Out-of-Plane Equilibrium Points in the Photogravitational Hill Three-Body Problem

Abstract

This paper investigates the movement of a negligible mass body (third body) in the vicinity of the out-of-plane equilibrium points of the Hill three-body problem under the effect of radiation pressure of the primaries. We study the effect of the radiation parameters through the factors $q_i,i=1,2$ on the existence, position, zero-velocity curves and stability of the out-of-plane equilibrium points. These equilibrium positions are derived analytically under the action of radiation pressure exerted by the radiating primary bodies. We determined that these points emerge in symmetrical pairs, and based on the values of the radiation parameters, there may be two along the $Oz$ axis and either none or two on the $Oxz$ plane (outside the axes). A thorough numerical investigation found that both radiation factors have a strong influence on the position of the out-of-plane equilibrium points. Our results also reveal that the parameters have impact on the geometry of the zero-velocity curves. Furthermore, the stability of these points is examined in the linear sense. To do so, the spatial distribution of the eigenvalues on the complex plane of the linearized system is visualized for a wide range of radiation parameter combinations. By a numerical investigation, it is found that all equilibrium points are unstable in general.

1. Introduction

One of the most commonly studied problems in Celestial Mechanics is Hill’s three-body problem (HTBP), which is a special case of the widespread restricted three-body problem (RTBP). The circular RTBP focuses on the motion of a tiny mass body (third body) within a system consisting of two larger masses (the primaries), which orbit in circular paths around their shared center of mass. The mass of the third body is insignificant in comparison to that of the primary bodies, so its impact on their motion is negligible. On the other hand, the HTBP, originally formulated by Hill [1], aimed to investigate lunar dynamics. In this simplified model, the massless body is also subject to gravitational forces from two primary bodies, but one vastly dominates the other in mass, which is exemplified by the Sun and Earth system. In a rotating reference frame, the smaller primary is fixed at the origin, while the Ox-axis extends toward the larger primary, which is treated as being infinitely distant from the secondary (see, [2,3]). One of the key benefits of the HTBP is that it possesses a significantly simpler Hamiltonian than that of the RTBP, which allows for obtaining easier analytical solutions in certain cases [4,5,6,7] while it also provides significant benefits for numerical computations [8,9,10,11,12,13,14,15].

The dynamics of a negligible mass body in orbit around a single star and planet or around a binary star system can be viewed as a generalization of the classical RTBP. This model is called the photogravitational restricted three-body problem [16,17,18] and has attracted much interest (see, e.g., [19,20,21,22,23,24]). When the radiation pressure exerted by the primaries (one or both) is also considered, apart from the five coplanar equilibrium points, additional stationary points positioned outside the physical plane (the plane which is defined by the two primary bodies) have been identified, and the dynamics of orbits around these new points have been discussed in the literature (see, e.g., [25,26,27,28,29]). Several perturbed versions of the RTBP have also been found to possess additional equilibrium points. Many works have been devoted to these modified models and presented both numerical and analytical investigations on the existence and location of the equilibrium points along with their stability towards understanding the dynamical behavior of orbits around them (see, e.g., [30,31,32,33]).

On the other hand, Markellos et al. [34] introduced the model of the photogravitational HTBP and offered some initial insights into the problem’s formulation, the position of the equilibrium points and zero-velocity curves with estimates of the size of accretion disks in binary stars. The derived model builds upon the classical Hill lunar problem and represents one of its possible extensions. Other models for the HTBP that involve lack of sphericity of body shape (oblate primary) and radiation pressure effects were studied by [35,36,37,38,39,40,41,42,43], among others. Furthermore, Yárnoz et al. [44] studied the evolution of families a and g of periodic orbits in the HTBP with radiation pressure and found that the main structure of family characteristics differs from the corresponding solution of the classical HTBP.

