# Novel energy level structure of Dirac oscillator in magnetic field

Md. Moniruzzaman and S. B. Faruque Corresponding Author: Email:
<EMAIL_ADDRESS>Permanent Address: Department of Physics, Mawlana
Bhashani Science and Technology University, Santosh, Tangail-1902, Bangladesh

###### Abstract

We have presented an elegant high energy quantum problem, namely, the full
Dirac oscillator under axial magnetic field with its full solution. We have
found the energy spectrum which is rich and at the same time has a novel
structure. The quantized energy levels show coupling of the oscillator
frequency with the Larmor frequency in the 2D surface where the electrons
under consideration follow a 2D oscillator. The axis in which magnetic field
is pointed, the electrons follow a 1D oscillator. There is also coupling
between spin and orbital motion and also a coupling between a resultant effect
of orbital and spin motion with Larmor precession.

Keywords: (3+1) dimensional Dirac oscillator; Magnetic field; Novel energy
level structure

Department of Physics, Shahjalal University of Science and Technology, Sylhet
3114, Bangladesh

##  1 Introduction

Moshinsky and Szczepaniak [1] have introduced an interesting interaction in
the Dirac equation such that the Dirac Hamiltonian is linear in both momentum
and the coordinates. The resulting Dirac equation is known as Dirac oscillator
because it turns into a harmonic oscillator with strong spin-orbit coupling in
the non-relativistic limit. The oscillator is achieved by introducing the
coupling in the Dirac equation as [1]

$i\hbar\frac{\partial\varPsi}{\partial
t}=\left[c\overrightarrow{\alpha}.\left(\overrightarrow{p}-im\beta\omega\overrightarrow{r}\right)+\beta
mc^{2}\right]\varPsi,$

where $\varPsi$ is the four component bispinor corresponding to spin-half
particle, like electron, $c$ is the speed of light, $\overrightarrow{\alpha}$
and $\mathbf{\beta}$ are standard representation of Dirac matrices, $\omega$
is the classical frequency of the oscillator. When $\omega=0$ , we recover the
ordinary Dirac equation[2].

The Dirac oscillator plays a vital role in the depiction of relativistic many
body problems [3, 4 and 5] and super symmetric relativistic quantum mechanics
[6, 7, and 8]. Dirac oscillator framework is also proposed in quantum
chromodynamics, mainly in connection with quark confinement models in baryons
and mesons [9]. Most thrilling aspects of this oscillator are its association
to quantum optics [10]. It also maps into Anti-Jaynes-Cummings model for
describing the interaction of a atomic transition in two level systems [11,
12, and 13]. Moreover, a lot of attention and many physical applications in
various branches of physics of this oscillator have been found, for example,
in semiconductor physics [14], graphene [15] etc. Recently an experimental
demonstration of this oscillator has been initiated [16].

In this context, we consider (3+1) dimensional Dirac oscillator in an external
homogeneous magnetic field. We investigated mathematical formulation of the
oscillator problem and solved it. The quantized energy spectrum for up and
down spin is derived. In the non-relativistic limit, this energy spectrum
reduces to the spectrum of three dimensional harmonic oscillator with a spin-
orbit contribution, when the magnetic field is absent. We present this problem
and give a solution in section 2 and in section 3, we give a brief conclusion
of the work.

##  2 Dirac oscillator in magnetic field

In presence of an external magnetic field, the (3+1) dimensional Dirac
oscillator is given by

$\left[c\overrightarrow{\alpha}.\left(\overrightarrow{p}+\frac{e}{c}\overrightarrow{A}-im\beta\omega\overrightarrow{r}\right)+\beta
mc^{2}\right]\psi=E\psi,$ (1)

where $\overrightarrow{A}$ is vector potential and $-e(e>0)$ is the charge of
the oscillator which is considered to be an electron here. We consider the
uniform magnetic field to be along the $z$\- direction and vector potential to
be in the symmetric gauge as
$\overrightarrow{A}=\left(-\frac{B}{2}y,\frac{B}{2}x,0\right)$, where $B$ is
the magnetic field strength. Using the following standard form of
$\overrightarrow{\alpha}$ and $\beta$:

$\overrightarrow{\alpha}=\left(\begin{array}[]{cc}0&\overrightarrow{\sigma}\\\
\overrightarrow{\sigma}&0\end{array}\right)$

and

$\beta=\left(\begin{array}[]{cc}I&0\\\ 0&-I\end{array}\right)$ (2)

and two-component form for $\psi$ as

$\psi=\left(\begin{array}[]{c}\psi_{a}\\\ \psi_{b}\end{array}\right),$ (3)

we obtain the following two simultaneous coupled equations:

$\left(c\overrightarrow{\sigma}.\overrightarrow{p}+e\overrightarrow{\sigma}.\overrightarrow{A}\right)\psi_{b}+imc\omega\overrightarrow{\sigma}.\overrightarrow{r}\psi_{b}=\left(E-mc^{2}\right)\psi_{a}$
(4)

and

$\left(c\overrightarrow{\sigma}.\overrightarrow{p}+e\overrightarrow{\sigma}.\overrightarrow{A}\right)\psi_{a}-imc\omega\overrightarrow{\sigma}.\overrightarrow{r}\psi_{a}=\left(E+mc^{2}\right)\psi_{b}.$
(5)

The system gives the following two equations

$[c^{2}\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)^{2}+ce\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)+ce\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)+e^{2}\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)^{2}+imc^{2}\omega\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\right]$
$+imce\omega\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\right]+m^{2}c^{2}\omega^{2}\left(\overrightarrow{\sigma}.\overrightarrow{r}\right)^{2}]\psi_{a}=\left(E^{2}-m^{2}c^{4}\right)\psi_{a}$
(6)

and

$[c^{2}\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)^{2}+ce\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)+ce\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)+e^{2}\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)^{2}-imc^{2}\omega\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\right]$
$-imce\omega\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\right]+m^{2}c^{2}\omega^{2}\left(\overrightarrow{\sigma}.\overrightarrow{r}\right)^{2}]\psi_{b}=\left(E^{2}-m^{2}c^{4}\right)\psi_{b}.$
(7)

One can easily show that

$\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)^{2}=p^{2},$ (8)
$\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)+\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)=BL_{z}+B\hbar\sigma_{z},$
(9) $\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)^{2}=A^{2},$ (10)

$\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{p}\right)\right]=\frac{4i}{\hbar}\overrightarrow{L}.\overrightarrow{S}+3i\hbar,$
(11)

where electron spin
$\overrightarrow{S}=\frac{\hbar}{2}\overrightarrow{\sigma}$,

$\left[\left(\overrightarrow{\sigma}.\overrightarrow{r}\right),\left(\overrightarrow{\sigma}.\overrightarrow{A}\right)\right]=ir^{2}[\overrightarrow{\sigma}.\overrightarrow{B}-\left(\overrightarrow{\sigma}.\hat{r}\right)\left(\overrightarrow{B}.\hat{r}\right)]$
(12)

and

$\left(\overrightarrow{\sigma}.\overrightarrow{r}\right)^{2}=r^{2}.$ (13)

Using these, Eqs. (6) and (7) can be cast into

$\left[c^{2}p^{2}+m^{2}c^{2}\omega^{2}r^{2}+ceB(L_{Z}+\hbar\sigma_{z})+e^{2}A^{2}-\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}-mce\omega
r^{2}[\sigma_{z}B-\left(\overrightarrow{\sigma}.\hat{r}\right)\left(\overrightarrow{B}.\hat{r}\right)]\right]\psi_{a}$
$=\left(E^{2}-m^{2}c^{4}+3mc^{2}\hbar\omega\right)\psi_{a}$ (14)

and

$\left[c^{2}p^{2}+m^{2}c^{2}\omega^{2}r^{2}+ceB(L_{Z}+\hbar\sigma_{z})+e^{2}A^{2}+\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}+mce\omega
r^{2}[\sigma_{z}B-\left(\overrightarrow{\sigma}.\hat{r}\right)\left(\overrightarrow{B}.\hat{r}\right)]\right]\psi_{b}$
$=\left(E^{2}-m^{2}c^{4}-3mc^{2}\hbar\omega\right)\psi_{b}.$ (15)

Since $\overrightarrow{B}$ is along the z-direction, and in magnetic field
spin of an electron precesses around $\overrightarrow{B}$, the effective
component of $\overrightarrow{\sigma}$ is $\sigma_{z}$. Hence, the second term
within the square bracket in the 6th term of the both equations should reduce
as follows:

$\left(\overrightarrow{\sigma}.\hat{r}\right)\left(\overrightarrow{B}.\hat{r}\right)=\sigma_{z}B\cos^{2}\theta,$

where $\theta$ is the azimuthal angle.

Then we have

$\left[c^{2}p^{2}+m^{2}c^{2}\omega^{2}r^{2}+ceB(L_{Z}+\hbar\sigma_{z})+\frac{e^{2}B^{2}}{4}r^{2}\sin^{2}\theta-\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}-mce\omega\sigma_{z}Br^{2}\sin^{2}\theta\right]\psi_{a}$
$=\left(E^{2}-m^{2}c^{4}+3mc^{2}\hbar\omega\right)\psi_{a}$ (16)

and

$\left[c^{2}p^{2}+m^{2}c^{2}\omega^{2}r^{2}+ceB(L_{Z}+\hbar\sigma_{z})+\frac{e^{2}B^{2}}{4}r^{2}\sin^{2}\theta+\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}+mce\omega\sigma_{z}Br^{2}\sin^{2}\theta\right]\psi_{b}$
$=\left(E^{2}-m^{2}c^{4}-3mc^{2}\hbar\omega\right)\psi_{b}.$ (17)

The first two terms of the left hand side of Eqs.(16) and (17) constitute
Hamiltonian of 3 dinensoinal harmonic oscillator. Third term reveals an
interaction between $z$ component of total magnetic moment and the magnetic
field and the fourth term reveals interaction between electric charge and the
magnetic field. The fifth term appears due to spin-orbit interaction and sixth
term due to an interaction between electron spin and magnetic field.

