Elliptic vector loci of average electron velocity of electron swarm in constant-collision-frequency model gas under ac electric and dc magnetic fields crossed at arbitrary angles

Abstract

The analytic solution of the average electron velocity vector $\mathbf {W}(t)$ of an electron swarm in gas under ac electric and dc magnetic fields ($\mathbf {E}(t)$ and $\mathbf {B}$, respectively) in a constant-collision-frequency model is extended to cover not only the known case of a right crossing angle ($\mathbf {E}(t) \perp \mathbf {B}$) but also the cases of arbitrary crossing angles ($\mathbf {E}(t) \not \perp \mathbf {B}$). The x, y, and z components of $\mathbf {W}(t)$ are obtained as explicit formulae with the following parameters: the amplitude E of $\mathbf {E}(t)$, the angle $\theta$ between $\mathbf {E}(t)$ and $\mathbf {B}$, the angular frequency $\omega _E$ of $\mathbf {E}(t)$, the cyclotron angular frequency $\omega _B$ (subject to the strength B of $\mathbf {B}$), and the electron collision frequency $\nu$. A basic feature that $\mathbf {W}(t)$ draws elliptic locus in velocity space is unchanged even under $\mathbf {E}(t) \not \perp \mathbf {B}$, but the plane including the locus may tilt when $\omega _E$, $\omega _B$, or $\nu$ varies. The derivation of $\mathbf {W}(t)$ based on the analytic solution of single electron motion is detailed and fundamental behavior of the $\mathbf {W}(t)$ loci is observed to understand the electron transport under $\mathbf {E}(t) \times \mathbf {B}$ fields.

Graphical abstract

Appendices

Appendix A: Coordinate conversion between the $\mathbf {E}$-fixed and the $\mathbf {B}$-fixed systems

When vectors $(x^{\mathbf {E}}, y^{\mathbf {E}}, z^{\mathbf {E}})$ in the $\mathbf {E}$-fixed system and $(x^{\mathbf {B}}, y^{\mathbf {B}}, z^{\mathbf {B}})$ in the $\mathbf {B}$-fixed system are equivalent to each other with respect to the direction relative to $\mathbf {E}$ and $\mathbf {B}$, the components of one can be obtained from those of the other by the following relations representing rotations around the y-axis by angles $\pm (\pi /2 - \theta )$:

$$\begin{aligned}&\left\{ \begin{array}{l} x^{\mathbf {B}} = x^{\mathbf {E}} \sin \theta - z^{\mathbf {E}} \cos \theta \\ y^{\mathbf {B}} = y^{\mathbf {E}} \\ z^{\mathbf {B}} = x^{\mathbf {E}} \cos \theta + z^{\mathbf {E}} \sin \theta \end{array} \right. , \end{aligned}$$

(A.1)

$$\begin{aligned}&\left\{ \begin{array}{l} x^{\mathbf {E}} = x^{\mathbf {B}} \sin \theta + z^{\mathbf {B}} \cos \theta \\ y^{\mathbf {E}} = y^{\mathbf {B}} \\ z^{\mathbf {E}} = - x^{\mathbf {B}} \cos \theta + z^{\mathbf {B}} \sin \theta \end{array} \right. . \end{aligned}$$

(A.2)

Appendix B: $\mathbf {v}^{\mathbf {B}}(t)$ in the $\mathbf {B}$-fixed system

Let the initial electron velocity at $t = 0$ be $\mathbf {v}^{\mathbf {B}}(0) = \mathbf {v}_0^{\mathbf {B}} = (v_{x0}^{\mathbf {B}}, v_{y0}^{\mathbf {B}}, v_{z0}^{\mathbf {B}})$. In non-ECR condition $\omega _E \not = \omega _B$, an analytic solution of (5) for $\mathbf {v}^{\mathbf {B}}(t)$ is

$$\begin{aligned} v_x^{\mathbf {B}}= & {} \left( - \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \phi + v_{x0}^{\mathbf {B}} \right) \cos \omega _B t \nonumber \\&+ \left( + \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \sin \phi - v_{y0}^{\mathbf {B}} \right) \sin \omega _B t \nonumber \\&+ \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos (\omega _E t + \phi ), \end{aligned}$$

(B.3)

$$\begin{aligned} v_y^{\mathbf {B}}= & {} \left( - \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \phi + v_{x0}^{\mathbf {B}} \right) \sin \omega _B t \nonumber \\&+ \left( - \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \sin \phi + v_{y0}^{\mathbf {B}} \right) \cos \omega _B t \nonumber \\&+ \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \sin (\omega _E t + \phi ), \end{aligned}$$

