SOME SINGLE AND BINARY SOURCES OF MULTIPLE PERIODICITY
IN PULSATING STARS
by MARTIN JOHNSON, Birmingham University
Summary: Mechanisms causing the long and short beats discovered by
the Budapest observers in single stars of RR Lyr type might become less
difficult to understand if a connection were found between two other
problems: firstly the conditions for resonance between several forces
due to binary structure and the non-radial free pulsations in a spinning
component of the system, secondly the conditions under which circulation
and apsidal motion of a satellite or ring of gas could give rise to
similar excitation in the atmosphere of a single star. The first problem
calls for further study in the extremes of fast rotation and low gravity
suggested by the binary phi Persei, the second calls for distinction
between the beat phenomena in single and binary examples of beta Cep or
beta CMa type, and also a distinction between periodicities in single Be
stars such as gamma Cas compared with phi Per binaries. Atmospheric
mechanisms may be the more important in beta Cep periods, and internal
sources the more important in RR Lyr, but some features of the external
excitation of an atmosphere may act as selectors for resonance even
among oscillations which were initiated internally.
The phenomenon of "beats", or combined frequencies in pulsating
stars of short period, has been investigated notably at Budapest [1], in the
Netherlands [2], on the Pacific coast of U. S. A. [3] and in U. S. S. R.
A distinction has emerged between long beat and short beat, for instance among
the stars with considerable amplitude in luminosity variation (RR Lyr type).
Although their primary periods are mostly between 1/4 and 3/4 day, in RS Boo
and XZ Dra the beat period is hundreds of times greater, whereas at the other
extreme in SX Phe and AI Vel and VZ Can the beat is of the order of only
three or four times the primary period. RR Lyr itself, with 1/2 day period
and 41 day beat, lies between the extremes.
Much has been done at Budapest, Leiden, and in California, towards
extracting the variability of amplitude and shape of light-curve used in
tracing the underlying frequencies, and determining a second period adjacent
to the primary for combination into a beat. For spherical pulsations applicable
to RR Lyr type, Kluyver [4], Schwarzschild [5], Rosseland [6] and
others have gone far towards calculating the modes of radial oscillation
which could yield these periods. Recently a significant attempt to correlate
length of beat with depth of unstable convection zone in a star has been
made by Fitch [7].
Even in the light of these advances, it is not yet certain how adequately
the emergence of surface luminosity at the observed phases can be accounted
for, in any postulated co-existence of adjacent frequencies in the anharmonic
oscillation of a stellar interior under adiabatic or other gradients. The agency
for initiating such internal oscillation is often disputed, even for Cepheids,
and the RR Lyr type show more bewildering complexity of "acoustic spectrum"
than Cepheids or red variables. Stellar magnetic fields are not yet
sufficiently understood to invoke as controllers of frequency.
I suggest in this note that one useful step might be a detailed
comparison of these RR Lyr phenomena with some features of multiple
periodicity in other cases, (a) in the B stars which pulsate in velocity
rather than in luminosity, generally called by beta Cep or beta CMa as
type star, (b) in binaries of high spin and therefore periodic tidal
ejection of gas, notably phi Per. The beta Cep stars have been most
fully investigated by Struve and his associates; at first sight their
velocity cycles exhibit a complexity as baffling as that of RR Lyr
stars, 2 or 3 or even 4 periods being sometimes identifiable in the same
star, often with non-repetitive maxima and length of period. However,
their pulsation in luminosity is of far smaller amplitude than in RR Lyr
stars, and may be atmospheric or even circumstellar rather than in deep
structure; the spectral cycles caused by binary association are also in
general atmospheric. But since it is the atmosphere in any variable star
which exhibits the Doppler displacements, excitation therein enhanced by
motions in circumstellar gas may act as an agent of selection among the
possible frequencies. This might be true also for the beats in an RR Lyr
star, even when its primary impulses originate in its interior. Since
among beta Cep type some seem single and some binary, there may be hope
of distinguishing in them between forced and free periods, if these
stars contain indications of any of the mechanisms ascribable to binary
gas flow, as in phi Per where spin is sufficiently fast.
A notable beginning was made by Ledoux [8] towards understanding the
velocity cycles of beta CMa, in terms of the non-radial oscillations
studied by Cowling [9] and others; but he himself doubted his
identification of a long period with an 80 day rotation, and when his
free periods seemed correct they disagreed with the observed phasing.
His only source of forced periods was the possibility of a satellite or
small companion, possibly White Dwarf, which Struve had suggested as
able to raise local eruption as it travelled round the equator. This
possibility, of non-radial oscillation excited by a satellite, Struve
[10] had also explored for the star 12 Lac of beta Cep type: his theory
of zonal structure in sigma Sco [11] another star with beta Cep
features, makes the suggestion more plausible. Recently I have suggested
[12] somewhat similar equatorial tidal excitation of circumferential
oscillation for phi Per, using Hynek's [13] elucidation of tidal jets in
this star, in an attempt to find whether the subharmonics suggested by
the 20 day impulses of the earlier observers are a combination of tide
with the extremely fast spin of phi Per.
