Long-term Correlations of Polytropic Indices with Kappa Distributions in Solar Wind Plasma near 1 au

The polytropic relationship applies along the same streamline plasma flow. Hence, for the calculation of γ, we need to analyze subintervals obtained within the same streamline. The analyzed subintervals obtained from a single spacecraft (S/C) should be sufficiently short in order to minimize the possibility of mixing measurements of different streamlines. Following previous studies (e.g., Kartalev et al. 2006; Nicolaou et al. 2014a; Livadiotis & Desai 2016; Livadiotis 2018a; Elliot et al. 2019), we analyze every 5 consecutive Wind measurements, which cover ∼8 minutes if there are no time gaps between them. In order to exclude subintervals with large time gaps we reject any subinterval that covers a time period larger than 10 minutes.

Taking the logarithm of Equation (1), we have

$$\begin{eqnarray}&&\mathrm{log}T=(\gamma -1)\mathrm{log}n+\mathrm{const}.,\end{eqnarray} \tag{ 3 }$$

where the constant is different for different streamlines. We use orthogonal regression to calculate γ and σ_γ from the slope of $\mathrm{log}T\propto \mathrm{log}n$, within each subinterval of 5 consecutive measurements. The left panel of Figure 4 shows the fitting of Equation (3) to the density–temperature data within the subinterval obtained in DOY: 1995 January 4, from ∼10:20 to ∼10:28 UT. For the specific example, the slope in the $\mathrm{log}n-\mathrm{log}T$ diagram is 0.58 ± 0.04, thus, γ = 1.58 ± 0.04.

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As explained in Livadiotis et al. (2018), and based on the work of Nicolaou & Livadiotis (2016), the invariant kappa index, for a 3D kappa distribution function, can be calculated by combining u_th,m and u_th,f:

$$\begin{eqnarray}&&{\kappa }_{0}=\displaystyle \frac{2.5{u}_{{\mathrm{th},{\rm{f}}}^{2}}}{{{u}_{\mathrm{th},{\rm{m}}}}^{2}-{{u}_{\mathrm{th},{\rm{f}}}}^{2}}\Rightarrow 2.5{{u}_{\mathrm{th},{\rm{f}}}}^{2}={\kappa }_{0}\left({{u}_{\mathrm{th},{\rm{m}}}}^{2}-{{u}_{\mathrm{th},{\rm{f}}}}^{2}\right).\end{eqnarray} \tag{ 4 }$$

In Equation (4), we assume that the bi-Maxwellian fitting to the observed proton distributions underestimates the actual thermal speed of the plasma because the fitted model does not account for the higher energy "tails" of the distribution function (for more details, see: Nicolaou & Livadiotis 2016; Livadiotis 2018b). This underestimation is reflected in the values of thermal speed and pressure, samples of which are plotted in Figures 2 and 3; u_th,m is higher than u_th,f and the thermal pressure calculated for u_th,m is higher than the thermal pressure calculated for u_th,f.

We use orthogonal regression to calculate κ₀ and σ_κ from the slope of the diagram of $2.5{{u}_{\mathrm{th},{\rm{f}}}}^{2}$ plotted against $\propto ({{u}_{\mathrm{th},{\rm{m}}}}^{2}-{{u}_{\mathrm{th},{\rm{f}}}}^{2})$. The kappa index is estimated along with the polytropic index for each selected subinterval (discussed in Section 3.1). Note that according to Equation (4), the fitted line passes from the origin (thus we apply linear regression with the intercept fixed to zero). In the middle panel of Figure 4 we show an example of this fitting to the subinterval obtained in 1995 January 4 between ∼10:20 and ∼10:28 UT. For the specific subinterval, our analysis derives κ₀ = 7.26 ± 0.08.

