disc with accurate instruments, so as to make the result independent
of our ordinary way of judging. When the moon is near the horizon,
it seems placed beyond all the objects on the surface of the earth
in that direction, and therefore farther off than at the zenith,
where no intervening objects enable us to judge of its distance. In
any case, an object which keeps the same apparent magnitude seems to
us, through the instinctive habits of the eye, the larger in
proportion as we judge it to be more distant.
91. _The Apparent Size of the Moon increased by Irradiation._--In
the case of the moon, the word _apparent_ means much more than it
does in the case of other celestial bodies. Indeed, its brightness
causes our eyes to play us false. As is well known, the crescent of
the new moon seems part of a much larger sphere than that which it
has been said, time out of mind, to "hold in its arms." The bright
portion of the moon as seen with our measuring instruments, as well
as when seen with the naked eye, covers a larger space in the field
of the telescope than it would if it were not so bright. This effect
of _irradiation_, as it is called, must be allowed for in exact
measurements of the diameter of the moon.
[Illustration: Fig. 107.]
92. _Apparent Size of the Moon in Different Parts of her Orbit._--Owing
to the eccentricity of the moon's orbit, her distance from the earth
varies somewhat from time to time. This variation causes a corresponding
variation in her apparent size, which is illustrated in Fig. 107.
93. _The Mass of the Moon._--The moon is considerably less dense than
the earth, its mass being only about one-eightieth of that of the earth;
that is, while it would take only about fifty moons to make the bulk of
the earth, it would take about eighty to make the mass of the earth.
One method of finding the mass of the moon is to compare her effect
in producing the _tides_ with that of the sun. We first calculate
what would be the moon's effect in producing the tides, were she as
far off as the sun. We then form the following proportion: as the
sun's effect in producing the tides is to the moon's effect at the
same distance, so is the mass of the sun to the mass of the moon.
The method of finding the mass of the sun will be given farther on.
94. _The Orbital Motion of the Moon._--If we watch the moon from night
to night, we see that she moves eastward quite rapidly among the stars.
When the new moon is first visible, it appears near the horizon in the
west, just after sunset. A week later the moon will be on the meridian
at the same hour, and about a week later still on the eastern horizon.
The moon completes the circuit of the heavens in a period of about
thirty days, moving eastward at the rate of about twelve degrees a day.
This eastward motion of the moon is due to the fact that she is
revolving around the earth from west to east.
[Illustration: Fig. 108.]
95. _The Aspects of the Moon._--As the moon revolves around the earth,
she comes into different positions with reference to the earth and sun.
These different positions of the moon are called the _aspects_ of the
moon. The four chief aspects of the moon are shown in Fig. 108. When the
moon is at _M_, she appears in the opposite part of the heavens to the
sun, and is said to be in _opposition_; when at _M'_ and at _M'''_, she
appears ninety degrees away from the sun, and is said to be in
_quadrature_; when at _M''_, she appears in the same part of the heavens
as the sun, and is said to be in _conjunction_.
96. _The Sidereal and Synodical Periods of the Moon._--The _sidereal
period_ of the moon is the time it takes her to pass around from a star
to that star again, or the time it takes her to _make a complete
revolution around the earth_. This is a period of about twenty-seven
days and a third. It is sometimes called the _sidereal month_.
The _synodical period_ of the moon is the time that it takes the moon to
_pass from one aspect around to the same aspect again_. This is a period
of about twenty-nine days and a half, and it is sometimes called the
_synodical month_.
[Illustration: Fig. 109.]
The reason why the synodical period is longer than the sidereal period
will appear from Fig. 109. _S_ represents the position of the sun, _E_
that of the earth, and the small circle the orbit of the moon around the
earth. The arrow in the small circle represents the direction the moon
is revolving around the earth, and the arrow in the arc between _E_ and
_E'_ indicates the direction of the earth's motion in its orbit. When
the moon is at _M_{1}_, she is in conjunction. As the moon revolves
around the earth, the earth moves forward in its orbit. When the moon
has come round to _m_{1}_, so that _m_{3}m_{1}_ is parallel with
_M_{3}M_{1}_, she will have made a complete or _sidereal_ revolution
around the earth; but she will not be in conjunction again till she has
come round to _M_, so as again to be between the earth and sun. That is
to say, the moon must make more than a complete revolution in a
synodical period.
