with a telescope, is found to present a succession of phases similar to
those of the moon. The reason of this is evident from Fig. 146. As an
inferior planet passes around the sun, it presents sometimes more and
sometimes less of its bright hemisphere to the earth. When the earth is
at _T_, and Venus at superior conjunction, the planet turns the whole of
its bright hemisphere towards the earth, and appears _full_; it then
becomes _gibbous_, _half_, and _crescent_. When it comes into _inferior
conjunction_, it turns its dark hemisphere towards the earth: it then
becomes _crescent_, _half_, _gibbous_, and _full_ again.
128. _The Sidereal and Synodical Periods of an Inferior Planet._--The
time it takes a planet to make a complete revolution around the sun is
called the _sidereal period_ of the planet; and the time it takes it to
pass from one aspect around to the same aspect again, its _synodical
period_.
[Illustration: Fig. 147.]
The synodical period of an inferior planet is longer than its sidereal
period. This will be evident from an examination of Fig. 147. _S_ is the
position of the sun, _E_ that of the earth, and _P_ that of the planet
at inferior conjunction. Before the planet can be in inferior
conjunction again, it must pass entirely around its orbit, and overtake
the earth, which has in the mean time passed on in its orbit to _E'_.
While the earth is passing from _E_ to _E'_, the planet passes entirely
around its orbit, and from _P_ to _P'_ in addition. Now the arc _PP'_ is
just equal to the arc _EE'_: hence the planet has to pass over the same
arc that the earth does, and 360° more. In other words, the planet has
to gain 360° on the earth.
The synodical period of the planet is found by direct observation.
129. _The Length of the Sidereal Period._--The length of the
sidereal period of an inferior planet may be found by the following
computation:--
Let _a_ denote the synodical period of the planet,
Let _b_ denote the sidereal period of the earth,
Let _x_ denote the sidereal period of the planet.
Then _360°/b_ = the daily motion of the earth,
And _360°/x_ = the daily motion of the planet,
And _360°/x - 360°/b_ = the daily gain of the planet:
Also _360°/a_ = the daily gain of the planet:
Hence _360°/x - 360°/b = 360°/a_.
Dividing by 360°, we have _1/x - 1/b = 1/a_;
Clearing of fractions, we have _ab - ax = bx_:
Transposing and collecting, we have _(a + b)x = ab_:
Therefore _x = ab/a+b_.
130. _The Relative Distance of an Inferior Planet._--By the
_relative distance_ of a planet, we mean its distance from the sun
compared with the earth's distance from the sun. The relative
distance of an inferior planet may be found by the following
method:--
[Illustration: Fig. 148.]
Let _V_, in Fig. 148, represent the position of Venus at its
greatest elongation from the sun, _S_ the position of the sun, and
_E_ that of the earth. The line _EV_ will evidently be tangent to a
circle described about the sun with a radius equal to the distance
of Venus from the sun at the time of this greatest elongation. Draw
the radius _SV_ and the line _SE_. Since _SV_ is a radius, the angle
at _V_ is a right angle. The angle at _E_ is known by measurement,
and the angle at _S_ is equal to 90°- the angle _E_. In the
right-angled triangle _EVS_, we then know the three angles, and we
wish to find the ratio of the side _SV_ to the side _SE_.
The ratio of these lines may be found by trigonometrical computation
as follows:--
_VS : ES = sin SEV : 1._
Substitute the value of the sine of SEV, and we have
_VS : ES = .723 : 1._
Hence the relative distances of Venus and of the earth from the sun
are .723 and 1.
Superior Planets.
131. _The Superior Planets._--The _superior planets_ are those which lie
beyond the earth. They are _Mars_, the _Asteroids_, _Jupiter_, _Saturn_,
_Uranus_, and _Neptune_.
[Illustration: Fig. 149.]
132. _Apparent Motion of a Superior Planet._--In order to deduce the
apparent motion of a superior planet from the real motions of the earth
and planet, let _S_ (Fig. 149) be the place of the sun; 1, 2, 3, etc.,
the orbit of the earth; _a_, _b_, _c_, etc., the orbit of Mars; and
_CGL_ a part of the starry firmament. Let the orbit of the earth be
divided into twelve equal parts, each described in one month; and let
_ab_, _bc_, _cd_, etc., be the spaces described by Mars in the same
time. Suppose the earth to be at the point 1 when Mars is at the point
_a_, Mars will then appear in the heavens in the direction of 1 _a_.
