ORIGIN OF ELECTRON CYCLOTRON MASER INDUCED RADIO EMISSIONS AT ULTRACOOL DWARFS: MAGNETOSPHERE–IONOSPHERE COUPLING CURRENTS

For the purposes of development of the model, we assume that a Jovian picture of a UCD is appropriate, i.e., that the UCD's magnetic field is generated deep within its interior, such that the external field generated by the body is mainly dipolar, and that a non-conducting atmosphere surrounds the body, the upper portion of which is maintained to some degree conductive by photon and/or corpuscular impact from the outside. The atmospheres of cool UCDs are not expected to be strongly ionized, although it has been suggested that dust clouds may induce some degree of ionization (Helling et al. 2011a, 2011b). At solar system planets a number of sources of conductivity are known besides solar X-ray and EUV radiation, such as galactic cosmic rays and auroral electron precipitation (Rycroft et al. 2008). The latter is particularly important, since it results in a positive feedback that greatly increases the intensity of the auroral currents. At Jupiter, for example, modeling work indicates that the feedback from auroral precipitation increases the conductance from a residual value of ∼0.01 mho (where 1 mho ≡ 1 siemens) generated, e.g., by solar X-ray and EUV irradiation to values of up to 10 mho (Strobel & Atreya 1983; Millward et al. 2002), such that the feedback plays a dominant role in influencing the location and intensity of the M–I coupling currents (Nichols & Cowley 2004). Thus, we suggest that given some small initial conductivity generated by a source mentioned above, the auroral current system would amplify itself via the feedback from the auroral precipitation. The exact details of such feedback requires ionospheric modeling that is beyond the scope of the present paper, but suffice it to say that due to this effect the low temperatures of UCDs do not necessarily prohibit the flow of strong M–I coupling currents. In the absence of such an ionospheric model for UCDs, in the present work we simply take the Pedersen conductance Σ_P to be a constant, discussed further below.

A principal assumption is that the spin and magnetic axes are roughly co-aligned, since this is the configuration that has been most studied for bodies in our solar system, although we note that it is probable that only ∼50% of UCDs exhibit this polarity. With reversed field polarity, the associated M–I coupling current system described below would also be reversed, such that the upward current, which in Figure 1 is strongly peaked at 15°, would instead be widely distributed at high latitudes, and the strong downward currents would be easily carried by upward-going ionospheric electrons, leading to little auroral and radio emissions. We also assume approximate axisymmetry about the magnetic axis, such that the flows in the ionosphere are considered to be wholly azimuthal about that axis and can thus be described in terms of departure from corotation with the neutral atmosphere as a function of colatitude θ_i from the magnetic axis. Such a simplification seems appropriate for representing to lowest order the dynamics of a fast-rotating magnetosphere, although we note that some processes related to an open magnetosphere will consequently be excluded from the model, such as "cusp-like" or "substorm-like" processes that would be ordered in azimuth with respect to the object's velocity through the surrounding medium. We further assume for simplicity that departures of the upper neutral atmosphere from rigid corotation due to ion-neutral drag also take the form of winds that are approximately axisymmetric about the magnetic axis, thus neglecting the effect of the Coriolis force on such motions in cases where the spin and magnetic axes are significantly inclined.

