Transcriber's Note: The angle symbol has been replaced
with [angle]. Greek letters have been replaced with their
transliterations in brackets, e. g. [a] for alpha. The
upside-down Omega symbol has been replaced with [mho].
TWENTIETH CENTURY TEXT-BOOKS
EDITED BY
A. F. NIGHTINGALE, PH.D., LL.D.
FORMERLY SUPERINTENDENT OF HIGH SCHOOLS, CHICAGO
[Illustration: A TOTAL SOLAR ECLIPSE.
After Burckhalter's photographs of the eclipse of May 28, 1900.]
TWENTIETH CENTURY TEXT-BOOKS
A TEXT-BOOK OF
ASTRONOMY
BY
GEORGE C. COMSTOCK
DIRECTOR OF THE WASHBURN OBSERVATORY AND
PROFESSOR OF ASTRONOMY IN THE
UNIVERSITY OF WISCONSIN
[Illustration]
NEW YORK
D. APPLETON AND COMPANY
1903
COPYRIGHT, 1901
BY D. APPLETON AND COMPANY
PREFACE
The present work is not a compendium of astronomy or an outline course
of popular reading in that science. It has been prepared as a text-book,
and the author has purposely omitted from it much matter interesting as
well as important to a complete view of the science, and has endeavored
to concentrate attention upon those parts of the subject that possess
special educational value. From this point of view matter which permits
of experimental treatment with simple apparatus is of peculiar value and
is given a prominence in the text beyond its just due in a well-balanced
exposition of the elements of astronomy, while topics, such as the
results of spectrum analysis, which depend upon elaborate apparatus, are
in the experimental part of the work accorded much less space than their
intrinsic importance would justify.
Teacher and student are alike urged to magnify the observational side of
the subject and to strive to obtain in their work the maximum degree of
precision of which their apparatus is capable. The instruments required
are few and easily obtained. With exception of a watch and a protractor,
all of the apparatus needed may be built by any one of fair mechanical
talent who will follow the illustrations and descriptions of the text.
In order that proper opportunity for observations may be had, the study
should be pursued during the milder portion of the year, between April
and November in northern latitudes, using clear weather for a direct
study of the sky and cloudy days for book work.
The illustrations contained in the present work are worthy of as careful
study as is the text, and many of them are intended as an aid to
experimental work and accurate measurement, e. g., the star maps, the
diagrams of the planetary orbits, pictures of the moon, sun, etc. If the
school possesses a projection lantern, a set of astronomical slides to
be used in connection with it may be made of great advantage, if the
pictures are studied as an auxiliary to Nature. Mere display and scenic
effect are of little value.
A brief bibliography of popular literature upon astronomy may be found
at the end of this book, and it will be well if at least a part of these
works can be placed in the school library and systematically used for
supplementary reading. An added interest may be given to the study if
one or more of the popular periodicals which deal with astronomy are
taken regularly by the school and kept within easy reach of the
students. From time to time the teacher may well assign topics treated
in these periodicals to be read by individual students and presented to
the class in the form of an essay.
The author is under obligations to many of his professional friends who
have contributed illustrative matter for his text, and his thanks are in
an especial manner due to the editors of the Astrophysical Journal,
Astronomy and Astrophysics, and Popular Astronomy for permission to
reproduce here plates which have appeared in those periodicals, and to
Dr. Charles Boynton, who has kindly read and criticised the proofs.
GEORGE C. COMSTOCK.
UNIVERSITY OF WISCONSIN, _February, 1901_.
CONTENTS
CHAPTER PAGE
I.--DIFFERENT KINDS OF MEASUREMENT 1
The measurement of angles and time.
II.--THE STARS AND THEIR DIURNAL MOTION 10
Finding the stars--Their apparent motion--
Latitude--Direction of the meridian--Sidereal
time--Definitions.
III.--FIXED AND WANDERING STARS 29
Apparent motion of the sun, moon, and
planets--Orbits of the planets--How to find
the planets.
IV.--CELESTIAL MECHANICS 46
Kepler's laws--Newton's laws of motion--The law
of gravitation--Orbital motion--Perturbations--
Masses of the planets--Discovery of Neptune--
The tides.
V.--THE EARTH AS A PLANET 70
Size--Mass--Precession--The warming of the
earth--The atmosphere--Twilight.
VI.--THE MEASUREMENT OF TIME 86
Solar and sidereal time--Longitude--The
calendar--Chronology.
VII.--ECLIPSES 101
Their cause and nature--Eclipse limits--Eclipse
maps--Recurrence and prediction of eclipses.
VIII.--INSTRUMENTS AND THE PRINCIPLES INVOLVED IN THEIR USE 121
The clock--Radiant energy--Mirrors and lenses--
The telescope--Camera--Spectroscope--Principles
of spectrum analysis.
IX.--THE MOON 150
Numerical data--Phases--Motion--Librations--Lunar
topography--Physical condition.
X.--THE SUN 178
Numerical data--Chemical nature--Temperature--
Visible and invisible parts--Photosphere--Spots--
Faculæ--Chromosphere--Prominences--Corona--The
sun-spot period--The sun's rotation--Mechanical
theory of the sun.
XI.--THE PLANETS 212
Arrangement of the solar system--Bode's law--
Physical condition of the planets--Jupiter--
Saturn--Uranus and Neptune--Venus--Mercury--
Mars--The asteroids.
XII.--COMETS AND METEORS 251
Motion, size, and mass of comets--Meteors--Their
number and distribution--Meteor showers--Relation
of comets and meteors--Periodic comets--Comet
families and groups--Comet tails--Physical nature
of comets--Collisions.
XIII.--THE FIXED STARS 291
Number of the stars--Brightness--Distance--Proper
motion--Motion in line of sight--Double stars--
Variable stars--New stars.
XIV.--STARS AND NEBULÆ 330
Stellar colors and spectra--Classes of stars--
Clusters--Nebulæ--Their spectra and physical
condition--The Milky Way--Construction of the
heavens--Extent of the stellar system.
XV.--GROWTH AND DECAY 358
Logical bases and limitations--Development of the
sun--The nebular hypothesis--Tidal friction--Roche's
limit--Development of the moon--Development of stars
and nebulæ--The future.
APPENDIX 383
INDEX 387
LIST OF LITHOGRAPHIC PLATES
FACING PAGE
I.--Northern Constellations 124
II.--Equatorial Constellations 190
III.--Map of Mars 246
IV.--The Pleiades 344
Protractor _In pocket at back of book_
LIST OF FULL-PAGE ILLUSTRATIONS
FACING PAGE
A Total Solar Eclipse _Frontispiece_
The Harvard College Observatory, Cambridge, Mass. 24
Isaac Newton 46
Galileo Galilei 52
The Lick Observatory, Mount Hamilton, Cal. 60
The Yerkes Observatory, Williams Bay, Wis. 100
The Moon, one day after First Quarter 150
William Herschel 234
Pierre Simon Laplace 364
ASTRONOMY
CHAPTER I
DIFFERENT KINDS OF MEASUREMENT
1. ACCURATE MEASUREMENT.--Accurate measurement is the foundation of
exact science, and at the very beginning of his study in astronomy the
student should learn something of the astronomer's kind of measurement.
He should practice measuring the stars with all possible care, and
should seek to attain the most accurate results of which his instruments
and apparatus are capable. The ordinary affairs of life furnish abundant
illustration of some of these measurements, such as finding the length
of a board in inches or the weight of a load of coal in pounds and
measurements of both length and weight are of importance in astronomy,
but of far greater astronomical importance than these are the
measurement of angles and the measurement of time. A kitchen clock or a
cheap watch is usually thought of as a machine to tell the "time of
day," but it may be used to time a horse or a bicycler upon a race
course, and then it becomes an instrument to measure the amount of time
required for covering the length of the course. Astronomers use a clock
in both of these ways--to tell the time at which something happens or is
done, and to measure the amount of time required for something; and in
using a clock for either purpose the student should learn to take the
time from it to the nearest second or better, if it has a seconds hand,
or to a small fraction of a minute, by estimating the position of the
minute hand between the minute marks on the dial. Estimate the fraction
in tenths of a minute, not in halves or quarters.
EXERCISE 1.--If several watches are available, let one person tap
sharply upon a desk with a pencil and let each of the others note the
time by the minute hand to the nearest tenth of a minute and record the
observations as follows:
2h. 44.5m. First tap. 2h. 46.4m. 1.9m.
2h. 44.9m. Second tap. 2h. 46.7m. 1.8m.
2h. 46.6m. Third tap. 2h. 48.6m. 2.0m.
The letters h and m are used as abbreviations for hour and minute. The
first and second columns of the table are the record made by one
student, and second and third the record made by another. After all the
observations have been made and recorded they should be brought together
and compared by taking the differences between the times recorded for
each tap, as is shown in the last column. This difference shows how much
faster one watch is than the other, and the agreement or disagreement of
these differences shows the degree of accuracy of the observations. Keep
up this practice until tenths of a minute can be estimated with fair
precision.
2. ANGLES AND THEIR USE.--An angle is the amount of opening or
difference of direction between two lines that cross each other. At
twelve o'clock the hour and minute hand of a watch point in the same
direction and the angle between them is zero. At one o'clock the minute
hand is again at XII, but the hour hand has moved to I, one twelfth part
of the circumference of the dial, and the angle between the hands is one
twelfth of a circumference. It is customary to imagine the circumference
of a dial to be cut up into 360 equal parts--i. e., each minute space of
an ordinary dial to be subdivided into six equal parts, each of which
is called a degree, and the measurement of an angle consists in finding
how many of these degrees are included in the opening between its sides.
At one o'clock the angle between the hands of a watch is thirty degrees,
which is usually written 30°, at three o'clock it is 90°, at six o'clock
180°, etc.
A watch may be used to measure angles. How? But a more convenient
instrument is the protractor, which is shown in Fig. 1, applied to the
angle _A B C_ and showing that _A B C_ = 85° as nearly as the protractor
scale can be read.
The student should have and use a protractor, such as is furnished with
this book, for the numerous exercises which are to follow.
[Illustration: FIG. 1.--A protractor.]
EXERCISE 2.--Draw neatly a triangle with sides about 100 millimeters
long, measure each of its angles and take their sum. No matter what may
be the shape of the triangle, this sum should be very nearly
180°--exactly 180° if the work were perfect--but perfection can seldom
be attained and one of the first lessons to be learned in any science
which deals with measurement is, that however careful we may be in our
work some minute error will cling to it and our results can be only
approximately correct. This, however, should not be taken as an excuse
for careless work, but rather as a stimulus to extra effort in order
that the unavoidable errors may be made as small as possible. In the
present case the measured angles may be improved a little by adding
(algebraically) to each of them one third of the amount by which their
sum falls short of 180°, as in the following example:
Measured angles. Correction. Corrected angles.
° ° °
A 73.4 + 0.1 73.5
B 49.3 + 0.1 49.4
C 57.0 + 0.1 57.1
----- -----
Sum 179.7 180.0
Defect + 0.3
This process is in very common use among astronomers, and is called
"adjusting" the observations.
[Illustration: FIG. 2.--Triangulation.]
3. TRIANGLES.--The instruments used by astronomers for the measurement
of angles are usually provided with a telescope, which may be pointed at
different objects, and with a scale, like that of the protractor, to
measure the angle through which the telescope is turned in passing from
one object to another. In this way it is possible to measure the angle
between lines drawn from the instrument to two distant objects, such as
two church steeples or the sun and moon, and this is usually called the
angle between the objects. By measuring angles in this way it is
possible to determine the distance to an inaccessible point, as shown in
Fig. 2. A surveyor at _A_ desires to know the distance to _C_, on the
opposite side of a river which he can not cross. He measures with a tape
line along his own side of the stream the distance _A B_ = 100 yards and
then, with a suitable instrument, measures the angle at _A_ between the
points _C_ and _B_, and the angle at _B_ between _C_ and _A_, finding _B
A C_ = 73.4°, _A B C_ = 49.3°. To determine the distance _A C_ he draws
upon paper a line 100 millimeters long, and marks the ends _a_ and _b_;
with a protractor he constructs at _a_ the angle _b a c_ = 73.4°, and at
_b_ the angle _a b c_ = 49.3°, and marks by _c_ the point where the two
lines thus drawn meet. With the millimeter scale he now measures the
distance _a c_ = 90.2 millimeters, which determines the distance _A C_
across the river to be 90.2 yards, since the triangle on paper has been
made similar to the one across the river, and millimeters on the one
correspond to yards on the other. What is the proposition of geometry
upon which this depends? The measured distance _A B_ in the surveyor's
problem is called a base line.
EXERCISE 3.--With a foot rule and a protractor measure a base line and
the angles necessary to determine the length of the schoolroom. After
the length has been thus found, measure it directly with the foot rule
and compare the measured length with the one found from the angles. If
any part of the work has been carelessly done, the student need not
expect the results to agree.
[Illustration: FIG. 3.--Finding the moon's distance from the earth.]
In the same manner, by sighting at the moon from widely different parts
of the earth, as in Fig. 3, the moon's distance from us is found to be
about a quarter of a million miles. What is the base line in this case?
4. THE HORIZON--ALTITUDES.--In their observations astronomers and
sailors make much use of the _plane of the horizon_, and practically any
flat and level surface, such as that of a smooth pond, may be regarded
as a part of this plane and used as such. A very common observation
relating to the plane of the horizon is called "taking the sun's
altitude," and consists in measuring the angle between the sun's rays
and the plane of the horizon upon which they fall. This angle between a
line and a plane appears slightly different from the angle between two
lines, but is really the same thing, since it means the angle between
the sun's rays and a line drawn in the plane of the horizon toward the
point directly under the sun. Compare this with the definition given in
the geographies, "The latitude of a point on the earth's surface is its
angular distance north or south of the equator," and note that the
latitude is the angle between the plane of the equator and a line drawn
from the earth's center to the given point on its surface.
A convenient method of obtaining a part of the plane of the horizon for
use in observation is as follows: Place a slate or a pane of glass upon
a table in the sunshine. Slightly moisten its whole surface and then
pour a little more water upon it near the center. If the water runs
toward one side, thrust the edge of a thin wooden wedge under this side
and block it up until the water shows no tendency to run one way rather
than another; it is then level and a part of the plane of the horizon.
Get several wedges ready before commencing the experiment. After they
have been properly placed, drive a pin or tack behind each one so that
it may not slip.
5. TAKING THE SUN'S ALTITUDE. EXERCISE 4.--Prepare a piece of board 20
centimeters, or more, square, planed smooth on one face and one edge.
Drive a pin perpendicularly into the face of the board, near the middle
of the planed edge. Set the board on edge on the horizon plane and turn
it edgewise toward the sun so that a shadow of the pin is cast on the
plane. Stick another pin into the board, near its upper edge, so that
its shadow shall fall exactly upon the shadow of the first pin, and with
a watch or clock observe the time at which the two shadows coincide.