(B.4)

$$\begin{aligned} v_z^{\mathbf {B}}= & {} v_E \cos \theta \cos (\omega _E t + \phi ) - v_E \cos \theta \cos \phi + v_{z0}^{\mathbf {B}}.\nonumber \\ \end{aligned}$$

(B.5)

In the ECR condition $\omega _E = \omega _B \equiv \omega$,

$$\begin{aligned} v_x^{\mathbf {B}}= & {} \left( - \frac{1}{2} v_E \sin \theta \sin \phi \cos \phi - \frac{1}{2} \omega t v_E \sin \theta \right. \nonumber \\&\left. + v_{x0}^{\mathbf {B}} \sin \phi - v_{y0}^{\mathbf {B}} \cos \phi \right) \sin (\omega t + \phi ) \nonumber \\&+ \left( + \frac{1}{2} v_E \sin \theta \sin ^2 \phi + v_{x0}^{\mathbf {B}} \cos \phi + v_{y0}^{\mathbf {B}} \sin \phi \right) \nonumber \\&\qquad \times \cos (\omega t + \phi ), \end{aligned}$$

(B.6)

$$\begin{aligned} v_y^{\mathbf {B}}= & {} \left( + \frac{1}{2} v_E \sin \theta \sin \phi \cos \phi + \frac{1}{2} \omega t v_E \sin \theta \right. \nonumber \\&\left. - v_{x0}^{\mathbf {B}} \sin \phi + v_{y0}^{\mathbf {B}} \cos \phi \right) \cos (\omega t + \phi ) \nonumber \\&+ \left( - \frac{1}{2} v_E \sin \theta \cos ^2 \phi + v_{x0}^{\mathbf {B}} \cos \phi + v_{y0}^{\mathbf {B}} \sin \phi \right) \nonumber \\&\qquad \times \sin (\omega t + \phi ), \end{aligned}$$

(B.7)

$$\begin{aligned} v_z^{\mathbf {B}}= & {} v_E \cos \theta \cos (\omega t + \phi ) - v_E \cos \theta \cos \phi + v_{z0}^{\mathbf {B}}.\nonumber \\ \end{aligned}$$

(B.8)

Appendix C: $\mathbf {v}^{\mathbf {E}}(t)$ in the $\mathbf {E}$-fixed system

With $\mathbf {v}^{\mathbf {E}}(0) = \mathbf {v}_0^{\mathbf {E}} = (v_{x0}^{\mathbf {E}}, v_{y0}^{\mathbf {E}}, v_{z0}^{\mathbf {E}})$, analytic solution of (5) for $\mathbf {v}^{\mathbf {E}}(t)$ in the non-ECR condition $\omega _E \not = \omega _B$ is

$$\begin{aligned} v_x^{\mathbf {E}}= & {} \left( - \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin ^2 \theta \cos \phi \right. \nonumber \\&\left. + v_{x0}^{\mathbf {E}} \sin ^2 \theta - v_{z0}^{\mathbf {E}} \sin \theta \cos \theta \right) \cos \omega _B t \nonumber \\&+ \left( + \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin ^2 \theta \sin \phi - v_{y0}^{\mathbf {E}} \sin \theta \right) \nonumber \\&\qquad \times \sin \omega _B t \nonumber \\&+ \left( 1 + \frac{\omega _B^2}{\omega _E^2 - \omega _B^2} \sin ^2 \theta \right) v_E \cos (\omega _E t + \phi ) \nonumber \\&- v_E \cos ^2 \theta \cos \phi + v_{x0}^{\mathbf {E}} \cos ^2 \theta + v_{z0}^{\mathbf {E}} \sin \theta \cos \theta , \nonumber \\ \end{aligned}$$

(C.9)

$$\begin{aligned} v_y^{\mathbf {E}}= & {} \left( - \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \phi + v_{x0}^{\mathbf {E}} \sin \theta - v_{z0}^{\mathbf {E}} \cos \theta \right) \nonumber \\&\qquad \times \sin \omega _B t \nonumber \\&+ \left( - \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \sin \phi + v_{y0}^{\mathbf {E}} \right) \cos \omega _B t \nonumber \\&+ \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \sin (\omega _E t + \phi ), \end{aligned}$$

