1 Introduction

The behavior of nonintegrable dynamical systems is complicated due to the presence of complex structures in their phase space (Chirikov 1979). In particular, stable trajectories wherever exist can affect properties of the whole system significantly (Harsoula and Contopoulos 2018). Although chaotic trajectories are ergodic, they may also show nontrivial behaviors (Lichtenberg and Lieberman 1992; Zaslavsky 2002). As one of the simplest nonintegrable Hamiltonian systems, motion of charged particles in the dipole magnetic field has been studied extensively during the past century (Störmer 1907, 1930; Dragt 1965; Kolesnikov 2017; Liu et al. 2022). In particular, significant efforts have been put to study periodic orbits in the meridian plane (De Vogelaere 1950; Markellos and Halioulias 1977; Markellos and Klimopoulos 1977; Markellos et al. 1978a, b; Epp and Pervukhina 2018), and it has been known that stable periodic orbits can affect statistical properties of the whole system significantly (Zaslavsky 2002; Harsoula and Contopoulos 2018). Stable periodic orbits also bear significance in physical systems since their associated stable quasi-periodic orbits can be realized and have distinct characteristics (Wang et al. 2021; Horn-Stanja et al. 2018).

In celestial mechanics, stability of relative equilibria of axisymmetric systems has been studied extensively (Howard 1999; Mohandas et al. 2019; Leghmouche and Mebarki 2021; Katsiaris and Psillakis 1986), and these studies have been expanded to reveal the global dynamics (Howard et al. 1999). In particular, some periodic orbits were uncovered numerically and their stability was analyzed (Gong and Gong 2018). These studies have been used to explain observations of charged particles trapped in the magnetosphere of different planets of the solar system. A systematic theme is needed to have a comprehensive study of similar systems to uncover numerous islets of stable orbits that may be observed in real physical systems (Chirikov 1979). Such studies may be generalized to more general cases where the system is not axisymmetric to have applications in actual planetary systems (Stevenson 1983; Connerney 1993; Stevenson 2003; Epp and Masterova 2013, 2014).

With the aid of Lyapunov characteristic exponents (LCE), stable orbits can be readily identified since the value of their maximum LCE (mLCE) is zero. However, the distribution of stable orbits in the phase space is complex, showing fractal structures (Umberger and Farmer 1985; Lee and Flach 2020; Liu et al. 2022). To understand this distribution, one may identify classes of periodic orbits (Markellos et al. 1978b; Piña and Jiménez Lara 1987). In this paper, we focus on studying orbits of charged particles in the dipole magnetic field with an azimuthal initial velocity, i.e., initially static in the meridian plane. We first show that periodic orbits in the meridian plane have two static points half a period apart and those symmetric with respect to the equatorial plane cross it perpendicularly twice in each period at the same distance from the origin. Classes of such symmetric open periodic orbits are then identified and used to explore distribution of stable orbits in the phase space. Such a study can be generalized to axisymmetric systems that are symmetric with respect to an equatorial plane and be used to study the dynamics of charged dust grains in magnetic planets.

The paper is organized as the following. In Sect. 2 we present the Hamiltonian equations for the meridian plane motion of a charged particle moving in a dipole magnetic field and prove general properties of periodic orbits with open ends in Sect. 3. Our main results are shown in Sect. 4. Conclusions are drawn in Sect. 5.

2 Motion of charged particles in the meridian plane of a dipole magnetic field

Since the dipole magnetic field is axisymmetric, the azimuthal canonical momentum \(p_\phi \) and the energy H of a charged particle moving in it are conserved if one ignores the radiation reaction force, which is a quantum phenomenon and cannot be treated self-consistently in classical mechanism (Price et al. 2022; Gralla et al. 2009). Following the standard procedure (Dragt 1965), one can introduce a characteristic length with \(p_\phi \), the charge of the particle e, and the magnetic moment of the dipole M and obtain the dimensionless Hamiltonian for motion in the meridian plane \([\rho , z]\) (Liu et al. 2022):

$$\begin{aligned} H= \frac{1}{2}\left( {p_z}^2+{p_\rho }^2\right) + V \end{aligned}$$
(1)

where z is the symmetric axis, \(\rho \) is the radius in the equatorial plane, and the effective potential

$$\begin{aligned} V=\frac{1}{2}\left[ \frac{1}{\rho }-\frac{\rho }{(z^2+\rho ^2)^\frac{3}{2}}\right] ^2 \end{aligned}$$
(2)

corresponds to the kinetic energy associated with the azimuthal motion, and \(p_\rho \) and \(p_z\) are the momentum in the \(\rho \) and z directions, respectively. As shown in (Dragt and Finn 1976), the motion of charged particles in a dipole magnetic field can be recovered once the motion in the meridian plane is determined. We then have the Hamiltonian differential equations of motion:

$$\begin{aligned} {\dot{z}}&= \frac{\partial H}{\partial p_z}=p_z\,, \end{aligned}$$
(3)
$$\begin{aligned} {\dot{\rho }}&=\frac{\partial H}{\partial p_\rho }=p_\rho \,, \end{aligned}$$
(4)
$$\begin{aligned} \dot{p_z}&= -\frac{\partial H}{\partial z}=-\frac{3z\rho \left[ \frac{1}{\rho } -\frac{\rho }{(z^2+\rho ^2)^\frac{3}{2}}\right] }{(z^2+\rho ^2)^\frac{5}{2}}\,, \end{aligned}$$
(5)
$$\begin{aligned} \dot{p_\rho }&=-\frac{\partial H}{\partial \rho }\ \nonumber ,\\&=\left[ \frac{1}{\rho ^2}-\frac{3\rho ^2}{(z^2+\rho ^2)^\frac{5}{2}} +\frac{1}{(z^2+\rho ^2)^\frac{3}{2}}\right] \left[ \frac{1}{\rho }-\frac{\rho }{(z^2+\rho ^2)^\frac{3}{2}}\right] , \end{aligned}$$
(6)

where an upper dot \(\dot{}\) indicates a derivative with respect to the time t.

3 General characteristics of open periodic orbits

Periodic orbits are the simplest orbits and are ubiquitous in the phase space (Zehnder 1973). Stable quasi-periodic orbits are always associated with stable periodic orbits (Piña and Jiménez Lara 1987; Lichtenberg and Lieberman 1992). It is therefore essential to understand properties of periodic orbits to uncover the phase space structure of a dynamical system.

Particles with an azimuthal initial velocity have a zero initial velocity with \(p_{z,0}=0,p_{\rho ,0}=0\) in the \([\rho , z]\) plane. Periodic orbits in this meridian plane have open paths and some general characteristics that are essential to search for such orbits numerically. To prove these characteristics, we define \(\ddot{z}\equiv f(z,\rho )\) and \(\ddot{\rho }\equiv g(z,\rho )\), and the minimum positive period T. Static orbits (equilibria) with \(T=0\) are trivial and will not be considered here. The Hamiltonian equations then lead to

$$\begin{aligned} \begin{aligned} f(-z,\rho )&= -f(z,\rho ),\\ g(-z,\rho )&= g(z,\rho ),\\ \end{aligned} \end{aligned}$$
(7)

and

$$\begin{aligned} \begin{aligned} z(t+T)=z(t),\ \ {\dot{z}}(t+T)={\dot{z}}(t),\\ \rho (t+T)=\rho (t),\ \ {\dot{\rho }}(t+T)={\dot{\rho }}(t).\\ \end{aligned} \end{aligned}$$
(8)

Since open orbits start with zero initial velocities, it is evident that \(z(t)=z(-t),\ \rho (t)=\rho (-t)\).

Proof

Let \(t=-t^*\), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z}{{dt^*}^2}=f(z,\rho )\\ \frac{d^2\rho }{{dt^*}^2}=g(z,\rho ) \end{array}\right. } \textrm{with}\ \ {\left\{ \begin{array}{ll} z|_{t^*=0}=z_0,\ \ \frac{dz}{dt^*}|_{t^*=0}=0,\\ \rho |_{t^*=0}=\rho _0,\ \ \frac{d\rho }{dt^*}|_{t^*=0}=0, \end{array}\right. } \end{aligned}$$
(9)

which are the same equations followed by z(t) and \(\rho (t)\). Then we have \(z(t)=z(t^*),\rho (t)=\rho (t^*)\), i.e.,

$$\begin{aligned} \begin{aligned} z(t)=z(-t),\\ \rho (t)=\rho (-t). \end{aligned} \end{aligned}$$
(10)

We note that Eq. (10) and discussions below are valid for any potential that is symmetric with respect to the z axis. \(\square \)

3.1 Open periodic orbits

For open periodic orbits, it can be shown that there are two static points in each period at times \(t=0\) and \(t=\frac{T}{2}\), respectively.

