Time
A thorough understanding of time, one of the most important elements in
navigation, clears the way to a better idea of the theory of finding
one’s position at sea; there is, in the minds of many, considerable fog
hanging about certain portions of this subject, and it is hoped that
this explanation will clear some of this away.
Worcester’s Dictionary defines time as measured duration. It is the
interval between events. It flows ceaselessly and with uniformity, yet
the mortal mind is unable to conceive its beginning or its end. Man,
in order to measure his activities, has blocked it off into different
denominations convenient for his uses. Of these, the navigator uses the
following in his determinations and reckonings: years, months, days,
hours, minutes and seconds.
Certain astronomical phenomena were naturally enlisted by the ancient
astronomers to furnish standards for time measurements; the value of
a year was determined by the time necessary for a complete revolution
of the earth around the sun, while the length of a day was fixed by
the time of a rotation of the earth on its axis. The precision with
which these evolutions are accomplished gives the required accuracy.
The revolution of the earth governs the change of seasons, while
the rotation is responsible for the alternating periods of day and
night. With the exception of the month, the other measurements of
time mentioned above are denominations of these standards. The month,
one-twelfth of a year, is measured by the revolution of the moon around
the earth.
Solar time, as its name implies, is measured by the apparent diurnal
movement of the sun. It is the variety of time in universal use by
which is regulated the daily activities of life; and this is indeed
quite natural, for of necessity the work and play of the world depend
upon the light and darkness that this body serves out to us.
While we are unconscious of the earth’s rotation, its effect is seen in
the apparent daily course of the sun across the heavens, caused by our
turning past it, yet in common practice the sun is assumed to revolve
around the earth, and is usually thus spoken of for the purpose of
simpler explanation.
The time at each meridian is necessarily different from that of every
other, as only one of them holds the same position relative to the
sun at the same time or putting it in another way, only one meridian
can cross the sun at the same time, determining local noon for those
places located upon it. It is forenoon for that part of the world
westward of the sun and afternoon for that portion eastward of it.
As the earth turns from west to east, the places or meridians to the
eastward are first favored with the sun’s light, and those meridians
cross this body before those to the westward. The sun apparently moves
from the eastward to the westward, crossing each meridian in succession
until in a few hours it is afternoon for places to the eastward and
noon with us. The sun is now in our meridian, and it is forenoon for
people to the westward of us. For example, at 7 A.M., 75th meridian
time, it is noon in England and dead of night in our Pacific Coast; at
our noon (75th), it is late afternoon in England and breakfast time in
California.
It requires 24 hours, solar time, for the sun to make its apparent
revolution around the earth, this course being a circle; it contains
360° of arc. It follows that in one hour it passes over 15° of arc,
while 4 minutes are required for 1° to be traveled. Thus it is evident
that any arc of the circumference of the earth, or difference of
longitude, which is the same thing, has an equivalent time value
and vice versa. That is, the arc comprised between the meridian of
Greenwich and the 60th meridian west, for instance, besides being
measured as 60° W., is equal to 4 hours of time. Again 4 hours of
Greenwich time indicates that the sun has crossed the Greenwich
meridian 4 hours ago and is at that particular instant crossing the
meridian 60° west of Greenwich. If the arc were between Greenwich and
a place 60° E., the equivalent time interval would also be 4 hours,
because 60° of arc is everywhere equal to 4 hours of time; but the time
at Greenwich, with sun on the 60th meridian east, is 20 hours of the
previous day, or 8 A.M. of the present day. Thus: May 14, 20 hours, or
May 15, 8 A.M.
The meridians extend from pole to pole, and it matters not what
parallel you may be on, whether north or south latitude, your distance
can always be measured to the Greenwich meridian in arc or time
precisely as well as though you were on the exact parallel of Greenwich
itself. If the time at Greenwich is carried, and the local time of
any other meridian is desired, turn the difference of longitude into
time and apply it with regard to signs: - if west of Greenwich and +
if east. The local time at any place can thus be calculated; or to go
farther, if the time of any meridian is at hand, the time of any other
place can be readily found.
Every meridian carries a time of its own, and the instant of the click
of a telegraph key may be recorded all over the world in the local time
of each locality, yet the interval between this and a subsequent click
has an absolute value which is the same at every place, regardless
of whether it is expressed in solar, sidereal or lunar time, and its
actual value is invariable.
For convenience, on land, our country is blocked off into belts
of standard time, 15° wide, each carrying the time of its central
meridian. For instance, 75th meridian time is used by the eastern
states, while just westward the clocks’ faces show an hour earlier
time, that of the 90° belt, and so on.
It is a good rule to remember in reckoning all kinds of time that the
clock’s _face_ shows earlier time to the westward, and from this it is
easy to deduce the proper application of a correction.
There are two kinds of solar time used in navigation; the first to
be considered is apparent time, the kind shown by the sun dial, or
measured by the sun as we see it. It is noon of the apparent day
when the sun is seen with the sextant to dip while taking a meridian
altitude. It is at the moment of dipping that the navigator announces
12 o’clock, and with the striking of eight bells begins a new apparent
day on shipboard.
