Controlling the Perturbations of Solar Radiation Pressure on the Lorentz Spacecraft

: The aim of the present paper is to analyze the viability of using Lorentz Force (LF) acting on a charged spacecraft to neutralize the effects of Solar Radiation Pressure (SRP) on the longitude of the ascending node and the argument of perigee of the spacecraft’s orbit. In this setting, the Gauss planetary equations for LF and SRP are presented and averaged over the true anomaly. The averaged variations for the longitude of the ascending node ( h ) and the argument of perigee ( g ) are invariant under the symmetry ( i , g ) −→ ( − i , − g ) due to Lorentz Force. The sum of change rates due to both perturbing forces of LF and SRP is assigned by zero to estimate the charge amount to balance the variation for the argument of perigee and longitude of ascending. Numerical investigations have been developed to show the evolution of the charge quantity for different orbital parameters at both Low Earth and Geosynchronous Orbits.


Introduction
The importance of taking into account the influence of radiation pressure on artificial satellite motion does not admit discussion when noting the discrepancies between theories computations and practical observations of balloon-type satellites. The influence that is related to solar radiation pressure exceeds that of drag atmospheric at a height of 800 km and the force equals 10 −5 dyn/cm 2 for an atmospherical model with an exospheric temperature of 1400 k [1].
Many studies are devoted to the study, analysis, and control perturbations of Solar Radiation Pressure (SRP) on the Lorentz Spacecraft (LS) from different views. In [2], the authors used the variation of vector elements approach to obtain first order relations for the change rates in the osculating elements, which are generated by SRP. In [3], the authors found first order solutions using Lagrange's Planetary Equations (LPE). While [4] studied canonically the effect of SRP on space craft with complex shape. Furthermore, in [5], the authors studied the resonance effect, which is introduced by the commensurability between the different mean motions. Some significant works have been performed to analyze the effect of SRP. Good examples between others are: In [1] the author discussed the effect as one of the non-gravitational force. In [6], the authors orbit. Lorentz Force is also used to eliminate the perturbation impact of the solar radiation pressure, which demands a low-charging level that is smaller than 10 −2 C/kg. This work includes eight sections: A literature review of controlling the perturbations of solar radiation pressure on the Lorentz spacecraft is covered in Section 1. In Section 2, the Gauss planetary equations are recalled. In Section 3, orbital averaged properties are presented while the effect of solar radiation pressure and Lorentz Force are discussed in Sections 4 and 5, respectively. In Section 6, the values of charge per mass unit to balance the variations on the longitude of the ascending node and the argument of perigee are found as functions in the orbital elements. Section 7 is devoted to address the numerical results. Finally, the conclusion section is stated for illustrating the results graphically for Low Earth and Geosynchronous Orbits.

Perturbations in Orbital Elements
Let the elements of the Kepler orbit of the spacecraft be a, e, i, h, g, and l in their usual notations. In addition, n is the mean motion and r is the polar radial distance from Earth's mass center. Furthermore, we impose that R, T, and W be the perturbing forces components in three directions orthogonal axes, which are called the radial, transverse, and normal component of the perturbing force. Then LPE for perturbations in the orbital elements in Gauss form are given by [27]: and cos u = (e + cos f )/(1 + e cos f ), f is the true anomaly, r = p/(1 + e cos f ), p = a(1 − e 2 ) is the semiilatus rectum,h = √ µp is the constant of angular momentum, and µ is the gravitational parameter. Further orbital elements are defined by: • a is the semi-major axis; • e is the eccentricity; • I is the inclination; • Ω is the longitude of ascending node; •ω is the longitude of the perigee; • is the mean longitude at epoch.
The above equations are first constructed by Gauss and used to estimate the first order perturbation of Jupiter with respect to Pallas. Gauss also used these equations for constructing secular variation in the orbital elements. In fact the importance of these equations is extended to compute the element perturbations of comets and minor planets by numerical integration. Furthermore, it can be used in some applications to find only approximate perturbations. In order to avoid the time outside of trigonometric arguments and have more convenience with canonical equations (see [27] for details) the variables I, Ω,ω, and ε will be replaced by i, h, g, and l respectively. We would like to referω = ω + Ω where ω is the angular distance from the ascending node to the perigee and l = n t + −ω. Utilizing these variables with Equation (1), we get: where and here l is called the mean anomaly, and n is the mean motion.
In what follows, we will study the effect of the SRP and LF in Sections 3 and 4. The components R, T, and W of the solar radiation pressure and Lorentz Force will be calculated. The variations of the argument of perigee and longitude of the node due to both forces will also be illustrated.