3.4.1. Error of Plasma Parameters

3.4.2. Bernoulli Integral

In order to enhance the possibility that the analyzed subintervals correspond to individual streamlines where the polytropic relation is valid, we analyze only subintervals with quasi-constant energy of the proton plasma, that is, the Bernoulli integral (Kartalev et al. 2006; Nicolaou et al. 2014a; Pang et al. 2015, 2016; Livadiotis 2018a, 2018b; Park et al. 2019). For the plasma near 1 au, the Bernoulli integral can be estimated from the plasma parameters as (e.g., Livadiotis 2016):

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{{V}_{\mathrm{SW}}}^{2}+\displaystyle \frac{\gamma }{\gamma -1}\displaystyle \frac{P}{\rho }+\displaystyle \frac{{B}^{2}}{{\mu }_{0}\rho }\ \mathrm{for}\ \gamma \ne 1,\ \mathrm{and}\end{eqnarray} \tag{ 5a }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{{V}_{\mathrm{SW}}}^{2}+\displaystyle \frac{P}{\rho }\mathrm{ln}\rho +\displaystyle \frac{{B}^{2}}{{\mu }_{0}\rho }\ \mathrm{for}\ \gamma =1,\end{eqnarray} \tag{ 5b }$$

where P/ρ = k_BT/m_p = 1/2 · ${{u}_{\mathrm{th}}}^{2}$.

In the presented analysis, we first calculate the Bernoulli integral for each subinterval (five consecutive data points) using Equation 5(a) if $\left|\gamma -1\right|\gt 10 \%$, and Equation 5(b) if $\left|\gamma -1\right|\lt 10 \%$. We then reject subintervals for which the standard deviation of the Bernoulli integral is larger than 10% of its average value. The right panel of Figure 4 shows the Bernoulli integral calculated for the subinterval obtained in 1995 January 4 from ∼10:20 and ∼10:28. The red dashed lines in the plot correspond to the mean value ±10%. The nearly constant Bernoulli integral within the specific subinterval enhances the possibility that the analyzed plasma belongs to the same streamline.

First, we examine the dependence of γ and κ₀ on the solar wind speed V_SW, and then determine the relationship between the two indices. For each year, we generate the 2D histograms of γ and log κ₀ as a function of V_SW (similarly to Livadiotis & Desai 2016; Livadiotis 2018a). Figure 5 shows such a histogram for 1997, while Figure 6 shows the specific panels for the years from 1995 to 2017. For each year, we show (left) the non-normalized and (right) normalized occurrence of (top) γ and (bottom) log κ₀.

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For each velocity bin (column), we calculate the mean values of γ, log κ₀ (white lines in Figures 5 and 6) and their standard errors δγ and δlog κ₀, which we use to calculate the mean values of the secondary polytropic index $\nu \equiv {\left(\gamma -1\right)}^{-1}$ and the kappa index ${\kappa }_{0}={10}^{\mathrm{log}{\kappa }_{0}}$, as well as their propagation errors δν and δκ₀, respectively. Figure 7 shows the estimated mean values of ν and log κ₀, as a function of the solar wind speed V_SW for each year from 1995 to 2017.

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According to Equation (2), the secondary polytropic index ν is linearly related to the kappa index κ₀, with the slope equal to −1, and y-axis intercept given by:

$$\begin{eqnarray}&&{y}_{0}=\displaystyle \frac{1}{2}{d}_{{\rm{\Phi }}}-1.\end{eqnarray} \tag{ 6 }$$

For each year, we select the data points that follow the expected linear relation. The selection is based on the χ² optimization between the selected data points and the theoretical line in Equation (2). Our analysis excludes data points with κ₀ larger than 4.5, which is the theoretical limit of kappa indices for 3D kappa distributions describing both the kinetic and potential energy with $\tfrac{1}{2}{d}_{{\rm{\Phi }}}=6$ (Livadiotis 2015b). We calculate the average y-intercept ${\bar{y}}_{0}$ from the N selected data points (κ_0i, ν_i), as follows:

$$\begin{eqnarray}&&{\bar{y}}_{0}=\sum _{i=1}^{N}\delta {{y}_{0i}}^{-2}{y}_{0i}/\sum _{i=1}^{N}\delta {{y}_{0i}}^{-2},\end{eqnarray} \tag{ 7 }$$

where ${y}_{0i}={\kappa }_{0i}+{\nu }_{i}$ and $\delta {y}_{0i}=\sqrt{\delta {{\kappa }_{0i}}^{2}+\delta {{\nu }_{i}}^{2}}$. The standard error of ${\bar{y}}_{0}$ is

