traversed in one year, dz' = 2πα sin i Cos u. The maximum value of 2π a sin i this expression occurs when u = 0° or π, and is If the orbit is elliptical, p and u may be deduced from the elements, and d z may be expressed as a function of the eccentricity, node, and time, multiplied by the factor, which is constant for each orbit, a sin i Let V denote the velocity of light, v the velocity of approach of a star, λ the wave-length of a given ray of light, and 7 the corresponding change it undergoes, due to the velocity. Then V+v:V=λ+1:λ or v = V; v and V are commonly expressed in kilometers per second, I and λ in ten-millionths of a millimeter; V 300000. The line F is frequently used in these measures, and for it λ= 4865. Substituting these values, v = 62 7. For the D line, à 5900, and since the interval between the two components equals 6, a velocity of 305 kilometers per second will be required to produce a deviation equal to the interval between these lines. It will be more convenient to measure the velocity of a star in terms of m, the annual motion, taking the distance from the Earth to the Sun as a unit. This may then be reduced to seconds of arc, if the distance of the star is known, by multiplying by the parallax p. Light traverses the distance from the Earth to the Sun in about 498 seconds, or would traverse 63300 times this distance in a year. Accordingly, v = 63300 ; for the F line v = 131, for the interval of the D lines, v = 64 l. If is positive or the line moves toward the red end, it denotes that the star is receding from the observer. We have thus two values of the relative motion of the stars in the line of sight; one, d z, deduced by computation from the micrometer measurements; the other, v p, or 13 l p, if the Fline is observed, found by the spectroscope. Equating these values, since p is the only unknown quantity, p = 137• The dimensions of the orbit are now found directly, since will equal the semi-axis major in terms of the distance of the Sun from the Earth. d z It not unfrequently happens that we have an estimate of the difference in magnitude of the two components of a double star by one observer using a telescope, and also an estimate of their combined light by another observer viewing them with the unassisted eye. From these data we wish to determine the brightness of either component alone. Sometimes we have the opposite problem, given the magnitude of the separate stars to find that of both, as seen by the eye or in a telescope not capable of separating them. Let denote the light of the fainter star in terms of the brighter, and m the magnitude of the fainter minus the magnitude of the brighter. Then, on Pogson's system, m = =-2.5 log l. If M is the magnitude of the brighter star minus that of a star equivalent to the two combined, or having the light (1+), then M-2.5 log (1+1). From these formula we can always find the corresponding values of M and m. M=0.75 when m is zero or the stars are equal. Table IV. enables us to determine M to the nearest tenth of a magnitude for any value of m. As an example, suppose two stars have magnitudes 2.0 and 3.0; then m = 3.0 - 2.0 1.0, and M, from the table, lies between 0.35 and The light of both combined will therefore equal 0.45 or equals 0.4. 2.00.4 1.6. The maximum value of It is sometimes convenient to know what would be the magnitude of a star whose mass was equal to that of the two components of a double star of the same density and brightness. Let m' equal the difference in magnitudes of the two components, and I and n, the light and mass of the fainter in terms of the brighter. Then m' -2.5 log ——2.5 log n3 — —1.67 log n, since the light is proportional to the square, and the mass to the cube, of the diameter. If then M equals the magnitude of the brighter component minus that of both combined, we shall have M= 1.67 log (1 + n), from which M is determined as before from any given value of m'. The third column of Table IV. gives the value of m' corresponding to every odd twentieth of a magnitude of M. The value of the latter may thus always be determined to the nearest tenth of a magnitude. The maximum value of M is 0.50, when m'0. Adopting the same magnitudes as in the last example, if two stars have the magnitudes of 2.0 and 3.0, m' will equal 1.0. This value from the third column of Table IV. will correspond to a value of M lying between 0.15 and 0.25, or will equal 0.2. The magnitude of a star having the same mass as the binary will therefore have a magnitude 2.0 — 0.2 1.8. -3 Most of the binary stars whose orbits have been computed are compared in Table V. The successive columns give a current number, the name of the star, the number of the Dorpat Catalogue, the right ascension and declination for 1880, the semi-axis major in seconds, the