ELECTRON-BEAM-INDUCED RADIO EMISSION FROM ULTRACOOL DWARFS

We assume that electron beams are generated by some intense events on UCDs, e.g., magnetic reconnection or jet events. When the electron beams move to the magnetosphere of the UCDs, they interact with the magnetic field and the surrounding plasma. In the present study, we neglect the influence of heavy ions. Here, we investigate the energy transfer, including the induced EM field energy, the drift kinetic energy, and thermal kinetic energy of electrons, plus the growth rate and polarization of the EM fields and the SED.

We start the simulations from the fundamental physical laws. The EM fields and the interaction between them and electrons can be described by Maxwell's equations, i.e., Amp$\grave{\rm e}$re's law, Faraday's law of induction, Gaussian's law for magnetism, Gaussian's law, and the definition of current:

$$\begin{equation} \nabla \times \textbf {B}=\mu _{0}\textbf {J}+\frac{1}{c^{2}}\frac{\partial \textbf {E}}{\partial t}, \end{equation} \tag{ 1 }$$

$$\begin{equation} \nabla \times \textbf {E}=-\frac{\partial \textbf {B}}{\partial t}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \nabla \cdot \textbf {B}=0, \end{equation} \tag{ 3 }$$

$$\begin{equation} \nabla \cdot \textbf {E}=\frac{\rho }{\epsilon _{0}}, \end{equation} \tag{ 4 }$$

$$\begin{equation} \nabla \cdot \textbf {J}=-\frac{\partial \rho }{\partial t}, \end{equation} \tag{ 5 }$$

where $\textbf {B}$ and $\textbf {E}$ are the magnetic field and electric field, respectively, $\textbf {J}$ is the current, t is the time, c is the speed of light, ρ is the charge density, ₀ is the permittivity, and μ₀ is the permeability.

The motion of electrons is governed by the Lorentz force, which can be written as

$$\begin{equation} q(\textbf {E}+\textbf {v}\times \textbf {B})=\frac{\hbox{d}m\textbf {v}}{\hbox{d}t}, \end{equation} \tag{ 6 }$$

where $\textbf {v}$ is the velocity of one individual particle, q is the charge of a single particle (here it is for an electron), and m is the electron mass. In the non-relativistic case, we have m = m_e, where m_e is the rest mass of the electron. In the relativistic case, we have m = γm_e, where $\gamma ={1}/({\sqrt{1-(\textbf {v}/c)^{2}}})$ is the Lorentz factor. The relativistic case is also a general case for the motion equation of particles.

These equations were solved self-consistently as a pure initial value problem using a particle-in-cell method in a two-dimensional space (x- and y-directions) and three velocity and field dimensions (x-, y-, z-directions). Some of the numerical methods used here are from Omura & Matsumoto (1993) and Omura (2005). The Buneman–Boris method was used to solve the equation of motion (Hockney & Eastwood 1981; Birdsall & Langdon 1985). The equation of continuity of charge was solved by a charge conservation method (Villasenor & Buneman 1992). The spacing grid for the EM field in the present simulations is 64 × 64. The time step in each simulation is Δt = 0.001ν⁻¹_pe, where ν_pe is the electron plasma frequency, while the space step is Δx = Δy = 0.125λ_D, where λ_D = v_th/(2πν_pe) is the Debye length, with v_th being the thermal velocity of the electrons. All the velocities in the simulations are normalized by the speed of light c. The charge-mass ratio of electrons is assumed to be −1. We use an open boundary in the simulated system. This simulation configuration and the intrinsic properties of the electrons do not vary in any of the simulations.

