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Tsallis Statistics of the Magnetic Field in the Heliosheath Lf Burlaga,1 Af Vin

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Tsallis Statistics of the Magnetic Field in the Heliosheath Lf Burlaga,1 Af Vin The Astrophysical Journal, 644:L83–L86, 2006 June 10 ᭧ 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A. TSALLIS STATISTICS OF THE MAGNETIC FIELD IN THE HELIOSHEATH L. F. Burlaga,1 A. F. Vin˜ as,1 N. F. Ness,2 and M. H. Acun˜a3 Received 2006 March 27; accepted 2006 May 3; published 2006 May 25 ABSTRACT The spacecraft Voyager 1 crossed the termination shock on 2004 December 16 at a distance of 94 AU from the Sun, and it has been moving through the heliosheath toward the interstellar medium since then. The distributions of magnetic field strength B observed in the heliosheath are Gaussian over a wide range of scales, yet the measured profile appears to be filamentary with occasional large jumps in the magnetic field strengthB(t) . All of the probability distributions of changes in B,dBn { B(t ϩ t) Ϫ B(t) on scales t from 1 to 128 days, can be fit with the symmetric Tsallis distribution of nonextensive statistical mechanics. At scales ≥32 days, the distributions are Gaussian, but on scales from 1 through 16 days the probability distribution functions have non-Gaussian tails, suggesting that the inner heliosheath is not in statistical equilibrium on scales from 1 to 16 days. Subject headings: magnetic fields — solar wind — turbulence — waves 1. INTRODUCTION is expressed by the form of u in the distribution function. In The solar wind moves supersonically beyond a few solar Tsallis statistical mechanics, the probability that a system is in a state with energy u is proportional toexp (Ϫ u) . In classical radii from the Sun to a “termination shock” at ≈94 AU, carrying q b statistical mechanics, the probability that a system is in a state solar magnetic fields with it. The heliosheath is a subsonic Ϫ region extending from the termination shock to the interstellar with energy u is proportional toexp ( bu) , the Boltzmann-Gibbs distribution. For example, in a gas of molecules, the probability medium (Axford 1972; Parker 1963; Hundhausen 1972). Voy- p ager 1 entered the heliosheath by crossing the termination of finding molecules with energyu E is proportional to exp (ϪE/kT), where T is the temperature and k is Boltzmann’s shock on 2004 December 16 (Stone et al. 2005; Gurnett & p Kurth 2005; Decker et al. 2005; Burlaga et al. 2005), and it constant (b 1/kT ). has been moving through the inner heliosheath since that time. This Letter demonstrates that the PDFs of the fluctuations Magnetic field observations have been made by the MAG ex- of B observed in the heliosheath from DOY 1 through 256, periment on Voyager 1 (Behannon et al. 1977) from the launch 2005 on scales from 1 to 128 days can be described by the in 1977 to the present. Tsallis distribution. We show that the PDF of the heliosheath A generalization of Boltzmann-Gibbs statistical mechanics magnetic field strength is Gaussian (consistent with statistical 116 days, but it is not in Gaussian at was introduced by Tsallis (1988, 2004a) and has been applied equilibrium) at scales scales between 1 and 4 days. The observation of the Tsallis to many complex physical systems that are not in thermal equi- distribution of fluctuations of B in the heliosheath and the mea- librium (Tsallis 2004b; Tsallis & Brigatti 2004; Gell-Mann & surement of the variation of the parameters of this distribution Tsallis 2004). The statistical mechanics of Tsallis introduces a with scale describe the multiscale state of magnetic strength new entropy function, which is appropriate for nonequilibrium, fluctuations in the heliosheath and place strong constraints on nonextensive systems in which phase space has a hierarchical any model of these fluctuations. multiscale structure (rather than being uniformly occupied as it is for an equilibrium Gaussian distribution). By extremizing the entropy function subject to a normalization condition and 2. OBSERVATIONS OF DAILY AVERAGES OF MAGNETIC FIELD a constraint on energy, Tsallis derived a probability distribution STRENGTH IN HELIOSHEATH AND THEIR DISTRIBUTIONS function that describes many types of nonlinear physical sys- Daily averages of the magnetic field strength B as a function tems, e.g., “chaotic systems,” turbulence, cosmic-ray spectra, of time (DOY, day of year) from DOY 1 to 256, 2005 are plasma physics, and solar wind physics. shown in the top panel of Figure 1. The error bars on B are The probability distribution function (PDF) derived by Tsal- ≈0.015 nT, the estimated 1 j error due to both systematic and lis is proportional to the “q-exponential function” random effects. Two components of B are determined