a dripping handrail?
The Quasi-Periodic Oscillations and Low-Frequency Noise of Scorpius X-1 as Transient Chaos: A Dripping Handrail? J. D. Scargle D. L. Donoho James P. Crutchfield T. Steiman-Cameron J. Imamura
SFI WORKING PAPER: 1993-02-005
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
THE QUASI-PERIODIC OSCILLATIONS AND LOW-FREQUENCY NOISE OF SCORPIUS X-I AS TRANSIENT CHAOS: A DRIPPING HANDRAIL? Jeffrey D. Scargle,l David L. Donoho,2 James P. Grutchfield,3 Thomas Steiman-Cameron,! James Imamura,4 Karl YOllng 1
Abstract: We present. evidence tha.t the quasi-pedodic oscillations (QPO) and low frequency noise (LFN) chara.cteristic of ma.ny accl'etion sources are different aspects of the same physical process. We analyzed a long, high time l'esolution EXOSAT observation of Sco X-I. The x-ray luminosity varies stochastically on time scales from milliseconds to hours. The nature of this variability - as quantified with both power spectrum analysis and a new wavelet technique, the scalegram - agrees well with the dripping handmil accretion model, a simple dynamical system which exhibits transient chaos. In this model both the QPO and LFN are produced by radiation from blobs with a wide size distribution, resulting from accretion and subsequent diffusion of hot gas, the density of which is limited by an unspecified instability to lie below a threshold.
! Theoretical Studies Branch, Space Science Division, National Aeronautics and Space Administration, Ames Research Center 2 Department of Statistics, Stanford University 3 Santa Fe Institute; Permanent Address: Department of Physics, University of California, Berkeley 4 Institute of Theoretical Sciencp ilnd Department of Physics. University of Oregon
Scargle, Donoho, Crutchfield, Steiman-Cameron, Imamura, Young,
2
1. STOCHASTIC FLUCTUATIONS IN ACCRETION SOURCES
Many galactic x-ray/optical soun:e~ arise frolll binary st.ars in which gas from one component accretes onto the other - a compact object, such as a white dwarf, neutron star, or black hole. For example, in low-mass x-ray binaries (LMXB) suchas Sco X-I, gas fro111 a Roche-lobe-filling star accretes onto a neutron star. Irregulari ties in this flow and the accompanying release of gravi tational potential energy somehow produce the strong ltllilinosity fluctuations observed in most such objects. This variability has recent.ly been found to include aperiodic phenomena, such as quasi-periodic oscillations (QPO, defined by the presence of a broad peak in the power spectrum) and even more disordered noise 5 components, consisting of stochastic f1uctuat.ions on various time scales - especially long ones (low frequency noise, or LFN; van del' 1\lis 1989). How these fluctuations arise is the subject of this Letter. The QPOs in the LMXBs and similar objects have been ascribed to various processes (Lewin, Paradijs and van del' 1\lis 1988, Wood, Imamura and Wolff 1992). We describe a new model, in which both the QPO and LFN arise from chaotic accretion flow. This model was suggested by our analysis of X-ray luminosity time series of one such accretion source (Sco X-I), using tools designed to model the underlying processes as random (Scargle 1981, henceforth S I) or chaotic (Scargle 1990, henceforth S IVj Donoho and Scargle, 1993, henceforth DS). 2. ANALYSIS OF THE SCORPIUS X-I TIME SERIES We analyzed the same 5-20 keV EXOSAT data in which Middleditch and Priedhorsky (1986, henceforth MP) discovered QPO in Sco X-I. This amazing time series (MP, Fig. 2) provides information about variability on time scales ranging over 7 decades - from 2 ms (the sampling rate) to nearly 10 hI'. The initial several hours was an active period, during which the mean count rate was --';.8.4 counts/bin (biiJ. = .002 sec)j then Sco X-I settled down to a more quiescent (~ 6.0 counts/bin), but still strongly f1uctuat.ing state, during which t.he QPO phenomenon was found. 2.1. Self-simila.rity VilL Wa:uelet AnlLlysi.•: The scalegram
An observable X(t) is self-simila,r (also ca.lled scaling) with similarity exponent a if (1) holds statistically. That is, the probability distribution of both sides is the same (and therefore so are the mean, variaJ1ce, etc.) Hence the statistical character of the variability examined at different time resolutions is the same. The constant factor Aex is simply an amplitude scale change. This term refers to actual va:riat.ions of the source luminosity, not to be confused with observational errors. These real f1uctuat,ions are probably connected with irregularities, either random or chaotic, in the accretion process. We use stochastic to indicate disordered behavior, due eit.her to true randomness or deterministic chaos. 5
Chaotic Seo X·I: A Dripping Handrail?
