degrees. His horizon reaches fifty degrees beyond the pole of the earth,
and gives the same apparent elevation to the pole of the heavens. It
cuts the equator and all the circles of daily motion, at an angle of
forty degrees,--being always equal to what the altitude of the pole
lacks of ninety degrees: that is, it is always equal to the co-altitude
of the pole. Thus, let H O, Fig. 15, represent the horizon, E Q the
equator, and P P´ the axis of the earth. Also, _l l, m m, n n_,
parallels of latitude. Then the horizon of a spectator at Z, in latitude
fifty degrees, reaches to fifty degrees beyond the pole; and the angle E
C H, which the equator makes with the horizon, is forty degrees,--the
complement of the latitude. As we advance still further north, the
elevation of the diurnal circle above the horizon grows less and less,
and consequently, the motions of the heavenly bodies more and more
oblique to the horizon, until finally, at the pole, where the latitude
is ninety degrees, the angle of elevation of the equator vanishes, and
the horizon and the equator coincide with each other, as before stated.
[Illustration Fig. 15.]
_The circle of perpetual apparition is the boundary of that space around
the elevated pole, where the stars never set._ Its distance from the
pole is equal to the latitude of the place. For, since the altitude of
the pole is equal to the latitude, a star, whose polar distance is just
equal to the latitude, will, when at its lowest point, only just reach
the horizon; and all the stars nearer the pole than this will evidently
not descend so far as the horizon. Thus _m m_, Fig. 15, is the circle of
perpetual apparition, between which and the north pole, the stars never
set, and its distance from the pole, O P, is evidently equal to the
elevation of the pole, and of course to the latitude.
In the opposite hemisphere, a similar part of the sphere adjacent to the
depressed pole never rises. Hence, _the circle of perpetual occultation
is the boundary of that space around the depressed pole, within which
the stars never rise._
Thus _m´ m´_, Fig. 15, is the circle of perpetual occultation, between
which and the south pole, the stars never rise.
In an oblique sphere, the horizon cuts the circles of daily motion
unequally. Towards the elevated pole, more than half the circle is above
the horizon, and a greater and greater portion, as the distance from the
equator is increased, until finally, within the circle of perpetual
apparition, the whole circle is above the horizon. Just the opposite
takes place in the hemisphere next the depressed pole. Accordingly, when
the sun is in the equator, as the equator and horizon, like all other
great circles of the sphere, bisect each other, the days and nights are
equal all over the globe. But when the sun is north of the equator, the
days become longer than the nights, but shorter, when the sun is south
of the equator. Moreover, the higher the latitude, the greater is the
inequality in the lengths of the days and nights. By examining Fig. 15,
you will easily see how each of these cases must hold good.
Most of the appearances of the diurnal revolution can be explained,
either on the supposition that the celestial sphere actually turns
around the earth once in twenty-four hours, or that this motion of the
heavens is merely apparent, arising from the revolution of the earth on
its axis, in the opposite direction,--a motion of which we are
insensible, as we sometimes lose the consciousness of our own motion in
a ship or steam-boat, and observe all external objects to be receding
from us, with a common motion. Proofs, entirely conclusive and
satisfactory, establish the fact, that it is the earth, and not the
celestial sphere, that turns; but these proofs are drawn from various
sources, and one is not prepared to appreciate their value, or even to
understand some of them, until he has made considerable proficiency in
the study of astronomy, and become familiar with a great variety of
astronomical phenomena. To such a period we will therefore postpone the
discussion of the earth's rotation on its axis.
While we retain the same place on the earth, the diurnal revolution
occasions no change in our horizon, but our horizon goes round, as well
as ourselves. Let us first take our station on the equator, at sunrise;
our horizon now passes through both the poles and through the sun, which
we are to conceive of as at a great distance from the earth, and
therefore as cut, not by the terrestrial, but by the celestial, horizon.
As the earth turns, the horizon dips more and more below the sun, at the
rate of fifteen degrees for every hour; and, as in the case of the polar
star, the sun appears to rise at the same rate. In six hours, therefore,
it is depressed ninety degrees below the sun, bringing us directly under
the sun, which, for our present purpose, we may consider as having all
the while maintained the same fixed position in space. The earth
continues to turn, and in six hours more, it completely reverses the
position of our horizon, so that the western part of the horizon, which
at sunrise was diametrically opposite to the sun, now cuts the sun, and
soon afterwards it rises above the level of the sun, and the sun sets.