=The Day Lost and the Day Gained.=--The fact that the sun seems to
travel from east to west, determining the local time for successive
meridians or places along the way, causes an interesting condition in
reckoning time aboard ship. A vessel steaming westward on a parallel
sails with the sun; in the forenoon she is sailing away from it, at
noon the sun overhauls the vessel and they race together, but it
becomes a hopeless chase for the steamer during the afternoon. In
consequence of their similar course, however, the vessel will hold the
sun longer, and the length of daylight will be increased over that
time allotted a stationary position in proportion to the speed of
the vessel. On the other hand, a vessel steaming eastward each hour
advances to meet the sun; at noon the effect is as if they pass each
other, and during the remainder of the day they are moving in opposite
directions, hence this vessel has a shorter term of light and is
deprived of its full share of sunshine.
In practice these facts require the continuous setting back of the
ship’s clock, keeping apparent time on a westbound vessel. Take a
concrete case for illustration: to-day assume we are at sea on the
45th meridian west and set the clock at the dipping of the sun,
apparent noon; the vessel is westbound, steaming along the equator,
and rolls along at a good 15-knot clip. In 24 hours by the clock we
will cover 360 miles, or 360´ of arc on the equator, which is equal
to 6° difference of longitude. (Should the easting or westing be made
in higher latitudes, the difference of longitude will be increased
proportionately.) So when the ship’s clock shows noon we will be
6° farther west than at the preceding noon, or in 51° west. The
navigator, should he observe the sun, would find it had not reached
its highest altitude (the meridian), and he would be obliged to wait
(approximately) 24 minutes, the equivalent in time of 6°, before the
sun would dip. The clock is carrying 45th meridian time, and we are now
determining noon for the 51st meridian. He sings out 8 bells, but the
clock shows 12.24 P.M. The ship has gained 24 minutes by sailing with
the sun, and the clock is set back and a fresh start is made.
A vessel sailing east has the opposite experience. The navigator, if
guided by the ship’s clock, would find that the sun had dipped some 24
minutes before noon if a run similar to the above mentioned was made
eastward. In this case the apparent time of the 51st meridian is shown
by the clock, while the ship has moved on to the 45th, and the time of
noon is correspondingly approximately 24 minutes earlier than the clock
admits.
In the above example, the clock in the first instance is 24 minutes
fast and is set back that amount to correct it for the time of the
51° meridian W.; but this time cannot be thus arbitrarily thrown away
without some subsequent reckoning. There is just so much time all over
the world, and there are no gaps or extra intervals; it is absolute in
its uniform flow. Therefore, there must be a way of squaring ourselves
with Old Father Time.
But let us follow the voyage farther and see what transpires:
Continuing the course westward and ignoring for convenience all
intervening land, each day it becomes necessary to set the clock back
24 minutes until we have circumnavigated the earth. Suppose we took
our departure from the Greenwich meridian and kept our log throughout
the voyage with great care, expecting, according to our reckoning, to
arrive on a Saturday, we would indeed be mystified on arrival to hear
the ringing of church bells and find that it was Sunday. We have lost
a whole day according to our log, by throwing away 24 minutes at a
time. The time of the world goes on just the same, regardless of how we
juggle the hands of the clock. Now, if we try a similar voyage eastward
around the earth, we will be setting the clock ahead 24 minutes each
day, and when the anchor is dropped on our return, we will discover
that it is Friday instead of Saturday. The ship’s clock has skipped
this 24 minutes each day, and our log is a day ahead of what it should
be.
In order to prevent this difference of date, it was decided years
ago to establish an international date line, which should correspond
approximately, with the 180th meridian. The logs of vessels going west
around the earth will be a day behind the calendar when they reach
Greenwich, so a day is dropped from the reckoning when crossing the
180th meridian; that is, if it is Monday, the next day in the log will
be Wednesday. On the contrary vessels bound eastward will be a day
ahead when they reach their destination of Greenwich, so the date of
crossing the date line is entered twice in the log, as for instance,
there will appear two Mondays. By this method the accumulated errors
of chasing local time, are in a measure straightened out, and ship’s
logs are kept in agreement with the calendar of those at home. Thus it
will be seen that it is the accumulation of time thus gained or lost
that obliges navigators to add or drop a day to or from their logs when
crossing the 180th meridian.
In slow cargo steamers and sailing vessels, particularly when the
course creates but little departure, the change of time due to
difference of longitude is not sufficiently large to cause much
inconvenience and can be taken care of by setting the clock back
or ahead at noon. But with the development of the modern steamer,
speed has increased to such an extent that the easting or westing of
certain day’s runs correspond to a considerable amount of time, and
to correct the clock to local time, all at once, would be a source
of inconvenience and a bother. This is especially true where a fast
steamer covers much easting and westing in high latitudes where the
convergence of the meridians has shortened the degrees of longitude,
thereby increasing the difference of longitude over a similar day’s
run in lower latitudes. Hence, in order to more equally distribute
its error, the longitude at noon is anticipated by the navigator and
the clocks set at 8 A.M. for the local time of the approaching noon
meridian.
When the clocks are set at noon, they are correct only for the moment
and then start an accumulating error, depending in amount upon the
rapidity of the easting or westing made. But by anticipating the
longitude at noon, the forenoon watch will experience a decreasing
error instead of one accumulated for twenty hours, and still
increasing. It serves to keep the time of day more nearly correct.