Proof

From Eq. (10), we have

$$\begin{aligned} \begin{aligned} {\dot{z}}(t)=-{\dot{z}}(-t),\\ {\dot{\rho }}(t)=-{\dot{\rho }}(-t). \end{aligned} \end{aligned}$$
(11)

Let \(t=\frac{T}{2}\), Eq. (8) then gives

$$\begin{aligned} \begin{aligned} \dot{z}\left( {\frac{T}{2}} \right) = - \dot{z}\left( { - \frac{T}{2}} \right) = - \dot{z} \left( {\frac{T}{2}} \right) ,\\ \dot{\rho }\left( {\frac{T}{2}} \right) = \dot{\rho }\left( { - \frac{T}{2}} \right) = - \dot{\rho }\left( {\frac{T}{2}} \right) .\end{aligned} \end{aligned}$$
(12)

Therefore,

$$\begin{aligned} \begin{aligned} \dot{z}\left( {\frac{T}{2}} \right) = 0, \\ \dot{\rho }\left( {\frac{T}{2}} \right) = 0 , \end{aligned} \end{aligned}$$
(13)

and we define \(z_1\equiv z(\frac{T}{2}),\ \rho _1\equiv \rho (\frac{T}{2})\). Clearly, \((z_1,\rho _1)\ne (z_0,\rho _0)\). Otherwise, T will no longer be the minimum positive period since \(T/2< T\) is also a positive period. \(\square \)

Furthermore, we can prove that \(t=\frac{T}{2}\) is the only t that satisfies \({\dot{z}}(t)=0,\ {\dot{\rho }}(t)=0\) in (0, T).

Proof

Suppose that there exists a \(t_1\in (0,T),t_1\ne \frac{T}{2}\) such that \({\dot{z}}(t_1)=0,\ {\dot{\rho }}(t_1)=0\), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z}{{dt}^2}=f(z,\rho )\\ \frac{d^2\rho }{{dt}^2}=g(z,\rho ) \end{array}\right. } \textrm{with}\ \ {\left\{ \begin{array}{ll} z|_{t=t_1}=z(t_1),\ \ \frac{dz}{dt}|_{t=t_1}=0,\\ \rho |_{t=t_1}=\rho (t_1),\ \ \frac{d\rho }{dt}|_{t=t_1}=0. \end{array}\right. } \end{aligned}$$
(14)

Let \(t=-t^{*}+2t_1\), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z}{{dt^*}^2}=f(z,\rho )\\ \frac{d^2\rho }{{dt^*}^2}=g(z,\rho ) \end{array}\right. } \textrm{with}\ \ {\left\{ \begin{array}{ll} z|_{t^*=t_1}=z(t_1),\ \ \frac{dz}{dt^*}|_{t^*=t_1}=0,\\ \rho |_{t^*=t_1}=\rho (t_1),\ \ \frac{d\rho }{dt^*}|_{t^*=t_1}=0. \end{array}\right. } \end{aligned}$$
(15)

Equations (14) and (15) have the same solutions: \(z(t)=z(t^*),\ \rho (t)=\rho (t^*)\). Therefore,

$$\begin{aligned} \begin{aligned} z(t)=z(-t+2t_1)=z(t-2t_1)=z(t+T-2t_1), \\ \rho (t)=\rho (-t+2t_1)=\rho (t-2t_1)=\rho (t+T-2t_1). \end{aligned} \end{aligned}$$
(16)

Then \(0<|T-2t_1|\ \textrm{or}\ 2t_1<T\) is a positive period. \(\square \)

3.2 Symmetric open periodic orbits

For open periodic orbits symmetric with respect to the equatorial plane, \((z_0,\rho _0) = (-z_1,\rho _1)\). When \(t=\frac{T}{4}\), it can be shown that they pass through the equatorial plane perpendicularly with \(p_\rho =0\) and \(z=0\) and there is only one such perpendicular crossing within \(\frac{T}{2}\).

Proof

Let \(z=-z^*,\ t=t^*+\frac{T}{2}\) and considering Eq. (7), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z^*}{{dt^*}^2}=f(z^*,\rho )\\ \frac{d^2\rho }{{dt^*}^2}=g(z^*,\rho ) \end{array}\right. } \textrm{with} \ \ {\left\{ \begin{array}{ll} z^*|_{t^*=0}=-z_1=z_0,\ \ \frac{dz^*}{dt^*}|_{t^*=0}=0,\\ \rho |_{t^*=0}=\rho _1=\rho _0,\ \ \frac{d\rho }{dt^*}|_{t^*=0}=0, \end{array}\right. } \end{aligned}$$
(17)

which are the same equations followed by \((z(t),\ \rho (t))\). Therefore, \(z(t)=z^*(t^*),\ \rho (t)=\rho (t^*)\), and we have

$$\begin{aligned} \begin{aligned} z(t) = - z\left( {t - \frac{T}{2}} \right) \\ \rho (t) = \rho \left( {t - \frac{T}{2}} \right) . \end{aligned} \end{aligned}$$
(18)

Taking the time derivative of \(\rho (t)\), one has

$$\begin{aligned} \begin{aligned} \dot{\rho }(t) = \dot{\rho }\left( {t - \frac{T}{2}} \right) . \end{aligned} \end{aligned}$$
(19)