[Illustration: Fig. 110.]
The greater length of the synodical period is also evident from Fig.
110. _T_ represents the earth, and _L_ the moon. The arrows indicate
the direction in which each is moving. When the earth is at _T_, and
the moon at _L_, the latter is in conjunction. When the earth has
reached _T'_, and the moon _L'_, the latter has made a sidereal
revolution; but she will not be in conjunction again till the earth
has reached _T''_, and the moon _L''_.
97. _The Phases of the Moon._--When the new moon appears in the west, it
has the form of a _crescent_, with its convex side towards the sun, and
its horns towards the east. As the moon advances towards quadrature, the
crescent grows thicker and thicker, till it becomes a _half-circle_ at
first quarter. When it passes quadrature, it begins to become convex
also on the side away from the sun, or _gibbous_ in form. As it
approaches opposition, it becomes more and more nearly circular, until
at opposition it is a _full_ circle. From full moon to last quarter it
is again gibbous, and at last quarter a half-circle. From last quarter
to new moon it is again crescent; but the horns of the crescent are now
turned towards the west. The successive phases of the moon are shown in
Fig. 111.
[Illustration: Fig. 111.]
98. _Cause of the Phases of the Moon._--Take a globe, half of which is
colored white and the other half black in such a way that the line which
separates the white and black portions shall be a great circle which
passes through the poles of the globe, and rotate the globe slowly, so
as to bring the white half gradually into view. When the white part
first comes into view, the line of separation between it and the black
part, which we may call the _terminator_, appears concave, and its
projection on a plane perpendicular to the line of vision is a concave
line. As more and more of the white portion comes into view, the
projection of the terminator becomes less and less concave. When half of
the white portion comes into view, the terminator is projected as a
straight line. When more than half of the white portion comes into view,
the terminator begins to appear as a convex line, and this line becomes
more and more convex till the whole of the white half comes into view,
when the terminator becomes circular.
[Illustration: Fig. 112.]
The moon is of itself a dark, opaque globe; but the half that is towards
the sun is always bright, as shown in Fig. 112. This bright half of the
moon corresponds to the white half of the globe in the preceding
illustration. As the moon revolves around the earth, different portions
of this illumined half are turned towards the earth. At new moon, when
the moon is in conjunction, the bright half is turned entirely away from
the earth, and the disc of the moon is black and invisible. Between new
moon and first quarter, less than half of the illumined side is turned
towards the earth, and we see this illumined portion projected as a
crescent. At first quarter, just half of the illumined side is turned
towards the earth, and we see this half projected as a half-circle.
Between first quarter and full, more than half of the illumined side is
turned towards the earth, and we see it as gibbous. At full, the whole
of the illumined side is turned towards us, and we see it as a full
circle. From full to new moon again, the phases occur in the reverse
order.
99. _The Form of the Moon's Orbit._--The orbit of the moon around the
earth is an ellipse of slight eccentricity. The form of this ellipse is
shown in Fig. 113. _C_ is the centre of the ellipse, and _E_ the
position of the earth at one of its foci. The eccentricity of the
ellipse is only about one-eighteenth. It is impossible for the eye to
distinguish such an ellipse from a circle.
[Illustration: Fig. 113.]
100. _The Inclination of the Moon's Orbit._--The plane of the moon's
orbit is inclined to the ecliptic by an angle of about five degrees. The
two points where the moon's orbit cuts the ecliptic are called her
_nodes_. The moon's nodes have a westward motion corresponding to that
of the equinoxes, but much more rapid. They complete the circuit of the
ecliptic in about nineteen years.
The moon's latitude ranges from 5° north to 5° south; and since, owing
to the motion of her nodes, the moon is, during a period of nineteen
years, 5° north and 5° south of every part of the ecliptic, her
declination will range from 23-1/2° + 5° = 28-1/2° north to 23-1/2° + 5°
= 28-1/2° south.
101. _The Meridian Altitude of the Moon._--The _meridian altitude_ of
any body is its altitude when on the meridian. In our latitude, the
meridian altitude of any point on the equinoctial is forty-nine degrees.
