
necessarily very slow. By computing its hour angle, we can locate its
position on this circle, and hence by applying a correction to its
altitude, subtracting or adding according to the position of the star
above or below the pole, we will obtain the altitude of the pole.
A rough estimate of the position of the pole may be secured by noting
the position of the Big Dipper, the second star in the handle, called
Mizar, is approximately in line with Polaris and the pole.
We will now proceed to show the method by which the hour angle is
obtained:
In the talk on Time, it was stated that the local (astronomical) mean
time plus the right ascension of the mean sun is equal to the local
sidereal time; and again, that the right ascension of a star plus its
hour angle equals local sidereal time. With these facts as a basis, the
formula for latitude by Polaris given in the Nautical Almanac will be
followed in explanation.
[Illustration: FIG. 6.]
The time of observation must be noted by chronometer and converted
into local (astronomical) mean time; this must be corrected by Table
III (Nautical Almanac) in order to change this solar interval into
a sidereal time interval; to this converted time must be added the
Greenwich sidereal time of mean noon (page 2); that is, the hour angle
of the First Point of Aries, or what is the same thing, the right
ascension of the mean sun; to this sum must be applied a correction for
longitude, in time, taken from the foot of page 2, N. A. The sum is the
local sidereal time.
The reason for the correction of longitude is this: The difference
between the right ascension of the mean sun at noon on two successive
days is 3 m. 56 s., the same as the accumulated difference between
solar and sidereal time in 1 day. Now we take from the Nautical Almanac
this element for Greenwich mean noon, yet the sun has since covered
the distance equal to the longitude, and during the interval required
to do this, the sidereal time has accelerated over the solar an amount
which bears the same ratio to the 3 m. 56 s., that the longitude in
time bears to 24 hours. The Nautical Almanac handles the terms of this
proportion in tabular form at the foot of page 2. It is stated that the
sun has traveled from the meridian of Greenwich to the local meridian,
and it might be suggested that at the time of observation the sun has
covered this amount plus the local hour angle or the local astronomical
mean time. This is true but the amount of local hour angle has been
previously accelerated to sidereal time by the correction to local
astronomical mean time.
With the local sidereal time enter Table I (Nautical Almanac) and pick
out the correction to be applied according to sign to the altitude.
It is probably needless to say that the observed altitude must be
corrected for index error, dip and refraction before applying this
latter correction, which converts it into latitude.
This is called the Nautical Almanac method and is sufficiently accurate
for navigational purposes, but should a greater refinement be desired
there are tables of further corrections given in the American Ephemeris
and Nautical Almanac.
It is always advantageous to get an observation of a star near twilight
or dawn, in order that a well-defined horizon may be available; but,
in taking a sight of Polaris, another important feature is to be
considered. When the star’s hour angle is at or near 6 or 18 hours,
that is, near that part of its orbit cut by a line passing through the
pole and parallel to the horizon, it is rising or falling most rapidly,
with the result that a small error in time will produce a considerable
error in the hour angle, an error of 3 minutes introducing a difference
of 1´ in the latitude.
It is quite worth while, therefore, to select a time for the
observation of Polaris when this star is near either of its
culminations, its highest or lowest positions, where the time need not
be especially accurate; but by carefully noting the time it is possible
to get good results at other times when the horizon is defined. By
using the position of the star Mizar, as suggested above, however, the
navigator will be greatly aided in selecting the most propitious time
for observing Polaris.