We can find closed form solution of Eqs.(16) and (17) using either $\mid
Y_{l}^{m_{l}}(\theta,\phi)\chi_{m_{s}}>$ basis or $\mid
e^{im_{l}\phi}\chi_{m_{s}}>$ basis, where the quantum number $m_{l}$ is
associated with the eigenfunctions $e^{im_{l}\phi}$ of $L_{z}$ and
$S_{z}\chi_{m_{s}}=m_{s}\hbar\chi_{m_{s}}$, $m_{s}=\pm\frac{1}{2}$. In both of
those bases, the eigen value of $\overrightarrow{L}.\overrightarrow{S}$ is
$\hbar^{2}m_{l}m_{s}$. We prefer now to work with the $\mid
e^{im_{l}\phi}\chi_{m_{s}}>$ basis.

Since the system possesses cylindrical symmetry, in cylindrical coordinates
$\rho$, $\phi$ and $z$ , where $\rho^{2}=x^{2}+y^{2}$, Eqs. (16) and (17) can
be written respectively as

$[-c^{2}\hbar^{2}\left\\{\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{\partial^{2}}{\partial
z^{2}}\right\\}+m^{2}c^{2}\omega^{2}\rho^{2}+m^{2}c^{2}\omega^{2}z^{2}+ceB(L_{Z}+\hbar\sigma_{z})+\frac{e^{2}B^{2}}{4}\rho^{2}-\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}$
$-mce\omega\sigma_{z}B\rho^{2}]\psi_{a}=\left(E^{2}-m^{2}c^{4}+3mc^{2}\hbar\omega\right)\psi_{a}$
(18)

and

$[-c^{2}\hbar^{2}\left\\{\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial}{\partial\rho}\right)+\frac{1}{\rho^{2}}\frac{\partial^{2}}{\partial\phi^{2}}+\frac{\partial^{2}}{\partial
z^{2}}\right\\}+m^{2}c^{2}\omega^{2}\rho^{2}+m^{2}c^{2}\omega^{2}z^{2}+ceB(L_{Z}+\hbar\sigma_{z})+\frac{e^{2}B^{2}}{4}\rho^{2}+\frac{4mc^{2}\omega}{\hbar}\overrightarrow{L}.\overrightarrow{S}$

$+mce\omega\sigma_{z}B\rho^{2}]\psi_{b}=\left(E^{2}-m^{2}c^{4}-3mc^{2}\hbar\omega\right)\psi_{b}.$
(19)

As the equations involve partial derivatives of $\rho$ and $z$ and operators
of z-component of angular momentum and spin, we take the solutions in the form
:

$\psi_{a}=F_{a}(\rho)G_{a}(z)e^{im_{l}\phi}\chi_{m_{s}=-\frac{1}{2}}$ (20)

and

$\psi_{b}=F_{b}(\rho)G_{b}(z)e^{im_{l}\phi}\chi_{m_{s}=\frac{1}{2}}.$ (21)

Then we get from Eq.(18)

$-\frac{1}{F_{a}(\rho)}c^{2}\hbar^{2}\frac{1}{\rho}\frac{\partial}{\partial\rho}\left(\rho\frac{\partial
F_{a}(\rho)}{\partial\rho}\right)+c^{2}\hbar^{2}\frac{m_{l}^{2}}{\rho^{2}}+\left(\frac{B^{2}e^{2}}{4}+mce\omega
B+m^{2}c^{2}\omega^{2}\right)\rho^{2}-c^{2}\hbar^{2}\frac{1}{G_{a}(z)}\frac{\partial^{2}G_{a}(z)}{\partial
z^{2}}+m^{2}c^{2}\omega^{2}z^{2}$ $=E^{2}-m^{2}c^{4}+3mc^{2}\hbar\omega-
ceB(m_{l}-1)\hbar-2mc^{2}\omega m_{l}\hbar.$ (22)

Let

$E^{2}-m^{2}c^{4}+3mc^{2}\hbar\omega-ceB(m_{l}-1)\hbar-2mc^{2}\omega
m_{l}\hbar=\lambda,$ (23)

where $\lambda$ is a constant. Hence, the Eq.(22) demands

$-c^{2}\hbar^{2}\frac{1}{G_{a}(z)}\frac{d^{2}G_{a}(z)}{dz^{2}}+m^{2}c^{2}\omega^{2}z^{2}=\epsilon,$
(24)

where $\epsilon$ is another constant. Then we have

$-c^{2}\hbar^{2}\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{dF_{a}(\rho)}{d\rho}\right)+c^{2}\hbar^{2}\frac{m_{l}^{2}}{\rho^{2}}F_{a}(\rho)+\left(\frac{B^{2}e^{2}}{4}+mce\omega
B+m^{2}c^{2}\omega^{2}\right)\rho^{2}F_{a}(\rho)+(\epsilon-\lambda)F_{a}(\rho)=0.$
(25)

Multiplying Eq.(24) by $\frac{1}{2mc^{2}}$, we have

$-\frac{\hbar^{2}}{2m}\frac{d^{2}G_{a}(z)}{dz^{2}}+\frac{1}{2}m\omega^{2}z^{2}G_{a}(z)=\frac{\epsilon}{2mc^{2}}G_{a}(z)$
(26)

This is the simple harmonic oscillator Schrogringer equation and hence,

$\epsilon=2\left(n+\frac{1}{2}\right)\hbar\omega mc^{2},n=0,1,2...$ (27)

and $G_{a}(z)$ is the wave function of simple harmonic oscillator.

Eq.(25) can be written as

$\frac{d^{2}F_{a}(\rho)}{d\rho^{2}}+\frac{1}{\rho}\frac{dF_{a}(\rho)}{d\rho}-\frac{m_{l}^{2}}{\rho^{2}}F_{a}(\rho)-\left(\frac{B^{2}e^{2}}{4c^{2}\hbar^{2}}+\frac{me\omega
B}{\hbar^{2}c}+\frac{m^{2}\omega^{2}}{\hbar^{2}}\right)\rho^{2}F_{a}(\rho)+\frac{1}{c^{2}\hbar^{2}}(\lambda-\epsilon)F_{a}(\rho)=0$
(28)

Now, performing a change of variable

$\xi=\sqrt[4]{\frac{B^{2}e^{2}}{4c^{2}\hbar^{2}}+\frac{me\omega
B}{\hbar^{2}c}+\frac{m^{2}\omega^{2}}{\hbar^{2}}}\varrho,$ (29)

we obtain from Eq.(28)

$\frac{d^{2}F_{a}(\xi)}{d\xi^{2}}+\frac{1}{\xi}\frac{dF_{a}(\xi)}{d\xi}-\frac{m_{l}^{2}}{\xi^{2}}F_{a}(\xi)-\xi^{2}F_{a}(\xi)+DF_{a}(\xi)=0,$
(30)

where

$D=\frac{\lambda-\epsilon}{c^{2}\hbar^{2}\sqrt{\frac{B^{2}e^{2}}{4c^{2}\hbar^{2}}+\frac{me\omega
B}{\hbar^{2}c}+\frac{m^{2}\omega^{2}}{\hbar^{2}}}}.$ (31)

According to [17], the corresponding solutions are

$D=2(N+1),\begin{array}[]{c}\end{array}N=0,1,2...$ (32)

and

$F_{a}(\xi)=Ae^{-\frac{1}{2}\xi^{2}}f_{m_{l}+k}^{m_{l}}(\xi),$ (33)

where $A$ is a normalization constant and $f_{m_{l}+k}^{m_{l}}(\xi)$ satisfy
the equation

$\xi\frac{d^{2}f_{m_{l}+k}^{m_{l}}(\xi)}{d\xi^{2}}+\left(m_{l}+1-\xi\right)\frac{df_{m_{l}+k}^{m_{l}}(\xi)}{d\xi}+kf_{m_{l}+k}(\xi)=0,$
(34)
$k=\frac{1}{2}\left(N-m_{l}\right)=\begin{cases}0,1,2.........\frac{N}{2},&\begin{array}[]{cccccc}for&N&is&an&even&integer\end{array}\\\
0,1,2.........\frac{N-1}{2},&\begin{array}[]{cccccc}for&N&is&an&odd&intiger\end{array}\end{cases},$
(35)

and $m_{l}$ can take $\left(N/2\right)+1$ values for an even intiger $N$ and
$N+1$ values for an odd intiger $N$.

Equation (32) gives the energy spectrum of the (3+1) dimensional Dirac
oscillator in presence of magnetic field:

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+1\right)\hbar\left(\omega+\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega+\left(m_{l}-\frac{3}{2}\right)\hbar\omega+(m_{l}-1)\hbar\omega_{L}\right]\begin{array}[]{ccccc}&for&m_{s}=-\frac{1}{2}\end{array},$
(36)

where the Larmor frequency $\omega_{L}$ is

$\omega_{L}=\frac{eB}{2mc}.$ (37)

The Larmor frequency occurs in the energy spectrum due to interaction between
electric charge and the magnetic field. The third term in the square bracket
of Eq.(36) appears for spin-orbit coupling and the fourth term for an
interaction between total magnetic moment and the magnetic field.