Without lifting the board from the plane, turn it around so that the
opposite edge is directed toward the sun and set a third pin just as the
second one was placed, and again take the time. Remove the pins and draw
fine pencil lines, connecting the holes, as shown in Fig. 4, and with
the protractor measure the angle thus marked. The student who has
studied elementary geometry should be able to demonstrate that at the
mean of the two recorded times the sun's altitude was equal to one half
of the angle measured in the figure.
[Illustration: FIG. 4.--Taking the sun's altitude.]
When the board is turned edgewise toward the sun so that its shadow is
as thin as possible, rule a pencil line alongside it on the horizon
plane. The angle which this line makes with a line pointing due south is
called the sun's _azimuth_. When the sun is south, its azimuth is zero;
when west, it is 90°; when east, 270°, etc.
EXERCISE 5.--Let a number of different students take the sun's altitude
during both the morning and afternoon session and note the time of each
observation, to the nearest minute. Verify the setting of the plane of
the horizon from time to time, to make sure that no change has occurred
in it.
6. GRAPHICAL REPRESENTATIONS.--Make a graph (drawing) of all the
observations, similar to Fig. 5, and find by bisecting a set of chords
_g_ to _g_, _e_ to _e_, _d_ to _d_, drawn parallel to _B B_, the time at
which the sun's altitude was greatest. In Fig. 5 we see from the
intersection of _M M_ with _B B_ that this time was 11h. 50m.
The method of graphs which is here introduced is of great importance in
physical science, and the student should carefully observe in Fig. 5
that the line _B B_ is a scale of times, which may be made long or
short, provided only the intervals between consecutive hours 9 to 10, 10
to 11, 11 to 12, etc., are equal. The distance of each little circle
from _B B_ is taken proportional to the sun's altitude, and may be upon
any desired scale--e. g., a millimeter to a degree--provided the same
scale is used for all observations. Each circle is placed accurately
over that part of the base line which corresponds to the time at which
the altitude was taken. Square ruled paper is very convenient, although
not necessary, for such diagrams. It is especially to be noted that from
the few observations which are represented in the figure a smooth curve
has been drawn through the circles which represent the sun's altitude,
and this curve shows the altitude of the sun at every moment between 9
A. M. and 3 P. M. In Fig. 5 the sun's altitude at noon was 57°. What was
it at half past two?
[Illustration: FIG. 5.--A graph of the sun's altitude.]
7. DIAMETER OF A DISTANT OBJECT.--By sighting over a protractor, measure
the angle between imaginary lines drawn from it to the opposite sides of
a window. Carry the protractor farther away from the window and repeat
the experiment, to see how much the angle changes. The angle thus
measured is called "the angle subtended" by the window at the place
where the measurement was made. If this place was squarely in front of
the window we may draw upon paper an angle equal to the measured one and
lay off from the vertex along its sides a distance proportional to the
distance of the window--e. g., a millimeter for each centimeter of real
distance. If a cross line be now drawn connecting the points thus found,
its length will be proportional to the width of the window, and the
width may be read off to scale, a centimeter for every millimeter in the
length of the cross line.
The astronomer who measures with an appropriate instrument the angle
subtended by the moon may in an entirely similar manner find the moon's
diameter and has, in fact, found it to be 2,163 miles. Can the same
method be used to find the diameter of the sun? A planet? The earth?
CHAPTER II
THE STARS AND THEIR DIURNAL MOTION
8. THE STARS.--From the very beginning of his study in astronomy, and as
frequently as possible, the student should practice watching the stars
by night, to become acquainted with the constellations and their
movements. As an introduction to this study he may face toward the
north, and compare the stars which he sees in that part of the sky with
the map of the northern heavens, given on Plate I, opposite page 124.
Turn the map around, upside down if necessary, until the stars upon it
match the brighter ones in the sky. Note how the stars are grouped in
such conspicuous constellations as the Big Dipper (Ursa Major), the
Little Dipper (Ursa Minor), and Cassiopeia. These three constellations
should be learned so that they can be recognized at any time.
_The names of the stars._--Facing the star map is a key which contains
the names of the more important constellations and the names of the
brighter stars in their constellations. These names are for the most
part a Greek letter prefixed to the genitive case of the Latin name of
the constellation. (See the Greek alphabet printed at the end of the
book.)
9. MAGNITUDES OF THE STARS.--Nearly nineteen centuries ago St. Paul
noted that "one star differeth from another star in glory," and no more
apt words can be found to mark the difference of brightness which the
stars present. Even prior to St. Paul's day the ancient Greek
astronomers had divided the stars in respect of brightness into six
groups, which the modern astronomers still use, calling each group a
_magnitude_. Thus a few of the brightest stars are said to be of the
first magnitude, the great mass of faint ones which are just visible to
the unaided eye are said to be of the sixth magnitude, and intermediate
degrees of brilliancy are represented by the intermediate magnitudes,
second, third, fourth, and fifth. The student must not be misled by the
word magnitude. It has no reference to the size of the stars, but only
to their brightness, and on the star maps of this book the larger and
smaller circles by which the stars are represented indicate only the
brightness of the stars according to the system of magnitudes. Following
the indications of these maps, the student should, in learning the
principal stars and constellations, learn also to recognize how bright
is a star of the second, fourth, or other magnitude.
10. OBSERVING THE STARS.--Find on the map and in the sky the stars
[a] Ursæ Minoris, [a] Ursæ Majoris, [b] Ursæ Majoris. What geometrical
figure will fit on to these stars? In addition to its regular name,
[a] Ursæ Minoris is frequently called by the special name Polaris, or
the pole star. Why are the other two stars called "the Pointers"? What
letter of the alphabet do the five bright stars in Cassiopeia suggest?
EXERCISE 6.--Stand in such a position that Polaris is just hidden behind
the corner of a building or some other vertical line, and mark upon the
key map as accurately as possible the position of this line with respect
to the other stars, showing which stars are to the right and which are
to the left of it. Record the time (date, hour, and minute) at which
this observation was made. An hour or two later repeat the observation
at the same place, draw the line and note the time, and you will find
that the line last drawn upon the map does not agree with the first one.
The stars have changed their positions, and with respect to the vertical
line the Pointers are now in a different direction from Polaris.
Measure with a protractor the angle between the two lines drawn in the
map, and use this angle and the recorded times of the observation to
find how many degrees per hour this direction is changing. It should be
about 15° per hour. If the observation were repeated 12 hours after the
first recorded time, what would be the position of the vertical line
among the stars? What would it be 24 hours later? A week later? Repeat
the observation on the next clear night, and allowing for the number of
whole revolutions made by the stars between the two dates, again
determine from the time interval a more accurate value of the rate at
which the stars move.
The motion of the stars which the student has here detected is called
their "diurnal" motion. What is the significance of the word diurnal?
In the preceding paragraph there is introduced a method of great
importance in astronomical practice--i. e., determining something--in
this case the rate per hour, from observations separated by a long
interval of time, in order to get a more accurate value than could be
found from a short interval. Why is it more accurate? To determine the
rate at which the planet Mars rotates about its axis, astronomers use
observations separated by an interval of more than 200 years, during
which the planet made more than 75,000 revolutions upon its axis. If we
were to write out in algebraic form an equation for determining the
length of one revolution of Mars about its axis, the large number,
75,000, would appear in the equation as a divisor, and in the final
result would greatly reduce whatever errors existed in the observations
employed.
Repeat Exercise 6 night after night, and note whether the stars come
back to the same position at the same hour and minute every night.
[Illustration: FIG. 6. The plumb-line apparatus.]
[Illustration: FIG. 7. The plumb-line apparatus.]
11. THE PLUMB-LINE APPARATUS.--This experiment, and many others, may be
conveniently and accurately made with no other apparatus than a plumb
line, and a device for sighting past it. In Figs. 6 and 7 there is
shown a simple form of such apparatus, consisting essentially of a board
which rests in a horizontal position upon the points of three screws
that pass through it. This board carries a small box, to one side of
which is nailed in vertical position another board 5 or 6 feet long to
carry the plumb line. This consists of a wire or fish line with any
heavy weight--e. g., a brick or flatiron--tied to its lower end and
immersed in a vessel of water placed inside the box, so as to check any
swinging motion of the weight. In the cover of the box is a small hole
through which the wire passes, and by turning the screws in the
baseboard the apparatus may be readily leveled, so that the wire shall
swing freely in the center of the hole without touching the cover of the
box. Guy wires, shown in the figure, are applied so as to stiffen the
whole apparatus. A board with a screw eye at each end may be pivoted to
the upright, as in Fig. 6, for measuring altitudes; or to the box, as in
Fig. 7, for observing the time at which a star in its diurnal motion
passes through the plane determined by the plumb line and the center of
the screw eye through which the observer looks.
The whole apparatus may be constructed by any person of ordinary
mechanical skill at a very small cost, and it or something equivalent
should be provided for every class beginning observational astronomy. To
use the apparatus for the experiment of § 10, it should be leveled, and
the board with the screw eyes, attached as in Fig. 7, should be turned
until the observer, looking through the screw eye, sees Polaris exactly
behind the wire. Use a bicycle lamp to illumine the wire by night. The
apparatus is now adjusted, and the observer has only to wait for the
stars which he desires to observe, and to note by his watch the time at
which they pass behind the wire. It will be seen that the wire takes the
place of the vertical edge of the building, and that the board with the
screw eyes is introduced solely to keep the observer in the right place
relative to the wire.
12. A SIDEREAL CLOCK.--Clocks are sometimes so made and regulated that
they show always the same hour and minute when the stars come back to
the same place, and such a timepiece is called a sidereal clock--i. e.,
a star-time clock. Would such a clock gain or lose in comparison with an
ordinary watch? Could an ordinary watch be turned into a sidereal watch
by moving the regulator?
[Illustration: FIG. 8.--Photographing the circumpolar stars.--BARNARD.]
13. PHOTOGRAPHING THE STARS.--EXERCISE 7.--For any student who uses a
camera. Upon some clear and moonless night point the camera, properly
focused, at Polaris, and expose a plate for three or four hours. Upon
developing the plate you should find a series of circular trails such as
are shown in Fig. 8, only longer. Each one of these is produced by a
star moving slowly over the plate, in consequence of its changing
position in the sky. The center indicated by these curved trails is
called the pole of the heavens. It is that part of the sky toward which
is pointed the axis about which the earth rotates, and the motion of the
stars around the center is only an apparent motion due to the rotation
of the earth which daily carries the observer and his camera around this
axis while the stars stand still, just as trees and fences and telegraph
poles stand still, although to the passenger upon a railway train they
appear to be in rapid motion. So far as simple observations are
concerned, there is no method by which the pupil can tell for himself
that the motion of the stars is an apparent rather than a real one, and,
following the custom of astronomers, we shall habitually speak as if it
were a real movement of the stars. How long was the plate exposed in
photographing Fig. 8?
14. FINDING THE STARS.--On Plate I, opposite page 124, the pole of the
heavens is at the center of the map, near Polaris, and the heavy trail
near the center of Fig. 8 is made by Polaris. See if you can identify
from the map any of the stars whose trails show in the photograph. The
brighter the star the bolder and heavier its trail.
Find from the map and locate in the sky the two bright stars Capella and
Vega, which are on opposite sides of Polaris and nearly equidistant from
it. Do these stars share in the motion around the pole? Are they visible
on every clear night, and all night?
Observe other bright stars farther from Polaris than are Vega and
Capella and note their movement. Do they move like the sun and moon? Do
they rise and set?
In what part of the sky do the stars move most rapidly, near the pole or
far from it?
How long does it take the fastest moving stars to make the circuit of
the sky and come back to the same place? How long does it take the slow
stars?
15. RISING AND SETTING OF THE STARS.--A study of the sky along the lines
indicated in these questions will show that there is a considerable part
of it surrounding the pole whose stars are visible on every clear night.
The same star is sometimes high in the sky, sometimes low, sometimes to
the east of the pole and at other times west of it, but is always above
the horizon. Such stars are said to be circumpolar. A little farther
from the pole each star, when at the lowest point of its circular path,
dips for a time below the horizon and is lost to view, and the farther
it is away from the pole the longer does it remain invisible, until, in
the case of stars 90° away from the pole, we find them hidden below the
horizon for twelve hours out of every twenty-four (see Fig. 9). The sun
is such a star, and in its rising and setting acts precisely as does
every other star at a similar distance from the pole--only, as we shall
find later, each star keeps always at (nearly) the same distance from
the pole, while the sun in the course of a year changes its distance
from the pole very greatly, and thus changes the amount of time it
spends above and below the horizon, producing in this way the long days
of summer and the short ones of winter.
[Illustration: FIG. 9.--Diurnal motion of the northern constellations.]
How much time do stars which are more than 90° from the pole spend above
the horizon?
We say in common speech that the sun rises in the east, but this is
strictly true only at the time when it is 90° distant from the
pole--i. e., in March and September. At other seasons it rises north or
south of east according as its distance from the pole is less or greater
than 90°, and the same is true for the stars.
16. THE GEOGRAPHY OF THE SKY.--Find from a map the latitude and
longitude of your schoolhouse. Find on the map the place whose latitude
is 39° and longitude 77° west of the meridian of Greenwich. Is there any
other place in the world which has the same latitude and longitude as
your schoolhouse?
The places of the stars in the sky are located in exactly the manner
which is illustrated by these geographical questions, only different
names are used. Instead of latitude the astronomer says _declination_,
in place of longitude he says _right ascension_, in place of meridian he
says _hour circle_, but he means by these new names the same ideas that
the geographer expresses by the old ones.
Imagine the earth swollen up until it fills the whole sky; the earth's
equator would meet the sky along a line (a great circle) everywhere 90°
distant from the pole, and this line is called the _celestial equator_.
Trace its position along the middle of the map opposite page 190 and
notice near what stars it runs. Every meridian of the swollen earth
would touch the sky along an hour circle--i. e., a great circle passing
through the pole and therefore perpendicular to the equator. Note that
in the map one of these hour circles is marked 0. It plays the same part
in measuring right ascensions as does the meridian of Greenwich in
measuring longitudes; it is the beginning, from which they are reckoned.
Note also, at the extreme left end of the map, the four bright stars in
the form of a square, one side of which is parallel and close to the
hour circle, which is marked 0. This is familiarly called the Great
Square in Pegasus, and may be found high up in the southern sky whenever
the Big Dipper lies below the pole. Why can it not be seen when Ursa
Major is above the pole?
Astronomers use the right ascensions of the stars not only to tell in
what part of the sky the star is placed, but also in time reckonings, to
regulate their sidereal clocks, and with regard to this use they find
it convenient to express right ascension not in degrees but in hours,
24 of which fill up the circuit of the sky and each of which is equal
to 15° of arc, 24 × 15 = 360. The right ascension of Capella is
5h. 9m. = 77.2°, but the student should accustom himself to using it
in hours and minutes as given and not to change it into degrees. He
should also note that some stars lie on the side of the celestial
equator toward Polaris, and others are on the opposite side, so that the
astronomer has to distinguish between north declinations and south
declinations, just as the geographer distinguishes between north
latitudes and south latitudes. This is done by the use of the + and -
signs, a + denoting that the star lies north of the celestial equator,
i. e., toward Polaris.