$$\begin{eqnarray}&&\delta {y}_{0}=\sqrt{{\left(\delta {{y}_{0}}^{\mathrm{prop}}\right)}^{2}+{\left(\delta {{y}_{0}}^{\mathrm{stat}}\right)}^{2}},\end{eqnarray} \tag{ 8 }$$

where $\delta {{y}_{0}}^{\mathrm{prop}}$ is the propagation error:

$$\begin{eqnarray}&&\delta {{y}_{0}}^{\mathrm{prop}}=\displaystyle \frac{1}{\sqrt{\sum _{i=1}^{N}\delta {{y}_{0i}}^{-2}}}.\end{eqnarray} \tag{ 9 }$$

$\delta {{y}_{0}}^{\mathrm{stat}}$ is the statistical error:

$$\begin{eqnarray}&&\delta {{y}_{0}}^{\mathrm{stat}}=\sqrt{\displaystyle \frac{\tfrac{1}{N}{\sum }_{i=1}^{N}\delta {{y}_{0i}}^{-2}{\left({\bar{y}}_{0}-{y}_{0i}\right)}^{2}}{{\sum }_{i=1}^{N}\delta {{y}_{0i}}^{-2}-{\left({\sum }_{i=1}^{N}\delta {{y}_{0i}}^{-2}\right)}^{-1}{\sum }_{i=1}^{N}\delta {y}_{0i}^{-4}}}.\end{eqnarray} \tag{ 10 }$$

Finally, we calculate the potential degrees of freedom:

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{d}_{{\rm{\Phi }}}={\bar{y}}_{0}+1.\end{eqnarray} \tag{ 11 }$$

Figure 8 shows the indices ν and κ₀ (averaged per V_SW-bin) plotted as a function of the solar wind speed V_SW for the annual interval in 1997; it also shows κ₀ plotted against ν, by canceling V_SW. The blue data points (ν_i, κ_0i,) are fitted with Equation (2) (dashed line) in order to estimate the value of the potential d.o.f., $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$. The analysis is completed for each year from 1995–2017, in order to examine the long-term variation of $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ over the last two solar cycles (see also Table 2). In Figure 9, we show the panels of κ₀ plotted against ν (similar to that in the top right panel of Figure 8), for all the years from 1995 to 2017.

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Table 2.  Annual Means of Polytropic and Kappa Indices, and the Related Potential d.o.f, for 1995–2017

Year	$\bar{\nu }$	$\bar{\gamma }$	σγ	$\overline{\mathrm{log}{\kappa }_{0}}$	σlog κ0	${\bar{\kappa }}_{0}$	σκ0	$\tfrac{1}{2}{d}_{{\rm{\Phi }}}$	$\delta \tfrac{1}{2}{d}_{{\rm{\Phi }}}$
1995	2.24	1.45	1.09	0.48	0.29	3.00	2.00	5.89	0.05
1996	2.25	1.45	1.04	0.46	0.25	2.91	1.70	n/a	n/a
1997	1.98	1.50	1.11	0.43	0.32	2.72	2.02	5.57	0.06
1998	1.46	1.69	1.07	0.34	0.35	2.18	1.76	4.75	0.04
1999	1.38	1.73	1.01	0.31	0.36	2.03	1.69	4.53	0.04
2000	1.45	1.69	1.03	0.22	0.38	1.66	1.43	3.94	0.03
2001	1.43	1.70	1.05	0.25	0.39	1.77	1.60	4.21	0.04
2002	1.30	1.77	1.02	0.30	0.33	1.98	1.52	n/a	n/a
2003	1.38	1.73	0.97	0.36	0.32	2.31	1.70	4.78	0.05
2004	1.35	1.74	1.02	0.36	0.33	2.30	1.75	4.70	0.03
2005	1.56	1.64	1.05	0.31	0.34	2.04	1.58	4.72	0.04
2006	1.48	1.68	1.03	0.43	0.28	2.68	1.72	5.07	0.03
2007	1.40	1.72	1.00	0.44	0.28	2.78	1.77	n/a	n/a
2008	1.32	1.76	0.95	0.52	0.28	3.32	2.17	5.51	0.05
2009	1.38	1.73	1.01	0.53	0.29	3.41	2.30	n/a	n/a
2010	1.34	1.74	1.03	0.40	0.32	2.52	1.87	4.82	0.04
2011	1.32	1.76	1.03	0.35	0.34	2.24	1.73	4.65	0.03
2012	1.43	1.70	1.06	0.25	0.35	1.76	1.43	4.19	0.04
2013	1.40	1.71	1.08	0.28	0.35	1.89	1.51	4.30	0.03
2014	1.40	1.71	1.05	0.28	0.34	1.91	1.47	4.26	0.03
2015	1.46	1.68	1.02	0.30	0.31	1.99	1.43	5.16	0.02
2016	1.44	1.70	1.02	0.39	0.30	2.47	1.69	5.38	0.04
2017	1.57	1.64	1.02	0.42	0.27	2.63	1.66	5.23	0.03