Now, the other component of $\psi$ is

$\psi_{b}=F_{b}(\rho)G_{b}(z)e^{im_{l}\phi}\chi_{m_{s}=\frac{1}{2}}=F_{b}(\xi)G_{b}(z)e^{im_{l}\phi}\chi_{m_{s}=\frac{1}{2}}=A^{\prime}e^{-\frac{1}{2}\xi^{2}}f_{m_{l}+k}^{m_{l}}(\xi)G_{b}(z)e^{im_{l}\phi}\chi_{m_{s}=\frac{1}{2}}$
(38)

and the corresponding energy spectrum is

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+1\right)\hbar\left(\omega+\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega+\left(m_{l}+\frac{3}{2}\right)\hbar\omega+(m_{l}+1)\hbar\omega_{L}\right]\begin{array}[]{ccccc}&for&m_{s}=\frac{1}{2}\end{array}.$
(39)

For $B=0$

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+n+\frac{3}{2}\right)\hbar\omega+\left(m_{l}-\frac{3}{2}\right)\hbar\omega\right]$
$=2mc^{2}\left[\left(n^{\prime}+\frac{3}{2}\right)\hbar\omega+\left(m_{l}-\frac{3}{2}\right)\hbar\omega\right]\begin{array}[]{ccccc}&for&m_{s}=-\frac{1}{2}\end{array}$
(40)

and

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+n+\frac{3}{2}\right)\hbar\omega+\left(m_{l}+\frac{3}{2}\right)\hbar\omega\right]$
$=2mc^{2}\left[\left(n^{\prime}+\frac{3}{2}\right)\hbar\omega+\left(m_{l}+\frac{3}{2}\right)\hbar\omega\right]\begin{array}[]{ccccc}&for&m_{s}=\frac{1}{2}\end{array},$
(41)

where

$n^{\prime}=N+n=0,1,2....$ (42)

These (Eqs.(40) and (41)) are of the same nature as in spherically symmetric
Dirac oscillator [18]; the difference occurred only because of the cylindrical
symmetry we have assumed from the very beginning of the solution procedure.

The non relativistic limit of the energy spectrum is obtained by setting
$E=mc^{2}+K$ with the consideration $K\ll mc^{2}$ and we get

$K=\left(N+1\right)\hbar\left(\omega+\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega+\left(m_{l}-\frac{3}{2}\right)\hbar\omega+(m_{l}-1)\hbar\omega_{L}\begin{array}[]{ccccc}&for&m_{s}=-\frac{1}{2}\end{array}$
(43)

and

$K=\left(N+1\right)\hbar\left(\omega+\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega+\left(m_{l}+\frac{3}{2}\right)\hbar\omega+(m_{l}+1)\hbar\omega_{L}\begin{array}[]{ccccc}&for&m_{s}=\frac{1}{2}.\end{array}$
(44)

For $B=0$

$K=\left(n^{\prime}+\frac{3}{2}\right)\hbar\omega+\left(m_{l}-\frac{3}{2}\right)\hbar\omega\begin{array}[]{ccccc}&for&m_{s}=-\frac{1}{2}\end{array}$
(45)

and

$K=\left(n^{\prime}+\frac{3}{2}\right)\hbar\omega+\left(m_{l}+\frac{3}{2}\right)\hbar\omega\begin{array}[]{ccccc}&for&m_{s}=\frac{1}{2}.\end{array}$
(46)

Except spin-orbit contribution terms
$\left(m_{l}-\frac{3}{2}\right)\hbar\omega$ in Eq.(45) and
$\left(m_{l}+\frac{3}{2}\right)\hbar\omega$ in Eq.(46), these are exactly the
energy spectrum of 3D non-relativistic harmonic oscillator.

If we take the solutions in the form :

$\psi_{a}=F_{a}(\rho)G_{a}(z)e^{im_{l}\phi}\chi_{m_{s}=\frac{1}{2}}$ (47)

and

$\psi_{b}=F_{b}(\rho)G_{b}(z)e^{im_{{}_{l}}\phi}\chi_{m_{s}=-\frac{1}{2}},$
(48)

the corresponding energy spectrum of $\psi_{a}$ and $\psi_{b}$ are given by
respectively as

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+1\right)\hbar\left(\omega-\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega-\left(m_{l}+\frac{3}{2}\right)\hbar\omega+(m_{l}+1)\hbar\omega_{L}\right]\begin{array}[]{ccccc}&for&m_{s}=\frac{1}{2}\end{array}$
(49)

and

$E^{2}-m^{2}c^{4}=2mc^{2}\left[\left(N+1\right)\hbar\left(\omega-\omega_{L}\right)+(n+\frac{1}{2})\hbar\omega-\left(m_{l}-\frac{3}{2}\right)\hbar\omega+(m_{l}-1)\hbar\omega_{L}\right]\begin{array}[]{ccccc}&for&m_{s}=-\frac{1}{2}.\end{array}$
(50)

##  3 Conclusion

We have solved the (3+1) Dirac oscillator with magnetic field and found the
energy spectrum. The system is cylindrically symmetric: there are basically
two oscillators, one is a 2D oscillator in the $x-y$ plane and the other is a
1D oscillator in the $z$\- direction. The energy spectrum shows this splitting
exactly. Moreover, there appears an oscillation which is basically a Larmor
precession, coupled (i) with the 2D oscillator in $x-y$ plane and (ii) coupled
with a net effect of orbital and spin angular momentum. The 2nd coupling is
shown in the last term in Eqs.(36) and (39) or (49) and (50). There is also
spin-orbit coupling coupled with the Dirac oscillator in the third term in
Eqs.(36) and (39) or (49) and (50). The non-relativistic limit of the energy
spectrum is rightly that of 3D harmonic oscillators plus spin-orbit coupling.
Everything appeared smoothly in our calculation because of the initial
decomposition of the problem by a cylindrical symmetry. In the absence of
magnetic field, the problem shows characteristics of a 3D Dirac oscillator as
can be verified with what is available in literature. The energy spectrum is
novel and rich. This oscillator might find application in quantum optics. The
system can easily be manipulated using photon beams or electromagnetic signal
and made to absorb and emit radiation with energy $\hbar\omega$ or
$\hbar\omega_{L}$ or $\hbar\left(\omega+\omega_{L}\right)$ or
$\hbar\left(\omega-\omega_{L}\right)$. This is clear from the expression of
the energy levels.

In conclusion, a very attractive quantum system has been elucidated which is a
3D Dirac oscillator immersed in axial magnetic field.

## Acknowledgment

We are grateful to the reviewer whose constructive suggestions were greatly
useful in correcting our calculation and in improving the presentation of the
work.

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# The Sun-Earth-Moon Connection:
II–Solar Wind and Lunar Surface Interaction

###### Abstract

Context: In the pursuit of lunar exploration and the investigation of water
presence on the lunar surface, a comprehensive understanding of plasma-surface
interactions is crucial since the regolith’s space weathering can create H2O.
However, the Moon is in the Earth’s magnetotail for nearly 20% of its orbit,
which could affect this water creation on the side facing the Earth if this
condition shields it from the solar wind.

Aims: The objective of this study is to understand how the passage of the Moon
in the Earth’s magnetotail affects the plasma delivery near the lunar surface.
Method: The Particle-In-Cell Electromagnetic (EM) Relativistic Global Model,
known as IAPIC, is employed to kinetically simulate the Solar Wind-
Magnetosphere-Ionosphere-Moon Coupling. Results: The Earth’s magnetotail does
not prevent the influx of solar wind ions and ionospheric ions into the solar
environment; therefore the space weathering of the regolith is not stopped in
these conditions. In addition, the charge separation of solar wind ions and
electrons happens is modeled, leading to electric fields and charging of the
lunar surface that can be validated by observations. Conclusion: The study of
the Sun-Earth-Moon system provides insight into the lunar environment while in
the magnetotail, which is essential to better interpret the results of future
Lunar missions. It also provides insights in the Lunar charging in different
conditions that could affect the human presence on the Moon.

JGR: Space Physics sb gg sh National Institute of Aerospace, 100 Exploration
Way, Hampton, VA 23666 Jet Propulsion Laboratory, California Institute of
Technology, M/S 183–401, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
Chemistry and Dynamics Branch, Science Directorate, NASA Langley Research
Center, Hampton, Virginia, USA Institut d’Astrophysique de Paris, UMR7095,
Sorbonne Université, Paris, France Department of Physics, University of
Alberta, Edmonton, Alberta, Canada, Department of Physics and Astronomy,
University of Calgary, Calgary, Alberta, Canada. Science Systems and
Application Inc, Hampton, Va, USA Suleiman M<EMAIL_ADDRESS>
A fully kinetic simulation of the Sun-Earth-moon system was performed

The Lunar surface charging and the wake dynamics were evaluated

The Earth’s magnetotail does not prevent solar-wind ions and ionospheric ions
from reaching the Lunar environment.