Let \(t=\frac{T}{4}\), Eqs. (10), (11), (18), and (19) give

$$\begin{aligned} \begin{aligned} z\left( {\frac{T}{4}} \right) = z\left( { - \frac{T}{4}} \right) = - z\left( { - \frac{{3T}}{4}} \right) = - z\left( {\frac{T}{4}} \right) \\ \dot{\rho }\left( {\frac{T}{4}} \right) = \dot{\rho }\left( { - \frac{T}{4}} \right) = - \dot{\rho }\left( {\frac{T}{4}} \right) . \end{aligned} \end{aligned}$$
(20)

Therefore,

$$\begin{aligned} \begin{aligned} z\left( {\frac{T}{4}} \right) = 0 \\ \dot{\rho }\left( {\frac{T}{4}} \right) = 0. \end{aligned} \end{aligned}$$
(21)

One can prove that \(t=\frac{T}{4}\) is the only \(t\in [0,\frac{T}{2})\) that satisfies \(z(t)=0,\ {\dot{\rho }}(t)=0\). \(\square \)

Proof

Suppose there exists a \(t_1\in [0,\frac{T}{2})\), and \(t_1\ne \frac{T}{4}\) such that \(z(t_1)=0,\ {\dot{\rho }}(t_1)=0\), we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z}{{dt}^2}=f(z,\rho )\\ \frac{d^2\rho }{{dt}^2}=g(z,\rho ) \end{array}\right. } \textrm{with}\ \ {\left\{ \begin{array}{ll} z|_{t=t_1}=0,\ \ \frac{dz}{dt}|_{t=t_1}=v_{z}(t_1),\\ \rho |_{t=t_1}=\rho _{t_1},\ \ \frac{d\rho }{dt}|_{t=t_1}=0. \end{array}\right. } \end{aligned}$$
(22)

Let \(t=-t^{*}+2t_1,\ z=-z^*\), Eq. (7) implies:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{d^2z^*}{{dt^*}^2}=f(z^*,\rho )\\ \frac{d^2\rho }{{dt^*}^2}=g(z^*,\rho ) \end{array}\right. } \textrm{with}\ \ {\left\{ \begin{array}{ll} z^*|_{t^*=t_1}=0,\ \ \frac{dz^*}{dt^*}|_{t^*=t_1}=v_{z}(t_1),\\ \rho |_{t^*=t_1}=\rho _{t_1},\ \ \frac{d\rho }{dt^*}|_{t^*=t_1}=0. \end{array}\right. } \end{aligned}$$
(23)

Equations (22) and (23) have the same solutions: \(z(t)=z^*(t^*),\ \rho (t)=\rho (t^*)\). Equations (10), (18), and (19) then lead to

$$\begin{aligned} \begin{aligned} z(t)=-z(-t+2t_1)=-z(t-2t_1)=z(t+\frac{T}{2}-2t_1),\\ \rho (t)=\rho (-t+2t_1)=\rho (t-2t_1)=\rho (t+\frac{T}{2}-2t_1). \end{aligned} \end{aligned}$$
(24)

Then \(0<|\frac{T}{2}-2t_1|\le \frac{T}{2}<T\) is a positive period.

One can actually show that any open orbit that crosses the equatorial plane perpendicularly must be symmetric open periodic orbits.

Equations (10), (22), and (23) imply

$$\begin{aligned} \begin{aligned} z(t)=-z(-t+2t_1)=-z(t-2t_1),\\ \rho (t)=\rho (-t+2t_1)=\rho (t-2t_1). \end{aligned} \end{aligned}$$
(25)

Then we have

$$\begin{aligned} \begin{aligned} z(t)=z(t-4t_1),\\ \rho (t)=\rho (t-4t_1). \end{aligned} \end{aligned}$$
(26)

Therefore, \(T=4t_1\) is a positive period and \(z(0)=-z(T/2)\), \(\rho (0)=\rho (T/2)\).

In summary, for an effective potential symmetric with respect to the z axis, the open-path orbits are symmetric with respect to the time t, and there are two static points, half a period apart, for each open-path orbit. Any open periodic orbit symmetric with respect to the z axis crosses the equatorial plane perpendicularly twice at the same distance from the origin in each period. \(\square \)

4 Results

Fig. 1
figure 1

The distribution of the mLCE (for \(H<1/32\)) and the escape time (for \(H>1/32\)) of particles with an azimuthal initial velocity. The black lines indicate the energy contours of \(H=1/32\). The cyan lines indicate the maximum energy of periodic orbits (Markellos et al. 1978a). The radial acceleration is positive beyond the outer boundary of the colored scanned region. Besides several blocks of stable quasi-periodic orbits associated with stable periodic orbits in the equatorial plane, there are many small isolated regions of stable quasi-periodic orbits. See text for details

Full size image

Using the same numerical codes used by Liu et al. (2022), i.e., RK45 formula being used to solve the relevant ordinary differential equations with default absolute and relative error tolerance of \(10^{-9}\), we have the main result shown in Fig. 1. For \(H<1/32\), the \([\rho ,\ z]\) plane is scanned to obtain the mLCE of each orbit starting with \(\rho =\rho _0,\ z=z_0,\ p_\rho =p_z=0\). It should be pointed out that at a given energy, most chaotic orbits should have the same value of the mLCE (Lee and Flach 2020). Finite integration time and stickiness of orbits near stable orbits make the numerical results not as uniform as expected (Meiss 2007; Harsoula and Contopoulos 2018; Harsoula et al. 2019).