In each of the simulations, we assume that a constant external magnetic field $\textbf {B}_{0}$ exists in the spatial x–y-plane, and the angle between $\textbf {B}_{0}$ and x-direction is defined as θ with 0° ⩽ θ ⩽ 90°. The charged particles are initially distributed in the x–y-plane randomly. We assume that background thermal electrons may exist in the radio emission region, i.e., the magnetosphere of UCDs, with number density n_th and thermal velocity v_th. The injected electrons have a number density n_d, thermal velocity v'_th, and drift velocity $\textbf {v}_{\rm d}$ along the x-direction. Figure 1 shows the spatial simulation box schematically. In the simulations, we determine the strength of the external magnetic field via its close relation with the cyclotron frequency ν_ce ≈ 2.8B₀ MHz. The relation between plasma frequency and the number of electrons is ν_pe ≈ 8.98 × 10⁻³(n_e cm⁻³)^1/2 MHz.

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Since we do not know the electron density in the radio emission region on UCDs, a range of values and related plasma parameters are listed in Table 1. From the values in the table, we see that the present simulations are in a relatively microregion and short timescale (0.1 ns to 10 μs).

In order to see the influence of the above parameters on the released EM waves, we set a group of standard values for them (see Table 2). In this model, we assume the direction of the external magnetic field parallel to the y-direction. The cyclotron frequency is set to be 10 times the plasma frequency. We take the thermal velocity of the background electrons and the drift electrons as 0.01c. This means that the temperature of the electrons is about 3 × 10⁵ K, determined by T = (1/2)m_ev²_th/k in the non-relativistic case, where k is the Boltzmann constant. We take v_d = 0.05c. We vary the value for one of the parameters, while the other parameters remain as the standard values. These parameters and their values are summarized in Table 2. We will interpret these parameters in Section 3.2. The standard values of these parameters are derived from estimations of solar bursts (typically 0.1c to 0.5c for the drift velocity and 0.002c to 0.05c for the thermal velocity; Dulk 1985) and the studies on auroral kilometric radiation on the Earth, Jovian millisecond bursts, and Saturnian kilometric radiation (∼1–10 keV for the energetic electrons and ∼100 eV for the thermal electrons; Zarka 1998; Hess et al. 2007a, 2007b; Zarka 2007; Lamy et al. 2010). The values of the electron velocities in the relativistic case (see Section 5) refer to the work of Louarn et al. (1986), and references therein. We choose the optional values over a wide range so that the approximate functions between energies and the parameters can be obtained.

Table 2. Various Parameters in the Simulations (Section 3) and Their Values

Parameters	Symbols	Standard	Optional
		Values	Values
Plasma frequency	νpe	1	...
Cyclotron frequency	νce	10	0, 5
Angle: $\textbf {B}_{0}$ and $\textbf {v}_{\rm d}$	θ	90°	0, 45°
Thermal velocity
of background e	vth	0.01c	0.005c, 0.05c
Thermal velocity
of drift e	v'th	0.01c	0.005c, 0.05c
Drift velocity	$\textbf {v}_{\rm d}$	0.05c	0.0c, 0.1c
Number of superparticles
of background e	nth	16384	0, 32768
Number of superparticles
of drift e	nd	16384	...

Note. Note that part of the values may vary in Sections 4 and 5.

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In this paper, we distinguish the irregular thermal motion of the electrons and their uniform motion. The thermal energy of the electrons in the non-relativistic case is defined and calculated from the thermal motion of the electrons by

$$\begin{eqnarray} E_{\rm th}&=& E_{\rm tk}-E_{\rm d}=\sum _{i=1}^{n}\left(\frac{1}{2}m_{\rm e}\textbf {v}_{i}^{2}\right)-\frac{1}{2}m_{\rm e}\left(\sum _{i=1}^{n}\textbf {v}_{i}\right)^{2}, \end{eqnarray} \tag{ 7 }$$

where n is n_th for the background electrons and n_d for the drift electrons, m_e is the electron mass, and $\textbf {v}_{i}$ is the velocity of the ith particle. On the right-hand side of this equation, the first term represents the total kinetic energy of the system E_tk, and the second term describes the drift energy of the electrons E_d.

