Laboratory astronomy

Chapter I The Sun's Diurnal Motion

PAGE

Path of the Shadow of a Pin-head cast by the Sun upon a Horizontal Plane 1

Altitude and Bearing 4

Representation of the Celestial Sphere upon a Spherical Surface ... 5 The Sun's Diurnal Path upon the Hemisphere is a Circle — a Small Circle

except about March 20 and September 21 8

Determination of the Pole of the Circle 9

Bearing of the Points of Sunrise and Sunset 11

The Meridian — the Cardinal Points 11

Magnetic Declination . . 12

Azimuth 12

The Equinoctial 14

Position of the Pole as seen from Different Places of Observation . . 15

Latitude equals Elevation of Pole 16

Hour-angle of the Sun 17

Uniform Increase of the Sun's Hour-angle — Apparent Solar Time . .18

Declination of the Sun — its Daily Change 20

CHAPTER II Position of the Moon by its Configuration with Neighboring Stars . .21

Plotting the Position of the Moon upon a Star Map .... 24

Position of the Moon by Measures of Distance from Neighboring Stars . 25

The Cross-staff 25

Length of the Month . . .29

Node of the Moon's Orbit 30

Errors of the Cross-staff 31

iii iv TABLE OF CONTENTS

Chapter III The Diurnal. Motion Of The Stars Page

Instrument for measuring Altitude and Azimuth 34

Adjustment of the Altazimuth 35

Determination of Meridian by Observations of the Sun .... 37

Determination of Apparent Noon by Equal Altitudes of the Sun . . 39

Meridian Mark 40

Selection of Stars — Magnitudes . . .41

Plotting Diurnal Paths of Stars on the Hemisphere .... 42

Paths of Stars compared with that of the Sun 42

Drawing of Hemisphere with its Circles 42

Rotation of the Sphere as a Whole 43

Declinations of Stars do not change like that of the Sun ... 43

Equable Description of Hour-angle by Stars 43

Hour-angle and Declination fix the Position of a Heavenly Body as well as Altitude and Azimuth — Comparison of the Two Systems of

Coordinates 44

Equatorial Instrument for measuring Hour-angle and Declination . . 45

Universal Equatorial — Advantages of the Equatorial Mounting . . 45

Chapter IV

Rotation of the Heavens about an Axis passing through the Pole explains

Diurnal Motions of Sun, Moon, and Stars 47

Relative Position of Two Stars determined by their Declinations and the

Difference of their Hour-angles 48

Use of Equatorial to determine Positions of Stars 49

Use of a Timepiece to improve the Foregoing Method .... 50 Map of Stars by Comparison with a Fundamental Star . . . .53

Extension of Use of Timepiece to reduce Labor of Observation . . 54

The Vernal Equinox to replace the Fundamental Star — Right Ascension 56

Sidereal Time — Sidereal Clock . . . 57

Right Ascension of a Star is the Sidereal Time of its Passage across the

Meridian 58

Right Ascension of any Body plus its Hour-angle at any Instant is Sidereal Time at that Instant 58

Finding Stars by the Use of a Sidereal Clock and the Circles of the Equatorial Instrument . . . 59

The Clock Correction , . 60

List of Stars for determining Clock Error 61

TABLE OF CONTENTS V

Chapter V Motion Of The Moon And Sun Among The Stars Page

Plotting Stars upon a Globe in their Proper Relative Positions . . 63 Plotting Positions of the Moon upon Map and Globe by Observations of Declination, and Difference of Right Ascension from Neighboring

Stars 64

Variable Rate of Motion of the Moon 65

Variable Semi-diameter of the Moon 65

Position of Greatest Semi-diameter and of Greatest Angular Motion . 65

Plotting Moon's Path on an Ecliptic Map 65

Observations of Sun's Place in Reference to a Fundamental Star by Equatorial and Sidereal Clock 66

Sun's Place referred to Stars by Comparison with the Moon or Venus . 68

Plotting the Sun's Path upon the Globe — the Ecliptic .... 70

CHAPTER VI v

Use of the Altazimuth or Equatorial in the Meridian 72

The Meridian Circle 73

Adjustments of the Meridian Circle 74

Level 74

Collimation 78

Azimuth 78

Determination of Declinations 80

Determination of the Polar Point 81

Absolute Determination of Declination 81

Determination of the Equinox 83

Absolute Right Ascensions 84

Autumnal Equinox of 1899 85

Autumnal Equinox of 1900 87

Length of the Year . . . . .88

PEEFATOEY NOTE

The following pages are the beginning of an attempt to formulate a course of observation in preparation for the Laboratory Examination in Astronomy for admission to Harvard University.

