MOVEMENTS OF PLANETS, SATELLITES, AND COMETS.
APPARENT MOVEMENTS OF PLANETS.—It has already been pointed out that like
the sun and moon, the planets also have an apparent movement with
respect to the more distant stars. Mercury and Venus are never seen very
far from the sun, while other planets, among which are Mars, Jupiter,
and Saturn, may be seen in the part of the heavens opposite to the sun.
One point, and that a very important one, which we notice from our
observations is that the planets never depart very far from the
ecliptic, so that the planes in which they perform their movements are
nearly coincident with the plane in which our own annual journey round
the sun is performed. The apparent movements of the planets are such
that it is quite impossible to regard these bodies as circulating in
regular orbits round the earth itself. If they revolve round any other
body it is manifest that their apparent or geocentric motions will be
compounded of the real movements of the planets and that of the earth.
It is not necessary here to trace the steps by which it has been
determined that the planets revolve in regular orbits around the sun.
Suffice it to say that their observed movements are simply and
sufficiently explained by supposing that, like the earth, which may now
be regarded as a planet, they travel in elliptic orbits with the sun at
one of the foci. Besides this revolution, the planets have a rotatory
motion about their axes, but this question cannot be studied apart from
the telescopic features, and will therefore be treated in Section III.
of the present work.
The circumstance that the planets Mercury and Venus are never seen long
after sunset or before sunrise, indicates that their orbits must lie
between us and the sun. Hence, they are distinguished as the _interior
planets_, while those outside the earth’s orbit are called the _exterior
planets_.
MOVEMENTS OF INTERIOR PLANETS.—Let us consider briefly the conditions
under which we observe the interior planets. If such a planet be
represented by M in Fig. 23, while the earth is represented by E
traversing a larger orbit, the planet is said to be in _inferior
conjunction_ with the sun, when it lies directly between the sun and
earth. The actual movements of the planets being direct—that is,
anticlockwise—the planet at M has an apparent westerly motion as seen by
an observer situated on the earth, and from this we gather that it moves
more rapidly than the earth. For simplicity let us regard the earth as
being at rest at the point E. Then, as the planet reaches the position
M′, where it is as far as possible to the west of the sun, it is said to
be at its _greatest western elongation_. Proceeding in its orbit, the
planet’s apparent movement is direct, and it eventually comes in line
with the sun on the further side as seen from the earth; it is then said
to be in _superior conjunction_. From this point the planet moves to the
east of the sun until it comes to the point M, after which the motion
becomes retrograde, and the planet proceeds to inferior conjunction
again. When at its greatest distance to the east of the sun, as at M‴,
the planet is said to be at its _greatest eastern elongation_. Taking
the term _elongation_ in general, it may be regarded as a measure of the
angular distance of a planet from the sun as observed from the earth.
[Illustration:
FIG. 23.—_Movement of an Interior Planet._
]
If the orbits of the planets were perfect circles, the greatest
elongation distances of an interior planet would always be the same;
sometimes, however, we are nearer to the sun than at the other times,
and the apparent separation of the planet from the sun would seem
greater than at other times, even if there were no other cause at work.
The variations of the elongation distances are greater than can be
accounted for by our own varying distance, and are naturally attributed
to the elliptical form of the orbits of the interior planets themselves.
Mercury, for example, sometimes only departs 18° from the sun, while at
other times it reaches as far as 28° east or west.
When we take account of the fact that the earth has also a movement
along its orbit, it will be seen that the same conditions hold good with
regard to elongations and conjunctions, except that the intervals
between them will be longer.
[Illustration:
FIG. 24.—_Morning and Evening Stars._
]
MORNING AND EVENING STARS.—From superior to inferior conjunction an
interior planet is to the east of the sun. It then rises after the sun,
and sets after the sun, so that it is visible for a short time in the
early evening; in other words, it is an _evening star_ during this part
of its path. Between inferior and superior conjunctions, the planet is
conversely a _morning star_. This is illustrated in Fig. 24, where the
position of an observer towards whom the sun is rising is shown at A. An
interior planet at P is above the horizon at sunrise, but will be below
at sunset, the observer having been carried to A′ by the earths
rotation; it will thus be a morning star. When the planet occupies the
position P′ it is below the horizon at sunrise, but will remain in sight
after the sun has set in the evening, the observer then having been
transferred to A′ by the earth’s rotation.