In particular, Kanavos et al. [35] analyzed the equilibrium points and associated periodic solutions in the photogravitational HTBP, concentrating on the 2D context. In this paper, we extend this analysis to the spatial case of this specific modification of the HTBP by investigating the out-of-plane equilibrium points (OoPEPs) and, in particular, the motion around them. Our objective is to determine the positions of the OoPEPs with respect to the radiation coefficients, the regions of motion as defined by the zero-velocity curves, and the linear stability of these points. We present the spatial distribution of the eigenvalues of the linearized system, showing how they scatter across a systematic grid of radiation factor values. This provides both a qualitative and quantitative overview of how the radiation from the two primaries influences the stability of the OoPEPs. To achieve this, we employ a new two-step technique. In the first step, we discretize the continuous space of the linearized system around the equilibrium points over a dense grid of radiation parameter combinations and numerically compute their eigenvalues. In the second step, we discretize the continuous space of the eigenvalue complex plane and compute the local spatial density. The combination of these two steps yields valuable information regarding the basins of attraction at which the eigenvalues are accumulated.

At this point it should be emphasized that from the above literature and also to the authors’ knowledge, no research has been performed on the existence and location of equilibrium points outside the orbital plane defined by the primaries and especially the motion around them on the HTBP with radiating primaries, hence, it sparked our interest in examining the impact of radiation pressure on the dynamical characteristics of these points. We identify two symmetrical pairs of equilibrium points situated along the $z$ axis and $Oxz$ plane, correspondingly. The presence of equilibrium points located outside the plane of the primaries, known as OoPEPs, is a new aspect of the model that had not been addressed in [35]. To tackle the current problem more effectively, we employ both an analytical approach and a numerical treatment.

Our work is organized as follows: In Section 2, we recall the equations of motion of the photogravitational HTBP along with its Jacobian integral. In Section 3, we investigate the OoPEPs of the system with variations in the radiation factors. In Section 4 and Section 5, the configurations of the zero velocity curves and linear stability of the OoPEPs are investigated, respectively, while the conclusion and discussion are presented in Section 6. The model under consideration can be utilized to study the motion of minor objects like cosmic dust and grains.

2. Photogravitational Hill’s Equations of Motion

In the present analysis, we adopt the terminology and notations associated with the photogravitational HTBP, which were discussed and analyzed in [34,35]. Therefore, the equations governing the motion of the massless body within the context of the spatial photogravitational HTBP are as follows:

$$\ddot{x}-2\dot{y}={\displaystyle \frac{\partial \mathsf{\Omega }}{\partial x}}={\mathsf{\Omega }}_x, \ddot{y}+2\dot{x}={\displaystyle \frac{\partial \mathsf{\Omega }}{\partial y}}={\mathsf{\Omega }}_y, \ddot{z}={\displaystyle \frac{\partial \mathsf{\Omega }}{\partial x}}={\mathsf{\Omega }}_z,$$

(1)

where the photogravitational potential is defined as

$$\mathsf{\Omega }=\mathsf{\Omega }(x,y,z)={\displaystyle \frac{3x^2}{2}}-Q_{1}x+{\displaystyle \frac{q_2}{r}}-{\displaystyle \frac{z^2}{2}},$$

(2)

and

$$r^2=x^2+y^2+z^2, Q_1={\displaystyle \frac{1-q_1}{{\mu }^{\frac{1}{3}}}}$$

where $r$ represents the distance between the particle and the smaller primary body. The Jacobian integral associated with this problem, is expressed by the equation

$$\Gamma =-{\dot{x}}^2-{\dot{y}}^2-{\dot{z}}^2+3x^2-2Q_{1}x+{\displaystyle \frac{2q_2}{r}}-z^2$$

(3)

where $\Gamma$ is the Jacobian constant. In this context, $\mu$ represents the fraction of the smaller primary’s mass relative to the total mass of the primary bodies and $0<\mu <{\displaystyle \frac{1}{2}}$. Also, $q_1$, $q_2$ are the radiation factors of the primary and secondary body, respectively. Let us recall that according to Schuerman [17], the radiation force effect $q$ can be represented by the relation $q=1-b$, where $b={\displaystyle \frac{F_r}{F_g}}$ signifies the ratio of the force $F_r$ generated by the radiation to the gravitational force $F_g$ resulting from the two primary bodies. In case where $q=1,$ namely $b=0$, it reduces to the classical problem that takes into account only gravitational forces while if $0<q<1,$ that is $b\in (0,1),$ gravitational force surpasses the radiation pressure. Furthermore, when $q=0,$ namely $b=1,$ the forces of radiation and gravity are in equilibrium and finally, when $q<0,$ i.e., $b>1,$ radiation pressure overrides the gravitational attraction. Consequently, it results that the allowable range of values for the radiation factor in the present problem could be taken $q_{1,2}\in (-\infty ,$ 1]. It is worth noting that in case of $Q_1=0,q_2=1$, the problem becomes the classical HTBP (see, e.g., [45]).