##  1 Introduction

As exploration efforts in the Lunar environment pick up pace, many questions
arise concerning the source of water, its potential movement towards the
poles, and its accessibility for human exploration. Among these questions is
the extent to which the solar wind (SW) contributes to water production by
implanting particles. The Lunar regolith, which comprises a large percentage
the surface of the Moon, is formed by a constant barrage of large and small
meteorites, as well as the unrelenting impact of solar wind. As a result, it
serves as the actual boundary between the Moon and the surrounding
interplanetary magnetic field and plasmas (McKay ., 1991, and references
therein). The investigation of the interaction of solar wind protons with the
Lunar regolith using data from the Lunar Prospector mission found that the
solar wind protons can penetrate up to several tens of centimeters into the
Lunar regolith, causing a significant chemical alteration of the surface
materials (Lucey ., 2006). Multiple investigations have demonstrated that the
$\mathrm{H^{+}}$ ions found in the solar wind have the propensity to react
with the upper layers of the Lunar regolith, as well as the rocks present on
the surface (Taylor, 1982; Halekas ., 2012; Poppe, 2019). When the energetic
ions interact with the crystalline structure, they prompt O atoms to respond,
which can eventually lead to the creation of OH or $\mathrm{H_{2}O}$. Recent
observations from the Lunar surface by Chang’e 5, as reported by Mamo .
(2022); Lin . (2022), indicate that a significant quantity of water is present
in undisturbed regolith. In contrast, the regolith that has been disturbed by
the impact of the lander’s rockets, i.e., the lower layers, was found to be
relatively $\mathrm{H_{2}O}$-depleted. Water molecules are a major component
of the Lunar exosphere, resulting from various processes such as chemical
reactions and the release of water from meteorite impacts. The studying of the
water amount variations in the Lunar exosphere during the Moon’s passage
through the Earth’s magnetotail could help understand the relative importance
of these processes (see e.g. Wang ., 2021). This method assumed that the solar
wind cannot directly produce water during these conditions. However, while the
magnetosphere is a barrier against the direct impact of the solar wind, recent
research has revealed that its behavior is more intricate than previously
believed (Egan ., 2019). Recent measurements have indeed been challenging the
belief that the magnetosphere was protecting against atmospheric escape.
Notably, polar outflow can trigger the escape of significant amounts of
$\mathrm{O^{+}}$ through the poles, comparable to the rate of escape for
unmagnetized planets (Gronoff ., 2020). As a result, it is crucial to revise
our understanding of the Earth’s magnetotail surroundings and its impact on
the Lunar environment. Previous researches have concentrated on the effect of
the passage of the Moon through the Earth’s magnetotail on the Lunar exosphere
(Wilson ., 2006; Sarantos ., 2008), showing that it affects its composition
and impacts the sodium plume.

The solar wind can also induce surface charging on the Lunar surface, which
can be a significant concern for human activities due to the risk of equipment
discharges that may jeopardize their life support and equipment. Earlier
studies (Stubbs ., 2007) have addressed the electrostatic charging of the
Lunar surface and the electrically driven dust transport. According to Poppe .
(2021), surface charging on the Lunar surface is attributed to tail plasma
charging and an additional source generated by currents produced by
micrometeorite impacts, which contribute 15%-40% to the total charging. Bale
(1997) discussed the charge imbalance resulting from the topological
differences that induce charge fields in the solar wind plasma near the Moon.
Kimura Nakagawa (2008) discovered that ions dominate the dayside while
electrons dominate the nightside/wake, causing a considerable potential drop.
Lunar Prospector confirmed the theoretically predicted prevalence of negative
charges on the shaded side of the Moon (Halekas ., 2005). Nonetheless, the
impact of the magnetotail on surface charging is still unclear.

Several techniques have been employed to better understand the Lunar plasma
environment and address these questions. For instance, space probes such as
WIND or NASA/THEMIS-ARTEMIS have been utilized in extensive research (Halekas
., 2011; Poppe ., 2014, 2018, 2021; Fatemi ., 2014; Xu ., 2019; Akay ., 2019;
A. Rasca ., 2021) to probe in-situ plasma parameters. The magnetic field
within the lunar wake is amplified by a factor of 1.4, and this enhancement is
accompanied by the presence of asymmetry. Furthermore, the MHD simulation
allowed for the calculation of the induced electric field resulting from
convection.(e.g. Wang ., 2011; Xie ., 2013). In addition to space probes,
kinetic simulations (e.g. Birch Chapman, 2001) have been extensively utilized
to supplement the understanding of the Lunar plasma environment.

The MHD simulation in Xie . (2013) provided insights into the interaction
between the solar wind and the Moon, with different IMF orientations including
north, radial, and tilted. The simulations were compared with NASA/WIND
observations in the Lunar wake, revealing the significant role of electron
dynamics in the SW-Moon interaction. Moreover, global hybrid simulations have
been employed to model the WIND data and the Lunar wake (e.g. Trávníček .,
2005; Fuqua Haviland ., 2019; Jin Pang, 2020; A. Rasca ., 2021; Omelchenko .,
2021; AP. Rasca ., 2022).

In this paper, we present a novel approach to studying the plasma environment
of the Sun-Earth-Moon system and its connections using the Particle-In-Cell
Electrodynamics Relativistic global Code (Baraka ., 2021; Ben-Jaffel ., 2021)
where the transport of plasma originating from the Earth has been included.
This paper follows the work started in Barakat et al. “The Sun-Earth-Moon
Connection: I–3D Global Kinetic simulation”, referred to here as Paper I.
Paper I concentrated on the simulation of the Earth’s environment and on the
modifications/validation of the model. The present paper concentrates on the
Lunar environment. In the following, Section 2 quickly summarizes the
parameters used in the simulation. Section 3 describes the Lunar plasma
environment and the ion transport from the Earth to the Moon as simulated by
the model. Section 4 discusses the results in light of previous observations,
before the conclusion.

##  2 Initial conditions and Simulation Model: IAPIC

The initial conditions are similar to the one in Paper I. More specific
details on the formulation can be found in (Baraka ., 2021). In order to
observe the variations in plasma parameters, three distinct regions were
selected within the simulation box to investigate how these parameters evolve.
The values presented in Table 1 are expressed in code units, and the
conversion from real-world values to code units is thoroughly documented in
(Baraka ., 2021)

Table 1: The plasma parameters were computed independently for two regions:
the undisturbed solar wind at -24 to -18 $R_{E}$ and the dayside
magnetosphere, and the regions $\pm 4R_{E}$ on either side of the moon were
considered to avoid positions where solar wind particles were eliminated. All
quantities were averaged over the total number of cells considered to account
for charge separation and species accumulation at Y=$1R_{E}$ and Z=$1R_{E}$.

parameters(averaged over $4R_{E})$ | Undisturbed SW | Lunar Dayside($-4R_{E}$) | Lunar Nightside($+4R_{E}$)
---|---|---|---
Alfven velocity-$v_{A}$ | 0.22 | 0.1 | 0.14
sound speed-$c_{S}$ | 0.05 | 0.05 | 0.06
magnetosonic mach number-$M_{ms}$ | 0.92 | 0.62 | 0.5
sonic mach number-$M_{S}$ | 3.84 | 1.35 | 1.33
alfven mach number-$M_{A}$ | 0.95 | 0.70 | 0.54
Plasma ion frequency-$\omega_{pi}$ | 0.02 | 0.02 | 0.03
Plasma electron frequency-$\omega_{pe}$ | 0.2 | 0.26 | 0.30
Plasma ion gyro frequency–$\omega_{ci}$ | 0.01 | 0.01 | 0.01
Plasma electron gyro frequency–$\omega_{ce}$ | 0.97 | 0.65 | 0.75
Ion Intertial Length-$d_{i}$ | 22.21 | 14.70 | 19.15
Electron Intertial Length-$d_{e}$ | 2.23 | 1.92 | 1.66
gyro radius-$r_{i}$ | 0.04 | 0.23 | 0.26
gyro radius-$r_{e}$ | 1.e-3 | 1.e-3 | 1.e-3
Ion Debye Length-$\lambda_{Di}$ | 0.02 | 0.04 | 0.07
Electron Debye Length-$\lambda_{De}$ | 3.e-3 | 3.e-3 | 3.e-3
Ion beta -$\beta_{i}$ | 0.07 | 3.41 | 2.79
Electron beta–$\beta_{e}$ | 0.23 | 0.6 | 0.83
ompi/omci-$\omega_{pi}/\omega_{ci}$ | 2.32 | 5.19 | 3.46
ompe/omce–$\omega_{pe}/\omega_{ce}$ | 0.23 | 0.40 | 0.40
$v_{thi}$/$v_{sw}$ | 3.5e-3 | 1.3e-3 | 1.8e-3
$v_{the}$/$v_{sw}$ | 6.2.e-3 | 7.1e-3 | 8.2e-3

##  3 Simulation Results

During the full moon phase, the solar wind impinging on the far magnetotail
impacts the lunar surface in various ways, including its electric charging.
The present study focuses on the accumulation of solar wind densities, which
are considered absorbed by the lunar surface by typical models. Not following
that hypothesis leads to the decoupling of ions and electrons, resulting in
charge separation and therefore in electric currents.

More precisely, the dayside of the lunar surface becomes positively charged,
while the nightside becomes negatively charged, as illustrated in Figure (1).
The UV radiation would increase the ionization rate on the dayside but are not
taken into account in the present study and will deserve further
investigation.

As depicted by the green arrows in Figure 1, anti-corrections of solar wind
densities occur in three distinct regions: the foreshock, in the dayside
magnetosphere, and on both the lit and dark sides of the moon. Figure 1 (b)
and (c) panels provide a close-up look at the ions and electrons drifting
towards the day and night sides of the moon (centered for clarity at zero),
respectively.