In the transatlantic service, where high speed is maintained and the
courses result in a large amount of easting and westing, another method
is used for convenience. The navigator estimates the noon position
of the next day and accordingly divides the error into thirds. The
amount of the first third is applied at 11 P.M., the second at 3 A.M.
and the last third at 5 A.M. By this method the error is distributed
between the “first,” “mid” and “morning” watches. It is a matter of
considerable moment, and no joke, to the hard-working stokers to have
the clock set back on them the full amount of the day’s run all at one
time; and likewise going east, it would be giving an unfair advantage
to those on duty to set the clock ahead nearly an hour during the
morning watch.
The apparent or real sun is not a very accurate timekeeper and its
days are unequal in length. The aborigines and even our ancestors were
content with the time of day indicated by the sun dial, but as the
generations have passed, each bringing increased development, time has
become valuable; the crude timepieces have been forced aside by more
reliable instruments until to-day we figure down at times to a one
hundredth part of a second.
It is impossible to construct a clock that will follow the
irregularities of the apparent sun, so an imaginary sun has been
devised which is assumed to make its revolution at a uniform speed
along the celestial equator with exactly 24 hours between its transits
of the same meridian. This interval is the average of all apparent days
in one year.
The varying rate of the sun’s apparent motion is due to several causes
which will be subsequently discussed under the Equation of Time.
The time measured by the transit and progress of the mean sun is called
mean time; if at Greenwich, it is Greenwich mean noon and Greenwich
mean time (G. M. T.); if it represents the time of the observer’s place
or meridian, it is local mean time (L. M. T.). It is mean time that is
shown by all clocks and chronometers used in every day life.
The distance between the apparent and mean suns, expressed in time,
is known as the Equation of Time, and the application of this
correction depends upon which sun is ahead. It is tabulated in the
nautical almanac for every two hours of Greenwich mean time, with
hourly differences, so it can be reduced for longitude in time to
any meridian, or corrected to any intermediate Greenwich time. It is
applied according to the sign accompanying it, and can be used to
change apparent into mean, or mean into apparent time.
The progress of the mean sun across the sky with reference to the
meridian is measured by the angle at the pole (expressed in time),
between the meridian and the hour circle passing through the mean sun.
This is the hour angle of the mean sun as well as the local mean time.
Civil Time is a variety of mean time, and is reckoned through 12 hours
from midnight to noon, and again 12 hours from noon to midnight,
dividing the day into the well-known periods of A.M. and P.M. With
this kind of time, the day begins at midnight and the hour angle until
noon is measured eastward through 180° of the revolution and westward
through the remaining half from noon to midnight. In other words, 4:00
P.M. signifies that the sun has a westerly hour angle of 4 hours, while
8:00 A.M. indicates that the sun is 4 hours eastward of the meridian.
Astronomical time is reckoned westward through the whole 24 hours of
the day, 0 hours being noon. From noon to noon is an astronomic day.
Thus 5 P.M. civil is the same as 5 hours astronomical time, while 5
A.M., May 14th is the same as May 13th, 17 hours.
In every solar observation for time the real or apparent sun is
observed and hence the time derived from the sight must be local
apparent; to which the equation of time must be applied to convert it
into local mean time. It has already been made clear that the longitude
is equal to the difference between the local mean time and Greenwich
mean time, or between local apparent time and Greenwich apparent time.
SIDEREAL TIME
Sidereal is derived from the Latin word sidus, meaning of or belonging
to the stars. Sidereal time is measured by the apparent diurnal
revolution of the stars, resulting from the rotation of the earth.
By their use the conditions which render the sun inaccurate as a
timekeeper are eliminated; for the period of rotation of the earth is
so regular that the passages of the stars across the meridian occur
with great precision. This exactness enables the astronomer to keep the
observatory clock checked to a remarkable degree of accuracy. These
observatory clocks carry sidereal time, and for convenience it is
customary to divide their faces into 24 instead of 12 hours.
Sidereal time is the bedrock of all time; for it is by converting it
into solar time and sending it throughout the country by telegraph and
radio that the people of the world get the standard by which to set
their clocks and chronometers. Sidereal time is not practicable for
every day use as its noon occurs, without regard to light or darkness,
at every hour of day or night during a year. In March, at the time of
the vernal equinox, it agrees with the solar clocks, but in September
at the autumnal equinox, its noon occurs at the solar midnight.
While the sun is employed as the object of reference in solar time, it
may appear strange that no particular star is thus used in sidereal;
but in lieu of a definite stellar object by which to measure the
sidereal movement of the heavens, we refer to the celestial vernal
equinox.
This point was located in the constellation of Aries centuries ago, and
hence its popular name--The First Point of Aries; but this has become
a misnomer, for the point has long ago moved westward into another
constellation, as discussed under the Precession of the Equinoxes.
Navigators still cling to the name, however, and the equinox continues
to serve its purpose, regardless of its slow drift westward.
This imaginary point of reference crosses the observer’s meridian
much as the stars do, with the difference that it is always on the
celestial equator and acquires no declination. The value of this point
becomes further enhanced by the fact that it always lies in the same
direction regardless of our position in the orbit. In other words, the
distance of the equinox being infinite, lines drawn from perihelion and
aphelion, respectively, to it, fail to produce an appreciable angle.
In explanation of this statement, it must be understood that for all
uses on the earth the terrestrial system of direction (that is, using
the bearing of the north pole as a standard, with east to the left
and west to the right when the back is toward the pole) is entirely
adequate, but when dealing with the direction of celestial bodies,
a broader standard must be considered. North and South both have a
definite place in the heavens, being the points of the extended axis
of the earth, but east and west are only relative expressions. To
demonstrate this: it is possible for a man, traveling westward on the
Trans Siberian Railroad, to see from the rear platform, in the evening,
a certain star bearing eastward. At the same moment it is possible
for an officer of a transpacific liner in the early morning, to be
taking a sight of this same star bearing westward. In terms of absolute
direction that star bore the same from both sides of the world.