(C.10)

$$\begin{aligned} v_z^{\mathbf {E}}= & {} \left( + \frac{\omega _E^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \theta \cos \phi \right. \nonumber \\&\left. - v_{x0}^{\mathbf {E}} \sin \theta \cos \theta + v_{z0}^{\mathbf {E}} \cos ^2 \theta \right) \cos \omega _B t \nonumber \\&+ \left( - \frac{\omega _E \omega _B}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \theta \sin \phi + v_{y0}^{\mathbf {E}} \cos \theta \right) \nonumber \\&\qquad \times \sin \omega _B t \nonumber \\&- \frac{\omega _B^2}{\omega _E^2 - \omega _B^2} v_E \sin \theta \cos \theta \cos (\omega _E t + \phi ) \nonumber \\&- v_E \sin \theta \cos \theta \cos \phi + v_{x0}^{\mathbf {E}} \sin \theta \cos \theta + v_{z0}^{\mathbf {E}} \sin ^2 \theta .\nonumber \\ \end{aligned}$$

(C.11)

In the ECR condition $\omega _E = \omega _B \equiv \omega$,

$$\begin{aligned} v_x^{\mathbf {E}}= & {} \left( + v_{x0}^{\mathbf {E}} \sin \theta \cos \phi + v_{y0}^{\mathbf {E}} \sin \phi - v_{z0}^{\mathbf {E}} \cos \theta \cos \phi \right) \nonumber \\&\qquad \times \sin \theta \cos (\omega t + \phi ) \nonumber \\&+ \frac{1}{2} v_E \left( 2 \cos ^2 \theta + \sin ^2 \theta \sin ^2 \phi \right) \cos (\omega t + \phi ) \nonumber \\&+ \bigg ( - \frac{1}{2} v_E \sin \theta \sin \phi \cos \phi + v_{x0}^{\mathbf {E}} \sin \theta \sin \phi \nonumber \\&- v_{y0}^{\mathbf {E}} \cos \phi - v_{z0}^{\mathbf {E}} \cos \theta \sin \phi \bigg ) \sin \theta \sin (\omega t + \phi ) \nonumber \\&- \frac{1}{2} \omega t v_E \sin ^2 \theta \sin (\omega t + \phi ) \nonumber \\&- v_E \cos ^2 \theta \cos \phi + v_{x0}^{\mathbf {E}} \cos ^2 \theta + v_{z0}^{\mathbf {E}} \sin \theta \cos \theta , \nonumber \\ \end{aligned}$$

(C.12)

$$\begin{aligned} v_y^{\mathbf {E}}= & {} \bigg ( - \frac{1}{2} v_E \sin \theta \cos ^2 \phi + v_{x0}^{\mathbf {E}} \sin \theta \cos \phi \nonumber \\&+ v_{y0}^{\mathbf {E}} \sin \phi - v_{z0}^{\mathbf {E}} \cos \theta \cos \phi \bigg ) \sin (\omega t + \phi ) \nonumber \\&+ \bigg ( + \frac{1}{2} v_E \sin \theta \sin \phi \cos \phi - v_{x0}^{\mathbf {E}} \sin \theta \sin \phi \nonumber \\&+ v_{y0}^{\mathbf {E}} \cos \phi + v_{z0}^{\mathbf {E}} \cos \theta \sin \phi \bigg ) \cos (\omega t + \phi ) \nonumber \\&+ \frac{1}{2} \omega t v_E \sin \theta \cos (\omega t + \phi ), \end{aligned}$$

(C.13)

$$\begin{aligned} v_z^{\mathbf {E}}= & {} \bigg ( + \frac{1}{2} v_E \sin \theta (1 + \cos ^2 \phi ) - v_{x0}^{\mathbf {E}} \sin \theta \cos \phi \nonumber \\&- v_{y0}^{\mathbf {E}} \sin \phi + v_{z0}^{\mathbf {E}} \cos \theta \cos \phi \bigg ) \cos \theta \cos (\omega t + \phi ) \nonumber \\&+ \bigg ( + \frac{1}{2} v_E \sin \theta \sin \phi \cos \phi - v_{x0}^{\mathbf {E}} \sin \theta \sin \phi \nonumber \\&+ v_{y0}^{\mathbf {E}} \cos \phi + v_{z0}^{\mathbf {E}} \cos \theta \sin \phi \bigg ) \cos \theta \sin (\omega t + \phi ) \nonumber \\&+ \frac{1}{2} \omega t v_E \sin \theta \cos \theta \sin (\omega t + \phi ) \nonumber \\&- v_E \sin \theta \cos \theta \cos \phi + v_{x0}^{\mathbf {E}} \sin \theta \cos \theta + v_{z0}^{\mathbf {E}} \sin ^2 \theta .\nonumber \\ \end{aligned}$$

(C.14)

Both of these ECR and non-ECR solutions are convertible between the $\mathbf {B}$-fixed and the $\mathbf {E}$-fixed systems with relations (A.1) and (A.2).