Orbital Averaging
The averaging is an effective and beneficial method to analyze the variational equations of orbital elements. The classical theory of averaging was primarily modified to simplify the nonlinearity of non-autonomous dynamical systems. If the dynamical system is expressed by: ) and δ is a very small parameter, which may represent some physical quantity. We also denote by Γ orΓ for the averaging of Γ. This averaging yields a mean value, thereby we get: The main objective of averaging is eliminating the time dependence of the original dynamical system. Thus giving more simplifications for the obtained dynamical system. Although, we have to investigate under what restrictions the averaged and original dynamical system match, the averaging has a realistic meaning. Then with Equations (3) and (4), the averaged dynamical system is:Ø Since only the secular and long periodic variations affect the orbit with time, and eventually these perturbations change the orbit from its unperturbed state, we average the equations over the fast variable f (true anomaly) to keep only secular and long-period perturbations. Thereby, the application of the averaging theory on the variational equations of orbit is controlled by: where Γ is a perturbing potential and P = 2π/n is the period.

Effect of Solar Radiation Pressure
Solar Radiation (SR) is the expression that is used to characterize the Sun's radiation. The exposure to this radiation is not a large issue on the Earth's surface as the atmosphere is a protective layer. However, at a high altitude there is less of this preventive layer. Thereby we are exposed to a higher level of radiation. In astronomy, SRP is the force exerted by SR on objects within its reach. SRP is of interest in astrodynamics as it is one source of orbital perturbations. While Radiation Pressure (RP) is the pressure exerted upon any exposed surface to the radiation of electromagnetic. If it is absorbed, the pressure is the flux density of power divided by the light speed. But in the case that this radiation is completely reflected, the RP is doubled. For example, the Sun's radiation towards the Earth has a power flux density of 1368 Watt/m 2 , so the RP is about 4.6 MPa (absorbed) [28].
The acceleration experienced by an object with mass m and cross-sectional area A, under the effect of SRP, when all of the solar radiation is absorbed is: where R(γ) = A S 0 m C (1 + α) cos 2 γ S = cos θ i + cos ε sin θ j + sin ε sin θ k (8) and • C is the light speed; • r 0 is the distance between the Earth and Sun; • r 1 is the distance between the satellite and Sun; • α is represents the reflection coefficient of the surface; • γ is an incident angle of falls ray at to the surface; • s 0 is a solar constant due to the mean distance between the Earth and Sun; • S is a unit vector in the direction of Satellite-Sun given in a geocentric equatorial frame; • i, j and k are unit vectors in the geocentric coordinates system; • θ is the true longitude of the Sun; • ε is the obliquity of ecliptic.
In addition to direct solar radiation pressure, there is an indirect radiation pressure emitted from the Earth, called Albedo, which is the measure of the diffuse reflection of solar radiation out of the total solar radiation received by Earth. It causes a small perturbing force on a satellite moving around the Earth. This perturbation decreases slightly with increasing altitude. Another indirect radiation pressure is perpendicular to the surface, related to specular reflection, that is the mirror-like reflection at the air-surface interface. We would like to refer that the components of indirect radiation pressure effects are neglected in Equations (7) and (8), see [4] for details. For simplicity, we impose that the Sun is moving in circular orbits, such that θ becomes its mean longitude. In addition the direction and distance of the satellite from the Sun are similar to those of the Earth (r 1 is the Earth-Sun mean distance). Hence r 0 ∼ = r 1 and R(γ) = R c is considered a constant. In this context, we also impose that R s , T s , and W s are the radial component, transverse, and the normal components of SRP, thereby these components will be controlled by: here s = sin i, c = cos i, s 1 = sin ε, and c 1 = cos ε.
Compiling Equations (10)- (12) with Equation (9), then we can evaluate the components of SRP for practical applications.

