\section{Introduction}
\label{Intro}

Electromagnetic metamaterials can be defined as materials with electromagnetic properties given by their geometrical structure which differs from that given by their basic constituent properties. We can for example design a metallic surface with an effective refractive index different from the refractive index of the metal itself giving more desirable properties.
Metamaterials have been successfully used in order to create invisibility cloaks at microwave frequencies\cite{Schurig} as well, bringing the "dream" of invisibility a step nearer. The use of electromagnetic metamaterials is today well established in industry. Miniaturized antennas commonly designed for mobile telecommunications applications (e.g., \cite{Sievenpiper}, \cite{Sievenpiper2}, \cite{Sievenpiper3}) are a typical instance.

Recent research has led us to believe that electromagnetic metamaterials can also be used in the manufacture of antennas for satellite communications \cite{Lier}, and our ongoing research indicates that they can also improve the performance of astronomical radio-telescopes by providing ultra wide-band meta-antennas. This feature will bring about a revolution in the state of the art in the field of feedhorn-antennas with low cross-polarization and sidelobes levels, with has been static for several decades due to insurmountable limitations in the bandwidth of the corrugated feedhorns \cite{Clarricoats}. This characteristic is explained in more detail in section \ref{Theory}.

However, given the geometric complexity and the small size of the metamaterials needed for  applications in feedhorns for satellite communications and astronomy, where the lengths, and especially the thicknesses of the three-dimensional metallic pieces are of the order of a fraction of the wavelength of the usable frequency\footnote{The usable frequency is the centre frequency of the transmission/reception band.}, their fabrication is very challenging.

In section \ref{Methods}, we present a mushroom type electromagnetic metamaterial \cite{ADA} manufactured by three different methods, i.e., metallic additive 3D-printing, stereolithography (SLA) 3D-prototyping and a simplified machinable version\footnote{Surprisingly finding a notable electromagnetic benefit, explained in section \ref{Theory}.}, and we perform a metrological study in order to verify that the fabricated prototypes achieve the mechanical tolerances we need for their application in radioastronomy. This tolerance is of the order of magnitude of $\lambda/100$, which is, around 100[$\mu$m] for microwave frequencies of tens of gigahertz.

Since the feedhorns do not do mechanical work, a stress test is not applicable. However, since the feedhorns may well be cooled in a cryostat to 4[K], a theoretical calculation of the differential contraction of the SLA 3D-printing prototype and its copper skin is relevant and is taken on in section \ref{Theory}.

In section \ref{Methods}, various fabrication techniques are described. In section \ref{Results}, we present a metrological study of the manufactured prototypes, while general conclusions are presented in section \ref{Conclusions}.

Success in this research could have important benefits for astronomical applications, because the sensitivity of broadband receivers scales as a factor $\sqrt{\Delta f }$, i.e., the square root of bandwidth obtained in comparison with a classic corrugated horn-antenna. Also this feature could imply a reduction in the complexity of a radiometer design if the same feedhorn could be used to cover what has been two bands up until now. In terms of sensitivity for our telescopes in operation and future radiotelescopes placed at the Observatorio del Teide (OT) \cite{Quijote}, we estimate that this benefit could be as much as 41\% overall sensitivity. Furthermore, a successful manufactured design of meta-horn will also permit a reduction in the number of horn-antennas per experiment for a given frequency band, leading to a reduction in the complexity of the instruments and saving both mechanical and electronic resources.

\section{Theory and calculations}
\label{Theory}
\subsection{Surface impedance calculations for periodic structures}
Radiotelescopes generally use parabolic reflectors. The field generated at the optical focus of a paraboloid with large $f/D$ ratio is composed of  two polarizations \cite{Minnet}. One P-polarization or also called transverse magnetic (TM) component is polarized in the plane of incidence and has its electric (E) component oriented normally to the focal plane and an S-polarization or transverse electric (TE) pattern, polarized normally to the plane of incidence. In this reference, the authors explain that the feedhorn which transmits both TE and TM polarization with null cross-polarization must have inner surfaces satisfying the following expression

\begin{equation}
Z^{S}Z^{P}=-Z_0^2 \;,
\label{equation_1}
\end{equation}

where $Z^{S,P}$ is the surface impedance for both polarizations  (or spatial directions from the surface) and $Z_0$ is the impedance of the vacuum. The relation in (\ref{equation_1}) is known as the {\it balanced hybrid-mode condition} and guaranties that the $HE_{11}$ hybrid mode is the lower cut-off frequency mode, that is, the \textit{fundamental} mode, and of a high cross polarization purity. This is a necessary but not a sufficient condition to attenuate cross-polarization to desirable levels. Other factors such as the profile of the antenna or the antenna-reflector array optics also influence the final levels of cross polarization.
In this work by Minnet\&Thomas it is even suggested that a radially corrugate cylindrical waveguide be used in order to support the hybrid-mode condition over a limited band. Other authors gave a more in-depth treatment of this corrugated feed-horn technology \cite{Clarricoats}, and that is the most extensive technology used in radioastronomy and satellite communications for the design of horn antennas up until now, presenting a limiting bandwidth factor of around 1.4:1. This bandwidth can be slightly increased by the use of ring-loaded slots in the throat up umtil a limit of 2:1 bandwidth factor, but the cross-polarization behaviour doesn't satisfy the standard requirements in radioastronomy or satellite communications over the entire band (e.g., return-loss better than -20[dB], side-lobes level better than -30[dB], cross-polarization level better than -35[dB] and maximum gain around 20[dB]).

In order to avoid these geometrical limitations  for a bandwidth factor over 2:1 with low cross-polarization levels over the entire band, i.e., satisfying the hybrid-condition, the use of novel metamaterials is key.

Thus, it is necessary to be able to calculate surface impedance to design new metamaterials with $Z_{S,P}$ taking on the critical value. This is not trivial. We have developed a similar theoretical approach to that presented in \cite{Lier} to correctly run calculations and simulations. This idea is summarized in the following lines.

It is well known that the reflection coefficient for a wave travelling through vacuum and perpendicularly to a plane surface is given by

\begin{equation}
\Gamma=\frac{E^-}{E^+}=\frac{Z_s-Z_0}{Z_s+Z_0} \;,
\label{equation_2}
\end{equation}

where $E^+$ represents the forward incident wave and $E^-$ represents the backward reflected wave, $Z_s$ is the impedance of the plane surface and $Z_0$ is the impedance of the free space. This is represented in Fig. \ref{fig3_1}.

\begin{figure}[ht!]
		\includegraphics[width=0.5\textwidth]{fig3_1.PNG}
		\centering
		\caption{Schematic of S and P polarization of electromagnetic modes}
		\label{fig3_1}
	\end{figure}

Thus, it is possible to obtain expressions for TE and TM modes being reflected by a surface with characteristic $Z^{S,P}$ impedance. In order to obtain these expressions, it must be noted that the impedance of free space ''depends'' on the incidence angle ($\theta$). The expressions are, for P-polarized modes

\begin{equation}
TM\equiv P
\begin{cases}
 |H\big|= H_0 \\
 |E\big|=E_0\mathrm{cos}\theta
\end{cases}
\longrightarrow Z_0^{P}(\theta)=\frac{E^{P}(\theta)}{H^{P}(\theta)}=Z_0\mathrm{cos}\theta,
\label{equation_3}
\end{equation}

and for S-polarized modes

\begin{equation}
TE\equiv S
\begin{cases}
 |H\big|= H_0 \mathrm{cos}\theta\\
 |E\big|=E_0
\end{cases}
\longrightarrow Z_0^{S}(\theta)=\frac{E^{S}(\theta)}{H^{S}(\theta)}=\frac{Z_0}{\mathrm{cos}\theta}.
\label{equation_4}
\end{equation}

From equations (\ref{equation_2}), (\ref{equation_3}) and (\ref{equation_4}) it is straight forward to obtain the following expression for the surface impedance of reflected S-modes

\begin{equation}
Z_s^{S}=\frac{Z_0}{\mathrm{cos} \theta} \frac{1+\Gamma^{S}}{1-\Gamma^{S}}\;,
\label{equation_5}
\end{equation}

and for reflected P-modes

\begin{equation}
Z_s^{P}=Z_0 \, \mathrm{cos} \theta \, \frac{1+\Gamma^{P}}{1-\Gamma^{P}}\;.
\label{equation_6}
\end{equation}

Thus, we are able to obtain calculations of the surface impedance once we have calculated (from simulation) the reflection coefficient $\Gamma$ of each mode, and then introduced both reflection coefficients in (\ref{equation_1}) in order to know if our metasurface satisfies the hybrid-mode condition. This procedure can be run by using CST Studio Suite$^{\textregistered}$. CST includes a finite element method (FEM) 3D Electromagnetic analysis software package for components and systems. In our simulations, a plane wave is reflected on a metamaterial containing an E-field probe over its surface. Comparing measurements from the E-field probe in the same position with and without the presence of the metamaterial under test, it is possible to subtract the incoming and the reflected signals, allowing us to obtain the reflection coefficient for each S or P polarization from the equations (\ref{equation_5}) and (\ref{equation_6}).

\begin{figure}[ht!]
		\includegraphics[width=1\textwidth]{fig3_2.PNG}
		\centering
		\caption{Mushroom type metamaterial based on \cite{ADA}. For covering the band 10-20[GHz], the parameters can adopt the values w=2, s=0.4, p=1, l=2 and d=5.2[mm]. The thickness (e) is 0.05 and the diameter of the cylinders is 0.2[mm]. The lattice period or space between consecutive lines is 0.4[mm].}
		\label{fig3_2}
	\end{figure}

These simulations have been taken for a mushroom 3D-type metamaterial based on \cite{ADA} and presented in Fig.\ref{fig3_2}, obtaining the results shown in the Fig.\ref{fig3_4}.  This metamaterial is named as \textit{model-1}. It is convenient to keep in mind now that the aim of this simulations is to design the inner surface of a horn-antenna, made up of tens or hundreds of rings such as the shown in the Fig. \ref{fig3_3} joined together.

\begin{figure}[ht!]
		\includegraphics[width=0.4\textwidth]{fig3_3.PNG}
		\centering
		\caption{Ring of the meta-horn antenna. Here the more realistic dimensions of the model-2 are used in a ring of 17.4[mm] exterior radius.}
		\label{fig3_3}
	\end{figure}

From the hybrid-mode condition in (\ref{equation_1}),  a metamaterial satisfies the hybrid condition shown in Fig.\ref{fig3_4}  when $-Z^{S} Z^{P}/Z_0^2$ is exactly 1. Thus, a metamaterial with values $-Z^{S} Z^{P}/Z_0^2=1$ would theoretically allow us to design inner walls for feedhorn antennas with null cross-polarization levels, while it is expected that values of $-Z^{S} Z^{P}/Z_0^2$ in the vicinity of 1 yields  relative cross-polarization attenuation. This is the case of this metamaterial, whose dimensions are given in Fig.\ref{fig3_2} and whose simulation is represented by the solid black line of the Fig.\ref{fig3_4}, showing good behaviour in the band 12-28[GHz] approximately, which is a 2.5:1 bandwidth factor.

However, the small dimensions for the sample of metamaterial in the Fig.\ref{fig3_2} are not possible to achieve in a mechanical workshop or by additive techniques. Thus, for the prototype presented later in the section \ref{Methods} we imposed several constraints. Firstly, since 0.05[mm] thickness for the plates is far from a realistic value a conservative value for this thickness was established as 1[mm]. The diameter of the cylinder is challenging to manufacture and it has been set at 0.85[mm]. Thirdly, the dimension of the parameter "s" had been increased to 0.9[mm]. The rest of parameters of model-2 have the same value as model-1. The effect of these modifications over the model-1 design with the original dimensions based on \cite{ADA} is studied with the help of the results of a new simulation shown in Fig.\ref{fig3_4}, where  this typology is named as \textit{model-2}. It is possible to see how the effect is not significant straying more from the value $-Z^{S} Z^{P}/Z_0^2\sim1$ for centre frequencies but slightly increasing the overall bandwidth to lower frequencies. The explanation for this increment of bandwidth maybe due to the capacitive impedance given by the increase in cross section of the plates and cylinders whilst simultaneously reducing the inductance.

We have compared the highly-complex topology of metamaterials based on \cite{ADA} (model-1) and its realistically manufacturable version (model-2) with a simpler version redesigned to be manufactured by ''traditional'' workshop techniques. Now, the cylindrical pattern is reduced to a solid wall, while the plates are also unified, forming a "tee" from a frontal view. This design is presented in Fig.\ref{fig3_5} and named as \textit{top-plated corrugations}.