When the earth is at 3, and Mars at _c_, he will appear in the heavens
at _C_. When the earth arrives at 4, Mars will arrive at _d_, and will
appear in the heavens at _D_. While the earth moves from 4 to 5 and from
5 to 6, Mars will appear to have advanced among the stars from _D_ to
_E_ and from _E_ to _F_, in the direction from west to east. During the
motion of the earth from 6 to 7 and from 7 to 8, Mars will appear to go
backward from _F_ to _G_ and from _G_ to _H_, in the direction from east
to west. During the motion of the earth from 8 to 9 and from 9 to 10,
Mars will appear to advance from _H_ to _I_ and from _I_ to _K_, in the
direction from west to east, and the motion will continue in the same
direction until near the succeeding opposition.
The apparent motion of a superior planet projected on the heavens is
thus seen to be similar to that of an inferior planet, except that, in
the latter case, the retrogression takes place near inferior
conjunction, and in the former it takes place near opposition.
[Illustration: Fig. 150.]
133. _Aspects of a Superior Planet._--The four aspects of a superior
planet are shown in Fig. 150, in which _S_ is the position of the sun,
_E_ that of the earth, and _P_ that of the planet.
When the planet is on the opposite side of the earth to the sun, as at
_P_, it is said to be in _opposition_. The sun and the planet will then
appear in opposite parts of the heavens, the sun appearing at _C_, and
the planet at _A_.
When the planet is on the opposite side of the sun to the earth, as at
_P''_, it is said to be in _superior conjunction_. It will then appear
in the same part of the heavens as the sun, both appearing at _C_.
When the planet is at _P'_ and _P'''_, so that a line drawn from the
earth through the planet will make a right angle with a line drawn from
the earth to the sun, it is said to be in _quadrature_. At _P'_ it is in
its western quadrature, and at _P'''_ in its eastern quadrature.
[Illustration: Fig. 151.]
134. _Phases of a Superior Planet._--Mars is the only one of the
superior planets that has appreciable phases. At quadrature, as will
appear from Fig. 151, Mars does not present quite the same side to the
earth as to the sun: hence, near these parts of its orbit, the planet
appears slightly gibbous. Elsewhere in its orbit, the planet appears
full.
All the other superior planets are so far away from the sun and earth,
that the sides which they turn towards the sun and the earth in every
part of their orbit are so nearly the same, that no change in the form
of their disks can be detected.
135. _The Synodical Period of a Superior Planet._--During a synodical
period of a superior planet the earth must gain one revolution, or 360°,
on the planet, as will be evident from an examination of Fig. 152, in
which _S_ represents the sun, _E_ the earth, and _P_ the planet at
opposition. Before the planet can be in opposition again, the earth must
make a complete revolution, and overtake the planet, which has in the
mean time passed on from _P_ to _P'_.
[Illustration: Fig. 152.]
In the case of most of the superior planets the synodical period is
shorter than the sidereal period; but in the case of Mars it is longer,
since Mars makes more than a complete revolution before the earth
overtakes it.
The synodical period of a superior planet is found by direct
observation.
136. _The Sidereal Period of a Superior Planet._--The sidereal
period of a superior planet is found by a method of computation
similar to that for finding the sidereal period of an inferior
planet:--
Let _a_ denote the synodical period of the planet,
Let _b_ denote the sidereal period of the earth,
Let _x_ denote the sidereal period of the planet.
Then will _360°/b_ = daily motion of the earth,
And _360°/x_ = daily motion of the planet;
Also _360°/b - 360°/x_ = daily gain of the earth.
But _360°/a_ = daily gain of the earth:
Hence _360°/b - 360°/x = 360°/a_
_1/b - 1/x = 1/a_
_ax - ab = bx_
_(a-b)x = ab_
_x = ab/(a-b)_.
[Illustration: Fig. 153.]
137. _The Relative Distance of a Superior Planet._--Let _S_, _e_,
and _m_, in Fig. 153, represent the relative positions of the sun,
the earth, and Mars, when the latter planet is in opposition. Let
_E_ and _M_ represent the relative positions of the earth and Mars
the day after opposition. At the first observation Mars will be seen
in the direction _emA_, and at the second observation in the
direction _EMA_.