The meridian altitude of the summer solstice is 49° + 23-1/2° = 72-1/2°,
and that of the winter solstice is 49° - 23-1/2° = 25-1/2°. The greatest
meridian altitude of the moon is 72-1/2° + 5° = 77-1/2°, and its least
meridian altitude, 25-1/2° - 5° = 20-1/2°.
When the moon's meridian altitude is greater than the elevation of the
equinoctial, it is said to run _high_, and when less, to run _low_. The
full moon runs high when the sun is south of the equinoctial, and low
when the sun is north of the equinoctial. This is because the full moon
is always in the opposite part of the heavens to the sun.
102. _Wet and Dry Moon._--At the time of new moon, the cusps of the
crescent sometimes lie in a line which is nearly perpendicular with
the horizon, and sometimes in a line which is nearly parallel with
the horizon. In the former case the moon is popularly described as a
_wet_ moon, and in the latter case as a _dry_ moon.
[Illustration: Fig. 114.]
The great circle which passes through the centre of the sun and moon
will pass through the centre of the crescent, and be perpendicular
to the line joining the cusps. Now the ecliptic makes the least
angle with the horizon when the vernal equinox is on the eastern
horizon and the autumnal equinox is on the western. In our latitude,
as we have seen, this angle is 25-1/2°: hence in our latitude, if
the moon were at new on the ecliptic when the sun is at the autumnal
equinox, as shown at _M_{3}_ (Fig. 114), the great circle passing
through the centre of the sun and moon would be the ecliptic, and at
New York would be inclined to the horizon at an angle of 25-1/2°. If
the moon happened to be 5° south of the ecliptic at this time, as at
_M_{4}_, the great circle passing through the centre of the sun and
moon would make an angle of only 20-1/2° with the horizon. In either
of these cases the line joining the cusps would be nearly
perpendicular to the horizon.
[Illustration: Fig. 115.]
If the moon were at new on the ecliptic when the sun is near the
vernal equinox, as shown at _M_{1}_ (Fig. 115), the great circle
passing through the centres of the sun and moon would make an angle
of 72-1/2° with the horizon at New York; and were the moon 5° north
of the ecliptic at that time, as shown at _M_{2}_, this great circle
would make an angle of 77-1/2° with the horizon. In either of these
cases, the line joining the cusps would be nearly parallel with the
horizon.
At different times, the line joining the cusps may have every
possible inclination to the horizon between the extreme cases shown
in Figs. 114 and 115.
103. _Daily Retardation of the Moon's Rising._--The moon rises, on the
average, about fifty minutes later each day. This is owing to her
eastward motion. As the moon makes a complete revolution around the
earth in about twenty-seven days, she moves eastward at the rate of
about thirteen degrees a day, or about twelve degrees a day faster than
the sun. Were the moon, therefore, on the horizon at any hour to-day,
she would be some twelve degrees below the horizon at the same hour
to-morrow. Now, as the horizon moves at the rate of one degree in four
minutes, it would take it some fifty minutes to come up to the moon so
as to bring her upon the horizon. Hence the daily retardation of the
moon's rising is about fifty minutes; but it varies considerably in
different parts of her orbit.
There are two reasons for this variation in the daily retardation:--
(1) The moon moves at a _varying rate in her orbit_; her speed being
greatest at perigee, and least at apogee: hence, other things being
equal, the retardation is greatest when the moon is at perigee, and
least when she is at apogee.
[Illustration: Fig. 116.]
[Illustration: Fig. 117.]
(2) The moon moves at a _varying angle to the horizon_. The moon
moves nearly in the plane of the ecliptic, and of course she passes
both equinoxes every lunation. When she is near the autumnal
equinox, her path makes the greatest angle with the eastern horizon,
and when she is near the vernal equinox, the least angle: hence the
moon moves away from the horizon fastest when she is near the
autumnal equinox, and slowest when she is near the vernal equinox.
This will be evident from Figs. 116 and 117. In each figure, _SN_
represents a portion of the eastern horizon, and _Ec_, _E'c'_, a
portion of the ecliptic. _AE_, in Fig. 116, represents the autumnal
equinox, and _AEM_ the daily motion of the moon. _VE_, in Fig. 117,
represents the vernal equinox, and _VEM'_ the motion of the moon for
one day. In the first case this motion would carry the moon away
from the horizon the distance _AM_, and in the second case the
distance _A'M'_. Now, it is evident that _AM_ is greater than
_A'M'_: hence, other things being equal, the greatest retardation of
the moon's rising will be when the moon is near the autumnal
equinox, and the least retardation when the moon is near the vernal
equinox.