On the 21st of March the earth, sun, and celestial vernal equinox are
in range, with the sun between the earth and the equinox. For a place
in north latitude on the meridian of the terrestrial equinox, the sun
as usual bears south at noon this day, and hence the range mentioned
above bears south at that time.
This coincidence of bearing is only momentary, for the earth with its
onward motion immediately moves out of range and forms an angle between
the sun and the celestial equinox. At noon on the day succeeding the
equinox the sun bears to the left of the so-called First Point of Aries
(celestial vernal equinox). The sun according to terrestrial direction
always bears south by true compass at noon, yet the First Point of
Aries being at an infinite distance always bears the same by absolute
direction. If this point could be seen and a bearing of it taken by
compass simultaneously with the sun, it would be, perhaps, S. 1° W. and
so on widening the angle, roughly speaking, a degree each day.
[Illustration: The acceleration of Sidereal over Mean Time
FIG. 2.]
The interval between two successive transits of the sun across the
meridian constitutes a solar day, and likewise the period required
for a certain star to return to the same meridian is a sidereal day,
but these two days are not of the same length. The solar day we know
is 24 hours long, but its sidereal contemporary has a length of only
23 hours, 56 minutes (approximately) solar time. The sidereal clocks,
however, are geared to show 24 hours, sidereal time, in 23 hours, 56
minutes, solar time. By this it will be seen that in any given period
the face of the sidereal clock will show more hours than the solar
clock.
Both the solar and sidereal clocks start even at the vernal equinox,
about March 21st, but from then on, the sidereal clock gains on the
solar time clock about 4 minutes a day until in a year it is a full 24
hours ahead, showing that there is one more day in a sidereal year than
in a solar year. The approximate relation of the times shown by these
clocks is readily calculated by allowing a gain in the sidereal clock
of one hour for each 15 days after March 21st, or two hours for each
month.
In order to aid in a simpler explanation, let us again follow the earth
around its orbit and note the conditions that distinguish sidereal from
solar time. Let us once more assume it to be the time of the vernal
equinox, the clocks, both sidereal and solar, now show 0 hours, and the
sun, the earth and the First Point of Aries are in range. The earth
immediately moves out of line by virtue of its onward motion, and the
sun correspondingly appears to move eastward; this is imperceptible at
first, however, and not noticeable without a careful measurement, as it
seems to be swallowed up in the contrary (westward) diurnal movement.
After 24 hours of rotation from the instant of the equinox the earth
turns the meridian until it causes the First Point of Aries to transit,
marking sidereal noon of the first day. The sidereal clock at this
moment reads 24 hours, but a glance at the solar clock shows 11 hours
56 minutes A.M., about 4 minutes short of (solar) noon. An observation
will show that the sun has apparently moved about a degree eastward
of the hour circle passing through the First Point of Aries since the
preceding noon, and the earth must turn this extra degree before the
sun will be brought to the meridian, thus occupying the 4 minutes
mentioned above. In other words, the earth turns 360° in a sidereal
day but must turn about 361° in a solar day.
Three months after the vernal equinox, the angle between the First
Point of Aries and the sun becomes, in round numbers, 90°, and it
requires 6 hours for the earth to bring the sun to the meridian after
the passage of the First Point of Aries. In plainer language, when the
First Point of Aries crosses the meridian (sidereal noon) the sun is
about 90° to the left--about rising in the eastern sky; the earth must
make a quarter turn, or 6 hours, before it will be solar noon. Thus it
will be seen that at this point sidereal time is 6 hours ahead of solar
time.
In six months, when the First Point of Aries is on the meridian the
noonday sun is shining on the antipodes, and it lacks 12 hours of solar
noon. The difference between the sidereal and solar clocks has now
reached 12 hours and through a continuation of the same process the
interval between their readings, widens throughout the remainder of the
year.
When the 21st of March comes around again, and the meridian presents
itself to the sun and the First Point of Aries in range, a careful
count of the number of times this latter point has crossed the meridian
during the year, discloses 366¼ transits. That is, the earth has
actually turned about its axis 366¼ times. The sun is found to have
passed the meridian only 365¼ times. Counting the rotations of the
earth by the number of the sun’s transits while we are revolving around
him, causes the apparent loss of a day due to the earth unwinding
itself once, so to speak, during the year. The accumulated difference
amounts to one sidereal day. Hence it will be seen that a year
contains 366¼ sidereal days of 23 hours, 56 minutes each, and 365¼
solar days of 24 hours each.
Now for a recapitulation of the subject of time.
The rotation of the earth is the real standard of measuring time
intervals; the period required for this rotation does not vary. It
has been suggested that the tide waves have a minute effect on the
regularity of this movement, but the construction of our clocks is
such, that if any variation exists we are unable to detect it. Whether
we use the passages of the stars, or the transits of the sun to reckon
our time, it falls back in either case upon the diurnal rotation of the
earth.
Apparent time is measured by the seeming progress of the actual sun.
The time of its transit of the meridian is irregular, but is always
shown by its “dip,” culmination, with the sextant.