Moreover, we reconverted our top-plated or ``T'' topology presented in fig.(\ref{fig3_5}) adapting it to a new thin version of the classic ring-loaded corrugations topology \cite{Clarricoats}, more frequently used in taper-converters for feedhorns. This thin-ring-loaded or inverted-``L'' topology is presented in Fig.\ref{fig3_6}. Simulations of this inverted-``L'' metamaterial indicate that both top-plated and inverted-``L'' topologies are equivalent from an electromagnetic point of view, so both could be combined or used for different applications depending on the constraints and requirements of each case. This electromagnetic similarity is logical because both typologies have identical surface inductance and capacitance. This idea can be qualitatively understood with the help of Fig.\ref{fig3_7}. The dotted black line in the Fig.\ref{fig3_4} which is named model-3 represents both top-plated and inverted-``L'' topologies. In this figure it can be seen that the results are better for this simplified model-3 than for model-1 or 2. This was initially surprising to the author. We think that this is due to several aspects of the design. Firstly, the reduction in the minimum thickness of the plates (adopting now the value 0.4[mm] for each thickness), but mainly because of the gain in radial symmetry in comparison with the metamaterial of models 1 and 2, where different points around the circumference can have different values of surface impedance due to the radial pattern. This has been confirmed by radially displacing the electric field probe in different simulations.

Thus, it can be seen in fig. \ref{fig3_4}, that the operating band is theoretically even wider for the top-plated and inverted-``L'' metamaterial topologies (model-3), covering the 6.5-35[GHz] band for an impressive 5:1 bandwidth factor, and also because the value of $-Z^{S} Z^{P}/Z_0^2$ is closer to the unity over the whole band, theoretically presenting a better behavior in terms of cross polarization attenuation and satisfaction of the hybrid mode condition.Thus it can be concluded that, at least in theory, the metamaterial presented in \cite{ADA} does not have a real advantage over a simpler and more ``easily'' manufacturable thin version of the ring-loaded topology, presented in articles around 30 years ago by several authors (e.g.,\cite{Clarricoats}). Anyway, our interest in this article is to explore methods of fabrication  and more advantageous metamaterials for the future.

\begin{figure}[ht!]
		\includegraphics[width=0.8\textwidth]{fig3_4.PNG}
		\centering
		\caption{Metamaterial FEM-simulations for 3 different models. The metamaterial satisfies the hybrid mode condition when the expression in the y-axis is exactly 1. The solid line represents the results for the model-1 mushroom topology metamaterial inspired on \cite{ADA}. The dashed line corresponds to the modified  model-1 (model-2) mushroom metamaterial, with cylinder diameter = 0.85, s=0.9 and plate thickness  equal 1[mm]. The dotted line represents both top-plated and inverted-``L'' topologies of model-3, since they present identical results. The dot-stripe gray line represents an equivalent model using corrugations instead of mushrooms to provide a reference. The material is a perfect electric conductor (PEC) for all simulations.}
		\label{fig3_4}
	\end{figure}

\begin{figure}[ht!]
		\includegraphics[width=0.75\textwidth]{fig3_5.PNG}
		\centering
		\caption{Top-plated corrugations in a simplified version of the original metamaterial where e=0.4, l=1.66 d=6.5[mm] and the free space between corrugations is 0.25[mm]. Note that a classic mechanization permits a reducible thickness (e).}
		\label{fig3_5}
	\end{figure}

	\begin{figure}[ht!]
		\includegraphics[width=0.75\textwidth]{fig3_6.PNG}
		\centering
		\caption{Inverted-``L'' in a simplified version of the original metamaterial where e=0.4, l=1.66 d=6.5[mm] and the free space between corrugations is 0.25[mm].}
		\label{fig3_6}
	\end{figure}

	\begin{figure}[ht!]
		\includegraphics[width=0.6\textwidth]{fig3_7.PNG}
		\centering
		\caption{Illustration of the Top-plated (up) and inverted-``L'' (down) topologies similarity in terms of impedance. Since $R \simeq 0$ for good conductors, a generic impedance approximately adopts the value $Z=j\omega L-j/(\omega C)$ in both models, where $L$ and $C$ are given values of inductance and capacitance respectively. }
		\label{fig3_7}
	\end{figure}

Finally, it is interesting to conclude this analysis comparing these new metamaterials with the more extended topology for the present-day design of feed-horns using corrugations. This reference helps in order to see how innovative these new metamaterials are. Thus, an equivalent version of a corrugated surface has been simulated in order to calculate its bandwidth. In this reference model, the depth of the slots is 6.9[mm], the thickness of the corrugations is 0.4[mm] and the lattice period or space between two consecutive lines of corrugations is an optimized value around 1.5[mm]. The result of the simulations of this model is represented by the dot-stripe gray line in Fig.\ref{fig3_4}, where it can be seen that the product of normalized impedances has a large slope so its bandwidth factor is much lower than the bandwidth factor of the metamaterial-based models, being around a 1.4:1 factor or 40\% of the band, as the bibliography predicts. \newline

The metamaterials presented in this article can satisfy the hybrid mode condition in wider bandwidths than that of the reference. This phenomenon can be explained qualitatively. The benefit in bandwidth with low levels of cross polarization of our meta-ring is based on two characteristics. Firstly, the gain in symmetry compared to those referenced geometries, i.e., the mushroom patterns are not symmetrical in their top plane. This lack of symmetry, caused by the teeth between consecutive mushroom plates, make the surface impedance to vary with the position in the upper plane of the structure so cannot be adjusted simultaneously through two consecutive zones, i.e., between plates and at the consecutive rectangular hollow space, yielding a global spurious effect and reducing the band. Secondly, in some cases, the reduction of thicknesses in comparison to the referenced mushroom geometries. The mechanical manufacturing in workshop allows to have much smaller thicknesses than those allowed by additive manufacturing or other techniques. This control over the thickness is transmitted to a control over the inductance and capacitance, which allow us to design a more precise structure in the fulfillment of the hybrid mode condition. On the other hand, regarding the classic corrugations, the bandwidth benefit of the metamaterials in the Fig.\ref{fig3_4} is produced due to the extra degree of freedom available to adjust the surface impedance values, given by the addition of top-plates.

\section{Material and methods}
\label{Methods}
\subsection{Materials justification}
Applications in feedhorn-antennas require the use of metals with high electric conductivity. Copper , titanium and aluminum were considered.

A horn-antenna is a cylindrical-conical free-standing structure, so the only limitation in mechanical terms is to be able to manipulate it without deforming or breaking it, and that it be able to support cryogenic-vacuum conditions (to have adequate plasticity, resistance, and resilience).  Copper has the advantage of a higher conductivity, but aluminum is easier to mechanize. For each technique of fabrication presented we have taken the best material option which is always a compromise between electrical and mechanical properties. Anyway, both metals satisfies the basic conditions and they have been used in similar applications for our group for over twenty five years and in seven different telescopes and instruments (\cite{1994Natur.367..333H}, \cite{1998ApJ...498..117F},  \cite{2001MNRAS.327.1178G}, \cite{2003MNRAS.341.1057W}, \cite{JBO}, \cite{2014A&A...566A..54P}, \cite{2017MNRAS.464.4107G}).

Furthermore, the penetration depth for both copper and aluminum at gigahertz frequencies is in the order of a few microns or lower. Because of this, we have considered the possibility of fabricating a metamaterial using SLA technology and covering with a metallic coating, because the microwaves do not the penetrate to the inner dielectric structure.

Table 1 compares the conductivity ($\sigma$) and penetration depth ($\delta$) of 3 pure metals (Al, Cu and Ti) at room temperature\footnote{Since $\sigma$ increase under cryogenics conditions, these characteristics are even more favorable at 4[K].} and 10[GHz] conditions \cite{Eskelinen}.

\begin{table}
\centering
\begin{tabular}{l*{3}{c}r}
\toprule Metal & $\sigma[S/m]$ & $\delta[\mu m]$ \\
\midrule
Cu               & $5.8\cdot10^7$ & 0.66 \\
Al            & $3.5\cdot10^7$ & 0.83 \\
Ti  & $2.5\cdot10^6$ & 3.7 \\

\bottomrule
\label{table_1}
\end{tabular}
\caption{Conductivity ($\sigma$) and penetration depth ($\delta$) for Cu, Al and Ti at 300[K] and 10[GHz].}
\end{table}

\subsection{Fabrication of advanced 3D metallic metamaterials}
 In this section we present two different ways of manufacturing the metamaterials presented in the section \ref{Theory}. We have focused our interest on two methods, i.e. Lost-wax printing and direct SLA 3D-printing \cite{Snyder}  being the most accurate techniques available and being the most promising for the geometries needed for  electromagnetic metamaterials at microwave frequencies.

 In the conceptual stage simulations were run on conical geometries with a diameter of 35[mm] and a height of 220[mm].
Due to the complexity of the geometry, 3D-printing presents itself as the most suitable method.
Direct metal manufacturing was considered, and in particular Electron Beam Melting (EBM) \cite{Murr} in Ti6Al4v-eli. EBM does not require too much support structure, but  has not the resolution needed since the tolerances are of $\pm$0.2[mm]. Anyway, this method can be used with titanium which is not the most ideal since it has not the best conductivity (see table 1).

  It is important to note that the use of a filling material instead of air was discarded early because in this case the metamaterial structure  must be scaled by a factor $\sqrt{\varepsilon_r}$, where $\varepsilon_r$ is the relative electric permittivity of the filling material. For example, in the case of polylactic acid or PLA, used commonly in 3D printing with $\varepsilon_r \sim 3$ at room temperature and 16[GHz] \cite{Dichtl}, the scale factor is around 1:1.7. That means all the geometrical dimensions of the metamaterial must be scaled by a factor 1:2 in order to operate in the same electromagnetic band, making the prototype, in principle, more difficult to manufacture. This solution will also complicate the use of the meta-feedhorn under cryogenic conditions, because of the differential contraction between metal and dielectric solids.

To conclude this section, we present a workshop mechanized version of the simplified metamaterial along with metrological studies of prototypes.

\subsubsection{Metamaterial ring Lost-wax 3D-printing}
Firstly, we tried Lost-wax printing, where the 3D-model is built as a positive object through Stereo lithography, and later on prepared for traditional casting. The model and its support are attached to a wax “tree”, placed in a flask and covered in fine plaster. When the plaster solidifies, it forms a mold for the copper, silver, or gold casting.
For SLA, generally a minimum thickness of 1[mm] is recommended and minimum features of 0.5[mm]. In the XY (in-plane) the material is fused with a 140$[\mu$m] laser spot and built up to a layer height of 50$[\mu$m]. This leads to adequate precision and the desired tolerances $\pm$0.1[mm] or better.
A first iteration was done at a 10:1 (i.e., ten times larger) scale, where we checked the tolerances on the scaled version as well as minimum limits. This prototype is shown in Fig.\ref{fig3_8}. To guarantee that the structure hold its shape reinforcements were added and the thickness adapted to manufacturing limits.

\begin{figure}[ht!]
		\includegraphics[width=0.75\textwidth]{fig3_8.JPG}
		\centering
		\caption{10:1 scaled version of the planar metamaterial of Fig.\ref{fig3_2}.}
		\label{fig3_8}
	\end{figure}

Following this, one ring of a horn was printed by the same technique, presented here in Fig.\ref{fig3_9}. Dividing the cone structure into discs that are reassembled afterwards turns out to beneficial. Smaller, planar objects result in better geometrical tolerances, and more importantly require less support-structure, therefore less manual manipulation.  This is because most of the geometry rests on the build-plate.

\begin{figure}[ht!]
		\includegraphics[width=0.75\textwidth]{fig3_9.jpg}
		\centering
		\caption{Ring structure metamaterial printed in copper by loss-wax technique.}
		\label{fig3_9}
	\end{figure}

Deformation over large scales was significant. The nature of the process has a lot of manual operations: manipulating the mold, support removal, recovering the cast, polishing, packaging, shipping. Small dents and bumps lead to deformations, that end up giving unreliable and inconsistent results.

