The numberof starsbrighterthanmagnitude1.0is11"""""2.0"39"""""3.0"142"""""4.0"463"""""5.0"1,483"""""6.0"4,326
It must not be inferred from this table that there are in the whole sky only 4,326 stars visible to the naked eye. The actual number is probably 50 or 60 per cent greater than this, and the normal human eye sees stars as faint as the magnitude 6.4 or 6.5, the discordance between this number and the previous statement, that the sixth magnitude is the limit of the naked-eye vision, having been introduced in the attempt to make precise and accurate a classification into magnitudes which was at first only rough and approximate. This same striving after accuracy leads to the introduction of fractional numbers to represent gradations of brightness intermediate between whole magnitudes. Thus of the 2,843 stars included between the fifth and sixth magnitudes a certain proportion are said to be of the 5.1 magnitude, 5.2 magnitude, and so on to the 5.9 magnitude, even hundredths of a magnitude being sometimes employed.
We have found the number of stars included between the fifth and sixth magnitudes by subtracting from the last number of the preceding table the number immediatelypreceding it, and similarly we may find the number included between each other pair of consecutive magnitudes, as follows:
Magnitude0123456Number of stars11281033211,0202,8434 × 3m12361083249722,916
In the last line each number after the first is found by multiplying the preceding one by 3, and the approximate agreement of each such number with that printed above it shows that on the whole, as far as the table goes, the fainter stars are approximately three times as numerous as those a magnitude brighter.
The magnitudes of the telescopic stars have not yet been measured completely, and their exact number is unknown; but if we apply our principle of a threefold increase for each successive magnitude, we shall find for the fainter stars—those of the tenth and twelfth magnitudes—prodigious numbers which run up into the millions, and even these are probably too small, since down to the ninth or tenth magnitude it is certain that the number of the telescopic stars increases from magnitude to magnitude in more than a threefold ratio. This is balanced in some degree by the less rapid increase which is known to exist in magnitudes still fainter; and applying our formula without regard to these variations in the rate of increase, we obtain as a rude approximation to the total number of stars down to the fifteenth magnitude, 86,000,000. The Herschels, father and son, actually counted the number of stars visible in nearly 8,000 sample regions of the sky, and, inferring the character of the whole sky from these samples, we find it to contain 58,500,000 stars; but the magnitude of the faintest star visible in their telescope, and included in their count, is rather uncertain.
How many first-magnitude stars would be needed to give as much light as do the 2,843 stars of magnitude 5.0to 6.0? How many tenth-magnitude stars are required to give the same amount of light?
To the modern man it seems natural to ascribe the different brilliancies of the stars to their different distances from us; but such was not the case 2,000 years ago, when each fixed star was commonly thought to be fastened to a "crystal sphere," which carried them with it, all at the same distance from us, as it turned about the earth. In breaking away from this erroneous idea and learning to think of the sky itself as only an atmospheric illusion through which we look to stars at very different distances beyond, it was easy to fall into the opposite error and to think of the stars as being much alike one with another, and, like pebbles on the beach, scattered throughout space with some rough degree of uniformity, so that in every direction there should be found in equal measure stars near at hand and stars far off, each shining with a luster proportioned to its remoteness.