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It has been suggested by Kuznetsov et al. (2012) that the short duration of the spikes in the CMI-induced radio emissions is indicative of an active sector in the magnetosphere of a UCD. We suggest, however, that the observed short duty cycles of ⩽0.15 for UCD radio emissions (Hallinan et al. 2006) are not inconsistent with an essentially axisymmetric field-aligned current distribution. Jupiter's main auroral oval is to first order described by an axisymmetric annulus of isotropic emission ∼1° wide located ∼15° from the magnetic pole, which is tilted by ∼10° from the spin axis (Grodent et al. 2003; Clarke et al. 2004; Nichols et al. 2009). Therefore, as the dipole axis cones around the spin axis, from a viewpoint near the planet's spin equatorial plane over half of the auroral oval is visible over roughly a third of the planet's rotation period (Nichols et al. 2007). The associated radio emission, however, is strongly beamed into ∼1.6 sr in the case of the hectometric (HOM) and decametric (DAM) emissions (Zarka et al. 2004). Hence, the radio emission peaks strongly as the beam rotates into view, and the FWHM of the occurrence probability distribution of the "Source A" DAM emission (i.e., that used to determine Jupiter's System-III rotation period of ∼9 hr 55 minutes) is ∼50°, i.e., an apparent duty cycle of ∼0.14 (Higgins et al. 1996). This is consistent with the duty cycles of ⩽0.15 for UCDs. Having said this, it is important to recognize that axisymmetry is simply a lowest order approximation. Saturn's radio emissions, for example, are observed from all local times but are brightest in the morning sector (Lamy et al. 2009), despite the internal planetary magnetic field being axisymmetric to within detection limits (Burton et al. 2010). Finally, as has also been suggested by Berger et al. (2009), it is probable that as is observed at Jupiter the magnetic poles may not lie on respective antipodes, due to the non-dipolar terms in the internal field, such that the radio emission from the two hemispheres would not necessarily be expected to be observed 180° apart in phase.

We consider three angular velocities in the computation of the M–I coupling currents. The first is the angular velocity of the deep interior of the UCD Ω_UCD, taken initially to be 8.7 × 10⁻⁴  rad s⁻¹ in conformity with the observed periods of ∼2 hr for radio-bright UCDs. The second is the angular velocity of the plasma above the ionosphere ω, taken to be constant on each flux tube, and the third is the angular velocity of the atmospheric neutrals in the Pedersen layer of the ionosphere Ω*_UCD, which is expected to lie somewhere between Ω_UCD and ω due to the frictional drag of ion-neutral collisions (Huang & Hill 1989; Millward et al. 2005). In such a case we have

$$\begin{equation} (\Omega _{{\rm UCD}}-\Omega _{{\rm UCD}}^*)=k(\Omega _{{\rm UCD}}-\omega), \end{equation} \tag{ 1 }$$

where 0 < k < 1. The actual value of k is somewhat uncertain for Jupiter, with modeling work indicating a value of ∼0.5 (Millward et al. 2005), which is employed in the previous Jovian modeling work discussed and also now adopted here, although the results are not expected to be sensitively dependent on reasonable choices. We should note that in the following work we consider only sub-rotation of the plasma with respect to the UCD. For Jupiter, super-rotation of the magnetospheric plasma may occur following strong solar-wind-induced compressions of the magnetosphere, with a corresponding reversal of the current system described below (Hanlon et al. 2004; Cowley et al. 2007; Nichols et al. 2007), but in the steady state the plasma sub-rotates, associated with the transfer of angular momentum from the atmosphere to the external medium.

The equatorward-directed height-integrated ionospheric Pedersen current is then given by

$$\begin{equation} i_P = \Sigma _P E_i=\Sigma _P\rho _i(\Omega _{{\rm UCD}}^* - \omega)B_i, \end{equation} \tag{ 2 }$$

where E_i is the electric field in the rest frame of the neutral atmosphere and ρ_i = R_UCDsin θ_i is the distance from the spin axis (where we assume that initially the UCD has a radius R_UCD equal to Jupiter's equatorial 1 bar level radius R_J = 71, 373 km since all bodies of roughly solar composition from giant planets through the very lowest mass stars have radii roughly similar to Jupiter). Employing Equation (1) in Equation (2) and introducing the "effective" Pedersen conductance Σ*_P, reduced from the true value by a factor of (1 − k) owing to the slippage of the neutral atmosphere from rigid corotation, yields

$$\begin{equation} i_P = \Sigma _P^*\rho _i(\Omega _{{\rm UCD}}-\omega)B_i. \end{equation} \tag{ 3 }$$

As mentioned above, we simply take a constant value for Σ*_P, initially 0.5 mho, which is similar to the values employed in previous Jupiter modeling works and, as shown below, yields radio power values in agreement with observations of UCDs. However, in the results below we also examine the effect of different values of this parameter. The total Pedersen current at a given colatitude is then found by integrating in azimuth, such that

$$\begin{equation} I_P = 2\pi \rho _ii_P = 2\pi \Sigma _P^* \rho _i^2(\Omega _{{\rm UCD}}-\omega)B_i, \end{equation} \tag{ 4 }$$

the divergence of which gives the field-aligned current density just above the ionosphere, i.e., for the northern hemisphere:

$$\begin{equation} j_{\Vert i} = -\frac{1}{2\pi \rho _i^2 \sin \theta _i}\frac{dI_p}{d\theta _i}. \end{equation} \tag{ 5 }$$