\subsubsection{SLA prototyping of metamaterial periodic structures}

A second iteration was carried out in SLA resin printing where for both plaques and struts the thickness and the diameter was repeated from 0.5 to 1[mm] in 0.05 increments within the 3D structure in order to show feasibility of simulations which were run on geometries inferior to 1[mm] thickness.
The results of the evaluation shown in Fig.\ref{fig3_11} reveal that  0.85[mm] and 0.9[mm] is consistently achievable. The viability of this technique is discussed in the following text.

\begin{figure}[ht!]
		\includegraphics[width=0.5\textwidth]{fig3_11.JPG}
		\centering
		\caption{SLA resin iterations. It can be noted that the declination of the model (top)  is beneficial for printing dimensions which are forbidden for the planar version (bottom).}
		\label{fig3_11}
	\end{figure}

The process of fabrication a film-substrate system generally generates stress in the mechanical pieces. Due to the differences in the coating and substrate properties, principally in the coefficient of thermal expansion (CTE), an additional thermal stress is generated on  cooling them. Different coating materials have been analyzed to find out feasibility of this process in our application. Despite the low conductivity of titanium alloys, these materials are evaluated as a starting point for this solution due to their mechanical properties. Ti-Al compound coatings are commonly applied in aerospace and automobile products. In fact, coatings obtained by Physical Vapor Disposition (PVD) techniques are widely applied to improve lifetime and performance  (e.g., \cite{Song}, \cite{Mo}).

In order to evaluate their behavior, it is necessary to make some simplifications that allow solving a parsimonious model. The simplified geometry to be evaluated consists of a dimensionally identical  ring of resin post-cured mixture of acrylate monomers, oligomers and photoinitiators with specific mechanical properties that are shown in table 2. The corners are eliminated avoiding stress concentrations. The base is covered with a metallic layer that is be prepared by a PVD technique. In addition, the study of different films is made by choosing different materials to evaluate the dependence of the mechanical properties of the film.

In this section, the structural interaction of the resin with a coating of 50 microns after decreasing the temperature to reach cryogenic conditions is analyzed using finite element method, more specifically the ANSYS Workbench$^{\textregistered}$. The results were correlated using reduced equations of the analytical solution in the case of thin-walled concentric cylinders to validate the results of the computational model.
In the analysis, the contact condition between components is rigid. It means that there is no slip in the surface region and the residual stress is not taken into account in the calculation. On the other hand, the relation of stiffness-thickness of the coating respect to the substrate is not significant. As a consequence of the fact that the thermal deformation is clearly more significant in the substrate ($\Delta r_s >> \Delta r_c$), we can assume that the coating does not provide any rigidity to the assembly and the deformations are mainly due to the cooling of the resin. Indeed, it can be considered that  the thermal displacements of the base are the sole course of the tensional state.

\begin{table}[]
\centering
\begin{tabular}{@{}ccccc@{}}
\toprule
Material   & UTS [MPa] & TM [GPa] & Elongation [\%] & TE [$^\circ C^{-1}$] \\ \midrule
Green      & 35                              & 1.4                   & 32.5            &    -               \\
Post-Cured & 61                              & 2.6                   & 13              & 7.85E-5                  \\ \bottomrule
\end{tabular}
\label{table_2}
\caption{Resin properties, where Ultimate Tensile Strength is UTS, Tensile Modulus is TM and TE is Thermal Expansion.}
\end{table}

\begin{table}[]
\begin{tabular}{@{}cccc@{}}
\toprule
Configuration      & Type      & Equiv. Lineal Strain [\%] & Equiv. Lineal Stress [MPa] \\ \midrule
Resin/Copper Alloy & Substrate & 0.0268                        & 68.86                          \\
Resin/Copper Alloy & Coating   & 0.0100                        & 1101.4                         \\
Resin/Ti6Al4V      & Substrate & 0.0276                        & 71.65                          \\
Resin/Ti6Al4V      & Coating   & 0.0063                        & 707.36                         \\
Resin/AA6061-T6    & Substrate & 0.0268                        & 69.38                          \\
Resin/ AA6061-T6   & Coating   & 0.0098                        & 684.12                         \\ \bottomrule

\label{table_3}

\end{tabular}
\caption{Stress State of film-Substrate.}
\end{table}

The results for the different materials in a cryogenic cool-down process are shown in table 3. From the results it is verified that the deformations caused by the variation of the temperature are practically identical with and without the coating, $\varepsilon \simeq 0.027$. The values obtained for the resin in each case are similar. As has been mentioned previously, the thermal contraction and the stiffness can be assumed null because the coating thickness is small when compared with the substrate.
The titanium alloy is the only material capable of withstanding the requirements in its elastic state. However, only one cold cycle has been evaluated, so the fatigue conditions and the concentration of tensions in the geometry of the antenna will eventually crack the coating. Furthermore, the rigidity of titanium is much higher than resin. The coating loads the substrate reaching its state of load-limit for just one temperature  cycle, which is why it would not stand a reasonable life span which consists of many cooldowns. In conclusion, it is not feasible to use this solution for this application without previously testing and proving the assembly.

\subsubsection{Mechanization of a simplified version of the metamaterial}

Two simplified prototypes of the meta-horn corrugation were manufactured by traditional machining processes, milling and turning, in order to compare their capabilities to the meta-material ones. Both rings are similar, they have the same wall thickness while the corrugation profile is different (see Fig.\ref{fig3_12}).

\begin{figure}[ht!]
		\includegraphics[width=0.75\textwidth]{fig3_12.jpg}
		\centering
		\caption{Simplified workshop mechanized prototype based on the model in Fig.\ref{fig3_6}.}
		\label{fig3_12}
	\end{figure}

The lathe and milling machine must apply a certain force to cut the material, so the minimum thickness that can be achieved is related to the material stiffness. Therefore, aluminium AW-6061-T6 was chosen instead of copper because it has better machinability.
Due to the thickness and the required tolerances, of the order of 0.4[mm] and 0.050[mm] respectively, some supporting tools were made prior to manufacturing in order to control the deformation during the process.
The examples presented are only a prototype for proving the capability of milling and turning when sub-millimetre thickness is required. Hence, pieces were made as rings without any assembly interface. Since the antenna will be made from several disks assembled together, an interface must be designed for this purpose, which supposes a challenging mechanical work in itself as it must ensure the concentricity of each disk with respect to any other for a good feedhorn performance. Anyway, practical ring junctions are well established and can be done, for example, with the technique published in \cite{Milano}.

Metrological results have been collected for all the manufactured metamaterials and are presented in section \ref{Results}.

\section{Results}
\label{Results}

In order to evaluate the acceptability of the different manufacturing processes, a metrology analysis was performed for each piece. In this analysis, both dimensional and geometric tolerances, i.e., roundness, parallelism and flatness, are measured. A schematic drawing of the pieces is shown in Fig.\ref{fig3_13}. The results are summarized in table \ref{table_4}.

\begin{figure}[ht!]
		\includegraphics[width=0.9\textwidth]{fig3_13.JPG}
		\centering
		\caption{Schematic of top-plated corrugations of model-3 (left), inverted-``L'' typology of model-3 (middle) and mushroom type metamaterial of model-2 (right). Note the dimensions presented in table \ref{table_4}}.
		\label{fig3_13}
	\end{figure}

\begin{table}[h]
\begin{tabular}{ccccccc} \toprule
            &         & \multicolumn{3}{c}{Model-3}                & \multicolumn{2}{c}{Model-2} \\
            &         &              & Top-plated   & Inverted-``L''  &              &              \\
Dimension   & Measure & Nom. {[}mm{]} & Dev. {[}$\mu$m{]} & Dev. {[}$\mu$m{]} & Nom.{[}mm{]}  & Dev.{[}$\mu$m{]}  \\ \midrule
A (inner D) & D       & 19.000       & -2           & -7           & 19.000       & +91           \\
            & R       &              & 0            & +3            &              & +100          \\
B           & D       & 19.800       & -30          & -15          & 21.000       & -262         \\
            & R       &              & +3            & +2            &              & +127          \\
C           & D       & 30.800       & +28           & +11           & 30.800       & -273         \\
            & R       &              & +3            & +7            &              & +42           \\
D (outer D) & D       & 34.800       & +15           & -13          & 34.800       & -407         \\
            & R       &              & +12           & +15           &              & +91           \\
E           & T       & 2.500        & -30          & -23          & 2.500        &              \\
            & P       &              & +14           & +10           &              &              \\
            & F       &              & +8            & +9            &              &              \\
F           & T       & 0.400        & -48          & -8           & 0.900        & +50           \\
            & P       &              & +7            & +20           & R            & +12           \\
            & F       &              & +16           & +12           &              &

\end{tabular}
\caption{Summary of metrology where the nominal (nom.) and deviation (dev.) in the dimensions highlited in the Fig.\ref{fig3_13} are taken and the measured properties are D=diameter, R=Roundness, T=thickness, P=parallelism referred to ``Y'' and F=flatness.}
\label{table_4}

\end{table}

Table 4 shows how the dimensions of the aluminum pieces manufactured by turning/milling have a deviation less than 0,050[mm] from the nominal values. For the meta-horn prototype manufactured by additive printing the deviation is higher and exceeds the acceptable values for a good feedhorn performance, established around 100[$\mu m$] for frequencies of tens of gigahertz.

\section{Discussion and Conclusions}
\label{Conclusions}

The key to the prototyping of metamaterials is the small high precision parts. 3D-printing processes require support structures and these can be difficult or impossible to remove without damaging the model.
Strategic design can drastically improve the situation and reduce the need for supports, which leads to better 3D-printed results, and tolerances. We have found that the lost-wax method doesn't seem to allow us to fabricate metamaterials at microwave frequencies with the required precision, and that the SLA+metallic-skin method suffers from breakage of the metamaterial in multiple cryogenic cycles. Workshop machining has been presented as a reliable option.\newline

\begin{figure}[ht!]
		\includegraphics[width=1\textwidth]{fig3_14.PNG}
		\centering
		\caption{Design of a meta-horn. General dimensions, in millimeters.}
		\label{fig_14}
	\end{figure}
\begin{figure}[ht!]
		\includegraphics[width=0.8\textwidth]{fig3_14.png}
		\centering
		\caption{Design of a meta-horn. Results of the simulations.}
		\label{fig_15}
	\end{figure}
Here we show a more practical and usable example using the meta-ring of Fig.\ref{fig3_12} to design a conical feedhorn composed of meta-rings. The design of the horn and the very promising results based on CST Studio Suite$^{\textregistered}$ simulations are shown in Figs.\ref{fig_14} and \ref{fig_15}. In Fig.\ref{fig_15} an excellent behaviour over a entire 2:1 bandwidth in terms of return-loss, side-lobe levels, gain and particularly cross-polarization, where levels better than -35[dB] over the entire 10-20[GHz] band are predicted.
The E and H planes demonstration is shown in the Fig.\ref{fig_16}, where an excellent behaviour is presented at the central frequency and the limits of the band of work. The polar farfield realized gain pattern diagrams are shown in Fig.\ref{fig_17} for both $\phi=90^{\circ}$ and $\phi=0^{\circ}$ planes, where the \textit {half power beam width} (HPBW) presents exceptional results as well.
\newline

\begin{figure}[ht!]
\centering
		\includegraphics[width=1\textwidth]{fig3_16.PNG}
		\centering
		\caption{E and H plane farfield antenna patterns at 10, 15 and 20[GHz].}
		\label{fig_16}
		\vspace{-\baselineskip
	\end{figure}

This design is very compact even using a simple conical profile because an easily manufacturing solution was required, but the characteristics could be improved further by using a more adequate profile \cite{Miguel_Hern_ndez_2019}.

\begin{figure}[ht!]
\centering
		\includegraphics[width=1\textwidth]{fig3_17.PNG}
		\centering
		\caption{Farfield realized gain patterns in decibels ($dB_G$) at 10, 15 and 20[GHz] for $\phi=0^{\circ}$ and $\phi=90^{\circ}$ planes. Angular width at -3[dB] (HPBW) given for each case.}
		\label{fig_17}
		\vspace{-\baselineskip
	\end{figure}

\bibliographystyle{elsarticle-num}


\section{Introduction}

The attitude stabilization of a spacecraft is subject to the perturbation
torques which produce turning moments about the center of mass of an
orbiting spacecraft. The significant effect of these torque disturbances on
the spacecraft is dependent on the configuration of the spacecraft.The
perturbation torques may be used to produce a persistent turning moment
about the center of mass of the spacecraft.