3
One indication of self-similarity in Sco X-I is that normalized plots of the data, smoot,hed to time resolutions differing by factors of 100 or more, look very much alike. Another indirect clue lay in the failure of our attempts to deconvolve 6 these data. We used a procedure (8 IV) that searches for a representation which has the simplest dyn.amical structure. But the results failed a consistency test, namely that the estimated pulse shape be independent of the time resolution of the data. Rather, estimated pulse widths were roughly proportional to the degree of smoothing. Simulations verified that this unphysical behavior can result if the actual pulse widths vary randomly over a wide range. Led by these preliminary indications, we used wavelet analysis (Daubechies 1992, D8) - a powerful tool for detecting self-similar behavior in time series, or for simply characterizing behavior over a wide range of time scale. Arbitrary time series can be represented as superpositions of elementary functions, wavelets, that are local in time (not extending over the entire interval, as do Fourier components). First one chooses a function !/J(t) called the analyzing wavelet. Because it is the simplest wavelet shape and avoids problems at the edges of the data, we use the Haar analyzing wavelet:
I,
!/J(t)
= { -1, 0,
O:Stc>ment, between the model scalegram and the corrected one for the quies
SFI WORKING PAPER: 1993-02-005
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu
SANTA FE INSTITUTE
THE QUASI-PERIODIC OSCILLATIONS AND LOW-FREQUENCY NOISE OF SCORPIUS X-I AS TRANSIENT CHAOS: A DRIPPING HANDRAIL? Jeffrey D. Scargle,l David L. Donoho,2 James P. Grutchfield,3 Thomas Steiman-Cameron,! James Imamura,4 Karl YOllng 1
Abstract: We present. evidence tha.t the quasi-pedodic oscillations (QPO) and low frequency noise (LFN) chara.cteristic of ma.ny accl'etion sources are different aspects of the same physical process. We analyzed a long, high time l'esolution EXOSAT observation of Sco X-I. The x-ray luminosity varies stochastically on time scales from milliseconds to hours. The nature of this variability - as quantified with both power spectrum analysis and a new wavelet technique, the scalegram - agrees well with the dripping handmil accretion model, a simple dynamical system which exhibits transient chaos. In this model both the QPO and LFN are produced by radiation from blobs with a wide size distribution, resulting from accretion and subsequent diffusion of hot gas, the density of which is limited by an unspecified instability to lie below a threshold.
! Theoretical Studies Branch, Space Science Division, National Aeronautics and Space Administration, Ames Research Center 2 Department of Statistics, Stanford University 3 Santa Fe Institute; Permanent Address: Department of Physics, University of California, Berkeley 4 Institute of Theoretical Sciencp ilnd Department of Physics. University of Oregon
Scargle, Donoho, Crutchfield, Steiman-Cameron, Imamura, Young,
2
1. STOCHASTIC FLUCTUATIONS IN ACCRETION SOURCES
Many galactic x-ray/optical soun:e~ arise frolll binary st.ars in which gas from one component accretes onto the other - a compact object, such as a white dwarf, neutron star, or black hole. For example, in low-mass x-ray binaries (LMXB) suchas Sco X-I, gas fro111 a Roche-lobe-filling star accretes onto a neutron star. Irregulari ties in this flow and the accompanying release of gravi tational potential energy somehow produce the strong ltllilinosity fluctuations observed in most such objects. This variability has recent.ly been found to include aperiodic phenomena, such as quasi-periodic oscillations (QPO, defined by the presence of a broad peak in the power spectrum) and even more disordered noise 5 components, consisting of stochastic f1uctuat.ions on various time scales - especially long ones (low frequency noise, or LFN; van del' 1\lis 1989). How these fluctuations arise is the subject of this Letter. The QPOs in the LMXBs and similar objects have been ascribed to various processes (Lewin, Paradijs and van del' 1\lis 1988, Wood, Imamura and Wolff 1992). We describe a new model, in which both the QPO and LFN arise from chaotic accretion flow. This model was suggested by our analysis of X-ray luminosity time series of one such accretion source (Sco X-I), using tools designed to model the underlying processes as random (Scargle 1981, henceforth S I) or chaotic (Scargle 1990, henceforth S IVj Donoho and Scargle, 1993, henceforth DS). 2. ANALYSIS OF THE SCORPIUS X-I TIME SERIES We analyzed the same 5-20 keV EXOSAT data in which Middleditch and Priedhorsky (1986, henceforth MP) discovered QPO in Sco X-I. This amazing time series (MP, Fig. 2) provides information about variability on time scales ranging over 7 decades - from 2 ms (the sampling rate) to nearly 10 hI'. The initial several hours was an active period, during which the mean count rate was --';.8.4 counts/bin (biiJ. = .002 sec)j then Sco X-I settled down to a more quiescent (~ 6.0 counts/bin), but still strongly f1uctuat.ing state, during which t.he QPO phenomenon was found. 2.1. Self-simila.rity VilL Wa:uelet AnlLlysi.•: The scalegram
An observable X(t) is self-simila,r (also ca.lled scaling) with similarity exponent a if (1) holds statistically. That is, the probability distribution of both sides is the same (and therefore so are the mean, variaJ1ce, etc.) Hence the statistical character of the variability examined at different time resolutions is the same. The constant factor Aex is simply an amplitude scale change. This term refers to actual va:riat.ions of the source luminosity, not to be confused with observational errors. These real f1uctuat,ions are probably connected with irregularities, either random or chaotic, in the accretion process. We use stochastic to indicate disordered behavior, due eit.her to true randomness or deterministic chaos. 5
Chaotic Seo X·I: A Dripping Handrail?
3
One indication of self-similarity in Sco X-I is that normalized plots of the data, smoot,hed to time resolutions differing by factors of 100 or more, look very much alike. Another indirect clue lay in the failure of our attempts to deconvolve 6 these data. We used a procedure (8 IV) that searches for a representation which has the simplest dyn.amical structure. But the results failed a consistency test, namely that the estimated pulse shape be independent of the time resolution of the data. Rather, estimated pulse widths were roughly proportional to the degree of smoothing. Simulations verified that this unphysical behavior can result if the actual pulse widths vary randomly over a wide range. Led by these preliminary indications, we used wavelet analysis (Daubechies 1992, D8) - a powerful tool for detecting self-similar behavior in time series, or for simply characterizing behavior over a wide range of time scale. Arbitrary time series can be represented as superpositions of elementary functions, wavelets, that are local in time (not extending over the entire interval, as do Fourier components). First one chooses a function !/J(t) called the analyzing wavelet. Because it is the simplest wavelet shape and avoids problems at the edges of the data, we use the Haar analyzing wavelet:
I,
!/J(t)
= { -1, 0,
O:Stc>ment, between the model scalegram and the corrected one for the quies