188.Distances of the stars.—Now, in order to separate the true from the false in this last mode of thinking about the stars, we need some knowledge of their real distances from the earth, and in seeking it we encounter what is perhaps the most delicate and difficult problem in the whole range of observational astronomy. As shown inFig. 121, the principles involved in determining these distances are not fundamentally different from those employed in determining the moon's distance from the earth. Thus, the ellipse at the left of the figure represents the earth's orbit and the position of the earth at different times of the year. The direction of the starAat these several times is shown by lines drawn throughAand prolonged to the background apparently furnished by the sky. A similar construction is made for the starB, and it is readily seen that owing to the changing position of the observer as he moves around the earth's orbit, bothAandBwill appear to move upon the background in orbitsshaped like that of the earth as seen from the star, but having their size dependent upon the star's distance, the apparent orbit ofAbeing larger than that ofB, becauseAis nearer the earth. By measuring the angular distance betweenAandBat opposite seasons of the year (e. g., the anglesA—Jan.—B, andA—July—B) the astronomer determines from the change in this angle how much larger is the one path than the other, and thus concludes how much nearer isAthanB. Strictly, the difference between the January and July angles is equal to the difference between the angles subtended atAandBby the diameter of the earth's orbit, and ifBwere so far away that the angleJan.—B—Julywere nothing at all we should get immediately from the observations the angleJan.—A—July, which would suffice to determine the stars' distance. Supposing the diameter of the earth's orbit and the angle atAto be known, can you make a graphical construction that will determine the distance ofAfrom the earth?
Fig. 121.—Determining a star's parallax.Fig. 121.—Determining a star's parallax.
The angle subtended atAby the radius of the earth's orbit—i. e., 1/2 (Jan.—A—July)—is called the star's parallax, and this is commonly used by astronomers as a measure of the star's distance instead of expressing it in linear units such as miles or radii of the earth's orbit. The distanceof a star is equal to the radius of the earth's orbit divided by the parallax, in seconds of arc, and multiplied by the number 206265.
A weak point of this method of measuring stellar distances is that it always gives what is called a relative parallax—i. e., the difference between the parallaxes ofAandB; and while it is customary to select forBa star or stars supposed to be much farther off thanA, it may happen, and sometimes does happen, that these comparison stars as they are called are as near or nearer thanA, and give a negative parallax—i. e., the difference between the angles atAandBproves to be negative, as it must whenever the starBis nearer thanA.
The first really successful determinations of stellar parallax were made by Struve and Bessel a little prior to 1840, and since that time the distances of perhaps 100 stars have been measured with some degree of reliability, although the parallaxes themselves are so small—never as great as 1''—that it is extremely difficult to avoid falling into error, since even for the nearest star the problem of its distance is equivalent to finding the distance of an object more than 5 miles away by looking at it first with one eye and then with the other. Too short a base line.
189.The sun and his neighbors.—The distances of the sun's nearer neighbors among the stars are shown inFig. 122, where the two circles having the sun at their center represent distances from it equal respectively to 1,000,000 and 2,000,000 times the distance between earth and sun. In the figure the direction of each star from the sun corresponds to its right ascension, as shown by the Roman numerals about the outer circle; the true direction of the star from the sun can not, of course, be shown upon the flat surface of the paper, but it may be found by elevating or depressing the star from the surface of the paper through an angle, as seen from the sun, equal to its declination, as shown in the fifth column of the following table,
No.Star.Magnitude.R. A.Dec.Parallax.Distance.1α Centauri0.714.5h.-60°0.75"0.272Ll. 21,1856.811.0+370.450.46361 Cygni5.021.0+380.400.514η Herculis3.616.7+390.400.515Sirius-1.46.7-170.370.566Σ 2,3988.218.7+590.350.587Procyon0.57.6+50.340.608γ Draconis4.817.5+550.300.689Gr. 347.90.2+430.290.7110Lac. 9,3527.523.0-360.280.7411σ Draconis4.819.5+690.250.8212A. O. 17,415-69.017.6+680.250.8213η Cassiopeiæ3.40.7+570.250.8214Altair1.019.8+90.210.9715ϵ Indi5.221.9-570.201.0316Gr. 1,6186.710.1+500.201.031710 Ursæ Majoris4.28.9+420.201.0318Castor1.57.5+320.201.0319Ll. 21,2588.511.0+440.201.0320ο2Eridani4.54.2-80.191.0821A. O. 11,6779.011.2+660.191.0822Ll. 18,1158.09.1+530.181.1423B. D. 36°, 3,8837.120.0+360.181.1424Gr. 1,6186.510.1+500.171.2125β Cassiopeiæ2.30.1+590.161.282670 Ophiuchi4.418.0+20.161.2827Σ 1,5166.511.2+740.151.3828Gr. 1,8306.611.8+390.151.3829μ Cassiopeiæ5.41.0+540.141.4730ϑ Eridani4.43.5-100.141.4731ι Ursæ Majoris3.28.9+480.131.5832β Hydri2.90.3-780.11.5833Fomalhaut1.022.9-300.131.5834Br. 3,0776.023.1+570.131.5835ϑ Cygni2.520.8+330.121.7136β Comæ4.513.1+280.111.8737ψ5Aurigæ8.86.6+440.111.8738π Herculis3.317.2+370.111.8739Aldebaran1.14.5+160.102.0640Capella0.15.1+460.102.0641B. D. 35°, 4,0039.220.1+350.102.0642Gr. 1,6466.310.3+490.102.0643γ Cygni2.320.3+400.102.0644Regulus1.210.0+120.102.0645Vega0.218.6+390.102.06
in which the numbers in the first column are those placed adjacent to the stars in the diagram to identify them.
