**3.4 Properties and peculiar characteristics of the dilute e-gas of strong magnetized matter**

We can further our observational understanding by doing a deeper analysis of the electron's distribution shown in **Figure 10**. And **Figure 11** reveals us more about this state of matter.

**Figure 11** illustrates for a steady matter MHD case with intense/dominating **B-**field a set of 8 'skymaps'showing an angular map of the electron's distribution. In addition,

**Figure 11.** *Parts of e-vel distribution function* F*(*v*,*θ*,*φ*), including Strahl aligned to B-field.*

point and arrow indicate in the plane the orientation of the **B-**field at the time interval considered, see Figure 1 in Figueroa Viñas et al. [44], see also [45].

The '*e' velocity* distribution function (VDF) in spherical coordinates is represented in **Figure 11** by a function f(v,θ, ϕ), where the independent variables are the velocity, polar angle (elevation in the figure), and azimuthal angle (Phase label, bottom, in the figure) respectively. **Figure 11** shows a subset not contiguous of the 30 energy steps of one VDF measurement, see instrument description in Ogilvie et al. [46]

The 2-D **Figure 11** choice of the longitude scale (from -180 to 180°) in each panel produces a split view of the selective one-dimensional distribution for MC (This distribution, along the **B-**field, is shown in **Figure 10**, right panel for the electrons (see ref. [38]). The location of the direction **B-**field is shown. The different panels below show from bottom up, on the right and continued on the left panels the intensity variation (from lowest values in lighter dark with increasing electron intensity through green, then yellow reaching the maximum intensity with red and most intense electron presence in dark red at the center of the distribution). Additionally, the plots show the orientation of the **B-**field direction through all panels (see arrowhead maintains the direction (θ = 30°, ϕ = 90°), and base of it seen in the right panes at (θ = 30°, ϕ = 90°), and indicates the current flowing in the direction opposite to the **B-**field intensity at energies of about more than 70 keV (all panels where the **B-**field electron aligned current Je appears and in this case study extending beyond the upper energy interval shown of 268.3 keV.

**Figure 12** shows that the *electrons* data are '*3s averages*,' also from the SWE instrument, but an instrument part built in GSFC (all in SC Wind) [47]. Fitting parameter

*Hydromagnetic Steady Magnetized Plasma Encountered by Voyager in the Interstellar Space DOI: http://dx.doi.org/10.5772/intechopen.112362*

**Figure 12.** *An example of anomalous gas properties in the SW for a B-field MHD medium.*

*γ = ½* is the overall value of the polytropic exponent index on a strongly **B-**field dominated structure (MC interval with date 6/2/1998 observed) which passes Wind in its motion away from the Sun.

The γ for the FR region observed with Wind (see enclosed in black values on the right) would indeed show a smaller value for the relationship between temperature *'T'* of electrons and density of the plasma (γ � 0.7).

From a case study, *see* [18]*,* obtained values of *T, vs, μ,* **J***free. And also:*

$$\mathbf{T} = \mathbf{1}.24 \cdot \mathbf{1}0^5 \,\mathrm{K} > \mathbf{1}/2 \,\mathrm{m}\_{\boldsymbol{\epsilon}} < (\mathbb{V}\_{\mathbf{e}} - < \mathbb{V}\_{\mathbf{e}} >)^2 > \text{which is proved right} \tag{11}$$


$$\mathbf{B} = \mathbf{B}\_0 \left(\mathbf{t}\_0/\mathbf{t}\right)^2 \left[\mathbf{H} \,\mathbf{J}\_1(\mathbf{a} \,\mathbf{r}) \,\mathbf{e}\_\Psi + \mathbf{J}\_0(\mathbf{a} \,\mathbf{r}) \,\mathbf{e}\_\mathbf{x}\right],\tag{12}$$

own overall polytropic behavior on June 2, 1998, as it is shown in **Figure 12**.

The characteristics of the gas being of so considered anomalous kind, as the graph in **Figure 13** bottom panel shows. The top panel in **Figure 13** shows the usual ideal gas dependence of *T* (temperature) on *ρ* (density) for an adiabatic branch. It corresponds to the entropy conserving adiabatic process with polytropic exponent *ϒ* = 5/3, see, e.g. [48].