The present work analyze the attitude stabilization of a charged spacecraft
by taking into account the effects of gravitational torque, geomagnetic
torque and Lorentz torque. In the case of electrostatically charged
spacecraft, due to the interaction with space plasma, the Lorentz force must
be taken into account as a perturbation on the orbital and attitude motions
of the spacecraft.
The nascent concept of Lorentz spacecraft which is an electrostatically
charged space vehicle may provide a new approach into the solution of the
attitude stabilization of a spacecraft moving around the Earth in low Earth
orbit (LEO). Recently a novel attitude orientation and formation flying
concept using electrostatic propulsion has been proposed by Pollock et al.
(2011), and Chad and Yang (2012). The charge of the spacecraft is controlled
to generate inter-spacecraft Coulomb forces in geostationary orbit. Lorentz
force is a possible means for charging and thus controlling the spacecraft
orbits without consuming propellant (Hiroshima et al. 2009). Peck (2005) was
the first to introduce a control scheme using Lorentz augmented orbits. The
spacecraft orbits accelerated by the Lorentz force are termed Lorentz
-augmented orbits, because Lorentz force cannot completely replace the
traditional rocket propulsion. Many authors introduced Lorentz force as
perturbations on the orbital motion and formation flying such as in
Vokrouhlicky (1989), Abdel-Aziz (2007a), Streetman and Peck (2007),
Hiroshima et al. (2009) , Gangestad et al. (2010), and Abdel-Aziz and Khalil
(2014).

Abdel-Aziz (2007b) studied the attitude stabilization of rigid spacecraft
moving in a circular orbit due to Lorentz torque in the case of uniform
magnetic field and cylindrical shape of spacecraft. Yamakawa et al. (2012)
investigated the attitude motion of a charged pendulum spacecraft moving in
circular orbit, having the shape of a dumbbell pendulum due to Lorentz
torque. Their analysis of the stability of the equilibrium points are
focused only on pitch direction within the equatorial plane. In a recent
study Abdel-Aziz and Shoaib (2014) studied the relation between the
magnitude of Lorentz torque and inclination of the orbits for certain
equilibrium positions where the spacecraft was considered to be in circular
orbit.

In this paper, we analyze the attitude stabilization of a charged spacecraft
moving in geomagnetic field in Low Earth Orbit (LEO). We developed a new
model for the torque due to the Lorentz force for the general shape of the
spacecraft using the Earth magnetic field, which is modeled as a non-tilted
diploe. The total Lorentz force and its torque are developed as a function
of orbital elements of the spacecraft. A dynamical model is built to
describe the attitude dynamics of Lorentz spacecraft. Therefore, based on
the dynamical model, the required control torque due to Lorentz force for
different configurations is developed. The Lorentz acceleration can't
compensate the total propellant but can be used to reduce the consumption of
propellant. Thus, this paper analyzes the attitude stability of the
spacecraft with the Lorentz acceleration and gives the corresponding
required specific charge to mass ratio for such attitude orientation. This
paper also analyzes the effects of charge to mass ratio on the position and
stability of equilibrium positions. We also numerically analyze the behavior
of orbits close to the equilibrium positions.

\subsection{Formulation of the Spacecraft}

We assume that the spacecraft is equipped with an electrostatically charged
protective shield, having an intrinsic magnetic moment. The attitude
orientation of the spacecraft about its center of mass is analyzed under the
influence of gravity gradient torque ${T}_{G}$ , Magnetic torque ${T}_{M}$
and the torque ${T}_{L}$ due to Lorentz force. The torque ${T}_{L}$ results
from the interaction of the geomagnetic field with the charged screen of the
electrostatic shield.

We consider the orbital coordinate system $C_{{x_{o}}{y_{o}}{z_{o}}}$ with $%
C_{x_{o}}$ tangent to the orbit in the direction of motion, $C_{y_{o}}$ lies
along the normal to the orbital plane, and $C_{z_{o}}$ lies along the radius
vector $r$ of the point $O_{E}$ relative to the center of the Earth. The
investigation is carried out assuming the rotation of the orbital coordinate
system relative to the inertial system with the angular velocity $\Omega $.
As an inertial coordinate system, the system $O_{XYZ}$ is taken, whose axis $%
OZ(k)$ is directed along the axis of the Earth's rotation, the axis $OX(i)$
is directed toward the ascending node of the orbit, and the plane coincides
with the equatorial plane. Also, we assume that the spacecraft's principal
axes of inertia $C_{{x_{b}}{y_{b}}{z_{b}}}$ are rigidly fixed to a
spacecraft $(i_{b},j_{b},k_{b})$. The spacecraft's attitude may be described
in several ways, in this paper the attitude will be described by the angle
of yaw $\psi \,$ the angle of pitch $\theta $ , and the angle of roll $%
\varphi $, between the spacecraft's $C_{{x_{b}}{y_{b}}{z_{b}}}$ and the set
of reference axes $O_{XYZ}$. The three angles are obtained by rotating
spacecraft axes from an attitude coinciding with the reference axes to
describe attitude in the following way:

- The angle of precession $\psi \,$is taken in plane orthogonal to $Z$-axis.

- $\theta $ is the rotation angle between the axes $Z$ $\ $and ${z_{0}.}$

- ${\phi }$ is angle of self -rotation around the $Z$-axis

We write the relationship between the reference frames $C_{{x_{b}}{y_{b}}{%
z_{b}}}$ and $C_{{x_{o}}{y_{o}}{z_{0}}}$ as below (Wertz, 1978):
\begin{equation}
A=\left(
\begin{array}{ccc}
\alpha _{1} & \alpha _{2} & \alpha _{3} \\
\beta _{1} & \beta _{2} & \beta _{3} \\
\gamma _{1} & \gamma _{2} & \gamma _{3}%
\end{array}%
\right) ,
\end{equation}

where

\begin{equation}
\begin{array}{l}
({\alpha _{1},\alpha }_{2},\alpha _{3})={(\cos \psi \cos \phi -\sin \psi
\sin \phi \cos \theta ,-\,\cos \psi \sin \phi -\cos \theta \sin \psi \cos
\phi ,\sin \theta \,\sin \psi ),} \\
({\beta _{1},\beta _{2},\beta _{3})=(\sin \psi \cos \varphi \,+\,\,\cos
\theta \cos \psi \sin \varphi ,-\sin \psi \sin \phi +\cos \theta \cos \psi
\cos \phi ,-\,\sin \theta \cos \psi ),} \\
{\,(\gamma _{1},\gamma _{2},\gamma _{3})=(\sin \theta \,\sin \phi ,\sin
\theta \cos \phi ,\cos \theta ),}%
\end{array}%
\end{equation}

and
\begin{equation}
\vec{\alpha}=\alpha _{1}i_{b}+\alpha _{2}j_{b}+\alpha _{3}k_{b}{,}\,\,\,{%
\vec{\beta}}=\beta _{1}i_{b}+\beta _{2}j_{b}+\beta _{3}k_{b}{,}\,\,{\vec{%
\gamma}}=\gamma _{1}i_{b}+\gamma _{2}j_{b}+\gamma _{3}k_{b}.\,
\end{equation}

\begin{figure}[t]
\centering
\resizebox{70mm}{!}{\includegraphics {fig1}}
\caption{Spherical coordinates used in the derivation of the equations of
motion}
\label{Fig1}
\end{figure}
\section{Total Torque due Lorentz Force}

We use spherical coordinates to describe the magnetic and gravitational
fields, and the spacecraft trajectory, as shown in Figure (1). The $X$, $Y,$
and $Z$ axes form a set of inertial cartesian coordinates. The Earth is
assumed to rotate about the $Z$-axes. The magnetic dipole is not tilted and
therefore, axi-symmetric. The spherical coordinates consist of radius r,
colatitude angle $\Phi $, and azimuth from the $X$ direction $\Theta $ (see
Figure. 1). The magnetic field is expressed as%
\begin{equation}
\vec{B}\mathbf{=}\frac{B_{0}}{r^{3}}\left[ 2\cos \Phi ~\hat{r}\mathbf{+}\sin
\Phi ~\hat{\Phi}\mathbf{+0}\hat{\Theta}\right] ,
\end{equation}%
where $B_{0}$ is the strength of the magnetic field in Wb m. The
acceleration in inertial coordinates is given by

\begin{equation}
\vec{a}\mathbf{=}\frac{\vec{F}}{m}=-\frac{\mu }{r^{3}}\vec{r}\mathbf{+}\frac{%
q}{m}(\vec{E}+\vec{V}_{rel}\times \vec{B}\mathbf{),}
\end{equation}%
where $\frac{q}{m}$is the charge-to-mass ratio of the spacecraft, $\vec{V}%
_{rel}$ is the velocity of the spacecraft relative to the magnetic field of
The Earth. The total Lorentz force (per unit mass) can be written as:

\begin{eqnarray}
\vec{F_{L}} &=&\frac{q}{m}\left[ \vec{E}+\vec{V}_{rel}\times \vec{B}\right] =%
\frac{q}{m}\vec{E}+\frac{q}{m}(\vec{V}_{rel}\times \vec{B}\mathbf{)}  \notag
\\
&=&\vec{F}_{elec}+\vec{F}_{mag},
\end{eqnarray}%
where $\vec{F}_{mag}$ is the Lorentz force experienced by magnetic field and
$F_{elec}$ is the Lorentz force experienced by an electric dipole moment in
the presence of electric field,

\begin{equation}
\vec{F}_{elec}=\frac{q}{m}\vec{E}.
\end{equation}%
Now we start with $\vec{F}_{mag}$ using Maxwell (1861), we can write

\begin{equation}
\vec{F}_{mag}=\frac{q}{m}(\vec{V}_{rel}\times \vec{B}\mathbf{),}\vec{V}%
_{rel}=\vec{V}-\vec{\omega _{e}}\times \vec{r},
\end{equation}%
where $\vec{V}$ is the inertial velocity of the spacecraft, $\mathbf{\omega }%
_{e}~$is the angular velocity vector of the Earth. According to Gangestad et
al. (2010), we used

\begin{equation*}
\vec{V}\mathbf{=}~\dot{{r}}~\hat{r}\mathbf{+}{r}\dot{\Phi}\hat{\Phi}\mathbf{+%
}{r}\mathbf{~}\dot{\Theta}\sin {\Phi }~\hat{\Theta},
\end{equation*}%
and

\begin{equation}
\vec{r}\mathbf{~=}r\mathbf{~}\hat{r},~\vec{\omega _{e}}=\omega _{e}~\hat{z},~%
\hat{z}~=\cos \Phi ~\hat{r}\mathbf{+}\sin \Phi \mathbf{~}\hat{\Phi}.
\end{equation}%
Therefore the acceleration in inertial coordinates is given by

\begin{equation}
\vec{F}_{mag}=\frac{qB_{0}}{m~r^{2}}\left[
\begin{array}{c}
-\left( \dot{\Theta}-\omega _{e}\right) \left( \sin ^{2}\Phi ~\hat{r}+\sin (2%
{\Phi })\hat{\Phi}\right) \\
+\left( \frac{\dot{r}}{r}\sin \Phi -2\dot{\Phi}~\cos \Phi \right) \hat{\Theta%
}%
\end{array}%
\right] .
\end{equation}%
In the case of Torque we need the perturbing force $\vec{F}_{L}$ decomposed
into radial, transverse, and normal direction. The unit vector $\hat{n}$
normal to the orbit is collinear with the angular momentum unit vector $\hat{%
h}$.

\begin{equation}
\hat{n}=\hat{h}=\left( \vec{r}\times \vec{V}\right) /\sqrt{\mu p}=r^{2}/%
\sqrt{\mu p}(-\dot{\Theta}\sin \Phi \hat{\Phi}+\dot{\Phi}\hat{\Theta}),
\end{equation}%
where $p=a(1-e^{2})$, $\mu $ is the Earth's gravitational parameter, $a$ is
the semi-major axis, $e$ is the eccentricity of the spacecraft orbit, and
the transverse unit vector $\hat{t}$ can be calculated from the right-handed
set, $\hat{t}=\hat{n}\times \hat{r}$. Decomposition of the Lorentz force
experienced by the geomagnetic field into the radial, transverse, and normal
components $\left( R_{mag},T_{mag},~N_{mag}\right) $ respectively yields,

\begin{equation}
R_{mag}=\vec{F}_{mag}\cdot \hat{r}~=\frac{q}{m}\frac{B_{0}}{~r^{2}}\left[
\omega _{e}-\dot{\Theta}\right] \sin ^{2}\Phi ,
\end{equation}%
\begin{equation}
T_{mag}=\vec{F}_{mag}\cdot \hat{t}~=\frac{q}{m}\frac{B_{0}}{~\sqrt{\mu p}}%
\left[
\begin{array}{c}
\frac{\dot{r}}{r}\dot{\Theta}\sin ^{2}\Phi \\
-2\omega _{e}~\dot{\Phi}\cos \Phi \sin \Phi%
\end{array}%
\right] ,
\end{equation}%
\begin{equation}
N_{mag}=\vec{F}_{mag}\cdot \hat{n}~=\frac{q}{m}\frac{B_{0}}{~\sqrt{\mu p}}%
\left[
\begin{array}{c}
2\dot{\Theta}(\omega _{e}-\dot{\Theta})\sin ^{2}\Phi \cos \Phi ~ \\
+\frac{\dot{r}}{r}\dot{\Theta}\sin \Phi -2\dot{\Phi}^{2}~\cos \Phi%
\end{array}%
\right] .
\end{equation}%
The relationship between the spherical coordinates and the orbital elements
is required to derive the components of Lorentz force experienced by
magnetic part as a function of orbital elements.