During the next twelve hours, the sun continues on the invisible side of
the sphere, until the horizon returns to the position from which it set
out, and a new day begins.
Let us next contemplate the similar phenomena at the _poles_. Here the
horizon, coinciding, as it does, with the equator, would cut the sun
through its centre and the sun would appear to revolve along the surface
of the sea, one half above and the other half below the horizon. This
supposes the sun in its annual revolution to be at one of the equinoxes.
When the sun is north of the equator, it revolves continually round in a
circle, which, during a single revolution, appears parallel to the
equator, and it is constantly day; and when the sun is south of the
equator, it is, for the same reason, continual night.
When we have gained a clear idea of the appearances of the diurnal
revolutions, as exhibited to a spectator at the equator and at the pole,
that is, in a right and in a parallel sphere, there will be little
difficulty in imagining how they must be in the intermediate latitudes,
which have an oblique sphere.
The appearances of the sun and stars, presented to the inhabitants of
different countries, are such as correspond to the sphere in which they
live. Thus, in the fervid climates of India, Africa, and South America,
the sun mounts up to the highest regions of the heavens, and descends
directly downwards, suddenly plunging beneath the horizon. His rays,
darting almost vertically upon the heads of the inhabitants, strike with
a force unknown to the people of the colder climates; while in places
remote from the equator, as in the north of Europe, the sun, in Summer,
rises very far in the north, takes a long circuit towards the south, and
sets as far northward in the west as the point where it rose on the
other side of the meridian. As we go still further north, to the
northern parts of Norway and Sweden, for example, to the confines of the
frigid zone, the Summer's sun just grazes the northern horizon, and at
noon appears only twenty-three and one half degrees above the southern.
On the other hand, in mid-winter, in the north of Europe, as at St.
Petersburgh, the day dwindles almost to nothing,--lasting only while the
sun describes a very short arc in the extreme south. In some parts of
Siberia and Iceland, the only day consists of a little glimmering of the
sun on the verge of the southern horizon, at noon.
LETTER IX.
PARALLAX AND REFRACTION.
"Go, wondrous creature! mount where science guides,
Go measure earth, weigh air, and state the tides;
Instruct the planets in what orbs to run,
Correct old Time, and regulate the sun."--_Pope._
I THINK you must have felt some astonishment, that astronomers are able
to calculate the exact distances and magnitudes of the sun, moon, and
planets. We should, at the first thought, imagine that such knowledge as
this must be beyond the reach of the human faculties, and we might be
inclined to suspect that astronomers practise some deception in this
matter, for the purpose of exciting the admiration of the unlearned. I
will therefore, in the present Letter, endeavor to give you some clear
and correct views respecting the manner in which astronomers acquire
this knowledge.
In our childhood, we all probably adopt the notion that the sky is a
real dome of definite surface, in which the heavenly bodies are fixed.
When any objects are beyond a certain distance from the eye, we lose all
power of distinguishing, by our sight alone, between different
distances, and cannot tell whether a given object is one million or a
thousand millions of miles off. Although the bodies seen in the sky are
in fact at distances extremely various,--some, as the clouds, only a few
miles off; others, as the moon, but a few thousand miles; and others, as
the fixed stars, innumerable millions of miles from us,--yet, as our eye
cannot distinguish these different distances, we acquire the habit of
referring all objects beyond a moderate height to one and the same
surface, namely, an imaginary spherical surface, denominated the
celestial vault. Thus, the various objects represented in the diagram on
next page, though differing very much in shape and diameter, would all
be _projected_ upon the sky alike, and compose a part, indeed, of the
imaginary vault itself. The place which each object occupies is
determined by lines drawn from the eye of the spectator through the
extremities of the body, to meet the imaginary concave sphere. Thus, to
a spectator at O, Fig 16, the several lines A B, C D, and E F, would all
be projected into arches on the face of the sky, and be seen as parts of
the sky itself, as represented by the lines A´ B´, C´ D´, and E´ F´. And
were a body actually to move in the several directions indicated by
these lines, they would appear to the spectator to describe portions of
the celestial vault. Thus, even when moving through the crooked line,
from _a_ to _b_, a body would appear to be moving along the face of the
sky, and of course in a regular curve line, from _c_ to _d_.