In the relativistic case, the energy of one electron is defined by the energy–momentum relation,

$$\begin{eqnarray} E_{i \rm e}^{2}&=& \textbf {p}_{i\rm e}^{2}c^{2}+m_{\rm e}^{2}c^{4}= (\gamma m_{\rm e}c^{2})^{2}, \end{eqnarray} \tag{ 8 }$$

where $\textbf {p}_{i\rm e}=\gamma m_{\rm e}\textbf {v}_{i}$ is the momentum of the ith electron and $\gamma ={1}/({\sqrt{1-(\textbf {v}_{i}/c)^{2}}})$ is the Lorentz factor. Therefore, the kinetic energy of one electron can be expressed as

$$\begin{eqnarray} E_{i \rm ek}&=& E_{i \rm e}-m_{\rm e}c^{2}= (\gamma -1) m_{\rm e}c^{2}. \end{eqnarray} \tag{ 9 }$$

Then the total kinetic energy of the system E_tk is

$$\begin{equation} E_{\rm tk}=\sum _{i=1}^{n} E_{i \rm ek}. \end{equation} \tag{ 10 }$$

For the drift energy of the system E_d, we first define $\gamma _{\rm d}={1}/({\sqrt{1-(\sum _{i=1}^{n}\textbf {v}_{i}/nc)^{2}}})$. Then we have

$$\begin{equation} E_{\rm d}=(\gamma _{\rm d}-1)\cdot n\cdot m_{\rm e}c^{2}. \end{equation} \tag{ 11 }$$

So the thermal energy is E_th = E_tk − E_d. The definition of the drift energy of the system in both the non-relativistic and relativistic cases realizes the uniform motion of the system along one direction via eliminating the irregular random motion of the electrons. Under these definitions, the non-relativistic case (i.e., Equation (7)) is a good approximation of the relativistic case (i.e., Equations (8)–(11)) when particles move with low velocities compared to the speed of light. Our calculations indicate that the equations for the relativistic case are valid for the velocity range v_i ∈ [0, c), while the non-relativistic case is only valid for low-velocity particles. A general definition of temperature then becomes kT = (γ − 1)m_ec².

The energy of the EM field w can be described by the Poynting theorem, so that we have $w=({1}/{2})\epsilon _{0}\textbf {E}^{2}+({\textbf {B}^{2}}/{2\mu _{0}})$, where we take ₀ = 1 and μ₀ = (1/c²₀). The first term on the right-hand side of this equation gives the electric field energy, while the second term represents the magnetic field energy. All kinds of energies are normalized by the initial total energy of the system to satisfy energy conservation. We exclude the energy of the external magnetic field $\textbf {B}_{0}$.

The growth rate is defined as

$$\begin{equation} \Gamma =\frac{\ln \big(E_{(t+\Delta t)}^{2}\big)-\ln \big(E_{(t)}^{2}\big)}{2\Delta t}. \end{equation} \tag{ 12 }$$

The degree of linear polarization in the EM fields Π is

$$\begin{equation} \Pi =\frac{w_{\perp }-w_{\parallel }}{w_{\perp }+w_{\parallel }}, \end{equation} \tag{ 13 }$$

where w_⊥ is the EM field energy in the direction perpendicular to the external magnetic field and w_∥ is the EM field energy in the direction parallel to the external magnetic field.

The background electrons are assumed to be in thermal equilibrium, so their velocity distribution obeys a Gaussian distribution. In this paper it is always taken as the following expression:

$$\begin{equation} f_{0\rm b} = n_{\rm th}{\rm exp}\left(-\frac{v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{v_{\rm th}^{2}}\right), \end{equation} \tag{ 14 }$$

where v_x, v_y, v_z are the components of the velocity $\textbf {v}_{\rm b}$ of the background electrons along the x-, y-, z- directions, respectively.