Mean time is reckoned by the revolutions of a fictitious sun, called
the mean sun, and the length of one of these revolutions is the average
of a year of apparent days. This, owing to its uniformity, is the time
used for the everyday purposes of life. The difference between apparent
and mean time is called the equation of time, and, by applying it
according to signs given in the Nautical Almanac, one can be converted
to the other as desired.
Sidereal time is indicated by the position of the First Point of Aries
relative to the meridian; it is star time. The stars make a complete
revolution of the heavens in 4 minutes less time than is required by
the mean sun. Therefore the sidereal day is that amount shorter than
the solar day.
The point of the celestial vernal equinox or First Point of Aries is
a sort of celestial “bench mark;” besides indicating sidereal time,
it serves as a point from which right ascension is measured eastward.
This subject has been discussed previously, but as it is intimately
associated with sidereal time, perhaps it may be made clearer since the
latter has been so fully explained.
Right ascension is measured on the celestial equator, precisely the
same as longitude on the earth, excepting that it is always measured
eastward through the full 24 sidereal hours, contrary to the diurnal
movement of heavenly bodies. Moreover, the meridian passing through the
vernal equinox is called the celestial prime meridian, and sometimes
the Greenwich of the heavens. There is another point of distinction,
however, between this prime meridian of the sky and our meridian of
Greenwich, which, while it does not effect practical navigation, has
to receive consideration in the long run; our longitude values on the
earth remain at all times constant, but owing to the precession of the
equinoxes the celestial prime meridian is slowly moving westward, thus
causing the right ascensions measured from it to become very slowly in
error (50´´ yearly).
The hour angle of a body is the angle formed at the pole between the
meridian and the hour circle passing through the body measured westward.
With all these important facts well in mind we will go ahead under a
slow bell, through a few more statements which may be found a little
perplexing. However, a careful study of the Time Diagram will, no
doubt, drive away the haziness so often surrounding the subject of
time.
[Illustration: This diagram represents the plane of the equator
looking down upon the North Pole. The 75° W meridian is chosen as that
of the observer and local time reckoned therefrom. The arrows on the
outer circumference indicate the directions of the earth’s rotation;
the other smaller arrows indicate the direction in which each element
is reckoned.
FIG. 3.]
The hour angle of the mean sun is the local mean time, and the hour
angle of the First Point of Aries is the local sidereal time. The local
mean time and longitude in time accelerated by Table III Nautical
Almanac plus right ascension of the mean sun is equal to the local
sidereal time. The right ascension of the meridian is the same thing,
exactly, as the hour angle of the First Point of Aries, and both of
these are identical with the local sidereal time. The sidereal time of
Greenwich mean noon is the same as the right ascension of the mean sun
at that time. The hour angle of a star plus the right ascension of the
same star is equal to the local sidereal time.
Difference of longitude can be represented by an interval of sidereal
time or by a difference of right ascension, precisely the same as by a
difference of solar time. Thus with the local sidereal time calculated
from an observation of a star, and the corresponding Greenwich sidereal
time taken from the Nautical Almanac, the longitude is at hand, by
turning their difference into arc. The fact that the actual time
interval is longer in the case of solar time than in an interval of
the same number of hours of sidereal time, has no influence on the
resulting difference of longitude. The number of degrees in any arc
can be the same, yet vary in linear measurement, but the same number
of hours of solar and sidereal time represent the same proportionate
part of a circle. It was just stated that the hour angle of a star plus
its right ascension is the same as the local sidereal time; now this
is also true of the sun. The hour angle of the mean sun plus the right
ascension of the mean sun is equal to the local sidereal time; by means
of this equality we are able to find the Greenwich sidereal time on any
occasion. It is necessary to have this element in order to compare it
with the local sidereal time, which we find by observation of a star,
to obtain difference of longitude (in time). In page 2 of the Nautical
Almanac will be found the Right Ascension of the Mean Sun at Greenwich
Mean Noon; this must always be taken out for the preceding noon. We now
have a measure of sidereal time to be added to a measure of mean time,
but it will be remembered in early arithmetic that an apple and a peach
can not be added together any more than ½ can be added to ⅓. The only
course to steer is to reduce the quantities to a common denominator,
or like quantities. So in handling these two varieties of time, solar
time must be accelerated by adding a correction to it, or sidereal
time retarded by subtracting an amount necessary to make it equal to
a corresponding value of solar time. The tables for the conversion of
one of these varieties of time to the other are found in the American
Ephemeris Tables II and III, Bowditch Tables 8 and 9, and at the foot
of pages 2 and 3, Nautical Almanac.
A practical illustration of this may make a clearer impression. In the
early evening of April 8th, secured a sight of Regulus; the chronometer
showed 8 days, 9 hours, 56 minutes, and 0 seconds (corrected), with
other necessary elements given, the sight is worked as usual to find
the star’s hour angle, which proves to be:
h. m. s.
H A Regulus 1 59 47 bearing west.
R A 10 3 02 RAMS 1 h. 6 m. 7 s.
-- -- -- GMT 9 h. 56 m. 0 s.
L. S. T. 12 02 49 Accel. 9 h. 1 m. 29 s.
G. S. T. 11 03 45 Accel. 56 m. 9 s.
-- -- -- ----- ----- -----
Long. in time 59 04 G. S. T. 11 h. 3 m. 45 s.
