Popular Science Monthly/Volume 57/July 1900/Chapters on the Stars I
|CHAPTERS ON THE STARS.|
By Professor SIMON NEWCOMB.
IT would be difficult to name any subject of investigation, the progress of which during our time has been more remarkable than that in the field of stellar astronomy. Several features of this progress are especially noteworthy. One of these is the mere extension of research. A natural result of the northern hemisphere being the home of civilized peoples was that, thirty years ago, the study of the southern heavens had been comparatively neglected. It is true that the curiosity of the inquiring astronomers of the past would not be satisfied without their knowing something of what was to be seen south of the equator. Various enterprises and establishments had therefore contributed to our knowledge of the region in question. As far back as 1667, during a voyage to St. Helena, Halley catalogued the brighter stars in the region near the South Pole. About 1750 Lacaille, of France, established an observing station at the Cape of Good Hope, and made a catalogue of several thousand stars which has remained a handy book for the astronomer up to the present time. In 1834-38 Sir John Herschel made a special voyage to the Cape of Good Hope, armed with the best telescopes which the genius of his father had shown him how to construct, for the purpose of doing for the southern heavens as much as possible of what his father had done for the northern. The work of this expedition forms one of the most important and interesting chapters in the history of astronomic science. Not only is Herschel's magnificent volume a classic of astronomy, but the observations which it contains are still as carefully and profitably studied as any that have since been made. They may be said to form the basis of our present knowledge of the region which they included in their scope.
Herschel's work may be described as principally in the nature of an exploration. He had no instruments for accurately determining the positions of stars. In the latter field the first important contributions after Lacaille were made by Sir Thomas Brisbane, Governor of New South Wales, and Rumker, his assistant, at Paramata. Johnson, of England, about 1830, introduced modern accuracy into the construction of a rather limited catalogue of stars which he observed at St. Helena. About the same time the British Government established the observatory at the Cape of Good Hope, which has maintained its activity to the present time, though, at first, its means were extremely limited. About the middle of the century the Government of New South Wales established, first at Williamstown and then at Melbourne, an observatory which has worked in the same field with marked success.
An American enterprise in the same direction was that of Captain James M. Gilliss, who, in 1849, organized an astronomical expedition to. The principal motive of this enterprise was the determining of the solar parallax by observations upon Venus and Mars near the time of their nearest approach to the earth. As these observations would take but a small part of his time, Gilliss determined to take with him instruments for determining the positions of the stars. He established his observatory at a point near Santiago, where he continued his observations for nearly three years. He was a practical observer, but an untoward circumstance detracted from the value of his work. His observatory was built upon a rocky eminence, a foundation which seemed to afford the best possible guarantee of the stability of his instruments. He made no attempt to reduce his observations till after his return home. Then it was found that the foundation, through the expansion and contraction due to the heat of the sun, was subject to a diurnal change which made it extremely difficult to derive good results from his careful work. It was not until 1896, more than thirty years after his death, that the catalogue of the stars observed by him was at last completed and published.
We do not derogate in any way from the merit of these efforts in saying that they could not lead to results comparable with those of the score of richly equipped northern observatories which the leading nations and universities of Europe had endowed and supported for more than a hundred years. Only within the last thirty years has it been possible to bring our knowledge of the southern heavens up to a satisfactory stage. Now, however, the progress of southern astronomy, if we may use the term, is such that in several points our knowledge of the southern heavens surpasses that of the northern ones. If we measure institutions by the importance of the work they are doing, there are several in the southern hemisphere which must to-day be placed in the first rank.
The history and work of the Cordova Observatory are of special interest. In 1870 Dr. B. A. Gould, who might fairly be considered as the father of modern American astronomy, conceived the idea of establishing an observatory of the first class in South America. He found the President and Governor of the Argentine Republic ready to support his scheme with a liberality well fitted to impress us with a high sense of their standard of civilization. In a year or two the observatory at Cordova was in active operation. A statement of its work belongs to a subsequent chapter. Suffice it to remark here that Dr. Gould continued in active charge until 1885, when he returned home, and was succeeded by Thome, the present director.
A few years after Gould went to Cordova, Gill was made director of the Royal Observatory at the Cape of Good Hope. The rapid growth of this institution to one of the first rank is due no less to the scientific ability of the new director than to the unflagging energy which he has devoted to the enlargement of the resources of the institution. The great fact which he sought to impress upon his supporters was that the southern celestial hemisphere was as large as the northern, and therefore equally worthy of study.