\begin{equation}
r=p/\left( 1+e~\cos f\right) ,~\dot{r}=e\sqrt{\mu /p}\sin f,
\end{equation}%
\begin{equation}
\cos \Phi =\sin i\sin \left( \omega ^{\ast }+f\right) ,\sin \Phi =\sqrt{%
1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast }+f\right) },
\end{equation}%
\begin{equation}
\dot{\Phi}=-\sqrt{\mu /p^{3}}\frac{\sin i\cos \left( \omega ^{\ast
}+f\right) }{\sqrt{1-\sin ^{2}i\cos ^{2}\left( \omega ^{\ast }+f\right) }}%
\left( 1+e~\cos f\right) ^{2},
\end{equation}%
\begin{equation}
\dot{\Theta}=\sqrt{\mu /p^{3}}\frac{\cos i~}{1-\sin ^{2}i\sin ^{2}\left(
\omega ^{\ast }+f\right) }\left( 1+e~\cos f\right) ^{2},
\end{equation}%
where $i$, $\omega ^{\ast }$ and $f$ are the inclination of the orbit on the
equator, argument of the perigee, and the true anomaly of the spacecraft
orbit respectively. Therefore, rewriting the components of the magnetic part
of the Lorentz force as a function of orbital elements, we obtain
\begin{equation}
R_{mag}=\frac{q}{m}\frac{B_{0}}{~r^{2}}\left[
\begin{array}{c}
\omega _{e}(1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast }+f\right) ) \\
-\sqrt{\mu /p^{3}}\cos i~\left( 1+e~\cos f\right) ^{2}%
\end{array}%
\right] ,
\end{equation}

\begin{equation}
T_{mag}=\frac{q}{m}\frac{B_{0}}{~\sqrt{\mu p}}\left[
\begin{array}{c}
\frac{\dot{r}}{r}\sqrt{\mu /p^{3}}\cos i~\left( 1+e~\cos f\right)
^{2}+2\omega _{e}~\sqrt{\mu /p^{3}}\sin ^{2}i\sin \left( \omega ^{\ast
}+f\right) \\
\times \cos \left( \omega ^{\ast }+f\right) \left( 1+e~\cos f\right) ^{2}%
\end{array}%
\right] ,  \label{26}
\end{equation}

\begin{eqnarray}
N_{mag} &=&\frac{q}{m}\frac{B_{0}}{~\sqrt{\mu p}}\times  \label{27} \\
&&\left(
\begin{array}{c}
2\left( \omega _{e}(1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast }+f\right) )-%
\sqrt{\mu /p^{3}}\cos i~\left( 1+e~\cos f\right) ^{2}\right) \\
\sqrt{\mu /p^{3}}\cos i~\left( 1+e~\cos f\right) ^{2}~+\frac{\dot{r}}{r}%
\sqrt{\mu /p^{3}}\frac{\cos i~}{\sqrt{1-\sin ^{2}i\sin ^{2}\left( \omega
^{\ast }+f\right) }}\times \\
\left( 1+e~\cos f\right) ^{2}-2\frac{\mu }{p^{3}}\frac{\sin ^{3}i\cos
^{2}\left( \omega ^{\ast }+f\right) \sin \left( \omega ^{\ast }+f\right) }{%
1-\sin ^{2}i\cos ^{2}\left( \omega ^{\ast }+f\right) }\left( 1+e~\cos
f\right) ^{4}%
\end{array}%
\right) .  \notag
\end{eqnarray}

Now we develop the Lorentz force experienced by electric field $F_{elec}.$

According to Ulaby (2005) and Heilmann et al. (2012) we can write the
electric force as follows.

\begin{equation}
\vec{F}_{elec}=-\nabla \vec{V}_{elec}=\left( \frac{\partial ~V_{elec}}{%
\partial r}\hat{r}\mathbf{+}\frac{1}{r}\frac{\partial ~V_{elec}}{\partial
\Phi }\hat{\Phi}\mathbf{+}\frac{1}{r\sin \Theta }\frac{\partial ~V_{elec}}{%
\partial \Theta }\hat{\Theta}\right) ,
\end{equation}%
where $\vec{V}_{elec}$ is the electric potential,

\begin{equation}
V_{elec}=\frac{\vec{P}\cdot \hat{r}}{4\pi ~\epsilon _{o}r^{2}}.
\end{equation}
$\vec{P}=q\vec{d}$ is called the electric dipole moment, $\vec{d}$ is the
distance vector from charge $-q$ to charge $+q$, $\epsilon _{o}=8.85\times
10^{-12}$ coul$^{2}/(N-m^{2})\equiv volt^{-1}meter^{-1}$ is the permittivity
of free space. Then the final form of the Lorentz force experienced by an
electric dipole moment in the presence of electric field is

\begin{equation}
\vec{F}_{elec}=\frac{qd}{4\pi ~\epsilon _{o}r^{3}}\left( 2\cos \Phi ~\hat{r}%
\mathbf{+}\sin \Phi ~\hat{\Phi}\mathbf{+0~}\hat{\Theta}\right) .
\end{equation}%
Similarly as we did for the magnetic force, we can write the radial,
transverse, and normal components $\left( R_{elec},T_{elec},~N_{elec}\right)
$ of the electric force,

\begin{equation}
R_{elec}=-\frac{q}{m}\frac{d}{4\pi ~\epsilon _{o}r^{3}}\left( \omega _{e}-%
\dot{\Theta}\right) \sin ^{2}\Phi .
\end{equation}

\begin{equation}
T_{elec}=\vec{F}_{elec}\cdot \hat{t}~=\frac{q}{m}\frac{d}{4\pi ~\epsilon
_{o}r^{3}}\frac{r^{3}}{\sqrt{\mu ~a\left( 1-e^{2}\right) }}\left( \omega
_{e}-\dot{\Theta}\right) ~\dot{\Phi}\sin \Phi ~\cos \Phi .
\end{equation}

\begin{eqnarray}
N_{elec} &=&\vec{F}_{elec}\cdot \hat{n}~ \\
&=&\frac{q}{m}\frac{d}{4\pi ~\epsilon _{o}r^{3}}\frac{r^{2}}{\sqrt{\mu
~a\left( 1-e^{2}\right) }}\left( \omega _{e}-\dot{\Theta}\right) ~\dot{\Theta%
}~\sin ^{2}\Phi ~\cos \Phi .  \notag
\end{eqnarray}%
Similarly, we can write the components of the Lorentz force experienced by
an electric field as a function of orbital elements as follows.

\begin{equation}
R_{elec}=-\frac{q}{m}\frac{d}{4\pi ~\epsilon _{o}r^{3}}\left[
\begin{array}{c}
\omega _{e}\left( 1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast }+f\right)
\right) - \\
\sqrt{\mu /a^{3}\left( 1-e^{2}\right) ^{3}}~\cos i~\left( 1+e~\cos f\right)
^{2}%
\end{array}%
\right] .
\end{equation}

\begin{eqnarray}
T_{elec} &=&\frac{q}{m}\frac{d}{4\pi ~\epsilon _{o}r^{2}}\times  \label{35}
\\
&&\left( \omega _{e}-\sqrt{\mu /a^{3}\left( 1-e^{2}\right) ^{3}}\frac{\cos
i~\left( 1+e~\cos f\right) ^{2}}{1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast
}+f\right) }\right) \sin ^{2}i\cos \left( \omega ^{\ast }+f\right) \sin
\left( \omega ^{\ast }+f\right) .  \notag
\end{eqnarray}

\begin{eqnarray}
N_{elec} &=&\frac{q}{m}\frac{d}{4\pi ~\epsilon _{o}r^{3}}\times  \label{36}
\\
&&\left( \omega _{e}-\sqrt{\mu /a^{3}\left( 1-e^{2}\right) ^{3}}\frac{\cos
i~\left( 1+e~\cos f\right) ^{2}}{1-\sin ^{2}i\sin ^{2}\left( \omega ^{\ast
}+f\right) }\right) \sin i~\cos i\sin \left( \omega ^{\ast }+f\right) ~.
\notag
\end{eqnarray}%
Assuming that the spacecraft is equipped with a charged surface (screen) of
area $S$ with the electric charge $q=\int_{{S}}\sigma \,dS$ distributed over
the surface with density $\sigma $. Therefore, as in Tikhonov et al. (2011),
we can write the torque of these forces relative to the spacecraft's center
of mass as follows.

\begin{equation}
{\vec{T}}_{{L}}=\vec{T}_{mag}+\vec{T}_{elec}=\int_{S}{\sigma }{\vec{\rho}}%
\times (\vec{E}+\vec{V}\times {\vec{B})}{dS},
\end{equation}%
where $\vec{\rho}$\ is the radius vector of the screen's element $dS$
relative to the spacecraft's center of mass and $\vec{V}$ is the velocity of
the element $dS$ relative to the geomagnetic field. Finally, the torque due
to Lorentz force can be written as follows

\begin{equation}
\vec{T}_{mag}=\vec{\rho}_{{0}}\times {A}^{T}\left(
R_{mag},T_{mag},~N_{mag}\right) ^{T},\vec{T}_{elec}=\vec{\rho}_{{0}}\times {A%
}^{T}\left( R_{elec},T_{elec},~N_{elec}\right) ^{T},
\end{equation}

\begin{equation}
{\vec{\rho}}_{{0}}{=x}_{{0}}{i}_{{b}}{+y}_{{0}}{j}_{{b}}{+z}_{{0}}k_{{b}}{=}%
q^{-1}\int_{{S}}\sigma \,\vec{\rho}\,dS
\end{equation}%
$\vec{\rho}_{{0}}$\ is the radius vector of the charged center of a
spacecraft relative to its center of mass and $A^{{T}}$ is the transpose of
the matrix ${A.}$

\subsection{ Geomagnetic field model and its Torque}

In this paper we are using non-tilted dipole for the geomagnetic field. Let
a dipole magnetic field be $\vec{B}=(B_{1},B_{2},B_{3}),$ and the magnetic
moment be $\vec{M}=(m_{1},m_{2},m_{3})$ of the spacecraft. Therefore the
torque due to the geomagnetic field is
\begin{equation}
\vec{T}_{M}=\vec{M}\times \vec{B}
\end{equation}%
As in Wertz (1978) we can write geomagnetic field and the total magnetic
moment of the orbital system directed to the tangent of the orbital plane,
normal to the orbit, and in the direction of the radius respectively as
below:
\begin{eqnarray}
B_{1} &=&\frac{B_{0}}{2r^{3}}\sin \theta _{m}^{\prime }\,[3\cos (2f-\alpha
_{m})+\cos \alpha _{m}],\text{ }B_{2}=-\frac{B_{0}}{2r^{3}}\,\cos \theta
_{m}^{\prime }\,, \\
B_{3} &=&\frac{B_{0}}{2r^{3}}\sin \theta _{m}^{\prime }\,[3\sin (2f-\alpha
_{m})+\sin \alpha _{m}].
\end{eqnarray}

\begin{equation}
m_{1}=m\,\sin \theta _{m}^{\prime }\cos \alpha _{m}\,{\beta _{1}}%
,~m_{2}=m\,\sin \theta _{m}^{\prime }\sin \alpha _{m}\,{\beta _{2}},\text{ }%
m_{3}=m\,\cos \theta _{m}^{\prime }\,{\beta _{3}},
\end{equation}%
where $B_{0}=-8\times 10^{-15}$, $\theta _{m}^{\prime }=168.6^{\circ }$ is
the co-elevation of the dipole, $\alpha _{m}=109.3^{\circ }$, is the east
longitude of the dipole and $f$ is the true anomaly measured from ascending
node and $m$ is the magnitude of the total magnetic moment.