Figure 2 shows the density when the moon is not positioned at 60$R_{E}$ along
the Sun-Earth line. The figure depicts ion densities (a) and electron
densities (b) of the solar wind at $\pm 2R_{L}$ in both dawn and dusk
directions. The asymmetries between dawn and dusk regions in the deep
magnetotail can be highlighted by comparing the two figures. If the moon is
absent, the average densities at both dawn and dusk locations should be
similar to those at the lunar location. It is important to mention that the
total count of solar wind components is determined at the lunar surface within
a spherical shell with a thickness of $0.1R_{M}$, approximately 174 km, on
both sides of the lunar surface.

The results of the simulation indicate that the number of ions on the dayside
of the moon outweighs that of electrons, whereas the opposite is true on the
night side of the moon. Table 2 presents an estimate of the overall count of
ions and electrons, providing a comparison of the respective charges on each
side of the moon. Previous research by (Deca ., 2015) suggested that the Lunar
Magnetic Anomalies, mini-magnetospheres, could exacerbate these charge
separations. However, this topic is beyond the scope of the current study. The
phenomenon of lunar surface charging has been widely documented and discussed
within the scientific community.(Stubbs ., 2007; Fatemi ., 2014; Poppe .,
2014; Halekas ., 2014; Lue ., 2018; Poppe ., 2021).

Due to insufficient time for plasma to refill the wake with thermal
velocities, typical lunar wakes often develop in the supersonic background
plasma of the solar wind or remote magnetosheath. THEMIS observations have
revealed that the field-aligned plasma flow within the lunar wake is highly
organized when it is subsonic, particularly in cases where the plasma beta is
low (Xu ., 2019). The results indicate a preference for electrons to enter the
wake from the dawn side of the Moon, while ions tend to refill the wake from
the dusk side. The computed charging percentages shown in Table 2 are
consistent with the charging percentage reported in (Poppe ., 2021).

The densities (Figure 3) and temperatures (Figure 4) of ions and electrons at
various distances from the lunar surface allow to obtain a comprehensive
understanding of the plasma parameters within the lunar wake. Specifically,
these values were extracted at the lunar surface, the terminator, and at
distances of 3, 5, 7, and 9 $R_{L}$ inside the wake. The wake had a width of
$\pm 6R_{L}$, extending equally on both sides of the dawn-dusk directions.

Figure 3-a highlights the tracking of ion densities along the OX direction,
starting from the lunar surface and extending up to 9 $R_{L}$. The assumption
of the plasma accumulating on the lunar surface causes the front flow of the
plasma to shift towards a less resistive direction in both the dawn and dusk
directions. Specifically, at the subsolar lunar surface, the plasma flow is
halted. At the terminator, the ion densities are significantly higher in the
dawn direction compared to the dusk direction.

Inside the lunar wake, at a distance of 3$R_{L}$, that plasma is slipping into
the wake more prominently in the dusk direction, with a ratio of $3:2$
compared to the dawn side. The average density of this slipping plasma is
approximately 100 ions cm-3. Moving closer to the lunar surface, at 2$R_{L}$
there is an increase in ion filling, which is drifting towards the dusk
direction. The rough density at this point is around 20 ion cm-3. However, as
we reach 7 $R_{L}$, the density begins to fade away, and the number of ions
reduces to approximately 6 ion cm-3. This value remains constant until 9
$R_{L}$.

In Figure 3-b, an interesting observation is made regarding the number of
electrons at the lunar surface’s subsolar point: their count is close to that
of ions. However, away from the terminator, a significant change in the ratio
occurs, reaching an order of magnitude difference, with electrons outnumbering
ions by a factor of 10.

Furthermore, it is noticed that electrons tend to slip inside the lunar wake,
showing a preference for the dawn side boundaries of the wake. The average
number of electrons decreases from 300 to 5 electrons cm-3 from 3$R_{L}$ to 9
$R_{L}$ within the lunar wake. The obtained results align with the analytical
and hybrid-kinetic model simulations, as well as observations from the ARTEMIS
mission, as reported in Gharaee . (2015)

In Figure 4 (a) and (b), the temperatures of solar wind ions and electrons
inside the lunar wake are depicted. A significant heating effect is observed,
beginning at a distance of 7 $R_{L}$ and continuing up to 9$R_{L}$ The
temperature remains elevated within this range.

Near the lunar surface, in Figure 5, the magnetic field components
($B_{x},B_{y},B_{z},andB_{tot}$) along the ion densities are analyzed at two
specific locations: the terminator (depicted in blue) and 3$R_{L}$ inside the
lunar wake (depicted in red). Ion densities are also shown at the lunar
surface (in green).

A minor discrepancy in the magnetic field components can be seen in Figure
5-(b), with the exception of $B_{y}$. This could be attributed to the
significant disparity in the electric field at a distance of 3 times the lunar
radius (3$R_{L}$, since the $B_{y}$ component at this distance exhibits
symmetry both at dawn and dusk. It is noteworthy that this component exhibits
a change in polarity inside the lunar wake, indicating a distinct magnetic
field behavior within that region. The Magnetic field values at 3$R_{L}$
[$B_{x},B_{y},B_{z},andB_{tot}$] reads [2,-0.2, 5,5]nT. Looking at the solar
wind ion density in Figure 5-(e), we observe an average density of $\approx$
10 ions cm-3. However, comparing the density at the lunar surface to the
density at the terminator, there is a noticeable flare-out of the plasma
towards the anti-sunward direction.

The magnetic field (a-d), electric fields (e-f), and solar wind velocity (i-m)
on the moon’s surfaces along three parallel planes: OX ($\pm 5R_{E}$), and at
2$R_{L}$ in both the dawn and dusk directions are also of interest to
understand the effect of the Moon. Regarding the X-components in Figure 6 for
magnetic field (B), electric field (E), and velocity (V), B is enhanced by a
factor of 2 on the dayside, indicating bipolarity of E(between -3$R_{E}$ and
-2$R_{E}$) followed by a polarity change from 0.2 to -0.2 at -1.5 $R_{E}$,
while V decreases until it reaches a stagnant state at the lunar surface. All
dawn-dusk components exhibit asymmetry, therefore showing a model prediction
that could be tested.

In terms of the Y-components, B shows enhancement, E remains nearly constant,
and duskward velocity decreases until stagnation. The asymmetry of solar wind
parameters is more pronounced on the dayside compared to the nightside.

As for the Z-components, B increases and exhibits almost symmetric behavior on
the dayside, while E shows asymmetry and polarity reversal between –4$R_{E}$
and -1$R_{E}$, and V decreases from south to north until it reaches a stagnant
point at the moon’s position.

Concerning the overall plasma parameters, B is enhanced on both sides of the
lunar surface, while a decrease is observed on the dusk side and constancy
along the dawn side within the lunar wake. There is a significant asymmetry
between dusk and dawn for E and V, while plasma velocity increases inside the
lunar wake from 0-150 km/s.

Furthermore, Figure 7 illustrates two contour plots depicting the total
magnetic field and total velocity in the equatorial plane, revealing the
magnetic field enhancement and the corresponding solar wind ion velocity. In
Figure 7-a, the magnetic field enhancement is displayed on both the dayside
and nightside of the lunar surface, with an average magnetic value reaching a
maximum of 6nT (increased by a factor of 1.2 i.e. (Wang ., 2011)). On the
other hand, Figure 7-b shows velocity stagnation at the lunar surface ranging
from 300 km/s to 0 km/s on the dayside, while inside the lunar wake in the
equatorial plane, there is an increase of approximately 150 km/s.

In Kimura Nakagawa (2008), two-dimensional electromagnetic full particle code
simulations were employed to successfully simulate the structure of the
electric field near the lunar surface. Here, a three-dimensional
electromagnetic relativistic global code (Baraka ., 2021; Ben-Jaffel ., 2021)
was used, by incorporating the formula in (Kimura Nakagawa, 2008) i.e.,
$\mathrm{E_{0}=m_{e}v_{e}\omega_{pe}/q_{e}}$, where $m_{e}$, is the electron
mass, $v_{e}$, is the electron velocity, $\omega_{pe}$, the electron plasma
frequency. and $q_{e}$, is the electron charge, to account for the background
electric field.

As depicted in Figure 1, and 2, the phenomenon of charge separation gives rise
to a potential difference, which we determined as an induced electric field
$E_{0}$. This electric field was utilized to standardize the solar wind
electric field along both surfaces of the moon, specifically within the range
of $\pm 5R_{E}$ Enlarging the moon’s size by a factor of 5, equivalent to
about 1.$R_{E}$, does not impact the physical microscale of the process, which
remains smaller than our grid size as shown in Figure 8.

In Figure 8-A, the total Electric Field (E.F.) is plotted along three planes
along OX, namely at $Y=0$, $Y=1RL$(dawn) and at $Y=-1RL$(dusk) directions. The
blue line represents the bulk E.F. The red line represents the background E.F.
generated solely due to the charge separation. In order to account for an
effective electric field, it is important to include the induced electric
field ($E_{0}$) as a crucial component in our kinetic modeling. It results in
a bulk E.F. of 2.6$E_{0}$ at $Y=0$, 2.1$E_{0}$ at $Y=+1R_{L}$, and 2.6$E_{0}$
at $Y=-1R_{L}$, particularly at the terminator ($Y=0$). These results are
consistent with those reported in Kimura Nakagawa (2008) and the references
therein.