But the fixed stars are so distant, that if a line, _eA_, were drawn
to a fixed star at the first observation, and a line, _EB_, drawn
from the earth to the same fixed star at the second observation,
these two lines would be sensibly parallel; that is, the fixed star
would be seen in the direction of the line _eA_ at the first
observation, and in the direction of the line _EB_, parallel to
_eA_, at the second observation. But if Mars were seen in the
direction of the fixed star at the first observation, it would
appear back, or west, of that star at the second observation by the
angular distance _BEA_; that is, the planet would have retrograded
that angular distance. Now, this retrogression of Mars during one
day, at the time of opposition, can be measured directly by
observation. This measurement gives us the value of the angle _BEA_;
but we know the rate at which both the earth and Mars are moving in
their orbits, and from this we can easily find the angular distance
passed over by each in one day. This gives us the angles _ESA_ and
_MSA_. We can now find the relative length of the lines _MS_ and
_ES_ (which represent the distances of Mars and of the earth from
the sun), both by construction and by trigonometrical computation.
Since _EB_ and _eA_ are parallel, the angle _EAS_ is equal to _BEA_.
_SEA = 180° - (ESA + EAS)_
_ESM = ESA - MSA_
_EMS = 180° - (SEA + ESM)_.
We have then
_MS : ES = sin SEA : sin EMS._
Substituting the values of the sines, and reducing the ratio to its
lowest terms, we have
_MS : ES = 1.524 : 1._
Thus we find that the relative distances of Mars and the earth from
the sun are 1.524 and 1. By the simple observation of its greatest
elongation, we are able to determine the relative distances of an
inferior planet and the earth from the sun; and, by the equally
simple observation of the daily retrogression of a superior planet,
we can find the relative distances of such a planet and the earth
from the sun.
IV. THE SUN.
I. MAGNITUDE AND DISTANCE OF THE SUN.
[Illustration: Fig. 154.]
138. _The Volume of the Sun._--The apparent diameter of the sun is about
32', being a little greater than that of the moon. The real diameter of
the sun is 866,400 miles, or about a hundred and nine times that of the
earth.
As the diameter of the moon's orbit is only about 480,000 miles, or some
sixty times the diameter of the earth, it follows that the diameter of
the sun is nearly double that of the moon's orbit: hence, were the
centre of the sun placed at the centre of the earth, the sun would
completely fill the moon's orbit, and reach nearly as far beyond it in
every direction as it is from the earth to the moon. The circumference
of the sun as compared with the moon's orbit is shown in Fig. 154.
The volume of the sun is 1,305,000 times that of the earth.
139. _The Mass of the Sun._--The sun is much less dense than the earth.
The mass of the sun is only 330,000 times that of the earth, and its
density only about a fourth that of the earth.
To find the mass of the sun, we first ascertain the distance the
earth would draw the moon towards itself in a given time, were the
moon at the distance of the sun, and then form the proportion: as
the distance the earth would draw the moon towards itself is to the
distance that the sun draws the earth towards itself in the same
time, so is the mass of the earth to the mass of the sun.
Although the mass of the sun is over three hundred thousand times that
of the earth, the pull of gravity at the surface of the sun is only
about twenty-eight times as great as at the surface of the earth. This
is because the distance from the surface of the sun to its centre is
much greater than from the surface to the centre of the earth.
[Illustration: Fig. 155.]
140. _Size of the Sun Compared with that of the Planets._--The size of
the sun compared with that of the larger planets is shown in Fig. 155.
The mass of the sun is more than seven hundred and fifty times that of
all of the planets and moons in the solar system. In Fig. 156 is shown
the apparent size of the sun as seen from the different planets. The
apparent diameter of the sun decreases as the distance from it
increases, and the disk of the sun decreases as the square of the
distance from it increases.
[Illustration: Fig. 156.]
141. _The Distance of the Sun._--The mean distance of the sun from the
earth is about 92,800,000 miles. Owing to the eccentricity of the
earth's orbit, the distance of the sun varies somewhat; being about
3,000,000 miles less in January, when the earth is at perihelion, than
in June, when the earth is at aphelion.
"But, though the distance of the sun can easily be stated in
figures, it is not possible to give any real idea of a space so
enormous: it is quite beyond our power of conception. If one were to
try to walk such a distance, supposing that he could walk four miles
an hour, and keep it up for ten hours every day, it would take
sixty-eight years and a half to make a single million of miles, and
more than sixty-three hundred years to traverse the whole.