In any general review of the progress of stellar astronomy during the past twenty years, we should find Harvard University before us at every turn. What it has done will be seen, perhaps in an imperfect way, in subsequent chapters. Not satisfied with the northern hemisphere, it has established a branch at Arequipa, Peru, in which its methods of observation and research are extended to the south celestial pole. Its principal specialties have been the continuous exploration of the heavens. Celestial photography, photometry and spectroscopy sum up its fields of activity. For more than ten years it might be almost said that a sleepless watch of the heavens has been kept up by an all-seeing photographic eye, with an accuracy of which the world has hardly had a conception. The completeness with which its work has been done has recently been shown in a striking way. Our readers are doubtless acquainted with the singular character of the minor planet Eros, whose orbit passes through that of Mars, as one link of a chain passes through another, and which comes nearer the earth at certain times than any other celestial body, the moon excepted. When the character of the orbit became established, it was of interest to know whether the planet had ever been observed as a fixed star at former oppositions. Chandler, having computed the path of the planet at the most important of the oppositions, beginning with 1892-94, communicated his results to Director Pickering, and suggested a search of the Harvard photographs to see if the planet could be found on them. The result was the discovery of the planet upon more than a score of plates taken at various times during the preceding ten years. New stars were formerly supposed to be of very rare occurrence, but since the Harvard system of photographing the heavens has been introduced, no less than three have been known to break out.
The great revelations of our times have come through the application of the spectroscope to the measurement of motions in the line of sight from us to a star. No achievement of the intellect of man would have seemed farther without the range of possibility to the thinker of half a century ago, than the discoveries of invisible bodies which are now being made with this instrument. The revelations of the telescope take us by surprise. But, if we consider what the thinker alluded to might regard as attainable, they are far surpassed by those of the spectroscope. The dark bodies, planets, we may call them, which are revolving round the stars, must be forever invisible in any telescope that it would be possible to construct. They would remain invisible if the power of the instrument were increased ten thousand times. And yet, if there are inhabitants on these planets, our astronomers could tell them more of the motions of the world on which they live than the human race knew of the motions of the earth before the time of Copernicus.
The men and institutions which have contributed to this result are so few in number that it will not be tedious to mention at least the principal actors. The possibility of measuring the motions of the stars in the line of sight by means of the spectroscope was first pointed out by Mr. now Sir William Huggins. He actually put the method into operation. As soon as its feasibility was demonstrated it was taken up at Greenwich. In these earlier attempts, eye methods alone were used, and the results were not always reliable. Then spectrum photography was applied at the astrophysical observatory at Potsdam by Vogel. Thence the photographic method soon spread to Meudon and Pulkova. But, as often happens when new fields of research are opened, we find them ablaze in quarters where we should least expect. The successful application of the method requires not only the best spectroscope, but the most powerful telescope at command. Ten years ago the most powerful telescope in the world was at the Lick Observatory. Mr. D. O. Mills put at its eye end the best spectograph that human art could make at that time, the work of Brashear. It is Campbell, who, with this instrument, has inaugurated a series of discoveries in the line in question which are without a parallel.
A mere survey of what has been done in the various lines we have mentioned would be far from giving an idea of the real significance of the advance we are considering. Cataloguing the stars, estimating their magnitudes, recording and comparing their spectra and determining their motions, might be considered as, after all, barren of results of the highest human interest. When we know the exact position of every star in the heavens, the direction in which it is moving and the character of its spectral lines, how much wiser are we?
What could hardly have been foreseen fifty years ago, is that these various classes of results are now made to combine and converge upon the greatest problem which the mind of man has ever attempted to grasp—that of the structure of the universe. The study of variable stars has suddenly fallen into line, so to speak, so that now, it is uniting itself to the study of all the other subjects to give us at least a faint conception of what the solution of this problem may be.
One of the principal objects of the present chapter is to make a comparison of these various researches, and discuss the views respecting the constitution of the stars individually, as well as of the universe as a whole, to which they lead us. But there are a number of details to be considered singly before we can combine results in this way. Our early chapters will therefore be devoted to the special features and individual problems of stellar astronomy which have occupied the minds of astronomers from the beginning of their work to the present time. Keeping these details in mind, we can profitably proceed to the consideration of the general conclusions to be drawn from them.