Fig. 122.—Stellar neighbors of the sun.Fig. 122.—Stellar neighbors of the sun.
190.Light years.—The radius of the inner circle inFig. 122, 1,000,000 times the earth's distance from the sun, is a convenient unit in which to express the stellar distances, and in the preceding table the distances of the stars from the sun are expressed in terms of this unit. To express them in miles the numbers in the table must be multiplied by 93,000,000,000,000. The nearest star, α Centauri, is 25,000,000,000,000 miles away. But there is another unit in more common use—i. e., the distance traveled overby light in the period of one year. We have already found (§ 141) that it requires light 8m. 18s. to come from the sun to the earth, and it is a simple matter to find from this datum that in a year light moves over a space equal to 63,368 radii of the earth's orbit. This distance is called alight year, and the distance of the same star, α Centauri, expressed in terms of this unit, is 4.26 years—i. e., it takes light that long to come from the star to the earth.
InFig. 122the stellar magnitudes of the stars are indicated by the size of the dots—the bigger the dot the brighter the star—and a mere inspection of the figure will serve to show that within a radius of 30 light years from the sun bright stars and faint ones are mixed up together, and that, so far as distance is concerned, the sun is only a member of this swarm of stars, whose distances apart, each from its nearest neighbor, are of the same order of magnitude as those which separate the sun from the three or four stars nearest it.
Fig. 122is not to be supposed complete. Doubtless other stars will be found whose distance from the sun is less than 2,000,000 radii of the earth's orbit, but it is not probable that they will ever suffice to more than double or perhaps treble the number here shown. The vast majority of the stars lie far beyond the limits of the figure.
191.Proper motions.—It is evident that these stars are too far apart for their mutual attractions to have much influence one upon another, and that we have here a case in which, according to§ 34, each star is free to keep unchanged its state of rest or motion with unvarying velocity along a straight line. Their very name,fixed stars, implies that they are at rest, and so astronomers long believed. Hipparchus (125B. C.) and Ptolemy (130A. D.) observed and recorded many allineations among the stars, in order to give to future generations a means of settling this very question of a possible motion of the stars and a resulting change in their relative positions upon the sky. For example, theyfound at the beginning of the Christian era that the four stars, Capella, ϑ Persei, α and β Arietis, stood in a straight line—i. e., upon a great circle of the sky. Verify this by direct reference to the sky, and see how nearly these stars have kept the same position for nearly twenty centuries. Three of them may be identified from the star maps, and the fourth, ϑ Persei, is a third-magnitude star between Capella and the other two.
