How many first-magnitude stars would be needed to give as much light as do the 2,843 stars of magnitude 5.0 to 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 in [Fig. 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 star A at these several times is shown by lines drawn through A and prolonged to the background apparently furnished by the sky. A similar construction is made for the star B, and it is readily seen that owing to the changing position of the observer as he moves around the earth's orbit, both A and B will appear to move upon the background in orbits shaped like that of the earth as seen from the star, but having their size dependent upon the star's distance, the apparent orbit of A being larger than that of B, because A is nearer the earth. By measuring the angular distance between A and B at opposite seasons of the year (e. g., the angles A—Jan.—B, and A—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 is A than B. Strictly, the difference between the January and July angles is equal to the difference between the angles subtended at A and B by the diameter of the earth's orbit, and if B were so far away that the angle Jan.—B—July were nothing at all we should get immediately from the observations the angle Jan.—A—July, which would suffice to determine the stars' distance. Supposing the diameter of the earth's orbit and the angle at A to be known, can you make a graphical construction that will determine the distance of A from the earth?
Fig. 121.—Determining a star's parallax.
The angle subtended at A by 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 distance of 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 of A and B; and while it is customary to select for B a star or stars supposed to be much farther off than A, it may happen, and sometimes does happen, that these comparison stars as they are called are as near or nearer than A, and give a negative parallax—i. e., the difference between the angles at A and B proves to be negative, as it must whenever the star B is nearer than A.
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 in [Fig. 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,