PHASES OF INTERIOR PLANETS.—From the conditions which have been stated
with regard to the movements of the interior planets, one is not
surprised to find that telescopic examination reveals that these bodies
put on phases similar to those of the moon. At superior conjunction the
planets exhibit a fully illuminated disc, at greatest elongations they
appear as a half moon, while at inferior conjunction their dark sides
alone are presented to us. The apparent sizes of the planets, as
measured with the aid of a telescope, are also found to vary according
to their positions; when at inferior conjunction, the planet is much
nearer to us than at other times, and it consequently appears larger.
The apparent brightness of an interior planet also varies. At superior
conjunction the whole of the disc is illuminated, but the planet is then
so far removed from us that its light is very feeble. On the other hand,
at inferior conjunction, when it is nearest to us, the dark side of the
planet is turned towards us. The greatest brightness thus occurs at some
intermediate point. In the case of Venus this is between the greatest
elongations and inferior conjunction, when it is 40° from the sun. It is
then bright enough to be seen with the naked eye in full sunshine, and
has sometimes, on such occasions, been erroneously regarded by ignorant
persons as the Star of Bethlehem.
TRANSIT OF VENUS.—If an inferior conjunction occurs when the planet is
very near to a node—this term having the same significance as in the
case of the moon (p. 94)—the planet, whether it be Mercury or Venus,
will be seen projected as a dark spot upon the bright disc of the sun.
Such an occurrence is called a _transit of Venus_ or of Mercury, as the
case may be. Just as we do not get an eclipse of the sun every month, so
we do not get a transit of Venus every time the earth and that planet
have the same heliocentric longitude, and for the same reason, namely,
that the plane of the orbit is inclined to the ecliptic. As we shall see
in another chapter, a transit of Venus has a most important application
in the determination of one of the fundamental constants of
astronomy—the sun’s distance. The conditions as to the recurrence of
transits are of great interest. In the case of Venus, the _synodic_
period is 584 days, this being the time which elapses between two
successive inferior conjunctions. Five synodic periods are thus very
nearly equal to eight years, and 152 synodic revolutions are even more
nearly equal to 243 years. As seen from the earth, the sun crosses the
nodes of the orbit of Venus on June 5 and December 7, and since there
can be no transit when the planet is more than 4½° from the node, the
transits will all occur about these dates. A transit will be followed by
another after the lapse of 8 years, if the planet is not too far from
the node; but there can be no other transit with the planet at the same
node until 243 years have elapsed. There are, however, transits
occurring at similar intervals when the planet is at the other node. The
following dates on which transits have occurred, or will occur, will
illustrate the foregoing statements:—
8 years│December 7, 1631,│243 years.│——————————
„ │December 4, 1639,│ „ │243 years.
8 years│December 9, 1874,│ „ │ „
„ │December 6, 1882,│——————————│ „
8 years│June 5, 1761, │243 years.│——————————
„ │June 3, 1769, │ „ │243 years.
8 years│June 8, 2004, │ „ │ „
„ │June 6, 2012, │——————————│ „
[Illustration:
FIG. 25.—_Movement of an Exterior Planet._
]
MOVEMENTS OF EXTERIOR PLANETS.—The exterior planets are at once
recognised as such by their occasional appearance in the part of the sky
opposite to that of the sun. They are then said to be in _opposition_.
When in the same line as the sun, and on the remote side of it, as at P′
in Fig. 25, the planet is in _conjunction_. The apparent movements of
such a planet are very complex. Neglecting for a moment the earth’s
motion, it is evident that the apparent rate of movement of the planet
with reference to the stars will vary very considerably according as the
planet is near opposition or near conjunction, the movement appearing to
be most rapid when the planet is nearest to us. Upon this unequal rate
of motion is superposed a varying direction of motion produced by the
changing position of the earth. When the planet is at P, and the earth
at E, both are moving in the same direction, but as the earth has the
greater angular velocity, the apparent motion of the planet will be
retrograde, that is, the planet will appear to go backwards in its path.