Appendix D: Acceleration acting on electrons in the $\mathbf {B}$-fixed system

In the $\mathbf {B}$-fixed system, the equation of motion (5) for an electron becomes

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} v_x^{\mathbf {B}}= & {} - \frac{e B}{m} v_y^{\mathbf {B}} - \frac{e E}{m} \sin \theta \sin (\omega _E t + \phi ), \qquad \end{aligned}$$

(D.15)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} v_y^{\mathbf {B}}= & {} \frac{e B}{m} v_x^{\mathbf {B}}, \end{aligned}$$

(D.16)

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t} v_z^{\mathbf {B}}= & {} - \frac{e E}{m} \cos \theta \sin (\omega _E t + \phi ). \end{aligned}$$

(D.17)

We obtain the following relations:

$$\begin{aligned}&\left( \frac{\mathrm{d}}{\mathrm{d}t} v_x^{\mathbf {B}}, \frac{\mathrm{d}}{\mathrm{d}t} v_y^{\mathbf {B}}, \frac{\mathrm{d}}{\mathrm{d}t} v_z^{\mathbf {B}} \right) \nonumber \\&\quad \cdot \left( v_x^{\mathbf {B}}, v_y^{\mathbf {B}} + \frac{E}{B} \sin \theta \sin (\omega _E t + \phi ), 0 \right) = 0, \end{aligned}$$

(D.18)

$$\begin{aligned}&\sqrt{ \left( \frac{\mathrm{d}}{\mathrm{d}t} v_x^{\mathbf {B}} \right) ^2 + \left( \frac{\mathrm{d}}{\mathrm{d}t} v_y^{\mathbf {B}} \right) ^2} \nonumber \\&\quad = \frac{e B}{m} \sqrt{ \left( v_x^{\mathbf {B}} \right) ^2 + \left( v_y^{\mathbf {B}} + \frac{E}{B} \sin \theta \sin (\omega _E t + \phi ) \right) ^2 }. \nonumber \\ \end{aligned}$$

(D.19)

Equation (D.18), which gives the inner product between two vectors, represents that the instantaneous acceleration acting on an electron having a velocity $\mathbf {v}^{\mathbf {B}} = (v_x^{\mathbf {B}}, v_y^{\mathbf {B}}, v_z^{\mathbf {B}})$ is always perpendicular to its position vector relative to a point $(0, - (E/B) \sin \theta \sin (\omega _E t + \phi ), v_z^{\mathbf {B}})$. This point is on a central axis of rotation C represented as $v_x = 0$ and $v_y = - (E/B) \sin \theta \sin (\omega _E t + \phi )$. C is parallel to the $v_z$-axis, and moves temporally up and down across the origin O of velocity space (see Fig. 5). Equation (D.19) represents that the acceleration is proportional to the distance of $\mathbf {v}^{\mathbf {B}}$ from C and that the angular frequency $\omega _B = e B/m$ of the rotation is constant irrespective of $\mathbf {v}^{\mathbf {B}}$. The acceleration for $v_z^{\mathbf {B}}$ induces a translational shift along the $v_z$-axis (i.e. along C) in velocity space because $\mathrm{d}v_z^{\mathbf {B}}/\mathrm{d}t$ is equal for all electrons.

In consequence, the infinitesimal motion of the electrons in velocity space by the instantaneous acceleration is helical; i.e. a composition of the rotation around C and the parallel shift along C. The shape of the EVDF of an electron swarm subset remains isotropic around its center in this acceleration.

