Stochastic excitation in semi-regular variable stars
R. M. Maderak1
1
Department of Physics and Astronomy, Benedictine College, Atchison, KS 66002
Abstract
Here, we examine the behavior of semi-regular (SR) variable stars. We develop a
model to simulate the behavior of these stars, based on stochastic excitation by
convective turbulence. We apply this model to a group of twenty SR variables of
the subclass SRb. Results are presented.
1. Introduction
The semi-regular (SR) class of variable stars are long period variables whose light
curves exhibit additional complexities beyond those of the classically behaved
variables. While similar to other long period variables, such as the Mira class, they
are neither regular, nor defined enough to be classified as such. Here we will
consider primarily SR variables of subclass SRb. These are variables with poorly
defined periodicity, as noted in the General Catalog of Variable Stars.
Additionally, these stars often have more than one pulsational mode, also as noted
in the GCVS.
We believe that stochastic excitation is responsible for the irregularities
observed in SRb variables. In this case, convective turbulence within the star
produces acoustic noise which is selectively amplified at frequencies near the
pulsational mode of the star (Kumar 1996). Although the effect of convection on
variable stars is not well understood, there is evidence that sun-like oscillations of
stochastic nature are present in some SR variables (Christensen-Dalsgaard et al.
2001; Bedding et al. 2002). Here, as the premise for our model, we assume that the
oscillation modes present in SRb variables are stochastic in nature.
2. Mode Envelopes
Using Fourier analysis on the differential light curves of the (SRb type) stars in our
study, we reveal that the oscillation modes appear in the power spectrum not as
single peaks, but as clusters of peaks under Lorentzian envelopes, as seen in the
power spectrum of Z UMa, presented in Figure 1 (see attached). In contrast, the
power spectrum of Mira type variables show a single peak. In addition, power
spectrums of SRa type variables also show single peaks, although these are not
quite as symmetrical as those of the Mira class. It should be noted here that these
stars are characterized by have persistent periodicity, as stated in the GCVS.
It is our belief that the broadening in the envelopes of the SRb type
variables in our study is due to stochastic excitation by convective turbulence
(occurring as described previously, with reference to Kumar 1996). The multiple
peak structure of the envelopes has also been interpreted as implying
multiperiodicity (Kiss et al. 1999; Kiss et al. 2000), and the behavior of the SR
variables explained in terms of beating between these so-called closely spaced
periods (Kiss et al. 2000). However, we believe that the peaks are too closely
spaced to be real signals of actual periods, again espousing a stochastic
explanation. Kiss et al. (2000) also cites "weak chaos" as an alternative explanation
for the observed behavior.
3. Stochastic Excitation
The model we have developed for stochastic excitation in SR variables works as
follows. The star is assumed to be pulsating in its usual mode(s). Acoustic noise
produced by the convective turbulence within the star is amplified at frequencies
near these modes (as previously described). At the beginning of the
convective/stochastic timescale, these frequencies are excited, the proximate
pulsational mode is reset, and undergoes a random phase change. Thereafter, the
energy in the pulsational mode decays until the next excitation. Note that only the
phase and amplitude of the mode are affected, but that the pulsational frequency
(and period) is not. Phase jumps of the kind included in this model are seen in the
phase diagrams of some SRb variables. The SRb variable R Dor exhibits sudden
phase jumps of over 190 degrees, as seen in Figure 2 (see attached).
We believe that these jumps in the pulsational mode are responsible for the
multiple peak structure of the mode envelope, rather than multiperiodicity (as
previously mentioned). Kiss et al. (2000) does note that these sudden jumps are
present in some stars, and that the similarity between their timescale and the
convective timescale suggests possible strong coupling between pulsation and
convection (here we have assumed that the stochastic and convective timescales
are equivalent). It follows from this model that stars which are reset more often
(e.i., have shorter stochastic timescales) should have envelopes which are more
broadened, with more sub-peaks (by corollary, stars with longer stochastic
timescales should have envelopes which are thinner, with less sub-structure).
In developing the model we have made the following assumption. Firstly,
given that stochastic/convective timescales result form a process which is
turbulent, we have assumed that the values of these time scales are taken from a
normal (Gaussian) distribution, centered on value which is on the order of the
convective timescale. However, the model must necessarily exclude negative
values; the distribution is therefore cut off at zero. Secondly, the phase jumps at the
start of each stochastic timescale are assume to be uniformly random. This
assumption is made simply because there is no reason to believe that the phase
jumps would have preferred values. Thirdly, each mode is subject to independent
stochastic effects. Finally, the decay of the modes after excitation is assumed to be
exponential.