eccentricity, the period in years, and the inclination of the plane of the orbit in degrees. The next two columns give the magnitudes of the components as estimated by Struve. Three of the stars are not contained in the Dorpat Catalogue, and for them the magnitudes given have been assumed. The next column gives the equivalent diameter 0.00933 a P or the magnitude of a star having the mass of the binary and the density and brightness of the Sun. From the magnitudes of the components we may compute, by the third column of Table IV., the brightness of a star having the same mass as the binary and the same brightness and density as its components. Subtracting from this quantity that given in the preceding column gives the next column. If these quantities were small, we might assume that they were due to errors in the assumed magnitudes of the stars. Their variations are, however, far too large to be explained in this way. As they are almost all negative, we may infer that the assumed light of the Sun is too small, or that a larger value should have been given on page 2 to S. A great part of the difference must be ascribed to variations in the density or brightness of the stars. We have at present no way of discriminating between these causes. has been proposed on page 3 for determining I would serve to distinguish them. Until then, it will be convenient to reduce this difference from magnitudes to the relative diameters of two stars of equal density and brightness, one having a mass, the other emitting a light equal to that of the binary. Assuming the diameter of the first of these stars as a unit, the diameter of the other is given in the next column, and may be denoted by C. In almost all cases this quantity is greater than unity, from which we should infer that most of the stars enumerated are either much brighter or much less dense than the Sun, unless, as suggested above, the measurements of the light of the Sun are largely in error. Let d denote the density, b the brightness of the components of the binary, and D the equivalent diameter of the binary in terms of the same unit as C. Then D2: C2 1: b, and D3: 1=1: d; eliminating D, C = Such a method as or the brightness is propor tional to the square of C and the density inversely as its cube. If then the star has the same density as the Sun, the square of C will give its brightness. Again, if the star has the same brightness as the Sun, its density will equal one divided by the cube of C. The product of the semi-axis major by the sine of the inclination and divided by the period is given in the last column but one. It serves as a measure of the annual approach or recession of the two components. Neglecting the eccentricity, the maximum motion in seconds will equal this quantity multiplied by 2 π = 6.28. The last column gives the name of the astronomer by whom the orbit was computed, which is adopted in this discussion. a sin i P a sin i π An inspection of the last column but one shows that the value of in several cases amounts to 0.03 or even more. Neglecting the eccentricity, the maximum motion would therefore equal 2 times this quantity, or nearly 0".2. The eccentricity in some cases would diminish the motion, but in other cases it would increase it. An eccentricity of 0.5 might vary it from 0.1 to 0".4, according to the position of the peri-astron. This value of would probably be even larger for some of the recently discovered stars, in which Pis still smaller than in the stars given in the table. It is commonly supposed that the parallax of an average first-magnitude star does not much exceed 0.1. That of a sixth-magnitude star would then be about 0.01 unless the fainter stars are really smaller than the brighter, or unless there is a perceptible absorption of light in space. Substituting the values d z = 0".2, p = 0".01, in the formula for the dz F line, p = given on page 9, we deduce l 137 ccordingly the difference in the positions of the F line would be 1.5 times as great as the deviation observed in the case of Sirius. As the spectra of the two components could be observed in turn (or perhaps simultaneously) without disturbing the spectroscope, many of the causes of uncertainty present in similar measures of single stars would be removed. In any case, if the F line could be seen in both components, we could assign a limit within which we could be certain that it was the same for both, and this would give a value of the parallax which must be less than the true parallax. A determination of the outside limit of distance of a star would appear to have nearly the same importance as the inside limit of distance found by micrometric distance. Moreover it does not seem probable that a star will be found whose parallax is very large, or previous observation might have detected it. |