The first two chapters had been written out nearly in their present form as part of a proposed text-book for that purpose. The later portions, though less fully developed, are still in a form that may prove useful to exceptional students and certainly to teachers.

In order to test the demand for such a presentation of the subject, it seemed worth while to print as much as is here given, for circulation among those who are interested in this method of teaching Astronomy, although the whole is not in suitable shape for use as a text-book.

Chapters VII and VIII treat of the Nautical Almanac and the solution of problems by means of the celestial globe.

These first eight chapters cover the subjects which are required of all candidates offering Astronomy for admission to Harvard University. The succeeding chapters will treat of those subjects from which selection may be made by each student in accordance with his tastes and opportunities.

The method and apparatus here described have been used since 1897 in the summer course in Astronomy for teachers at Harvard, and in part in a few high schools. All the apparatus may be obtained from at least one of the firms supplying laboratory apparatus.

Some copies of this edition have been interleaved in the binding for the convenience of teachers who may wish to make use of it in its present form, and in the hope of drawing suggestions from those interested.

Chapter I The Diurnal Motion Of The Sun

The most obvious and important astronomical phenomenon that men observe is the succession of day and night, and the motion of the sun which causes this succession is naturally the first object of astronomical study. Every one knows that the sun rises in the east and sets in the west, but very many educated people know little more of the course of the sun than this. The first task of the beginner in astronomy should be to observe, as carefully as possible, the motion of the sun for a day. What is to be observed then ? A little thought shows that it can only be the direction in which we have to look to see it at different times ; that is, toward what point of the compass — how far above the ground. All astronomical observation, indeed, comes down ultimately to this ■ — the direction in which we see things. The strong light of the sun enables us to make use of a very simple method depending on the principle that the shadow of a body lies in the same straight line with the body and the source of light.

Path of the Shadow of a Pin-head. — If we place a pin upright. on a horizontal plane in the sunlight and mark the position of the shadow of its head at any time, we thus fix the position of the sun at that time, since it is in the prolongation of the line drawn from the shadow to the pin-head. In order to carry out systematic observations by this method in such a form that the results may be easily discussed, it will be convenient to have the following apparatus : (1) A firm table in such a position as to receive sunlight for as long a period as possible. It is better that it should be in the open air, in which case it may be made by driving small posts into

Fig. 1

the ground and securely fastening a stout plank about 18 inches square as a top. (2) A board, 18 inches long and 8 inches broad, furnished with leveling screws and smoothly covered with white paper fastened down by (3) thumb tacks. (4) A level for leveling the board. (5) A compass. (6) A glass plate, 6 inches long and 2 inches broad, along the median line of which a straight black line is drawn. (7) A pin, 5 cm. long, with a spherical head and an accurately turned base for setting it vertical. (8) A timepiece. Draw a straight pencil line across the center of the paper as

Fig. 2 nearly as possible perpendicular to the length of the board. Place the board upon the table and level approximately. Put the compass on the middle of the pencil line and put the glass plate on the compass with its central line over the center of the needle ; turn the plate till its median line is parallel to the pencil line (Fig. 2),

TIIE DIUKNAL MOTION OF THE SUN 6

and swing the whole board horizontally, till the needle is parallel to the two lines, which are then said to be in the magnetic meridian. Press the leveling screws firmly into the table, and thus make dents by which the board may at any future time be placed in the same position without the renewed use of the compass. Level the board

Fig. 3 carefully, placing the level first east and west, then north and south. Place the pin in the pencil line, — in the center if the observation is made between March 20 and September 20, but near the southern edge of the board at any other time of the year, — pressing it firmly down till the base is close to the paper, so that the pin is perpendicular to the paper. Mark with a hard pencil the estimated center of the shadow of the pin-head, A (Fig. 3), noting the time by the watch to the nearest minute, affix a number or letter, and affix the same number to the recorded time of the observation in the note-book. It is a good plan to use pencil for notes made while fig. 4 observing, and ink for computations or notes added afterward in discussing them. Repeat at hourly, or better half-hourly, intervals, thus fixing a set of points (Pig. 4), through which a continuous curve may be drawn showing the path of the shadow for several hours. The same observation should be repeated two weeks later.

ALTITUDE AND BEARING

By the foregoing process we obtain a diagram on which is shown the position of the pin point, a magnetic meridian line through this point, and a series of numbered points showing the position of the shadow of the pin-head at different times ; the height of the pin is known and also the fact that its head was in the same vertical line with its point.