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The occurrences of γ and log κ₀ as a function of V_SW (Figures 5 and 6) show that γ ranges between −2.5 and 5 and log κ₀ ranges between −1 and 1, but they mostly lie around their mean values, that is, ∼1.68 and ∼0.37, respectively. The average γ and log κ₀ (per V_SW bin, white lines in Figures 5 and 6) do not exhibit any characteristic or systematic variation with V_SW. The standard error of both indices is of the same order for the entire solar wind speed range (see also Figure 1). We note that the standard error increases with the spread of the values around their mean and decreases when increasing the amount of samples within each V_SW bin. Therefore, as the majority of the data reside in the intermediate speeds, our result indicates that the spread of γ and κ₀ is greater within that speed range.

The γ values are roughly symmetrically distributed; log κ₀ values occasionally exhibit asymmetric occurrences with "tails" toward the lower log κ₀ values. We observe these "tails" to be more pronounced during years with stronger solar activity. For instance, in the years 1995–1997 and 2007–2009, corresponding to minimum solar activity periods, the occurrence of log κ₀ does not extend significantly below zero. On the other hand, in the years 2000–2002 and 2012–2014, corresponding to maximum solar activity, the occurrence exhibits tails extending to ∼−1. These tails shift the average log κ₀ (white line on top of the occurrence plot) toward lower values.

Figure 5, shows the occurrences of the polytropic and kappa indices as a function of the solar wind speed for 1997. The average log κ₀ is higher for lower speeds, V_SW < 300 km s⁻¹, whichmay be caused by less reliable measurements that characterize these speeds; for higher speeds and a wider range of speeds log κ₀ almost flattens, while it slightly increases for the fast solar wind (V_SW > 550 km s⁻¹). The behavior of the polytropic and kappa indices as a function of V_SW is slightly different for different years of the last two solar cycles (Figure 6).

Figure 7 shows the mean values of ν and log κ₀ as a function of V_SW, for each year between 1995 and 2017. Interestingly, while there is no systematic variation of both indices as a function of V_SW, most of the time, the two indices are anti-correlated, as expected from theory (Livadiotis 2017, 2018b, 2019a, 2019b, 2019c). For instance, in 1997, ν ∼ 1.5 within the low speed range (V_SW < 300 km s⁻¹), reaches a maximum value ν ∼ 2.5 in the intermediate speed range (450 km s⁻¹ < V_SW < 550 km s⁻¹) and drops again down to ν ∼ 1.5 as the speed increases (V_SW > 550 km s⁻¹). Within the same year, log κ₀ follows an opposite trend; log κ₀ ∼ 0.5 within the low speed range (V_SW < 300 km s⁻¹), reaches a minimum value log κ₀ ∼ 0.25 in the intermediate speed range (450 km s⁻¹ < V_SW < 550 km s⁻¹), and increases again to log κ₀ ∼ 0.5 as the speed increases (V_SW > 550 km s⁻¹). In another characteristic example, in 2004, the two indices exhibit a different behavior as a function V_SW but similar to 1997, they are anti-correlated. For this specific year, ν ∼ 1 in the low speed range (V_SW < 300 km s⁻¹) and ∼2 in the high speed range (V_SW > 500 km s⁻¹), exhibiting a local maximum (ν ∼ 1.5) for V_SW ∼ 400 km s⁻¹ and a local minimum (ν ∼ 1.3) for V_SW ∼ 500 km s⁻¹. In the same year, log κ₀ ranges between 0.5 and 0, with a global minimum (log κ₀ ∼ 0.3) at V_SW ∼ 400 km s⁻¹ and a local maximum (logκ₀ ∼ 0.4) at V_SW ∼ 500 km s⁻¹.

