{'fraction_non_alphanumeric': 0.0426399388431156, 'fraction_numerical': 0.01987598742886265, 'mean_word_length': 4.143730886850153, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On the Chermnykh-Like Problems: I. The Mass Parameter \mu=0.5

Following Papadakis (2005)'s numerical exploration of the Chermnykh's problem, we here study a Chermnykh-like problem motivated by the astrophysical applications. We find that both the equilibrium points and solution curves become quite different from the ones of the classical planar restricted three-body problem. In addition to the usual Lagrangian points, there are new equilibrium points in our system. We also calculate the Lyapunov Exponents for some example orbits. We conclude that it seems there are more chaotic orbits for the system when there is a belt to interact with.


Introduction
Chermnykh's problem concerns the motion of a test particle in the orbital plane of a dumb-bell. This dumb-bell rotates with constant angular velocity n around its mass center.
This interesting problem was first analysed in Chermnykh (1987) on the Lyapunov stability of the triangular solutions. Gozdziewski and Maciejewski (1998) studied the Lyapunov stability of the Lagrangian points in the entire range of angular velocity n and mass parameter µ. Recently, Papadakis (2005) investigated the equilibrium points and the zero-velocity curves of this problem. In particular, some families of periodic orbits are studied in great detail. The stability zones are also discussed in that paper.
In general, this important problem has many applications in celestial mechanics (see Gozdziewski and Maciejewski 1999) and chemistry (see Strand and Reinhardt 1979). .
When the angular velocity n = 1, it is the classical planar restricted three-body problem.
When n = 0, we get the Euler's problem of two fixed gravitational centers.
On the other hand, because (i) there are many discovered extra-solar planetary systems (Laughlin & Adams 1999, Rivera & Lissauer 2000, Jiang & Ip 2001, Ji et al. 2002, in which about five of them are in binary stars and, (ii) the effect of belts might be important according to some of our previous studies (Yeh & Jiang 2001, Jiang & Yeh 2003, 2004, we here construct a Chermnykh-like problem motivated by these astrophysical applications. In principle, we consider the angular velocity n > 1 and its value is determined by the belt's gravitational potential, which is considered as part of the total potential to govern the test particle's motion.
Therefore, we construct our basic model in Section 2. In Section 3, we study the existence of new equilibrium points. We discuss the behavior of the orbits with initial conditions near the new equilibrium points in Section 4 and calculate their values of Lyapunov Exponent in Section 5. Section 6 concludes the paper.

The Model
We consider the motion of a test particle influenced by the gravitational force from the central binary and the circumbinary disc.
For convenience, we set the gravitational constant G = 1. We assume that two masses of the central binary are µ 1 and µ 2 and choose the unit of mass to make µ 1 + µ 2 = 1. The separation between two stars in the central binary is set to be unity. The time unit is therefore determined from the above choice.
Under the influence from the belt, when µ 1 = µ 2 = 0.5, both components of the central binary move on circular orbits at r = 0.5 with mean motion, i.e. angular velocity The gravitational force f b from the belt is F (ξ) and E(ξ) are elliptic integral of the first kind and the second kind and (Please see Appendix A for the derivation of Eq.(3).) Hence, the x and y components of where f b is in Eq.
(3) and V is the potential from the belt.
where the potential U * is given by Eq.
The Jacobi integral for this system is therefore The density profile of the belt is where r = √ x 2 + y 2 , c is a constant completely determined by the total mass of the belt and p is a natural number. In this paper, we set p = 2, r a = r i + 0.1, r b = r i + 0.9, and r o = r i + 1.0 for all numerical results (Lizano & Shu 1989). It is a power-law profile with smooth edges. Hence, the total mass of the belt is We use r i and M b as the controlling parameters of the belt profile. Using Eq.(8), we determine the zero-velocity curves of our system when r i = 0.7 and M b = 0.3, as shown in Figure 1. Because there are elliptic integrals in the potential V , this has to be done numerically and the points are not uniformly distributed on the curves. For the results in Section 4 and Section 5, we always set r i = 0.7 but M b = 0.3 or 0. Following Tancredi et al.
(2001), we will integrate the orbits up to 1000 binary period.

The Equilibrium Points
We know that there are five equilibrium points (Lagrange points) for the classical restricted three-body problem. Thus, our system shall have five equilibrium points when M b = 0. It would be interesting to investigate the number of equilibrium points when M b > 0. When there are more than five equilibrium points, we claim that the new equilibrium points exist.
For convenience, we define and We have the following properties for the case of equal-mass binary: If (x e , y e ) is an equilibrium point of System (7), then we have: either (1) x e = 0 and y e satisfies h(y) = 0 or (2) x e satisfies k(x) = 0 and y e = 0.
Proof: Suppose that (x e , y e ) is an equilibrium point, thus it satisfies f (x, y) = 0 and g(x, y) = 0. From Eq. (14), we have (xe,ye) = 0. We now discuss these two cases separately.
(II) y e = 0: Thus, (x e , 0) is the equilibrium point of System (7). ♦ Because of the above two properties, in stead of searching the roots for two-variable functions f (x, y) and g(x, y) to determine the equilibrium points on the x − y plane, we only need to find the roots of one variable functions h(y) and k(x) to get all the equilibrium points of System (7) when the mass parameter µ = 0.5. Therefore, we numerically solve both h(y) = 0 and k(x) = 0 and find out the number of equilibrium points for different given parameters. The numerical scheme of root finding is the Van Wijngaarden-Dekker-Brent Method (Brent 1973). This is an excellent algorithm recommended by Press et al. (1992).
We set a high level of accuracy that the maximum error is 10 −8 for the locations of equilibrium points on both x-axis and y-axis. We find that the region with new equilibrium points is itself a triangle. To have new equilibrium points, the mass of the belt has to be larger than 0.15 and r i is somewhere between 0.7 and 0.8.
It is not surprising that there are more equilibrium points in our system because the potential field of whole system is more complicated than the classical restricted three-body problems and thus there are more possibilities that gravitational forces from different components can balance each other.