3. Existence and Location of the Equilibrium Points of the System

For the existence of any equilibrium point, the necessary and sufficient conditions that must be fulfilled are $\dot{x}=\dot{y}=\dot{z}=\ddot{x}=\ddot{y}=\ddot{z}=0,$ i.e., the velocity and acceleration components must both be zero. As such, the coordinates $(x,y,z)$ that define the equilibrium points can be established through the solution of the equations

$${\mathsf{\Omega }}_x=0, {\mathsf{\Omega }}_y=0, {\mathsf{\Omega }}_z=0,$$

(4)

where ${\mathsf{\Omega }}_x,$${\mathsf{\Omega }}_y,$${\mathsf{\Omega }}_z$ are the first-order partial derivatives of $\mathsf{\Omega }(x,y,z)$ with respect to $x,y,$ and $z,$ respectively. In particular, we have

$$\begin{gathered} {\mathsf{\Omega }}_x=3x-Q_1-{\displaystyle \frac{q_{2}x}{r^3}}, \\ {\mathsf{\Omega }}_y=-{\displaystyle \frac{q_{2}y}{r^3}}, \\ {\mathsf{\Omega }}_z=-z\left[1+{\displaystyle \frac{q_2}{r^3}}\right]. \end{gathered}$$

(5)

Since in this paper, we are mainly interested in the OoPEPs, we focus on the third equation of System (4). So, this system could potentially possess stationary points outside the orbital plane when solved for $x\ne 0$, $y=0$ and $z\ne 0$. Based in this premise, the third Equation (4) is satisfied if

$$1+{\displaystyle \frac{q_2}{r^3}}=0.$$

(6)

By substituting Equation (6) into second of Equations (4), it follows that $y=0$, that is to say, the equilibrium solution will be on the $Oxz$ plane.

Equation (6) can be expressed as

$$r={\left(-q_2\right)}^{{\displaystyle \frac{1}{3}}}\equiv k_0$$

(7)

with $r^2=x^2+z^2.$ For $q_2<0$, we get from Equation (7) that $r>0$ and the locus of these points describes an Apollonius circle surrounding the body. According to Equation (7), the existence of any real solution of the form $(x,0,z)$ requires that at least one of the following conditions is met (some values in this region might not yield OoPEPs):

$$q_2<0 \mathrm{or} q_2=0.$$

(8)

The second condition will not be considered in this paper in order System (4) to have real solutions of the form ($x,0,z$) as for $y=0$ its second equation must be fulfilled. As a result, the equilibrium points outside the orbital plane are positioned on the $Oxz$ plane, necessitating the solution of the other two equations of System (4) for $(x,z)\ne (0,0).$ That is, the OoPEPs result from the solution of the system

$$\begin{gathered} 3x_0-Q_1-{\displaystyle \frac{q_2{x}_0}{r_0^3}}=0, \\ 1+{\displaystyle \frac{q_2}{r_0^3}}=0 \end{gathered}$$

(9)

with

$$r_0^2=x_0^2+z_0^2.$$

(10)

These solutions depend on the radiation factors $Q_1$ and $q_2$, which in turn depend on the problem considered. From the second equation of Equation (9) combined with the first equation, we have

$$x_0={\displaystyle \frac{Q_1}{4}}.$$

(11)

Substituting Equation (11) in Equation (10) in combination with Equation (7), we have

$$z_0=\pm \sqrt{{(-q_2)}^{{\displaystyle \frac{2}{3}}}-{\displaystyle \frac{Q_1^2}{16}}}, \left({(-q_2)}^{{\displaystyle \frac{2}{3}}}\ge {\displaystyle \frac{Q_1^2}{16}}\right).$$

(12)

This proves the existence of OoPEPs in pair, namely $L_1^z$ for $z>0$ and its symmetric point $L_2^z$ for $z<0$ with respect to the axis Ox. We point out here the symmetry property $z\to -z$ in Equation (9). As such, if the point $L_1^z$ is an equilibrium point, also $L_2^z$ is. We remark that the points $L_{1,2}^z$ and $L_{3,4}^z$ do not have any real existence on the $xz$ plane simultaneously. So, Equations (11) and (12) are the locations of the OoPEPs of the HTBP under radiating primaries and they solely depend on the radiation factors.