The injected electrons have an intrinsic thermal velocity distribution, but due to the acceleration, they will obtain a drift velocity along some direction. For the purpose of simplicity, we assume that all of the injected electrons are accelerated along one direction. We take the direction as the x-direction in our simulations, so that the initial velocity distribution for the injected electrons can be expressed as

$$\begin{equation} f_{0\rm d} = n_{\rm d}\left(\frac{v^{\prime }_{x}-v_{\rm d}}{v^{\prime }_{\rm th}}\right)^{2l}{\rm exp}\left(-\frac{(v^{\prime }_{x}-v_{\rm d})^{2}+v_{y}^{{\prime }2}+v_{z}^{{\prime }2}}{v_{\rm th}^{{\prime }2}}\right), \end{equation} \tag{ 15 }$$

where l = 0, 1, 2, 3, ⋅⋅⋅, and v'_x, v_y', v'_z are the components of the velocity $\textbf {v}_{\rm in}$ of the injected electrons along the x-, y-, z-directions, respectively. l is the parameter that describes the size of the loss cone in velocity space. In the present simulations, we take l = 0 and 3. All the particles are assumed to be randomly distributed in space.

We first present the simulation with standard parameters in detail; this we call the standard model. This standard configuration means that the background electrons and the injected electrons have the same temperature (10⁵ K) but the density of the latter is higher. (Note that we use the definition of the temperature in Section 2.2, which reflects the measure of the velocity dispersion). This is for the purpose of modeling a dense electron beam injected into the magnetosphere of a UCD.

Figure 2 shows the spatial evolution of the electrons (a movie is available for the spatial evolution of the system at http://www.arm.ac.uk/highlights/2012/600/beam/beam_space/). The time in each snapshot is t = 0, t = 0.3, t = 1.04, t = 2.86, t = 4.9, t = 6.52ν⁻¹_pe from the top left to bottom right. Due to the existence of the external magnetic field, the motion of the electrons along the x-direction is confined and they can only freely move along the y-direction, which is parallel to the magnetic field. The motion of the injected electrons as a whole should be in a helical orbit with the radius determined by r_d = (m_ev_d)/qB since the induced magnetic field is very small. In this simulation, r_d is about 0.715. Since we only perform a two-dimensional simulation, we see the oscillation of the injected electrons in the x–y-plane instead of a helical motion.

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The mix of background electrons and injected electrons in the velocity space triggered by the EM field is rather interesting. Figure 3 shows snapshots of the velocity distribution of the electrons at the same time as in Figure 2 (a movie is available for the velocity evolution at http://www.arm.ac.uk/highlights/2012/600/beam/beam_velocity/). As seen in the top left panel in the figure, the velocities of both background electrons and injected electrons are initially a Gaussian distribution, with the injected electrons having a drift velocity of 0.05c. When time evolves, the current generated by the injected electrons induces a strong electric field along the direction perpendicular to the external $\textbf {B}_{0}$, which only alters the direction of the flow. This electric field accelerates a fraction of the background electrons so that a tail at high velocity is developed in v_x and v_z (see the velocity distribution in each time snapshot), while the injected electrons are decelerated by the electric field, losing their drift velocity along the x-direction gradually and wavily. Also because of the perpendicular electric field, the injected electrons gain a drift velocity in v_z that oscillates between −0.1c and 0.1c. The appearance of a double peak in the distribution of v_y is due to the magnetic constraint and the acceleration of the induced electric field on the perpendicular velocity v_x − v_y. One can imagine that if the evolutionary time is sufficiently long, the electrons may separate into two groups. One group has a velocity along the direction of the external $\textbf {B}_{0}$, while the other group has the opposite velocity direction.

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The consequence of this process is that the velocities of the electrons evolve from a concentrated Gaussian distribution to an expanded quasi-Gaussian distribution with a high-velocity tail. During the diffusion process of the particles in velocity space, an energy transfer occurs between the kinetic energy of the electrons and the induced EM field energy. As the injected electrons move in the external $\textbf {B}_{0}$ at time t = 0, they start releasing their kinetic energy to the EM wave energy in the manner of an increase in the induced EM field strength. After some time, the EM field energy will reach its maximum while E_tk approaches a minimum. Then the EM field energy may be absorbed by the electrons to compensate for lost kinetic energy. The E_tk will increase after a short time as the field energy decreases. Oscillations in E_tk and the field energy will last for some time until the system is balanced and there is an anti-phase relation between the two kinds of energies.