Although we observe anti-correlation between the values of ν and κ₀ in most of the annual intervals, there are examples in which the two indices do not exhibit any systematic correlation. We identify these cases in years 2002, 2007, and 2009. Interestingly, we also observe that in 1996, the two indices are positively correlated. Within this year, ν and log κ₀ exhibit similar behavior within a wide range of V_SW. Specifically, for V_SW between 350 km s⁻¹ and 500 km s⁻¹, both indices decrease; ν drops from a local maximum ν ∼ 2.5, to a local minimum ν ∼ 1.5, while log κ₀ decreases from local maximum log κ₀ ∼ 0.6, to a local minimum log κ₀ ∼ 0.35. The linear relation of Equation (2) is not applicable to any of these annual intervals (see also Figure 9). We speculate that during these annual intervals there is more than one potential affecting the distribution function of the particles. In such a case, we need to use a superposition of potentials to describe the observed relationship, which is not the scope of this current work.

We fit ν and κ₀ with Equation (2), within each of the annual intervals that the two indices are anti-correlated (19 out of the 23 annual intervals) in order to determine the y-intercept ${\bar{y}}_{0}$, and the potential degrees of freedom $\tfrac{1}{2}{d}_{{\rm{\Phi }}}={\bar{y}}_{0}+1$. In Figure 8, we show ν and κ₀ as a function V_SW and ν as a function of κ₀ for the year 1997. In this typical example, we fit the majority of the data points with the linear model of Equation (2) and we calculate ${\bar{y}}_{0}=4.55$, therefore, $\tfrac{1}{2}{d}_{{\rm{\Phi }}}=5.55$. In Figure 9 we show ν as a function of κ₀ for each year from 1995 to 2017. Although there is a clear scattering of the data points, probably due to the turbulent nature and the multiple structures within the solar wind, the linear relationship between the polytropic and kappa indices can be statistically identified in most of the annual subintervals.

We investigate the annual averages of the key parameters as a function of time, for the last two solar cycles. Table 2 shows the annual averages of the polytropic index $\bar{\gamma }$ and $\bar{\nu }$, the logarithm of the invariant kappa index $\overline{\mathrm{log}{\kappa }_{0}}$, the invariant kappa index ${\bar{\kappa }}_{0}$, the potential degrees of freedom $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$, and their errors, as calculated for each year from 1995 to 2017. For the years in which the linear model does not apply (that is the cases of the four years mentioned above, i.e., 1996, 2002, 2007, and 2009), we do not derive ${\bar{y}}_{0}$ and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ (listed as n/a in Table 2). Figure 10 shows the time series of the calculated average parameters along with the annual average sunspot number S_n and the interplanetary magnetic field strength B. We identify the following characteristics in the long-term behavior of the parameters:

• 1.  
  The annual average $\overline{\mathrm{log}{\kappa }_{0}}$ and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ are correlated with each other, while both these parameters are anti-correlated with the solar activity. From 1995 to 2000, both $\overline{\mathrm{log}{\kappa }_{0}}$ and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ decrease monotonically as S_n increases. During the solar maximum in 2000, $\overline{\mathrm{log}{\kappa }_{0}}$ and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ reach their minimum values, ∼0.2 and ∼4, respectively. During the decay phase of the 23rd solar cycle, both $\overline{\mathrm{log}{\kappa }_{0}}$ and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ increase; during the solar minimum in 2008–2009, log κ₀ reaches a maximum value ∼0.5 and $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ reaches a local maximum ∼5.5. During the ascending phase of the 24th solar cycle between 2009 and 2012, both $\overline{\mathrm{log}{\kappa }_{0}}$ and ${\bar{y}}_{0}$ drop systematically and reach local minima in 2012, before they increase again.
• 2.  
  The annual average polytropic index $\bar{\gamma }$ exhibits weak correlation with the solar activity. For the first three years, the plasma is sub-adiabatic, with $\bar{\gamma }\sim 1.4$, with a slide increase in 1997, as $\bar{\gamma }\sim 1.5$. From 1998 until 2017, the plasma is nearly adiabatic with $\bar{\gamma }$ ranging between 1.65 and 1.8. The results of this study do not precisely follow the characteristic long-term behavior of $\bar{\gamma }$ and its anti-correlation with the solar activity shown by Nicolaou et al. (2014a). The differences may be caused by the different number of data points per subinterval and/or by the different data sets and processing, as Nicolaou et al. (2014a) used timeshifted, cross-calibrated data sets of plasma parameters, measured by different S/C and processed with different time resolutions.

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We further quantify the relation between $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ and S_n. We seek a power-law function that relates the two parameters using the fitting method based on the correlation coefficient maximization (Livadiotis & McComas 2013b). We specifically search for the positive exponent p for which the absolute correlation coefficient between $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ and ${{S}_{{\rm{n}}}}^{p}$ is maximized. Because $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ and ${{S}_{{\rm{n}}}}^{p}$ are anti-correlated, we optimize the exponent p for the minimum Pearson correlation coefficient value. We find that the minimum correlation (Pearson coefficient −0.82) is achieved for p ∼ 0.6 (Figure 11, left). We achieve an approximate fit (Figure 11, right) with

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{d}_{{\rm{\Phi }}}=5.75-0.08{{S}_{{\rm{n}}}}^{0.6}.\end{eqnarray} \tag{ 12 }$$

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We examine 23 annual intervals, from 1995 to 2017, to investigate the long-term behavior of the polytropic $\nu \equiv {(\gamma -1)}^{-1}$ and kappa κ₀ indices, and quantify the linear relationship between them (averaged in each solar wind speed bin) over the last two solar cycles. In 19 of the 23 examined intervals (∼83%) we determine a significant amount of data points that are linearly correlated. The analysis of these intervals derives the potential d.o.f., $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$. This may be caused by the fact that the annually based data analysis is not sensitive for the various solar wind structures, thus we were averaging features for which the relationship between the two indices is different—for reasons that are not fully understood at the moment. A deviation from the theoretical model in Equation (2) may be expected if more than one potential is acting on the plasma particles within the specific intervals. Such intervals could be analyzed in future studies with finer resolutions, in order to resolve the existing potentials. Moreover, different solar wind structures such as magnetic clouds, CMEs, and CIRs may introduce scattering of the data points around the theoretical expectation. Also, the presence of enhanced turbulence can introduce additional data-point scattering and prevent the accurate determination of the potential d.o.f. Indeed, the anti-correlation is not persistently strong within all the analyzed annual intervals and the whole range of V_SW, nonetheless the result still constitutes strong evidence of the specific correlation, verifying the results and the theoretical expectations (Livadiotis 2018a, 2018b, 2019a).

We found a clear variation of the annual $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$, which is correlated with the annual average $\overline{\mathrm{log}{\kappa }_{0}}$. Both $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ and $\overline{\mathrm{log}{\kappa }_{0}}$ are anti-correlated with S_n (Figure 10). Further analysis quantifies the relationship between $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ and S_n (Figure 11). The results indicate that, during periods of strong activity, the enhanced magnetic field induces stronger correlations among particles (Livadiotis et al. 2018), resulting in kappa distributions with lower κ₀. This is specifically shown in Figure 12, where κ₀ decreases with increasing S_n and B. In the right panel of Figure 12, we show the data points within the two solar cycles. The anti-correlation between κ₀ and B is more pronounced during the 23rd solar cycle, for which κ₀ drops from ∼3.5 to ∼1.5 as B increases from ∼3 nT to ∼8 nT. In addition, we observe anti-correlation between the magnetic field and the potential d.o.f.; we discuss this behavior in the next subsection.