SRa variables have not been used in our study. As mentioned, Fourier
analysis of these stars reveals that their mode peaks show little or no broadening,
as in the case of V Boo (see Figure 3, attached) owing to their relatively well
defined periodicity. If stochastic effects occur in these stars, the stochastic
timescales would have to much longer than the available data sets, as those
currently available due not exhibit any significant indication of such effects. We
believe that it is therefore more likely that stochastic effects are not present in these
stars. Furthermore, the phase diagrams of these stars show that their phases are
very stable, and relatively constant, as seen in Figure 4 for the case of R UMi (see
attached). As previously mentioned, the phase diagrams of SRb variables are very
unstable, and exhibit sudden, large phase jumps. This total inconsistency, along
with that of mode peaks versus mode envelopes, leads us to believe that SRa's due
not experience stochastic effects. We therefore have considered only SRb's as
viable canidates for the studt of stochastic effects.
4. Computer Modeling
4.1 Modeling Program
The modeling program is written in BASIC, and can model a maximum of two
modes. The parameters of the model are as follows: P1, P2, the periods of the
primary and secondary modes (in days); A1, A2, the amplitudes of the primary and
secondary modes (in units of differential magnitude); M, the mean stochastic
timescale (in days); σ, the standard deviation of the distribution of stochastic
timescales (in days); N, the length of the synthetic data set to be generated (in
days). The modeler generates data for a synthetic, differential light curve, based on
the supplied parameters. The data is averaged into ten day bins.
The polar form of the Box-Muller transformation is used to transform
BASIC's uniform random number generator into a Gaussian distribution. This
transform takes a uniform random number distribution between 0 and 1, transforms
it into a Gaussian distribution centered on 0 with a standard deviation of 1.
Multiplying by the inputed σ and adding the inputed M shifts and expands the
distribution as needed. This Gaussian is used to supply the stochastic timescale
values. Note that the transform produces pairs of values, with one value on each
side of the distribution. The uniform random generator is then used to select one of
the values.
The uniform random number generator is used to select a phase value
between 0 and 2π. Differential magnitude values are then generated for t = 1 to the
stochastic timescale value, according to the expression
dif. mag. = A1 e-(t / τ) sin (2π / P1 + φ)
where τ = M2 / σ . We selected this value of τ in order to eliminate the need for
additional free parameters, and to ensure a sufficiently long decay time scale (we
found that a decay time scale of M was too short). The values generated are put
into an array of dimension N. Counting loops are employed so that t is stepped
appropriately. Stochastic timescale values and phase values are generated until t =
N. If a second mode is to be modeled, then it is independently generated as just
described, and added to the values of the output array. The output values are then
averaged into ten day bins, and written to an output file. Note that it is possible to
disable the generation of a second mode.
The random number generator can be seeded prior to beginning the model,
allowing the model to start at a different point in the random number sequence.
This allows multiple realizations of the same parameters to be created (otherwise, a
given set of parameters will always generate the same output). The complete code
of the modeling program is attached in Appendix A.
4.2 Modeling Method
We have based the modeling method on finding the best fit to the base widths of
the mode envelopes in the Fourier Transform (FT) power spectrums of the stars
studied. The base width is used as opposed to the full width at half maximum
(FWHM), because we found that the other envelope attributes (FWHM, and
power) vary substantially with different random realizations of each model. We
found that the base width, however, varies less, and seems to be the more
fundamental quantity (as far as the models are concerned). However, we have not
quantified the observed variance in any of the three envelope attributes.
It is possible that the envelopes of the stars themselves might vary
significantly if multiple data windows were available. However, given that the data
sets available for the stars span up to 95 years, before which no records are
available, multiple data windows on the order of those used here are not available.
It would be possible to sub-divide the available data, but this would compromise
the resolution of the power spectrum. Though this might reveal the amount of
variance in the actual stars, it would compromise the resolution of the power
spectrums. As it is, we have made no systematic study of this possible effect.
Applying this possible variance back to the models, we believe that an exact match
to the stellar data would be coincidencial, and would not necessarily be
meaningful.
We produced the FT power spectrums of the stars and the models from the
differential lights curve of each. We have taken the periods of the oscillation
modes used in the modeling from the power spectrums of the actual stars, using
either the highest peak, or the envelope center. We chose N to be roughly equal to
the length of the actual data sets. M was varied until the widths of the synthetic
envelopes most closely matched those of the real ones. Although σ is an
independent quantity, we have found that σ = M / 2 works sufficiently well, and
have used this value in the models for all of the stars studied. However, we have
not actually made a systematic study of the σ dependence of the envelope
attributes. Fitting the amplitude of the modes by means of the envelope height
(power) was done only approximately, given the substantial variance previously
stated.