Notably, conditions (9) result in two distinct types of equilibrium points within the $Oxz$ plane. Particularly, one set of these points positioned along the Oz axis ($x_0=0$) of the synodic coordinate system symmetrically located with respect to the origin and the physical plane $(Oxy),$ which continue to exist for all values of the coefficients $q_2<0$ and $Q_1=0$, in contrast to the equilibrium points located on the $Oxz$ plane outside the $Ox$ and $Oz$ axis $(0<x_0<<\left|z_0\right|$ or $x_0>>\left|z_0\right|).$ The latter stationary points can exist only in the region ${\left\{(q\right.}_2,Q_1) \left| q_2\right.<0, Q_1>\left.0\right\},$ nonetheless, this is a necessary condition, meaning that there may be values within this region where these points fail to exist. All these pairs of points where exist have the same Jacobian constant.

From Equation (9), it could be obtained that the existence condition of $L_{1,2}^z$ on the $Oxz$ plane is that the radiation factors of the two primaries must satisfy

$$q_2<\mathsf{\Phi }(Q_1)<0 \mathrm{where} \mathsf{\Phi }(Q_1)=-{\displaystyle \frac{Q_1^3}{64}}, Q_1>0.$$

(13)

It must be pointed out that the equilibrium points on the axis $Oz$ and on the $Oxz$ plane will not exist together at the same time (simultaneously) in the photogravitational HTBP, which validates findings of [37] in the photogravitational HTBP with oblateness.

To support the conclusion drawn above, Figure 1 illustrates the planar curves defined by the implicit equations ${\mathsf{\Omega }}_x=0$ (green) and ${\mathsf{\Omega }}_z=0$ (red) curves. The points where these curves intersect reveal the $(x_0,z_0)$ coordinates of the equilibrium points within the $xz$-plane, for two different cases of $Q_1$, (a) $Q_1=0$ and (b) $Q_1=0.5.$ The radiation factor $q_2$ of the smaller primary in both examples have the same value $q_2=-2$. As previously noted, in the left frame of this figure the equilibrium points are situated along the $Oz$ axis, whereas in frame (b), they are found on the $Oxz$ plane. In both cases, their positions are denoted by blue dots. These numerical results support our conclusion drawn above.

Next, we will examine the locations of the OoPEPs for the infinitesimal mass body when one or both primaries are radiating. The respective coordinates are displayed in the second column of

Table 1. The locations ($x_0,\pm z_0$) of the OoPEPs $L_1^z$ and $L_2^z$ of the problem when both primaries radiate.

${\mathit{q}}_{\mathbf{2}}$	${\mathit{Q}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$	${\mathit{Q}}_{\mathbf{1}}\mathbf{=}\mathbf{0.35}$	${\mathit{Q}}_{\mathbf{1}}\mathbf{=}\mathbf{0.5}$
−0.05	(0, ±0.36840315)	(0.08750000, ±0.35786119)	(0.12500000, ±0.34654853)
−0.04	(0, ±0.34199519)	(0.08750000, ±0.33061225)	(0.12500000, ±0.31833270)
−0.03	(0, ±0.31072325)	(0.08750000, ±0.29814877)	(0.12500000, ±0.28447133)
−0.02	(0, ±0.27144176)	(0.08750000, ±0.25695210)	(0.12500000, ±0.24094736)
−0.01	(0, ±0.21544347)	(0.08750000, ±0.19687468)	(0.12500000, ±0.17547333)

for a case of only the smaller primary body $q_2$ radiating when the value $Q_1=0.$ In this case, the problem admits two out-of-plane critical points on the axis $Oz$ along with the variation $q_2$. It is noted that an increase in the radiation factor $q_2$ leads to a decrease in the z-coordinates of the equilibrium points towards the origin. This effect is shown in Figure 2. It is worth noting that we choose relatively large values of radiation factor $q_2$ so it can be seen clearly in the figures. In the third and fourth columns of this table we present how the positions of the equilibrium points vary with changes in $q_2$ but for $Q_1=0.35$ and $Q_1=0.5,$ respectively. In both cases where the primaries radiating, the problem admits two OoPEPs on the $xz$-plane. As the radiation factor $q_2$ increases, for fixed $Q_1=0.35,$ a decrease in the z-coordinates of the equilibrium points toward the origin is noted, while the x-coordinates stay unchanged (maintain their positions). Therefore, it is apparent that the variation in the corresponding positions for $Q_1=0.5$ is comparable to the situation previously discussed for $Q_1=0.35.$