The electric field (Figure 8-B) in the present study has been normalized to
the induced E.F. that results from the charge separation of both surfaces of
the moon. Additionally, correlation can be performed between the total E.F.
and the induced electric field: it reveals anti-correlations at $Y=0$ with a
correlation coefficient of C.C.=-0.06. At $Y=+1R_{L}$, there is a positive
correlation coefficient of C.C.=0.81. In contrast, at $Y=-1R_{L}$, there is a
weaker positive correlation with a correlation coefficient of C.C.=0.22.

These findings have a remarkable consistency with Kimura Nakagawa (2008),
indicating a complete agreement when there is a drop of the potential inside
the lunar wake at around 2$\mathrm{R_{E}}$ as in Figure 8b. Furthermore, the
potential drop caused by surface charging is more pronounced in terms of the
potential drop associated with the lunar wake.

Finally, the utilization of 3D simulation allows to obtain the values of
$E_{x},E_{y},E_{z},andE_{tot}$ enabling a comprehensive three-dimensional
analysis of the magnetic field enhancement on both sides of the lunar surface.
This enhancement is clearly illustrated in the left and middle panels of
Figure 6.

Table 2:  An estimation is made of the total number of positive and negative charges confined within a spherical shell with a thickness of 0.1$R_{m}$, approximately 174 km above the lunar surface. The density per unit volume is then calculated for both sides of the lunar surface.  Charge sign | Lunar dayside | Lunar nightside |  $\pm$ charge ratio
---|---|---|---
Total # of positive charges(ions) | 4.83$\times$104 | 2.74$\times$104 | 171%
Total # of negative charges(electrons) | 1.99$\times$104 | 7.04$\times$104 | 39.4%
Relative charge | 2.84$\times$104 | -4.30$\times$104 | 66%
Number density per unit volume on lunar surface
Number density | Lunar dayside | Lunar nightside | average density
$\rho_{i}$=$N_{i}$/V | 58$cc^{-1}$ | 33$cc^{-1}$ |  $\overline{\rho_{i}}$=45.5$cc^{-1}$
$\rho_{e}$=$N_{e}$/V | 34$cc^{-1}$ | 85$cc^{-1}$ | $\overline{\rho_{e}}=\textbf{59.5}cc^{-1}$

These simulations reveal a novel point regarding backstreaming ions at $\pm
4R_{E}$ on both the lit and dark sides of the Moon. The present
characterization of the backstreaming ions is consistent with previous studies
(Bonifazi Moreno, 1981; Baraka ., 2021). However, here, the backstreaming ions
are not primarily reflected ions but instead diffuse ions with a small
percentage of intermediate ions. To prevent any confusion, the reflection
presented here is from the nightside of the Moon to its dayside; this is not a
reflection from the surface of the Moon which has be demonstrated to be
negligible (Holmström ., 2012). Figure (10 -a) presents the co-plotted ion
perpendicular temperature ($Ti_{\bot}$) and ion parallel temperature
($Ti_{\parallel}$) to study the temperature anisotropy in the near Moon during
the full Moon phase, in the range of $\pm 5R_{E}$. Similarly, Figure 10 -b
presents the electron perpendicular temperature ($Te_{\bot}$) and electron
parallel temperature ($Te_{\parallel}$) to study the temperature anisotropy.
Previous studies, such as (Gary Karimabadi, 2006; Chandran ., 2011; Treumann
Baumjohann, 2013; Karimabadi ., 2014; Gingell ., 2015), have also reported
temperature anisotropy in the Moon’s vicinity. We find that
$Ti_{\bot}$/$Ti_{\parallel}$ and $Te_{\bot}$/$Te_{\parallel}$ equal 9.2 and 7,
respectively. The electron correlation coefficient is 0.9, and the ion
correlation coefficient is 0.75. It has been reported in studies such as
Samsonov . (2012); Grygorov . (2017) that the backstreaming ions (Figure 9)
can affect the temperature anisotropy configuration.

Table 3 presents the bulk speed of the backstreaming ions, along with their
corresponding thermal velocities and the solar wind kinetic inflow. Notably,
the thermal velocity of the backscattered ions on both sides of the Moon is
higher than the bulk flow and the solar wind inflow speed. Additionally, the
speeds are slower at the day side of the Moon than at night.

Table 3: The values of the solar wind ions for inflow, thermal, and bulk backstreaming speeds are listed for both the dayside lunar surface and inside the lunar wake.  Speed |  $V_{SW}$-Lunar dayside |  $V_{SW}$-Lunar nightside | Night/day $V_{SW}$ ratio
---|---|---|---
$V_{backstreaming_{(bulk)}}$ | 58.6$~{}~{}km.sec^{-1}$ | 136.4$~{}~{}km.sec^{-1}$ | 2.3
$V_{backstreaming_{(thermal)}}$ | 99$~{}~{}km.sec^{-1}$ | 186$~{}~{}km.sec^{-1}$ | 1.9
$V_{SW_{(inflow)}}$ | 80.6$~{}~{}km.sec^{-1}$ | 110$~{}~{}km.sec^{-1}$ | 1.4

#### In Summary:

The charge separation on the lunar surface gives rise to an induced electric
field, which is closely associated with ions backstreaming. This backstreaming
phenomenon, in turn, is linked to the temperature anisotropy observed in the
lunar environment. Figure 11 depicts the findings related to the connection
between the Sun and Earth, the coupling between the magnetosphere and
ionosphere, as well as the interaction between the solar wind and the Moon.
This sketch offers a visual depiction of the findings from the study and
illustrates their interconnections.

##  4 Discussion on observational implications

This discussion focuses on the Lunar surface charging and on the plasma
parameters within the Lunar wake when the Moon is positioned solely along the
Sun-Earth line, i.e., during a Lunar eclipse and inside the extended
magnetotail of the Earth.

###  4.1 Lunar Surface Charging

Contrarily to previous studies such as Poppe . (2014), the present self-
consistent simulation did not remove or absorb solar wind particles upon
collision with the lunar surface. These particles are instead allowed to
accumulate and build up on the lunar unmagnetized barrier. By doing so, we
observed charge separation(see Figure 1), which led to a dominance of
positively charged particles on the dayside of the Moon and a prevalence of
dominance of negatively charged particles on the nightside. One should be
careful with this approach, because measurements show that protons are
absorbed by the lunar surface (Holmström ., 2012), which is what partially
leads to the creation of water with the regolith, future work will improve on
this interaction.

This approach aimed to account for the total number of charged particles on
both Lunar surfaces, as demonstrated in Table 2. Typically, the Debye length
determines the distance at which charges are separated or screened. In our
case, the charges of piled-up solar wind plasma considered at step 3700
$\Delta t$ were shielded from the influence of distant particles closer to
them. As a result, the total Debye length in the piled-up plasma is given by

$\lambda_{D_{l}}=\sqrt{\dfrac{K_{B}}{4\pi}\Sigma_{l}\dfrac{T_{l}}{q_{l}^{2}n_{l}}}$
(1)

as reported in Verscharen . (2019, Eq. 120).

The charging of the Lunar surface in this complex and non-uniform environment
can be attributed to various complex current systems, including photoemission
of electrons, plasma electrons, plasma ions, and secondary electrons resulting
from surface ionization. (and references therein Halekas ., 2005; Stubbs .,
2007; Collier ., 2014; Verscharen ., 2019; Rakesh Chandran ., 2022). The
amount of charging depends on several factors, including the density of the
species present ($n_{0}$), as well as the temperatures of the ions and
electrons ($T_{i},T_{e}$, respectively), and the bulk flow velocity of the
solar wind ($v_{i,e}$), as discussed in Stubbs . (2014).

The present simulation results, obtained using the PIC code, can be compared
with other simulations and observations conducted at local and global scales
((Halekas ., 2011); (Poppe ., 2014); (Deca ., 2015)). Since the present study
aims to explore the charging of the lunar surface by analyzing the
accumulation of solar wind particles on both sides of the Moon, the density at
which the correlation between ions and electrons breaks up can be used to
estimate the total number of charges on both lunar surfaces Figures 1 and 2.
This breaking of correlations between ions and electrons densities indeed
leads to charge separations due to topological differences((Bale, 1997). The
total number of positive and negative charges in a spherical shell with a
thickness equivalent to 0.1 $R_{L}$ ($\approx$175 km) above the lunar surface
has been computed (Table 2). This suggests that the dayside of the Moon is
positively charged, in agreement with Kimura Nakagawa (2008), while the
nightside of the Moon is negatively charged (Stubbs ., 2014). These results
are also comparable with measurements (Halekas ., 2002). It is worth noting
that the linear density at both the dawn and dusk positions in Figure 2
provides an estimate of the density measured along the OX direction of the
Moon when it is not in its current position.

Despite previous studies and references indicating a predominance of negative
charge in the lunar nightside which could implies that no positive ions are
transported to it, the SELENE mission (Nishino ., 2010) demonstrated the entry
of ions in the lunar wake. The present simulation aligns with these
observation: Table 9 shows that approximately 39% of ions detected within the
lunar wake are solar wind particle. This result highlights the versatile and
inclusive nature of the kinetic simulation of this complex interaction. The
potential differences between the two surfaces of the Moon cause a charge
separation that leads to the creation of an induced electric field. To
determine the background electric field, we utilized the formula outlined in
Kimura Nakagawa (2008), which is given by $E_{0}=m_{e}v_{e}\omega_{pe}/q_{e}$.

