Lévy Model for Interstellar Scintillations

Abstract

Observations of radio signals from distant pulsars provide a valuable tool for investigation of interstellar turbulence. The time-shapes of the signals are the result of pulse broadening by the fluctuating electron density in the interstellar medium. While the scaling of the shapes with the signal frequency is well understood, the observed anomalous scaling with respect to the pulsar distance has remained a puzzle for more than 30 years. We propose a new model for interstellar electron density fluctuations, which explains the observed scaling relations. We suggest that these fluctuations obey Lévy statistics rather than Gaussian statistics, as assumed in previous treatments of interstellar scintillations. PACS numbers: 95.30.Qd, 98.38.Am, 98.38.Dq, 1. Introduction. Electron density fluctuations in the interstellar medium (ISM) cause scintillations of the intensity of signals arriving from distant pulsars. If the medium were completely transparent, the shape of the arriving signal would coincide with the shape of the signal emitted by the pulsar. However, the observed pulse is much broader, and this effect is attributed to the random refraction the waves experience while they travel through the medium [1-6]. To investigate pulse broadening one can assume that the pulsar intrinsic signal is narrow in time, I 0 (t) ∝ δ(t − t 0 ), where I 0 (t) is the signal intensity. The observed signal is broad and asymmetric, with a sharp rise and a slow decay; see Fig. 1. Observations show that broadened shapes of the pulses are similar for different pulsars (after proper rescaling), suggesting that the density fluctuations statistics along different lines of sight are to some extent universal.For estimates assume that the pulsar distance is d ∼ 10 kpc, the typical electron density is n ∼ 0.03 cm −3 , and the observational wave frequency is ν ∼ 500 MHz. Then the plasma electron frequency ω pe = (4πne 2 /m e ) 1/2 is much smaller than ν, and density fluctuations change the wave phase only slightly. To estimate the time delay one can use the approach of geometric optics, where the propagating ray is refracted (scattered) by small prisms of density inhomogeneities [8][9][10][11][12]. At each scatter event occurring at a mean-free path l, the propagation angle changes by a small amount, ∆θ ∼ λ 2 r 0 ∆n [see below], where λ is the wavelength, r 0 = e 2 /m e c 2 is the classical radius of the electron, and ∆n is the density difference at characteristic separation l. Using the standard assumption that ∆θ is random and Gaussian, one finds that the path direction deviates from a straight line by θ ∼ λ 2 r 0 ∆n 0 (d/l) 1/2 , where ∆n 0 is the characteristic amplitude of density-difference fluctuations, and the path length deviates from the distance d by ∆d ∼ dθ 2 ∝ λ 4 d 2 . The broadening time can be estimated as τ d ∼ ∆d/c, which gives the standard scaling τ d ∝ λ 4 d 2 .

Interstellar scintillation of PSR J0437–4715

“…Alternatively, the wide range of scales might reflect an interstellar Levy flight as proposed by Boldyrev & Gwinn (2003). We respectfully request that observers preserve a greater portion of the rich information carried by the autocorrelation function of intensity in future observations.…”

“…We observe that there can be at least two explanations for such behavior. One is that multiple scales of scintillation are present; one might expect such a model in the interstellar Levy flight proposed by Boldyrev & Gwinn (2003), for example. In this case we might see fine frequency structure caused by occasional large-angle deflections, leading to large propagation times, as well as the coarser structure originating from more typical deflections.…”

Random flows and diagnostics of turbulence in the high latitude cirrus

“…Recently it was proposed that a Lèvy distribution produces the electron density fluctuations needed to explain the anomalous λ 4 DM 4 scaling for temporal pulse broadening of pulsars as a function of wavelength λ, with dispersion measures, DM, exceeding 20 pc cm 3 . Boldyrev & Gwinn (2003a,b, 2005 have derived the observed scaling assuming a power law electron density PDF.…”

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

“…with stationary , independent increments), the assumption about its Gaussianity can be easily violated. The examples range from the description of the dynamics in plasmas [5], diffusion in energy space, self-diffusion in micelle systems, exciton and charge transport in polymers under conformational motion and incoherent atomic radiation trapping -to the spectral analysis of paleoclimatic [6] or economic data [7], motion in optimal search strategies among randomly distributed target sites [8], fluorophore diffusion as studied in photo-bleaching experiments, interstellar scintillations [9], ratcheting devices [10,11] and many others [12].…”