"If some celestial railway could be imagined, the journey to the
sun, even if our trains ran sixty miles an hour day and night and
without a stop, would require over a hundred and seventy-five years.
Sensation, even, would not travel so far in a human lifetime. To
borrow the curious illustration of Professor Mendenhall, if we could
imagine an infant with an arm long enough to enable him to touch the
sun and burn himself, he would die of old age before the pain could
reach him; since, according to the experiments of Helmholtz and
others, a nervous shock is communicated only at the rate of about a
hundred feet per second, or 1,637 miles a day, and would need more
than a hundred and fifty years to make the journey. Sound would do
it in about fourteen years, if it could be transmitted through
celestial space; and a cannon-ball in about nine, if it were to move
uniformly with the same speed as when it left the muzzle of the gun.
If the earth could be suddenly stopped in her orbit, and allowed to
fall unobstructed toward the sun, under the accelerating influence
of his attraction, she would reach the centre in about four months.
I have said if she could be stopped; but such is the compass of her
orbit, that, to make its circuit in a year, she has to move nearly
nineteen miles a second, or more than fifty times faster than the
swiftest rifle-ball; and, in moving twenty miles, her path deviates
from perfect straightness by less than an eighth of an inch. And
yet, over all the circumference of this tremendous orbit, the sun
exercises his dominion, and every pulsation of his surface receives
its response from the subject earth." (Professor C. A. Young: The
Sun.)
142. _Method of Finding the Sun's Distance._--There are several
methods of finding the sun's distance. The simplest method is that
of finding the actual distance of one of the nearer planets by
observing its displacement in the sky as seen from widely separated
points on the earth. As the _relative_ distances of the planets from
each other and from the sun are well known, we can easily deduce the
actual distance of the sun if we can find that of any of the
planets. The two planets usually chosen for this method are Mars and
Venus.
(1) The displacement of Mars in the sky, as seen from two
observatories which differ considerably in latitude, is, of course,
greatest when Mars is nearest the earth. Now, it is evident than
Mars will be nearer the earth when in opposition than when in any
other part of its orbit; and the planet will be least distant from
the earth when it is at its perihelion point, and the earth is at
its aphelion point, at the time of opposition. This method, then,
can be used to the best advantage, when, at the time of opposition,
Mars is near its perihelion, and the earth near its aphelion. These
favorable oppositions occur about once in fifteen years, and the
last one was in 1877.
[Illustration: Fig. 157.]
Suppose two observers situated at _N'_ and _S'_ (Fig. 157), near the
poles of the earth. The one at _N'_ would see Mars in the sky at
_N_, and the one at _S'_ would see it at _S_. The displacement would
be the angle _NMS_. Each observer measures carefully the distance of
Mars from the same fixed star near it. The difference of these
distances gives the displacement of the planet, or the angle _NMS_.
These observations were made with the greatest care in 1877.
(2) Venus is nearest the earth at the time of inferior conjunction;
but it can then be seen only in the daytime. It is, therefore,
impossible to ascertain the displacement of Venus, as seen from
different stations, by comparing her distances from a fixed star.
Occasionally, at the time of inferior conjunction, Venus passes
directly across the sun's disk. The last of these _transits_ of
Venus occurred in 1874, and the next will occur in 1882. It will
then be over a hundred years before another will occur.
[Illustration: Fig. 158.]
Suppose two observers, _A_ and _B_ (Fig. 158), near the poles of the
earth at the time of a transit of Venus. The observer at _A_ would
see Venus crossing the sun at _V_{2}_, and the one at _B_ would see
it crossing at _V_{1}_. Any observation made upon Venus, which would
give the distance and direction of Venus from the centre of the sun,
as seen from each station, would enable us to calculate the angular
distance between the two chords described across the sun. This, of
course, would give the displacement of Venus on the sun's disk. This
method was first employed at the last transits of Venus which
occurred before 1874; namely, those of 1761 and 1769.
There are three methods of observation employed to ascertain the
apparent direction and distance of Venus from the centre of the sun,
called respectively the _contact method_, the _micrometric method_,
and the _photographic method_.