[Illustration Fig. 16.]
But, although all objects, beyond a certain moderate height, are
projected on the imaginary surface of the sky, yet different spectators
will project the same object on _different parts_ of the sky. Thus, a
spectator at A, Fig. 17, would see a body, C, at M, while a spectator at
B would see the same body at N. This change of place in a body, as seen
from different points, is called parallax, which is thus defined:
_parallax is the apparent change of place which bodies undergo by being
viewed from different points_. [Illustration Fig. 17.]
The arc M N is called the _parallactic arc_, and the angle A C B, the
_parallactic angle_.
It is plain, from the figure, that near objects are much more affected
by parallax than distant ones. Thus, the body C, Fig. 17, makes a much
greater parallax than the more distant body D,--the former being
measured by the arc M N, and the latter by the arc O P. We may easily
imagine bodies to be so distant, that they would appear projected at
very nearly the same point of the heavens, when viewed from places very
remote from each other. Indeed, the fixed stars, as we shall see more
fully hereafter, are so distant, that spectators, a hundred millions of
miles apart, see each star in one and the same place in the heavens.
It is by means of parallax, that astronomers find the distances and
magnitudes of the heavenly bodies. In order fully to understand this
subject, one requires to know something of trigonometry, which science
enables us to find certain unknown parts of a triangle from certain
other parts which are known. Although you may not be acquainted with the
principles of trigonometry, yet you will readily understand, from your
knowledge of arithmetic, that from certain things given in a problem
others may be found. Every triangle has of course three sides and three
angles; and, if we know two of the angles and one of the sides, we can
find all the other parts, namely, the remaining angle and the two
unknown sides. Thus, in the triangle A B C, Fig. 18, if we know the
length of the side A B, and how many degrees each of the angles A B C
and B C A contains, we can find the length of the side B C, or of the
side A C, and the remaining angle at A. Now, let us apply these
principles to the measurements of some of the heavenly bodies.
[Illustration Fig. 18.]
[Illustration Fig. 19.]
In Fig. 19, let A represent the earth, C H the horizon, and H Z a
quadrant of a great circle of the heavens, extending from the horizon to
the zenith; and let E, F, G, O, be successive positions of the moon, at
different elevations, from the horizon to the meridian. Now, a spectator
on the surface of the earth, at A, would refer the moon, when at E, to
_h_, on the face of the sky, whereas, if seen from the centre of the
earth, it would appear at H. So, when the moon was at F, a spectator at
A would see it at _p_, while, if seen from the centre, it would have
appeared at P. The parallactic arcs, H _h_, P _p_, R _r_, grow
continually smaller and smaller, as a body is situated higher above the
horizon; and when the body is in the zenith, then the parallax vanishes
altogether, for at O the moon would be seen at Z, whether viewed from A
or C.
Since, then, a heavenly body is liable to be referred to different
points on the celestial vault, when seen from different parts of the
earth, and thus some confusion be occasioned in the determination of
points on the celestial sphere, astronomers have agreed to consider the
true place of a celestial object to be that where it would appear, if
seen from the centre of the earth; and the doctrine of parallax teaches
how to reduce observations made at any place on the surface of the
earth, to such as they would be, if made from the centre.
When the moon, or any heavenly body, is seen in the horizon, as at E,
the change of place is called the horizontal parallax. Thus, the angle A
E C, measures the horizontal parallax of the moon. Were a spectator to
view the earth from the centre of the moon, he would see the
semidiameter of the earth under this same angle; hence, _the horizontal
parallax of any body is the angle subtended by the semidiameter of the
earth, as seen from the body_. Please to remember this fact.
It is evident from the figure, that the effect of parallax upon the
place of a celestial body is to _depress_ it. Thus, in consequence of
parallax, E is depressed by the arc H _h_; F, by the arc P _p_; G, by
the arc R _r_; while O sustains no change. Hence, in all calculations
respecting the altitude of the sun, moon, or planets, the amount of
parallax is to be added: the stars, as we shall see hereafter, have no
sensible parallax.