We observe a similar phenomenon (rows 2 through 6) in Table 1 for fixed values of $q_2$ and varying $Q_1$. The positions of the OoPEPs on the $xz$-plane are presented in Figure 3, where we show their evolution for $Q_1=0.5$ while varying $q_2$ since the behavior remains consistent for different values of these parameters. Another difference, with respect to the case of only the smaller primary radiating is that the OoPEPs, $L_1^z$ and $L_2^z$ are consistently found to be located in the second and fourth quadrants, respectively, for every value of $q_2.$ It is also remarkable that both the existence and number of these points are dependent on the radiation parameters $q_1$ and $q_2.$ In the case where $q_2<0$ and $Q_1\ne 0 (q_1\ne 1),$ our numerical computations show that we may have zero or two such points on the $xz$-plane in contrast to the photogravitational restricted three-body problem case (see, e.g., [16,18,28,40]), where either one or two pairs of equilibrium points may be present in the $Oxz$ plane in case where the parameters satisfy the conditions $q_2>0$ and $q_1<0$ or $q_2<0$ and $q_1>0$, correspondingly.

Towards an attempt to investigate and visualize, with a systematic way, the effect of $Q_1$ and $q_2$ on the positions of OoPEPs, we created the surfaces of Figure 4. In these plots, we evaluated Equations (11) and (12) with a dense two-dimensional grid of a wide range of the radiation parameters, $Q_1\in \left(\mathrm{0,1}\right]$ and $q_2\in [-\mathrm{5.5,0})$. The top subfigures show the effect on the position in a decouple way, while the bottom subfigure is a four-dimensional plot. Here, we used a color-code to depict the effect of $q_2$ on the equilibrium point position. It is clearly observed that lower values of $q_2$ displace the z direction of the equilibrium points, and $Q_1$ has more effect on the x direction, while their combination is a nearly plane surface.

4. Zero-Velocity Curves in the ($\mathit{x},\mathit{z}$) Plane

In this section, we present the contours of the surface (3) on the ($x,z$) plane for zero velocity, which define the zero velocity curves (ZVCs). In this surface projection, trapping regions—where the motion of the massless body is constrained—form within specific ranges of the Jacobian constant. As the value of $\Gamma$ varies, these regions can either expand or contract until they eventually reduce to individual points. Figure 5 shows the ZVCs for a fixed value of $Q_1=0.5$ and for three different values of the radiation factor, namely, $q_2=-2.5$, $q_2=-1$ and $q_2=-0.05$, respectively.

We have plotted only the curves (dotted lines) for Jacobian constant values corresponding to the OoPEP $L_1^z$ (or $L_2^z).$ Blue dots mark the positions of the OoPEPs in the system. In this case, between the OoPEPs, the zero velocity curves form small oval regions where motion is not allowed. These regions shrink as the radiation factor $q_2$ approaches zero (i.e., as $q_2$ increases), while the regions of allowed motion expand with the increase in $q_2$. This behavior is similar to the restricted three-body problem with radiating primaries and angular velocity variation described in [28]. It is clear from this figure that the radiation factors significantly influence the regions of possible and forbidden motion for the third body.

5. Stability of the Out-of-Plane Equilibrium Points

To examine the linear stability of the OoPEPs $L_1^z$ (or $L_2^z$), we position the origin at the equilibrium point by setting:

$$x=x_0+\xi , y=\eta , z=z_0+\zeta$$

(14)

let $\xi$, $\eta$, and $\zeta$ represent the perturbations along the $Ox,$ $Oy$ and $Oz,$ axis, respectively. By substituting Equation (14) into System (1) and performing a Taylor series expansion around the equilibrium point, keeping only the first-order terms, we derive the linearized equations of motion as follows:

$$\dot{\chi }=A\chi , \chi ={(\xi ,\eta ,\zeta ,\dot{\xi },\dot{\eta },\dot{\zeta })}^T.$$