In the discussion of these results, it will be convenient to proceed as follows :

Kemove the pin and draw with a hard pencil a fine line, AB (Fig. 5), through the pinhole and the point marked at the first observation. This line is called a line of bearing, and the angle which

Fig. 5 it makes with the magnetic meridian is called the magnetic Rearing of the line. This angle, which may be directly measured on the diagram by a protractor, fixes the position of the vertical plane which contains the observed point and passes also through the center of the pin-head and the sun. If this point bears KW. from the pin, the sun evidently bears S.E.

Imagine a line, A C (Fig. 3), connecting the observed point with the sun's center and passing also through the center of the pin-head. The position of the sun in the vertical plane is evidently fixed by this line. The angle between the line of bearing and this line, BA C, is called the altitude of the sun ; it measures, by the ordinary convention of solid geometry, the angle between the sun's direction and the plane of the horizon.

THE DIURNAL MOTION OP 1 THE SUN 5

To determine this angle, lay off the line B'C (Fig. 6), equal in length to the pin, 5 cm., draw a perpendicular through B' ; and by means of a pair of compasses or scale laid between the two points A and B (Fig. 5), lay off the line A'B' on the perpendicular, draw A'C, and measure the angle B'A'C by a protractor. We now have the bearing and altitude of the sun at the time of the first observation, the bearing of the sun from the pin being opposite to that of the point from the pin. , In this manner the altitude and bearing are determined for each observed point upon the path of the shadow, and noted against the corresponding time, in the note-book (to avoid confusion, it is convenient to«make a separate figure for the morning and afternoon observations, as shown in Fig. 6). We have thus obtained a series of values which will enable us to study more easily the path of the sun upon the concave of the sky.

Plotting the Sun's Path on a Spherical Surface. — Probably the most evident method of accomplishing this object would be to

Fig. 7 construct a small concave portion of a sphere, as in the accompanying figure, which suggests how the position of the sun might be referred to the inside of a glass shell.

LABORATOKY ASTRONOMY

But the hollow surface offers difficulty in construction and manipulation, and it requires but little stretch of the imagination to pass to the convex surface as follows. The glass shell, as seen from the other side, would appear thus :

Fig. 8 and we can more readily get at it to measure it, and moreover can more easily recognize the properties of the lines which we shall come to draw upon it, since we are used to looking upon spheres from the outside rather than from the inside, except in the case of the celestial sphere.

On both Figs. 7 and 8 is shown a group of dots which have nearly the configuration of a group of stars conspicuous in the southern heavens in midsummer and called the constellation of Scorpio. It is evident that the constellation has the same shape in both cases, except that in Tig. 8 it is turned right for left or semi-inverted, as is the image of an object seen in a mirror. This property obviously belongs to all figures drawn on the concave surface as seen from the center, when they are looked at from the outside directly toward the center.

So also the diurnal motion of the sun, which as we see it from the center is from left to right, would be from right to left as viewed from the outside of such a surface. This latter is so slight an inconvenience that it is customary to represent the motions of the heavenly bodies in the sky upon an opaque globe, and to determine the angles which these bodies describe about the center, by measuring the corresponding arcs upon the convex surface.

Plotting on a Hemisphere. — The apparatus required for plotting the sun's path consists of : a hemisphere, a, 4J- inches in diameter ; a circular protractor, b, a quadrantal protractor, c, of 2£ inches

Fig. 9 radius, and a pair of compasses, d, whose legs may be bent and one of which carries a hard pencil point.

Determine by trial with, the compasses the center of the base of the hemisphere, and mark two diameters by drawing straight lines upon the base at right angles through the center. Prolong these by marks about -J inch in length upon the convex surface. Place the

Fig. 10 hemisphere exactly central upon the circular protractor, by bringing the marked ends of one of the diameters upon those divisions of the protractor which are numbered 0° and 180°, and the other on the divisions numbered 90° and 270°. Determine and mark the highest point of the hemisphere by placing the quadrant with its base upon the circular protractor, and its arc closely against the sphere, and marking the end of the scale (Fig. 10). Eepeat this with the arc in four positions, 90° apart on the base. The points thus determined should coincide ; if they do not, estimate and mark the center of the four points thus obtained. This point represents the highest point of the dome of the heavens — the point directly overhead, called the zenith, and the zero and 180° points on the base protractor may be taken as representing the south and north points respectively of the magnetic meridian.

The Sun's Path a Circle. — To plot the altitude and bearing of the first observation, place the foot of the quadrant or altitude arc close against the sphere, the foot of its graduated face on the degree of the protractor which corresponds to the bearing. Mark a fine point on the sphere at that degree of the altitude arc corresponding to the altitude at the first observation. This point fixes the direction in which the sun would have been seen from the center of the hemisphere at the time of observation if the zero line had been truly in the magnetic meridian. Proceed in the same manner with the other observations of bearing and altitude, and thus obtain

Fig. 11 a series of points (Fig. 11), through which may be drawn a continuous line representiDg the sun's path upon that day.