As depicted in Figure 8, the electric field intensity along the subsolar Moon
is normalized to the background electric field and found to be $E=2.6E_{0}$ .
Similarly, at dawn and dusk (terminator) directions, the values are $2.1E_{0}$
and $2.6E_{0}$, respectively. These results align with those reported in
Kimura Nakagawa (2008), which indicate a measurement of $2.2E_{0}$ at the
terminator.

This result should be put in perspective with other magnetospheric
environments: in planets such as Jupiter, the magnetotail current system is
influenced by the planetary magnetic field. However, in the cases of Venus and
Mars, the magnetotail current system is primarily formed by the solar wind’s
magnetic field. The Earth’s magnetotail, on the other hand, can be considered
an intermediate case, where the specific configuration depends on the
geomagnetic conditions and the dynamics of the magnetotail itself. It is a
region connected to the Earth and is populated by hot, rarefied plasma. This
is a complex intermediate case study (i.e. Xu2018). Therefore, to accurately
account for the effective electric field near the lunar surface, it is
essential to employ kinetic modeling, which considers the backstreaming ions
and effectively captures the induced electric fields resulting from charge
separation. The present results are therefore one of the rare examples of the
whole consistent simulation of the system.

###  4.2 Plasma Parameters Within the Lunar Wake

Figure 5 presents the magnetic field components, total magnetic field, and
density of solar wind ions observed at the Lunar terminator and $3R_{E}$
inside the Lunar wake. This figure is evaluated against Poppe . (2014, Fig.
4), where the two events captured by ARTEMIS are compared with a model. The
present simulation results are shown for a single time step (3700$\Delta
t\approx 40min$ ), whereas, in Figure 4 of Poppe . (2014), the data was
captured over 50 minutes. The magnetic field components inside the lunar wake
at coordinates [63,0,0]$\mathrm{R_{E}}$ are computed to be
[bx,by,bz]=[2.30,-0.07,4.80]nT. A comparison with Table 1 of Poppe . (2014) is
unfortunately limited to one specific event due to the size of the present
simulation box. The March 2011 events has the spacecraft’s location within the
simulated region. The model at coordinates [56.9, 19.8, 3.1]$\mathrm{R_{E}}$,
gives a magnetic field of [-2.86,-0.87,2.17]nT to be compared with the
measured value of [-1.5, 0, -1]nT (Poppe ., 2014). The modeled values are
almost double the observation; the reason for this discrepancy is related to
the different solar wind speed, and initial IMF values. A more detailed study
of this exact observation would involve running the model with the exact
parameters of that day and is out of the scope of the present study. The Lunar
Prospector data show an increase of the magnetic field at the center of the
wake followed by a decrease near its boundaries(Akimov Dyatel, 2012; Poppe .,
2014). These results do not compare directly to our simulations since it takes
only a phase of the Moon and do not consider the present alignment. However,
Figure 6 shows such an increased magnetic field near the center of the wake,
at a total value of 5.5 nT near the center while it goes to 4 nT four Moon
radius’s away during the full moon phase. Figure 6-a,e,&i presents an increase
in the magnetic field on the dayside Lunar surface (depicted in red) along the
X-components of the plasma parameters. This increase corresponds to bipolarity
(normalized to induced $E_{0}=m_{e}v_{e}\omega_{pi}/q_{e}$) in the electric
field, which is a direct consequence of charge separation and induced
potential, as well as the stagnated solar wind velocity.

The remaining panels in Figure 6 demonstrate the variations in the magnetic
and electric fields, as well as solar wind velocities, across the Y, Z, and
total components. The Moon is situated at a distance of $60R_{E}$. Magnetic
field measurements are presented in units of nT, while velocities are given in
$km/s$, The electric field values are normalized based on the induced
background field that arises due to charge separations.

Higher solar wind ion velocity/temperature is observed at $\pm$5 RE as shown
in Table 3. This suggest that external forces, energetic inputs, or non-
equilibrium conditions increase these velocities (i.e., Futaana ., 2012). It
is to be noted that the solar wind not only provides electrons and ions but
also 20% of energetic neutral atoms (Vorburger ., 2016) whose effects are out
of the scope of the present study. In order to address the asymmetry between
dawn(blue) and dusk(green), the correlation coefficients were computed for the
magnetic and electric fields, as well as solar wind velocities, along the
dawn-dusk directions. The analysis revealed that the correlation coefficients
for $B_{x}$ and $B_{y}$ were negative at both dawn and dusk, with values of
-0.76 and -0.11, respectively. In contrast, the correlation coefficient for
$B_{z}$ C.C. is 0.43 was positive at both dawn and dusk, with a value of 0.43.
Furthermore, the total magnetic field was found to be positively correlated at
both dawn and dusk, with a coefficient of 0.44.

It is noteworthy that the significant disturbances observed in the magnetic
field components near the surface of the Moon are of crucial importance since
they are directly associated with the interaction between the lunar surface
and the solar wind. This can aid in distinguishing the effects of various
factors, such as photoelectron radiation and lunar magnetic anomalies, as
reported in previous studies (Zhang . (2020); Deca . (2015)).

The correlation coefficients for the electric field components were calculated
as follows: $E_{x}$, $E_{z}$, and $E_{tot}$ exhibited negative correlation,
with coefficients of -0.06, -0.78, and -0.12, respectively. However, the
$E_{y}$ component showed positive correlation, with a coefficient of 0.7. In
addition, the velocities measured at both dawn and dusk were found to be
positively correlated, with correlation coefficients of
[$v_{x},v_{y},v_{z},v_{tot}$] equal to 0.3, 0.82, 0.57, and 0.09,
respectively.

Apart from highlighting the dawn-dusk asymmetry in the region surrounding the
Moon, the dawn (green) and dusk (blue) magnetic and electric fields, as well
as velocities, can also serve as a reference level for investigating plasma
parameters in the magnetotail, assuming that the Moon is absent from its
current location. Thus, these dawn-dusk asymmetric parameters can be employed
to analyze the intricate current system in the magnetotail, up to a distance
of 65 $R_{E}$.

To better visualize the magnetic field enhancement on the lunar dayside and
within the wake, we have presented 2D contours in equatorial planes taken at
$\pm 5R_{E}$ in both the day/night and dawn-dusk directions of the Moon. The
Moon is centered at zero as shown in Figure 7. In Figure 7-a, it is evident
that the magnetic field is amplified on the illuminated side of the Moon, and
its value can be calculated using the formula $\Delta
B=(B_{moon}-B_{sw}/B_{sw}$) i.e. (Liuzzo ., 2021). This formula yields a value
of ($>$1.) for the enhancement factor.

Figure 8-A shows the total Electric Field (E.F.) plotted along three planes,
OX (at Y=0, Y=$+1R_{L}$(dawn), and at Y=$-1R_{L}$(dusk) directions. The bulk
E.F. is shown in blue, while the background E.F. generated only due to charge
separations is shown in red. The E.F. is considered at $\pm 5R_{E}$ at both
the day and night sides of the lunar surface. It is demonstrated in Figure 8
that the charge separation takes place at the lunar surface. The charge
separation results in the induced additional E.F. ($E_{0}$), which is equal to
$E_{0}=m_{e}v_{e}\omega_{pe}/q_{0}$. To consider the effective E.F.,$E_{0}$
should not be ignored, which can only be obtained by kinetic modeling. It is
shown that the bulk E.F.=2.6$E_{0}$ at Y=0, 2.1 $E_{0}$ at $Y=+1R_{L}$, and
2.6$E_{0}$ at $Y=-1R_{L}$ averaged over in the vicinity of the lunar surface,
especially at the terminator $Y=0$. These results are consistent with those
reported in Kimura Nakagawa (and references therein 2008) . As seen in Figure
1, positive charges dominate the lunar dayside, and the nightside is dominated
by negative charges, which also agrees with (Kimura Nakagawa, 2008). It is
noteworthy that the Correlation Coefficients (C.C.) between these two field
components are as follows: $C.C.=0.71$ along $Y=0$, 0.83 at $+R_{L}$, and 0.54
at $Y=-1R_{L}$, respectively

Backstreaming ions are present at the lunar surface and within the Lunar wake,
according to Bamford . (2012). This study found that the Electric Field (EF)
deflects the incoming solar wind ions and that Chandrayaan-1 has observed
backstreaming ions. In Figure 8, we present the effective electric field at
both lunar surfaces, which may be responsible for deflecting the incoming
solar wind.

Additionally, it has been reported that the solar wind can be absorbed by the
lunar surface, backscattered, or cause atoms to be removed from the Lunar
regolith by sputtering or desorption (Dandouras ., 2023). Some studies have
reported that a large percentage of backscattered solar wind ions can result
in the formation of Energetic Neutral Atoms (ENA), which have been observed by
the Interstellar Boundary Explorer (IBEX) (Allegrini ., 2013). Our
Electromagnetic PIC Model does not account for ENA emissions, so the
information presented in Figure 9 only pertains to charged solar wind ions.

It should be highlighted that the backstreaming ions were not only observed in
our simulations, but we were also able to analyze and categorize them based on
the ratio between the thermal speed and bulk speed of the solar wind.
Specifically, the backstreaming ions observed at both lunar surfaces were
identified as being diffuse rather than reflective, which aligns with the
criteria previously reported by Baraka . (2021)

Our simulation resulted in a solar wind bulk flow comprising roughly 23% of
backstreaming ions; the analysis accounted for the kinetic effects of these
ions. Figure 9 showcases the identification of backstreaming ions in the
vicinity of the Moon, which are attributed to temperature anisotropy, as
depicted in Figure 10. In brief, the present simulation allowed the solar wind
to accumulate on the lunar surface. This resulted in charge separations that
generated a potential difference and induced an additional electric field
component. In the absence of the lunar magnetosphere, the electric field
deflected the incoming solar wind, leading to backstreaming ions at the lunar
surface, which was associated with temperature anisotropy. This approach
provides complementary information to potential future microscale studies of
the Moon that investigate the influence of Lunar surface roughness on volatile
sources and sinks, abundance, and evolution in the Lunar environment (as
discussed in Davidsson Hosseini (2021) and Grumpe . (2019)), as well as
photoelectric emission (Mishra Bhardwaj, 2020).

