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 of [Fig. 124], while Pollux is a solar star presenting in [Fig. 125] a spectrum like that of the sun, as do the other stars of this class.
Two thirds of the sun's near neighbors, shown in [Fig. 122], have spectra of the solar type, and in general stars of this 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 in [Fig. 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 which may have "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/2 |
| For 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 with Vega 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 called binary systems, and in cases of this kind the principles of celestial mechanics set forth in [Chapter IV] hold 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 their distance, and their direction one from the other, which is called position angle.