Figure 4 illustrates the evolution of the total kinetic energy E_tk, drift energy E_d, and thermal energy E_th of the electrons and the field energy w. We plot the same time points for the space and velocity distribution of the electrons in this figure as solid black circles.

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As we can see from this figure, E_tk and w are exactly anti-phase as expected. The multi-peaks of w should be associated with the different wave mode in frequency space, which will be shown later. The relation of the fine structure of E_d and E_th is not as obvious as the relation between E_tk and w since the interaction between E_d and E_th is via the EM field as a time and space delay can affect the phase relation. However, we see that E_d decreases dramatically at the starting time when E_th increases rapidly. After about 1.5ν⁻¹_pe, when E_d and E_th have approximately equal values, the decrease of E_d slows, and so does the increase in E_th. At time 7.1ν⁻¹_pe, at least 70% of E_d is converted to E_th. This is interesting because it means that the transverse motion of electrons in an external magnetic field may be an efficient way to heat the ambient electrons.

The maximum growth rate of the EM wave in this simulation is about 9.68 × 10²ν_pe. The polarization of the released EM waves is highly linear or circular, depending on the initial configuration. More details on the growth rate and polarization in the standard model will be shown in the next section. Figure 5 shows the evolution history of the EM wave energy perpendicular and parallel to the external magnetic field. Again, we use the solid black circles to denote the time points for the space and velocity distribution of the electrons shown in Figures 2 and 3. From this figure, we clearly see that most of the EM waves are polarized in the direction perpendicular to $\textbf {B}_{0}$, while only a very small fraction of the EM waves are released parallel to $\textbf {B}_{0}$ (∼10⁻³ of the perpendicular energy).

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In this section, we investigate the influence of the following parameters on the energy history, growth rate, and polarization of the EM waves:

• 1.  
  The strength of the external magnetic field B₀. B₀ can be determined by the cyclotron frequency,

  $$\begin{equation} B_{0}\approx 0.357\times 10^{3}\frac{\nu _{\rm ce}}{{\rm GHz}}\: {\rm G}. \end{equation} \tag{ 16 }$$

  To date, all detected radio emission of UCDs is in the GHz band, while observations performed with the NRAO very large array in 2007 show no trace of radio emission from two UCDs at 325 MHz, placing an upper flux limit of ∼900 μJy at the 2.5σ level (Jaeger et al. 2011). From this, we infer that the magnitude of B₀ is a few kilo gauss (if the radio emission is truly at the cyclotron frequency). Here, we take ν_ce/ν_pe = 0, 5, and 10 to see how the magnetic field can affect the simulation results. When ν_ce/ν_pe = 0, there is no external magnetic field.
• 2.  
  The angle θ between the magnetic field and the drift velocity. θ is one of the crucial parameters to influence the direction of the radiated EM waves, i.e., the polarization. When θ = 0°, the injected electrons move uniformly parallel to the external $\textbf {B}_{0}$ in addition to the irregular thermal motion. When θ = 90°, the motion of the injected electrons is perpendicular to $\textbf {B}_{0}$.
• 3.  
  The drift velocity v_d. We expect that the radio emission may be enhanced by increasing the value of v_d. In this section, we take v_d = 0c, 0.005c, and 0.01c to avoid the relativistic effect where gyrosychrotron emission plays an important role. When v_d = 0c, we see the effect of pure thermal electrons moving in the external $\textbf {B}_{0}$ on the induced EM waves.
• 4.  
  The temperature T. We investigate the response of the radiated EM waves and the transfer between the energies by varying v_th. We take v_th = 0.005c, 0.01c, and 0.05c, for which the corresponding T is 7.5 × 10⁴, 3 × 10⁵, and 7.5 × 10⁶ K.
• 5.  
  The background electrons by changing n_th/n_d. Since our computation capability is limited, we only investigate the effect of the existence of the background electrons on the EM waves and the energy transport. We take n_th/n_d = 0, 1, 2. When n_th/n_d = 0, there are no background electrons.