This being a star sight we obtain from it the sidereal time at place of
observation and as the chronometer carries Greenwich mean time we seek
the corresponding sidereal time by adding this and its acceleration for
a sidereal interval to the right ascension of the mean sun taken from
Nautical Almanac. The result is the Greenwich Sidereal Time.
It is occasionally required to find the sidereal time at the ship in
which case it is only necessary to apply the longitude in time to the
Greenwich Sidereal Time.
As Greenwich mean time is the most used and is the best understood it
is a very convenient practice to carry G. M. T. on the navigator’s
watch. It is readily converted into any other time with ease but serves
more purposes as it is without conversion. A stop watch is an excellent
instrument for taking time sights where great accuracy is essential. By
setting it at 0 minutes a man can observe alone starting the watch as
he makes contact with the horizon and when subsequently comparing with
the chronometer subtract the reading of the watch to get the G. T. of
observation.
The most expeditious way to convert time into arc is to multiply the
hours by 15 and add the number of minutes divided by 4 to get the
degrees; multiply the remaining minutes by 15 and add the seconds
divided by 4 to get the minutes; multiply the remaining seconds by 15
to get the seconds in arc:
Thus:
Long. in time 2 hrs. 42 m. 23 s.
30 30 45
10 5
--------------------
Long. in arc 40° 35´ 45´´
To change arc into time divide the degrees by 15 to get the hours;
multiply the remainder by 4 and divide the minutes by 15 and add to get
the number of minutes (m.); multiply the remainder of minutes (´) by 4
and divide the seconds (´´) by 15 carrying the division of tenths if
desired, adding the result to get the seconds (s.):
Thus:
Long. in arc 40° 35´ 45´´
2 hrs. 40 m. 20 s.
2 3
-------------------
Long. in time 2 hrs. 42 m. 23 s.
This may appear complicated at first but is much the quickest way of
conversion. However, Table No. 7, Bowditch is always available if
desired.
In getting an understanding of any time problem, that is such as
changing mean time into sidereal time; obtaining the hour angle of a
star or planet; in seeking the local time from the chronometer, or any
time values that are found perplexing, always draw a diagram. Make the
circle on the plane of the equator, with the pole as the center, and
meridians radiating from it towards the circumference of the circle.
Now imagine for the moment that you are at the north pole, and the
date is the 21st of September. The sun is traveling in the horizon,
and if the direction of the Greenwich meridian is known, this body
serves as a time piece, for the angle between this meridian (direction)
and the sun is the Greenwich time. This angle corresponds with twice
the angle between XII hours on the watch and the hour hand; or would
(disregarding equation of time) coincide with it if the watch’s face
was divided into 24 hours. Likewise when the vernal equinox or First
Point of Aries lies in the direction of Greenwich it is Greenwich
sidereal noon, and the subsequent angle that appears through the
rotation of the earth, shows the Greenwich sidereal time.
EQUATION OF TIME
It is necessary in considering this subject to reiterate some of the
statements made in the preceding talk on time, but, as they are very
important, no time is wasted by further impressing them on the mind.
Let it be understood that the apparent orbit of the sun is actually due
to the earth’s revolution around him, yet for simpler explanation it is
considered to be the sun’s own revolution.
The apparent movement of the real sun is not of uniform speed and, in
consequence, it has become necessary to devise a fictitious sun whose
assumed revolutions around the earth are at all times regular in their
rate.
The equation of time is the difference between these two suns and, as
they are at times in conjunction and at other times attain a distance
from each other of 16 minutes 20 seconds, and, moreover, as the real
sun is sometimes ahead and again in the wake of the mean sun, it
becomes evident that the equation of time is an ever varying quantity.
The irregularity of the sun’s apparent movement as compared with the
uniformity of the mean sun, is subject to two causes: First, the earth
travels in an ellipse, and, as the length of a degree varies in the
different parts of the circumference, the motion would appear to be
irregular, that is, if the sun actually traveled at a uniform rate, it
would, from the above fact, appear to us to be variable in its motion;
furthermore, the laws of forces only allow a body traveling in a circle
the privilege of a uniform speed so the earth, owing to its varying
distance from the sun, experiences a corresponding change in the amount
of attraction exerted upon it by the sun and its velocity, actually
becomes variable. Thus, during the winter, December and January, when
they are nearest each other, the attraction is strongest and the earth
increases its speed in revolution; while in June and July the earth
is at its greatest distance from the sun and the attraction is less,
resulting in a slowing down in the rate of the onward movement. As the
sun appears to us to take on movements corresponding to those of the
earth, these variable movements of the latter are seen in the apparent
motion of the sun. Second, the plane of the earth’s orbit is inclined
at an angle to that of the equator, which makes the sun appear to be
traveling at a variable speed along the ecliptic.
With these two errors combined influencing the apparent sun, he
becomes unreliable for regulating timepieces. The mean sun, which was
originated to obviate these irregularities, is assumed to travel in
a circle with the earth located in the centre, which disposes of the
first reason for an apparent variable motion; and again, the mean sun
revolves in the plane of the equator, thus eliminating the second
obstacle in the way of uniform time.