It is now very easy to see how, when the parallax of a body is known, we
may find its distance from the centre of the earth. Thus, in the
triangle A C E, Fig. 19, the side A C is known, being the semidiameter
of the earth; the angle C A E, being a right angle, is also known; and
the parallactic angle, A E C, is found from observation; and it is a
well-known principle of trigonometry, that when we have any two angles
of a triangle, we may find the remaining angle by subtracting the sum of
these two from one hundred and eighty degrees. Consequently, in the
triangle A E C, we know all the angles and one side, namely, the side A
C; hence, we have the means of finding the side C E, which is the
distance from the centre of the earth to the centre of the moon.
[Illustration Fig. 20.]
When the distance of a heavenly body is known, and we can measure, with
instruments, its angular breadth, we can easily determine its
_magnitude_. Thus, if we have the distance of the moon, E S, Fig. 20,
and half the breadth of its disk S C, (which is measured by the angle S
E C,) we can find the length of the line, S C, in miles. Twice this line
is the diameter of the body; and when we know the diameter of a sphere,
we can, by well-known rules, find the contents of the surface, and its
solidity.
You will perhaps be curious to know, _how the moon's horizontal parallax
is found_; for it must have been previously ascertained, before we could
apply this method to finding the distance of the moon from the earth.
Suppose that two astronomers take their stations on the same meridian,
but one south of the equator, as at the Cape of Good Hope, and another
north of the equator, as at Berlin, in Prussia, which two places lie
nearly on the same meridian. The observers would severally refer the
moon to different points on the face of the sky,--the southern observer
carrying it further north, and the northern observer further south,
than its true place, as seen from the centre of the earth. This will be
plain from the diagram, Fig. 21. If A and B represent the positions of
the spectators, M the moon, and C D an arc of the sky, then it is
evident, that C D would be the parallactic arc.
[Illustration Fig. 21.]
These observations furnish materials for calculating, by the aid of
trigonometry, the moon's horizontal parallax, and we have before seen
how, when we know the parallax of a heavenly body, we can find both its
distance from the earth and its magnitude.
Beside the change of place which these heavenly bodies undergo, in
consequence of parallax, there is another, of an opposite kind, arising
from the effect of the atmosphere, called _refraction_. Refraction
elevates the apparent place of a body, while parallax depresses it. It
affects alike the most distant as well as nearer bodies.
In order to understand the nature of refraction, we must consider, that
an object always appears in the direction in which the _last_ ray of
light comes to the eye. If the light which comes from a star were bent
into fifty directions before it reached the eye, the star would
nevertheless appear in the line described by the ray nearest the eye.
The operation of this principle is seen when an oar, or any stick, is
thrust into water. As the rays of light by which the oar is seen have
their direction changed as they pass out of water into air, the apparent
direction in which the body is seen is changed in the same degree,
giving it a bent appearance,--the part below the water having apparently
a different direction from the part above. Thus, in Fig. 22, page 96, if
S _a x_ be the oar, S _a b_ will be the bent appearance, as affected by
refraction. The transparent substance through which any ray of light
passes is called a _medium_. It is a general fact in optics, that, when
light passes out of a rarer into a denser medium, as out of air into
water, or out of space into air, it is turned _towards_ a perpendicular
to the surface of the medium; and when it passes out of a denser into a
rarer medium, as out of water into air, it is turned _from_ the
perpendicular. In the above case, the light, passing out of space into
air, is turned towards the radius of the earth, this being perpendicular
to the surface of the atmosphere; and it is turned more and more towards
that radius the nearer it approaches to the earth, because the density
of the air rapidly increases near the earth.
[Illustration Fig. 22.]
Let us now conceive of the atmosphere as made up of a great number of
parallel strata, as A A, B B, C C, and D D, increasing rapidly in
density (as is known to be the fact) in approaching near to the surface
of the earth. Let S be a star, from which a ray of light, S _a_, enters
the atmosphere at _a_, where, being much turned towards the radius of
the convex surface, it would change its direction into the line _a b_,
and again into _b c_, and _c_ O, reaching the eye at O. Now, since an
object always appears in the direction in which the light finally
strikes the eye, the star would be seen in the direction O _c_, and,
consequently, the star would apparently change its place, by
refraction, from S to S´, being elevated out of its true position.