Figure 6 shows the energy evolution in each simulation. We see that the EM field energy (left panels) is in a very low level in all three specific cases (see the colored lines). In the first case, there is no magnetic field (which can be understood as the injected electrons just passing by very quickly without sufficient energy transfer via the EM field). In the second case, the motion of the injected electrons is parallel to the external magnetic field. In this case, the induced electrons cannot remain within the simulation box since the magnetic field cannot constrain them. In the third case, the drift velocity of the injected electrons is 0c; although the injected electrons can remain and interact with the background, there is no coherent current, i.e., the field energy is still small although higher than in the other two cases.

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A common expression of these cases is $\textbf {v}\times \textbf {B}_{0}=0$, which is the Lorentz force induced by the external magnetic field. This force makes the initially coherent current bend (i.e., the electrons with drift velocity), leading to spatial curled EM fields that are the medium to accomplish energy transport among different kinds of energies.

In other cases except the above three cases, the field energy can maintain a much higher level after they reach the maximum, typically orders of 2–3 that of the above cases. As shown in Figure 6, the field energies in all cases oscillate with large amplitude caused by the transfer between the kinetic energy and field energy. In other words, these oscillations reflect the emission and absorption of electrons to the EM waves. With the achievement of the diffusion process of the electrons in the velocity space, a dynamic balance is approached. This results in a gradual decrease in the amplitudes of the oscillations with time. The time for relaxation of the field energy is >10ν⁻¹_pe which is much longer than the time taken for the field energy to reach maximum, ∼0.14ν⁻¹_pe.

As expected, increasing B₀ by a factor of two, i.e., ν_ce/ν_pe from 5 to 10, leads to a rise in the field energy. It seems that the mean value of the field energy is not sensitive to θ when θ = 45° and 90°. However, the modes (or direction) of the EM waves are affected. In the case of θ = 45°, the EM waves have a similar energy level in the direction perpendicular and parallel to $\textbf {B}_{0}$, which is shown in Figure 7 where we discuss polarization. The increase of v_d from 0.05c to 0.1c only raises the energy level slightly in the present simulations. In fact, when v_d is sufficiently high, we have to consider the relativistic effect and hence gyrosynchrotron radiation, which will be addressed in Section 5. The mean energy level of the EM waves is not sensitive to the thermal velocity and background electron number density. Note that in the standard model, the initial density of the background electrons is ∼100 times less than that of the injected electrons.

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The polarization of the EM waves, or their energy distribution with respect to the magnetic field direction, is shown in Figure 7 quantitatively. The EM waves in the standard model are 100% linearly polarized. When $\textbf {v}\times \textbf {B}_{0}=0$, the EM waves frequently switch their direction from parallel-dominant to perpendicular-dominant, and rarely do they have up to 50% linear polarization. This behavior indicates that the waves have a significant linearly polarized component. In addition to θ, thermal motion (i.e., temperature) of the electrons can influence the level of polarization. As we see from the bottom second panel in Figure 7, the electrons with temperature 7.5 × 10⁶ K can generate the EM waves with ∼50%–90% linear polarization, while 100% linearly polarized waves are generated by the electrons with temperature 7.5 × 10⁴ and 3 × 10⁵ K. A low density of background electrons does not alter the highly linear polarization in the present simulations.

The dissipation of the drift energy in the injected electrons (when they move in the external magnetic field) is important because this may be sufficient to increase the thermal energy of the system. We show the history of the drift energy and thermal energy in Figure 6. Comparing the right panels in Figure 6, we find that (1) when $\textbf {v}\times \textbf {B}_{0}=0$, the energy transfer is the least efficient. In this case, there is almost no energy exchange; (2) when initially $\textbf {v}\times \textbf {B}_{0}\ne 0$ and E_d > E_th, E_d can be transported to E_th rapidly; the timescale in which E_d and E_th reach the same value is about ∼1–5ν⁻¹_pe.

Figure 8 illustrates the growth rate of the EM waves in the simulations. The increase of the system temperature (3 × 10⁵–7.5 × 10⁶ K) will suppress the growth rate significantly. The number of the electrons also affects the growth rate, but their relation is not so obvious.