Since \({\dot{r}}>0\) with \(r = (\rho ^2+z^2)^{1/2}\) for \(r>(4H)^{-1/3}\)(Markellos et al. 1978a)  we only consider the domain with \(r\le (4H)^{-1/3}\), beyond which all particles escape to infinity. In the meridian plane, this upper limit is equivalent to the magnetic field line passing through the point [\(\rho =2,\ z=0\)], then we have \(r\le 2\cos ^2\lambda \), where \(\cos \lambda = \rho /r\). For \(H\ge 1/32\), we calculate the escape time \(t_{\textrm{esc}}\) that it takes for the particle to reach \(\rho =2\). Particles with \(t_\textrm{esc}>10^4\) are considered as trapped particles, whose orbits are likely stable. It is interesting to note that there are spiral structures in the escape time and most isolated regions of stable orbits locate at some turning points of spiral structures. Such structures also appear to emerge repeatedly approaching the origin. We also indicate the contours for the maximum energy \(H\simeq 0.80819\) that periodic orbits can have (Markellos et al. 1978a) and \(H=1/32\) in Fig. 1.

The distribution of stable orbits (dark red) shows complex structures (Xie and Liu 2020). Besides stable orbits at low energies and those associated stable orbits in the equatorial plane (De Vogelaere 1950; Markellos et al. 1978b), there are many small islands of stable orbits spread over the whole phase space. The boundaries of regions of stable orbits appear to be fractal, and there are apparent self-similarities with the decrease of energy. We will try to understand these structures by identifying classes of symmetric open periodic orbits and analyzing their relations to regions of stable orbits.

Fig. 2
figure 2

Same as Fig. 1 with the open-path periodic orbits \(s_1\) indicated by gray lines. From left and right to middle, we have the families \(f_2\), \(f_6\),... and \(f_0\), \(f_4\),... defined by Markellos et al. (1978a), respectively. Regions of stable orbits at the low energy end of each family have similar structures

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4.1 Classes of symmetric open periodic orbits

Fig. 3
figure 3

Same as Fig. 1 with the open-path periodic orbits \(s_2\) and \(s_1\) indicated by red and gray lines, respectively

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Fig. 4
figure 4

Upper: Same as Fig. 1 with the open-path periodic orbits \(s_3\) indicated by red lines. Lower: Same as the upper panel except that the color for \(H<1/32\) indicates the crossing time \(t_{\textrm{cross}}\) defined in the text

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\(s_1\): Given the general characteristics of symmetric open periodic orbits discussed in Sect. 3.2, one can readily identify classes of such orbits numerically. We first search for the principle families introduced by Markellos et al. (1978a), members of which cross the equatorial plane perpendicularly and once in each half a period. For each set of initial conditions [\(\rho _0\), \(z_0\), 0, 0], we search for such orbits by following the radial velocity \(p_\rho \) at the first crossing of the equatorial plane of each orbit. Principle symmetric open-path periodic orbits have a null radial velocity at the first crossing of the equatorial plane.

Figure 2 shows families of such orbits (gray lines), which we labeled as “\(s_1\)". They corresponds to families identified by Markellos et al. (1978a) with even indexes. However, in contrast to these families in the \([\rho ,\ p_z]\) plane, each family locates on one side of the “thalweg", i.e., the bottom of the potential V. From high to low energies, we have \(f_0\), \(f_4\),... and \(f_2\), \(f_6\),.. on the right and the left side of the“thalweg", respectively. The bulk of stable orbits associated with the low energy part of each family have similar features at the boundary, and \(f_0\) goes through a prominent region of stable orbits with \(H>1/32\), which agrees to the results of Markellos et al. (1978b). Most stable orbits associated with the high-energy stable periodic orbits in the equatorial plane (De Vogelaere 1950), e.g., those at \(\rho \simeq 1.7\) and 1.9, are not associated with families in \(s_1\), except for \(f_2\), both of whose ends on the equatorial plane are stable.