It will appear at once that the arcs between the successive points are of nearly equal length if the times of observation were equidistant, and otherwise are proportional to the intervals of time

THE DIURNAL MOTION OF THE SUN V

between the corresponding observations — a property which does not at all belong to the shadow curve from which the points are derived. We thus have a noteworthy simplification in referring our observations to the sphere. It will also appear that a sheet of

Fig. 12 stiff paper or cardboard may be held edgewise between the hemisphere and the eye, so as to cover all the points ; that is, they all lie in the same plane. This fact shows that the sun's path is a circle on the sphere. It is shown by the principles of solid geometry that all sections of the sphere by a plane are circles. If the plane of the circle passes through the center, it is the largest possible, its radius being equal to that of the sphere ; it is then called a great circle. Near the 20th of March and 22d of September it will be found that the path of the shadow is nearly a straight line on the diagram, and that the path of the sun is nearly a great circle ; that is, the plane of this circle passes nearly through the center of the sphere. In general, the shadow path is a curve, with its concave side toward the pin in summer and its convex side toward it in winter, while the path on the sphere is a small circle, that is, its plane does not pass through the center of the sphere.

Determining the Pole of the Circle. — It is proved by solid geometry that all points of any circle on the sphere are equidistant from two points on the sphere, called the poles of the circle. It is important to determine the pole of the sun's diurnal path.

Estimate as closely as possible the position on the sphere of a point which is at the same distance from all the observed points of the sun's path and open the compasses to nearly this distance. For a closer approximation to the position of the pole, place the steel point of the compasses at the point on the hemisphere corresponding to the first observation, a, and with the other (pencil) point draw a short arc, m (Fig. 12), near the estimated pole. Draw the arc n

from the point of the last observation, e, and join these two arcs by a third drawn from an observed point, b, as near as possible to the middle of the path; the pole of the sun's diurnal circle will lie nearly on the great circle drawn from b to the middle point o of the arc last drawn.

Place the steel point at o, and the pencil point at b, and try the distance of the pencil point from the sun's , ., T „' ., path at either ex-

tremity. If the pencil point lies above (or below) the path at both extremities the compasses must be opened (or closed) slightly and the assumed pole shifted directly away from (or toward) the middle of the path. '

The proper opening of the compasses is thns quickly determined as well as a close approximation to the position of the pole Place the steel point at this new position,^, the pencil point at b, and again test the extreme points. If the west end of the path is below the pencil point (Fig. 13), the latter should be brought directly down

Fig. 13 to the path by shifting the steel point on the sphere in the plane of the compass legs, that is, along the great circle from p to s.

From the point thus found a circle can be described with the compasses so as to pass approximately through all the observed points ; that is, this point is the pole of the sun's path, and when it is fixed as exactly as possible a circle is to be drawn from horizon to horizon which will represent the sun's path from the point of sunrise to that of sunset, and passing very nearly through all the observed points. The bearing of the points of sunrise and sunset may then be read off on the horizontal circle.

THE MERIDIAN

The pole as thus determined marks a very interesting and important point in the heavens. We will draw a great circle through the zenith and the pole. To do this, place the altitude arc against the sphere, as if to measure the altitude of the pole ; and

Fig. 14

using it as a guide, draw the northern quadrant of the vertical circle through the zenith and the pole. Note the bearing of this vertical circle. Place the altitude arc at the opposite bearing, and draw another or southern quadrant of the same great circle till it meets the south horizon. This great circle (Fig. 14) is called the meridian of the place of observation, and its plane is called the plane of the meridian of the place of observation, — sometimes the true meridian, to distinguish it from the magnetic meridian.

12 LABOKATOKY ASTRONOMY

The line in which it cuts the base of the hemisphere represents the meridian line or true meridian line, just as the line first drawn represents the line of the magnetic meridian. If the observations are made in the United States, near a line drawn from Detroit to Savannah, it will be found that the true meridian coincides very nearly with the magnetic meridian. East of the line joining these cities, the north end of the magnet points to the west of the true meridian by the amounts given in the following table :

21° at the extreme N.E. boundary of Maine. 15 at Portland. 10 at Albany and New Haven. 5 at Washington and Buffalo.

While on the west the declination, as it is called, is to the east of the true meridian.

5° at St. Louis and New Orleans. 10 at Omaha and El Paso. 15 at Dead wood and Los Angeles. 20 at Helena, Montana, and C. Blanco. 23 at the extreme N.W. boundary of the United States.