The upward-directed field-aligned current computed in the previous section cannot, in general, be carried simply by precipitating magnetospheric electrons alone and must be driven by downward acceleration of those electrons by a field-aligned voltage. The maximum field-aligned current density that can be carried by an unaccelerated isotropic Maxwellian population is

$$\begin{equation} j_{\Vert i0} = en\left(\frac{W_{{\rm th}}}{2\pi m_e}\right)^{1/2}, \end{equation} \tag{ 6 }$$

where e, m_e, n, and W_th are the charge, mass, number density, and thermal energy of the electron source population, respectively, the latter being equal to k_BT, where k_B is Boltzmann's constant and T is the temperature. In common with previous works, we use the values for the high-latitude hot plasma at Jupiter, i.e., n = 0.01 cm⁻³ and W_th = 2.5 keV (Scudder et al. 1981), which, as shown below, yield results consistent with observations, but we also examine the effects of wide ranges of values for these parameters. The corresponding unaccelerated kinetic energy flux is

$$\begin{equation} E_{f0} = 2enW_{{\rm th}}\left(\frac{W_{{\rm th}}}{2\pi m_e}\right)^{1/2}. \end{equation} \tag{ 7 }$$

For field-aligned current densities larger than j_||i0, a field-aligned voltage is required. For Earth, the current–voltage relation derived using the kinetic theory of Knight (1973) is applicable, but for more powerful systems in which the source electron population either is very energetic initially or becomes so following acceleration by voltages comparable to or exceeding the electron rest-mass energy (∼511 keV), Cowley's (2006) relativistic extension to Knight's theory becomes appropriate. In the results shown in this paper it is clear that the relativistic theory is required in the case of M–I coupling current systems at UCDs, such that we employ the relativistic current–voltage relation given by

$$\begin{equation} \left(\frac{j_{\Vert i}}{j_{\Vert i\circ }}\right)= 1 + \left(\frac{e\Phi }{W_{{\rm th}}}\right)+ \frac{\left(\frac{e\Phi }{W_{{\rm th}}}\right)^2}{2\left[\left(\frac{m_ec^2}{W_{{\rm th}}}\right)+1\right]}, \end{equation} \tag{ 8 }$$

where c is the speed of light and Φ is the minimum voltage required to drive the current j_||i at ionospheric altitude, applicable in the limit that the accelerator is located at high altitude along the field line (where B ≪ B_i). The corresponding precipitating electron energy flux is given by

$$\begin{equation} \left(\frac{E_f}{E_{f0}}\right) = 1 + \left(\frac{e\Phi }{W_{{\rm th}}}\right) + \frac{1}{2}\left(\frac{e\Phi }{W_{{\rm th}}}\right)^2 + \frac{\left(\frac{e\Phi }{W_{{\rm th}}}\right)^3}{2\left[2\left(\frac{m_ec^2}{W_{{\rm th}}}\right)+3\right]}. \end{equation} \tag{ 9 }$$

Equations (6)–(9), along with values for the electron source population number density and temperature as discussed further below, are thus used to determine the precipitating energy flux resulting from the current system discussed above. The total precipitating power for each hemisphere P_e is then obtained by integration of E_f over the hemisphere, i.e.,

$$\begin{equation} P_e=\int _0^{90}2\pi R_{{\rm UCD}}^2\sin \theta _i \;E_f \;d\,\theta _i, \end{equation} \tag{ 10 }$$

and we finally obtain the spectral luminosity using

$$\begin{equation} L_r=\frac{P_e}{100 \: \Delta \nu }, \end{equation} \tag{ 11 }$$