The actual observations summarized in **Figure 14** are based on a direct observation of the effect inside strongly magnetized matter in the form of a MC, already introduced earlier, suddenly ejected from the Sun and its interpretation based on data shown below in **Figure 14**. Our complete illustration in **Figure 14** of the **B-**field and bulk plasma conditions in an *electron* 3 s time series of a duration of a little more than two minutes of the properties present in this ideal **B**-field dominated matter includes from the top to bottom a sequence of eight panels ordered 1, 2, 3, 4, 5, 6, 7, and 8. In the present description: (a) panels 1 and 2 show the estimated convection velocity, in (1) the bulk velocity magnitude, named *VB*, and in (2) on the left labels the longitude, and on the right the elevation of **V***<sup>B</sup>* in cthe orthogonal coordinate system GSE (e.g. [49]); (b) panels 3, 4, and 5 show the vector **B-**field, where panel 3

**Figure 13.** *Normal (top) vs. anomalous (bottom) gas.*

*Hydromagnetic Steady Magnetized Plasma Encountered by Voyager in the Interstellar Space DOI: http://dx.doi.org/10.5772/intechopen.112362*

**Figure 14.** *Illustration of key constitutive properties of the MHD-dominated structure.*

illustrates the steadiness of the intensity of the magnetic field, and the direction minimal changes are emphasized in panels 4 for elevation and 5 for longitude. Meanwhile, (c) panels 6–8 show the features of the medium in which panel 6 on the right shows the coherent variations showing anti-correlation between the 3 s mean **B-**field magnitude (on the left) and the electron density (on the right). Panel 7 on the right shows the clear anti-correlation between the electron *Te* (left label) and density (right label); These panels 6 and 7 are at the core of the interpretation of a diamagnetic medium, consistent with past identification of this MHD constitutive property at interplanetary magnetic holes (see [50, 51]). Finally, (d) the bottom panel 8 checks the thermodynamic equilibrium condition showing negligible anisotropy in the internal electrons energy (right side of panel 8) while substantial changes, also shown in panel 7, (also illustrated in the left side of panel 8) take place indicating the diamagnetic nature of the medium, i.e. thermal equilibrium holds and further entropy conservation is well-established supporting the conservation of the energy of this matter–magnetic field structure. Further, the Figure illustrates the ends of the described coherence effects discussed at the passing of the time/location of the observation suggesting the presence of domains in the structure that would interact solely through their vibrational contact keeping the observed gas (wide part of the electrons distribution, **Figure 10** right panel, typical characteristic of the SW (rooted in the presence of a dilute gas in each domain), which constitutes a key element to understand the thermal short equilibration times in the low solar corona in our consistent interpretation of the here discussed medium, see, e.g. [52].

**Figure 14** coherence of the diamagnetic coupling between plasma and **B-**field suggests again a presence in electron dilute gas of two works, which we discuss in the next paragraphs.

The presence of an anomalous polytropic index is the result of the two works done by the electron gas in the domains of the magnetized plasma structure (see [18, 52, 53]):

• ideal gas work only:

$$V^{\Upsilon} = \text{Constant} \qquad \text{with } \Upsilon = \mathfrak{c}\_p/\mathfrak{c}\_v = \mathfrak{1} + \mathbb{R}/\mathfrak{c}\_v = \mathfrak{5}/\mathfrak{3} \tag{13}$$

$$\text{If } V = N \, R \, T \text{ (equation of state) } \text{If } N \text{ constant, then}$$

$$\mathbf{T}/\rho^{r-1} = \mathbf{K} \bullet \text{with constant } \mathbf{K} = \text{constant}/\left(\mathbf{N}^T \mathbf{R}\right) \tag{15}$$

Ideal gas and magnetic work: *ϒ = 1+ R/cv(1 – α) where α = [magnetization work/gas work].*

Then it can be shown the relationship between *e-gas* and magnetic work that enables anomalous *γ (γ<sup>a</sup> = ½)* at position *P* in the FR in **Figure 8**. … for a noninteracting *e-gas* with internal energy *dU = cV dT,* and its equation of state *PV = RT* (for 1 mole, and *R* being the universal gas constant) the relationship for the reversible adiabatic exponent *γ<sup>a</sup>* is obtained from

$$c\_V \, dT + R \, (\mathbf{1} - a) \, T \, dV / V = \mathbf{0}. \tag{16}$$

and the condition of point-like *e-gas* with three degrees of freedom, where *cV/ R = 3/2,* then *α = 7/4.* (In Eqs. (13)–(16), *V* is the internal *energy* velocity of the gas and *not* the bulk velocity (*VB*).)