\(s_2\): With the principle families identified above, one can readily identify the second class of symmetric open period orbits, which cross the equatorial plane perpendicularly at the second crossing from the initial location. Numerically, when scanning the \([\rho _0, z_0]\) plane with null initial velocities, we trace the radial velocity \(p_\rho \) at the second crossing of the equatorial plane of each orbit. Periodic orbits in \(s_2\) have \(p_\rho =0\) at the second crossing of the equatorial plane. The orbits obtained this way include those in \(s_1\), which have been identified above and can be readily separated from those in \(s_2\).

Figure 3 shows \(s_2\) (red) together with \(s_1\) (gray). For each family in \(s_1\), there are 2 pairs of families in \(s_2\), with two families on each side of the thalweg, that go through the origin and terminate at 2 pairs of stable orbits in the equatorial plane at the other ends. Each pair of these stable orbits actually corresponds to the two ends of one orbit in the equatorial plane. One of these families associated with \(f_0\) in \(s_2\) crosses \(f_0\) in the highest energy region of stable orbits. Another crosses \(f_0\) at a lower energy. The orbit in \(s_2\) reduces to \(s_1\) at the crossing point of these two families. There are fine structures at the crossings between families in \(s_1\) and \(s_2\). The other two families on the other side of the thalweg do not go through families in \(s_1\).

The other families appear as loops and locate between these family pairs, and the number of loops increases quickly with the decrease of energy. These loops appear to trace spiral structures in the escape time and do not cross families in \(s_1\). Moreover, families in one class do not go through each other.

In Table 1, some families of different classes are listed and the families that terminate at the equatorial plane are indicated with “e" and the loops are indicated with “l”. Since families in each class do not cross each other, we list them from the outer region in the parameter space of their initial conditions [\(\rho _0\), \(z_0\), 0, 0] to the inner region. The number before “l” indicates the number of loops that can be seen with the current resolution of scanning of the parameter space [\(\rho _0\), \(z_0\)]. The “+" indicates that there may be more loops than the number before. When this number exceeds 10 or cannot be seen clearly, we use the upper case “L" to indicate these families of loops.

Table 1 Classes of symmetric open periodic orbits
Full size table

\(s_3\): Similarly, one can identify the third class of symmetric open periodic orbits that cross the equatorial plane perpendicularly at the third crossing from each of their initial location. Families in this class are shown in Fig. 4. Similar to the class \(s_2\), for each family in \(s_1\), there are four pairs of families in \(s_3\) that go through the origin and terminate at four pairs of stable periodic orbits in the equatorial plane at the other ends. Two of these families go through \(f_0\) in the highest energy region of stable orbits, while the other two cross \(f_0\) at lower energies. Fractal structures are seen near these crossing points. The other four families at the inner side of the potential do not cross families in \(s_1\).

Compared to \(s_2\), there are many more loops between these families, that clearly trace spiral structures in the escape time and approach to their centers. To reveal similar spiral structures for \(H<1/32\), one can introduce \(\rho _\textrm{max}=[1-(1-4(2\,H)^{1/2})^{1/2}]/[2(2\,H)^{1/2}]\) and \(\rho _\textrm{min}=[(1+4(2\,H)^{1/2})^{1/2}-1]/[2(2\,H)^{1/2}]\) that correspond to the maximum and the minimum distance to the origin for motion in the equatorial plane with an energy H, respectively. We define the crossing time \(t_{\textrm{cross}}\) as the first crossing of \(\rho = \rho _{\textrm{max}}-(\rho _{\textrm{max}}-\rho _{\textrm{min}})/50\) with a positive \(p_\rho \). This crossing time is indicated in the lower panel of Fig. 4. Similar to the escape time, initial conditions with the crossing time greater than \(10^4\) are indicated with dark red color and are well correlated with stable orbits in the upper panel of Fig. 4. There are spiral structures in the crossing time that are associated with family loops in \(s_3\). Similar to \(s_2\), the family loops do not appear to cross any other families, and families in \(s_2\) do not appear to cross families in \(s_3\) either, which is different from those in the [\(\rho ,\ p_z\)] plane, where families of symmetric periodic orbits may cross each other (Markellos et al. 1978b).

4.2 Classes of asymmetric open periodic orbits

Fig. 5
figure 5

Examples of stable periodic orbits (thin solid lines) with the corresponding energy contours (thick solid lines), and the bottom of the potential V (dotted) is also shown in the middle and bottom panels

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Fig. 6
figure 6

More examples of stable periodic orbits

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Although the \(s_3\) class of symmetric open periodic orbits go through the most prominent regions of stable orbits, the chain of isolated regions of stable orbits at the turns of the main spiral structure are clearly not associated with \(s_3\) (see Fig. 1). We therefore randomly plot a periodic orbit from each of these regions to reveal their properties. Figure 5 shows some of these orbits. The upper panel gives examples of symmetric periodic orbits in the highest energy region of stable orbits. Since all classes of symmetric periodic orbits go through this region, one can readily find orbits for each of these three classes presented above. Besides these trajectories, we also plot the corresponding energy contours to better demonstrate the orbital behavior.