The least retardation at New York is twenty-three minutes, and the
greatest an hour and seventeen minutes. The greatest and least
retardations vary somewhat from month to month; since they depend not
only upon the position of the moon in her orbit with reference to the
equinoxes, but also upon the latitude of the moon, and upon her nearness
to the earth.
[Illustration: Fig. 118.]
The direction of the moon's motion with reference to the ecliptic is
shown in Fig. 118, which shows the moon's motion for one day in July,
1876.
104. _The Harvest Moon_--The long and short retardations in the rising
of the moon, though they occur every month, are not likely to attract
attention unless they occur at the time of full moon. The long
retardations for full moon occur when the moon is near the autumnal
equinox at full. As the full moon is always opposite to the sun, the sun
must in this case be near the vernal equinox: hence the long
retardations for full moon occur in the spring, the greatest retardation
being in March.
The least retardations for full moon occur when the moon is near the
vernal equinox at full: the sun must then be near the autumnal equinox.
Hence the least retardations for full moon occur in the months of
August, September, and October. The retardation is, of course, least for
September; and the full moon of this month rises night after night less
than half an hour later than the previous night. The full moon of
September is called the "Harvest Moon," and that of October the
"Hunter's Moon."
105. _The Rotation of the Moon._--A careful examination of the spots on
the disc of the moon reveals the fact that she always presents the same
side to the earth. In order to do this, she must rotate on her axis
while making a revolution around the earth, or in about twenty-seven
days.
106. _Librations of the Moon._--The moon appears to rock slowly to and
fro, so as to allow us to see alternately a little farther around to the
right and the left, or above and below, than we otherwise could. This
apparent rocking of the moon is called _libration_. The moon has three
librations:--
(1) _Libration in Latitude._--This libration enables us to see
alternately a little way around on the northern and southern limbs of
the moon.
This libration is due to the fact that the axis of the moon is not
quite perpendicular to the plane of her orbit. The deviation from
the perpendicular is six degrees and a half. As the axis of the
moon, like that of the earth, maintains the same direction, the
poles of the moon will be turned alternately six degrees and a half
toward and from the earth.
(2) _Libration in Longitude._--This libration enables us to see
alternately a little farther around on the eastern and western limbs of
the moon.
[Illustration: Fig. 119.]
It is due to the fact that the moon's axial motion is uniform, while
her orbital motion is not. At perigee her orbital motion will be in
advance of her axial motion, while at apogee the axial motion will
be in advance of the orbital. In Fig. 119, _E_ represents the earth,
_M_ the moon, the large arrow the direction of the moon's motion in
her orbit, and the small arrow the direction of her motion of
rotation. When the moon is at _M_, the line _AB_, drawn
perpendicular to _EM_, represents the circle which divides the
visible from the invisible portion of the moon. While the moon is
passing from _M_ to _M'_, the moon performs less than a quarter of a
rotation, so that _AB_ is no longer perpendicular to _EM'_. An
observer on the earth can now see somewhat beyond _A_ on the western
limb of the moon, and not quite up to _B_ on the eastern limb. While
the moon is passing from _M'_ to _M''_, her axial motion again
overtakes her orbital motion, so that the line _AB_ again becomes
perpendicular to the line joining the centre of the moon to the
centre of the earth. Exactly the same side is now turned towards the
earth as when the moon was at _M_. While the moon passes from _M''_
to _M'''_, her axial motion gets in advance of her orbital motion,
so that _AB_ is again inclined to the line joining the centres of
the earth and moon. A portion of the eastern limb of the moon beyond
_B_ is now brought into view to the earth, and a portion of the
western limb at _A_ is carried out of view. While the moon is
passing from _M'''_ to _M_, the orbital motion again overtakes the
axial motion, and _AB_ is again perpendicular to _ME_.