Other allineations given by Ptolemy are: Spica, Arcturus and β Bootis; Spica, δ Corvi and γ Corvi; α Libræ, Arcturus and ζ Ursæ Majoris. Arcturus does not now fit very well to these alignments, and nearly two centuries ago it, together with Aldebaran and Sirius, was on other grounds suspected to have changed its place in the sky since the days of Ptolemy. This discovery, long since fully confirmed, gave a great impetus to observing with all possible accuracy the right ascensions and declinations of the stars, with a view to finding other cases of what was calledproper motion—i. e., a motion peculiar to the individual star as contrasted with the change of right ascension and declination produced for all stars by the precession.
Since the middle of the eighteenth century there have been made many thousands of observations of this kind, whose results have gone into star charts and star catalogues, and which are now being supplemented by a photographic survey of the sky that is intended to record permanently upon photographic plates the position and magnitude of every star in the heavens down to the fourteenth magnitude, with a view to ultimately determining all their proper motions.
The complete achievement of this result is, of course, a thing of the remote future, but sufficient progress in determining these motions has been made during the past century and a half to show that nearly every lucid star possesses some proper motion, although in most cases it is very small, there being less than 100 known stars in which itamounts to so much as 1" per annum—i. e., a rate of motion across the sky which would require nearly the whole Christian era to alter a star's direction from us by so much as the moon's angular diameter. The most rapid known proper motion is that of a telescopic star midway between the equator and the south pole, which changes its position at the rate of nearly 9" per annum, and the next greatest is that of another telescopic star, in the northern sky, No. 28 ofFig. 122. It is not until we reach the tenth place in a list of large proper motions that we find a bright lucid star, No. 1 ofFig. 122. It is a significant fact that for the most part the stars with large proper motions are precisely the ones shown inFig. 122, which is designed to show stars near the earth. This connection between nearness and rapidity of proper motions is indeed what we should expect to find, since a given amount of real motion of the star along its orbit will produce a larger angular displacement, proper motion, the nearer the star is to the earth, and this fact has guided astronomers in selecting the stars to be observed for parallax, the proper motion being determined first and the parallax afterward.
192.The paths of the stars.—We have already seen reason for thinking that the orbit along which a star moves is practically a straight line, and from a study of proper motions, particularly their directions across the sky, it appears that these orbits point in all possible ways—north, south, east, and west—so that some of them are doubtless directed nearly toward or from the sun; others are square to the line joining sun and star; while the vast majority occupy some position intermediate between these two. Now, our relation to these real motions of the stars is well illustrated inFig. 112, where the observer finds in some of the shooting stars a tremendous proper motion across the sky, but sees nothing of their rapid approach to him, while others appear to stand motionless, although, in fact, they are moving quite as rapidly as are their fellows. The fixedstar resembles the shooting star in this respect, that its proper motion is only that part of its real motion which lies at right angles to the line of sight, and this needs to be supplemented by that other part of the motion which lies parallel to the line of sight, in order to give us any knowledge of the star's real orbit.
Fig. 123.—Motion of Polaris in the line of sight as determined by the spectroscope. Frost.Fig. 123.—Motion of Polaris in the line of sight as determined by the spectroscope.Frost.
193.Motion in the line of sight.—It is only within the last 25 years that anything whatever has been accomplished in determining these stellar motions of approach or recession, but within that time much progress has been made by applying the Doppler principle (§ 89) to the study of stellar spectra, and at the present time nearly every great telescope in the world is engaged upon work of this kind. The shifting of the lines of the spectrum toward the violet or toward the red end of the spectrum indicates with certainty the approach or recession of the star, but this shifting, which must be determined by comparing the star's spectrum with that of some artificial light showing corresponding lines, is so small in amount that its accurate measurement is a matter of extreme difficulty, as may be seen fromFig. 123. This cut shows along its central line a part of the spectrum of Polaris, between wave lengths 4,450 and 4,600 tenth meters, while above and below are the corresponding parts of the spectrum of an electric spark whose light passed through the same spectroscope and was photographed upon the same plate with that of Polaris. This comparison spectrum is, as it should be, a discontinuous or bright-line one, while the spectrum of the star is a continuousone, broken only by dark gaps or lines, many of which have no corresponding lines in the comparison spectrum. But a certain number of lines in the two spectra do correspond, save that the dark line is always pushed a very little toward the direction of shorter wave lengths, showing that this star is approaching the earth. This spectrum was photographed for the express purpose of determining the star's motion in the line of sight, and with it there should be compared Figs.124and125, which show in the upper part of each a photograph obtained without comparison spectra by allowing the star's light to pass through some prisms placed just in front of the telescope. The lower section of each figure shows an enlargement of the original photograph, bringing out its details in a way not visible to the unaided eye. In the enlarged spectrum of β Aurigæ a rate of motion equal to that of the earth in its orbit would be represented by a shifting of 0.03 of a millimeter in the position of the broad, hazy lines.