(15)

Let $\chi$ represent the state vector of the third particle relative to the equilibrium points, and the time-independent coefficient matrix $A$ is given by

$$A={\left(\begin{array}{cc}{0}_{3\times 3}& I_{3\times 3}\\ {\mathsf{\Omega }}_{3\times 3}& B_{3\times 3}\end{array}\right)}_{6\times 6}.$$

(16)

Here, $0_{3\times 3}$ and $I_{3\times 3}$ represent the third-order zero matrix and the identity matrix. The two additional matrices are defined as follows:

$$\mathsf{\Omega }=\left(\begin{array}{ccc}{\mathsf{\Omega }}_{xx}^{(0)}& {\mathsf{\Omega }}_{xy}^{(0)}& {\mathsf{\Omega }}_{xz}^{(0)}\\ {\mathsf{\Omega }}_{yx}^{(0)}& {\mathsf{\Omega }}_{yy}^{(0)}& {\mathsf{\Omega }}_{yz}^{(0)}\\ {\mathsf{\Omega }}_{zx}^{(0)}& {\mathsf{\Omega }}_{zy}^{(0)}& {\mathsf{\Omega }}_{zz}^{(0)}\end{array}\right), B=\left(\begin{array}{ccc}0& 2& 0\\ -2& 0& 0\\ 0& 0& 0\end{array}\right)$$

(17)

where ${\mathsf{\Omega }}_{xx}^{(0)},{\mathsf{\Omega }}_{xy}^{(0)},{\mathsf{\Omega }}_{xz}^{(0)},\dots ,{\mathsf{\Omega }}_{zz}^{(0)}$ denote the second partial derivative of the system potential of the OoPEP ($x_0,z_0$).

The characteristic equation of system described in Equation (15) is thus written as

$${\lambda }^6+a_1{\lambda }^4+a_2{\lambda }^2+a_3=0$$

(18)

where

$$\begin{array}{l}{a}_1=4-{\mathsf{\Omega }}_{xx}^{(0)}-{\mathsf{\Omega }}_{yy}^{(0)}-{\mathsf{\Omega }}_{zz}^{(0)},\\ a_2={\mathsf{\Omega }}_{xx}^{(0)}{\mathsf{\Omega }}_{yy}^{(0)}+{\mathsf{\Omega }}_{yy}^{(0)}{\mathsf{\Omega }}_{zz}^{(0)}+{\mathsf{\Omega }}_{zz}^{(0)}{\mathsf{\Omega }}_{xx}^{(0)}-4{\mathsf{\Omega }}_{zz}^{(0)}-{({\mathsf{\Omega }}_{xz}^{(0)})}^2,\\ a_3={({\mathsf{\Omega }}_{xz}^{(0)})}^2{\mathsf{\Omega }}_{yy}^{(0)}-{\mathsf{\Omega }}_{xx}^{(0)}{\mathsf{\Omega }}_{yy}^{(0)}{\mathsf{\Omega }}_{zz}^{(0)}.\end{array}$$

(19)

By calculating the second-order derivatives of $\mathsf{\Omega }$ at the equilibrium point, we obtain the following:

$$\begin{array}{l}{\mathsf{\Omega }}_{xy}^{(0)}={\mathsf{\Omega }}_{yx}^{(0)}={\mathsf{\Omega }}_{yz}^{(0)}={\mathsf{\Omega }}_{zy}^{(0)}=0, {\mathsf{\Omega }}_{xz}^{(0)}={\mathsf{\Omega }}_{zx}^{(0)},\\ {\mathsf{\Omega }}_{xx}^{(0)}=3-{\displaystyle \frac{q_2}{r_0^3}}+{\displaystyle \frac{3q_2{x}_0^2}{r_0^5}},\\ {\mathsf{\Omega }}_{zz}^{(0)}=-1-{\displaystyle \frac{q_2}{r_0^3}}+{\displaystyle \frac{3q_2{z}_0^2}{r_0^5}},\\ {\mathsf{\Omega }}_{zx}^{(0)}={\displaystyle \frac{3q_2{x}_0{z}_0}{r_0^5}},\\ {\mathsf{\Omega }}_{yy}^{(0)}=-{\displaystyle \frac{q_2}{r_0^3}}.\end{array}$$

(20)

Equation (18) is a sixth-degree polynomial in $\lambda$, and the resulting eigenvalues indicate the linear stability or instability of the corresponding OoPEPs. An equilibrium point is considered linearly stable if all six roots of the characteristic equation are purely imaginary or have non-positive real parts; otherwise, the point is deemed unstable.