If the earth be near the point E′, its orbital movement will be directed
away from the planet, and will scarcely affect its apparent position;
accordingly, about this time the planet has a direct movement in the
heavens. Between these two points the direction of the apparent movement
of the planet has changed, so that at some intermediate position it
would seem to have suspended its wanderings; here we have a _stationary
point_. For a certain time, before and after conjunction, the linear
directions of movements of the earth and planet will be opposed to each
other, and on this account the _direct_ apparent motion of the planet
will be accelerated. Presently, as the earth gains on the planet,
another stationary point will be reached, and with the approach to
opposition the planet will again retrograde.
If both orbits were in the same plane, these apparent movements would
all be backwards and forwards along a great circle of the celestial
sphere coincident with the ecliptic, the eastward movement
predominating. The planes in which the planets perform their revolutions
are, however, inclined to the ecliptic, and the result is that they
appear to us to travel in loops, some of which are illustrated in Fig.
26.
[Illustration:
FIG. 26.—_Apparent Paths of Ceres, Pallas, Juno and Vesta, in 1896._
]
From the fact that we are constantly within the orbit of an outer
planet, it is evident that we must always see more than half of the
planetary hemisphere on which the sun is shining. Consequently, an
exterior planet never puts on a crescent phase, or presents the
appearance of a half moon. The nearer the planet the greater will be the
dark area which it is possible for us to observe. In the case of Mars,
for example, we sometimes see it gibbous like the moon about three days
from full, but in the more distant planets this gibbosity is scarcely
perceptible. The greatest phase of an exterior planet occurs when it is
at _quadrature_, that is, when a line joining the earth and sun is
perpendicular to one joining the earth with the planet.
FAVOURABLE AND UNFAVOURABLE OPPOSITIONS.—A little consideration of Fig.
25 will make it perfectly clear that an exterior planet is very much
nearer to us at a time of opposition than at a conjunction. We are, in
fact, then, nearer to the planet by the diameter of the earth’s orbit, a
matter of some 186 millions of miles. Accordingly, the planets, more
especially our neighbour Mars, are best studied in the telescope about a
time of opposition. Now, if we had to deal with circular orbits, the
distance of a planet at opposition would remain constant, and we should
see the planet equally well at all oppositions. It is found, however,
that this is not the case, and the ellipticity of the orbits of the
earth and planets supplies a simple and sufficient explanation. Sir
Robert Ball illustrates this in the case of Mars by a diagram similar to
Fig. 27. It will be seen that, when the opposition occurs in August, the
earth is much nearer to Mars than when it happens at other times. The
least favourable oppositions are those which occur in February, the
planet then being nearly twice as far removed from us as at the nearest
approach during an August opposition.
[Illustration:
FIG. 27.—_Opposition of Mars._
]
As regards the more distant planets, the diameter of the earth’s orbit
and the variations of opposition distance are of less importance, since
they form a much smaller proportion of the distances of those planets
from the sun.
ELEMENTS OF A PLANETARY ORBIT.—A complete study of the apparent
movements of the planets with which we are acquainted shows that their
real movements are performed round the sun in ellipses, the sun being
placed at a focus. Each orbit, like that of the earth, has its
perihelion and aphelion points, and its apse line; not being coincident
with the ecliptic, it will have a line of nodes, and an ascending and
descending node. Each planet will further have a particular inclination
to the ecliptic, and a period of revolution peculiar to itself.
Consequently, to systematise our knowledge of any particular orbit,
certain conventions are adopted, and the seven things we must know, in
order that we may specify the size of the orbit, its position in space,
and the situation of the planet in its orbit, are as follows:—
_a_ = Semi axis major of elliptic orbit.
_e_ = Eccentricity.
_i_ = Inclination to ecliptic.
Ω = Longitude of ascending node.
π = Longitude of perihelion.
P = Period of revolution. (_u_, the mean daily motion, sometimes
replaces P.)
E = The epoch, giving the longitude of the planet at some particular
time.[2]
The first two quantities indicate the size and shape of the orbit, the
next three its position with regard to the ecliptic, and the last two
are required to determine the situation of the planet in its orbit. Some
of the elements are illustrated in Fig. 28.