\section{ Equations of the attitude motion}

The nonlinear differential equation called Euler-Poisson \ equations are
used to describe the attitude orientation of the spacecraft.

\begin{equation}
{\ }\dot{\vec{\omega}}I+\vec{\omega}\times \vec{\omega}I=\vec{T}_{G}+\vec{T}%
_{M}+\vec{T}_{L},  \label{54}
\end{equation}%
\begin{equation}
\dot{\vec{\alpha}}+\vec{\alpha}\times \vec{\omega}\mathbf{=-}\Omega \vec{%
\gamma},\dot{\vec{\beta}}+\vec{\beta}\times \vec{\omega}=0,\,\,\,\,\,\,\,%
\dot{\vec{\gamma}}+\vec{\gamma}\times \vec{\omega}=\Omega \vec{\alpha}
\end{equation}%
where $\vec{T}_{G}=3\Omega ^{2}\vec{\gamma}\times \vec{\gamma}I$ is the well
known formula of the gravity gradient torque,

$I={diag}(A,B,C)$ is the inertia matrix of the spacecraft, $\Omega $ is the
orbital angular velocity, $\vec{\omega}$\textbf{\ }is the angular velocity
vector of the spacecraft. According to Wertz (1978) the angular velocity of
the spacecraft in the inertial reference frame is $\vec{\omega}=(p,q,r)%
\mathrm{,}$ where
\begin{equation}
p={\dot{\psi}\sin \theta \sin \phi +\dot{\theta}\,\cos \phi ,}q={\dot{\psi}%
\sin \theta \cos \phi -\dot{\theta}\,\sin \phi ,}r={\dot{\psi}\cos \theta +%
\dot{\phi}.}
\end{equation}

\subsection{Equations of motion in the pitch direction}

In this section the attitude motion of the spacecraft in the pitch direction
is considered, i.e. $\psi =\phi =0,~\theta \neq 0.$ Applying this condition
in equation (\ref{54}), we can derive the second order differential equation
of the motion in pitch direction.

\begin{eqnarray}
{A}\frac{d^{2}\theta }{dt^{2}} &=&(C-B)(3\Omega ^{2}-1){\sin \theta \cos
\theta }+z_{0}{\sin \theta }(~N_{mag}-kT_{mag}) \\
&&+z_{0}{\cos \theta }(kN_{mag}-T_{mag}){+}z_{0}{\sin \theta }%
(~N_{elec}-kT_{elec})  \notag \\
&&+z_{0}{\cos \theta }(kN_{elec}-T_{elec})+{m_{2}B}_{3}{\sin \theta +m_{3}B}%
_{2}{\cos \theta .}  \notag
\end{eqnarray}

Let $y_{0}=k~z_{0},$where $k$ is arbitrary number.%
\begin{eqnarray}
A\ddot{\theta} &=&\left( 3\Omega ^{2}-1\right) (C-B)\sin \theta \cos \theta
+z_{0}\sin \theta (N_{mag}-kT_{mag})  \notag \\
&&+z_{0}\cos \theta (kN_{mag}-T_{mag})+z_{0}\sin \theta (N_{elec}-kT_{elec})
\notag \\
&&+z_{0}\cos \theta (kN_{elec}-T_{elec})+B_{2}m_{3}\sin \theta
+B_{3}m_{2}\cos \theta  \notag \\
&=&g(k,z0,\alpha ^{\ast }=q/m,\theta ),  \label{pitchEq}
\end{eqnarray}%
where $N_{mag},T_{mag},N_{elec},T_{elec}$ are given in equations (\ref{26}),
(\ref{27}), (\ref{35}) and (\ref{36}) respectively.

A comparison in the oscillation of {\ }$\overset{.}{{\theta }}$ is given in
figures (\ref{Oscillationthetadot1}) to (\ref{Oscillationthetadot3}). It is
obvious from the first two figures that the most significant amount of
torque is coming from the magnetic part of the Lorentz torque which is of
the order $10^{-3}$. The effect from electric part of the Lorentz torque is
of the order $10^{-11}$ which is very small. The contribution from
geomagnetic torque is of the order $10^{-6}$. \ The {oscillation in }$%
\overset{.}{{\theta }}${\ due to total Torque is of the order }$10^{-3},$ as
shown in figure (\ref{Oscillationthetadot3} right). As the contribution from
the electric part of Lorentz force and geomagnetic field is very small
compared to the magnetic part of Lorentz force therefore it doesn't show up
in figure (\ref{Oscillationthetadot3} left). These figures are drawn for
fixed values of $B=0.7$, $C=0.1$, $\alpha ^{\ast }=\pm 1$.
\begin{figure}[tbp]
\centering
\resizebox{130mm}{!}{\includegraphics {fig2}}
\caption{{Oscillation in $\frac{d\protect\theta}{dt}$ due to (\textbf{left})
magnetic part from Lorentz Torque and (\textbf{right}) due to electric part
from Lorentz Torque. The dotted line corresponds to $\protect\alpha^* =-0.1$
and the continuous line corresponds to $\protect\alpha^*=0.1$}}
\label{Oscillationthetadot1}
\end{figure}

\begin{figure}[t]
\centering
\resizebox{130mm}{!}{\includegraphics {fig3}}
\caption{{Oscillation in $\frac{d\protect\theta }{dt}$ due to (\textbf{left}%
) geomagnetic torque  and (\textbf{right}) Torque due to total Lorentz force
and geomagnetic field. The dotted line corresponds to $\protect\alpha ^{\ast
}=-0.1$ and the continuous line corresponds to $\protect\alpha ^{\ast }=0.1$%
. Both the lines for $\protect\alpha ^{\ast }=\pm 0.1$ are overlapping.}}
\label{Oscillationthetadot3}
\end{figure}

\subsection{Derivation of equilibrium solutions in the pitch direction and
their linear stability analysis}

In this section the existence and stability of equilibrium position in the
pitch direction of a general shape spacecraft under the influence of
gravitational torque, Lorentz torque, and geomagnetic torque will be
discussed. The stability of the equilibrium solutions derived will be
discussed both analytically and numerically. To find the equilibrium
solutions, take the right hand side of equation (\ref{pitchEq}) equal to
zero which reduces to the following equation for $B=0.7,C=0.1,$ $%
a=6900km,i=51^{\circ },e=0.001,$ and $f=60^{\circ }.$

\begin{eqnarray}
g(k,z0,\alpha ^{\ast },\theta ) &=&(2.34\times
10^{-7}+(0.015+0.008k)z_{0}\alpha ^{\ast })\cos \theta  \label{pitcheq2} \\
&&+(1.56\times 10^{-6}+(0.008+0.015k)z_{0}\alpha ^{\ast })\sin \theta
+0.3\sin 2\theta =0.  \notag
\end{eqnarray}%
It is not possible to solve equation (\ref{pitcheq2}) in closed form as $%
\theta =f(k,z_{0},\alpha ^{\ast })$ therefore numerical techniques are used
to identify all the roots of equation (\ref{pitcheq2}). As equation (\ref%
{pitchEq}) is derived by taking$\ y=kz_{0},$ therefore without loss of
generality we take $z_{0}=1.$ For $0<\alpha ^{\ast }<1$ and $0<k<1,$ we have
five equilibrium solutions at $\theta \approx \frac{n\pi }{2},n=0,1,2,3,4$
when $\theta \in \lbrack 0,2\pi ].$ As $g(k,\alpha ^{\ast },\theta )$ is a
periodic function of period $2\pi $ therefore it is sufficient to
investigate the equilibrium solutions from $0$ to $2\pi .$ For $0<\alpha
^{\ast }<1$ and $k\leq 100$ there are five equilibrium solutions which
reduces to three or two when $k>100$. For sufficiently high values of $%
\alpha ^{\ast },$ the number of equilibrium solutions can be reduced to
three for even smaller values of $\alpha ^{\ast }.$ To see the progression
of roots from five to three see figures (\ref{rootprogressionTHETA-1}) where
$g(k,\alpha ^{\ast },\theta )$ is plotted for various fixed values of $k$
and $\alpha ^{\ast }.$ To completely describe the progression of the number
of equilibrium positions in $[0,2\pi ]$ from five to two a 3D implicit plot
of $g(k,\alpha ^{\ast },\theta )=0$ is given in figure (\ref{allrootstheta}%
). It can easily be seen that for high enough values of $\alpha ^{\ast }$
and $k$, the number of equilibrium points reduces to two. It is also obvious
from these figures that the equilibrium positions does not always remain at $%
\theta \approx \frac{n\pi }{2},n=0,1,2...$. By the comparison of figure (\ref%
{rootprogressionTHETA-1} left) and figure (\ref{rootprogressionTHETA-1}
right) it is evident that the equilibrium positions are not the same for
positively and negatively charged spacecrafts. It remains to be seen if this
or the other parameters such as $\alpha ^{\ast }$ or $k$ effect the
stability of the equilibrium points.

\begin{figure}[tbp]
\centering%
\resizebox{130mm}{!}{\includegraphics
{fig4}}
\caption{Progression of equilibrium solutions when (\textbf{left}) g1) $(%
\protect\alpha ^{\ast },k)=(0.1,1)$, g2) $(\protect\alpha ^{\ast },k)=(2,2)$%
, g3) $(\protect\alpha ^{\ast },k)=(5,7)$, g4) $(\protect\alpha ^{\ast
},k)=(7,7)$, g5) $(\protect\alpha ^{\ast },k)=(7,10)$, g6) $(\protect\alpha %
^{\ast },k)=(10,15)$ and (\textbf{right}) when ng1) $(\protect\alpha ^{\ast
},k)=(-0.1,1)$, ng2) $(\protect\alpha ^{\ast },k)=(-2,2)$, ng3) $(\protect%
\alpha ^{\ast },k)=(-5,7)$, ng4) $(\protect\alpha ^{\ast },k)=(-7,7)$, ng5) $%
(\protect\alpha ^{\ast },k)=(-7,10)$, ng6) $(\protect\alpha ^{\ast
},k)=(-10,15)$ }
\label{rootprogressionTHETA-1}
\end{figure}

\begin{figure}[tbp]
\centering\resizebox{70mm}{!}{\includegraphics {fig5}}
\caption{Implicit plot of $g(k,\protect\alpha ^{\ast },\protect\theta )=0$
when $z_{0}=1$.}
\label{allrootstheta}
\end{figure}

To discus the linear stability of the equilibrium points identified above we
use the standard procedure of linearization and convert equation (\ref%
{pitchEq}) to a system of two first order equations. We then find the
eigenvalues of the jacobian matrix from the equation given below.

\begin{equation}
\lambda ^{2}-\frac{\partial g(k,z0,\alpha ^{\ast },\theta )}{\partial \theta
}=0,  \label{Charactristic}
\end{equation}%
where%
\begin{eqnarray}
\frac{\partial g(k,z0,\alpha ^{\ast },\theta )}{\partial \theta } &=&A^{-1}({%
\cos \theta (1.56\times 10^{-6}+10^{-3}(8+}{15k)z_{0}\alpha ^{\ast })}
\label{G-theta} \\
&&{+(B-C)\cos (2\theta )}{-\sin \theta (2.34\times 10^{-7}+10^{-3}(15}{%
+8k)z_{0}\alpha ^{\ast })}.  \notag
\end{eqnarray}%
It is clear from equation (\ref{Charactristic}) that there are only two
types of eigenvalues possible. If $g_{\theta }(k,z0,\alpha ^{\ast },\theta
)>0$ there will be two eigenvalues one of which is negative and one
positive. A positive eigenvalue always imply instability. If $g_{\theta
}(k,z0,\alpha ^{\ast },\theta )<0$ the eigenvalues obtained will be
imaginary with a zero real part which means the equilibrium point in
question will be spectrally stable. Initially we will investigate the
equilibrium points obtained above for $A=1,B=0.7,C=0.1,z_{0}=1.$

\begin{eqnarray}
\frac{\partial g(k,\alpha ^{\ast },\theta )}{\partial \theta } &=&g_{\theta
}={\cos \theta (1.56\times 10^{-6}+10^{-3}(8+15k)\alpha ^{\ast })}{+0.6\cos
(2\theta )}  \notag \\
&&-{\sin \theta (2.34\times 10^{-7})}{+10^{-3}\sin \theta (15+8k)\alpha
^{\ast }.}
\end{eqnarray}