We analyze the nature of the roots of Equation (18) and find that for all admissible values $Q_1,q_2$, one pair of eigenvalues is purely imaginary, while the other two pairs are complex conjugates with positive real parts. The presence of complex roots with positive real parts indicates that motion around the OoPEPs is unstable, similar to cases previously examined within the framework of the restricted three-body problem (see, e.g., [28,37,46] and references therein).

To clearly illustrate this fact, we compute the stability of the equilibrium points along the $Oz$ axis and those on the $Oxz$ plane, as shown in

Table 2. The stability of $L_{1,2}^z$ on the $Oz$ axis as a function of $q_2$ for $Q_1=0$.

${\mathit{q}}_{\mathbf{2}}$	${\mathit{L}}_{\mathbf{1}\mathbf{,}\mathbf{2}}^{\mathit{z}}$(${\mathit{x}}_{\mathbf{0}}\mathbf{,}\mathbf{\pm }{\mathit{z}}_{\mathbf{0}}$)	${\mathsf{\lambda }}_{\mathbf{1}\mathbf{,}\mathbf{2}}$	${\mathsf{\lambda }}_{\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}}$
−5.5	(0, ±1.765174168)	±1.732050807$i$	±(1.118033988 ± 0.8660254037$i$)
−4.0	(0, ±1.587401052)	$\pm$1.732050808$i$	±(1.118033989 ± 0.8660254038$i$)
−2.0	(0, ±1.259921050)	±1.732050807$i$	±(1.118033989 ± 0.8660254038$i$)
−1.0	(0, ±1.000000000)	±1.732050808$i$	±(1.118033989 ± 0.8660254038$i$)
−0.01	(0, ±0.215443469)	±1.732050808$i$	±(1.118033989 ± 0.8660254038$i$)
−0.005	(0, ±0.170997595)	±1.732050807$i$	±(1.118033988 ± 0.8660254037$i$)

and

Table 3. The stability of $L_{1,2}^z$ on the $Oxz$ plane as a function of $q_2$ for $Q_1=0.5$.

${\mathit{q}}_{\mathbf{2}}$	${\mathit{L}}_{\mathbf{1}\mathbf{,}\mathbf{2}}^{\mathit{z}}$(${\mathit{x}}_{\mathbf{0}}\mathbf{,}\mathbf{\pm }{\mathit{z}}_{\mathbf{0}}$)	${\mathsf{\lambda }}_{\mathbf{1}\mathbf{,}\mathbf{2}}$	${\mathsf{\lambda }}_{\mathbf{3}\mathbf{,}\mathbf{4}\mathbf{,}\mathbf{5}\mathbf{,}\mathbf{6}}$
−5.5	(0.12500000, ±1.760742696)	±1.730962960$i$	±(1.116769707 ± 0.8654814802$i$)
−4.0	(0.12500000, ±1.582471832)	±1.730705035$i$	±(1.116469855 ±0.8653525173$i$)
−2.0	(0.12500000, ±1.253704930)	±1.729911465$i$	±(1.115547067 ± 0.8649557326$i$)
−1.0	(0.12500000, ±0.992156742)	$\pm$1.728647057$i$	±(1.114076068 ± 0.8643235285$i$)
−0.01	(0.12500000, ±0.175473327)	±1.647106082$i$	±(1.017211303 ± 0.8235530413$i$)
−0.005	(0.12500000, ±0.116684092)	±1.579922392$i$	±(0.933871550 ± 0.7899611956$i$)

, respectively, over a wide range of radiation parameters. Our analysis reveals that there is no instance where all the eigenvalues are purely imaginary. Therefore, we conclude that the positive real parts of the complex roots contribute to the instability of the OoPEPs.