Moreover, since, on account of the continual increase of density in
descending through the atmosphere, the light would be continually turned
out of its course more and more, it would therefore move, not in the
polygon represented in the figure, but in a corresponding curve line,
whose curvature is rapidly increased near the surface of the earth.
When a body is in the zenith, since a ray of light from it enters the
atmosphere at right angles to the refracting medium, it suffers no
refraction. Consequently, the position of the heavenly bodies, when in
the zenith, is not changed by refraction, while, near the horizon, where
a ray of light strikes the medium very obliquely, and traverses the
atmosphere through its densest part, the refraction is greatest. The
whole amount of refraction, when a body is in the horizon, is
thirty-four minutes; while, at only an elevation of one degree, the
refraction is but twenty-four minutes; and at forty-five degrees, it is
scarcely a single minute. Hence it is always important to make our
observations on the heavenly bodies when they are at as great an
elevation as possible above the horizon, being then less affected by
refraction than at lower altitudes.
Since the whole amount of refraction near the horizon exceeds
thirty-three minutes, and the diameters of the sun and moon are
severally less than this, these luminaries are in view both before they
have actually risen and after they have set.
The rapid increase of refraction near the horizon is strikingly evinced
by the _oval_ figure which the sun assumes when near the horizon, and
which is seen to the greatest advantage when light clouds enable us to
view the solar disk. Were all parts of the sun equally raised by
refraction, there would be no change of figure; but, since the lower
side is more refracted than the upper, the effect is to shorten the
vertical diameter, and thus to give the disk an oval form. This effect
is particularly remarkable when the sun, at his rising or setting, is
observed from the top of a mountain, or at an elevation near the
seashore; for in such situations, the rays of light make a greater angle
than ordinary with a perpendicular to the refracting medium, and the
amount of refraction is proportionally greater. In some cases of this
kind, the shortening of the vertical diameter of the sun has been
observed to amount to six minutes, or about one fifth of the whole.
The apparent enlargement of the sun and moon, when near the horizon,
arises from an optical illusion. These bodies, in fact, are not seen
under so great an angle when in the horizon as when on the meridian, for
they are nearer to us in the latter case than in the former. The
distance of the sun, indeed, is so great, that it makes very little
difference in his apparent diameter whether he is viewed in the horizon
or on the meridian; but with the moon, the case is otherwise; its
angular diameter, when measured with instruments, is perceptibly larger
when at its culmination, or highest elevation above the horizon. Why,
then, do the sun and moon appear so much larger when near the horizon?
It is owing to a habit of the mind, by which we judge of the magnitudes
of distant objects, not merely by the angle they subtend at the eye, but
also by our impressions respecting their distance, allowing, under a
given angle, a greater magnitude as we imagine the distance of a body to
be greater. Now, on account of the numerous objects usually in sight
between us and the sun, when he is near the horizon, he appears much
further removed from us than when on the meridian; and we unconsciously
assign to him a proportionally greater magnitude. If we view the sun, in
the two positions, through a smoked glass, no such difference of size is
observed; for here no objects are seen but the sun himself.
_Twilight_ is another phenomenon depending on the agency of the earth's
atmosphere. It is that illumination of the sky which takes place just
before sunrise and which continues after sunset. It is owing partly to
refraction, and partly to reflection, but mostly to the latter. While
the sun is within eighteen degrees of the horizon, before it rises or
after it sets, some portion of its light is conveyed to us, by means of
numerous reflections from the atmosphere. At the equator, where the
circles of daily motion are perpendicular to the horizon, the sun
descends through eighteen degrees in an hour and twelve minutes. The
light of day, therefore, declines rapidly, and as rapidly advances after
daybreak in the morning. At the pole, a constant twilight is enjoyed
while the sun is within eighteen degrees of the horizon, occupying
nearly two thirds of the half year when the direct light of the sun is
withdrawn, so that the progress from continual day to constant night is
exceedingly gradual. To an inhabitant of an oblique sphere, the twilight
is longer in proportion as the place is nearer the elevated pole.