(_a_) In the _contact_ method, the observation consists in noting
the exact time when Venus crosses the sun's limb. To ascertain this
it is necessary to observe the exact time of external and internal
contact. This observation, though apparently simple, is really very
difficult. With reference to this method Professor Young says,--
"The difficulties depend in part upon the imperfections of optical
instruments and the human eye, partly upon the essential nature of
light leading to what is known as diffraction, and partly upon the
action of the planet's atmosphere. The two first-named causes
produce what is called irradiation, and operate to make the apparent
diameter of the planet, as seen on the solar disk, smaller than it
really is; smaller, too, by an amount which varies with the size of
the telescope, the perfection of its lenses, and the tint and
brightness of the sun's image. The edge of the planet's image is
also rendered slightly hazy and indistinct.
[Illustration: Fig. 159.]
"The planet's atmosphere also causes its disk to be surrounded by a
narrow ring of light, which becomes visible long before the planet
touches the sun, and, at the moment of internal contact, produces an
appearance, of which the accompanying figure is intended to give an
idea, though on an exaggerated scale. The planet moves so slowly as
to occupy more than twenty minutes in crossing the sun's limb; so
that even if the planet's edge were perfectly sharp and definite,
and the sun's limb undistorted, it would be very difficult to
determine the precise second at which contact occurs. But, as things
are, observers with precisely similar telescopes, and side by side,
often differ from each other five or six seconds; and, where the
telescopes are not similar, the differences and uncertainties are
much greater.... Astronomers, therefore, at present are pretty much
agreed that such observations can be of little value in removing the
remaining uncertainty of the parallax, and are disposed to put more
reliance upon the micrometric and photographic methods, which are
free from these peculiar difficulties, though, of course, beset with
others, which, however, it is hoped will prove less formidable."
(_b_) Of the _micrometric_ method, as employed at the last transit,
Professor Young speaks as follows:--
"The micrometric method requires the use of a heliometer,--an
instrument common only in Germany, and requiring much skill and
practice in its use in order to obtain with it accurate measures. At
the late transit, a single English party, two or three of the
Russian parties, and all five of the German, were equipped with
these instruments; and at some of the stations extensive series of
measures were made. None of the results, however, have appeared as
yet; so that it is impossible to say how greatly, if at all, this
method will have the advantage in precision over the contact
observations."
(_c_) The following observations, with reference to the
_photographic_ method, are also taken from Professor Young:--
"The Americans and French placed their main reliance upon the
photographic method, while the English and Germans also provided for
its use to a certain extent. The great advantage of this method is,
that it makes it possible to perform the necessary measurements
(upon whose accuracy every thing depends) at leisure after the
transit, without hurry, and with all possible precautions. The
field-work consists merely in obtaining as many and as good pictures
as possible. A principal objection to the method lies in the
difficulty of obtaining good pictures, i.e., pictures free from
distortion, and so distinct and sharp as to bear high magnifying
power in the microscopic apparatus used for their measurement. The
most serious difficulty, however, is involved in the accurate
determination of the scale of the picture; that is, of the number of
seconds of arc corresponding to a linear inch upon the plate.
Besides this, we must know the exact Greenwich time at which each
picture is taken, and it is also extremely desirable that the
_orientation_ of the picture should be accurately determined; that
is, the north and south, the east and west points of the solar image
on the finished plate. There has been a good deal of anxiety lest
the image, however accurate and sharp when first produced, should
alter, in course of time, through the contraction of the collodion
film on the glass plate; but the experiments of Rutherfurd, Huggins,
and Paschen, seem to show that this danger is imaginary.... The
Americans placed the photographic telescope exactly in line with a
meridian instrument, and so determined, with the extremest
precision, the direction in which it was pointed. Knowing this and
the time at which any picture was taken, it becomes possible, with
the help of the plumb-line image, to determine precisely the
orientation of the picture,--an advantage possessed by the American
pictures alone, and making their value nearly twice as great as
otherwise it would have been.
"The figure below is a representation of one of the American
photographs reduced about one-half. _V_ is the image of Venus,
which, on the actual plate, is about a seventh of an inch in
diameter; _aa'_ is the image of the plumb-line. The centre of the
reticle is marked with a cross."
[Illustration: Fig. 160.]
The English photographs proved to be of little value, and the
results of the measurements and calculations upon the American
pictures have not yet been published. There is a growing
apprehension that no photographic method can be relied upon.
The most recent determinations by various methods indicate that the
sun's distance is such that his parallax is about eighty-eight seconds.
This would make the linear value of a second at the surface of the sun
about four hundred and fifty miles.
[Illustration: Plate 1.]
II. PHYSICAL AND CHEMICAL CONDITION OF THE SUN.
Physical Condition of the Sun.