The Orbits
Since we have discovered new equilibrium points for particular regions of r i − M b plane for our system, it would be interesting to investigate the orbital behavior around these new equilibrium points.
From Figure 2, we notice that when M b = 0.3, r i = 0.7, we could have new equilibrium points. We thus use this as a standard case. At the y-axis, we explicitly determine the locations of all equilibrium points, which are at (x e , y e ) = (0, ±0.73), (0, ±0.77), (0, ±1.05), in addition to the usual L1 point at (0, 0). (We find that all new equilibrium points are at y-axis because of the choice µ 1 = µ 2 = 0.5.) To understand the properties of these equilibrium points at the y-axis, we numerically calculate the orbits with initial conditions very close to these points. There are four initial conditions in this paper, i.e.  For Initial Condition L1, Figure 3(a) shows that the orbit fills the Roche lobe, i.e. the zero-velocity curve passing the L1 point. The test particle is gravitational bound to one of the star because it probably has less initial total energy than those ones in Figure 3  For Initial Condition E1, Figure 3(b) shows that the test particle moves randomly at the beginning and then steadily move out of the region of central binary. Figure 4(b) shows that the test particle drifts away from the initial location regularly.
For Initial Condition E2, Figure 3(c) is similar to Figure 3(b) but the test particles move more chaotically in the central region and then move out slowly at the first stage and more quickly finally. Figure 4(c) is very similar to Figure 4(b).
For Initial Condition E3, Figure 3(d) shows that the test particle is always close to the equilibrium point and moving on a regular periodical orbit. The equilibrium point could be a neutral stable point. Figure 4(d) shows that the test particle is on a complicated orbit and drifts away from the central region slowly.
We notice that the orbits in Figure 3 Most other orbits show that the test particles move spirally outward from the central region except the one in Figure 3(d).

Lyapunov Exponent
In addition to analyze the orbits themselves, we calculate these orbits' Lyapunov Exponents to understand how sensitively dependent on the initial conditions. In general, the larger value of Lyapunov Exponent means more sensitively dependent on the initial conditions.
Our procedure to calculate the Lyapunov Exponent was from Wolf et al. (1985). The maximum Lyapunov Exponent is defined by γ = lim t→∞ χ(t), where χ(t) is the Lyapunov Exponent Indicator. The system is chaotic if γ > 0 or otherwise regular. However, it is not possible to take t → ∞ in practice. We numerically determine the Lyapunov Exponent Indicator χ(t) and usually plot ln χ(t) as function of ln t after the evolution is followed up to some time. If the curve (or the envelope of the curve) on the ln t − ln χ(t) plane shows a negative constant slope, the system is regular, otherwise it is chaotic. From these results, we notice that the dotted curves in Figure 5(b)-(c) and the solid curve in Figure 5(d) show negative constant slopes. Thus, when M b = 0, the orbits of Initial Condition E1 and E2 are likely to be regular. When M b = 0.3, only the orbit of Initial Condition E3 is likely to be regular. This is probably due to that the equilibrium point close to Initial Condition E3 is neutral stable. Note that the solid curves in Figure 5 are shorter because the test particles were ejected and the calculations were stopped when the distances were larger than 10 4 from the central region.
We can also notice that, in Figure 5(a), the solid curve is always higher than the dotted curve. That is, the values of Lyapunov Exponent Indicator of the orbit with Initial Condition L1 when M b = 0.3 are always larger than the one of the orbit with Initial Condition L1 when M b = 0. To summarize, the existence of belt seems to make the system to have more chaotic orbits.

Concluding Remarks
We have provided the equations for a model which includes the gravitational influence from a belt around the central binary. We find that, in addition to the usual Lagrange points, there are new equilibrium points on the y-axis when each mass of the components of the binary is equal to 0.5. To study the orbits around these new equilibrium points, we calculate the orbits and their Lyapunov Exponents with four different initial conditions. It seems that the system could have more chaotic orbits when there is a belt.

A. Gravitational Force from the Belt f b
Let F (ξ) be the elliptic integral of the first kind and assume By sin 2 A = 1 2 (1 − cos 2A), we have Therefore, from Equation (A2), we have The gravitational potential is where Equation (A3) has been used. Now we differentiate this potential with respect to r to get the gravitational force, so Let E(ξ) be the elliptic integral of the second kind and ξ ′ = √ 1 − ξ 2 . Since F (ξ) is the elliptic integral of the first kind, from Byrd & Friedman (1971), we have By Equation (4) and (A6), we calculate We substitute Equation (A7) into Equation (A5), so we have