1. Varad Deshmukh1, Liz Bradley1, Fran Bagenal2
Understanding the Dynamics of Voyager 2 and New Horizons Solar Wind Data
Varad Deshmukh1, Liz Bradley1, Fran Bagenal2
1 Department of Computer Science, University of Colorado, Boulder, CO, USA
2 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder, CO, USA
The Problem
Time-series data
Delay reconstruction
The standard strategy for state-space reconstruction is to use a series of past values of a single scalar measurement y from a dynamical system to form a vector
that defines a point in a new (“reconstruction”) space.
For appropriate choices of the delay t and the dimension m, the reconstructed dynamics are diffeomorphic to the original state-space dynamics [2].
We employ standard heuristics—the average mutual information of [3] and the false near neighbor method of [4], both as implemented in the TISEAN package [5] — to select values for these two free parameters.
Since properties like the Lyapunov exponent are invariant under diffeomorphism, one can calculate values for those properties from the reconstructed dynamics.
The stream of protons, electrons and alpha particles ejected from the Sun’s corona constitutes the Solar Wind. As these high-energy particles travel outwards through the Solar System, their density, temperature, and velocity vary in complicated ways.
One of the important questions that the space research community would like to address is whether those dynamics change over long time scales.
Nonlinear time-series analysis of the solar-wind data [1] gathered by the New Horizons and Voyager 2 missions — launched in 1977 and 2007, respectively — offers a way to address this question.
antarcticarctic.wordpress.com/2013/07/04/aurora-australis
Challenges:
Want to compare dynamics at same radial heliospheric distance.
But the sample rates were non-uniform across both missions.
This severely restricted the length of the data sets for the comparative analysis (~400 hours from AU).
There are other conditions as well; y must be a smooth generic function on the state space, for example. In practice, there are additional requirements for successful application of l calculation algorithms to a data set.
Delay reconstructions of solar-wind data
Lyapunov exponents
Conclusions and future work
Delay-reconstructions of the 32.7AU solar-wind dynamics from New Horizons data appear to have higher Lyapunov exponents than similar reconstructions from Voyager 2 data.
However, these data sets are quite limited—well below the sizes required for successful calculation of dynamical invariants [7].
We are working with the space-physics community to obtain expanded data sets and to understand the implications of these results with respect to the physics of the solar wind.
Estimate largest Lyapunov exponent using the algorithm of Kantz [6], as implemented in the TISEAN package [5].
This involves fitting a line to the scaling region of a plot of the stretching factor Ds versus time.
Vary the lyap_k algorithm parameters to assure convergence:
Dimension (m)
Theiler window
Scale factor …
Citations
[1] H. Elliott et al., Astrophysical Journal Supplement Series, 223:19 (2016); J. D. Richardson et al., Proceedings Eighth Intl. Solar Wind Conf., 483 (1996)
[2] J. Crutchfield, senior undergraduate thesis (University of California, Santa Cruz, 1979); N. Packard et al., Phys. Rev. Lett. 45, 712 (1980); F. Takens, in Dynamical Systems and Turbulence (Springer, Berlin, 1981)
[3] A. Fraser and H. Swinney, Phys. Rev. A 33, 1134–1140 (1986)
[4] M. B. Kennel, R. Brown, and H. D. I. Abarbanel, Phys. Rev. A 45, 3403–3411 (1992). [5] R. Hegger, H. Kantz, and T. Schreiber, Chaos 9, 413–435 (1999). [6] H. Kantz, Phys. Lett. A 185, 77 (1994).
[7] A. Tsonis, J. Elsner, and K. Georgakakos, J. Atmos. Sci. 50, 2549–2555 (1993); L. Smith, Phys. Lett. A 133, 283–288 (1988).
Largest Lyapunov exponent in units of s-1
(2D projections shown)