##  5 Conclusion and Future Work

In these two studies, we have conducted a comprehensive analysis of the solar
wind interaction with the lunar surface, taking into account complex current
systems originating from various sources. This research presents the first-
ever kinetic simulation of the Sun-Earth-Moon system, focusing specifically on
a single scenario where the Moon is situated within Earth’s magnetotail.

Our findings reveal that the dayside of the lunar surface is predominantly
influenced by positively charged solar wind ions, while the night side
experiences a higher concentration of negatively charged electrons, arising
from charge separation. This relationship is clearly illustrated in the flow
chart provided in Figure 11.

Importantly, our results, more detailed in Paper I, demonstrate that, under
these conditions, Earth’s magnetosphere does not protect the Moon from the
solar wind, suggesting that the potential water creation process on the lunar
surface, driven by the solar wind, is not hindered by Earth’s magnetospheric
presence.

Moreover, our investigation has enabled us to explore numerous plasma
parameters within the lunar wake. These include plasma ion and electron
densities (Figure 1), plasma ion and electron temperatures (Figure 4), the
Interplanetary Magnetic Field (IMF) at the terminator and within the wake
(Figure 5), backstreaming ion characteristics (Figure 9), solar wind
temperature anisotropy (Figure 10), and the effective electric field
accounting for the induced electric field resulting from charge separation
(Figure 8). These critical aspects of kinetic simulations offer valuable
insights into the environmental conditions surrounding the Moon.

Future work will involve placing the Moon in various locations, including
those outside of Earth’s magnetotail, examining the effects of different solar
activity levels, and improving the modeling of Earth’s polar escape to better
understand the transport of oxygen ions to the Moon in these conditions. A
more in-depth understanding of the lunar plasma environment will significantly
benefit upcoming missions, particularly those enabled by the Artemis program
and the Moon to Mars initiative (Dandouras ., 2023).

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##  6 Acknowledgment

This research was carried out at the National Institute of Aerospace, Science
Sytems and Applications Inc., Langley Research Center, and partially at the
Jet Propulsion Laboratory, California Institute of Technology, under a
contract with the National Aeronautics and Space Administration Contract
Number 80NM0018D0004. We express our gratitude for the valuable code
contribution provided by Lotfi ben Jaffel. We gratefully acknowledge the
assistance provided by Bjorn Davidsson, Larry Paxton, Richard Barkus, Iannis
Dandouras, and Enrico Piazza for their immeasurable support and valuable
discussions on the issues and progress of this research.

##  7 Figures

Figure 1:  The densities of ions and electrons in the axis of the Sun-Earth-Moon system are illustrated in the figure below. The areas where the potential is created are also indicated. The zoomed-in panels (b) and (c) show the ion and electron densities in the region around $\pm 5R_{E}$ from the Moon(centered at zero). The difference in charge accumulation between the Lunar dayside and nightside is highlighted in these panels. Figure 2: The distribution of SW (solar wind) particles is shown in the figure below, plotted along OX at $\pm 5R_{E}$ on the day and night sides of the Moon. The distribution is also shown along three planes: the plane of the planet’s position and the dawn and dusk planes at $\pm 2R_{M}$ laterally. Panel (a) illustrates that ions outnumber electrons on the lunar dayside, while panel (b) shows that electrons outnumber ions on the night side. This phenomenon is caused by charge separation on both sides of the lunar surface. Specifically, the dayside lunar surface has a relative positive charge of $2.84\times 10^{4}$, whereas the dark side of the Moon has a negative charge of $-4.3\times 10^{4}$ (see Table 2).) |
---|---
A | B
Figure 3: The figure depicts a comparison between the profiles of ion (A) and electron (B) densities at the position of the Moon ($60R_{E}$) at various cuts along OX. These cuts are taken at the lit surface of the Moon, the dawn-dusk terminator, and 3, 5, 7, and $9R_{M}$ from the Moon, respectively. The densities are averaged over $1R_{E}$ in the OZ-direction and measured in the dusk-dawn direction at both sideways by $\pm 6R_{M}$. The figure shows the refilling of the lunar wake by solar wind particles. Ions mostly enter from the dusk side, while electrons mainly enter from the dawn side. This phenomenon is directly caused by the charge separation at the lunar surface and the resulting potential difference. |
---|---
A | B
Figure 4: The figure displays a comparison between the temperature profiles of
ions (A) and electrons (B) at the position of the Moon ($60R_{E}$) at various
cuts along OX. These cuts are taken at the lit surface of the Moon, the dawn-
dusk terminator, and 3, 5, 7, and $9R_{M}$ from the Moon, respectively. The
temperatures are averaged over $1R_{E}$ in the OZ-direction and measured in
the dusk-dawn direction at both sideways by $\pm 6R_{M}$. The figure depicts
the refilling of the lunar wake by solar wind particles. Ions mainly enter
from the dusk side, while electrons mainly enter from the dawn side. This
phenomenon directly results from the charge separation at the lunar surface
and the resulting potential difference. This figure corresponds to Fig. 3.
Figure 5: This Figure depicts the components of the magnetic field, the total
magnetic field, and the density at the lunar terminator and within the Lunar
wake. The boundaries of the Moon are marked by vertical green dashed lines. A
difference in the density plots is noticeable between the linear graphs shown
in green (taken at the lunar surface) and blue (taken at the terminator),
revealing the flaring out of the density flow around the Moon. The findings
are evaluated against the outcomes shown (Poppe ., 2014, Fig. 4). Figure 6:
This figure depicts the magnetic(a-d), electric field(e-h), and solar wind
velocity(i-m) components along three different planes, namely OX (with Y=Z=0
and X=$\pm 5R_{E}$ in red), dawn (with Y=$+2R_{L}$ and Z=0 in blue), and dusk
(with Y=$-2R_{M}$ and Z=0 in green). The plots are taken along OX at $\pm
5R_{E}$ on both the day and night sides of the moon. The linear plots at dawn
and dusk account for the asymmetry of the solar wind parameters. Figure 7:
This figure shows two contours for magnetic field and plasma velocity taken in
equatorial plane around the moon at $\pm 5R_{E}$ in both directions. color
codes are scaled to real values.

|
---|---
A | B

Figure 8:  This figure presents a zoomed in depiction of the moon, where the radius is illustrated as being five times greater than the ratio of the moon’s radius to the Earth’s radius ($R_{L}/R_{E}$) .The total electric field ($E.F.$) of the solar wind plasma is plotted in blue, and the induced electric field from charge separation ($E_{0}=m_{e}v_{e}\omega_{pe}/q_{0}$) is plotted in red. These fields are plotted along the OX direction at Y=0 and Y=$\pm 1R_{L}$ in the dawn and dusk regions, respectively. Panel-B shows the effective E.F. normalized over $E_{0}$ to represent the total E.F. in the vicinity of the lunar surface along the Sun-Earth-Moon Axis. |
---|---
Figure 9: Previous observations Poppe . (2014) and PIC simulations Deca .
(2015) have revealed that some solar wind ions backstream against the solar
wind inflow. This figure presents the characteristics of the backstreaming
ions on both sides of the Moon along the Sun-Earth-Moon direction. The
backstreaming ions are characterized based on their $\mathrm{A-factor}$,
defined as the ratio of backscattered velocity to ion thermal velocity
Bonifazi Moreno (1981). If $\mathrm{A}>1.2$, the net flow is a reflection; if
$0.8<\mathrm{A}<1.2$, the flow is intermediate, and finally, if
$\mathrm{A}<0.8$, the flow is diffuse. The comparison is made at both sides of
the Moon at $\pm$4$R_{E}$ (approximately 15$R_{M}$). Most backstreaming ions
are characterized as diffuse ions on both sides of the Moon. Figure 10: The
figure depicts ion temperature anisotropy (a) and electron temperature
anisotropy (b), along with cone angle $\theta_{nB}$ ) represented in (c). The
measurements were taken in the magnetotail region near the lunar position as
shown in this figure at a distance of $\pm 5R_{E}$. The data reveals that the
ratio of perpendicular to parallel ion temperature anisotropy
($Ti_{\bot}/Ti_{\parallel}$) is 2.6, whereas the ratio of perpendicular to
parallel electron temperature anisotropy ($Te_{\bot}/Te_{\parallel}$) is 2.1.
Figure 11: This figure succinctly summarizes our findings. It showcases the
interaction between the solar wind (SW) and the Earth’s magnetosphere (MS),
leading to the derivation of the magnetopause (MP) shape (represented by the
curved gray line). The figure clearly illustrates the dusk-dawn asymmetry.
Furthermore, it depicts the coupling between the magnetosheath (MS) and the
ionosphere (IS) through the escape of $\mathrm{H^{+}}$ ions (blue) and
$\mathrm{O^{+}}$ ions (green), which are superimposed on the background color
representing the plasma distribution of SW ions. Additionally, the figure
demonstrates the charging of the lunar surface on both its day and night
sides.