Variations of the Longitude of Ascending Node Due to Solar Radiation Pressure
The variation of h due to the SRP can be determined by substituting Equation (9c) into the fourth equation from Equation (2). Then the variation of h as a function of the variable of true anomaly f is given by: where and with definitions in Equations (5) and (6), we can rewrite Equation (13) after taking the averaging over the true anomaly as: Equation (14) gives the averaged variation of the longitude of the node due to the force of the solar radiation pressure.

Variations of the Argument of Perigee Due to Solar Radiation Pressure
Substituting Equations (9a), (9b) into the fifth equation of Equation (2) we get the variation of g as a function of the true anomaly f in the form: where Substituting Equations (16a)-(16c) into Equation (15), and take averaging over the true anomaly f , we get: where Utilizing Equations (18a)-(18c) with Equation (17), we can then estimate the averaged variation for the argument of perigee with respect to the force of the solar radiation pressure.

Effect of Lorentz Force
Following [21], the radial, R L , transverse, T L , and normal, W L components of LF affecting a satellite orbiting Earth with a magnetic field are given: respectively by where • B is the moment of magnetic dipole of planet; • q is the charge per unit mass of satellite; • υ is the planetary rotation speed.

Variations of the Longitude of Ascending Node Due to Lorentz Force
Substituting Equation (19c) into the fourth equation from Equation (2), after simplification, then the variation of h due to LF is given by: where Substituting Equation (21) into Equation (20), and averaging over the true anomaly f , then the averaged of the longitude of ascending node h due to Lorentz Force is given by: where

Variations of the Argument of Perigee Due to Lorentz Force
Substituting Equations (19a), (19b) into the fifth equation from Equation (2), after simplification, then the variation of g due to LF is given by: where and Substituting Equations (25)-(27) into Equation (24), and averaging over the true anomaly f , then the averaged of the variations of the argument of perigee g due to Lorentz force is also given by: where From Equations (22) and (28), it is clear that the averaged of variations for longitude of ascending node (h) and argument of perigee (g) are invariant under the symmetry (i, g) −→ (−i, −g) due to Lorentz Force, where s = sin i and c = cos i.

Balancing of Solar Radiation Pressure Effect
In this section, the variations on the elements with respect to the SRP and LF are combined to find a zero value for the variations depending on the control parameter q, which is the charge per mass unit for the spacecraft. Since we are interested only with the position of the orbit, the variations of the longitude of the node h and the argument of perigee g are the only variations considered, however any other element of the orbit can be controlled similarly.

Controlling on the Longitude of Ascending Node
Solving ḣ R + ḣ L = 0, for the charge per mass unit q, we get the values of q in terms of the required values. The solution is: where ḣ R is given by Equation (22) while A 1 and A 2 are given by Equation (23). However q h is the required charge value to balance the variation on h due to the solar radiation pressure.

Controlling on the Argument of Perigee
Solving ġ R + ġ L = 0, for the charge q, we get the values of q in terms of the required values. The solution is: where ġ R is given by Equation (28) while the B 1 and B 2 are given by Equation (29). Furthermore, q h is the required charge value to balance the variation on g with respect to the SRP.

Numerical Results
In this section, the required values of the charge per mass unit (q), to balance the variations on the longitude of ascending node (h) and the argument of perigee (g) due to the solar radiation pressure and Lorentz Force will be investigated numerically through Figures 1-16, which is measured by the Coulomb/Kg unit. These investigations are given for showing the evaluation of the charge (q) versus different orbital parameters, both for Low Earth Orbit (LEO) and a Geosynchronous Orbit (GSO). The numerical values of the included parameters are B = 8 × 10 6 T km 3 , µ =398,600 km 3 /s 2 , υ = π/12 h −1 , ε = 23.439 • , and i = π/4. The calculations have been estimated when the semi-major axes (a = 7100 km) and the constant R C = 9.2 × 10 −5 for Low Earth Orbit while the same quantities are (a = 42,160 km) and R C = 4.2 × 10 −5 for Geosynchronous Orbit along with the true longitude of the Sun (θ = 0) in both two cases.
The charge per mass unit, which is required to balance the longitude of the node or the argument of perigee, is represented by a function that displays three variables of orbital elements, which are the argument of perigee (g), the longitude of node (h), and the eccentricity (e). In this regard, we estimate the charge per mass unit, when one of these elements is displayed as variable while the other two elements are taken as fixed parameters.