5. Results
The values of the model parameters fitted to each star are presented in Table 1 (see
attached). Envelope width values are given (in days) as the inverse of the
frequency width. The frequencies corresponding to the positions of the modes in
the power spectrum of the star are also given. In the case of L2 Pup, the period used
in the model (139d) is slightly different than the value taken from the actual value
(138d) as taken from the FT power spectrum. We did this to obtain a slightly better
fit (the difference is less than 1%). In the case of PX Aql, a second mode is
present, but has not been modeled. We have taken the width value for each
envelope from the power spectrums of the actual stars. We found the typical
uncertainty in the width values to be 30%, and the typical uncertainty in the
stochastic timescale values to be 150d. The FT power spectrums of L2 Pup and its
fitted model are presented in Figure 5. The synthetic light curve generated for T
Cet is presented in Figure 6. The actual light curve of T Cet is presented in Figure
7. The fitted stochastic timescale versus the the inverse of average envelope width
is plotted in Figure 8. The data indicate a correlation between increasing stochastic
timescale and decreasing envelope width, as expected. The coefficient of
correlation of the data is 0.80 .
6. Conclusions
The high degree of correlation between the stochastic timescale and the mode
envelope width suggests that the model of stochastic excitation developed here is
valid for SRb variables. Furthermore, this process does not appear to occur in SRa
variables. This may reflect on the level of coupling between pulsation and
convection in these stars.
The model developed here will need to be refined as our understanding of the
effects of convection in variable stars increases. Systematic studies of the variance
in the FT power spectrums of the stars studied here, and of the models used here
are needed. The σ dependence of the mode envelope with in these stars also needs
to be established. Finally, more stars are needed to test and refine the correlation
established here.
ACKNOWLEDGMENTS
We are grateful to Steve Kawaler for extensive guidance and supervision. We are
also grateful to the AAVSO and VSOLJ data services for providing the data for
this study. Additional thanks goes to Tim Bedding, for collaboration, and for
providing additional data. Finally, we would like to thank the NSF for providing
the REU grants which made this research possible, and Iowa State University for
facilities, and various services.
REFERENCES
Bedding, T.R., Kawaler, S.D., Kjeldsen, H., & Zijlstra, A.A., 2002, MNRAS
Christensen-Dalsgaard, J., Kjeldsen, H. & Mattei, J.A., 2001, ApJ, 562, L141
Kiss, L.L., Szatmary, K., Cadmus, R.R., & Mattei, J.A., 1999, A&A, 346, 542
Kiss, L.L., Szatmary, K., Szabo, G., & Mattei, J.A., 2000, A&AS, 145, 283
Kumar, Pawan, 1996, ARXIV
Appendix A: Modeling Code
The complete code of the modeling program used in this study is given here, as
written in BASIC.
1 K = 0
2 I = 0
3 PRINT "P (pulsational period)"
4 INPUT P
5 IF P = 0 AND K = 1 THEN GOTO 296 ELSE GOTO 12
12 PRINT "A (amplitude)"
13 INPUT A
15 IF K = 1 THEN GOTO 105 ELSE GOTO 18
18 PRINT "M (mean stochastic period)"
19
20
21
22
23
30
31
40
INPUT M
PRINT "S (standard deviation)"
INPUT S
PRINT "N (observation period)"
INPUT N
PRINT "R (seed rnd R times before starting model)"
INPUT R
FOR X = 1 TO R
41 D = RND(1)
42 O = RND(2)
43 I = I + 1
45 NEXT X
100 DIM G(N)
101 FOR X = 1 TO N
102 G(N) = 0
103 NEXT X
105 W = 1
106 T = 0
110 WHILE T < N
120 WHILE W >= 1.0
140 R = (2.0 * RND(1)) - 1.0
150 D = (2.0 * RND(1)) - 1.0
155 I = I + 2
160 W = R^2 + D^2
170 WEND W < 1.0
180 W = SQR((-2.0 * LOG(W)) / W)
190 U = (R * W * S) + M
200 V = (D * W * S) + M
210 O = RND(2)
220 IF O = 2 THEN LET SP = V ELSE LET SP = U
225 IF SP < 0 THEN GOTO 120 ELSE GOTO 230
230 SP = INT(SP,0)
235 F = 2 * PI * RND(1)
236 I = I + 1
240 FOR X = 1 TO SP
250 Y = A * EXP(-X/(M^2 / S)) * SIN((2 * PI * X/P) + F)
255 T = T + 1
255 T = T + 1
270 G(T) = G(T) + Y
272 IF T = N THEN GOTO 290 ELSE GOTO 275
275 NEXT X
290 WEND T = N
291 IF K = 1 THEN GOTO 296 ELSE GOTO 293
293 K = 1
295 GOTO 3
296 T = 0
297 DIM C(N)
298 OPEN "starm.out" FOR OUTPUT AS #1
300 WHILE T < N
310 U = 0
320 FOR Z = (T + 1) TO (T + 10)
349 U = U + G(Z)
350 NEXT Z
360 T = T + 10
370 W = U / 10
375 C(T) = W
380 PRINT #1,T,C(T)
390 WEND T = N
400 CLOSE #1
410 PRINT I