These developments might be extended by asking what periodic properties
comparable with those of binaries could arise in a single star, if caused
by Struve's circulating satellite or by rotating rings of extruded gas such as
possessed by gamma Cas and other Be stars. Is an agency of forced atmospheric
oscillation possible in single as well as binary structures, and has it any
significance for the recent distinction of beta Cep stars into single
bodies (12 Lac, beta CMa, beta Cep, nu Eri, delta Sct) and binaries
(16 Lac, sigma Sco) ?
The final question would then become accessible: if gas circulating
in a binary system has any counterpart in deciding the periods of atmospheric
pulsation in a single star, is it also relevant to the deeper oscillations of
RR Lyr type? Since the mass of circumstellar gas is small, it cannot initiate
an internal oscillation, but it might for instance aid in selecting the
distribution of amplitudes among such natural frequencies as emerge at the
surface, and so affect the choice of modes to combine into a beat.
A first step, towards such mutual relevance of phi Per, beta Cep, gamma Cas,
and RR Lyr types, would require a classification of non-radial oscillations
into (i) natural, (ii) binary forced, (iii) single forced, which I
accordingly proceed tentatively to suggest.
(i) Natural frequencies of single star:
(a) In Cowling's frequencies, sigma^2 = 4/3 pi G rho multiplied by a
term > 1 < q for different modes, or in one case multiplied by (n-N)
where n is the polytrope index and 1/N = gamma -1 in the usual notations.
These allow periods of 3-12 hours in stars of suitable type, but can tend
to zero or infinity if central condensation and compressibility are adjusted.
(b) If the star has angular spin Omega, these frequencies split into pairs
sigma +- 2 beta Omega where beta is of the order 0.85. Hence when a wave
circulates against the spin, the period referred to stellar co-ordinates
may become much longer, for instance comparable with observed beat periods.
This was applied by Ledoux to beta CMa pulsation, by combining stationary
and progressive waves in sigma and sigma - 2 beta Omega, with the uncertain
results already quoted.
(c) Among "natural" periods in the gases ejected by a hot star must be
classified the time to complete a cycle of growing and diminishing opacity,
studied by Gerasimovic [14] for Be stars, and possibly underlying
the investigations of Be electron scattering with non-spherical symmetry
by the Burbidges [15] and even Baldwin's [16] studies of gamma Cas.
The time for ejected gas to reach a maximum excursion before falling
back to the star is also to be listed here.
(ii) Frequencies enforced by binary structure:
(a) Except in the closest binary pairs, resonance of sigma with orbital period
is unlikely, but even the longer orbital periods might excite frequencies
comparable with sigma - 2 beta Omega if spin and orbit are unequal
as in phi Per.
(b) In coordinates rotating with the star, the "semi-diurnal" tide interval
may resonate with sigma. I found this significant in deductions from the
very rapid rotation of phi Per discovered by Slettebak [17].
(c) Corresponding to the natural period of time for gases to reach a limit
in spiralling round a single star, there will be a period decided by the
time taken for ejected gas to reach the Roche point of escape from a binary,
as studied by Kuiper [18] and by Kopal [19]. This might impose
a cycle of increasing and decreasing density between the components of
fairly close binaries, a pulsation in a disc-shaped envelope.
(d) Of very long period will be the rotation of apse of a binary orbit; with
large eccentricities this will introduce a period in spectral line
displacements, but in general too lengthy for the problems discussed here.
(iii) Agencies of forced periods in a single star:
(a) Struve's theory of a satellite, applied by him to 12 Lac and by Ledoux
to beta CMa, would impose a periodicity upon spectral line structure
through its excitation of a local "sunspot" or eruptive area travelling
round the star's equator under the satellite.
(b) Since the satellite would have a very short period, the rotation of its
apse might not be too slow, but the small eccentricity might prevent this
from constituting another period of any detectable amplitude.
(c) The apse rotation of a gaseous ring might be much more potent in the
observed spectrum inspite of small mass, since ejection of the constituent
gases from the star may have had an almost parabolic velocity giving
high eccentricity. Struve [20] suggested this as a possible source
of spectral variability, but the topic seems not to have been pursued.
A star must be supposed intrinsically capable of radial and nonradial
oscillation, each in a wide range of possible frequencies dependent upon
density and gradients. Whether any particular periods are detectable will
depend on the manner and intensity of their excitation. If disturbance from
stable equilibrium is spherically symmetrical and sub-photospheric, pulsation
of RR Lyr type will be created and no other. But if an external agency exists,
for instance a binary tide, in a star whose gravity is low because of spin, or
a perturbation due to a satellite or the apse motion of a ring round a single
star, atmospheric pulsation of beta Cep type may be set up equatorially, with
luminosity effects smaller than the effects upon line velocities. Beats between
free and forcing periods will result, or the energy might be lost from
the atmosphere to bias the selection of spherical oscillations initiated from
inside. It would therefore be interesting to discover transition cases between
binary beta Cep pulsation, single beta Cep pulsation with ring or satellite,
and single RR Lyr pulsation; in these, clues from spinning gas-enveloped
binaries such as phi Per and single spinning stars such as gamma Cas would be
valuable as indicating how frequencies can select themselves for combination
into beats, or how overtones and fundamentals are suppressed or intensified
by resonance.
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