[Illustration: FIG. 10.--From a photograph of the Pleiades.]
Find on Plate II, opposite page 190, the Pleiades (Pl[=e]ad[=e]s),
R. A. = 3h. 42m., Dec. = +23.8°. Why do they not show on Plate I,
opposite page 124? In what direction are they from Polaris? This is one
of the finest star clusters in the sky, but it needs a telescope to
bring out its richness. See how many stars you can count in it with the
naked eye, and afterward examine it with an opera glass. Compare what
you see with Fig. 10. Find Antares, R. A. = 16h. 23m. Dec. = -26.2°. How
far is it, in degrees, from the pole? Is it visible in your sky? If so,
what is its color?
Find the R. A. and Dec. of [a] Ursæ Majoris; of [b] Ursæ Majoris; of
Polaris. Find the Northern Crown, _Corona Borealis_, R. A. = 15h. 30m.,
Dec. = +27.0°; the Beehive, _Præsepe_, R. A. = 8h. 33m., Dec. = +20.4°.
These should be looked up, not only on the map, but also in the sky.
17. REFERENCE LINES AND CIRCLES.--As the stars move across the sky in
their diurnal motion, they carry the framework of hour circles and
equator with them, so that the right ascension and declination of each
star remain unchanged by this motion, just as longitudes and latitudes
remain unchanged by the earth's rotation. They are the same when a star
is rising and when it is setting; when it is above the pole and when it
is below it. During each day the hour circle of every star in the
heavens passes overhead, and at the moment when any particular hour
circle is exactly overhead all the stars which lie upon it are said to
be "on the meridian"--i. e., at that particular moment they stand
directly over the observer's geographical meridian and upon the
corresponding celestial meridian.
An eye placed at the center of the earth and capable of looking through
its solid substance would see your geographical meridian against the
background of the sky exactly covering your celestial meridian and
passing from one pole through your zenith to the other pole. In Fig. 11
the inner circle represents the terrestrial meridian of a certain
place, _O_, as seen from the center of the earth, _C_, and the outer
circle represents the celestial meridian of _O_ as seen from _C_, only
we must imagine, what can not be shown on the figure, that the outer
circle is so large that the inner one shrinks to a mere point in
comparison with it. If _C P_ represents the direction in which the
earth's axis passes through the center, then _C E_ at right angles to it
must be the direction of the equator which we suppose to be turned
edgewise toward us; and if _C O_ is the direction of some particular
point on the earth's surface, then _Z_ directly overhead is called the
_zenith_ of that point, upon the celestial sphere. The line _C H_
represents a direction parallel to the horizon plane at _O_, and _H C P_
is the angle which the axis of the earth makes with this horizon plane.
The arc _O E_ measures the latitude of _O_, and the arc _Z E_ measures
the declination of _Z_, and since by elementary geometry each of these
arcs contains the same number of degrees as the angle _E C Z_, we have
the
_Theorem._--The latitude of any place is equal to the declination of its
zenith.
_Corollary._--Any star whose declination is equal to your latitude will
once in each day pass through your zenith.
[Illustration: FIG. 11.--Reference lines and circles.]
18. LATITUDE.--From the construction of the figure
[angle] _E C Z_ + [angle] _Z C P_ = 90°
[angle] _H C P_ + [angle] _Z C P_ = 90°
from which we find by subtraction and transposition
[angle] _E C Z_ = [angle] _H C P_
and this gives the further
_Theorem._--The latitude of any place is equal to the elevation of the
pole above its horizon plane.
An observer who travels north or south over the earth changes his
latitude, and therefore changes the angle between his horizon plane and
the axis of the earth. What effect will this have upon the position of
stars in his sky? If you were to go to the earth's equator, in what part
of the sky would you look for Polaris? Can Polaris be seen from
Australia? From South America? If you were to go from Minnesota to
Texas, in what respect would the appearance of stars in the northern sky
be changed? How would the appearance of stars in the southern sky be
changed?
[Illustration: FIG. 12.--Diurnal path of Polaris.]
EXERCISE 8.--Determine your latitude by taking the altitude of Polaris
when it is at some one of the four points of its diurnal path, shown in
Fig. 12. When it is at _1_ it is said to be at upper culmination, and
the star [z] Ursæ Majoris in the handle of the Big Dipper will be
directly below it. When at _2_ it is at western elongation, and the
star Castor is near the meridian. When it is at _3_ it is at lower
culmination, and the star Spica is on the meridian. When it is at _4_ it
is at eastern elongation, and Altair is near the meridian. All of these
stars are conspicuous ones, which the student should find upon the map
and learn to recognize in the sky. The altitude observed at either _2_
or _4_ may be considered equal to the latitude of the place, but the
altitude observed when Polaris is at the positions marked _1_ and _3_
must be corrected for the star's distance from the pole, which may be
assumed equal to 1.3°.
The plumb-line apparatus described at page 12 is shown in Fig. 6
slightly modified, so as to adapt it to measuring the altitudes of
stars. Note that the board with the screw eye at one end has been
transferred from the box to the vertical standard, and has a screw eye
at each end. When the apparatus has been properly leveled, so that the
plumb line hangs at the middle of the hole in the box cover, the board
is to be pointed at the star by sighting through the centers of the two
screw eyes, and a pencil line is to be ruled along its edge upon the
face of the vertical standard. After this has been done turn the
apparatus halfway around so that what was the north side now points
south, level it again and revolve the board about the screw which holds
it to the vertical standard, until the screw eyes again point to the
star. Rule another line along the same edge of the board as before and
with a protractor measure the angle between these lines. Use a bicycle
lamp if you need artificial light for your work. The student who has
studied plane geometry should be able to prove that one half of the
angle between these lines is equal to the altitude of the star.
After you have determined your latitude from Polaris, compare the result
with your position as shown upon the best map available. With a little
practice and considerable care the latitude may be thus determined
within one tenth of a degree, which is equivalent to about 7 miles. If
you go 10 miles north or south from your first station you should find
the pole higher up or lower down in the sky by an amount which can be
measured with your apparatus.
19. THE MERIDIAN LINE.--To establish a true north and south line upon
the ground, use the apparatus as described at page 13, and when Polaris
is at upper or lower culmination drive into the ground two stakes in
line with the star and the plumb line. Such a meridian line is of great
convenience in observing the stars and should be laid out and
permanently marked in some convenient open space from which, if
possible, all parts of the sky are visible. June and November are
convenient months for this exercise, since Polaris then comes to
culmination early in the evening.
20. TIME.--What is _the time_ at which school begins in the morning?
What do you mean by "_the time_"?
The sidereal time at any moment is the right ascension of the hour
circle which at that moment coincides with the meridian. When the hour
circle passing through Sirius coincides with the meridian, the sidereal
time is 6h. 40m., since that is the right ascension of Sirius, and in
astronomical language Sirius is "_on the meridian_" at 6h. 40m. sidereal
time. As may be seen from the map, this 6h. 40m. is the right ascension
of Sirius, and if a clock be set to indicate 6h. 40m. when Sirius
crosses the meridian, it will show sidereal time. If the clock is
properly regulated, every other star in the heavens will come to the
meridian at the moment when the time shown by the clock is equal to the
right ascension of the star. A clock properly regulated for this purpose
will gain about four minutes per day in comparison with ordinary clocks,
and when so regulated it is called a sidereal clock. The student should
be provided with such a clock for his future work, but one such clock
will serve for several persons, and a nutmeg clock or a watch of the
cheapest kind is quite sufficient.
[Illustration: THE HARVARD COLLEGE OBSERVATORY, CAMBRIDGE, MASS.]
EXERCISE 9.--Set such a clock to sidereal time by means of the transit
of a star over your meridian. For this experiment it is presupposed that
a meridian line has been marked out on the ground as in § 19, and the
simplest mode of performing the experiment required is for the observer,
having chosen a suitable star in the southern part of the sky, to place
his eye accurately over the northern end of the meridian line and to
estimate as nearly as possible the beginning and end of the period
during which the star appears to stand exactly above the southern end of
the line. The middle of this period may be taken as the time at which
the star crossed the meridian and at this moment the sidereal time is
equal to the right ascension of the star. The difference between this
right ascension and the observed middle instant is the error of the
clock or the amount by which its hands must be set back or forward in
order to indicate true sidereal time.
A more accurate mode of performing the experiment consists in using the
plumb-line apparatus carefully adjusted, as in Fig. 7, so that the line
joining the wire to the center of the screw eye shall be parallel to the
meridian line. Observe the time by the clock at which the star
disappears behind the wire as seen through the center of the screw eye.
If the star is too high up in the sky for convenient observation, place
a mirror, face up, just north of the screw eye and observe star, wire
and screw eye by reflection in it.
The numerical right ascension of the observed star is needed for this
experiment, and it may be measured from the star map, but it will
usually be best to observe one of the stars of the table at the end of
the book, and to obtain its right ascension as follows: The table gives
the right ascension and declination of each star as they were at the
beginning of the year 1900, but on account of the precession (see
Chapter V), these numbers all change slowly with the lapse of time, and
on the average the right ascension of each star of the table must be
increased by one twentieth of a minute for each year after 1900--i. e.,
in 1910 the right ascension of the first star of the table will be
0h. 38.6m. + (10/20)m. = 0h. 39.1m. The declinations also change
slightly, but as they are only intended to help in finding the star on
the star maps, their change may be ignored.
Having set the clock approximately to sidereal time, observe one or two
more stars in the same way as above. The difference between the observed
time and the right ascension, if any is found, is the "correction" of
the clock. This correction ought not to exceed a minute if due care has
been taken in the several operations prescribed. The relation of the
clock to the right ascension of the stars is expressed in the following
equation, with which the student should become thoroughly familiar:
A = T ± U
_T_ stands for the time by the clock at which the star crossed the
meridian. _A_ is the right ascension of the star, and _U_ is the
correction of the clock. Use the + sign in the equation whenever the
clock is too slow, and the - sign when it is too fast. _U_ may be found
from this equation when _A_ and _T_ are given, or _A_ may be found when
_T_ and _U_ are given. It is in this way that astronomers measure the
right ascensions of the stars and planets.
Determine _U_ from each star you have observed, and note how the several
results agree one with another.
21. DEFINITIONS.--To define a thing or an idea is to give a description
sufficient to identify it and distinguish it from every other possible
thing or idea. If a definition does not come up to this standard it is
insufficient. Anything beyond this requirement is certainly useless and
probably mischievous.
Let the student define the following geographical terms, and let him
also criticise the definitions offered by his fellow-students: Equator,
poles, meridian, latitude, longitude, north, south, east, west.
Compare the following astronomical definitions with your geographical
definitions, and criticise them in the same way. If you are not able to
improve upon them, commit them to memory:
_The Poles_ of the heavens are those points in the sky toward which the
earth's axis points. How many are there? The one near Polaris is called
the north pole.
_The Celestial Equator_ is a great circle of the sky distant 90° from
the poles.
_The Zenith_ is that point of the sky, overhead, toward which a plumb
line points. Why is the word overhead placed in the definition? Is there
more than one zenith?
_The Horizon_ is a great circle of the sky 90° distant from the zenith.
_An Hour Circle_ is any great circle of the sky which passes through the
poles. Every star has its own hour circle.
_The Meridian_ is that hour circle which passes through the zenith.
_A Vertical Circle_ is any great circle that passes through the zenith.
Is the meridian a vertical circle?
_The Declination_ of a star is its angular distance north or south of
the celestial equator.
_The Right Ascension_ of a star is the angle included between its hour
circle and the hour circle of a certain point on the equator which is
called the _Vernal Equinox_. From spherical geometry we learn that this
angle is to be measured either at the pole where the two hour circles
intersect, as is done in the star map opposite page 124, or along the
equator, as is done in the map opposite page 190. Right ascension is
always measured from the vernal equinox in the direction opposite to
that in which the stars appear to travel in their diurnal motion--i. e.,
from west toward east.
_The Altitude_ of a star is its angular distance above the horizon.
_The Azimuth_ of a star is the angle between the meridian and the
vertical circle passing through the star. A star due south has an
azimuth of 0°. Due west, 90°. Due north, 180°. Due east, 270°.
What is the azimuth of Polaris in degrees?
What is the azimuth of the sun at sunrise? At sunset? At noon? Are these
azimuths the same on different days?
_The Hour Angle_ of a star is the angle between its hour circle and the
meridian. It is measured from the meridian in the direction in which the
stars appear to travel in their diurnal motion--i. e., from east toward
west.
What is the hour angle of the sun at noon? What is the hour angle of
Polaris when it is at the lowest point in its daily motion?
22. EXERCISES.--The student must not be satisfied with merely learning
these definitions. He must learn to see these points and lines in his
mind as if they were visibly painted upon the sky. To this end it will
help him to note that the poles, the zenith, the meridian, the horizon,
and the equator seem to stand still in the sky, always in the same place
with respect to the observer, while the hour circles and the vernal
equinox move with the stars and keep the same place among them. Does the
apparent motion of a star change its declination or right ascension?
What is the hour angle of the sun when it has the greatest altitude?
Will your answer to the preceding question be true for a star? What is
the altitude of the sun after sunset? In what direction is the north
pole from the zenith? From the vernal equinox? Where are the points in
which the meridian and equator respectively intersect the horizon?
CHAPTER III
FIXED AND WANDERING STARS
23. STAR MAPS.--Select from the map some conspicuous constellation that
will be conveniently placed for observation in the evening, and make on
a large scale a copy of all the stars of the constellation that are
shown upon the map. At night compare this copy with the sky, and mark in
upon your paper all the stars of the constellation which are not already
there. Both the original drawing and the additions made to it by night
should be carefully done, and for the latter purpose what is called the
method of allineations may be used with advantage--i. e., the new star
is in line with two already on the drawing and is midway between them,
or it makes an equilateral triangle with two others, or a square with
three others, etc.
A series of maps of the more prominent constellations, such as Ursa
Major, Cassiopea, Pegasus, Taurus, Orion, Gemini, Canis Major, Leo,
Corvus, Bootes, Virgo, Hercules, Lyra, Aquila, Scorpius, should be
constructed in this manner upon a uniform scale and preserved as a part
of the student's work. Let the magnitude of the stars be represented on
the maps as accurately as may be, and note the peculiarity of color
which some stars present. For the most part their color is a very pale
yellow, but occasionally one may be found of a decidedly ruddy
hue--e. g., Aldebaran or Antares. Such a star map, not quite complete,
is shown in Fig. 13.
So, too, a sharp eye may detect that some stars do not remain always of
the same magnitude, but change their brightness from night to night,
and this not on account of cloud or mist in the atmosphere, but from
something in the star itself. Algol is one of the most conspicuous of
these _variable stars_, as they are called.
[Illustration: FIG. 13.--Star map of the region about Orion.]
24. THE MOON'S MOTION AMONG THE STARS.--Whenever the moon is visible
note its position among the stars by allineations, and plot it on the
key map opposite page 190. Keep a record of the day and hour
corresponding to each such observation. You will find, if the work is
correctly done, that the positions of the moon all fall near the curved
line shown on the map. This line is called the ecliptic.