Now we will continue a little farther into the explanation of the
reasons for the irregular movement of the real sun. A law discovered by
Kepler, and named for him, provides that a radius from the sun to the
earth covers sectors of equal areas in equal times; a sector equal in
area to any sector covered in the same time. That is, when the earth is
in that part of the orbit near the vernal equinox, the radius of the
orbit will in a given time, say a week, sweep over a certain area; the
earth proceeds toward aphelion and when in the vicinity of that point,
the radius becomes greatly increased in length. Now in a week with
this longer radius, a far greater area would be covered if the earth
maintained the same rate of speed as at the equinox, but Kepler’s law
says, “equal areas in equal times,” so in order to conform with the
law, the earth’s speed of revolution must be reduced. The earth does
not slow down just for the sake of obeying Kepler, but at this part of
the orbit it is at its greatest distance from the sun and hence the
reduced attraction causes the earth to lag a little.
At the time of the autumnal equinox, an area similar to that of the
vernal equinox is covered. As the earth approaches perihelion, the
radius is gradually shortened by the eccentricity of the sun in this
part of the orbit and the increased attraction causes the earth to
speed up correspondingly. At the increased speed, the shorter radius
sweeps the same area in a week as at other parts of the orbit and
Kepler’s law still holds good.
As the apparent motion of the real sun corresponds exactly with the
real motion of the earth, it is evident from the above that the real
sun apparently moves at different rates of speed along the ecliptic,
faster in winter and slower in summer than the mean sun.
The value of a mean solar day is the average of a year of apparent
days, or in other words, there is the same number of mean solar days in
a year as there are apparent days.
In considering the effect of the variable motion of the earth in its
orbit, we will recall the conditions used when defining sidereal and
solar days. The former comprises the interval between two successive
passages of a certain star across the meridian, or perhaps better,
between two successive passages of the meridian over a star. This is
the true length of the earth’s rotation and is the standard to which we
may refer the length of the mean or apparent solar days.
Now it requires about 3 minutes 56 seconds longer for the meridian to
sweep around from sun to sun than from star to star, owing to the fact
that the mean sun moves uniformly eastward that amount daily, thereby
requiring the meridian, after reaching its position of yesterday
noon, to overhaul the mean sun this 3 minutes 56 seconds of eastward
movement. The mean sun maintains this uniform difference between its
days and the length of the sidereal day. Without this daily easting of
the sun, the sidereal and solar day would be the same.
But, in considering the apparent sun, we find the length of its days
continually varies from both that of the sidereal and mean day. This is
explained by the fact that the eastward movement of the apparent sun
is due to the movement of the earth in its orbit, and as this movement
becomes faster or slower the eastward movement of the sun becomes
correspondingly faster or slower. Thus, we readily see that with the
apparent sun moving eastward faster or slower at times, the length
of the apparent day must vary accordingly and we cannot establish a
uniform difference between it and the sidereal day, as in the case of
the latter and the mean day. The apparent days exceed the mean days
in length, between September and March, while the earth is traveling
fastest in its orbit. Beginning at the autumnal equinox with the
apparent sun eight minutes behind the mean sun, the former gains slowly
at first but with increasing rapidity. About the end of December, at
perihelion, it overhauls the mean sun and they are coincident _as
regards this correction only_. Leaving perihelion, the apparent sun
rapidly takes the lead but with a gradually decreasing amount until
at the equinox in March, reaches its maximum lead of 8 minutes.
Entering that portion of the year March to September, we find the earth
traveling slower and the mean sun gaining on the apparent sun; between
the vernal equinox and aphelion, the mean sun gains until both are
together at the summer solstice and then forging ahead the mean sun
attains a lead of 8 minutes in September.
It must be borne in mind that this error is caused only by the
eccentricity of the orbit and is but a component part of the whole
correction of the equation of time. The other portion is due to the
obliquity of the orbit, or its inclination to the equator.
This error is introduced through the fact that the apparent sun
moves in the ecliptic and the mean sun is assumed to proceed along
the celestial equator. In considering this phase of the question, we
will ignore entirely, for the time being, the error of eccentricity
described above.
The error of equation of time due to the obliquity of the orbit is a
simple one to see, but like many simple things it is easier to show it
by a diagram than to explain in words, so the reader is referred to the
accompanying figure, that a study of it may be made before proceeding.
[Illustration: FIG. 4.]
At the equinoxes, where the ecliptic and equator cross and the
solstices--the vertices of the ecliptic--that is, four times a year,
the true and mean sun are together, but departing from these points
they do not travel with the same right ascension, remembering that
right ascension is measured on the equator. Taking, for example, the
earth in that quadrant of the orbit comprised between the vernal
equinox and summer solstice, the apparent sun in the heavens would
be by cause of obliquity alone, to the right or to the westward of
the mean sun, and thus it will be seen that with the earth rotating
from right to left the apparent sun will cross the meridian first;
consequently between March 21st and June 21st that part of the equation
of time due to the obliquity of the orbit bears a minus sign when mean
time is desired from the apparent time. This correction reaches its
maximum half-way or 45° from the equinox, amounting at that point to
nearly 10 minutes.
[Illustration: FIG. 5.]
Now the reason for this difference between the
mean and apparent sun when each (so far as this problem is concerned)
moves along its respective path--the equator and the ecliptic--at the
same rate, is this: suppose the equator between the equinox and the
solstice is divided into an equal number of parts and an hour circle
drawn through each point of division. Beginning from the equinox (the
common apex of the triangle) the arc of each hour circle between
the equator and the ecliptic, forms the altitude of a right-angled
triangle, while the equator and the ecliptic are base and hypothenuse
respectively. Thus it will be seen that each portion of the equator
(base) is shorter than the corresponding part of the ecliptic as
defined the hour circle, to the extent of the ratio of the base to the
hypothenuse.