Fig. 5

Schematic of electron acceleration under $\mathbf {E}(t) = (E \sin \theta \sin (\omega _E t + \phi ), 0, E \cos \theta \sin (\omega _E t + \phi ))$ and $\mathbf {B} = (0, 0, B)$ projected to the $v_x^{\mathbf {B}} v_y^{\mathbf {B}}$-plane in velocity space. The instantaneous direction of acceleration $(\mathrm{d}v_x^{\mathbf {B}}/\mathrm{d}t, \mathrm{d}v_y^{\mathbf {B}}/\mathrm{d}t)$ (broken arrows) acting on electrons with velocity $\mathbf {v}^{\mathbf {B}}$ (thick arrows) is rotational around a central axis C: $(v_x^{\mathbf {B}}, v_y^{\mathbf {B}}) = (0, -(E/B) \sin \theta \sin (\omega _E t + \phi ))$ (full circle)

Appendix E: Details on electron collision cross sections of CCF model gas

The CCF model gas used for the Monte Carlo simulations has the following electron collision cross sections [55], where $q_\mathrm{el}$ is for elastic collision, $q_\mathrm{inel,1}$ and $q_\mathrm{inel,2}$ are for inelastic collisions with loss energies of 10 and 15 eV, respectively, and $\varepsilon _\mathrm{1eV} = 1$ eV:

$$\begin{aligned} q_\mathrm{total}(\varepsilon )= & {} q_0/\sqrt{\varepsilon /\varepsilon _\mathrm{1eV}}, \end{aligned}$$

(E.20)

$$\begin{aligned} q_0= & {} 4.0 \times 10^{-19} \text{ m}^2, \end{aligned}$$

(E.21)

$$\begin{aligned} q_\mathrm{inel,1}(\varepsilon )= & {} \left\{ \!\! \begin{array}{llll} 0 &{} \text{ for } \varepsilon \le 10\,\text{ eV } \\ &{} q_1 [1 - \exp (-0.25 \times \varepsilon /\varepsilon _\mathrm{1eV} ) ]^{10} \\ &{} \;\; \times \exp [-0.01712 \times (\varepsilon /\varepsilon _\mathrm{1eV} - 10) ] \\ &{} \text{ for } 10 \text{ eV } < \varepsilon \le 300\,\text{ eV } \\ &{}q_1 \exp [-0.01712 \times (\varepsilon /\varepsilon _\mathrm{1eV} - 10) ] \\ &{}\text{ for } \varepsilon > 300\,\text{ eV } \end{array} \right. \!\!\! , \nonumber \\\end{aligned}$$

(E.22)

$$\begin{aligned} q_1= & {} 1.5 \times 10^{-20} \text{ m}^2, \end{aligned}$$

(E.23)

$$\begin{aligned} q_\mathrm{inel,2}(\varepsilon )= & {} \left\{ \!\! \begin{array}{lllll} 0 &{} \text{ for } \varepsilon \le 15\,\text{ eV } \\ &{} q_2 \{ 1 - \exp [ -(\varepsilon /\varepsilon _\mathrm{1eV})^{0.1} ] \}^6 \\ &{} \;\; \times \exp [ -0.0015 \times (\varepsilon /\varepsilon _\mathrm{1eV} - 15) ] \\ &{} \text{ for } 15 \text{ eV } < \varepsilon \le 500\,\text{ eV } \\ &{} q_2 \exp [ -0.0015 \times (\varepsilon /\varepsilon _\mathrm{1eV} - 15) ] \\ &{} \text{ for }\, \varepsilon > 500\,\text{ eV } \end{array} \right. \!\!\! ,\nonumber \\ \end{aligned}$$

(E.24)

$$\begin{aligned} q_2= & {} 2.0 \times 10^{-20} \text{ m}^2, \end{aligned}$$

(E.25)

$$\begin{aligned} q_\mathrm{el}(\varepsilon )= & {} q_\mathrm{total} (\varepsilon ) - q_\mathrm{inel,1}(\varepsilon ) - q_{\mathrm{inel,2}}(\varepsilon ). \end{aligned}$$

(E.26)

The original $q_\mathrm{inel,2}$ [55] is an ionization cross section, but it is treated as an excitation cross section in this work when the electron-conservative case is examined. In the demonstration of the electron-nonconservative case, $q_\mathrm{inel,2}$ is restored to be the ionization cross section. The residual energy after the production of a secondary electron by a primary electron is divided into two portions to be assigned to the two electrons at a ratio $\delta : (1 - \delta )$ with an energy division ratio $\delta$, and $\delta$ is assumed to be equiprobable in a range $0 \le \delta \le 1$.