\begin{figure}[tbp]
\centering%
\resizebox{130mm}{!}{
        \includegraphics{fig6}}
\caption{Trajectory in the $\protect\theta $-$\frac{d\protect\theta }{dt}$
phase plane when \textbf{(left) }$\protect\alpha ^{\ast
}=0.01,k=1,z_{0}=1,A=1,B=0.7,C=0.1$,\textbf{(right) }$\protect\alpha ^{\ast
}=-1,k=100,z_{0}=1,A=1,B=0.7,C=0.1$. }
\label{theta1}
\end{figure}
The values of $g_{\theta }|_{\theta =0}$ and $g_{\theta }|_{\theta =2\pi }$
remain positive for all positive values of $k$ and $\alpha ^{\ast }$ which
implies that the equilibrium position at $\theta =0$ and $\theta =2\pi $ are
unstable. The value of $g_{\theta }|_{\theta =\pi /2}$ is negative for all
positive values of $k$ and $\alpha ^{\ast }$ which implies that the
equilibrium position at $\theta =\pi /2$ will be stable. By similar
argument, the equilibrium position at $\theta =\pi $ will be stable if $%
\alpha ^{\ast }$ satisfy the following inequality.%
\begin{equation*}
\alpha ^{\ast }>\frac{0.6}{0.008+0.015k}=\alpha _{1}^{\ast }.
\end{equation*}%
This also means that $\theta =\pi $ will always be unstable if the
spacecraft is negatively charged as the right hand side of the above
inequality is always positive. To check the stability of the remaining four
equilibrium positions when $\alpha ^{\ast }<0,$ Let $\alpha ^{\ast }=-\alpha
_{p}$ such that $\alpha _{p}>0.$ It can easily be shown that the equilibrium
position at $\theta =0$ and $\theta =2\pi $ will be stable if $\alpha
_{p}<\alpha _{1}^{\ast }.$ For example when $k=1,$ $\alpha _{p}$ must be
smaller than $26.09.$ Similarly, for the equilibrium position at $\theta
=\pi /2$ to be stable for negatively charged spacecraft $\alpha _{p}$ must
satisfy the following inequality.%
\begin{equation*}
\alpha _{p}<\frac{0.6}{0.008k+0.015}.
\end{equation*}%
It can be safely concluded from this discussion that the sign and amount of
charge on the spacecraft plays a significant role in the stability of the
equilibrium positions. A typical trajectory in the $\theta $-$\frac{d\theta
}{dt}$ phase plane around $\theta =0$ is given in figure ( \ref{theta1}) for
$\alpha ^{\ast }=0.01,-1,$ $k=1,100,$ $z_{0}=1,$ $A=1,$ $B=0.7,$ $C=0.1$. It
can be seen that all the trajectories are moving away from $\theta =0$ when $%
\alpha ^{\ast }=0.01,$ which indicate instability. In the second case it is
stable.

\begin{figure}[tbp]
\centering
\resizebox{130mm}{!}{
        \includegraphics{fig7}}
\caption{A set of orbits with initial positions close to $(\protect\theta ,%
\dot{\protect\theta )}\rightarrow (0+,0+)$ in the $\protect\theta $-$\frac{d%
\protect\theta }{dt}$ phase plane when $\protect\alpha ^{\ast
}=0.01,k=1,z_{0}=1,A=1,B=0.7,C=0.1$ }
\label{theta2}
\end{figure}
\begin{figure}[tbp]
\centering
\resizebox{130mm}{!}{
        \includegraphics{fig8}  }
\caption{Same orbits as in figure (\protect\ref{theta2}) but integrated for
much longer time.}
\label{theta3}
\end{figure}

\begin{figure}[tbp]
\centering
\resizebox{130mm}{!}{
        \includegraphics{fig9} }
\caption{A set of orbits with initial positions close to $(\protect\theta ,%
\dot{\protect\theta})\rightarrow (0-,0-)$ in the $\protect\theta $-$\frac{d%
\protect\theta }{dt}$ phase plane when $\protect\alpha ^{\ast
}=0.01,k=1,z_{0}=1,A=1,B=0.7,C=0.1$ }
\label{theta4}
\end{figure}
\begin{figure}[tbp]
\centering
\resizebox{140mm}{!}{
        \includegraphics{fig10}}
\caption{Same orbits as in figure (\protect\ref{theta4}) but integrated for
much longer time.}
\label{theta5}
\end{figure}

To understand the long term behavior of orbits around the equilibrium
positions a set of orbits with initial positions close to $(\theta ,\dot{%
\theta )}\rightarrow (0+,0+)$ and $(\theta ,\dot{\theta )}\rightarrow (\frac{%
\pi }{2},0)$ are given in figures ( \ref{theta2}, \ref{theta3}) in the $%
\theta $-$\frac{d\theta }{dt}$ phase plane when $\alpha ^{\ast
}=0.01,k=1,z_{0}=1,A=1,B=0.7,C=0.1$. These orbits are allowed to evolve for
a short period of time and their trajectories are traced in figure ( \ref%
{theta2}). It can be seen that the orbits starting close to $0+$ (close to 0
and positive) are immediately captured by the nearby stable equilibrium at $%
\frac{\pi }{2}.$ The orbits which start near $(\theta ,\dot{\theta )}%
\rightarrow (\frac{\pi }{2},0)$ remain in elliptic orbit around $(\frac{\pi
}{2},0).$ When these orbits are allowed to evolve for a longer period of
time some of the orbits near $(0+,0+)$ are being captured by the nearby
stable equilibrium at $\frac{3\pi }{2}$ and a couple of orbits escapes. The
orbits close to $(\frac{\pi }{2},0)$ remain in near circular orbit about the
center which is a strong evidence of the existence of periodic orbits around
$(\frac{\pi }{2},0).$ Similar analysis is performed for orbits with initial
positions close to $(\theta ,\dot{\theta )}\rightarrow (0-,0-)$ and $(\theta
,\dot{\theta )}\rightarrow (-\frac{\pi }{2},0).$ It can be seen in figures ( %
\ref{theta4}, \ref{theta5}) that the orbits starting close to $0-$ (close to
0 and negative) are immediately captured by the nearby center at $-\frac{\pi
}{2}.$ Some of them when integrated for a much longer period of time gets
captured by the center at $\frac{-3\pi }{2}.$ The orbits which start around $%
(-\frac{\pi }{2},0)$ remain in near circular orbits around $(-\frac{\pi }{2}%
,0).$ Similar behavior is observed around all the spectrally stable
equilibriums. Therefore we can safely conjecture that around each stable
equilibrium position there is a family of periodic orbits.

As mentioned earlier and shown in figures (\ref{rootprogressionTHETA-1}) and
(\ref{allrootstheta}) the number of equilibrium points \ when $0<\theta
<2\pi $ reduce from five to three and in some cases two for higher values of
$\alpha ^{\ast }$ and $k.$ For example when $B=0.7,C=0.1,k=10,z_{0}=1,$ and $%
\alpha ^{\ast }=7$ there are two equilibrium points at $\theta =2.32$
(stable)$,5.89$ (unstable)$.$ If $\alpha ^{\ast }=-7$ i.e. the spacecraft is
negatively charged, the position of the two equilibriums are changed and the
stability reversed. When $|B-C|<1$, the positions of the equilibrium points
are almost identical to what we have shown above. Its effect on stability is
explained below.

\begin{enumerate}
\item $\theta =0,2\pi $: When $B\geq C$ and $\alpha ^{\ast }>0,$ the
equilibrium positions at $\theta =0,2\pi $ will be stable. But when $\alpha
^{\ast }<0$ these two equilibrium positions are unstable.

\item $\theta =\frac{\pi }{2}$: For $B\geq C$ the equilibrium position at $%
\theta =\frac{\pi }{2}$ is always stable$.$ However, when $B<C$ the value of
$\alpha ^{\ast }$ have to be significantly high in which case $\theta =\frac{%
\pi }{2}$ will no more be an equilibrium position.

\item $\theta =\pi :$When $B\leq C$ and $\alpha ^{\ast }>0,$ $\theta =\pi $
is stable. For $\ $\ a negatively charged spacecraft $B<C$ is a necessary
condition for the stability of the equilibrium position at $\theta =\pi .$
Therefore when $B>C,$ the value of $\alpha ^{\ast }$ have to be
significantly high in which case $\theta =\pi $ will no more be an
equilibrium position.

\item $\theta =\frac{3\pi }{2}:B>C$ is a necessary and sufficient condition
for the stability of the equilibrium position at $\theta =\frac{3\pi }{2}$
unless $\alpha ^{\ast }$ is very large and negative in which case $\theta =%
\frac{3\pi }{2}$ will no more be at equilibrium position.
\end{enumerate}

In summary, when $0<\alpha ^{\ast }<1$ and $B<C,$ the equilibrium positions
at $\theta =0,\pi ,2\pi $ are stable and at $\theta =\frac{\pi }{2},\frac{%
3\pi }{2}$ are unstable and when $B>C$ the nature of the five equilibrium
positions is reversed. To demonstrate this behavior a typical example is
given in figure (\ref{theta6})
\begin{figure*}[tbp]
\centering%
\resizebox{130mm}{!}{
        \includegraphics{fig11}}
\caption{Trajectory in the $\protect\theta $-$\frac{d\protect\theta }{dt}$
phase plane when $\protect\alpha =0.01,k=1,z_{0}=1$ and \textbf{\ (Left)}: $%
B=0.5,C=0.9$. \textbf{(Right)}: $B=0.9,C=0.5$. Black dots correspond to
stable equilibrium points and red dots correspond to unstable equilibrium
points.}
\label{theta6}
\end{figure*}

\subsection{Equations of motion in the roll direction}

In this section we study the attitude motion of the spacecraft in the roll
direction, i.e. $\psi =\theta =0,~\phi \neq 0.$ Applying this condition in
Euler equation of the attitude motion of the spacecraft, we obtain the
second order differential equation of the motion in roll direction.%
\begin{eqnarray*}
{C}\frac{d^{2}{\phi }}{dt^{2}} &=&\Omega ^{2}(A-B){\sin \phi \cos \phi }%
+x_{0}{\sin \phi }(kT_{mag}-~R_{mag})+x_{0}{\cos \phi }(T_{mag}-kR_{mag}) \\
&&+x_{0}{\sin \phi }(kT_{elec}-~R_{elec})+x_{0}{\cos \phi (T_{elec}}{%
-~kR_{elec})+m_{1}B}_{2}{\cos \phi -{m_{2}B}_{1}\sin \phi .}
\end{eqnarray*}

Let ${y}_{{0}}=k{x}_{{0}}~$meter then,

\begin{eqnarray}
{\ C}\frac{d^{2}{\phi }}{dt^{2}} &=&\Omega ^{2}(A-B){\sin \phi \cos \phi }%
+x_{0}{\sin \phi }(kT_{mag}-~R_{mag})+x_{0}{\cos \phi }(T_{mag}-kR_{mag})
\notag \\
&&{+}x_{0}{\sin \phi }(kT_{elec}-~R_{elec})+x_{0}{\cos \phi (T_{elec}}{%
-~kR_{elec})\cos \phi +m_{1}B}_{2}{\cos \phi -{m_{2}B}_{1}\sin \phi }  \notag
\\
&=&h(\alpha ^{\ast },k,{\phi ,x}_{0},{A,B).}  \label{rollEq}
\end{eqnarray}

\subsection{Derivation of equilibrium solutions in Roll direction and their
linear stability analysis}

In this section the existence and stability of equilibrium positions in the
roll direction of a general shape spacecraft under the influence of
gravitational torque, Lorentz torque, and geomagnetic torque will be
discussed. The stability of the equilibrium positions derived will be
discussed both analytically and numerically. To find the equilibrium
positions, take the right hand side of equation (\ref{rollEq}) equal to zero
which reduces to the following equation for $A=0.1,B=0.7,C=1,$ $x_{0}=1,$ $%
a=6900km,i=51^{\circ },e=0.001,$ and $f=60^{\circ }.$
\begin{eqnarray}
h_{1}(k,\alpha ^{\ast },\phi )=(1.036\times 10^{-7}+(-0.012+9.41k)\alpha
^{\ast }) &&\times \cos \phi +(-1.13\times 10^{-7}  \notag \\
+9.41\times 10^{-10}\alpha ^{\ast }-0.012k\alpha ^{\ast })\sin \phi
&&-3.63\times 10^{-7}\sin (2\phi )=0.
\end{eqnarray}

