Note.--The elements refer to the brighter component of each system, exceptin
the case of Mizar
(ζ Ursae Majoris), where the value of K is that of the relative orbit.
The table includes all stars (so far as known to the writer) for which the "velocity-curve"
appears to be certainly unsymmetrical. It contains, for each star, the computed elements
P, e, ω, and K, except in a few cases where such data are not available.*
* For α Andromedae and γ Geminorum only the general form
of the oscillation [Continued at bottom of next page.]
72 J. Miller Barr
Of the thirty stars included in this table there are but four for which the
values of ω (as calculated) lie between 180° and 360°. For these four
stars we have in each case,
D > I,
where D denotes the time-interval during which the star's "radial velocity" is
decreasing ; I the interval during which it is increasing
(algebraically). For the remaining 26 stars D < I, except in the case of
η Virginis (bright component), where D = I, corresponding to ω
The apparent grouping of the periastron about certain values of ω is a yet
more striking feature, which is clearly shown in the annexed diagram. That such a
distribution of the apses really exists is, of course, very improbable--so improbable that
we are certainly justified in seeking a different explanation of the observed facts. In
other words, the elliptic elements, e and ω, as computed and published
for the orbits under notice, are probably illusory ; the "observed radial velocities,"
upon which they are based, being vitiated by some neglected source of systematic error.†
It now remains to point out the probable nature of this source of error. Two distinct
hypotheses are suggested, viz.:
(1) The spectrum-lines, for the stars under notice, are
curve has been found. The elements of U Aquilae and X Sagittarii are as yet
unpublished. In the case of δ Cephei, Belopolsky's value for ω
(272°.3) as printed in the
Astrophysical Journal (February 1895) is erroneous. For the benefit of readers
unversed in this subject it may be added that P denotes the period of revolution;
e the eccentricity; ω the longitude of periastron, reckoned from the
ascending node; K the "single amplitude," zK the total range in the star's
* Excluding those stars for which e < 0.10, we find:
D = I for one star,
D > I '' 2 stars,
D < I '' 21 ''
†The period P may of course be relied upon; and it is satisfactory to note that this
element has in many cases been determined with a high degree of precison. In cases such
as that of Capella, for which the oscillation-curve is almost symmetrical, the computed
values of K and a sin i are doubtless nearly correct.
Orbits and Velocity Curves of Spectroscopic Binaries 73
periodically shifted from their normal positions, owing to exceptional conditions
of pressure or temperature in the star's photosphere, or its surrounding atmosphere.
(2) The disks of the stars under notice are not uniformly bright. The distribution
of surface-brightness in longitude is for each star, unequal, and for some
stars, distinctly unsymmetrical. Such conditions, combined with rapid axial rotation,
would result in a more or less unsymmetrical broadening of the spectral lines. The
effective result would be a periodic shift of these lines, as measured on the
The possible source of error referred to in (1) has been made the subject of careful
investigation.* In a few cases, such as that of Mira Ceti, there is evidence
that a "physical shift" of certain lines in the star-spectrum does actually occur.† On
any rational theory, however, it is very unlikely that physical causes would give rise
to periodic displacements, affecting in a similar degree the positions
of several or many lines (due to various elements) in the spectrum of a star.
The second hypothesis rests upon a much more substantial basis. It was suggested by a
perusal of Dr. Albrecht's paper on "A Spectrographic Study of the Fourth-Class Variable
Stars Y Ophiuchi and T Vulpeculae."‡ In that paper, Dr. Albrecht calls
attention to a most important relation which exists between the light and velocity
curves of δ Cephei variables. "In every observed case," he remarks, "light-
maximum and greatest velocity of approach occur within one-fifteenth of the period of
each other. Likewise minimum brightness and greatest velocity of recession occur at
the same time."
* Among recent papers dealing with this subject are those of Humphreys
Astrophysical Journal, 26, 18, 297, 1907), Larmor
(Ibid., 26, 120) and
(Ibid., 26, 375).