A numerical study of Equation (16), conducted on a dense grid with parameters $Q_1\in \left(\mathrm{0,1}\right]$ and $q_2\in [-\mathrm{5.5,0})$, provides insights into the stability of the OoPEPs. We numerically evaluate matrix $A$ and compute its eigenvalues. The distribution of these eigenvalues for all parameter combinations is presented in Figure 6. This plot reveals a broad range of corresponding eigenvalues; while some individual eigenvalues fall within the unit disk, none of the parameter combinations result in stable equilibrium points (i.e., at least one pair of eigenvalues indicates instability).

Although this figure provides an overview of the range of the eigenvalues, it lacks information regarding their density on the complex plane, an important piece of information about the role of the radiation parameters. An estimation of the density is computed by using a honeycomb-like pattern of regular hexagons as shown in Figure 7 (left subplot). This way, we obtain a discretization of the complex plane that acts as a reference for the estimation of the spatial density of the eigenvalues. Each computed eigenvalue of $A$ for every parameter combination is assigned to a single hexagon. By counting the total number of eigenvalues that belong to each hexagon with respect to the total number of eigenvalues, we estimate the local density. In the right subplot of Figure 7, we observe a clear high density on six locations (bright yellow hexagons), i.e., nearly all matrices have those six particular eigenvalues except a very small fraction.

This observation calls for a further investigation. Here, we try to answer the following question: Which combinations of $Q_1$ and $q_2$ cause this deviation? In Figure 8, with red color, we show the combinations of the radiation parameters that lead to at least one-pair eigenvalue deviation. We observe the following patterns: (a) values of $Q_1\approx 1$ in combination with $q_2\approx 0$ and (b) values of $Q_1\approx 0$ lead to different eigenvalues than all other combinations.

6. Discussion and Conclusions

We investigated the Hill three-body problem (HTBP) with radiating primaries, utilizing the equations of motion established by Kanavos et al. [35]. Our focus was on the existence and location of the out-of-plane equilibrium points (OoPEPs), revealing that the spatial form of the dynamical equations admits this kind of point when the radiation factor $q_2$ of the secondary body is negative. We examined two scenarios: first, when the smaller primary radiates while the larger primary does not, and second, when both primaries are radiating. Notably, these points do not have corresponding ones in the classical HTBP.

Initially, we analyzed a pair of equilibrium points positioned along the z-axis when the larger primary does not radiate. In the case when both primaries radiate, we found two such points in the $Oxz$ plane. The radiation factors significantly influence the existence of equilibrium points, leading to situations where either zero or two OoPEPs exist in symmetrical positions relative to the $(x,y)$ plane. The locations of these points are affected by the radiation factors that express the perturbing forces. Our findings indicate that as the radiation factor of the smaller primary approaches zero (from negative values) while the radiation factor of the larger primary remains fixed, the OoPEPs converge toward the location of the secondary, which is the origin.

We examined the regions of motion for the particle as defined by the zero-velocity surface, yielding corresponding equipotential curves. Our analysis revealed various closed or trapped regions where the infinitesimal mass can move freely, depending on the values of the Jacobian constant $\Gamma$. It was noted that the regions permitting motion for the infinitesimal particle expand as the radiation factor $q_2$ increases. The trapped motion regions (non-permissible for motion) diminish in size as the radiation factor $q_2$ increases for fixed values of the radiation factor $Q_1$. Thus, radiation pressure forces significantly impact the shape of the zero-velocity curves in the $(x,z)$ plane.

We also assessed the linear stability of these equilibrium points across a broad range of radiation parameter values and determined that the motion of the infinitesimal mass around the OoPEPs is unstable. This finding reinforces the conclusions of [28,29,37,46], and others regarding the circular restricted three-body problem under the influence of radiation pressure from the primaries. A thorough numerical analysis of matrix A (Equation (16)) revealed that, in general, the eigenvalues are concentrated around six dominant complex values, while a narrow area of radiation parameter value combinations results in divergent eigenvalues.

Finally, we note that the simpler form of the Hamiltonian for the HTBP enables us to analytically study the positions of the stationary points as well as their stability. In contrast, such analyses in the restricted three-body problem are feasible only through numerical methods (see, e.g., [18,28,46]). A similar investigation into the photogravitational HTBP with oblateness has been conducted by Markakis et al. [37]. In this context, the oblateness of the secondary body allows for the existence of OoPEPs, which are situated solely in the ($x-z$) plane with $x\ne 0$.