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The observed d.o.f. $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ decreases from ∼6 during solar minimum to ∼4 during solar maximum. For the power-law central potential,

$$\begin{eqnarray}&&{\rm{\Phi }}(r)\propto -{r}^{-b},\end{eqnarray} \tag{ 13 }$$

(negative ${\rm{\Phi }}(r)\lt 0$ and attractive $\partial {\rm{\Phi }}(r)/\partial r\gt 0$), it has been shown that the d.o.f. $\tfrac{1}{2}{d}_{{\rm{\Phi }}}$ can be expressed as a function of the exponent b and the potential dimensionality d_r (Livadiotis 2015b, 2017, Chapters 3 and 4)

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}{d}_{{\rm{\Phi }}}={b}^{-1}{d}_{{\rm{r}}}.\end{eqnarray} \tag{ 14 }$$

For a typical interplanetary electric field with b = 1/2 (e.g., Cuperman & Harten 1971; Lacombe et al. 2002; Livadiotis 2018b), our findings indicate that the dimensionality d_r reduces from 3 during solar minimum and a weak magnetic field to 2 during solar maximum and a strong magnetic field. Table 3 shows the corresponding values of the potential dimensionality d_r, for such a typical interplanetary electric potential (b = 0.5), and the annual S_n and B. Figure 13 shows d_r as a function of S_n and B. The linear fitting to the data shows that during low activity and a weak interplanetary magnetic field, the interplanetary electric potential is characterized by spherical symmetry, which reduces to cylindrical symmetry during high activity and a strong interplanetary magnetic field.

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Table 3.  Annual Means of the Potential d.o.f., Space Dimensionality, and Magnetic Field for 1995–2017

Year	$\tfrac{1}{2}{d}_{{\rm{\Phi }}}$	$\delta \left(\tfrac{1}{2}{d}_{{\rm{\Phi }}}\right)$	dr	δdr	B	Sn
1995	5.89	0.05	2.95	0.03	5.71	25.1
1996	n/a	n/a	n/a	n/a	5.08	11.6
1997	5.57	0.06	2.79	0.03	5.50	28.9
1998	4.75	0.04	2.38	0.02	6.75	88.3
1999	4.53	0.04	2.27	0.02	6.63	136.3
2000	3.94	0.03	1.97	0.02	6.90	173.9
2001	4.21	0.04	2.11	0.02	6.77	170.4
2002	n/a	n/a	n/a	n/a	7.46	163.6
2003	4.78	0.05	2.39	0.03	7.13	99.3
2004	4.70	0.03	2.35	0.02	6.19	65.3
2005	4.72	0.04	2.36	0.02	6.11	45.8
2006	5.07	0.03	2.54	0.02	4.95	24.7
2007	n/a	n/a	n/a	n/a	4.35	12.6
2008	5.51	0.05	2.76	0.03	4.15	4.2
2009	n/a	n/a	n/a	n/a	3.85	4.8
2010	4.82	0.04	2.41	0.02	4.61	24.9
2011	4.65	0.03	2.33	0.02	5.21	80.8
2012	4.19	0.04	2.10	0.02	5.64	84.5
2013	4.30	0.03	2.15	0.02	5.16	94.0
2014	4.26	0.03	2.13	0.02	5.79	113.3
2015	5.16	0.02	2.58	0.01	6.54	69.8
2016	5.38	0.04	2.69	0.02	5.96	39.8
2017	5.23	0.03	2.62	0.02	5.20	21.7

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It is possible that both d_r and b may vary with the solar activity and the interplanetary magnetic field. The diagram in Figure 14 plots the exponent b as a function of d_r within the observed range of values of 1/2d_Φ. In the same plot we indicate the solutions for b = 0.5, which are discussed above (intersection with the horizontal magenta), and the solutions of d_r = 2 and d_r = 3 (intersections with the vertical gray lines). For example, in the case of potential dimensionality fixed to d_r = 3 (right gray dashed in Figure 14), our findings indicate that the exponent b would vary from 0.75 during solar maximum to 0.5 during solar minimum.

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