where we assume that the beam from only one hemisphere is observable at any one time and we take the electron cyclotron maser instability to have a generation efficiency of ∼1%, a value observed at both Jupiter and Saturn. Specifically, at Saturn this value was directly measured during the traversal of the Cassini spacecraft through the source region of the Saturn Kilometric Radiation (SKR; Lamy et al. 2011), and at Jupiter the radio power output is ∼1% of the total precipitating electron power, as measured from observations of the UV aurora and computed theoretically (Gustin et al. 2004; Clarke et al. 2009; Cowley et al. 2005). The bandwidth Δν is assumed to be equal to the electron cyclotron frequency in the polar ionosphere, i.e.,

$$\begin{equation} \Delta \nu =\frac{eB_i}{2\pi m_e}, \end{equation} \tag{ 12 }$$

an approximation validated by observations of solar system planets. For example, Zarka (1998) argues that the Jovian b-KOM, HOM, and DAM emissions are a single radio component, thus with a frequency range of ∼10 kHz to ∼40 MHz, and at Saturn the SKR extends from a few kHz to ∼1200 kHz. This arises because the CMI occurs at all points below the field-aligned voltage drop, situated at least a few planetary radii up the field line, and the top of the ionosphere at 1 planetary radius. The emission is generated at the local electron cyclotron frequency, such that the bandwidth is then determined by the difference between the field strengths at these two locations. The strength of a dipole field drops off approximately as the cube of the distance along the field line, such that at altitudes of only ∼1 planetary radius up the field line the frequency drops by an order of magnitude. The bandwidth is thus reasonably well represented by the high-frequency cutoff, which is also the frequency used to measure the magnetic field strength in the polar region.

We now discuss the model for the plasma angular velocity used in this paper. Following previous work (Cowley et al. 2005), the gradient in the angular velocity is conveniently represented by a tanh function with respect to colatitude, given by

$$\begin{eqnarray} \left(\frac{\omega }{\Omega _{{\rm UCD}}}\right) &=& \left(\frac{\omega }{\Omega _{{\rm UCD}}}\right)_L + \frac{1}{2}\left(1 - \left(\frac{\omega }{\Omega _{{\rm UCD}}}\right)_L\right) \nonumber\\ &&\times \left(1+\tanh \left(\frac{\theta _i - \theta _{ic}}{\Delta \theta _i}\right)\right), \end{eqnarray} \tag{ 13 }$$

where (ω/Ω_UCD)_L is the low value to which the angular velocity transitions near the magnetic pole from rigid corotation at lower latitudes, and θ_ic and Δθ_i are the latitudinal center and half-width of the transition region, respectively. We initially employ values for these parameters that give a Jupiter-like angular velocity profile, which on that planet generates the field-aligned currents that cause the main auroral oval and the HOM, b-KOM, and non-Io-DAM radio emissions, i.e., (ω/Ω_p)_L = 0.25, θ_ic = 15°, and Δθ_i = 05. The actual form of the angular velocity gradient depends on the physical processes causing the shear, as is examined further in the discussion, and in the results below we also examine the effects on the radio power of taking different values for these parameters.

In this section, we present results of the model using appropriate spot values for the various model parameters as discussed above, before going on in the next section to examine ranges in the parameter space. Figure 1 shows representative results, in which we employ a uniform ionospheric field strength of 0.3 T and rotation period of 2 hr, in conformity with typical values obtained from the bandwidth and modulation of the UCD radio emissions (Hallinan et al. 2008). First, Figure 1(a) shows the angular velocity profile employed given by Equation (13). Moving from large to small colatitudes (or equivalently on field lines that thread the equatorial plane at increasing radial distances), the plasma angular velocity initially near-rigidly corotates and transitions over ∼1° at 15° colatitude to a quarter of rigid corotation at smaller colatitudes. The resulting equatorward azimuth-integrated ionospheric field-perpendicular (Pedersen) current given by Equation (4) is shown in Figure 1(b), increasing from small values near the pole to a value of ∼58 GA, before falling rapidly due to the plasma angular velocity gradient shown in Figure 1(a). The resulting field-aligned current given by Equation (5) is shown in Figure 1(c), which is negative, i.e., downward, near the pole and switches to a significant peak of positive, i.e., upward, field-aligned current values, reaching amplitude ∼0.7 mA m⁻² and centered on 15°. This is the region of auroral field-aligned current that sustains the unstable electron distributions responsible for the CMI and thus the radio emissions. The required field-aligned voltage given by Equation (8) is then shown in Figure 1(d), peaking at ∼11 MV, and the corresponding precipitating electron energy flux given by Equation (9) is shown in Figure 1(e), peaking at ∼9 kW m⁻². Integrating this energy flux and converting to spectral luminosity as discussed above yield a radio power output of ∼1.1 MW Hz⁻¹, i.e., consistent with the typical observed radio luminosities of UCDs. We also note that the bandwidth of the radio emission, equal to the electron cyclotron frequency of ∼8.4 GHz, is perforce similar to that observed, since the ionospheric field used is determined from the measured radio frequencies.

