Tsallis distribution of the interplanetary magnetic field at 0.72 AU: Venus Express observation



[1] Previous work shows that Probability Distribution Functions (PDFs) of the interplanetary magnetic field strength differences can be described by a single function - Tsallis distribution at Earth and beyond. Launch of Venus Express enables us to extend the application of Tsallis distribution to the inner heliosphere at 0.72 AU. This paper analyzes the distributions of increments of interplanetary magnetic field magnitude on scales from 1 hour to 211 hours (∼85.3 days), and fit all these PDFs to Tsallis distribution. The entropy index q value of all the PDFs on these scales at 0.72 AU are in the range of 1.5 to 1.7, which implies the non-Gaussianality of the PDFs. The variation of the statistical parameters such as cumulants, variance and kurtosis with scales is also discussed.

1. Introduction

[2] The Probability Distribution Function (PDF) of field differences provide a direct approach to describe the statistical properties of magnetic field fluctuations in solar wind. The traditional analyses were based on Boltzmann-Gibbs statistical mechanics which cannot describe physical systems which are not in thermal equilibrium. Though many studies have clearly presented the non-Gaussian feature of distributions of magnetic difference on relatively minor scales [Marsch and Tu, 1994, 1997], a quantitative way to characterize the fluctuations is still required. By introducing the nonextensive entropy Sq = k∑(1 − piq)/(q − 1) (with qR, and S1 = SBG), Tsallis [1988] extended the B-G statistical mechanics, where pi is the probability of the microstate, and q is a scale-dependent constant measuring the nonextensivity. To obtain the thermal equilibrium distribution, Sq is optimized with two energy constraint, and the Tsallis q-distribution function is defined as:

equation image

where x is physical quantity such as magnetic magnitude; A, B, and q are scale dependent parameters [Tsallis and Brigatti, 2004]. In the limit q → 1, the statistical mechanics of Tsallis recovers to the usual Boltzmann-Gibbs mechanics, and Gaussian distribution is recovered as a limiting case [Tsallis et al., 2004]. The Tsallis distribution has been applied to describe many physical system in which long-range forces or long-term memory effects are dominated, such as fluxes of cosmic ray [Tsallis et al., 2003], turbulence [Beck, 2001], and solar wind.

[3] Burlaga and Viñas [2004] first used Tsallis statistical mechanics to analyze the magnetic strength differences. They found that the PDF of magnetic strength increment at 1 AU on scales from 1 hour to 171 days can be described by Tsallis distribution. They also used Tsallis distribution to described fluctuations in daily observations of B between 7 and 87 AU on scales from 1 to 128 days [Burlaga and Viñas, 2005]. Burlaga et al. [2006] observed the Tsallis distribution of magnetic strength differences in the heliosheath. In the recent works, Burlaga et al. [2007] and Burlaga and Viñas [2007] stated that The PDFs of the increments of B are Tsallis distributions on scales from 1 hr-128 days between 1 and 90 AU and they applied a deterministic MHD model to predict these results. The predictions of the model agree with the observations from Voyager 1 at 80 AU. Thus, It has been proved that the fluctuations of B from 1 AU to heliosheath can be described by Tsallis distribution on a very wide range of scales.

[4] Venus Express was launched on 9 November 2005, and placed in elliptical orbits about the Venus [Titov et al., 2006]. Despite of aiming at exploring the Venusian atmosphere, it spends majority of it 24-hour period in solar wind, thus it provides good opportunity to research the interplanetary magnetic field at 0.72 AU, and to extend the new statistical conclusion to inner heliosphere.

2. Observation of Interplanetary Magnetic Field Strength at 0.72 AU

[5] In this study we use the magnetic field measurement made in the entire year of 2007 during the solar declining phase, by the Venus Express magnetometer [Zhang et al., 2007]. We calculate the hourly averages from 10-minute observations and then focus on the scale from 1 hour to 211 hours. We choose the hourly averages and this scale range because hourly data is a typical format widely used in the studies before, and so it is easy to compare results with previous work on the same scale range at 1 AU.

[6] Figure 1 shows the hourly averages of magnetic field strength B versus time (in the form of Data/Month) observed by VEX during 2007. The two large gaps appeared in June and August were caused by adjustment of the spacecraft orbit. There are also some regular small gaps owing to the remove of the data down stream of Venus bow shock, as we consider interplanetary magnetic field only. A continuous data set is ideal for the statistical analysis adopted in this paper. There are 8.64% data missing in this data set, and it does not affect the statistical study in a significant way. All the missing data are not included when computing the PDF.

Figure 1.

Hourly averages of the IMF versus time (date of the year) measured at 0.72 AU by VEX through the year 2007.

[7] Burlaga [2001] concluded that the distribution of the magnetic field strength B in the heliosphere is approximately lognormal and discussed the implications. In a recent work, Burlaga and Viñas [2007] restated that the distributions of the fluctuations of magnetic field B(t) observed on a scale of a year in the heliosphere between 1 AU and 90 AU are approximately lognormal. Here we demonstrate that the PDF of magnetic strength B at 0.72 AU can also be described by lognormal distribution. Figure 2 shows the PDF of magnetic strength B and its fit to lognormal distribution defined as the follow function:

equation image
Figure 2.

