Longitude
The longitude of any position on the earth is its distance east or west
from the meridian of Greenwich, which has been chosen as the meridian
of origin. Longitude is measured on the equator eastward and westward
through 180°, completing in this way the whole circumference of the
earth.
The circumference of every circle comprises 360°, whether it is a great
circle of the earth or any of the parallels which range in size from a
point at the poles to a great circle at the equator. There are always
360° but the length of each degree is determined by the size of the
circle. Thus a degree of longitude on the equator is 60 miles, while
on the 50th parallel of latitude it is only about 39 miles, owing to
the decreasing size of the parallels of latitude. A minute of longitude
on the equator, like a minute of latitude, is equal to one mile, but
the difference between the meridians in actual distance decreases
toward the poles gradually lessening the linear value of a degree of
longitude. Thus it will be seen that when it is desired to represent
a difference of longitude in distance, it must be done in terms of
departure (miles) corresponding to the particular parallel of latitude
of the position.
The sun apparently moves around the earth in its diurnal motion,
covering 360° in 24 hours, whether the declination is north or south,
and a little simple division shows that in one hour he passes over
15° of longitude, whatever the latitude. This reduced shows that 1°
is passed over every 4 minutes. As the standard time, the world over,
is reckoned by the movements of the sun, it is plainly seen that when
considering longitude, a definite relation exists between time and arc
(°-´-´´). Owing to this relation, time and arc become interchangeable
by a simple process of conversion.
So it follows that if we have the time at Greenwich by a chronometer,
and through a trigonometrical calculation we determine the local mean
time at the ship, the difference in time between Greenwich and the
ship’s meridian represents the longitude in time, which is readily
converted into arc.
The calculation involved in determining the local mean time is the
solution of the astronomical triangle, or in other words it is a
problem in spherical trigonometry. This triangle has its apex at the
pole with one side as the polar distance (90° - declination of the
observed body), another side the co-latitude (90° - dead reckoning
latitude) and the third side the zenith distance (90° - the corrected
altitude of the body).
It is one of the principles of trigonometry that with any three
elements given in a triangle any of the remaining elements may be
computed; that is, any angle or side is obtainable. The solution of
the astronomical triangle for various elements includes the finding of
the zenith distance and from this the altitude, which forms the main
feature of the problem involved in the New Navigation. It also provides
us with the angle between co-latitude and the zenith distance, which
is the azimuth of the body, by which the mariner is able to ascertain
the error of his compass.
The most important feature of the astronomical triangle is the angle
at the pole, known as the hour angle, which when found secures for
the navigator his local time. The problem presents itself in the form
of three sides being given to find one angle. It is found by the time
sight formula, which is too well-known to need any discussion here.
The shape of the triangle is determined by the declination of the
body, its altitude and the latitude of the vessel, and the polar or
hour angle; and it stands to reason that a formula will not produce
the same accuracy in the hour angle with every shape of the triangle.
For instance, in high latitudes or when the body has a declination
approaching 90°, the accuracy of the time sight formula becomes
effected.
Another very important point to bear in mind when observing a body with
the view of computing its hour angle, is its azimuth. When the bearing
is nearly east or west, on the prime vertical, the body is rising or
falling faster than at any other time, and an error in altitude or
latitude will produce the least error in the resulting longitude. The
necessity for close attention to this point is increased with the
latitude. Observations for time taken when the body has an azimuth of
less than 45° or over 135° are wholly unreliable.
The sun does not always cross the prime vertical in his daily track
across the heavens, for under certain conditions, say during the
northern winter, he will rise southward of east and set southward of
west. Under these adverse conditions, the calculation of longitude is
not dependable, and the best a navigator can do when using the sun is
to observe as soon as he has an altitude sufficient to clear the excess
refraction existing near the horizon.
It is under such circumstances that star sights are of incalculable
value, for a star can always be found in a suitable position with but
little waiting, or we may employ the New Navigation method, where the
azimuth of the sun is as good one place as another.
In order that a body will cross the prime vertical, the latitude must
be of the same name and greater than the declination. In conditions
cited above the declination of the sun is south and the latitude is
north, hence the body will never be on the prime vertical. If the
latitude is less than the declination, the sun’s diurnal track is
tilted toward the zenith, instead of away from it as when the latitude
is greater, and the result is that the sun, while never on the prime
vertical, approaches it for a time after rising, then recedes again. It
should be observed when at its nearest point to the bearing of east of
west.
The bearing of various bodies can be readily found by an inspection of
Hydrographic Office Azimuth Tables Nos. 71 and 120, the declination and
latitude being used as arguments.
There is a method of finding the longitude known as the equal
altitude method, but it is not valuable. The conditions are exacting
where accurate results are required and when these conditions exist
the ordinary time sight is available and at its best advantage, so
longitude by equal altitudes is not popular. To secure good results,
the body must have an altitude above 70° and near the prime vertical;
and, furthermore, the ship must be kept on an east or west course or
remain stationary. The theory of the problem is simplicity itself, and
for this reason is very alluring, but the best use that equal altitudes
can be put to is the determination of chronometer error ashore, and
in these days of radio time signals even this use is almost obsolete.
The rule is as follows: Observe the sun’s altitude, simultaneously
noticing the time by chronometer and clamp the sextant to prevent any
chance of the altitude becoming disturbed. When the sun has fallen to
the same altitude as of the forenoon sight, note the time again by
the chronometer. The mean of the two times, corrected for chronometer
error, equation of time, and the equation of equal altitudes due to
change in declination, in case of the sun, is the Greenwich apparent
time corresponding to our local noon or our longitude in time, which
should then be converted into arc.
The stars and planets are available as well as the sun for the finding
of longitude and when there is a distinct horizon, stellar sights
have many advantages. The problem depends upon the solution of the
astronomical triangle by the same formula as with the sun.
There are a few points of difference between a time sight of the sun
and one of a star or planet needing explanation. In the case of the
former body, we naturally compare the solar time of Greenwich with
the solar time of the local meridian, but in stellar work we employ
for this comparison stellar time, or, as it is more popularly called,
sidereal time. So it becomes necessary to turn the Greenwich mean time
of the chronometer into Greenwich sidereal time and compare it with
local sidereal time. The difference, as in mean time, is the longitude
in time, which is converted into arc in precisely the same way.
The Greenwich mean time is turned into sidereal time by adding to it
the right ascension of the mean sun, taken from the Nautical Almanac
and the acceleration for the Greenwich mean time (Table 9, Bowditch).
The local sidereal time is the result of an addition of the star’s
right ascension and the star’s hour angle, the right ascension is
taken from the Nautical Almanac without correction if a fixed star is
being considered and the computation of a time sight gives the star’s
hour angle. The local sidereal time being the right ascension of the
meridian, it follows that the angle from the vernal equinox to the star
plus the angle from the star to the meridian is what we desire; hence
the above rule for obtaining the local sidereal time.
Should the star bear east of the meridian, the local sidereal time may
be found by subtracting the (easterly) hour angle from the star’s right
ascension or adding them as above and subtracting 24 hours. Reference
to the Time Diagram, Fig. 3, will make these points clear also.
It is customary to add up the familiar logs of time sight--sec. lat.,
cosec. p. d., cos ½ sum, sin, remainder--divide by 2 and seek the H. A.
(hour angle) in the A.M. or P.M. column of Table 44, Bowditch, using
the log as a sin; but a more expeditious way is to use the sum of the
logs as the log haversine in Table 45 and pick out the hour angle
directly.