Appendix F: Semi-major axis and vertices of elliptic vector locus

The calculation of the vector length $L = |\mathbf {W}(t)|$ can be rearranged into the following form:

$$\begin{aligned} L^2= & {} [W_x(t)]^2 + [W_y(t)]^2 + [W_z(t)]^2 \end{aligned}$$

(F.27)

$$\begin{aligned}= & {} P \cos ^2 \alpha + Q \cos \alpha \sin \alpha + R \sin ^2 \alpha \end{aligned}$$

(F.28)

$$\begin{aligned}= & {} \frac{P + R}{2} + \frac{Q}{2} \sin 2 \alpha + \frac{P - R}{2} \cos 2 \alpha \end{aligned}$$

(F.29)

$$\begin{aligned}= & {} \frac{P + R}{2} + \frac{\sqrt{Q^2 + (P - R)^2}}{2} \sin (2 \alpha + \beta ),\nonumber \\ \end{aligned}$$

(F.30)

where

$$\begin{aligned} P= & {} v_E^2 \omega _E^2 \left[ \frac{\omega _E^2 \sin ^2 \theta }{\varOmega } + \frac{\omega _E^2 \cos ^2 \theta }{\varXi ^2} \right] , \end{aligned}$$

(F.31)

$$\begin{aligned} Q= & {} - 2 v_E^2 \omega _E^2 \left[ \frac{\nu \omega _E \sin ^2 \theta }{\varOmega } + \frac{\nu \omega _E \cos ^2 \theta }{\varXi ^2} \right] , \end{aligned}$$

(F.32)

$$\begin{aligned} R= & {} v_E^2 \omega _E^2 \left[ \frac{(\omega _B^2 + \nu ^2) \sin ^2 \theta }{\varOmega } + \frac{\nu ^2 \cos ^2 \theta }{\varXi ^2} \right] , \end{aligned}$$

(F.33)

$$\begin{aligned} \varOmega= & {} \left[ (\omega _E - \omega _B)^2 + \nu ^2 \right] \left[ (\omega _E + \omega _B)^2 + \nu ^2 \right] , \end{aligned}$$

(F.34)

$$\begin{aligned} \varXi= & {} \omega _E^2 + \nu ^2, \end{aligned}$$

(F.35)

$$\begin{aligned} \alpha= & {} \omega _E t + \phi , \end{aligned}$$

(F.36)

$$\begin{aligned}&(\cos \beta , \sin \beta ) \nonumber \\= & {} \left( \frac{Q}{\sqrt{Q^2 + (P - R)^2}}, \frac{P - R}{\sqrt{Q^2 + (P - R)^2}} \right) . \end{aligned}$$

(F.37)

$L^2$ can be rewritten as

$$\begin{aligned} L^2= & {} \frac{1}{2} v_E^2 \omega _E^2 \left[ \frac{ (\omega _E^2 + \omega _B^2 + \nu ^2) \sin ^2 \theta }{ \varOmega } + \frac{ \cos ^2 \theta }{ \varXi } \right] \nonumber \\&+ \frac{1}{2} v_E^2 \omega _E^2 \sqrt{ \frac{ \sin ^2 \theta }{ \varOmega } - \frac{ \omega _B^4 \sin ^2 \theta \cos ^2 \theta }{ \varOmega \varXi ^2 } + \frac{ \cos ^2 \theta }{ \varXi ^2 } } \nonumber \\&\times \sin (2 \alpha + \beta ). \end{aligned}$$

(F.38)

L becomes its maximum (i.e. the length of the semi-major axis of the elliptic vector locus) when $\sin (2 \alpha + \beta ) = 1$ twice in a domain $0 \le \alpha < 2 \pi$, and $\mathbf {W}(t)$ points the vertices of the ellipse at that time. Such $\mathbf {W}(t)$ is specified by (14)–(16) with the $\sin \alpha$ and $\cos \alpha$ values corresponding to $\alpha = \pi /4 - \beta /2$ and $5 \pi /4 - \beta /2$:

$$\begin{aligned} \sin \alpha= & {} \pm \frac{\sqrt{2}}{2} \cos \frac{\beta }{2} \mp \frac{\sqrt{2}}{2} \sin \frac{\beta }{2} \nonumber \\= & {} \pm \frac{1}{2} \sqrt{1 + \cos \beta } \mp \frac{1}{2} \sqrt{1 - \cos \beta }, \end{aligned}$$

(F.39)

$$\begin{aligned} \cos \alpha= & {} \pm \frac{\sqrt{2}}{2} \cos \frac{\beta }{2} \pm \frac{\sqrt{2}}{2} \sin \frac{\beta }{2} \nonumber \\= & {} \pm \frac{1}{2} \sqrt{1 + \cos \beta } \pm \frac{1}{2} \sqrt{1 - \cos \beta }. \end{aligned}$$

(F.40)

X in (25) is $\cos \beta$ in (F.37) calculated with P, Q, and R in (F.31)–(F.33).