PDF of magnetic strength B and its fit to lognormal distribution.

[8] The quality of the fit is given by R2 = 0.989. The best-fit parameters are:σ = 0.401 ± 0.0059, C = 4.799 ± 0.031. This result is in accordance with the early studies.

3. PDF of Magnetic Field differences and Fit to Tsallis Distribution

[9] The time series of magnetic strength differences dB(t) = (B(t + τ) − B(t))/〈B〉 was calculated from VEX hourly average observations. 〈B〉 is the average of the whole interval; τ is constant for a set of dB, denoting the scale (or referred as lag) of this certain dB(t) series. We make τ = 2n hour (n = 0, 1, 2, …, 11), and compute the respective dBn time series, thus we are considering the scales from 1 hour to 211 hours (about 85.3 days).

[10] We choose seven representative scales of PDFs plotted in Figure 3. Here n = 0, 2, 4, 6, 7, 9, 11, corresponding to the scales τ0 = 1 hour, τ2 = 4 hours, τ4 = 16 hours, τ6 = 2.7 days, τ7 = 5.3 days, τ9 = 21.3 days, τ11 = 85.3 days, respectively. The filled black squares represent histograms of the fraction of counts in bins on the semilog scale. The lowest PDF is dB0, corresponding to the time lag 20 hour. The other six PDFs is shifted by a factor of 100 from the one below it.

Figure 3.

PDFs of dBn and their fits to Tsallis distribution.

[11] PDFs on these scales are all symmetric relatively, so the skewness of these distribution should be close to zero. The shapes of distribution are narrow at small scales, and broader at larger scales. But PDFs at all scales are obviously kurtotic, and have large tails. The statistical features expressed quantitatively by parameters are discussed in next section.

[12] We fit all the PDFs to the Tsallis distribution as follow:

equation image

where w = 1/equation image in this function. The weighted nonlinear regression we performed is based on Levenberg-Marquardt arithmetic. The solid curve show the fits of dBn/〈B〉. The goodness for all the fits satisfy: R2 ≥ 0.95

4. Best-Fit Parameters of the Tsallis Distributions Versus Scale

[13] Here we discuss four parameters: SD, Kurtosis, q, and w. SD(Standard Deviation) and Kurtosis are the second and fourth cumulants of a distribution, derived from dBn(t) time series; q and w are parameters of the Tsallis distribution, computed from the fits. They are all scale dependent. Figure 4 shows the four parameters as a function of n (the scaleτ = 2n hours). Error bars are plotted by the 95% confidence interval of the best-fit value.

Figure 4.

Parameters of the Tsallis distribution derived from fits and high order moments derived from dBn time series versus scale. The scale τ = 2nhours.

[14] Entropic Index q is the key parameter of Tsallis Function. It measures the degree of nonextensivity, thus it denotes the complexity of a system. In the limit q → 1, the Tsallis distribution reduces to Gaussian distribution. It also describes the kurtosis and tails of a distribution. A distribution has large tails with a high q, indicating large jumps on this scale. Burlaga and Viñas [2004] has proved that the dB distributions are non-Gaussian on all the scales ranging from 20 hour to 211 hours at 1 AU, during the declining phase. The results in this paper are similar: q maintains a relatively high value varying between 1.5 and 1.7, considering the uncertainties. The lowest value of q appears at time lag 28 hours, about 10.7 days. It does not continue to decrease with increasing scale.

[15] The kurtosis value measures of the flatness (negative value) or peakedness (positive value) of the distribution relative to a Gaussian one with the same mean and standard deviation. The kurtosis should be zero for a Gaussian distribution. The kurtosis is computed from the time series of dBn. It is defined as a normalized form of the fourth central moment of a distribution.

equation image

where σ is the variance and equation image indicates averaging, xi are the observed values of the variables, and N is the number of data points.

[16] The parameter w = 1/equation image was first introduced by Burlaga and Viñas [2004] instead of parameter B for measuring the width of Tsallis distribution. The w value tends to increase with increasing scale before n = 6 (lag = 2.7 days), and reaches a plateau after this scale. The variations of w and SD with time lag n have very similar shapes.

5. Summary and Discussion

[17] We have discussed the PDFs of magnetic strength B and the PDFs of magnetic strength differences dB at 0.72 AU observed by Venus Express in 2007 on scales from 1 hour to 85.3 days. Magnetic field strength distribution is well fit to lognormal distribution. We can also draw a conclusion that Tsallis distribution function can describe the fluctuations of magnetic strength at 0.72 AU quantitatively over varying scales. As Tsallis distribution is derived from the nonextensive, nonadditional entropy, it also can reflect the non-Gaussianality of the PDFs on these scales. When the entropy index q approaches 1, the distribution approaches a Gaussian distribution. The q values of all the PDFs on these scales at 0.72 AU are in the range of 1.5 to 1.7, and the PDFs have large non-Gaussian tails on these scales. They remain non-Gaussian even at the largest scale τ = 85.3 days. This result is in accordance with early studies at 1 AU.


[18] This work was supported by CAS grant KJCX2-YW-T13 and NNSFC grant 40621003 and in part by the Specialized Research Fund for State Key Laboratories of China. We are grateful to L. Burlaga and J. Kan for their helpful suggestions and comments.