Pictorial Wavelet Demo

Here we demonstrate how wavelets find quasi-periodicity, using our analysis of OJ 287.

Good Publicity!

Amara Graps featured our early wavelet analysis in an article for the April 1998 Scientific Computing World.

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Analysis of OJ 287

Participants: In Extraordinary Activity in the BL Lacertae Object OJ 287 we use a continuous wavelet transform to analyze more than two decades of data for the BL Lac object OJ 287 acquired as part of the University of Michigan Radio Astronomy Observatory (UMRAO) variability program. We find clear evidence for a persistent modulation of the total flux and polarization with period ~1.66 yr and for another signal that dominates activity in the 1980s with period ~1.12 yr. The relationship between these two variations can be understood in terms of a "shock-in-jet" model, in which the longer timescale periodicity is associated with an otherwise quiescent jet and the shorter timescale activity is associated with the passage of a shock. The different periodicities of these two components may reflect different internal conditions of the two flow domains leading to different wave speeds or different contractions of a single underlying periodicity due to the different Doppler factors of the two flow components. We suggest that the modulation arises from a wave driven by some asymmetric disturbance close to the central engine. The periodic behavior in polarization exhibits excursions in U that correspond to a direction ~45 deg from the VLBI jet axis. This behavior is not explained by the random walk in the Q-U plane that is expected from models in which a pattern of randomly aligned magnetic field elements propagate across the visible portion of the flow and suggests a small amplitude, cyclic variation in the flow direction in that part of the flow that dominates centimeter wavelength emission.

Analysis of 0048-097

Participants: In A Quasi-Periodic Modulation of the Radio Light Curve of the Blazar PKS B0048-097 we present the results of a wavelet analysis of the radio light curve of the BL Lac Object PKS B0048-097 from the University of Michigan Radio Astronomy Observatory monitoring program at 8GHz during twenty-five years, from 1979 to 2004. The results show a remarkable periodicity of 450-470 days in the early 1980s that changed to a ~585 day periodicity in the late 1980s to early 1990s. A less pronounced ~400 day periodicity is found after ~1995. Very-long-baseline interferometry imaging at 15GHz shows dramatic structural changes in the usually unresolved source between two epochs, 1995.57 and 2002.38. The pronounced northward directed jet seen in the 2002 image differs by more than 90 degrees in direction from the source structure found in earlier epochs. These findings make PKS B0048-097 a primary target for multi-wavelength observations and intensive radio monitoring to decipher the blazar-variability phenomenon.

Analysis of X-Ray Data for NGC 628

Participants: Quasi-periodic oscillations and X-ray spectroscopy are powerful probes of black hole masses and accretion disks, and in A 2 Hour Quasi Period in an Ultraluminous X-Ray Source in NGC 628 we apply these diagnostics to an ultraluminous X-ray source (ULX) in the spiral galaxy NGC 628 (M74). This object was observed four times over 2 years with the Chandra X-Ray Observatory and XMM-Newton, with three long observations showing dramatic variability, distinguished by a series of outbursts with a quasi period of 4000-7000 s. This is unique behavior among ULXs and Galactic X-ray binaries because of the combination of its burstlike peaks and deep troughs, its long quasi periods, its high variation amplitudes of >90%, and its substantial variability between observations. The X-ray spectra are fitted by an absorbed accretion disk plus a power-law component, suggesting the ULX was in a spectral state analogous to the low/hard state or the very high state of Galactic black hole X-ray binaries. A black hole mass of ~(2-20)x103Msun is estimated from the fb-M scaling relation found in the Galactic X-ray binaries and active galactic nuclei.

Analysis of X-Ray Data for 3C 273

Participants: Quasi-periodic signals have yielded important constraints on the masses of black holes in galactic X-ray binaries, and in Wavelet Analysis of AGN X-Ray Time Series: A QPO in 3C 273? we have extended this to active galactic nuclei (AGN). We have employed a wavelet technique to analyze 19 observations of 10 AGN obtained with the XMM-Newton EPIC-PN camera. We report the detection of a statistically significant 3.3 kilosecond quasi-period in 3C 273. If this period represents an orbital time scale originating near a last stable orbit of 3 Rs, it implies a central black hole mass of 7.3 x 106 Msun. For a maximally rotating black hole with a last stable orbit of 0.6 Rs, a central black hole mass of 8.1 x 107 Msun is implied. Both of these estimates are substantially lower than previous reverberation mapping results which place the central black hole mass of 3C 273 at about 2.35 x 10 8 Msun. Assuming that this reverberation mass is correct, the X-ray quasi-period must be caused by a higher order oscillatory mode of the accretion disk.

Analysis of Pearson-Readhead Sources

Participants: In The Cross-Wavelet Transform and Analysis of Quasi-periodic Behavior in the Pearson-Readhead VLBI Survey Sources we introduce an algorithm for applying a cross-wavelet transform to analysis of quasi-periodic variations in a time series and introduce significance tests for the technique. We apply a continuous wavelet transform and the cross-wavelet algorithm to the Pearson-Readhead VLBI survey sources using data obtained from the University of Michigan 26 m paraboloid at observing frequencies of 14.5, 8.0, and 4.8 GHz. Thirty of the 62 sources were chosen to have sufficient data for analysis, having at least 100 data points for a given time series. Of these 30 sources, a little more than half exhibited evidence for quasi-periodic behavior in at least one observing frequency, with a mean characteristic period of 2.4 yr and standard deviation of 1.3 yr. We find that out of the 30 sources, there were about four timescales for every 10 time series, and about half of those sources showing quasi-periodic behavior repeated the behavior in at least one other observing frequency.

Analysis of RMHD Crab Simulation

Participants: In Observations of "wisps" in magnetohydrodynamic simulations of the Crab Nebula we describe results of new high-resolution axisymmetric relativistic MHD simulations of Pulsar Wind Nebulae. The simulations reveal strong breakdown of the equatorial symmetry and highly variable structure of the pulsar wind termination shock. The synthetic synchrotron maps, constructed using a new more accurate approach, show striking similarity with the well known images of the Crab Nebula obtained by Chandra, and the Hubble Space Telescope. In addition to the jet-torus structure, these maps reproduce the Crab's famous moving wisps whose speed and rate of production agree with the observations. The variability is then analyzed using various statistical methods, including the method of structure function and wavelet transform. The results point towards the quasi-periodic behaviour with the periods of 1.5-3 yr and MHD turbulence on scales below 1 yr.

Analysis of UMRAO Sources

Participants: We have used a combination of wavelet and cross-wavelet transforms to quantify the quasiperiodicity in the total flux light curves of AGN monitored as part of the ( University of Michigan Radio Astronomy Observatory ) program between 1980.0 and 2005.0. Statistically significant quasiperiodic variations have already been reported for the sources OJ 287 and 0048-097. No other sources are found to exhibit such clear quasiperiodicity, but ~20% of all sources exhibit such variations at the 99% confidence level, for at least part of the time spanned by the data. We interpret these flux variations as arising from the propagation of the normal modes of a relativistic jet, leading to a periodic variation in the lateral flow displacement, and thus to a concomitant variation in local flow direction and Doppler boost. The need to occasionally re-excite such modes, and small realignment of the overall flow direction, might be significantly modulating the observed amplitude of this behavior, giving it a somewhat transient appearance. This behavior is normally `masked' by the larger amplitude, more random fluctuations in the flow, evident on superficial inspection of light curves.

See Wiki Wavelets for a good introduction and lots of links. For the application of spectral techniques, rather than wavelets, see: Spectral Methods