This amount increases with the increasing size of the triangle, but
a new element enters to counteract its effect. With the increasing
divergence of the ecliptic and equator, the divergence of the hour
circles becomes a factor and as the solstice is approached the
divisions on the equator, are represented on the ecliptic by gradually
decreasing spaces between the hour circles.
The combination of these effects produces the error due to the
obliquity of the orbit. The error has the opposite effect in the next
quadrant, that is, from June to September; and in opposite quadrants it
is the same.
So it will be seen that error due to the eccentricity causes the
apparent sun to lead the mean sun from December 31st to June 30th,
reaching a maximum of 8 minutes about April 2d. This sun then falls
astern until December, again attaining a maximum of -8 minutes about
October 1st. The error due to the obliquity of the orbit accumulates
between the equinoxes and solstices for at these points the two suns
are together and there is no error, but about the 6th of May, August,
November, and February, it reaches a maximum of 10 minutes; in August
and February, the mean sun takes the lead.
These two errors of equation of time combined algebraically will result
in the plain line of the diagram.
CALENDAR
The ancients, in order to keep track of past events and to anticipate
the future, devised a calendar, which, while not identical with the
one now in use, was of itself a remarkable production. They chose the
revolutions of the moon as their basis of measuring the passage of
time, and, as a lunar month contains only 27⅓ days, the 12 months used
in this early calendar comprised a year of about 354 days.
It became apparent as the world progressed that use of the moon
was very unsatisfactory for this purpose, as the calendar became
complicated and confusion resulted, owing to the difference between
the lunar and solar years. This condition remained until the reign of
Caesar, when that monarch determined to establish a more satisfactory
method of reckoning time. With the aid of an eminent astronomer,
he completely revised the calendar, using the revolution of the
earth around the sun as the standard for the length of a year. The
time required for this is 365¼ days, approximately, and as it was
inconvenient to include the ¼ day in the year, it was allowed to
accumulate for 4 years, when as a whole day, it was added to the end of
February.
Caesar, evidently proud of this accomplishment, honored himself by
naming his astronomer’s invention the Julian Calendar and in order to
further immortalize himself he changed the name of the seventh month
to July. Augustus, his successor, apparently envious of the honor his
predecessor derived from this source, and determined not to be thus
outdone in perpetuating his name, changed the month Sextilis to August.
The commonly accepted 365¼ days as the length of year is only an
approximation, however, and the small difference between this and the
actual length of a year began to accumulate until this weak point in
the Julian Calendar became a matter of moment. The exact length of
a year from vernal equinox to vernal equinox is 365 days 5 hours 48
minutes and 46 seconds, which lacks just 11 minutes 14 seconds of 365¼
days. This caused the dates of the equinoxes and solstices to keep
slipping back 11 minutes each year and when considerable time had
passed the difference became large enough to cause inconvenience; the
date of the vernal equinox having dropped back to March 11th in 1582.
In this year Pope Gregory, acting under the advice of an astronomer by
the name of Clavius, modified the Julian Calendar. He first added 10
days to restore the date, and then to forestall a further retrogressing
of the calendar, provided that only those century years divisible by
400 should be considered leap years. In this way the 11 minutes 14
seconds was prevented from causing further mischief. This calendar
known as the Gregorian Calendar is now in almost universal use, though
at first it was adopted only by Catholic countries.
It is interesting to note that the time consumed by the sun in making
his apparent yearly revolution from a certain star back to that star
again is a sidereal year of 365 days 6 hours 9 minutes 9 seconds.
The tropical year--the one in common use--is shorter, being the time
required for the sun to leave and return to the vernal equinox, or
First Point of Aries. This point, it will be remembered, is moving
westward about 50´´ annually, and it will be seen that while the sun
starts its eastward revolution among the stars, the equinox is very
slowly moving westward to meet him, thus making the tropical year about
20 minutes shorter than the sidereal year.
While discussing the calendar it is an opportune time to explain a
matter concerning the dates of the equinoxes and solstices. It has of
course been noticed by everyone that the vernal equinox occurs one year
on March 20th and another on March 21st, or the summer solstice on June
22d and yet another year on June 21st and so on.
Aside from the slight change due to the dropping back of the seasons in
the orbit by the precession of the equinoxes, the actual time of the
equinoxes and solstices may be considered as constant, yet the dates
vary a few hours.
The year in common use--the tropical year--contains approximately 365¼
days, yet we take account of only 365 days, the extra ¼ day being laid
aside for future reckoning. During the next year this 6 hours will be
augmented by 6 more; the next by another 6, making 18 hours ahead of
the calendar. The fourth year this amount reaches 24 hours, and a full
day, the 29th of February, is added to the calendar for that year, and
we are square again. But the different equinoxes and solstices occur
at just 364¼ days; taking for example the vernal equinox, it occurs on
the following (approximate) dates, which it will be noticed are 6 hours
later each year:
1908--March 20, 12 hours (leap year).
1909--March 20, 18 hours.
1910--March 20, 24 hours or March 21st noon.
1911--March 21, 6 hours.
1912--March 20, 12 hours (leap year).
It is evident by the above that the insertion of the extra day just
previous to the equinox in the leap years, sets the date of the equinox
back a day by the calendar. Juggling the ¼ day as shown above causes
the change in the calendar dates of these phenomena.