The middle and lower panels show the upper and lower chains of islands of stable orbits, respectively. Based on the number of crossing through the equatorial plane and through the bottom of the potential V: \(r=\cos ^2\lambda \) in each half a period, we label these orbits with the letter “a” (for asymmetric) with the lower and upper indexes indicating these numbers, respectively. It is evident that orbits with an odd number of the upper index have a static point on each side of the thalweg, while those with an even number of the upper index have their static points on one side of the thalweg. These two kinds of asymmetric orbits belong to different families. The two orbits \(a_1^3\) in the lower panel show that for stable orbits near \(H=1/32\), orbits with \(H<1/32\) are very similar to those with \(H>1/32\). The critical energy \(H=1/32\) is not an important parameter determining properties of these orbits.

On the other hand, along families of symmetric open periodic orbits, the number of crossing of the bottom of the potential may increase with the decrease of energy (Markellos et al. 1978a). Moreover, Markellos and Klimopoulos (Markellos and Klimopoulos 1977) discovered a principle asymmetric class (surface) in the three-dimensional sub-phase-space of \(z=0\) that contains both open and close types of periodic orbits, and they both can oscillate around the bottom of the potential V many times. They also found more than 20 regimes of stable orbits in this asymmetric class. In the four-dimensional phase space, this asymmetric class occupy a three-dimensional volume and its intersections with the plane of static points for the Meridian motion should be one-dimensional. The stable asymmetric orbits in Fig. 5 all belong to the principle asymmetric class identified by Markellos and Klimopoulos (Markellos and Klimopoulos 1977).

It is difficult to identify the asymmetric open periodic orbits numerically. As shown in Sect. 3.1, the defining characteristic of asymmetric open periodic orbits is their two static ends half a period apart. However, there are numerous chaotic or quasi-periodic orbits nearby with velocities very close to zero besides the initial static point. There is no simple numerical method so that the asymmetric periodic orbits can be identified with sufficient accuracy.

Nevertheless, based on the discussion above, in the plane of static points, the principle asymmetric class can be divided into two families, one with their two static ends on one side of the thalweg and the other with their two static ends on each side of the thalweg. Figure 5 shows that these two families follow the main spiral structure and the number of oscillations around the bottom of the potential increases as these families approach the center of this spiral structure. Since these two families cross the equatorial plane once in each half a period, their two ends locate on two sides of the equatorial plane.

The family with an even upper indexes crosses \(f_0\) in the low energy end and extends toward the origin. Since both ends of this family are on the outer side of the thalweg, there is a one to one map for points on both sides of the crossing point with \(f_0\). As this family approaches the center of the main spiral structure, it approaches the origin at the other end (below the equatorial plane). The principle asymmetric class terminates at the equatorial plane (Markellos and Klimopoulos 1977). The family with an odd upper index should terminate at the equatorial plane as well. The other ends of this family locate on the inner side of the thalweg (below the equatorial plane). This family therefore has another branch on the inner side of the potential. As this family approaches the center of the spiral structure, its other end approaches the origin.

Fig. 7
figure 7

Upper: The maximum (left: \(\lambda _{\textrm{max}}\)) and minimum (right: \(\lambda _{\textrm{min}}\)) angles to the equatorial plane for \(H<1/32\). Lower: the sum of the maximum and the minimum of \(\lambda \)

Full size image

Besides the main spiral structure, there are many other smaller spiral structures. The upper left panel of Fig. 6 shows some stable asymmetric periodic orbits around a spiral structure below the main spiral structure and closer to the equatorial plane (see Fig. 4). Compared with the principle asymmetric class discussed above, it crosses the equatorial plane twice in half a period. Otherwise, their behaviors are very similar to those around the main spiral structure. There therefore should be two families of asymmetric orbits around each spiral structure, and one of them crosses \(f_0\) approaching the origin and the other terminates at the equatorial plane. Since both of their open ends are on one side of the equatorial plane, as the family with an even upper index crosses \(f_0\), its two ends merge into one and another static point appears near one quarter of its period (i.e., half a period of the corresponding symmetric orbit). In contrast to the principle asymmetric class that bifurcates from \(f_0\) with the same period, the period doubles as this class bifurcates from \(f_0\).

4.3 Other regions of stable orbits

We also notice that there are other tiny islands of stable orbits. In the upper right panel of Fig. 6, the upper orbit is similar to the families in the upper left panel but belongs to the main spiral structure since it locates above a family in \(s_3\) that surrounds the main spiral structure. The lower orbit belongs to \(s_3\). It likely belongs to a branch bifurcating from the family of the principle asymmetric class that approaches the origin. The period also doubles at the point of bifurcation.