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The EM field (wave) energy is the integral (or sum if the signal is discrete) of the contribution from different frequencies. In order to obtain the energy distribution of the EM field in frequency space (i.e., the SED), we perform a fast Fourier transform to the EM field energy history. Figure 9 illustrates the SED.

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We clearly see many emission and absorption lines in Figure 9 that represent the EM field energy distribution at different frequencies. We find that the majority of the field energy is from frequencies <80ν_pe with bandwidth at half-maxima ∼6ν_pe.

Figure 10 illustrates the EM field in the left panels, and in the right panel we have the thermal and drift energy of the electrons from the loss-cone-driven ECM. From this figure, we see that only a very small fraction of kinetic energy is converted to EM field energy if there is no external magnetic field or the loss-cone velocity distribution is along the direction parallel to the external magnetic field.

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In other cases, an increase in any one of the parameters B₀, θ, and v'_th leads to an increase in the induced EM field energy. Raising the temperature of the background electrons (i.e., the thermal level) can suppress the growth of the field energy. Since we only vary the number of background electrons in a very small range, it does not affect the field energy significantly. As seen from the left bottom panel in Figure 10, increasing the number of background electrons decreases the induced EM field energy. These results are consistent with the growth rate of the EM wave illustrated in Figure 11. From the right panels in Figure 10, we see that the drift energies are rapidly dissipated, eventually leading to irregular motion of the electrons in the external magnetic field.

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The history of the degree of the polarization of the EM waves in each simulation is shown in Figure 12. The conditions for the circularly polarized EM waves are notable, ranging from (1) decreasing the magnetic field, (2) varying the angle θ from perpendicular to non-perpendicular, to (3) increasing the thermal level of the background electrons. We suggest that these conditions may be associated with the circularly polarized components of the radio emission from UCDs.

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The influence of the parameters on the spectrum is shown in Figure 13, in which we find that the magnetic field and the angel θ can affect the SED significantly, while the thermal effect and the number of background electrons only play a minor role. Some frequency bands are notable, e.g., from 10 to 70ν_pe.

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Comparison between the different initial velocity distributions can help us understand the roles of the initial parameters in the process of releasing EM wave energy and the transfer between drift energy and thermal energy. Combining Figures 6 and 7 (beam-driven) and Figures 10 and 12 (loss-cone-driven), we see that

• 1.  
  the existence of the drift kinetic energy of the electrons can be considered as a coherent current (which is necessary to generate the intense EM field energy), while the form of the initial velocity distribution is not important;
• 2.  
  in order to efficiently obtain the intense EM field energy, the external magnetic field plays a crucial role. The angle between the magnetic field and the coherent current significantly affects the strength of the released EM field energy in a nonlinear relation (see Figures 8 and 11) and also the propagation direction of the EM waves;
• 3.  
  pure thermal motion of the injected electrons in the external magnetic field may play a role in generating the EM waves (e.g., when v_d = 0 in the beam-driven ECM, the middle panel in Figure 6), while the thermal level of the background electrons mainly suppresses the generation of the EM waves.

The SED of the beam-driven and loss-cone-driven ECM indicates that all the parameters (except the number of background electrons in the present simulations) can affect the SED; however, certain harmonic frequency bands will appear if the coherent current is sufficiently strong, for example, see the region from 10 to 70ν_pe. There is a negligible signal in the very high frequency band >100ν_pe. Also, it seems that the SED weakly depends on the number of background electrons, but this needs further investigation since we do not vary the number of particles over a sufficiently wide range in the simulations.

The history of the three kinds of energies, the polarization of the induced EM field energy, and their growth rates are illustrated in Figures 14–16. It is seen that the background electrons in the simulations have a negligible influence on the induced EM field energy and the linear polarization, while the thermal energy level of the background electrons can suppress the growth of the EM waves.

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From the left panels in Figure 14, we find that the direction and values of the magnetic field and the drift velocity can significantly affect the induced EM field. The existence of the magnetic field is necessary in order to obtain the fast growth of the EM field energy, and the energy conversion efficiency seems to have a nonlinear relation with the magnetic field strength.