We may begin by refreshing our memories on some points, an understanding of which must be taken for granted. What are familiarly known as the heavenly bodies belong to two classes. Those nearest to us form a sort of colony far removed from all the others, called the solar system. The principal bodies of this system are the sun and eight great planets with their moons, revolving round it. On one of the planets, small when compared with the great bodies of the universe, but large to our every-day conceptions, we dwell. The other planets appear to us as stars. Four of them, Venus, Mars, Jupiter and Saturn, are distinguished from the fixed stars by their superior brightness and characteristic motions. Of the remaining three, Mercury will only rarely excite notice, while Uranus and Neptune are as good as invisible to the naked eye.
The dimensions of the solar system are vast when compared with any terrestrial standard. A cannon shot going incessantly at its utmost speed would be a thousand years in crossing the orbit of Neptune from side to side. But vast as the dimensions are, they sink into insignificance when compared with the distance of the stars. Outside the solar system are spaces which, so far as we know, are absolutely void, save here and there a comet or a meteor, until we look far outside the region which a cannon shot would cross in a million of years.
The nearest star is thousands of times farther away than the most distant planet. Scattered at these inconceivable distances are the bodies to which our attention is directed in the present work. If we are asked what they are, we may reply that the stars are suns. But we might equally well say that the sun is one of the stars; a small star, indeed, surrounded by countless others, many of which are much larger and brighter than itself. We shall treat our theme as far as possible by what we may call the natural method, beginning with what, being most obvious to the eye, was first noticed by man, or will be first noticed by an observer, and tracing knowledge up step by step to its present state.
Several features of the universe of stars will be evident at a glance. One of these is the diversity of the apparent brightness, or, in technical language, of the magnitudes of the stars. A few far outshine the great mass of their companions. A greater number are of what we may call medium brightness; there is a yet larger number of fainter ones, and about one half of all those seen by a keen eye under favorable conditions are so near the limit of visibility as to escape ordinary notice. Moreover, those which we see are but an insignificant fraction of the number revealed by the telescope. The more we increase our optical power, the greater the number that come into view. How many millions may exist in the heavens it is scarcely possible even to guess. The photographic maps of the heavens now being made probably show fifty millions, perhaps one hundred millions or more.
Another evident feature is the tendency of the brighter stars to cluster into groups, known as constellations. The latter are extremely irregular, so that it is impossible to decide where one constellation should end and another begin, or to which constellation a certain star may belong. Hence, we can neither define the constellations nor say what is their number, and the division of the stars among them is a somewhat arbitrary proceeding.
A third feature is the Milky Way or Galaxy, which, to ordinary vision, appears as an irregular succession of cloud-like forms spanning the heavens. We now know that these seeming clouds are really congeries of stars too small to be individually visible to the naked eye. We shall hereafter see that the stars of the Galaxy form, so to speak, the base on which the universe appears to be constructed. Each of these three features will be considered in its proper place.
II. Magnitudes of the Stars.
The apparent brightness of a star, as we see it from the earth, depends upon two causes—its intrinsic brilliancy or the quantity of light which it actually emits, and its distance from us. It follows that if all the stars were of equal intrinsic brightness we could determine their relative distances by measuring the respective amounts of light which we receive from them. The quantity of light in such a case varies inversely as the square of the distance. This will be made evident by Fig. 1, where S represents the position of a star, regarded as a luminous point, while A and B are screens placed at such a distance that each will receive the same amount of light from the star. If the screen B is twice as far as the screen A, its sides must be twice as large as those of A in order that it shall receive all the light that would fall on A. In this case its surface will be four times the surface of A. It is then evident that any small portion of the surface of B will receive one fourth as much light as an equal portion of surface A. Thus an eye or a telescope in the position B will receive from the star one fourth as much light as in the position A, and the star will seem one fourth as bright.
The fact is, however, that the stars are very unequal in their actual brightness, and in consequence the apparent magnitude of a star gives us no clue to its distance. Among the nearer of the stars are some scarcely, if at all. visible to the naked eye, while among the brighter
ones are several whose distances are immeasurably great. A remarkable example is that of, the second brightest star in the heavens.