It is not possible to solve equation $h_{1}(k,\alpha ^{\ast },\phi )=0$ in
closed form as $\phi =f(k,\alpha ^{\ast })$ therefore numerical techniques
are used to identify all the roots of equation $h_{1}(k,\alpha ^{\ast },\phi
)=0$ which are the desired equilibrium solutions. Let $k=1.$ For $\alpha
^{\ast }\in (-2.15\times 10^{-5},2.15\times 10^{-5})$ i.e for a very small
amount of charge, there are four equilibrium solutions and for higher values
of $\alpha ^{\ast }$ there are two equilibrium solutions.

\begin{enumerate}
\item $\phi _{1}\in (0,0.4)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{2}\in (1.47,1.89)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{3}\in (2.80,3.18)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{4}\in (4.37,4.73)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5})$

\item $\phi _{5}=2.36$ when $\left\vert \alpha ^{\ast }\right\vert
>2.15\times 10^{-5}$

\item $\phi _{6}=5.5$ when $\left\vert \alpha ^{\ast }\right\vert
>2.15\times 10^{-5}$
\end{enumerate}

In the above example $B>A.$ Now we switch the values of $A$ and $B$ to have $%
B<A$ and find the location of the equilibrium positions. In this case we
still have four equilibrium solutions when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5})$ and two when $\left\vert \alpha ^{\ast
}\right\vert >2.15\times 10^{-5}.$ All the equilibrium positions when $%
A=0.7, $ $B=0.1,k=1,C=1,x_{0}=1,$ $a=6900km,i=51^{\circ },e=0.001,$ and $%
f=60^{\circ }$ are listed below.

\begin{enumerate}
\item $\phi _{7}\in (1.23,1.58)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{8}\in (3.12,3.47)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{9}\in (4.69,5.04)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5}).$

\item $\phi _{10}\in (5.97,6.32)$ when $\alpha ^{\ast }\in (-2.15\times
10^{-5},2.15\times 10^{-5})$

\item $\phi _{11}=2.36$ when $\left\vert \alpha ^{\ast }\right\vert
>2.15\times 10^{-5}$

\item $\phi _{12}=5.5$ when $\left\vert \alpha ^{\ast }\right\vert
>2.15\times 10^{-5}$
\end{enumerate}

As ${y}_{0}=kx_{0}$ we can take $x_{0}=1.$ To reduce the dimensions, without
loss of generality, we define $ab=A-B$ and rewrite $h_{1}(k,\alpha ^{\ast
},\phi ,ab)$ as below.%
\begin{eqnarray*}
h_{1}(k,\alpha ^{\ast },\phi ,ab) &=&\left( -1.13\times
10^{-7}+10^{-3}(6k-15)\alpha ^{\ast }\right) {\cos }\phi +1.03\times 10^{-7}
\\
&&+10^{-3}(6-15k)\alpha ^{\ast }){\sin }\phi +6.05\times 10^{-7}ab{\sin }%
(2\phi ).
\end{eqnarray*}%
It can be seen from figure (\ref{roll1}) that there are four equilibrium
solutions for small values of $\alpha ^{\ast }$ and all values of $ab$ when $%
k=1$. For higher values of $\alpha ^{\ast }$ there are only two equilibrium
solutions at $\phi =2.3$ and $\phi =5.5$ for all values of $ab$. We have
seen above that for $k=1$, the changing values of $ab$ and $\alpha ^{\ast }$
have significant effect on the existence of equilibrium solutions in the
roll direction. To see this for $k\neq 1$, we plot $h_{1}(k,\alpha ^{\ast
},\phi ,ab)=0$ for fixed values of $ab=0.3,$ and $ab=-0.3$ in figure (\ref%
{roll2b}). It is clear from figure (\ref{roll2b}) that with the changing
value of $k$ the position of equilibrium changes significantly but the
numbers of equilibrium positions remains four as before both for negative
and positive values of $ab$ when $\alpha ^{\ast }$ is small. The effect of $%
ab$ is significant when $k<1.$ For higher values $\alpha ^{\ast }$ the
number of equilibrium positions remains to be two but their positions change
with the changing values of $ab$ and $k,$ see figures (\ref{roll1} and \ref%
{roll2b}).

\begin{figure}[tbp]
\centering
\resizebox{65mm}{!}{
        \includegraphics{fig12} }
\caption{Implicit plots of ( $h(\protect\alpha ,\protect\phi ,ab)=0$. }
\label{roll1}
\end{figure}

\begin{figure}[tbp]
\centering
\resizebox{130mm}{!}{
        \includegraphics{fig13}}
\caption{Implicit plots of $h(\protect\alpha ,\protect\phi ,k,ab)=0$ when (%
\textbf{left}) $ab=-0.3$ and (\textbf{right}) $ab=0.3$.}
\label{roll2b}
\end{figure}

To study the stability of the equilibrium position derived we use the same
method which was used for the pitch direction. We write equation (\ref%
{rollEq}) as a system of two first order equations, linearize them, and find
the eigenvalues of the jacobian matrix from the equation given below.

\begin{equation}
\lambda ^{2}-h_{\phi }(\alpha ^{\ast },k,\phi ,ab)=0.  \label{CharEqphi}
\end{equation}%
It can be seen from equation (\ref{CharEqphi}) that there are only two types
of eigenvalues possible. If $h_{\phi }(\alpha ^{\ast },k,\phi ,ab)>0$ there
exist two eigenvalues one of which is negative and one is positive.
Therefore $h_{\phi }(\alpha ^{\ast },k,\phi ,ab)>0$ becomes a sufficient
condition for instability. If $h_{\phi }(\alpha ^{\ast },k,\phi ,ab)<0$ the
equilibrium point in question will be spectrally stable or a stable center.
We will investigate the equilibrium points obtained above for $%
ab=0.6,ab=-0.6,k=1,x_{0}=1$,$~a=6900km,i=51^{\circ },e=0.001,f=60^{\circ },$
and write $h_{\phi }(\alpha ^{\ast },\phi ,ab)$ as below.%
\begin{eqnarray*}
h_{\phi }(\alpha ^{\ast },\phi )|_{ab=-0.6} &=&(-1.13\times
10^{-7}-0.01\alpha ^{\ast })\cos \phi -7.26\times 10^{-7}(\cos \phi )^{2} \\
&&+(-1.04\times 10^{-7}+0.01\alpha ^{\ast }+7.26\times 10^{-7}\sin \phi
)\sin \phi .
\end{eqnarray*}%
The equilibrium positions at $\phi _{1}$ and $\phi _{3}$ are stable as in
these cases $h_{\phi }(\alpha ^{\ast },\phi )|_{ab=-0.6}<0$. Similarly $\phi
_{2}$ and $\phi _{4}$ are unstable as in these cases $h_{\phi }(\alpha
^{\ast },\phi )|_{ab=-0.6}>0.$ By similar arguments $\phi _{5}$ will be an
unstable equilibrium if the spacecraft is positively charged and $\phi _{6}$
will be unstable if the spacecraft is negatively charged. Similarly, when $%
ab=0.6,$ $\phi _{7}$ and $\phi _{9}$ are stable, $\phi _{8}$ and $\phi _{10}$
are unstable, $\phi _{11}$ is stable when $\alpha ^{\ast }<-1.11\times
10^{-6}$ and $\phi _{12}$ is stable when $\alpha ^{\ast }>3.25\times
10^{-7}. $ A typical example is given in figure (\ref{phiphase11}) when $%
ab=\pm 0.6$. The equilibrium at $\phi =2.36$ is stable when $\alpha ^{\ast
}=-0.1$ and unstable when $\alpha ^{\ast }=0.1$. Similarly, the equilibrium
at $\phi =5.5 $ is unstable when $\alpha ^{\ast }=-0.1$ and stable when $%
\alpha ^{\ast }=0.1$.
\begin{figure*}[tbp]
\centerin
\resizebox{130mm}{!}{
        \includegraphics{fig14}        }
\caption{Trajectory in the $\protect\phi $-$\frac{d\protect\phi }{dt}$ phase
plane when \textbf{(left) }$\protect\alpha ^{\ast }=0.1,k=1,z_{0}=1,ab=-0.6$,%
\textbf{(right) }$\protect\alpha ^{\ast }=-0.1,k=1,z_{0}=1,ab=-0.6$. }
\label{phiphase11}
\end{figure*}
\begin{figure*}[tbp]
\centerin
\resizebox{130mm}{!}{
        \includegraphics{fig15}        }
\caption{A set of orbits with initial positions close to the equilibrium
positions when $e=0.1,ab=0.6$ and (left) $\protect\alpha ^{\ast }=-0.1$,
(right) $\protect\alpha ^{\ast }=0.1$ }
\label{phiorbit1}
\end{figure*}

\begin{figure*}[tbp]
\centerin
\resizebox{130mm}{!}{
        \includegraphics{fig16}        }
\caption{A set of orbits with initial positions close to the equilibrium
positions when $e=0.1,ab=0.6$ and (left) $\protect\alpha ^{\ast }=-1$,
(right) $\protect\alpha ^{\ast }=1$ }
\label{phiorbit2}
\end{figure*}

To understand the long term behavior of orbits around the equilibrium
positions, a set of orbits with initial positions close to the equilibrium
are given in figures ( \ref{phiorbit1}, \ref{phiorbit2}) in the $\phi $-$%
\frac{d\phi }{dt}$ phase plane when $\alpha ^{\ast }=\pm 0.1,\pm 1,$ and $%
k=1,z_{0}=1,ab=0.6,e=0.1$. These orbits are allowed to evolve for a long
period of time and their trajectories are traced in figures (\ref{phiorbit1}%
, \ref{phiorbit2}). The orbits in figure (\ref{phiorbit1}left) are given for
$\alpha ^{\ast }=-0.1$ and it can be seen that all the orbits are captured
by the equilibrium position at $\phi =2.36$ which is a stable equilibrium.
The orbits which are closer to the stable equilibrium position remain in
perfect periodic orbit while the orbits which are not so close have an
elliptic orbit in the vicinity of the equilibrium position but are not
necessarily periodic. For $\alpha ^{\ast }=0.1$ in figure ( \ref{phiorbit1}%
right), the equilibrium position at $\phi =2.36$ is unstable. Hence the same
orbits are captured by another nearby stable equilibrium at $\phi =-0.723$
which is a mirror image of the stable equilibrium at $\phi =5.56$. When the
same orbits are integrated for $\alpha ^{\ast }=\pm 1$, similar behavior is
observed. Also, similar behavior is observed around all the stable
equilibriums. Therefore we can safely conjecture that around each stable
equilibrium position there is a family of periodic orbits.

\section{Conclusions}

The paper discussed the attitude stabilization of a charged spacecraft
moving in an elliptic orbit using Lorentz torque. The Lorentz torque is
developed in two parts $T_{mag}$ and $T_{elec}.$ $T_{mag}$ is the Lorentz
torque which is experienced by magnetic field and $T_{elec}$ is the Lorentz
Torque experienced by an electric dipole moment in the presence of electric
field. The model we developed incorporates all Lorentz torques as a function
of orbital elements and the radius vector of the charged center of the
spacecraft relative to it's center of mass. We investigated, both
analytically and numerically, the existence and stability of equilibrium
positions both in pitch and roll directions. In the pitch direction there
are a total of five equilibrium points at $\theta =n\pi /2,n=0,1,2,3,4$ when
$-1<\alpha ^{\ast }=q/m<1$, $0<k<1$ and $\theta \in \lbrack 0,2\pi ]$ .
Their stability is analyzed for changing values of the charge to mass ratio,
$\alpha ^{\ast }$, and it is shown that $\alpha ^{\ast }$ effect the
stability and existence of equilibrium positions. The equilibrium positions
at $\theta =0,2\pi $ are unstable for $\alpha ^{\ast }>0$ when $B=0.7$ and $%
C=0.1.$ These two equilibrium positions are stable when $B=0.1$ and $C=0.7.$
These equilibrium positions are also stable for $\alpha ^{\ast }<0.$ We have
shown that the sign and amount of charge play a significant role in
determining the equilibrium positions and their stability. In the case of
roll direction we have four equilibrium points when $\alpha ^{\ast }\in
(-2.15\times 10^{-5},2.15\times 10^{-5})$ and only two equilibrium positions
when $\alpha ^{\ast }\notin (-2.15\times 10^{-5},2.15\times 10^{-5}).$ It is
demonstrated both analytically and numerically that almost all the
equilibrium positions depend on the values and sign of charge to mass ratio
both in terms of existence and stability. In the same way as in pitch
direction, the equilibrium positions which are stable for $A<B$ becomes
unstable when $A>B$ and vice versa. This is not true in general but this
happens in most of the cases. Here $A,B$ and $C,$ refers to the components
of moment of inertia of the spacecraft.