By drawing these lines on the map, as in Fig. 15, it is easy to estimate the declinations at intermediate points within one or two degrees, — at the present time west declinations in the United States are increasing and east declinations decreasing by about 1° in fifteen years.

A great circle perpendicular to the meridian may be drawn by placing the altitude protractor at readings 90° and 270° from the meridian reading and drawing arcs to the zenith in each case. This circle is the prime vertical, and intersects the horizon in the east and west points ; thus all the cardinal points are fixed by the meridian determined from our plotting of the sun's path.

Azimuth. — Place the hemisphere upon the circular protractor in such a position that the line of the true meridian on the hemisphere coincides with the zero line of the protractor.

Place the altitude arc so as to measure the altitude at any part of the sun's path west of the meridian (Fig. 16). The reading of the foot of the arc will give the angle between the true meridian and the vertical plane containing the sun at that point of its diurnal circle. This angle is its true bearing and differs from its magnetic

Fig. 15 bearing by the declination of the compass, being evidently less than the magnetic bearing, if the declination is west of north. It is also called the azimuth of the sun's vertical circle, or, briefly, of the sun.

Fig. 16

Formerly azimuth was usually reckoned from north through the west or east, to 180° at the south point. It is now customary to measure it from south through west up to 360°, so that the azimuth of a body when east of the meridian lies between 180° and 360°. The present method is more convenient because the given angle fixes the position of the vertical circle without the addition of the letters E. and W. It is worthy of notice that with this notation the azimuth of the sun as seen in northern latitudes outside of the tropics always increases with the time ; and indeed this is true of most of the bodies we shall have occasion to observe.

Now place the altitude quadrant so that its foot is at a point on the circular protractor where the reading is 360° minus the azimuth of the point just measured ; the sun at this point of its path is just as far east of the meridian as it was west of the meridian at the point last considered, and it will be found that the altitude of the two points is the same. On the path shown in Fig. 16 the altitude is 45° at the points whose azimuths are 60° and 300° (60 E. of S.).

This fact, that equal altitudes of the sun correspond to equal azimuths east and west of the true meridian, is an important one, and will presently be made use of to enable us to determine the position of the true meridian with a greater degree of precision.

THE EQUINOCTIAL

We shall find it convenient to draw upon the hemisphere another line, which plays an important role in astronomy, the great circle 90° from the pole. Placing the steel point of the compasses at the zenith, open the legs until the pencil point just comes to the horizon plane where the spherical surface meets it, so that if it were revolved about the zenith, the pencil point would move in the horizon. The compass points now span an arc of 90° upon the hemisphere. Place the steel point at the pole, and draw as much of a great circle as can be described on the sphere above the horizon. This will be just one-half of the great circle, and will cut the horizon in the east and west points. The new circle is called the equinoctial or celestial equator (Eig. 17).

We have seen that the path of the sun over the dome of the •heavens appears to be a small circle described from east to west about a fixed point in the dome as a pole. The ancient explanation of this fact was that the sun is fixed in a transparent spherical shell of immense size revolving daily about an axis, the earth being a plane in the center of unknown extent, but whose known regions are so small compared to the shell that from points even widely separated on the earth the appearance is the same ; just as the

Fig. 17

apparent direction and motion of the sun would be practically the same on our hemisphere to a microscopic observer at the center, and to another anywhere within one-hundredth of an inch of the center. When observations were made, however, at points some hundreds of miles apart on the same meridian, very perceptible differences were found, whose nature will be understood from a comparison of the hemisphere (Fig. 18 a), plotted from

Fig. 18 observations made Aug. 8, 1897, at a point in Canada, not far from Quebec, with a second hemisphere (Fig. lSb), on which is shown the path of the sun on the same date derived from observation of the shadow of a pin-head at Polf os in Norway. It appears on comparison that the distance of the pole above the north horizon

16 LABOKATOKY ASTKONOMY

is considerably greater in the latter, while the equator is just as much nearer the southern horizon ; the sun is at the same distance from the equator in each case. This fact cannot be explained on the supposition that the horizon planes of the two places are the same, for in that case we should have the spherical shell which contains the sun revolving at the same time about two different fixed axes, which is impossible. It is not, however, improbable that the earth's surface should be curved, if we can admit as a possibility that the direction of gravity, which is perpendicular to a horizontal plane, may be different at different places. That the earth's surface in the east and west direction is curved, we know; for men have traversed it from east to west and returned to the starting point, so that we have good reason to believe that its surface is everywhere curved. Long before this conclusive proof was obtained, however, the globular form of the earth was inferred on good grounds.