For these reasons astronomers are obliged to content themselves, in the first place, with determinations of the actual amount of light that the various stars send to us, or their apparent brilliancy, without regard to their distance or actual brilliancy. The ancient astronomers divided all the stars they could see into six classes, the number expressing the apparent brightness being called the magnitude of the star. The brightest ones, numbering in all about fourteen, were said to be of the first magnitude. The fifty next in brightness were said to be of the second magnitude. Three times as many, an order fainter, were of the third magnitude. The progression was continued up to the sixth magnitude, which included those which were barely visible.
As the stars are actually of every degree of apparent brilliancy, no sharp line of demarkation could be drawn between those of one magnitude and those of the magnitude next higher. Hence, different observers made different estimates, some calling a star of the second magnitude which others would call of the first, while others would designate a star of the third magnitude which others would call of the second. It is therefore impossible to state with absolute numerical precision what number of stars should be regarded of one magnitude and what of another.
An idea of the magnitude of a star can be readily gained by the casual observer. Looking at the heavens on almost any cloudless evening, we may assume that the two, three or more brightest stars which we see are of the first magnitude. As examples of those of the second magnitude, may be taken the five brightest stars of the Dipper, the Pole Star and the brighter stars of Cassiopeia. Some or all of these objects can be seen on any clear night of the year in our latitude. Stars of the third magnitude are so numerous that it is difficult to select any one for comparison. The brightest star of the Pleiades is really of this magnitude, but it does not appear so in consequence of the five other stars by which it is surrounded. At a distance of 15° from the Pole Star, Beta Ursa Minoris is always visible, and may be distinguished by being slightly redder than the Pole Star: it lies between two fainter stars, the brighter of which is of the third and the other of the fourth magnitude. The five readily visible but fainter stars of the Pleiades are about of the fourth magnitude. Of the fifth magnitude are the faintest stars which are easily visible to the naked eye, while the sixth comprises those which are barely visible with good eyes.
Modern astronomers, while adhering to the general system which has come down to them from ancient times, have sought to give it greater definiteness. Careful study showed that the actual amount of light corresponding to the different magnitudes varied nearly in geometrical progression from one magnitude to another, a conclusion which accords with the well-known psychological law that the intensity of sensation varies by equal amounts when the exciting cause varies in geometrical progression. It was found that an average star of the fifth magnitude gave between two and three times as much light as an average one of the sixth; one of the fourth gave between two and three times as much light as one of the fifth; and so on to the second. In the case of the first magnitude, the diversity is so great that it is scarcely possible to fix an average ratio. Sirius, for example, is really six times as bright as Altair, which is commonly taken as a standard for a first magnitude star. To give precision to their estimates, modern astronomers are gradually seeking to lay the subject of magnitudes on an exact basis by defining a change of one unit in the magnitude as corresponding to an increase of about two and one half times in the amount of light.
If the practice of separating the visible stars into only six orders of magnitude were continued without change, we should still have the anomaly of including in one class stars of markedly different degrees of brightness. Some more than twice as bright as others would be designated of the same magnitude. Hence, to give quantitative exactness to the results, a magnitude is regarded as a quantity which may have any value whatever, and may be expressed by decimals—tenths or even hundredths. Thus, we may have stars of magnitude 5.0, 5.1, 5.2, etc., or we may even subdivide yet farther and speak of stars having magnitudes 5.11, 5.12, etc. Unfortunately, however, there is as yet no way known of determining the amount of light received from a star except by an estimate of its effect upon the eye. Two stars are regarded as equal when they appear to the eye of equal brilliancy. In such a case the judgment is very uncertain. Hence, observers have endeavored to give greater precision to it by the use of photometers,—instruments for measuring quantities of light. But even with this instrument the observer must depend upon an estimated equality of light as judged by the eye. The light from one star is increased or diminished in a known proportion until it appears equal to that of another star, which may be an artificial one produced by the flame of a candle. The proportion of increase or diminution shows the difference of magnitude between the two stars.
As we proceed to place the subject of photometric measures of star light on this precise basis we find the problem to be a complex one. In the first place not all the rays which come from a star are visible to our eyes as light. But all the radiance, visible or invisible, may be absorbed by a dark surface, and will then show its effect by heating that surface. The most perfect measure of the radiance of a star would therefore be the amount of heat which it conveys, because this expresses what is going on in the body better than the amount of visible light can do. But unfortunately the heating effect of the rays from a star is far below what can be measured or even indicated by any known instrument. We are therefore obliged to abandon any thought of determining the total amount of radiation and confine ourselves to that portion which we call light.