In the above section, we have used reasonable values for a number of system parameters based on observations at Jupiter in order to demonstrate the model and indicate how an angular velocity gradient leads to radio luminosity that, for the case of fast-rotating UCDs, is comparable to that observed. However, the only parameters appropriate to this model that are known from observations, besides the radio power, are the polar magnetic field strength and rotation rate of such objects. The other parameters, such as the form of the angular velocity profile, the density and temperature of the source electron population, and the ionospheric conductance, are not known. Here we therefore examine ranges of these parameters to determine what values are consistent with the observed radio emission. We first examine the effects of the conductance and source electron population parameters, given the angular velocity profile given by Equation (13), and we go on to investigate the effect of changing the form of the angular velocity profile.

In each panel of Figure 2, we show the radio spectral luminosities obtained from the model using different pairs of values for the source electron thermal energy W_th and number density n. The values adopted above obtained from Voyager observations of the plasma outside the current sheet at Jupiter, i.e., 2.5 keV and ∼0.01 cm⁻³, respectively, are shown by the crosses in each panel. The low density arises since the cold plasma is centrifugally confined to the equatorial plane, and only the hot, rarefied plasma population has a centrifugal confinement scale height large enough to populate the entire flux tube (Hill & Michel 1976; Caudal 1986). At high latitudes, the low beta plasma, combined with the field-aligned voltage and loss of particles to the ionosphere, is then favorable for generation of the CMI. We note that previous estimates of the electron number density and temperature on the flux tubes producing the CMI at UCDs have been obtained, i.e., (1.25–5) × 10⁵ cm⁻³ and (1–5) × 10⁷ K (equivalent to ∼0.86 keV) by Yu et al. (2011), but these assume that the CMI is produced by an unaccelerated loss-cone population. We suggest, however, that at fast-rotating UCDs, at which the magnetospheric plasma is likely to be significantly more centrifugally confined than at Jupiter, the high-latitude plasma parameters may possibly be closer to the Jovian case. In light of the uncertainty surrounding the source plasma population, we show the results of the model using the angular velocity profile given by Equation (13) for wide ranges of these parameters, i.e., five orders of magnitude either side of the Jovian values. In panels (a)–(d) we employ Σ*_P = 0.005, 0.05, 0.5, and 5 mho, respectively, reflecting the uncertainty in the Pedersen conductance. The contour labeled "6" indicates where the radio luminosity is equal to 1 MW Hz⁻¹, representative of the observed luminosities.

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It is apparent that over the majority of the parameter space plotted, the luminosity increases with decreasing n and increasing W_th, with a change of slope as the relativistic terms dominate above W_th ≃ 511 keV. Where the slope reverses in the top right corner of each panel, the field-aligned current can be carried by an unaccelerated population. For simplicity, in this region we have assumed that in the auroral region the loss cone is filled by strong pitch angle diffusion and that the required net current is then created by a corresponding upward flux of electrons. We have also taken the same CMI generation efficiency of 1% as for the accelerated population, despite the less favorable loss-cone distribution in this case. Both these assumptions will likely lead to overestimates of the radio flux from this region. We also note that the previous estimates of the source population parameter values discussed above (Yu et al. 2011) are off the top of each panel, thus clearly in the unaccelerated regime. In any case, it is apparent in the accelerated region that a wide range of parameter values are able to produce the required radio luminosity, in general requiring hotter, less dense plasma for lower Pedersen conductance. The plot also confirms the results shown above, i.e., that typically Jovian values for the source plasma population parameters and the Pedersen conductance lead to observed UCD radio luminosities.

