It will be seen that the periods are very nearly doubled for each diminution of the brilliancy of the star by one magnitude. Moreover, the value of the photometric ratio for two consecutive magnitudes is a little uncertain, so that we may, without adding to the error of our results, suppose the period to be exactly double for each addition of unity to the magnitude. A computation of the period for any magnitude may be made with all necessary precision by the formula:
| P = Oy.88 × 2m; | |
| or,log. | P = 9.994 + 0.3m. |
It will now be of interest to compare the results of this theory with the observed periods of binary systems with a view to comparing their constitution with that of our Sun. There are, however, two difficulties in the way of doing this with rigorous precision.
The first difficulty is that there are very few binary systems of which the apparent dimensions of the orbit and the periods are known with any approach to exactness. This would not be a serious matter were it not that the short, and, therefore, known periods belong to a special class, that having the greatest density. Hence, when we derive our results from the systems of known periods we shall be making a biased selection from this particular class of stars.
The next difficulty is that the theory which we have set forth assumes the mass of the satellite either to be very small compared with that of the star, or the two bodies to be of the same constitution. If we apply the theory to systems in which this is not the case, the results which we shall get will be, in a certain way, those corresponding to the mean of the two components. Were it a question of masses, we should get with entire precision the sum of the masses of the two bodies. The best we can do, therefore, is to suppose the two companions fused into one having the combined brilliancy of the two. Then, if the result is too small for one, it will be too large for the other.
To show the method of proceeding, I have taken the six systems of shortest period found in Dr. See's 'Researches on Stellar Evolution.' The principal numbers are shown in the table below.
The first column, a", after the name of the star, gives the apparent semi-major axis of the orbit in seconds of arc. The next column gives the period in years. Column Mag. gives the apparent magnitude which the system would have were the two bodies fused into one.
Column P gives the period in years as it would be were the radius of the orbit equal to one second. It is formed by dividing the actual period by A. The next column gives the period as it would be were the stars of similar constitution to the Sun. The last column gives the square of the ratio of the two bodies, which, if the stars had the same