Here, when we aim at precision, we find that light, as we understand it, is properly measured only by its effect on the optic nerve, and there is no way of measuring this effect except by estimation. Thus, all the photometer can do is to give us the means of increasing or diminishing the light from one star, so that we can make it equal by estimation to that from some other star or source of light.
The difficulty of reaching strict results in this way is increased by the fact that stars are different in color. Two lights can be estimated as equal with greater precision when they are of the same color than when their colors are different. An additional source of uncertainty is brought in by what is known as the Purkinje phenomenon, after the physicist who first observed it. He found that if we took two lights of equal apparent brightness, the one red and the other green, and then increased or diminished them in the same proportion, they would no longer appear equal. In other words, the geometrical axiom that halves or quarters of equal quantities are themselves equal, does not apply to the effect of light on the eye. If we diminish the two equal lights, we find that the green will look brighter than the red. If we increase them in the same proportion, the red will look brighter than the green. In other words, the red light will, to our vision, increase or fade away more rapidly with a given amount of change than the green light will.
It is found in recent times that this law of change does not extend progressively through all spectral colors. It is true that as we pass from the red to the violet end of the spectrum the yellow fades away less rapidly with a given diminution than does the red, and the green still less rapidly than the yellow. But when we pass from the green to the blue, it is said that the latter does not fade out quite so fast as the green.
One obvious conclusion from all this is that two stars of different colors which look equal to the naked eye will not look equal in the telescope. The red or yellow star will look relatively brighter in a telescope; the green or bluish one relatively brighter to the naked eye.
In recent times stars have been photographed on a large scale. Their magnitudes can then be determined by the effect of the light on the photographic plate, the impression of the star, as seen in a microscope, being larger and more intense as the star is brighter. But the magnitude thus determined is not proportional to the apparent brightness as seen by the eye, because the photographic effect of blue light is much greater than that of red light having the same apparent brightness. In fact, the difference is so great that, with the chemicals formerly used, red light was almost without photographic effect. Even now, what we measure in taking the photograph of a star is almost entirely the light in the more refrangible portions of the spectrum. It appears, therefore, that when a blue and a yellow star, equally bright to the naked eye, are photographed, the impression made on the negative by the blue star will be greater than that made by the yellow one. A distinction is therefore recognized between photographic and visual magnitudes.
The photographic magnitudes of the stars are now being investigated and catalogued on a scale even larger than that on which we have studied the visual magnitudes. Yet we have to admit the noncorrespondence of the two systems. The bluer the star, the brighter will be its photographic as compared with its visual magnitude. The most that can be done is to bring abont the best attainable agreement between the two systems in the general average of all the stars.
Fortunately the differences between the colors of the stars are by no means so great as those between the colors of natural objects around us. All the stars radiate light of all colors; and although the difference is quite appreciable either by the eye or by the photograph, it is not so great as it would have been were the variations in color as wide as in the case of terrestrial objects.
Two comprehensive surveys of the heavens, intended to determine as accurately as possible the magnitudes of all the brighter stars, have recently been undertaken. One of these is the Harvard photometry, commenced by Professor Pickering at the Harvard Observatory, and now extended to the Southern Hemisphere by the aid of a branch establishment at Arequipa, Peru.
The instrument designed by Professor Pickering for his purpose is termed a meridian photometer, and is so arranged that the observer can see in the field of his telescope a reflected image of the Pole Star, and, at the same time, the image of some other star while it is passing the meridian. By a polarizing apparatus the image of the star to be measured is made to appear of equal brightness with that of the Pole Star, and the position of a Nicol prism, which brings out this equality, shows the ratio between the magnitudes of the two stars.