(3) _Parallactic Libration._--While an observer at the centre of the
earth would get the same view of the moon, whether she were on the
eastern horizon, in the zenith, or on the western horizon, an observer
on the surface of the earth does not get exactly the same view in these
three cases. When the moon is on the eastern horizon, an observer on the
surface of the earth would see a little farther around on the western
limb of the moon than when she is in the zenith, and not quite so far
around on the eastern limb. On the contrary, when the moon is on the
western horizon, an observer on the surface of the earth sees a little
farther around on the eastern limb of the moon than when she is in the
zenith, and not quite so far around on her western limb.
[Illustration: Fig. 120.]
This will be evident from Fig. 120. _E_ is the centre of the earth,
and _O_ a point on its surface. _AB_ is a line drawn through the
centre of the moon, perpendicular to a line joining the centres of
the moon and the earth. This line marks off the part of the moon
turned towards the centre of the earth, and remains essentially the
same during the day. _CD_ is a line drawn through the centre of the
moon perpendicular to a line joining the centre of the moon and the
point of observation. This line marks off the part of the moon
turned towards _O_. When the moon is in the zenith, _CD_ coincides
with _AB_; but, when the moon is on the horizon, _CD_ is inclined to
_AB_. When the moon is on the eastern horizon, an observer at _O_
sees a little beyond _B_, and not quite to _A_; and, when she is on
the western horizon, he sees a little beyond _A_, and not quite to
_B_. _B_ is on the western limb of the moon, and _A_ on her eastern
limb.
Since this libration is due to the point from which the moon is
viewed, it is called _parallactic_ libration; and, since it occurs
daily, it is called _diurnal_ libration.
[Illustration: Fig. 121.]
107. _Portion of the Lunar Surface brought into View by
Libration._--The area brought into view by the first two librations
is between one-twelfth and one-thirteenth of the whole lunar
surface, or nearly one-sixth of the hemisphere of the moon which is
turned away from the earth when the moon is at her state of mean
libration. Of course a precisely equal portion of the hemisphere
turned towards us during mean libration is carried out of view by
the lunar librations.
If we add to each of these areas a fringe about one degree wide, due
to the diurnal libration, and which we may call the _parallactic_
fringe, we shall find that the total area brought into view is
almost exactly one-eleventh part of the whole surface of the moon. A
similar area is carried out of view; so that the whole region thus
swayed out of and into view amounts to two-elevenths of the moon's
surface. This area is shown in Fig. 121, which is a side view of the
moon.
[Illustration: Fig. 122.]
108. _The Moon's Path through Space._--Were the earth stationary,
the moon would describe an ellipse around it similar to that of Fig.
113; but, as the earth moves forward in her orbit at the same time
that the moon revolves around it, the moon is made to describe a
sinuous path, as shown by the continuous line in Fig. 122. This
feature of the moon's path is greatly exaggerated in the upper
portion of the diagram. The form of her path is given with a greater
degree of accuracy in the lower part of the figure (the broken line
represents the path of the earth); but even here there is
considerable exaggeration. The complete serpentine path of the moon
around the sun is shown, greatly exaggerated, in Fig. 123, the
broken line being the path of the earth.
[Illustration: Fig. 123.]
The path described by the moon through space is much the same as
that described by a point on the circumference of a wheel which is
rolled over another wheel. If we place a circular disk against the
wall, and carefully roll along its edge another circular disk (to
which a piece of lead pencil has been fastened so as to mark upon
the wall), the curve described will somewhat resemble that described
by the moon. This curve is called an _epicycloid_, and it will be
seen that at every point it is concave towards the centre of the
larger disk. In the same way the moon's orbit is _at every point
concave towards the sun_.
[Illustration: Fig. 124.]
The exaggeration of the sinuosity in Fig. 123 will be more evident
when it is stated, that, on the scale of Fig. 124, the whole of the
serpentine curve would lie _within the breadth_ of the fine circular
line _MM'_.
109. _The Lunar Day._--The lunar day is twenty-nine times and a half as
long as the terrestrial day. Near the moon's equator the sun shines
without intermission nearly fifteen of our days, and is absent for the
same length of time. Consequently, the vicissitudes of temperature to
which the surface is exposed must be very great. During the long lunar
night the temperature of a body on the moon's surface would probably
fall lower than is ever known on the earth, while during the day it must
rise higher than anywhere on our planet.
[Illustration: Fig. 125.]