The lower left panel of Fig. 6 shows two stable orbits that belong to families in \(s_2\) that approach the origin. The lower one actually belongs to a family that is associated with \(f_2\). The top one in the lower right panel of Fig. 6 shows a stable asymmetric open orbit that appears to be bifurcating from the family in the principle asymmetric class that terminates at the equatorial plane. The top, middle, and bottom panels are associated with \(f_0\), \(f_2\), and \(f_4\), respectively, confirming self-similarity among different families in \(s_1\).

4.4 Asymmetry of open orbits

Calculation of the mLCE is time-consuming. There is a more efficient way to obtain region of stable asymmetric orbits. Given the asymmetry of the corresponding periodic orbits, one may just obtain the maximum and minimum values of z/r for a given long enough computation time. Figure 7 shows the maximum and minimum of \(\lambda = \arcsin {z/r}\) and the sum of these two for a dimensionless computation time of \(10^4\). The most prominent isolated region of stable asymmetric orbits near \([\rho =1.62,\ z=0.205]\) can be readily seen. Moreover, we find some asymmetric stable orbits near the boundary of the region for the bulk of low energy stable orbits centered at the thalweg. It is interesting to note that the center of the spiral structure below the main spiral and closer to the equator is also highly asymmetric.

5 Conclusions and discussions

In this paper, we systematically study orbits of charged particles in a dipole magnetic field with an azimuthal initial velocity that correspond to the open-path type orbits in the meridian plane starting with a zero velocity. To understand the distribution of stable orbits, we identified three classes of open periodic orbits symmetric with respect to the equatorial plane. We found that orbits in the class \(s_1\) that cross the equatorial plane once in half a period locate on several separated families and there are compelling evidence of similarities among these families. We therefore focus on exploring stable orbits associated with the highest energy one \(f_0\).

We found that families in \(s_2\) and \(s_3\) either approach the origin with the other end terminating at the equatorial plane or have a loop shape. The former appears in pairs and one branch of the inner pairs cross \(f_0\) at a stable orbit. These families separate the phase space into several regions with spiral structures in the escape and crossing times. The loops trace these spiral structures. Families in \(s_2\) and \(s_3\) do not seem to cross each other.

Although we did not identify classes of asymmetric open periodic orbits explicitly, by plotting orbits in isolated regions of stable orbits, we found that except for the highest energy one and those associated with stable orbits in the equatorial plane, the most prominent isolated regions of stable orbits are associated with a class of asymmetric open periodic orbits, that is the principle asymmetric family identified by Markellos and Klimopoulos (1977). In the plane with zero meridian velocities, this class can be divided into two families: one crossing \(f_0\) and approaching the origin with the two open ends of each orbit locating on one side of V and the other family terminating at the equatorial plane with the other open end of each orbit in a branch on the other side of the bottom of the potential V. The other ends of these two families trace the main spiral structure and approach to its center. The number of oscillations around the bottom of the potential V increases as one moves along these families toward the center of the main spiral structure with the other open end of the corresponding orbit approaching the origin.

Loops in the symmetric open periodic orbit classes also approach the center of the main spiral structure with the increase in number of crossing of the equatorial plane in half a period. The center of the main spiral structure appears to be singular in the sense that both symmetric periodic orbits oscillating around the equatorial plane and asymmetric periodic orbits oscillating around the bottom of the potential with their two open ends locating in regions near the origin and this central point, respectively, approach to this same point with the increase in the number of oscillations. The nature of the orbit at the center of spiral structure is ambiguous. One can imagine that similar families of asymmetric orbits exist around each spiral structures in the escape and crossing times. There are therefore many singular centers. These results imply that there are discrete energies associated with the centers of spiral structures whose orbits may go through the origin of the dipole, challenging the uniqueness of the Stormer orbit (Braun 1971). Further explorations are warranted.

We found evidence that, besides the principle asymmetric family bifurcating from \(f_0\) (Markellos et al. 1978b) and tracing the main spiral structure, there are many other asymmetric families bifurcating from \(f_0\) and tracing other spiral structures. There are also branches bifurcating from the asymmetric families tracing these spiral structures. Such kinds of bifurcation exist in all families in \(s_1\).

The results can be readily generalized to take into the effects of gravity and rotation of the magnetic dipole (Howard et al. 1999; Howard 1999; Epp and Masterova 2013, 2014). They can then be used to understand properties of charged particles and dust grains trapped in magnetic planets of the solar system (Kolesnikov 2017; Leghmouche and Mebarki 2021). They may also be used to explore the origin of extreme particle acceleration efficiency in pulsars (Epp and Pervukhina 2018; Mohandas et al. 2019; Lhaaso et al. 2021).