[Illustration:
FIG. 28.—_Elements of an Elliptic Orbit._
]
DETERMINATION OF A PLANET’S PERIOD.—Observations enable us to determine
the synodic period of a planet, and knowing that the earth’s period is a
year, it is a simple matter to determine that of the planet. In the case
of an exterior planet, the interval from opposition to opposition
furnishes the best means of determining the synodic period. The exact
moment of an opposition cannot usually be directly observed, and what
one actually does is to measure the R.A. and declination of the sun on
several days about the time of opposition, as also those of the planet;
then, by reducing these co-ordinates to celestial longitude and
latitude, it is not difficult to determine at what moment the longitudes
differed by 180°, that is, the moment at which opposition took place.
The problem of finding the planet’s sidereal period, then, amounts to
this: at what rate must the planet be moving in order that the earth may
make a complete revolution, and move, in addition, through the same
angle as the planet? In other words, what must be the period of the
planet in order that the earth may gain a whole revolution in the
interval corresponding to the synodic period? The daily movement of the
planet will be 360°/P, and that of the earth 360°/365¼, if P denote the
number of days in the planet’s sidereal period. The earth’s gain per day
will thus be the difference between these two quantities, and since a
whole revolution is gained in the synodic period, the gain per day can
be expressed as 360°/S, where S represents the synodic period; thus we
get
360°/365¼ − 360°/P = 360°/S
or
1/365¼ − 1/P = 1/S
The synodic period of Mars is 780 days, and the application of the
foregoing formula leads us to 687 days as the time of its revolution
round the sun.
A single determination of a synodic period does not give precise
results, for the reason that the orbits of the planets are elliptical,
and the intervals consequently dependent upon whether the planet is near
perihelion, or far removed from it when an opposition is observed. It
is, therefore, necessary to determine the time of opposition at long
intervals, and so reduce the errors in measuring the length of a single
period.
MOVEMENTS OF SATELLITES.—Telescopic observations show that some of the
planets are accompanied by _satellites_, which revolve round their
primaries as the moon revolves round the earth. The apparent movements
of these bodies, with regard to the planets, are very similar to those
of the interior planets with regard to the sun, having similar points of
greatest eastern and western elongations. The facts which have been
collected show that each satellite, like our own moon, moves in an
elliptical orbit, with the planet in one of its foci. With one
exception, the satellites attending the planets of our system have a
direct movement; those of Uranus, however, have apparently a movement in
the same direction as the hands of a watch, but this can be regarded as
direct, if we consider the plane of the orbit to be inclined more than
90° to the plane of the ecliptic.
THE ORBITS OF COMETS.—Another class of bodies which circulate round the
sun now claims our attention. These are the _comets_, some of which are
never seen without the aid of telescopes, while others have been
brilliant enough to excite a widespread wonder and interest. They
usually have a very rapid movement relatively to the stars; and to learn
something as to their real motions, we commence by measuring their right
ascensions and declinations as frequently as possible. When such
observations are plotted, they give us the geocentric movement of a
comet, which generally seems very irregular, and gives one the idea that
it is subject to no law. Unlike the planets, comets do not usually keep
near the ecliptic, but move in planes inclined at all angles to it.
Their rates of apparent movement also change very rapidly.
When the effect of the earth’s movement upon that of a comet is
eliminated, it is found that the movement of the comet is performed
either in an ellipse, a parabola, or an hyperbola, the sun in each case
occupying one of the foci.
From our definition of the eccentricity of an ellipse, it will be seen
that, when the eccentricity is zero, we have a circle. When the
eccentricity becomes unity, the ellipse becomes a parabola, so that the
latter curve may be regarded as part of an ellipse, of which the foci
are at an infinite distance apart. In the case of the hyperbola, the
eccentricity is greater than unity.
Comets which move round the sun in ellipses are called _periodic
comets_, for the reason that they return regularly into the sun’s
neighbourhood. Those which traverse parabolic or hyperbolic paths will
pass once round the sun and continue to journey into the depths of
interstellar space until their movements are changed by the proximity of
other bodies into the neighbourhood of which their wanderings may take
them.
When a new comet is observed, one of the things which astronomers
endeavour to do is to determine its orbit, so that its path may be
predicted with sufficient accuracy to enable it to be picked up readily
with a telescope when it becomes so feeble that it is no longer visible
to the naked eye. In the first instance, the motion is assumed to be
parabolic, and any deviation from such an orbit forms the subject of a
rigorous calculation by means of which the precise form is determined.