Results of Longitude of Ascending Node
The estimation of the charge per mass unit (q), which is required to balance the longitude of ascending node (h) of the spacecraft's orbit will be shown numerically through multiple diagrams in Figures 1-10. This estimation will be investigated by using Equation (30) in the following three subsections.    It is observed that the variation of charge values is sine periodic with g. Although the amplitude changes of curves are increasing with the increase in the longitude of ascending node and eccentricity in the case of LEO. The same results are satisfied in the case of GSO too, except for that the amplitude changes of curves are fluctuating due to the changes of the eccentricity, see Figure 3.

Evolution of Charge Quantity Versus Longitude of Ascending Node
In this subsection, the required values of charge per mass unit required to balance the longitude of ascending node is taken as a function of the longitude of ascending node (h). The values of this charge are shown at Low Earth Orbit in Figures 5 and 6 for both different values of the argument of perigee (g) and the eccentricity (e), and at Geosynchronous Orbit in Figures 7 and 8.  With the analysis of diagrams, we obtain a similar result as in previous sub-section, where the variation of charge values is also sine periodic with h. The amplitude changes of curves are increasing with an increase in the argument of perigee and eccentricity in the case of LEO. The same results are also satisfied in the case of GSO too, except for when the amplitude changes of curves are fluctuating due to the changes of the eccentricity, see Figure 7.

Evolution of Charge Quantity Versus Eccentricity
The required values of charge per mass unit required to balance the longitude of ascending node is taken as a function of the eccentricity (e). The values of this charge are shown in Figure 9 at Low Earth Orbit and at Figure 10 for both different values of the argument of perigee (g) at the fixed value of the longitude of the ascending node (h). After analyzing these figures, we show that the variation of charge values versus the eccentricity (e) is linear for Low Earth Orbits. But in the Geosynchronous Orbit the variation of charge values versus the eccentricity (e) is nonlinearly increasing till a certain maximum (different according to different values of h and g). Then q starts to decrease with increasing e. In a simple description, the variation of the charge value may be similar to quadratic curve behavior. In addition, the same results are satisfied for both different values of the longitude of ascending node (h) at the fixed value of the argument of perigee (g).

Results of Argument of Perigee
In the following three subsections, the estimation of the charge (q), which is needed to balance the argument of perigee (g) of the spacecraft's orbit will also be shown numerically through multiple diagrams in Figures 11-16.

Evolution of Charge Quantity Versus Argument of Perigee
The required values of charge per mass unit is taken as a function of the argument of perigee (g) to balance the variation on the argument of perigee for the spacecraft's orbit. These values will be illustrated by using Equation (31) through Figures 11 and 12 in the case of Low Earth Orbits.
It is shown that the variation of charge per mass unit is cosine periodic and the amplitude of the curve is decreasing with an increase in both the values of the longitude of node and eccentricity, but with a continuation in the increasing longitude of node value, the variation will be sine periodic, see Figure 11d. We also emphasize that the same results are satisfied in the case of Geosynchronous Orbits.

Evolution of Charge Quantity Versus Longitude of Ascending Node
The charge values (q) which is needed to balance the argument of perigee (g) as a function of the longitude of perigee (h) are presented in Figure 13 and 14 for Low Earth Orbit. The obtained results here are similar to those of the previous sub-section, the variation of the required charge is cosine periodic and the amplitude of the curve is decreasing with an increase in both the values of the argument of perigee and eccentricity, but with a continuation increasing in the argument of perigee value, the variation will be sine periodic, see Figure 13d. The same results will satisfy the Geosynchronous Orbit.

