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The present article studies the stability conditions of central control artificial equilibrium generalized restricted problem of three bodies. It is generalized in the sense that here we have taken the larger primary body to be in shape of an oblate spheroid. The equilibrium points are sought by the application of the propellant for which it would just balance the gravitational forces. The launching flight of such a satellite is seen to be applicable for having arbitrary space stations for these different missions. Specialty of the result of the investigation lies in the fact that an arbitrary space station can be formed to attain any specified mission.

An important recent publication on total sailing is a book by Mclnnes [

At the initial stage an extremely large (800 m × 800 m) three axis stabilized square solar sail with four deployable booms was considered but was dropped in 1977 due to the perceived risks associated with boom. Nowadays a square solar sail configuration is seen as optimum for these smaller solar sails.

The mission using solar-electric propulsion was initially strongly supported but later on due to escalating cost the propulsion was dropped [

Solar sail is a proposed form of spacecraft propulsion that takes advantage of the radiation pressure to propel a spacecraft by means of a large membrane mirror. The impact of the photons emitted by the sun on the surface of the sail and its further reflection accelerates the spacecraft. Although this acceleration produced by the solar radiation pressure is smaller than the one achieved by the traditional propulsion systems, this one is continuous and unlimited. This makes long-time missions more accessible [

Sun-sail and other hybrid sails have been recently proposed to explore various cosmic research programmes. An electric sail was proposed to be designed which could be capable of guarantying the fulfillment of trajectories that would be otherwise unfeasible through conventional system. In particular, the proposal [

The Lagrangian points of the circular restricted problem of three bodies (CR3BP) are known to be the five positions in an orbital system rotating with the two massive bodies where a small object (e.g. a spacecraft) affected by their gravity and centripetal forces can be stationary relative to the two larger bodies. Now if some propellant forces such as solar sail or electrodynamics or magnetic are introduced, then we have some other or more equilibrium points [

The paper has been concluded with the appendices giving the calculations of the various terms occurring in the paper and in the last section a final conclusion of our studies has been given.

Using dimensionless variables and a system the equations of motion referred to the solution [5,6] when the origin is taken at the smaller mass M_{2} may be written as:

where,

R_{e} = Equatorial radius of the larger mass, R_{p} = Polar radius of the larger mass, R = 1 = the mutual distance between the primaries.

[The larger mass is placed at M_{1} and the smaller one at. is their centre of mass and and. The space-craft is taken at M. The line is taken for the X-axis with for the origin and perpendicular at for Y-axis and the axes are taken to be rotating along a normal to - plane with the angular velocity n].

The inertial kinetic energy per unit mass is given by

and the total potential due to the gravitational and the control acceleration is:

where are the x and y components of the control acceleration. Thus the Equations (2.1) may be written as:

Then the equilibrium point for the control acceleration will be given by

where are the radial distance of the equilibrium point from the primaries.

In this section we shall find out the conditions for the linear stability of the equilibrium points. We have for the Lagrangian function as:

and the canonical variables p_{x} & p_{y} for Hamiltonian variables

Hence the Hamiltonian H may be written as.

Yielding H as given by

Now let us transfer the origin to the equilibrium point and the new coordinates be written as, so that

Hence the transformed Hamiltonian may be written after eliminating the constant term as

where,

Now expanding H in powers of, we shall have

where,

and = functions of, , or, and they are given by (Refer to Appendix Section I(a))

Hence the Linearized equations of motion may be written as

The corresponding characteristic equation will be

where f and g may be written as

If be the two oscillation frequencies, then they will be given as

and for linear stability we must have

Putting the value for in the expression for, we get

To minimize the objective function, let us consider and writing the value for a_{x} & a_{y}, we get

whence we have

Now putting and equating the corresponding coefficients of μ, μ^{2} & A_{1} to zero, we get

and the value of corresponding to these δ_{0}, δ_{1}, δ_{2} & δ_{3} will be the maximum value of.

Hence we have,

where, we have taken

will be the corresponding dimensional acceleration, being taken to be the distance between the primaries. It may be verified that when and, the result (5.1) & (5.2) are constant giving the control acceleration to be zero.

Let us look for an analytical estimation of minimum distance from the second primary (ρ) allowing the linear stability conditions. We have

We have ignored the term of the in the expression for and and those in the expression for, hence we get equation for ρ as

From the 1^{st} two conditions we obtain our stability boundary limit, i.e.;

[Since]

For earth-moon and

and so it may be checked as.

Thus the value coincides with the value attained by [Bombardelli] and the corresponding control acceleration at the boundary will be

In dimensional units the minimum distance from the second primary in order to have stability will be

and the required dimensional control acceleration will be

.

So it shows that with the presence of obliquity the amount of acceleration decreases and a lesser power of thrust will be needed.

We have investigated the properties of minimum control artificial equilibrium points in the planar circular restricted three-body problem while the effect of the oblateness of the bigger primary body is also taken into account. We have found the analytical expressions which characterize their location, control acceleration and stability properties. It is seen that due to the presence of oblateness in the expression all the properties are likely to be affected and the disturbance is more natural since the study of effect of oblateness is quite necessary for exhaustive study of the effect. The specialty of the presence of the oblateness lies in the fact that the amount of acceleration is less and consequently a less power of thrust is required for the mission.

(a) Expression for a, b and c:

Let the suffix (0) denote the terms corresponding to the equilibrium position and the suffix denote the contribution due to the presence of the oblateness of the larger primary body.

Onwards we drop the suffix (0) with.

Now are the coefficients of in the expression for

Expression for and: We shall now onwards ignore second and higher powers of A_{1} and also multiple μA_{1} and so on. Thus

So,

Calculation of

We have

where we have dropped the suffix (0) with x_{0}, y_{0}, δ_{0}, ,

Putting and equating the different corresponding coefficients to zero, we shall get

Expression for, & for minimum value of δ given by

We shall ignore the terms of the order of and in the expression for but the terms of will be retained in the expression for. thus

Ignoring the terms of the order of and also taking into account, we have

Similarly, we shall have the expression for as follows:

Thus neglecting the terms as mentioned above, we have