The researches of Julius on "dispersion-bands," should be considered; also the possibility
that a shift of the spectral lines may result from electrical or magnetic conditions in the
†See papers by Campbell
(Astrophysical Journal, 9, 31, 1899),
(Ibid., 18, 341) and
(Jour. Roy.Ast. Soc. Can., 1, 45, 1907).
‡Astrophysical Journal, 25, 330, 1907
74 J. Miller Barr
An inspection of the accompanying table will show that it includes eight variables of the
δ Cephei type. Their orbits, according to the published elements, are more
or less decidedly elliptic--the computed values for the eccentricity e varying from
0.10 in the case of Y Ophiuchi to 0.489 in that of &eta Aquilae. If we assume
that these orbits are in reality nearly circular,† it would appear that the observed facts--
as graphically summarized in the light- and "velocity"-curves--may be explained on the
second hypothesis outlined above. The unsymmetrical distribuiton of light on the discs, as
postulated for these stars, is probably due to tidal action, modified by an unequal angular
rotation in different latitudes, such as exists in the solar photosphere.
Accepting this theory, we must suppose that the preceding side of the revolving star
is, on the whole, more luminous than the opposite hemisphere.‡ A similar state of things
seems to exist in certan variable stars of the Algol type--notably S Cancri, U Coronae,
δ Librae. For such stars, the rise from minimum to maximum brightness is less
rapid than the fall from maximum to minimum. This fact would seem to admit of only one
probable explanation, which is in harmony with our present theory, viz., that the
advancing front of such a star, as it traverses its orbit, is more luminous than the
Further evidence tending in the same direction is afforded by certain facts of observation,
which are here summarized :
*The light curve of one of these stars, viz., W Sagittiarii, is apparently subject to
distinct changes in form. According to Schmidt's observations (1866-76) it was formerly of
the δ Cephei type; but the Harvard observations of 1898 give a light-curve in
which the decrease of light is more rapid than the increase (H. C. O. Annals,
46, Part II).
†This assumption, for theoretical reasons, seems highly probable when the shortness of the
periods is considered. (Cf. Darwin and See on the theory of tidal evolution).
(Astrophysical Journal, 20, 186, 1904),
(Ibid, 25, 330, 1907),
(Ibid, 26, 369, 1907).
[For Curtiss, the page number is given as 186. It may be
§Strictly speaking, this remark applies only to that hemisphere of the star which is turned
towards the earth about the time of minimum brightness.
Orbits and Velocity Curves of Spectroscopic Binaries 75
(a) Distinct irregularities occur in the "velocity-curves" of certain variable stars, and
these correspond with inequalities in the light-curves of the stars.
(b) Broad, unsymmetrical lines have been noted in the spectra of several binaries--notably
(c) In the interesting case of η Virginis,* Ichinohe has obtained dissimilar
velocity curves from measures of spectrograms taken, respectively, with full dispersion and with a
single-prism spectrograph. This remark applies to the brighter member of the system. For the faint
component, a double curve having the same period (71.9 days) has been found. Moreover, the
deduced "radial velocity of the center of gravity" is, for the bright component, -0.4 km., and for
the fainter star +30 km. --a most significant difference.
The further discussion of this interesting subject is reserved for a future paper. In the latter I
hope to deal with certain details of the theory here advocated.† and to offer some hints concerning
practical methods for the separation of effects due, respectively, to axial rotation and orbital
revolution of the stars under notice.
ST. CATHARINES, ONT.,
February 10, 1908.
NOTES ON MR. BARR'S PAPER
I. By W. F. King
II. By J.S. Plaskett,
followed by a
SUPPLEMENTARY NOTE BY MR. BARR
Barr's graph of computed ωs with the eight Cepheid variables and three stars, with unknown eccentricity and orbit speed omitted, can be seen at: barr.htm.