Evolution of Charge Quantity versus Eccentricity
The charge per mass unit values (q) required to balance the argument of perigee (g) as a function of the eccentricity e are presented in Figures 15 and 16   After analyzing these figures, we demonstrate that the effect of increasing argument of perigee (g) or the longitude of node (h) is to reduce the values of the charge per mass unit (q), furthermore this value tends to zero as the eccentricity (e) tend to unity. In addition the same results are satisfied in the case of Geosynchronous Orbit, when e ∈]0, 0.1[ in the case of Low Earth Orbit, while e ∈]0, 1[ for Geosynchronous Orbit.

Implications on Spacecraft Attitude
In [29], the author proved that the spacecraft attitude is stimulated by three magnetic torques and subjected to a magnetic field that varies with time. These are accessible if the magnetic field and its first two time derivatives are linearly indented. Moreover, they are controllable if the magnetic field is periodic in time. Since in our study we are interested in controlling the perturbation due to solar radiation pressure, on the argument of perigee and the argument of ascending node, using LF, then the pointing of the spacecraft in the direction of the local magnetic field or the Sun is out of scope of the subject. However, it should be mentioned that controlling the Lorentz Force depends mainly on the electric charge magnitude and whether is is positive or negative, which is much easier than attitude control. In addition, solar radiation pressure effects on different orientations of the satellite can be handled with controlling the charge instead of attitude control.
A spacecraft is charged because of the fact that outer space contains high energy-charged particles, a spacecraft is naturally charged. Although these charged particles cause serious problems to the spacecraft, e.g., damage to electrical equipment and interference in measurements, but we can use them to produce electric charge that allow for the use of the Earth magnetic field and thus getting use of the Lorentz Force as an effective force in the control dynamic of the satellite whether is is orbit or attitude control. Generally, the devices that produce electric power for different uses of the satellite can also be used to charge the surface of the satellite. These devices usually have a wight of about 10% of the satellite, e.g., OCEAN satellites whose electric power supply system has the weight of 584 kg, which is less than about 10% of the total weight of the satellite, with a 3-year-orbital resource time [30].

Conclusions
In this article, a treatment was made to control the orbit position against the perturbation effect of solar radiation pressure. This was achieved through balancing the argument of perigee and longitude of ascending node of the spacecraft, by using the specific charge density (charge per mass unit). This charge was used to control the required value of Lorentz Force to provide the desired balance. The variations of the orbital elements due to both forces were calculated. The variations of g and h were set equal to zero, under the combined effect of both forces, to find the required values of the charge per mass unit q.
Multiple graphical investigations were given to illustrate the range of possible values for the charge per mass unit q, which is needed to establish the control. These investigations showed that the variation was periodic versus the argument of perigee and the longitude of ascending node in both cases of Low Earth and Geosynchronous Orbits. The obtained results also showed that for balancing the longitude of node h, the change of charge per mass unit value was periodic due to the argument of perigee g and the longitude of node with peaks varying from 1.5 × 10 −5 C/kg to 4 × 10 −5 C/kg at Low Earth Orbit, and from 5 × 10 −3 C/kg to 30 × 10 −3 C/kg at Geosynchronous Orbit.
Furthermore, the variation of charge per mass unit values required that balance of the longitude of ascending node versus the eccentricity (e) quadratic curve behavior for different values of the argument of perigee (g) and the longitude of node (h). While the same values to balance the argument of perigee were reduced with the increase in both values of h and g.
The maximum charge that can be obtained by current technology is 0.03 C/kg as mentioned in the introduction section. However, the results of the calculations show that in many cases (especially for GSO) the required charge for balancing exceeds this value. Thus we would like to remark that the graphs of high altitude at Geosynchronous Orbit show very large values for q rather than for Low Earth Orbits. This is because the Lorentz Force variations on h and g depend on (1/a 3 ), see Equations (22) and (24). Thus moving from a = 7100 to about 42,160 means an increase by about 216 greater required values of q. In addition, the computations and graphics of the paper were produced using the latest version 12.0 of Mathematica Wolfram. Finally we emphasize that the proposed technical model is considered in the modern literature as closely related to both the electrodynamic sweeping of outer space around the Earth and the nano-electrodynamic spacecraft of future space exploration.