Fig. 124.—Spectrum of β Aurigæ.—Pickering.Fig. 124.—Spectrum of β Aurigæ.—Pickering.
Despite the difficulty of dealing with such small quantities as the above, very satisfactory results are now obtained, and from them it is known that the velocities of stars in the line of sight are of the same order of magnitude as the velocities of the planets in their orbits, ranging all the way from 0 to 60 miles per second—more than 200,000 miles per hour—which latter velocity, according to Campbell, is the rate at which μ Cassiopeiæ is approaching the sun.
The student should not fail to note one important difference between proper motions and the motions determined spectroscopically: the latter are given directly in miles per second, or per hour, while the former are expressed in angular measure, seconds of arc, and there can be no direct comparison between the two until by means of the known distances of the stars their proper motions are converted from angular into linear measure. We are brought thus to the very heart of the matter; parallax, proper motion, and motion in the line of sight are intimately related quantities, all of which are essential to a knowledge of the real motions of the stars.
Fig. 125.—Spectrum of Pollux.—Pickering.Fig. 125.—Spectrum of Pollux.—Pickering.
194.Star drift.—An illustration of how they may be made to work together is furnished by some of the stars—which make up the Great Dipper—β, γ, ϑ, and ζ Ursæ Majoris, whose proper motions have long been known to point in nearly the same direction across the sky and to be nearly equal in amount. More recently it has been found that these stars are all moving toward the sun with approximately the same velocity—18 miles per second. One other star of the Dipper, δ Ursæ Majoris, shares in the common proper motion, but its velocity in the line of sight has not yet been determined with the spectroscope. These similar motions make it probable that the stars are really traveling together through space along parallel lines; and on thesupposition that such is the case it is quite possible to write out a set of equations which shall involve their known proper motions and motions in the line of sight, together with their unknown distances and the unknown direction and velocity of their real motion along their orbits. Solving these equations for the values of the unknown quantities, it is found that the five stars probably lie in a plane which is turned nearly edgewise toward us, and that in this plane they are moving about twice as fast as the earth moves around the sun, and are at a distance from us represented by a parallax of less than 0.02"—i. e., six times as great as the outermost circle inFig. 122. A most extraordinary system of stars which, although separated from each other by distances as great as the whole breadth ofFig. 122, yet move along in parallel paths which it is difficult to regard as the result of chance, and for which it is equally difficult to frame an explanation.
Fig. 126.—The Great Dipper, past, present, and future.Fig. 126.—The Great Dipper, past, present, and future.
The stars α and η of the Great Dipper do not share in this motion, and must ultimately part company with the other five, to the complete destruction of the Dipper's shape.Fig. 126illustrates this change of shape, the upper part of the figure (a) showing these seven stars as they were grouped at a remote epoch in the past,while the lower section (c) shows their position for an equally remote epoch in the future. There is no resemblance to a dipper in either of these configurations, but it should be observed that in each of them the stars α and η keep their relative position unaltered, and the other five stars also keep together, the entire change of appearance being due to the changing positions of these two groups with respect to each other.