After several such observations have been made and plotted, find by
measurement from the map how many degrees per day the moon moves. How
long would it require to make the circuit of the heavens and come back
to the starting point?
On each night when you observe the moon, make on a separate piece of
paper a drawing of it about 10 centimeters in diameter and show in the
drawing every feature of the moon's face which you can see--e. g., the
shape of the illuminated surface (phase); the direction among the stars
of the line joining the horns; any spots which you can see upon the
moon's face, etc. An opera glass will prove of great assistance in this
work.
Use your drawings and the positions of the moon plotted upon the map to
answer the following questions: Does the direction of the line joining
the horns have any special relation to the ecliptic? Does the amount of
illuminated surface of the moon have any relation to the moon's angular
distance from the sun? Does it have any relation to the time at which
the moon sets? Do the spots on the moon when visible remain always in
the same place? Do they come and go? Do they change their position with
relation to each other? Can you determine from these spots that the moon
rotates about an axis, as the earth does? In what direction does its
axis point? How long does it take to make one revolution about the axis?
Is there any day and night upon the moon?
Each of these questions can be correctly answered from the student's own
observations without recourse to any book.
25. THE SUN AND ITS MOTION.--Examine the face of the sun through a
smoked glass to see if there is anything there that you can sketch.
By day as well as by night the sky is studded with stars, only they can
not be seen by day on account of the overwhelming glare of sunlight, but
the position of the sun among the stars may be found quite as
accurately as was that of the moon, by observing from day to day its
right ascension and declination, and this should be practiced at noon on
clear days by different members of the class.
EXERCISE 10.--The right ascension of the sun may be found by observing
with the sidereal clock the time of its transit over the meridian. Use
the equation in § 20, and substitute in place of _U_ the value of the
clock correction found from observations of stars on a preceding or
following night. If the clock gains or loses _with respect to sidereal
time_, take this into account in the value of _U_.
EXERCISE 11.--To determine the sun's declination, measure its altitude
at the time it crosses the meridian. Use either the method of Exercise
4, or that used with Polaris in Exercise 8. The student should be able
to show from Fig. 11 that the declination is equal to the sum of the
altitude and the latitude of the place diminished by 90°, or in an
equation
Declination = Altitude + Latitude - 90°.
If the declination as found from this equation is a negative number it
indicates that the sun is on the south side of the equator.
The right ascension and declination of the sun as observed on each day
should be plotted on the map and the date, written opposite it. If the
work has been correctly done, the plotted points should fall upon the
curved line (ecliptic) which runs lengthwise of the map. This line, in
fact, represents the sun's path among the stars.
Note that the hours of right ascension increase from 0 up to 24, while
the numbers on the clock dial go only from 0 to 12, and then repeat 0 to
12 again during the same day. When the sidereal time is 13 hours, 14
hours, etc., the clock will indicate 1 hour, 2 hours, etc., and 12 hours
must then be added to the time shown on the dial.
If observations of the sun's right ascension and declination are made
in the latter part of either March or September the student will find
that the sun crosses the equator at these times, and he should determine
from his observations, as accurately as possible, the date and hour of
this crossing and the point on the equator at which the sun crosses it.
These points are called the equinoxes, Vernal Equinox and Autumnal
Equinox for the spring and autumn crossings respectively, and the
student will recall that the vernal equinox is the point from which
right ascensions are measured. Its position among the stars is found by
astronomers from observations like those above described, only made with
much more elaborate apparatus.
Similar observations made in June and December show that the sun's
midday altitude is about 47° greater in summer than in winter. They show
also that the sun is as far north of the equator in June as he is south
of it in December, from which it is easily inferred that his path, the
ecliptic, is inclined to the equator at an angle of 23°.5, one half of
47°. This angle is called the obliquity of the ecliptic. The student may
recall that in the geographies the torrid zone is said to extend 23°.5
on either side of the earth's equator. Is there any connection between
these limits and the obliquity of the ecliptic? Would it be correct to
define the torrid zone as that part of the earth's surface within which
the sun may at some season of the year pass through the zenith?
EXERCISE 12.--After a half dozen observations of the sun have been
plotted upon the map, find by measurement the rate, in degrees per day,
at which the sun moves along the ecliptic. How many days will be
required for it to move completely around the ecliptic from vernal
equinox back to vernal equinox again? Accurate observations with the
elaborate apparatus used by professional astronomers show that this
period, which is called a _tropical year_, is 365 days 5 hours 48
minutes 46 seconds. Is this the same as the ordinary year of our
calendars?
26. THE PLANETS.--Any one who has watched the sky and who has made the
drawings prescribed in this chapter can hardly fail to have found in the
course of his observations some bright stars not set down on the printed
star maps, and to have found also that these stars do not remain fixed
in position among their fellows, but wander about from one constellation
to another. Observe the motion of one of these planets from night to
night and plot its positions on the star map, precisely as was done for
the moon. What kind of path does it follow?
Both the ancient Greeks and the modern Germans have called these bodies
wandering stars, and in English we name them planets, which is simply
the Greek word for wanderer, bent to our use. Besides the sun and moon
there are in the heavens five planets easily visible to the naked eye
and, as we shall see later, a great number of smaller ones visible only
in the telescope. More than 2,000 years ago astronomers began observing
the motion of sun, moon, and planets among the stars, and endeavored to
account for these motions by the theory that each wandering star moved
in an orbit about the earth. Classical and mediæval literature are
permeated with this idea, which was displaced only after a long struggle
begun by Copernicus (1543 A. D.), who taught that the moon alone of
these bodies revolves about the earth, while the earth and the other
planets revolve around the sun. The ecliptic is the intersection of the
plane of the earth's orbit with the sky, and the sun appears to move
along the ecliptic because, as the earth moves around its orbit, the sun
is always seen projected against the opposite side of it. The moon and
planets all appear to move near the ecliptic because the planes of their
orbits nearly coincide with the plane of the earth's orbit, and a narrow
strip on either side of the ecliptic, following its course completely
around the sky, is called the _zodiac_, a word which may be regarded as
the name of a narrow street (16° wide) within which all the wanderings
of the visible planets are confined and outside of which they never
venture. Indeed, Mars is the only planet which ever approaches the edge
of the street, the others traveling near the middle of the road.
[Illustration: FIG. 14.--The apparent motion of a planet.]
27. A TYPICAL CASE OF PLANETARY MOTION.--The Copernican theory,
enormously extended and developed through the Newtonian law of
gravitation (see Chapter IV), has completely supplanted the older
Ptolemaic doctrine, and an illustration of the simple manner in which it
accounts for the apparently complicated motions of a planet among the
stars is found in Figs. 14 and 15, the first of which represents the
apparent motion of the planet Mars through the constellations Aries and
Pisces during the latter part of the year 1894, while the second shows
the true motions of Mars and the earth in their orbits about the sun
during the same period. The straight line in Fig. 14, with cross ruling
upon it, is a part of the ecliptic, and the numbers placed opposite it
represent the distance, in degrees, from the vernal equinox. In Fig. 15
the straight line represents the direction from the sun toward the
vernal equinox, and the angle which this line makes with the line
joining earth and sun is called the earth's longitude. The imaginary
line joining the earth and sun is called the earth's radius vector, and
the pupil should note that the longitude and length of the radius vector
taken together show the direction and distance of the earth from the
sun--i. e., they fix the relative positions of the two bodies. The same
is nearly true for Mars and would be wholly true if the orbit of Mars
lay in the same plane with that of the earth. How does Fig. 14 show that
the orbit of Mars does not lie exactly in the same plane with the orbit
of the earth?
EXERCISE 13.--Find from Fig. 15 what ought to have been the apparent
course of Mars among the stars during the period shown in the two
figures, and compare what you find with Fig. 14. The apparent position
of Mars among the stars is merely its direction from the earth, and this
direction is represented in Fig. 14 by the distance of the planet from
the ecliptic and by its longitude.
[Illustration: FIG. 15.--The real motion of a planet.]
The longitude of Mars for each date can be found from Fig. 15 by
measuring the angle between the straight line _S V_ and the line drawn
from the earth to Mars. Thus for October 12th we may find with the
protractor that the angle between the line _S V_ and the line joining
the earth to Mars is a little more than 30°, and in Fig. 14 the position
of Mars for this date is shown nearly opposite the cross line
corresponding to 30° on the ecliptic. Just how far below the ecliptic
this position of Mars should fall can not be told from Fig. 15, which
from necessity is constructed as if the orbits of Mars and the earth lay
in the same plane, and Mars in this case would always appear to stand
exactly on the ecliptic and to oscillate back and forth as shown in Fig.
14, but without the up-and-down motion there shown. In this way plot in
Fig. 14 the longitudes of Mars as seen from the earth for other dates
and observe how the forward motion of the two planets in their orbits
accounts for the apparently capricious motion of Mars to and fro among
the stars.
[Illustration: FIG. 16.--The orbits of Jupiter and Saturn.]
28. THE ORBITS OF THE PLANETS.--Each planet, great or small, moves in
its own appropriate orbit about the sun, and the exact determination of
these orbits, their sizes, shapes, positions, etc., has been one of the
great problems of astronomy for more than 2,000 years, in which
successive generations of astronomers have striven to push to a still
higher degree of accuracy the knowledge attained by their predecessors.
Without attempting to enter into the details of this problem we may say,
generally, that every planet moves in a plane passing through the sun,
and for the six planets visible to the naked eye these planes nearly
coincide, so that the six orbits may all be shown without much error as
lying in the flat surface of one map. It is, however, more convenient to
use two maps, such as Figs. 16 and 17, one of which shows the group of
planets, Mercury, Venus, the earth, and Mars, which are near the sun,
and on this account are sometimes called the inner planets, while the
other shows the more distant planets, Jupiter and Saturn, together with
the earth, whose orbit is thus made to serve as a connecting link
between the two diagrams. These diagrams are accurately drawn to scale,
and are intended to be used by the student for accurate measurement in
connection with the exercises and problems which follow.
In addition to the six planets shown in the figures the solar system
contains two large planets and several hundred small ones, for the most
part invisible to the naked eye, which are omitted in order to avoid
confusing the diagrams.
29. JUPITER AND SATURN.--In Fig. 16 the sun at the center is encircled
by the orbits of the three planets, and inclosing all of these is a
circular border showing the directions from the sun of the
constellations which lie along the zodiac. The student must note
carefully that it is only the directions of these constellations that
are correctly shown, and that in order to show them at all they have
been placed very much too close to the sun. The cross lines extending
from the orbit of the earth toward the sun with Roman numerals opposite
them show the positions of the earth in its orbit on the first day of
January (_I_), first day of February (_II_), etc., and the similar lines
attached to the orbits of Jupiter and Saturn with Arabic numerals show
the positions of those planets on the first day of January of each year
indicated, so that the figure serves to show not only the orbits of the
planets, but their actual positions in their orbits for something more
than the first decade of the twentieth century.
The line drawn from the sun toward the right of the figure shows the
direction to the vernal equinox. It forms one side of the angle which
measures a planet's longitude.
[Illustration: FIG. 17.--The orbits of the inner planets.]
EXERCISE 14.--Measure with your protractor the longitude of the earth on
January 1st. Is this longitude the same in all years? Measure the
longitude of Jupiter on January 1, 1900; on July 1, 1900; on September
25, 1906.
Draw neatly on the map a pencil line connecting the position of the
earth for January 1, 1900, with the position of Jupiter for the same
date, and produce the line beyond Jupiter until it meets the circle of
the constellations. This line represents the direction of Jupiter from
the earth, and points toward the constellation in which the planet
appears at that date. But this representation of the place of Jupiter in
the sky is not a very accurate one, since on the scale of the diagram
the stars are in fact more than 100,000 times as far off as they are
shown in the figure, and the pencil mark does not meet the line of
constellations at the same intersection it would have if this line were
pushed back to its true position. To remedy this defect we must draw
another line from the sun parallel to the one first drawn, and its
intersection with the constellations will give very approximately the
true position of Jupiter in the sky.
EXERCISE 15.--Find the present positions of Jupiter and Saturn, and look
them up in the sky by means of your star maps. The planets will appear
in the indicated constellations as very bright stars not shown on the
map.
Which of the planets, Jupiter and Saturn, changes its direction from the
sun more rapidly? Which travels the greater number of miles per day?
When will Jupiter and Saturn be in the same constellation? Does the
earth move faster or slower than Jupiter?
The distance of Jupiter or Saturn from the earth at any time may be
readily obtained from the figure. Thus, by direct measurement with the
millimeter scale we find for January 1, 1900, the distance of Jupiter
from the earth is 6.1 times the distance of the sun from the earth, and
this may be turned into miles by multiplying it by 93,000,000, which is
approximately the distance of the sun from the earth. For most purposes
it is quite as well to dispense with this multiplication and call the
distance 6.1 astronomical units, remembering that the astronomical unit
is the distance of the sun from the earth.
EXERCISE 16.--What is Jupiter's distance from the earth at its nearest
approach? What is the greatest distance it ever attains? Is Jupiter's
least distance from the earth greater or less than its least distance
from Saturn?
On what day in the year 1906 will the earth be on line between Jupiter
and the sun? On this day Jupiter is said to be in _opposition_--i. e.,
the planet and the sun are on opposite sides of the earth, and Jupiter
then comes to the meridian of any and every place at midnight. When the
sun is between the earth and Jupiter (at what date in 1906?) the planet
is said to be in _conjunction_ with the sun, and of course passes the
meridian with the sun at noon. Can you determine from the figure the
time at which Jupiter comes to the meridian at other dates than
opposition and conjunction? Can you determine when it is visible in the
evening hours? Tell from the figure what constellation is on the
meridian at midnight on January 1st. Will it be the same constellation
in every year?
30. MERCURY, VENUS, AND MARS.--Fig. 17, which represents the orbits of
the inner planets, differs from Fig. 16 only in the method of fixing the
positions of the planets in their orbits at any given date. The motion
of these planets is so rapid, on account of their proximity to the sun,
that it would not do to mark their positions as was done for Jupiter and
Saturn, and with the exception of the earth they do not always return to
the same place on the same day in each year. It is therefore necessary
to adopt a slightly different method, as follows: The straight line
extending from the sun toward the vernal equinox, _V_, is called the
prime radius, and we know from past observations that the earth in its
motion around the sun crosses this line on September 23d in each year,
and to fix the earth's position for September 23d in the diagram we have
only to take the point at which the prime radius intersects the earth's
orbit. A month later, on October 23d, the earth will no longer be at
this point, but will have moved on along its orbit to the point marked
30 (thirty days after September 23d). Sixty days after September 23d it
will be at the point marked 60, etc., and for any date we have only to
find the number of days intervening between it and the preceding
September 23d, and this number will show at once the position of the
earth in its orbit. Thus for the date July 4, 1900, we find
1900, July 4 - 1899, September 23 = 284 days,
and the little circle marked upon the earth's orbit between the numbers
270 and 300 shows the position of the earth on that date.