The right panels in Figure 14 illustrate the drift energy and the thermal energy of the electrons. If the magnetic field strength is sufficiently high to constrain the motion (or spatial position) of the injected electrons, the drift energy can be rapidly converted to the thermal energy in a timescale that is similar to the non-relativistic case. If the magnetic field is not very strong, we see that it is possible for the system to still retain some residual drift energy when the electrons escape from the local region.

The influence of the parameters on the polarization in the relativistic case (Figure 15) is different from the non-relativistic case. The strength of the magnetic field in the present simulations affects the degree of the linear polarization. The existence of a strong magnetic field is essential to generate the linear polarization. The non-perpendicular angle between the magnetic field and the drift velocity plays an important role in the generation of the linear polarization. In fact, at high frequency, linear polarization becomes dominant in the relativistic case. The drift velocity, the thermal level of the background electrons, and the relative number of drift electrons and background electrons do not affect the polarization significantly in the present simulations. As shown in Figure 16, the growth rate of the EM field rapidly increases with the magnetic field strength and injection angle. However, it seems that in the relativistic case the increase of the drift velocity slightly decreases the growth of the EM field, perhaps because of a strong interaction between the induced EM field and the high-energy electron beam. The thermal level of the background electrons can suppress the growth of the EM field, and there is an ambiguous relation between the number of background electrons and the growth rate of the EM field in the present simulations.

The influence of the direction between the magnetic field and the drift velocity is very interesting since the maximum energy conversion efficiency occurs in the non-perpendicular injection, which differs from the non-relativistic case. In order to determine the direction where we can obtain the maximum energy conversion efficiency, we have done some extra computations by varying the direction of the magnetic field. Figure 17 illustrates the results in which we see that the maximum energy conversion efficiency occurs at ∼75°.

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An interesting phenomenon in our simulations is that the velocity distribution of the beam electrons may evolve from an initially drifted Gaussian (or Gaussian-like) distribution to ring distribution (or incomplete-ring or spiral-ring or horseshoe, e.g., see the standard models in both the relativistic case and non-relativistic case; a movie is available for the velocity evolution of the electron beam in the relativistic case at http://www.arm.ac.uk/highlights/2012/600/relativistic_beam/relativistic_beam_velocity/) or spherical-shell distribution (or incomplete shell, e.g., if we change the angle between the magnetic field and the beam electron from perpendicular, e.g., 90°, to non-perpendicular, e.g., 45° or 75°). This may imply that the ring and shell (or ring-like and shell-like) velocity distribution would have a common origin–beam distribution. The influence of the angle on the evolution of the velocity of electrons is mainly caused by the external magnetic field and the self-induced EM fields. The spatial current induced by the motion of the charged particles plays an important role in the process, which initially causes the variation of a spatial magnetic field and time-dependent electric field, i.e., Equation (1). This electric field will accelerate or decelerate the electrons to deform the distribution values of the electron velocity.

The SED of the radiation in the relativistic electron beam apparently differs from that in the non-relativistic case. Figure 18 illustrates the SEDs in the relativistic case. In the standard model, we clearly see that the energy may not only distribute in the range of 10–100ν_pe but extend up to a frequency of ∼500ν_pe and some negligible signal at an even higher frequency harmonic.

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For a single relativistic electron, the peak value of its synchrotron radiation may be at ν_peak ≈ γ²ν_cesin θ (Rybicki & Lightman 1979). For a many-electron system, the frequencies where we can obtain emission lines are strongly associated with the distribution of the Lorentz factor γ. Figure 19 illustrates the distribution of γ in different parameters. In the standard model, we find that when the instability reaches saturation, the distribution of γ expands only a little in the high-energy part. However, when varying the angle θ from 90° to 75°, we see a distinct high-energy electron tail around γ = 9.5, which corresponds to the kinetic energy E_iek = (γ − 1)m_ec² ≈ 4.35 MeV. These high-energy electrons may contribute to possible X-ray emission from UCDs via thermal or non-thermal bremsstrahlung or even inverse Compton scattering.

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