This phenomenon of groups of stars moving together is calledstar drift, and quite a number of cases of it are found in different parts of the sky. The Pleiades are perhaps the most conspicuous one, for here some sixty or more stars are found traveling together along similar paths. Repeated careful measurements of the relative positions of stars in this cluster show that one of the lucid stars and four or five of the telescopic ones do not share in this motion, and therefore are not to be considered as members of the group, but rather as isolated stars which, for a time, chance to be nearly on line with the Pleiades, and probably farther off, since their proper motions are smaller.
To rightly appreciate the extreme slowness with which proper motions alter the constellations, the student should bear in mind that the changes shown in passing from one section ofFig. 126to the next represent the effect of the present proper motions of the stars accumulated for a period of 200,000 years. Will the stars continue to move in straight paths for so long a time?
195.The sun's way.—Another and even more interesting application of proper motions and motions in the line of sight is the determination from them of the sun's orbit among the stars. The principle involved is simple enough. If the sun moves with respect to the stars and carries the earth and the other planets year after year into new regions of space, our changing point of view must displace in some measure every star in the sky save those which happen to be exactly on the line of the sun's motion, and even thesewill show its effect by their apparent motion of approach or recession along the line of sight. So far as their own orbital motions are concerned, there is no reason to suppose that more stars move north than south, or that more go east than west; and when we find in their proper motions a distinct tendency to radiate from a point somewhere near the bright star Vega and to converge toward a point on the opposite side of the sky, we infer that this does not come from any general drift of the stars in that direction, but that it marks the course of the sun among them. That it is moving along a straight line pointing toward Vega, and that at least a part of the velocities which the spectroscope shows in the line of sight, comes from the motion of the sun and earth. Working along these lines, Kapteyn finds that the sun is moving through space with a velocity of 11 miles per second, which is decidedly below the average rate of stellar motion—19 miles per second.
196.Distance of Sirian and solar stars.—By combining this rate of motion of the sun with the average proper motions of the stars of different magnitudes, it is possible to obtain some idea of the average distance from us of a first-magnitude star or a sixth-magnitude star, which, while it gives no information about the actual distance of any particular star, does show that on the whole the fainter stars are more remote. But here a broad distinction must be drawn. By far the larger part of the stars belong to one of two well-marked classes, called respectively Sirian and solar stars, which are readily distinguished from each other by the kind of spectrum they furnish. Thus β Aurigæ belongs to the Sirian class, as does every other star which has a spectrum like that ofFig. 124, while Pollux is a solar star presenting inFig. 125a spectrum like that of the sun, as do the other stars of this class.
Two thirds of the sun's near neighbors, shown inFig. 122, have spectra of the solar type, and in general stars ofthis class are nearer to us than are the stars with spectra unlike that of the sun. The average distance of a solar star of the first magnitude is very approximately represented by the outer circle inFig. 122, 2,000,000 times the distance of the sun from the earth; while the corresponding distance for a Sirian star of the first magnitude is represented by the number 4,600,000.
A third-magnitude star is on the average twice as far away as one of the first magnitude, a fifth-magnitude star four times as far off, etc., each additional two magnitudes doubling the average distance of the stars, at least down to the eighth magnitude and possibly farther, although beyond this limit we have no certain knowledge. Put in another way, the naked eye sees many Sirian stars whichmayhave "gone out" and ceased to shine centuries ago, for the light by which we now see them left those stars before the discovery of America by Columbus. For the student of mathematical tastes we note that the results of Kapteyn's investigation of the mean distances (D) of the stars of magnitude (m) may be put into two equations:
For Solar Stars,D= 23 × 2m/2For Sirian Stars,D= 52 × 2m/2
where the coefficients 23 and 52 are expressed in light years. How long a time is required for light to come from an average solar star of the sixth magnitude?