In what constellation was the sun on July 4, 1900? What zodiacal
constellation came to the meridian at midnight on that date? What other
constellations came to the meridian at the same time?
The positions of the other planets in their orbits are found in the same
manner, save that they do not cross the prime radius on the same date in
each year, and the times at which they do cross it must be taken from
the following table:
TABLE OF EPOCHS
-----------------------------------------------------------
A. D. | Mercury. | Venus. | Earth. | Mars.
--------+------------+-----------+------------+------------
Period | 88.0 days. |224.7 days.|365.25 days.| 687.1 days.
1900 | Feb. 18th. | Jan. 11th.| Sept. 23d. | April 28th.
1901 | Feb. 5th. | April 5th.| Sept. 23d. | ...
1902 | Jan. 23d. | June 29th.| Sept. 23d. | March 16th.
1903 | April 8th. | Feb. 8th. | Sept. 23d. | ...
1904 | March 25th.| May 3d. | Sept. 23d. | Feb. 1st.
1905 | March 12th.| July 26th.| Sept. 23d. | Dec. 19th.
1906 | Feb. 27th. | March 8th.| Sept. 23d. | ...
1907 | Feb. 14th. | May 31st. | Sept. 23d. | Nov. 6th.
1908 | Feb. 1st. | Jan. 11th.| Sept. 23d. | ...
1909 | Jan. 18th. | April 4th.| Sept. 23d. | Sept. 23d.
1910 | Jan. 5th. | June 28th.| Sept. 23d. | ...
-----------------------------------------------------------
The first line of figures in this table shows the number of days that
each of these planets requires to make a complete revolution about the
sun, and it appears from these numbers that Mercury makes about four
revolutions in its orbit per year, and therefore crosses the prime
radius four times in each year, while the other planets are decidedly
slower in their movements. The following lines of the table show for
each year the date at which each planet first crossed the prime radius
in that year; the dates of subsequent crossings in any year can be found
by adding once, twice, or three times the period to the given date, and
the table may be extended to later years, if need be, by continuously
adding multiples of the period. In the case of Mars it appears that
there is only about one year out of two in which this planet crosses the
prime radius.
After the date at which the planet crosses the prime radius has been
determined its position for any required date is found exactly as in the
case of the earth, and the constellation in which the planet will appear
from the earth is found as explained above in connection with Jupiter
and Saturn.
The broken lines in the figure represent the construction for finding
the places in the sky occupied by Mercury, Venus, and Mars on July 4,
1900. Let the student make a similar construction and find the positions
of these planets at the present time. Look them up in the sky and see if
they are where your work puts them.
31. EXERCISES.--The "evening star" is a term loosely applied to any
planet which is visible in the western sky soon after sunset. It is easy
to see that such a planet must be farther toward the east in the sky
than is the sun, and in either Fig. 16 or Fig. 17 any planet which
viewed from the position of the earth lies to the left of the sun and
not more than 50° away from it will be an evening star. If to the right
of the sun it is a morning star, and may be seen in the eastern sky
shortly before sunrise.
What planet is the evening star _now_? Is there more than one evening
star at a time? What is the morning star now?
Do Mercury, Venus, or Mars ever appear in opposition? What is the
maximum angular distance from the sun at which Venus can ever be seen?
Why is Mercury a more difficult planet to see than Venus? In what month
of the year does Mars come nearest to the earth? Will it always be
brighter in this month than in any other? Which of all the planets comes
nearest to the earth?
The earth always comes to the same longitude on the same day of each
year. Why is not this true of the other planets?
The student should remember that in one respect Figs. 16 and 17 are not
altogether correct representations, since they show the orbits as all
lying in the same plane. If this were strictly true, every planet would
move, like the sun, always along the ecliptic; but in fact all of the
orbits are tilted a little out of the plane of the ecliptic and every
planet in its motion deviates a little from the ecliptic, first to one
side then to the other; but not even Mars, which is the most erratic in
this respect, ever gets more than eight degrees away from the ecliptic,
and for the most part all of them are much closer to the ecliptic than
this limit.
CHAPTER IV
CELESTIAL MECHANICS
32. THE BEGINNINGS OF CELESTIAL MECHANICS.--From the earliest dawn of
civilization, long before the beginnings of written history, the motions
of sun and moon and planets among the stars from constellation to
constellation had commanded the attention of thinking men, particularly
of the class of priests. The religions of which they were the guardians
and teachers stood in closest relations with the movements of the stars,
and their own power and influence were increased by a knowledge of them.
[Illustration: ISAAC NEWTON (1643-1727).]
Out of these professional needs, as well as from a spirit of scientific
research, there grew up and flourished for many centuries a study of the
motions of the planets, simple and crude at first, because the
observations that could then be made were at best but rough ones, but
growing more accurate and more complex as the development of the
mechanic arts put better and more precise instruments into the hands of
astronomers and enabled them to observe with increasing accuracy the
movements of these bodies. It was early seen that while for the most
part the planets, including the sun and moon, traveled through the
constellations from west to east, some of them sometimes reversed their
motion and for a time traveled in the opposite way. This clearly can not
be explained by the simple theory which had early been adopted that a
planet moves always in the same direction around a circular orbit having
the earth at its center, and so it was said to move around in a small
circular orbit, called an epicycle, whose center was situated upon
and moved along a circular orbit, called the deferent, within which the
earth was placed, as is shown in Fig. 18, where the small circle is the
epicycle, the large circle is the deferent, _P_ is the planet, and _E_
the earth. When this proved inadequate to account for the really
complicated movements of the planets, another epicycle was put on top of
the first one, and then another and another, until the supposed system
became so complicated that Copernicus, a Polish astronomer, repudiated
its fundamental theorem and taught that the motions of the planets take
place in circles around the sun instead of about the earth, and that the
earth itself is only one of the planets moving around the sun in its own
appropriate orbit and itself largely responsible for the seemingly
erratic movements of the other planets, since from day to day we see
them and observe their positions from different points of view.
[Illustration: FIG. 18.--Epicycle and deferent.]
33. KEPLER'S LAWS.--Two generations later came Kepler with his three
famous laws of planetary motion:
I. Every planet moves in an ellipse which has the sun at one of its
foci.
II. The radius vector of each planet moves over equal areas in equal
times.
III. The squares of the periodic times of the planets are proportional
to the cubes of their mean distances from the sun.
These laws are the crowning glory, not only of Kepler's career, but of
all astronomical discovery from the beginning up to his time, and they
well deserve careful study and explanation, although more modern
progress has shown that they are only approximately true.
EXERCISE 17.--Drive two pins into a smooth board an inch apart and
fasten to them the ends of a string a foot long. Take up the slack of
the string with the point of a lead pencil and, keeping the string drawn
taut, move the pencil point over the board into every possible position.
The curve thus traced will be an ellipse having the pins at the two
points which are called its foci.
In the case of the planetary orbits one focus of the ellipse is vacant,
and, in accordance with the first law, the center of the sun is at the
other focus. In Fig. 17 the dot, inside the orbit of Mercury, which is
marked _a_, shows the position of the vacant focus of the orbit of Mars,
and the dot _b_ is the vacant focus of Mercury's orbit. The orbits of
Venus and the earth are so nearly circular that their vacant foci lie
very close to the sun and are not marked in the figure. The line drawn
from the sun to any point of the orbit (the string from pin to pencil
point) is a _radius vector_. The point midway between the pins is the
_center_ of the ellipse, and the distance of either pin from the center
measures the _eccentricity_ of the ellipse.
Draw several ellipses with the same length of string, but with the pins
at different distances apart, and note that the greater the eccentricity
the flatter is the ellipse, but that all of them have the same length.
If both pins were driven into the same hole, what kind of an ellipse
would you get?
The Second Law was worked out by Kepler as his answer to a problem
suggested by the first law. In Fig. 17 it is apparent from a mere
inspection of the orbit of Mercury that this planet travels much faster
on one side of its orbit than on the other, the distance covered in ten
days between the numbers 10 and 20 being more than fifty per cent
greater than that between 50 and 60. The same difference is found,
though usually in less degree, for every other planet, and Kepler's
problem was to discover a means by which to mark upon the orbit the
figures showing the positions of the planet at the end of equal
intervals of time. His solution of this problem, contained in the second
law, asserts that if we draw radii vectors from the sun to each of the
marked points taken at equal time intervals around the orbit, then the
area of the sector formed by two adjacent radii vectores and the arc
included between them is equal to the area of each and every other such
sector, the short radii vectores being spread apart so as to include a
long arc between them while the long radii vectores have a short arc. In
Kepler's form of stating the law the radius vector is supposed to travel
with the planet and in each day to sweep over the same fractional part
of the total area of the orbit. The spacing of the numbers in Fig. 17
was done by means of this law.
For the proper understanding of Kepler's Third Law we must note that the
"mean distance" which appears in it is one half of the long diameter of
the orbit and that the "periodic time" means the number of days or years
required by the planet to make a complete circuit in its orbit.
Representing the first of these by _a_ and the second by _T_, we have,
as the mathematical equivalent of the law,
a^{3} ÷ T^{2} = C
where the quotient, _C_, is a number which, as Kepler found, is the same
for every planet of the solar system. If we take the mean distance of
the earth from the sun as the unit of distance, and the year as the unit
of time, we shall find by applying the equation to the earth's motion,
_C_ = 1. Applying this value to any other planet we shall find in the
same units, _a_ = _T_^{2/3}, by means of which we may determine the
distance of any planet from the sun when its periodic time, _T_, has
been learned from observation.
EXERCISE 18.--Uranus requires 84 years to make a revolution in its
orbit. What is its mean distance from the sun? What are the mean
distances of Mercury, Venus, and Mars? (See Chapter III for their
periodic times.) Would it be possible for two planets at different
distances from the sun to move around their orbits in the same time?
A circle is an ellipse in which the two foci have been brought together.
Would Kepler's laws hold true for such an orbit?
34. NEWTON'S LAWS OF MOTION.--Kepler studied and described the motion of
the planets. Newton, three generations later (1727 A. D.), studied and
described the mechanism which controls that motion. To Kepler and his
age the heavens were supernatural, while to Newton and his successors
they are a part of Nature, governed by the same laws which obtain upon
the earth, and we turn to the ordinary things of everyday life as the
foundation of celestial mechanics.
Every one who has ridden a bicycle knows that he can coast farther upon
a level road if it is smooth than if it is rough; but however smooth and
hard the road may be and however fast the wheel may have been started,
it is sooner or later stopped by the resistance which the road and the
air offer to its motion, and when once stopped or checked it can be
started again only by applying fresh power. We have here a familiar
illustration of what is called
THE FIRST LAW OF MOTION.--"Every body continues in its state of rest or
of uniform motion in a straight line except in so far as it may be
compelled by force to change that state." A gust of wind, a stone, a
careless movement of the rider may turn the bicycle to the right or the
left, but unless some disturbing force is applied it will go straight
ahead, and if all resistance to its motion could be removed it would go
always at the speed given it by the last power applied, swerving neither
to the one hand nor the other.
When a slow rider increases his speed we recognize at once that he has
applied additional power to the wheel, and when this speed is slackened
it equally shows that force has been applied against the motion. It is
force alone which can produce a change in either velocity or direction
of motion; but simple as this law now appears it required the genius of
Galileo to discover it and of Newton to give it the form in which it is
stated above.
35. THE SECOND LAW OF MOTION, which is also due to Galileo and Newton,
is:
"Change of motion is proportional to force applied and takes place in
the direction of the straight line in which the force acts." Suppose a
man to fall from a balloon at some great elevation in the air; his own
weight is the force which pulls him down, and that force operating at
every instant is sufficient to give him at the end of the first second
of his fall a downward velocity of 32 feet per second--i. e., it has
changed his state from rest, to motion at this rate, and the motion is
toward the earth because the force acts in that direction. During the
next second the ceaseless operation of this force will have the same
effect as in the first second and will add another 32 feet to his
velocity, so that two seconds from the time he commenced to fall he will
be moving at the rate of 64 feet per second, etc. The column of figures
marked _v_ in the table below shows what his velocity will be at the end
of subsequent seconds. The changing velocity here shown is the change of
motion to which the law refers, and the velocity is proportional to the
time shown in the first column of the table, because the amount of force
exerted in this case is proportional to the time during which it
operated. The distance through which the man will fall in each second is
shown in the column marked _d_, and is found by taking the average of
his velocity at the beginning and end of this second, and the total
distance through which he has fallen at the end of each second, marked
_s_ in the table, is found by taking the sum of all the preceding values
of _d_. The velocity, 32 feet per second, which measures the change of
motion in each second, also measures the _accelerating force_ which
produces this motion, and it is usually represented in formulæ by the
letter _g_. Let the student show from the numbers in the table that the
accelerating force, the time, _t_, during which it operates, and the
space, _s_, fallen through, satisfy the relation
s = 1/2 gt^{2},
which is usually called the law of falling bodies. How does the table
show that _g_ is equal to 32?
TABLE
_t_ _v_ _d_ _s_
0 0 0 0
1 32 16 16
2 64 48 64
3 96 80 144
4 128 112 256
5 160 144 400
etc. etc. etc. etc.
If the balloon were half a mile high how long would it take to fall to
the ground? What would be the velocity just before reaching the ground?
[Illustration: GALILEO GALILEI (1564-1642).]
Fig. 19 shows the path through the air of a ball which has been struck
by a bat at the point _A_, and started off in the direction _A B_ with a
velocity of 200 feet per second. In accordance with the first law of
motion, if it were acted upon by no other force than the impulse given
by the bat, it should travel along the straight line _A B_ at the
uniform rate of 200 feet per second, and at the end of the fourth second
it should be 800 feet from _A_, at the point marked 4, but during these
four seconds its weight has caused it to fall 256 feet, and its actual
position, 4', is 256 feet below the point 4. In this way we find its
position at the end of each second, 1', 2', 3', 4', etc., and drawing a
line through these points we shall find the actual path of the ball
under the influence of the two forces to be the curved line _A C_. No
matter how far the ball may go before striking the ground, it can not
get back to the point _A_, and the curve _A C_ therefore can not be a
part of a circle, since that curve returns into itself. It is, in fact,
a part of a _parabola_, which, as we shall see later, is a kind of orbit
in which comets and some other heavenly bodies move. A skyrocket moves
in the same kind of a path, and so does a stone, a bullet, or any other
object hurled through the air.
[Illustration: FIG. 19.--The path of a ball.]
36. THE THIRD LAW OF MOTION.--"To every action there is always an equal
and contrary reaction; or the mutual actions of any two bodies are
always equal and oppositely directed." This is well illustrated in the
case of a man climbing a rope hand over hand. The direct force or action
which he exerts is a downward pull upon the rope, and it is the reaction
of the rope to this pull which lifts him along it. We shall find in a
later chapter a curious application of this law to the history of the
earth and moon.