The other survey, with the same object, is now being made at the Potsdam Astrophysical Observatory, near Berlin. In the photometer used by the German astronomers the image of one star is compared with an artificial star formed by the flame of a candle. The work is performed in a more elaborate way than at the Harvard Observatory, and in consequence, only that part of the heavens, extending from the equator to 40° north declination, has been completed and published. A comparison of the results thus obtained with those of Professor Pickering, shows a curious difference depending on the color of the star. In the case of the reddest stars, the estimates are found to be in fairly close agreement, Pickering's being a little the fainter. But in the case of the white or bluish stars, the estimates of the German astronomers are more than one fourth of a magnitude greater than those of Pickering. This corresponds to an increase of nearly one fifth in the brightness. Whether this difference is to be regarded as purely psychological or due to the instruments used, is an interesting question which has not yet been settled. It is difficult to conceive how different instruments should give results so different. On the other hand, the comparisons made by the Germans make it difficult to accept the view that the difference is due purely to the personality of the observers. There are two German observers, Drs. Müller and Kempf, whose results agree with each other exactly. On the other hand, Pritchard, at Oxford, made quite an extensive photometric survey, using an instrument by which the light of one star was cut down by a wedge-shaped dark glass, whereby any gradation of light could be produced. A comparison shows that the results of Pritchard agree substantially with those of Pickering. It is quite possible that the Purkinje phenomenon may be the cause of the difference, the source of which is eminently worthy of investigation.
This fact simply emphasizes the lack of mathematical precision in photometric measurements of star light. Even apart from this difference of color, the estimates of two observers will frequently differ by 0.2 and sometimes by even 0.3 of a magnitude. These differences correspond roughly to 20 or 30 per cent in the amount of light.
It must not be supposed from this that such estimates are of no value for scientific purposes. Very important conclusions, based on great numbers of stars, may be drawn even from these uncertain quantities. Yet, it can hardly be doubted that if the light of a star could be measured from time to time to its thousandth part, conclusions of yet greater value and interest might be drawn from the measures.
We have said that in our modern system the aim has been to so designate the magnitudes of the stars that a series of magnitudes in arithmetical progression shall correspond to quantities of light ranging in geometrical progression. We have also said that a change of one unit of magnitude corresponds to a multiplication or division of the light by about 2.5. On any scale of magnitude this factor of multiplication constitutes the light-ratio of the scale. In recent times, after much discussion of the subject and many comparisons of photometric measures with estimates made in the old-fashioned way, there is a general agreement among observers to fix the light ratio at the number whose logarithm is 0.4. This is such that an increase of five units in the number expressing the magnitude corresponds to a division of the light by 100. If, for example, we take a standard star of magnitude one and another of magnitude six, the first would be 100 times as bright as the second. This corresponds to a light ratio slightly greater than 2.5.
When this scale is adopted, the series of magnitudes may extend indefinitely in both directions so that to every apparent brightness there will be a certain magnitude. For example, if we assign the magnitude 1.0 to a certain star, taken as a standard, which would formerly have been called a star of the first magnitude, then a star a little more than 2.5 times as bright would be of magnitude one less in number, that is, of magnitude 0. The one next brighter in the series would be of magnitude -1. So great is the diversity in the brightness of the stars formerly called of the first magnitude that Sirius is still brighter than the imaginary star just mentioned, the number expressing its magnitude being -1.4.
This suggests what we may regard as one of the capital questions in celestial photometry. There being no limit to the extent of the scale, what would be the stellar magnitude of the sun as we see it when expressed this way on the photometric scale? Such a number is readily derivable when we know the ratio between the light of the sun and that of a star of known magnitude. Many attempts have been made by observers to obtain this ratio; but the problem is one of great difficulty, and the results have been extremely discordant. Amongst them there are three which seem less liable to error than others; those of Wollaston, Bond and Zöllner. Their results for the stellar magnitude of the sun are as follow:
Of these, Zöllner's seems to be the best, and may, therefore, in taking the mean, be entitled to double weight. The result will then be:
Stellar magnitude of sun—26.4
From this number may be readily computed the ratio of sunlight to that of a star of any given magnitude. We thus find:
The sun gives us:
|10,000,000,000,||the light of Sirius.|
|91,000,000,000,||the light of a star of magnitude 1.|
|9,100,000,000,000,||the light of one of magnitude 6.|
The square roots of these numbers show the number of times we should increase the actual distance of the sun in order that it might shine as a star of the corresponding magnitude. These numbers and the corresponding parallax are as follows:
|Sirius;||Distance =||100,000:||Parallax =||2".06|
These parallaxes are those that the sun would have if placed at such a distance as to shine with the brightness indicated in the first column. They are generally larger than those of stars of the corresponding magnitudes, from which we conclude that the sun is smaller than the brighter of the stars.