We now consider the effect of the form of the angular velocity profile and show in Figure 3 the radio luminosities obtained when the parameters in Equation (13) are varied individually. Specifically, from top to bottom the parameters varied are (a) the low value to which the angular velocity transitions from rigid corotation (ω/Ω_p)_L, (b) the latitudinal half-width of the transition region Δθ_i, and (c) the center of the transition region θ_ic. As a guide, the horizontal dotted lines highlight 1 MW Hz⁻¹, which delimits the parameter ranges reasonably consistent with observations. Thus, an angular velocity profile that transitions to ≲ 70% of rigid corotation across a region of half-width ≲ 25 at a colatitude ≳ 9° will produce radio powers consistent with observations. These are not particularly tight constraints, indicating that the model does not require fine-tuning to produce the required emissions. Future modeling of the radio beams, along the lines of previous work that has concentrated on localized active regions (Kuznetsov et al. 2012), from angular velocity transition regions of different geometries would be useful to constrain the angular velocity profiles further.

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We now compare the model spectral luminosities with the confirmed sources of CMI-induced radio emissions discussed by Hallinan et al. (2008), i.e., TVLM 513-46546, LSR J1835+3259, and 2MASS J00361617+1821104 (hereafter TVLM 513, LSR J1835, and 2M J0036), which have relevant properties listed in Table 1. We estimate the spectral luminosities of the CMI-induced radio emission from the peak amplitudes and widths plotted by Hallinan et al. (2007, 2008). As discussed above, CMI-induced radio emission is strongly beamed, into 1.6 sr for the case of Jupiter's HOM and DAM emissions, resulting in a duty cycle of ∼0.14 for these emissions. We estimate the beamwidth σ for these UCDs by simply scaling to the Jovian emissions via

$$\begin{equation} \sigma = \left(\frac{\Delta \tau / \tau }{0.14}\right) 1.6, \end{equation} \tag{ 14 }$$

where the pulse duty cycle Δτ/τ is given in Table 1. This, however, does assume that the angular velocity profile, and thus that of the field-aligned currents, is the same for each dwarf, which may not be the case. The spectral luminosity then follows from the peak flux density F via

$$\begin{equation} L_r = F\sigma d^2, \end{equation} \tag{ 15 }$$

where d is the distance to the object from Earth. This is shown by the crosses in Figure 4. In comparison, the modeled spectral luminosity, shown by the pluses, is computed using the radii, rotation periods, polar ionospheric field strengths, and Pedersen conductances as listed in Table 1. The Pedersen conductances employed for TVLM 513 and 2M J0036 are 0.5 mho, as is also used in Figure 1, but it was found that 0.3 mho yields better agreement for LSR J1835. It is also worth noting that employing 0.8 mho for all three dwarfs yields spectral luminosities consistent to within ∼7% of the quiescent values (computed assuming isotropic emission) given by Hallinan et al. (2008).

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We also note the potential CMI-induced emissions from the L dwarf binary 2MASSW J0746425+200032 (Berger et al. 2009) and the T6.5 brown dwarf 2MASS J10475385+2124234 (Route & Wolszczan 2012). For simplicity, we do not consider the dwarf binary since the behavior of such a system could be significantly more complex than is represented by our simple axisymmetric model. Regarding the latter dwarf, two highly polarized bursts were observed at 4.75 GHz (implying a field strength of ∼0.17 T), with spectral flux densities of 2.7 ± 0.2 mJy (where 1 Jy = 10⁻²⁶ W m⁻² Hz⁻¹) and 1.3 ± 0.2 mJy, respectively. Assuming that the radio emission is beamed into 1.6 sr and noting the object's 10.3 pc distance, these flux densities imply a mean spectral luminosity of 3.3 ± 0.7 MW Hz⁻¹. This observation is not included in Figure 2 since no rotation period is known, although it must be at least 2 hr since only one burst was observed in each ∼2 hr observing interval. Using the parameter values employed in this paper, our model indeed requires rotation periods of 2.1–2.8 hr to produce radio powers consistent with the above luminosities, such that further observations of this object are required to test this scenario.