It is the great glory of Sir Isaac Newton that he first of all men
recognized that these simple laws of motion hold true in the heavens as
well as upon the earth; that the complicated motion of a planet, a
comet, or a star is determined in accordance with these laws by the
forces which act upon the bodies, and that these forces are essentially
the same as that which we call weight. The formal statement of the
principle last named is included in--
37. NEWTON'S LAW OF GRAVITATION.--"Every particle of matter in the
universe attracts every other particle with a force whose direction is
that of a line joining the two, and whose magnitude is directly as the
product of their masses, and inversely as the square of their distance
from each other." We know that we ourselves and the things about us are
pulled toward the earth by a force (weight) which is called, in the
Latin that Newton wrote, _gravitas_, and the word marks well the true
significance of the law of gravitation. Newton did not discover a new
force in the heavens, but he extended an old and familiar one from a
limited terrestrial sphere of action to an unlimited and celestial one,
and furnished a precise statement of the way in which the force
operates. Whether a body be hot or cold, wet or dry, solid, liquid, or
gaseous, is of no account in determining the force which it exerts,
since this depends solely upon mass and distance.
The student should perhaps be warned against straining too far the
language which it is customary to employ in this connection. The law of
gravitation is certainly a far-reaching one, and it may operate in every
remotest corner of the universe precisely as stated above, but
additional information about those corners would be welcome to
supplement our rather scanty stock of knowledge concerning what happens
there. We may not controvert the words of a popular preacher who says,
"When I lift my hand I move the stars in Ursa Major," but we should not
wish to stand sponsor for them, even though they are justified by a
rigorous interpretation of the Newtonian law.
The word _mass_, in the statement of the law of gravitation, means the
quantity of matter contained in the body, and if we represent by the
letters _m´_ and _m´´_ the respective quantities of matter contained in
the two bodies whose distance from each other is _r_, we shall have, in
accordance with the law of gravitation, the following mathematical
expression for the force, _F_, which acts between them:
F = k {m´m´´/r^{2}}.
This equation, which is the general mathematical expression for the law
of gravitation, may be made to yield some curious results. Thus, if we
select two bullets, each having a mass of 1 gram, and place them so that
their centers are 1 centimeter apart, the above expression for the force
exerted between them becomes
F = k {(1 × 1)/1^{2}} = k,
from which it appears that the coefficient _k_ is the force exerted
between these bodies. This is called the gravitation constant, and it
evidently furnishes a measure of the specific intensity with which one
particle of matter attracts another. Elaborate experiments which have
been made to determine the amount of this force show that it is
surprisingly small, for in the case of the two bullets whose mass of 1
gram each is supposed to be concentrated into an indefinitely small
space, gravity would have to operate between them continuously for more
than forty minutes in order to pull them together, although they were
separated by only 1 centimeter to start with, and nothing save their own
inertia opposed their movements. It is only when one or both of the
masses _m´_, _m´´_ are very great that the force of gravity becomes
large, and the weight of bodies at the surface of the earth is
considerable because of the great quantity of matter which goes to make
up the earth. Many of the heavenly bodies are much more massive than the
earth, as the mathematical astronomers have found by applying the law of
gravitation to determine numerically their masses, or, in more popular
language, to "weigh" them.
The student should observe that the two terms mass and weight are not
synonymous; mass is defined above as the quantity of matter contained in
a body, while weight is the force with which the earth attracts that
body, and in accordance with the law of gravitation its weight depends
upon its distance from the center of the earth, while its mass is quite
independent of its position with respect to the earth.
By the third law of motion the earth is pulled toward a falling body
just as strongly as the body is pulled toward the earth--i. e., by a
force equal to the weight of the body. How much does the earth rise
toward the body?
38. THE MOTION OF A PLANET.--In Fig. 20 _S_ represents the sun and _P_ a
planet or other celestial body, which for the moment is moving along the
straight line _P 1_. In accordance with the first law of motion it would
continue to move along this line with uniform velocity if no external
force acted upon it; but such a force, the sun's attraction, is acting,
and by virtue of this attraction the body is pulled aside from the line
_P 1_.
Knowing the velocity and direction of the body's motion and the force
with which the sun attracts it, the mathematician is able to apply
Newton's laws of motion so as to determine the path of the body, and a
few of the possible orbits are shown in the figure where the short cross
stroke marks the point of each orbit which is nearest to the sun. This
point is called the _perihelion_.
Without any formal application of mathematics we may readily see that
the swifter the motion of the body at _P_ the shorter will be the time
during which it is subjected to the sun's attraction at close range, and
therefore the force exerted by the sun, and the resulting change of
motion, will be small, as in the orbits _P 1_ and _P 2_.
On the other hand, _P 5_ and _P 6_ represent orbits in which the
velocity at _P_ was comparatively small, and the resulting change of
motion greater than would be possible for a more swiftly moving body.
What would be the orbit if the velocity at _P_ were reduced to nothing
at all?
What would be the effect if the body starting at _P_ moved directly away
from _1_?
[Illustration: FIG. 20.--Different kinds of orbits.]
The student should not fail to observe that the sun's attraction tends
to pull the body at _P_ forward along its path, and therefore increases
its velocity, and that this influence continues until the planet reaches
perihelion, at which point it attains its greatest velocity, and the
force of the sun's attraction is wholly expended in changing the
direction of its motion. After the planet has passed perihelion the sun
begins to pull backward and to retard the motion in just the same
measure that before perihelion passage it increased it, so that the two
halves of the orbit on opposite sides of a line drawn from the
perihelion through the sun are exactly alike. We may here note the
explanation of Kepler's second law: when the planet is near the sun it
moves faster, and the radius vector changes its direction more rapidly
than when the planet is remote from the sun on account of the greater
force with which it is attracted, and the exact relation between the
rates at which the radius vector turns in different parts of the orbit,
as given by the second law, depends upon the changes in this force.
When the velocity is not too great, the sun's backward pull, after a
planet has passed perihelion, finally overcomes it and turns the planet
toward the sun again, in such a way that it comes back to the point _P_,
moving in the same direction and with the same speed as before--i. e.,
it has gone around the sun in an orbit like _P 6_ or _P 4_, an ellipse,
along which it will continue to move ever after. But we must not fail to
note that this return into the same orbit is a consequence of the last
line in the statement of the law of gravitation (p. 54), and that, if
the magnitude of this force were inversely as the cube of the distance
or any other proportion than the square, the orbit would be something
very different. If the velocity is too great for the sun's attraction to
overcome, the orbit will be a hyperbola, like _P 2_, along which the
body will move away never to return, while a velocity just at the limit
of what the sun can control gives an orbit like _P 3_, a parabola, along
which the body moves with _parabolic velocity_, which is ever
diminishing as the body gets farther from the sun, but is always just
sufficient to keep it from returning. If the earth's velocity could be
increased 41 per cent, from 19 up to 27 miles per second, it would have
parabolic velocity, and would quit the sun's company.
The summation of the whole matter is that the orbit in which a body
moves around the sun, or past the sun, depends upon its velocity and if
this velocity and the direction of the motion at any one point in the
orbit are known the whole orbit is determined by them, and the position
of the planet in its orbit for past as well as future times can be
determined through the application of Newton's laws; and the same is
true for any other heavenly body--moon, comet, meteor, etc. It is in
this way that astronomers are able to predict, years in advance, in what
particular part of the sky a given planet will appear at a given time.
It is sometimes a source of wonder that the planets move in ellipses
instead of circles, but it is easily seen from Fig. 20 that the planet,
_P_, could not by any possibility move in a circle, since the direction
of its motion at _P_ is not at right angles with the line joining it to
the sun as it must be in a circular orbit, and even if it were
perpendicular to the radius vector the planet must needs have exactly
the right velocity given to it at this point, since either more or less
speed would change the circle into an ellipse. In order to produce
circular motion there must be a balancing of conditions as nice as is
required to make a pin stand upon its point, and the really surprising
thing is that the orbits of the planets should be so nearly circular as
they are. If the orbit of the earth were drawn accurately to scale, the
untrained eye would not detect the slightest deviation from a true
circle, and even the orbit of Mercury (Fig. 17), which is much more
eccentric than that of the earth, might almost pass for a circle.
[Illustration: FIG. 21. An impossible orbit.]
The orbit _P 2_, which lies between the parabola and the straight line,
is called in geometry a hyperbola, and Newton succeeded in proving from
the law of gravitation that a body might move under the sun's attraction
in a hyperbola as well as in a parabola or ellipse; but it must move in
some one of these curves; no other orbit is possible.[1] Thus it would
not be possible for a body moving under the law of gravitation to
describe about the sun any such orbit as is shown in Fig. 21. If the
body passes a second time through any point of its orbit, such as _P_ in
the figure, then it must retrace, time after time, the whole path that
it first traversed in getting from _P_ around to _P_ again--i. e., the
orbit must be an ellipse.
[1] The circle and straight line are considered to be special cases
of these curves, which, taken collectively, are called the conic
sections.
Newton also proved that Kepler's three laws are mere corollaries from
the law of gravitation, and that to be strictly correct the third law
must be slightly altered so as to take into account the masses of the
planets. These are, however, so small in comparison with that of the
sun, that the correction is of comparatively little moment.
39. PERTURBATIONS.--In what precedes we have considered the motion of a
planet under the influence of no other force than the sun's attraction,
while in fact, as the law of gravitation asserts, every other body in
the universe is in some measure attracting it and changing its motion.
The resulting disturbances in the motion of the attracted body are
called _perturbations_, but for the most part these are insignificant,
because the bodies by whose disturbing attractions they are caused are
either very small or very remote, and it is only when our moving planet,
_P_, comes under the influence of some great disturbing power like
Jupiter or one of the other planets that the perturbations caused by
their influence need to be taken into account.
The problem of the motion of three bodies--sun, Jupiter, planet--which
must then be dealt with is vastly more complicated than that which we
have considered, and the ablest mathematicians and astronomers have not
been able to furnish a complete solution for it, although they have
worked upon the problem for two centuries, and have developed an immense
amount of detailed information concerning it.
[Illustration: THE LICK OBSERVATORY, MOUNT HAMILTON, CAL.]
In general each planet works ceaselessly upon the orbit of every other,
changing its size and shape and position, backward and forward in
accordance with the law of gravitation, and it is a question of serious
moment how far this process may extend. If the diameter of the earth's
orbit were very much increased or diminished by the perturbing action of
the other planets, the amount of heat received from the sun would be
correspondingly changed, and the earth, perhaps, be rendered unfit
for the support of life. The tipping of the plane of the earth's orbit
into a new position might also produce serious consequences; but the
great French mathematician of a century ago, Laplace, succeeded in
proving from the law of gravitation that although both of these changes
are actually in progress they can not, at least for millions of years,
go far enough to prove of serious consequence, and the same is true for
all the other planets, unless here and there an asteroid may prove an
exception to the rule.
The precession (Chapter V) is a striking illustration of a perturbation
of slightly different character from the above, and another is found in
connection with the plane of the moon's orbit. It will be remembered
that the moon in its motion among the stars never goes far from the
ecliptic, but in a complete circuit of the heavens crosses it twice,
once in going from south to north and once in the opposite direction.
The points at which it crosses the ecliptic are called the _nodes_, and
under the perturbing influence of the sun these nodes move westward
along the ecliptic about twenty degrees per year, an extraordinarily
rapid perturbation, and one of great consequence in the theory of
eclipses.
[Illustration: FIG. 22.--A planet subject to great perturbations by
Jupiter.]
40. WEIGHING THE PLANETS.--Although these perturbations can not be
considered dangerous, they are interesting since they furnish a method
for weighing the planets which produce them. From the law of gravitation
we learn that the ability of a planet to produce perturbations depends
directly upon its mass, since the force _F_ which it exerts contains
this mass, _m´_, as a factor. So, too, the divisor _r^{2}_ in the
expression for the force shows that the distance between the disturbing
and disturbed bodies is a matter of great consequence, for the smaller
the distance the greater the force. When, therefore, the mass of a
planet such as Jupiter is to be determined from the perturbations it
produces, it is customary to select some such opportunity as is
presented in Fig. 22, where one of the small planets, called asteroids,
is represented as moving in a very eccentric orbit, which at one point
approaches close to the orbit of Jupiter, and at another place comes
near to the orbit of the earth. For the most part Jupiter will not exert
any very great disturbing influence upon a planet moving in such an
orbit as this, since it is only at rare intervals that the asteroid and
Jupiter approach so close to each other, as is shown in the figure. The
time during which the asteroid is little affected by the attraction of
Jupiter is used to study the motion given to it by the sun's
attraction--that is, to determine carefully the undisturbed orbit in
which it moves; but there comes a time at which the asteroid passes
close to Jupiter, as shown in the figure, and the orbital motion which
the sun imparts to it will then be greatly disturbed, and when the
planet next comes round to the part of its orbit near the earth the
effect of these disturbances upon its apparent position in the sky will
be exaggerated by its close proximity to the earth. If now the
astronomer observes the actual position of the asteroid in the sky, its
right ascension and declination, and compares these with the position
assigned to the planet by the law of gravitation when the attraction of
Jupiter is ignored, the differences between the observed right
ascensions and declinations and those computed upon the theory of
undisturbed motion will measure the influence that Jupiter has had upon
the asteroid, and the amount by which Jupiter has shifted it, compared
with the amount by which the sun has moved it--that is, with the motion
in its orbit--furnishes the mass of Jupiter expressed as a fractional
part of the mass of the sun.
There has been determined in this manner the mass of every planet in the
solar system which is large enough to produce any appreciable
perturbation, and all these masses prove to be exceedingly small
fractions of the mass of the sun, as may be seen from the following
table, in which is given opposite the name of each planet the number by
which the mass of the sun must be divided in order to get the mass of
the planet:
Mercury 7,000,000 (?)
Venus 408,000
Earth 329,000
Mars 3,093,500
Jupiter 1,047.4
Saturn 3,502
Uranus 22,800
Neptune 19,700
It is to be especially noted that the mass given for each planet
includes the mass of all the satellites which attend it, since their
influence was felt in the perturbations from which the mass was derived.
Thus the mass assigned to the earth is the combined mass of earth and
moon.
41. DISCOVERY OF NEPTUNE.--The most famous example of perturbations is
found in connection with the discovery, in the year 1846, of Neptune,
the outermost planet of the solar system. For many years the motion of
Uranus, his next neighbor, had proved a puzzle to astronomers. In
accordance with Kepler's first law this planet should move in an ellipse
having the sun at one of its foci, but no ellipse could be found which
exactly fitted its observed path among the stars, although, to be sure,
the misfit was not very pronounced. Astronomers surmised that the small
deviations of Uranus from the best path which theory combined with
observation could assign, were due to perturbations in its motion
caused by an unknown planet more remote from the sun--a thing easy to
conjecture but hard to prove, and harder still to find the unknown
disturber. But almost simultaneously two young men, Adams in England and
Le Verrier in France, attacked the problem quite independently of each
other, and carried it to a successful solution, showing that if the
irregularities in the motion of Uranus were indeed caused by an unknown
planet, then that planet must, in September, 1846, be in the direction
of the constellation Aquarius; and there it was found on September 23d
by the astronomers of the Berlin Observatory whom Le Verrier had invited
to search for it, and found within a degree of the exact point which the
law of gravitation in his hands had assigned to it.