197.Consequences of stellar distance.—The amount of light which comes to us from any luminous body varies inversely as the square of its distance, and since many of the stars are changing their distance from us quite rapidly, it must be that with the lapse of time they will grow brighter or fainter by reason of this altered distance. But the distances themselves are so great that the most rapid known motion in the line of sight would require more than 1,000 years (probably several thousand) to produce any perceptible change in brilliancy.
The law in accordance with which this change of brilliancy takes place is that the distance must be increased or diminished tenfold in order to produce a change of five magnitudes in the brightness of the object, and we may apply this law to determine the sun's rank among the stars. If it were removed to the distance of an average first-, or second-, or third-magnitude star, how would its light compare with that of the stars? The average distance of a third-magnitude star of the solar type is, as we have seen above, 4,000,000 times the sun's distance from the earth, and since 4,000,000 = 106.6, we find that at this distance the sun's stellar magnitude would be altered by 6.6 × 5 magnitudes, and would therefore be -26.5 + 33.0 = 6.5—i. e., the sun if removed to the average distance of the third-magnitude stars of its type would be reduced to the very limit of naked-eye visibility. It must therefore be relatively small and feeble as compared with the brightness of the average star. It is only its close proximity to us that makes the sun look brighter than the stars.
The fixed stars may have planets circling around them, but an application of the same principles will show how hopeless is the prospect of ever seeing them in a telescope. If the sun's nearest neighbor, α Centauri, were attended by a planet like Jupiter, this planet would furnish to us no more light than does a star of the twenty-second magnitude—i. e., it would be absolutely invisible, and would remain invisible in the most powerful telescope yet built, even though its bulk were increased to equal that of the sun. Let the student make the computation leading to this result, assuming the stellar magnitude of Jupiter to be -1.7.
198.Double stars.—In the constellation Taurus, not far from Aldebaran, is the fourth-magnitude star θ Tauri, which can readily be seen to consist of two stars close together. The star α Capricorni is plainly double, and a sharp eye can detect that one of the faint stars which withVega make a small equilateral triangle, is also a double star. Look for them in the sky.
In the strict language of astronomy the term double star would not be applied to the first two of these objects, since it is usually restricted to those stars whose angular distance from each other is so small that in the telescope they appear much as do the stars named above to the naked eye—i. e., their angular separation is measured by a few seconds or fractions of a single second, instead of the six minutes which separate the component stars of θ Tauri or α Capricorni. There are found in the sky many thousands of these close double stars, of which some are only optically double—i. e., two stars nearly on line with the earth but at very different distances from it—while more of them are really what they seem, stars near each other, and in many cases near enough to influence each other's motion. These are calledbinarysystems, and in cases of this kind the principles of celestial mechanics set forth inChapter IVhold true, and we may expect to find each component of a double star moving in a conic section of some kind, having its focus at the common center of gravity of the two stars. We are thus presented with problems of orbital motion quite similar to those which occur in the solar system, and careful telescopic observations are required year after year to fix the relative positions of the two stars—i. e., their angular separation, which it is customary to call theirdistance, and their direction one from the other, which is calledposition angle.
199.Orbits of double stars.—The sun's nearest neighbor, α Centauri, is such a double star, whose position angle and distance have been measured by successive generations of astronomers for more than a century, andFig. 127shows the result of plotting their observations. Each black dot that lies on or near the circumference of the long ellipse stands for an observed direction and distance of the fainter of the two stars from the brighter one, which is representedby the small circle at the intersection of the lines inside the ellipse. It appears from the figure that during this time the one star has gone completely around the other, as a planet goes around the sun, and the true orbit must therefore be an ellipse having one of its foci at the center of gravity of the two stars. The other star moves in an ellipse of precisely similar shape, but probably smaller size, since the dimensions of the two orbits are inversely proportional to the masses of the two bodies, but it is customary to neglect this motion of the larger star and to give to the smaller one an orbit whose diameter is equal to the sum of the diameters of the two real orbits. This practice, which has been followed inFig. 127, gives correctly the relative positions of the two stars, and makes one orbit do the work of two.