by rolls about the z-axis (directed toward the Sun) at the times indicated Ϫ { ϩ Ϫ 1/(1Ϫq) expq ( bu) [1 (q 1)bu] , (1) by the asterisks in the top panel of Figure 1. The third com- ponent (along the z-axis) is estimated at the time of a roll by p where b and q are constants. This function reduces to an ex- assumingABz S 0 in an interval of 52 days centered on a roll. ponential in the variable u when q approaches 1, and it ap- Corrections for varying contributions to B from the spacecraft proaches a power law in u when u is large. When q approaches and instrument are made between the rolls, using the infor- p 22Ϫ Ϫ 1 andu y ,expq ( bu)exp( approachesby ) , which is mation from both the primary and secondary instruments (Be- proportional to a Gaussian distribution. hannon et al. 1977). The physics for a particular application of the theory of Tsallis The fluctuations in daily averages of B from DOY 1 to 256 have a distribution that is approximately Gaussian with a mean 1 Laboratory for Solar and Space Physics, Code 612.2, NASA Goddard ABS p 0.11 nT, as shown in the bottom panel of Figure 1. The Space Flight Center, Greenbelt, MD 20771; [email protected]. p ϩ 2 solid curve is a fit to the Gaussian function B B0 Institute for Astrophysics and Computational Sciences, Catholic University 1/2Ϫ Ϫ 2 of America, Washington, DC 20064; [email protected]. {A/[w(p/2) ]} exp { 2[(B ABS)/w]}; the dotted curves are 3 Planetary Magnetospheres Laboratory, Code 695, NASA Goddard Space the 95% confidence bands based on a t-test computed by the Flight Center, Greenbelt, MD 20771; [email protected]. plotting program Origin. L83 L84 BURLAGA ET AL. Vol. 644 Fig. 1.—Top: Daily averages of the magnetic field strength B vs. time measured by Voyager 1 from DOY 1 to 256, 2005 in the heliosheath. Bottom: Distribution of B (filled squares) as well as a fit to a Gaussian distribution function (solid curve) and the 95% confidence bands (dotted curves). Fig. 2.—The symbols are PDFs of relative changes in the magnetic field strength measured by Voyager 1 in the heliosheath from DOY 1 to 256, 2005 The quality of the fit is given by the coefficient of determi- on scales from 1 to 128 days. The solid curves are fits of the data to the Tsallis nationR 2 p 0.89 , and the parameters derived from the fit are distribution function. p p ע p ע p Ϫ B0 1.8 6.0 nT,ABS 0.099 0.004 nT, w 2 j ע p ע 0.10 0.02 nT, andA 5.6 1.6 . bins that were used in constructing the histograms; each his- Gaussian distributions of B were also found at smaller scales togram is plotted a factor of 100 above the one below it for and in different intervals in the heliosheath (Burlaga et al. clarity. The error bars, the short horizontal segments above and 2005). However, the magnetic field strength fluctuations in the below the observed points in Figure 2, were computed as the top panel of Figure 1 have a filamentary structure rather than square root of the countsCi in bin i; thus, the error bars are 1/2 ע simple Gaussian fluctuations. For example, there are relatively plotted atCii(C ) with a shift of 100 for adjacent distri- large jumps at steps and peaks inB(t) , which we shall describe butions introduced for clarity. quantitatively and examine in some detail. These jumps and Using the Levenberg (1944) and Marquardt (1963) method, other changes related to various heliosheath features can be we fit these observed PDFs with the symmetric Tsallis described quantitatively by the probability distributions, as dis- distribution cussed in § 3. They are also associated with the multifractal structure observed in the interval under consideration (L. F. Ϫ Ϫ A exp [Ϫ (dBn)]22{ A [1 ϩ (q Ϫ 1) dBn ]1/(q 1) , (2) Burlaga et al. 2006, in preparation). qqb q qb q where q is the “entropic index” or “nonextensivity” factor, 3. MULTISCALE PROBABILITY DISTRIBUTION FUNCTIONS Ϫϱ ! q ≤ 3, which depends on scale (Tsallis 1988; Burlaga & From the daily averages of observations of B in the helio- Vin˜as 2004, 2005).ABqq and refer to a Tsallis distribution with sheath (Fig. 1), we computed a set of PDFs (histograms) de- a particular q; they too depend on scale. Forq p 1 , the Tsallis { ϩ Ϫ scribing increments in B,ondBn(tin; t ) B(t it n) B(t i) distribution reduces to the Boltzmann-Gibbs (Gaussian) distri- p n p scalestn 2 days, wheren 0 , 1, 2, 3, 4, 5, 6, and 7. Thus, bution. Burlaga & Vin˜as (2004, 2005, 2006) showed that this we consider scales from 1 to 128 days. The plasma is being Tsallis distribution (eq. [2]) provides good fits to PDFs of the convected past the spacecraft, but the speed is not measured magnetic field strength B observed in the supersonic solar wind directly; hence, we cannot determine the length scale associated on scales from 1 to 128 days from 1 to ≈87 AU.
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