This working backward from the perturbations experienced by Uranus to
the cause which produced them is justly regarded as one of the greatest
scientific achievements of the human intellect, and it is worthy of note
that we are approaching the time at which it may be repeated, for
Neptune now behaves much as did Uranus three quarters of a century ago,
and the most plausible explanation which can be offered for these
anomalies in its path is that the bounds of the solar system must be
again enlarged to include another disturbing planet.
42. THE SHAPE OF A PLANET.--There is an effect of gravitation not yet
touched upon, which is of considerable interest and wide application in
astronomy--viz., its influence in determining the shape of the heavenly
bodies. The earth is a globe because every part of it is drawn toward
the center by the attraction of the other parts, and if this attraction
on its surface were everywhere of equal force the material of the earth
would be crushed by it into a truly spherical form, no matter what may
have been the shape in which it was originally made. But such is not the
real condition of the earth, for its diurnal rotation develops in every
particle of its body a force which is sometimes called _centrifugal_,
but which is really nothing more than the inertia of its particles,
which tend at every moment to keep unchanged the direction of their
motion and which thus resist the attraction that pulls them into a
circular path marked out by the earth's rotation, just as a stone tied
at the end of a string and swung swiftly in a circle pulls upon the
string and opposes the constraint which keeps it moving in a circle. A
few experiments with such a stone will show that the faster it goes the
harder does it pull upon the string, and the same is true of each
particle of the earth, the swiftly moving ones near the equator having a
greater centrifugal force than the slow ones near the poles. At the
equator the centrifugal force is directly opposed to the force of
gravity, and in effect diminishes it, so that, comparatively, there is
an excess of gravity at the poles which compresses the earth along its
axis and causes it to bulge out at the equator until a balance is thus
restored. As we have learned from the study of geography, in the case of
the earth, this compression amounts to about 27 miles, but in the larger
planets, Jupiter and Saturn, it is much greater, amounting to several
thousand miles.
But rotation is not the only influence that tends to pull a planet out
of shape. The attraction which the earth exerts upon the moon is
stronger on the near side and weaker on the far side of our satellite
than at its center, and this difference of attraction tends to warp the
moon, as is illustrated in Fig. 23 where _1_, _2_, and _3_ represent
pieces of iron of equal mass placed in line on a table near a horseshoe
magnet, _H_. Each piece of iron is attracted by the magnet and is held
back by a weight to which it is fastened by means of a cord running over
a pulley, _P_, at the edge of the table. These weights are all to be
supposed equally heavy and each of them pulls upon its piece of iron
with a force just sufficient to balance the attraction of the magnet for
the middle piece, No. _2_. It is clear that under this arrangement No.
_2_ will move neither to the right nor to the left, since the forces
exerted upon it by the magnet and the weight just balance each other.
Upon No. _1_, however, the magnet pulls harder than upon No. _2_,
because it is nearer and its pull therefore more than balances the force
exerted by the weight, so that No. _1_ will be pulled away from No. _2_
and will stretch the elastic cords, which are represented by the lines
joining _1_ and _2_, until their tension, together with the force
exerted by the weight, just balances the attraction of the magnet. For
No. _3_, the force exerted by the magnet is less than that of the
weight, and it will also be pulled away from No. _2_ until its elastic
cords are stretched to the proper tension. The net result is that the
three blocks which, without the magnet's influence, would be held close
together by the elastic cords, are pulled apart by this outside force as
far as the resistance of the cords will permit.
[Illustration: FIG. 23.--Tide-raising forces.]
An entirely analogous set of forces produces a similar effect upon the
shape of the moon. The elastic cords of Fig. 23 stand for the attraction
of gravitation by which all the parts of the moon are bound together.
The magnet represents the earth pulling with unequal force upon
different parts of the moon. The weights are the inertia of the moon in
its orbital motion which, as we have seen in a previous section, upon
the whole just balances the earth's attraction and keeps the moon from
falling into it. The effect of these forces is to stretch out the moon
along a line pointing toward the earth, just as the blocks were
stretched out along the line of the magnet, and to make this diameter of
the moon slightly but permanently longer than the others.
[Illustration: FIG. 24.--The tides.]
THE TIDES.--Similarly the moon and the sun attract opposite sides of the
earth with different forces and feebly tend to pull it out of shape. But
here a new element comes into play: the earth turns so rapidly upon its
axis that its solid parts have no time in which to yield sensibly to the
strains, which shift rapidly from one diameter to another as different
parts of the earth are turned toward the moon, and it is chiefly the
waters of the sea which respond to the distorting effect of the sun's
and moon's attraction. These are heaped up on opposite sides of the
earth so as to produce a slight elongation of its diameter, and Fig. 24
shows how by the earth's rotation this swelling of the waters is swept
out from under the moon and is pulled back by the moon until it finally
takes up some such position as that shown in the figure where the effect
of the earth's rotation in carrying it one way is just balanced by the
moon's attraction urging it back on line with the moon. This heaping up
of the waters is called a _tide_. If _I_ in the figure represents a
little island in the sea the waters which surround it will of course
accompany it in its diurnal rotation about the earth's axis, but
whenever the island comes back to the position _I_, the waters will
swell up as a part of the tidal wave and will encroach upon the land in
what is called high tide or flood tide. So too when they reach _I´´_,
half a day later, they will again rise in flood tide, and midway between
these points, at _I´_, the waters must subside, giving low or ebb tide.
The height of the tide raised by the moon in the open sea is only a very
few feet, and the tide raised by the sun is even less, but along the
coast of a continent, in bays and angles of the shore, it often happens
that a broad but low tidal wave is forced into a narrow corner, and then
the rise of the water may be many feet, especially when the solar tide
and the lunar tide come in together, as they do twice in every month, at
new and full moon. Why do they come together at these times instead of
some other?
Small as are these tidal effects, it is worth noting that they may in
certain cases be very much greater--e. g., if the moon were as massive
as is the sun its tidal effect would be some millions of times greater
than it now is and would suffice to grind the earth into fragments.
Although the earth escapes this fate, some other bodies are not so
fortunate, and we shall see in later chapters some evidence of their
disintegration.
43. THE SCOPE OF THE LAW OF GRAVITATION.--In all the domain of physical
science there is no other law so famous as the Newtonian law of
gravitation; none other that has been so dwelt upon, studied, and
elaborated by astronomers and mathematicians, and perhaps none that can
be considered so indisputably proved. Over and over again mathematical
analysis, based upon this law, has pointed out conclusions which, though
hitherto unsuspected, have afterward been found true, as when Newton
himself derived as a corollary from this law that the earth ought to be
flattened at the poles--a thing not known at that time, and not proved
by actual measurement until long afterward. It is, in fact, this
capacity for predicting the unknown and for explaining in minutest
detail the complicated phenomena of the heavens and the earth that
constitutes the real proof of the law of gravitation, and it is
therefore worth while to note that at the present time there are a very
few points at which the law fails to furnish a satisfactory account of
things observed. Chief among these is the case of the planet Mercury,
the long diameter of whose orbit is slowly turning around in a way for
which the law of gravitation as yet furnishes no explanation. Whether
this is because the law itself is inaccurate or incomplete, or whether
it only marks a case in which astronomers have not yet properly applied
the law and traced out its consequences, we do not know; but whether it
be the one or the other, this and other similar cases show that even
here, in its most perfect chapter, astronomy still remains an incomplete
science.
CHAPTER V
THE EARTH AS A PLANET
44. THE SIZE OF THE EARTH.--The student is presumed to have learned, in
his study of geography, that the earth is a globe about 8,000 miles in
diameter and, without dwelling upon the "proofs" which are commonly
given for these statements, we proceed to consider the principles upon
which the measurement of the earth's size and shape are based.
[Illustration: FIG. 25.--Measuring the size of the earth.]
In Fig. 25 the circle represents a meridian section of the earth; _P P´_
is the axis about which it rotates, and the dotted lines represent a
beam of light coming from a star in the plane of the meridian, and so
distant that the dotted lines are all practically parallel to each
other. The several radii drawn through the points _1_, _2_, _3_,
represent the direction of the vertical at these points, and the angles
which these radii produced, make with the rays of starlight are each
equal to the angular distance of the star from the zenith of the place
at the moment the star crosses the meridian. We have already seen, in
Chapter II, how these angles may be measured, and it is apparent from
the figure that the difference between any two of these angles--e. g.,
the angles at _1_ and _2_--is equal to the angle at the center, _O_,
between the points _1_ and _2_. By measuring these angular distances of
the star from the zenith, the astronomer finds the angles at the center
of the earth between the stations _1_, _2_, _3_, etc., at which his
observations are made. If the meridian were a perfect circle the change
of zenith distance of the star, as one traveled along a meridian from
the equator to the pole, would be perfectly uniform--the same number of
degrees for each hundred miles traveled--and observations made in many
parts of the earth show that this is very nearly true, but that, on the
whole, as we approach the pole it is necessary to travel a little
greater distance than is required for a given change in the angle at the
equator. The earth is, in fact, flattened at the poles to the amount of
about 27 miles in the length of its diameter, and by this amount, as
well as by smaller variations due to mountains and valleys, the shape of
the earth differs from a perfect sphere. These astronomical measurements
of the curvature of the earth's surface furnish by far the most
satisfactory proof that it is very approximately a sphere, and furnish
as its equatorial diameter 7,926 miles.
Neglecting the _compression_, as it is called, i. e., the 27 miles by
which the equatorial diameter exceeds the polar, the size of the earth
may easily be found by measuring the distance _1_--_2_ along the
surface and by combining with this the angle _1 O 2_ obtained through
measuring the meridian altitudes of any star as seen from _1_ and _2_.
Draw on paper an angle equal to the measured difference of altitude and
find how far you must go from its vertex in order to have the distance
between the sides, measured along an arc of a circle, equal to the
measured distance between _1_ and _2_. This distance from the vertex
will be the earth's radius.
EXERCISE 19.--Measure the diameter of the earth by the method given
above. In order that this may be done satisfactorily, the two stations
at which observations are made must be separated by a considerable
distance--i. e., 200 miles. They need not be on the same meridian, but
if they are on different meridians in place of the actual distance
between them, there must be used the projection of that distance upon
the meridian--i. e., the north and south part of the distance.
By co-operation between schools in the Northern and Southern States,
using a good map to obtain the required distances, the diameter of the
earth may be measured with the plumb-line apparatus described in Chapter
II and determined within a small percentage of its true value.
45. THE MASS OF THE EARTH.--We have seen in Chapter IV the possibility
of determining the masses of the planets as fractional parts of the
sun's mass, but nothing was there shown, or could be shown, about
measuring these masses after the common fashion in kilogrammes or tons.
To do this we must first get the mass of the earth in tons or
kilogrammes, and while the principles involved in this determination are
simple enough, their actual application is delicate and difficult.
[Illustration: FIG. 26.--Illustrating the principles involved in
weighing the earth.]
In Fig. 26 we suppose a long plumb line to be suspended above the
surface of the earth and to be attracted toward the center of the earth,
_C_, by a force whose intensity is (Chapter IV)
F = k mE/R^{2},
where _E_ denotes the mass of the earth, which is to be determined by
experiment, and _R_ is the radius of the earth, 3,963 miles. If there is
no disturbing influence present, the plumb line will point directly
downward, but if a massive ball of lead or other heavy substance is
placed at one side, _1_, it will attract the plumb line with a force
equal to
f = k mB/r^{2},
where _r_ is the distance of its center from the plumb bob and _B_ is
its mass which we may suppose, for illustration, to be a ton. In
consequence of this attraction the plumb line will be pulled a little to
one side, as shown by the dotted line, and if we represent by _l_ the
length of the plumb line and by _d_ the distance between the original
and the disturbed positions of the plumb bob we may write the proportion
F : f :: l : d;
and introducing the values of _F_ and _f_ given above, and solving for
_E_ the proportion thus transformed, we find
E = B × l/d × (R/r)^{2}.
In this equation the mass of the ball, _B_, the length of the plumb
line, _l_, the distance between the center of the ball and the center of
the plumb bob, _r_, and the radius of the earth, _R_, can all be
measured directly, and _d_, the amount by which the plumb bob is pulled
to one side by the ball, is readily found by shifting the ball over to
the other side, at _2_, and measuring with a microscope how far the
plumb bob moves. This distance will, of course, be equal to _2 d_.
By methods involving these principles, but applied in a manner more
complicated as well as more precise, the mass of the earth is found to
be, in tons, 6,642 × 10^{18}--i. e., 6,642 followed by 18 ciphers, or in
kilogrammes 60,258 × 10^{20}. The earth's atmosphere makes up about a
millionth part of this mass.
If the length of the plumb line were 100 feet, the weight of the ball a
ton, and the distance between the two positions of the ball, _1_ and
_2_, six feet, how many inches, _d_, would the plumb bob be pulled out
of place?
Find from the mass of the earth and the data of § 40 the mass of the sun
in tons. Find also the mass of Mars. The computation can be very greatly
abridged by the use of logarithms.
46. PRECESSION.--That the earth is isolated in space and has no support
upon which to rest, is sufficiently shown by the fact that the stars are
visible upon every side of it, and no support can be seen stretching out
toward them. We must then consider the earth to be a globe traveling
freely about the sun in a circuit which it completes once every year,
and rotating once in every twenty-four hours about an axis which remains
at all seasons directed very nearly toward the star Polaris. The student
should be able to show from his own observations of the sun that, with
reference to the stars, the direction of the sun from the earth changes
about a degree a day. Does this prove that the earth revolves about the
sun?
But it is only in appearance that the pole maintains its fixed position
among the stars. If photographs are taken year after year, after the
manner of Exercise 7, it will be found that slowly the pole is moving
(nearly) toward Polaris, and making this star describe a smaller and
smaller circle in its diurnal path, while stars on the other side of the
pole (in right ascension 12h.) become more distant from it and describe
larger circles in their diurnal motion; but the process takes place so
slowly that the space of a lifetime is required for the motion of the
pole to equal the angular diameter of the full moon.
Spin a top and note how its rapid whirl about its axis corresponds to
the earth's diurnal rotation. When the axis about which the top spins is
truly vertical the top "sleeps"; but if the axis is tipped ever so
little away from the vertical it begins to wobble, so that if we imagine
the axis prolonged out to the sky and provided with a pencil point as a
marker, this would trace a circle around the zenith, along which the
pole of the top would move, and a little observation will show that the
more the top is tipped from the vertical the larger does this circle
become and the more rapidly does the wobbling take place. Were it not
for the spinning of the top about its axis, it would promptly fall over
when tipped from the vertical position, but the spin combines with the
force which pulls the top over and produces the wobbling motion. Spin
the top in opposite directions, with the hands of a watch and contrary
to the hands of a watch, and note the effect which is produced upon the
wobbling.
The earth presents many points of resemblance to the top. Its diurnal
rotation is the spin about the axis. This axis is tipped 23.5° away from
the perpendicular to its orbit (obliquity of the ecliptic) just as the
axis of the top is tipped away from the vertical line. In consequence of
its rapid spin, the body of the earth bulges out at the equator (27
miles), and the sun and moon, by virtue of their attraction (see Chapter
IV), lay hold of this protuberance and pull it down toward the plane of
the earth's orbit, so that if it were not for the spin this force would
straighten the axis up and set it perpendicular to the orbit plane. But
here, as in the case of the top, the spin and the tipping force combine
to produce a wobble which is called precession, and whose effect we
recognize in the shifting position of the pole among the stars. The
motion of precession is very much slower than the wobbling of the top,
since the tipping force for the earth is relatively very small, and a
period of nearly 26,000 years is required for a complete circuit of the
pole about its center of motion. Friction ultimately stops both the spin
and the wobble of the top, but this influence seems wholly absent in the
case of the earth, and both rotation and precession go on unchanged from
century to century, save for certain minor forces which for a time
change the direction or rate of the precessional motion, first in one
way and then in another, without in the long run producing any results
of consequence.
The center of motion, about which the pole travels in a small circle
having an angular radius of 23.5°, is at that point of the heavens
toward which a perpendicular to the plane of the earth's orbit points,
and may be found on the star map in right ascension 18h. 0m. and
declination 66.5°.
EXERCISE 20.--Find this point on the map, and draw as well as you can
the path of the pole about it. The motion of the pole along its path is
toward the constellation Cepheus. Mark the position of the pole along
this path at intervals of 1,000 years, and refer to these positions in
dealing with some of the following questions:
Does the wobbling of the top occur in the same direction as the motion
of precession? Do the tipping forces applied to the earth and top act in
the same direction? What will be the polar star 12,000 years hence? The
Great Pyramid of Egypt is thought to have been used as an observatory
when Alpha Draconis was the bright star nearest the pole. How long ago
was that?
The motion of the pole of course carries the equator and the equinoxes
with it, and thus slowly changes the right ascensions and declinations
of all the stars. On this account it is frequently called the precession
of the equinoxes, and this motion of the equinox, slow though it is, is
a matter of some consequence in connection with chronology and the
length of the year.
Will the precession ever bring back the right ascensions and
declinations to be again what they now are?
In what direction is the pole moving with respect to the Big Dipper?
Will its motion ever bring it exactly to Polaris? How far away from
Polaris will the precession carry the pole? What other bright stars will
be brought near the pole by the precession?
47. THE WARMING OF THE EARTH.--Winter and summer alike the day is on the
average warmer than the night, and it is easy to see that this surplus
of heat comes from the sun by day and is lost by night through radiation
into the void which surrounds the earth; just as the heat contained in a
mass of molten iron is radiated away and the iron cooled when it is
taken out from the furnace and placed amid colder surroundings. The
earth's loss of heat by radiation goes on ceaselessly day and night,
and were it not for the influx of solar heat this radiation would
steadily diminish the temperature toward what is called the "absolute
zero"--i. e., a state in which all heat has been taken away and beyond
which there can be no greater degree of cold. This must not be
confounded with the zero temperatures shown by our thermometers,
since it lies nearly 500° below the zero of the Fahrenheit scale (-273°
Centigrade), a temperature which by comparison makes the coldest winter
weather seem warm, although the ordinary thermometer may register
many degrees below its zero. The heat radiated by the sun into the
surrounding space on every side of it is another example of the same
cooling process, a hot body giving up its heat to the colder space about
it, and it is the minute fraction of this heat poured out by the sun,
and in small part intercepted by the earth, which warms the latter and
produces what we call weather, climate, the seasons, etc.
Observe the fluctuations, the ebb and flow, which are inherent in this
process. From sunset to sunrise there is nothing to compensate the
steady outflow of heat, and air and ground grow steadily colder, but
with the sunrise there comes an influx of solar heat, feeble at first
because it strikes the earth's surface very obliquely, but becoming more
and more efficient as the sun rises higher in the sky. But as the air
and the ground grow warm during the morning hours they part more and
more readily and rapidly with their store of heat, just as a steam pipe
or a cup of coffee radiates heat more rapidly when very hot. The warmest
hour of the day is reached when these opposing tendencies of income and
expenditure of heat are just balanced; and barring such disturbing
factors as wind and clouds, the gain in temperature usually extends to
the time--an hour or two beyond noon--at which the diminishing altitude
of the sun renders his rays less efficient, when radiation gains the
upper hand and the temperature becomes for a short time stationary, and
then commences to fall steadily until the next sunrise.
We have here an example of what is called a periodic change--i. e., one
which, within a definite and uniform period (24 hours), oscillates from
a minimum up to a maximum temperature and then back again to a minimum,
repeating substantially the same variation day after day. But it must be
understood that minor causes not taken into account above, such as
winds, water, etc., produce other fluctuations from day to day which
sometimes obscure or even obliterate the diurnal variation of
temperature caused by the sun.
Expose the back of your hand to the sun, holding the hand in such a
position that the sunlight strikes perpendicularly upon it; then turn
the hand so that the light falls quite obliquely upon it and note how
much more vigorous is the warming effect of the sun in the first
position than in the second. It is chiefly this difference of angle that
makes the sun's warmth more effective when he is high up in the sky than
when he is near the horizon, and more effective in summer than in
winter.
We have seen in Chapter III that the sun's motion among the stars takes
place along a path which carries it alternately north and south of the
equator to a distance of 23.5°, and the stars show by their earlier
risings and later settings, as we pass from the equator toward the north
pole of the heavens, that as the sun moves northward from the equator,
each day in the northern hemisphere will become a little longer, each
night a little shorter, and every day the sun will rise higher toward
the zenith until this process culminates toward the end of June, when
the sun begins to move southward, bringing shorter days and smaller
altitudes until the Christmas season, when again it is reversed and the
sun moves northward. We have here another periodic variation, which runs
its complete course in a period of a year, and it is easy to see that
this variation must have a marked effect on the warming of the earth,
the long days and great altitudes of summer producing the greater warmth
of that season, while the shorter days and lower altitudes of December,
by diminishing the daily supply of solar heat, bring on the winter's
cold. The succession of the seasons, winter following summer and summer
winter, is caused by the varying altitude of the sun, and this in turn
is due to the obliquity of the ecliptic, or, what is the same thing, the
amount by which the axis of the earth is tipped from being perpendicular
to the plane of its orbit, and the seasons are simply a periodic change
in the warming of the earth, quite comparable with the diurnal change
but of longer period.
It is evident that the period within which the succession of winter and
summer is completed, the year, as we commonly call it, must equal the
time required by the sun to go from the vernal equinox around to the
vernal equinox again, since this furnishes a complete cycle of the sun's
motions north and south from the equator. On account of the westward
motion of the equinox (precession) this is not quite the same as the
time required for a complete revolution of the earth in its orbit, but
is a little shorter (20m. 23s.), since the equinox moves back to meet
the sun.
48. RELATION OF THE SUN TO CLIMATE.--It is clear that both the northern
and southern hemispheres of the earth must have substantially the same
kind of seasons, since the motion of the sun north and south affects
both alike; but when the sun is north of the equator and warming our
hemisphere most effectively, his light falls more obliquely upon the
other hemisphere, the days there are short and winter reigns at the
time we are enjoying summer, while six months later the conditions are
reversed.
In those parts of the earth near the equator--the torrid zone--there is
no such marked change from cold to warm as we experience, because, as
the sun never gets more than 23.5° away from the celestial equator, on
every day of the year he mounts high in the tropic skies, always coming
within 23.5° of the zenith, and usually closer than this, so that there
is no such periodic change in the heat supply as is experienced in
higher latitudes, and within the tropics the temperature is therefore
both higher and more uniform than in our latitude.
In the frigid zones, on the contrary, the sun never rises high in the
sky; at the poles his greatest altitude is only 23.5°, and during the
winter season he does not rise at all, so that the temperature is here
low the whole year round, and during the winter season, when for weeks
or months at a time the supply of solar light is entirely cut off, the
temperature falls to a degree unknown in more favored climes.
If the obliquity of the ecliptic were made 10° greater, what would be
the effect upon the seasons in the temperate zones? What if it were made
10° less?
Does the precession of the equinoxes have any effect upon the seasons or
upon the climate of different parts of the earth?
If the axis of the earth pointed toward Arcturus instead of Polaris,
would the seasons be any different from what they are now?
49. THE ATMOSPHERE.--Although we live upon its surface, we are not
outside the earth, but at the bottom of a sea of air which forms the
earth's outermost layer and extends above our heads to a height of many
miles. The study of most of the phenomena of the atmosphere belongs to
that branch of physics called meteorology, but there are a few matters
which fairly come within our consideration of the earth as a planet. We
can not see the stars save as we look through this atmosphere, and the
light which comes through it is bent and oftentimes distorted so as to
present serious obstacles to any accurate telescopic study of the
heavenly bodies. Frequently this disturbance is visible to the naked
eye, and the stars are said to twinkle--i. e., to quiver and change
color many times per second, solely in consequence of a disturbed
condition of the air and not from anything which goes on in the star.
This effect is more marked low down in the sky than near the zenith, and
it is worth noting that the planets show very little of it because the
light they send to the earth comes from a disk of sensible area, while a
star, being much smaller and farther from the earth, has its disk
reduced practically to a mere point whose light is more easily affected
by local disturbances in the atmosphere than is the broader beam which
comes from the planets' disk.
50. REFRACTION.--At all times, whether the stars twinkle or not, their
light is bent in its passage through the atmosphere, so that the stars
appear to stand higher up in the sky than their true positions. This
effect, which the astronomer calls refraction, must be allowed for in
observations of the more precise class, although save at low altitudes
its amount is a very small fraction of a degree, but near the horizon it
is much exaggerated in amount and becomes easily visible to the naked
eye by distorting the disks of the sun and moon from circles into ovals
with their long diameters horizontal. The refraction lifts both upper
and lower edge of the sun, but lifts the lower edge more than the upper,
thus shortening the vertical diameter. See Fig. 27, which shows not only
this effect, but also the reflection of the sun from the curved surface
of the sea, still further flattening the image. If the surface of the
water were flat, the reflected image would have the same shape as the
sun's disk, and its altered appearance is sometimes cited as a proof
that the earth's surface is curved.
The total amount of the refraction at the horizon is a little more than
half a degree, and since the diameters of the sun and moon subtend an
angle of about half a degree, we have the remarkable result that in
reality the whole disk of either sun or moon is below the horizon at the
instant that the lower edge appears to touch the horizon and sunset or
moonset begins. The same effect exists at sunrise, and as a consequence
the duration of sunshine or of moonshine is on the average about six
minutes longer each day than it would be if there were no atmosphere and
no refraction. A partial offset to this benefit is found in the fact
that the atmosphere absorbs the light of the heavenly bodies, so that
stars appear much less bright when near the horizon than when they are
higher up in the sky, and by reason of this absorption the setting sun
can be looked at with the naked eye without the discomfort which its
dazzling luster causes at noon.
[Illustration: FIG. 27.--Flattening of the sun's disk by refraction and
by reflection from the surface of the sea.]
51. THE TWILIGHT.--Another effect of the atmosphere, even more marked
than the preceding, is the twilight. As at sunrise the mountain top
catches the rays of the coming sun before they reach the lowland, and at
sunset it keeps them after they have faded from the regions below, so
the particles of dust and vapor, which always float in the atmosphere,
catch the sunlight and reflect it to the surface of the earth while the
sun is still below the horizon, giving at the beginning and end of day
that vague and diffuse light which we call twilight.
[Illustration: FIG. 28.--Twilight phenomena.]
Fig. 28 shows a part of the earth surrounded by such a dust-laden
atmosphere, which is illuminated on the left by the rays of the sun, but
which, on the right of the figure, lies in the shadow cast by the earth.
To an observer placed at _1_ the sun is just setting, and all the
atmosphere above him is illumined with its rays, which furnish a bright
twilight. When, by the earth's rotation, this observer has been carried
to _2_, all the region to the east of his zenith lies in the shadow,
while to the west there is a part of the atmosphere from which there
still comes a twilight, but now comparatively faint, because the lower
part of the atmosphere about our observer lies in the shadow, and it is
mainly its upper regions from which the light comes, and here the dust
and moisture are much less abundant than in the lower strata. Still
later, when the observer has been carried by the earth's rotation to the
point _3_, every vestige of twilight will have vanished from his sky,
because all of the illuminated part of the atmosphere is now below his
horizon, which is represented by the line _3 L_. In the figure the sun
is represented to be 78° below this horizon line at the end of twilight,
but this is a gross exaggeration, made for the sake of clearness in the
drawing--in fact, twilight is usually said to end when the sun is 18°
below the horizon.
Let the student redraw Fig. 28 on a large scale, so that the points _1_
and _3_ shall be only 18° apart, as seen from the earth's center. He
will find that the point _L_ is brought down much closer to the surface
of the earth, and measuring the length of the line _2 L_, he should find
for the "height of the atmosphere" about one-eightieth part of the
radius of the earth--i. e., a little less than 50 miles. This, however,
is not the true height of the atmosphere. The air extends far beyond
this, but the particles of dust and vapor which are capable of sending
sunlight down to the earth seem all to lie below this limit.
The student should not fail to watch the eastern sky after sunset, and
see the shadow of the earth rise up and fill it while the twilight arch
retreats steadily toward the west.
[Illustration: FIG. 29.--The cause of long and short twilights.]
_Duration of twilight._--Since twilight ends when the sun is 18° below
the horizon, any circumstance which makes the sun go down rapidly will
shorten the duration of twilight, and anything which retards the
downward motion of the sun will correspondingly prolong it. Chief among
influences of this kind is the angle which the sun's course makes with
the horizon. If it goes straight down, as at _a_, Fig. 29, a much
shorter time will suffice to carry it to a depression of 18° than is
needed in the case shown at _b_ in the same figure, where the motion is
very oblique to the horizon. If we consider different latitudes and
different seasons of the year, we shall find every possible variety of
circumstance from _a_ to _b_, and corresponding to these, the duration
of twilight varies from an all-night duration in the summers of Scotland
and more northern lands to an hour or less in the mountains of Peru. For
the sake of graphical effect, the shortness of tropical twilight is
somewhat exaggerated by Coleridge in the lines,
"The sun's rim dips; the stars rush out:
At one stride comes the dark."
_The Ancient Mariner._
In the United States the longest twilights come at the end of June, and
last for a little more than two hours, while the shortest ones are in
March and September, amounting to a little more than an hour and a half;
but at all times the last half hour of twilight is hardly to be
distinguished from night, so small is the quantity of reflecting matter
in the upper regions of the atmosphere. For practical convenience it is
customary to assume in the courts of law that twilight ends an hour
after sunset.
How long does twilight last at the north pole?
_The Aurora._--One other phenomenon of the atmosphere may be mentioned,
only to point out that it is not of an astronomical character. The
Aurora, or northern lights, is as purely an affair of the earth as is a
thunderstorm, and its explanation belongs to the subject of terrestrial
magnetism.