© ESO 2018

1. Introduction

Radio and X-ray emission from ultra-cool objects has been reported by Berger (2002), Stelzer (2004), Hallinan et al. (2006), Osten et al. (2009), Route & Wolszczan (2012), Burgasser et al. (2015), Williams et al. (2014) and Kao et al. (2016) for example, and surveys are being conducted (Antonova et al. 2013; Pineda et al. 2016, 2017). The white-flare observations by Schmidt et al. (2016; for ASASSN-16AE) and Gizis et al. (2013; for W1906+40) provide evidence for stellar-type magnetic activity to occur also in brown dwarfs, and hence, Schmidt et al. (2015) and Sorahana et al. (2014) suggested that brown dwarfs have chromospheres. Stelzer (2004) studied the old brown dwarf G569Bab and concluded that coronal emission remains powerful also beyond young ages. The presence of a magnetised atmospheric plasma, including a sufficient number of electrons, is required to allow the formation of a chromosphere/corona through magnetohydrodynamics (MHD) processes such as wave heating, similar to the Sun (Brady & Arber 2016; Mullan & MacDonald 2016; Reep & Russell 2016). Seed electrons are also required to understand aurorae on moon-less brown dwarfs.

Rodríguez-Barrera et al. (2015) showed that thermal ionisation can produce a partially ionised gas in a substantial volume of cool brown dwarfs and giant gas planet atmospheres and a highly ionised gas in the hotter/younger brown dwarfs or M dwarfs. However, for low effective temperatures and low surface gravity, additional mechanisms are required to ionise the upper atmosphere to a degree that radio and X-ray observation become feasible, as was also suggested by Mohanty et al. (2002). Rimmer & Helling (2013) demonstrated that cosmic-ray (CR) irradiation significantly enhances the electron fraction compared to thermal ionisation in the upper atmospheres of brown dwarfs, but the local degree of ionisation does not exceed 10⁻⁷ in these low-density regions. Helling et al. (2013) and Rimmer & Helling (2013) further pointed out that CRs can affect the upper portion of the mineral clouds that form in brown dwarf atmospheres (Helling & Casewell 2014), an effect that is well established for solar system objects (Helling et al. 2016a). The present paper takes the idea of environmental effects on the ionisation of atmospheres of very low-mass, ultra-cool objects one step further.

Ultra-cool objects, that is, brown dwarfs and free-floating planets, are observed in a great variety of environments. Brown dwarfs and free-floating planets in star-forming regions are exposed to a stronger radiation field than objects that are situated in the interstellar medium (ISM). Star-forming regions host O and B stars that produce a substantial fraction of high-energy radiation that may lead to the ionisation of the outer atmosphere of brown dwarfs. Sicilia-Aguilar et al. (2008) and Forbrich & Preibisch (2007) showed that non-accreting brown dwarfs in star-forming regions have X-ray luminosities of the order of log₁₀L_x ≈28 [erg s⁻¹] for 0.5 ... 8 KeV. Kashyap et al. (2008) presented X-ray observations for planet-host stars, including M dwarfs and brown dwarfs (M2 ... M8.5) with X-ray luminosities of log₁₀ L_x = 26.36 ... 31.22 erg s⁻¹. This X-ray emission is therefore likely to originate from chromospheric activity of the brown dwarfs and low-mass stars.

In this paper we investigate the effect of Lyman-continuum radiation (from photons with energies above the Lyman limit of E > 13.6 eV, hence of wavelengths λ < 912 Å) in different Galactic environments on the ionisation of the atmospheric gas of brown dwarfs. We study three different cases: (i) irradiation from the interstellar radiation field (ISRF), (ii) irradiation from O and B stars within a star-forming region, and (iii) irradiation from a white dwarf companion. Our interest in case (iii) is supported by the recent finding of atomic line emission from a brown dwarf in the white dwarf and brown dwarf system (WD-BD) binary WD0137-349 (Longstaff et al. 2017). We use the analysis framework from Rodríguez-Barrera et al. (2015) to study the resulting plasma parameters and the possible magnetic coupling of the atmosphere. Our results support the expectation that brown dwarfs form chromospheres, and we make the first suggestions for the X-ray flux possibly emerging from a shell of optically thin, but hot gas that may form the outer part of brown dwarf atmospheres. Our approach is summarised in Sect. 2. We present our results in Sect. 3. Section 4 contains our discussion, and Sect. 5 contains our conclusions.

2. Approach

We used DRIFT-PHOENIX atmosphere model structures (T_eff = 2800 K, 2000 K, 1000 K, log(g) = 3.0, 5.0, [M/H] = 0.0; Helling et al. 2008a, 2008b; Witte et al. 2009, 2011) as input for Monte Carlo radiative transfer photoionisation calculations. Using global parameters that describe young and old brown dwarfs (T_eff [K], log(g) [cm s⁻²], [M/H]), the model atmosphere simulations DRIFT-PHOENIX provide the local gas properties (T_gas[K], p_gas [bar], p_e [bar]; Fig. 1). DRIFT-PHOENIX also calculates the detailed cloud structure in ultra-cool atmospheres (Woitke & Helling 2004; Helling & Woitke 2006; Helling & Casewell 2014). For this paper, we used the cloud particle number density, n_d (for more details on clouds, see e.g. Helling et al. 2016b), to determine whether Lyman-continuum photons might reach deep enough into the atmosphere to also be a source for cloud particle ionisation similar to cosmic ray ionisation. This is, however, only a side-line of the present paper.

We evaluated the effect of external Lyman-continuum (LyC) irradiation in three different environments:

• (i)

  ISM. This includes old brown dwarfs and free-floating planes irradiated by the interstellar radiation field (ISRF). We used the ISRF as a lower limit for the Lyman-continuum irradiation causing possible atmospheric ionisation in brown dwarfs.

• (ii)

  Star-forming region. Brown dwarfs reside in environments with a strong radiation field, such as star-forming regions with O and B stars. O stars allow us to explore an upper limit of the irradiation effect that is due to LyC, while B stars are more common members of star-forming regions, as in the Orion OB1 stellar association.

• (iii)

  White dwarf - brown dwarf binaries (WD-BD). Old brown dwarfs can be companions of a white dwarf. As an example, we use the binary system WD0837+185, which is composed of a cool white dwarf and an old brown dwarf with an orbital separation of d = 0.006 AU (Casewell et al. 2015)¹. Despite the low occurrence rate of WD-BD binaries of only 0.5% (Steele et al. 2013), these systems are an important link to giant gas planets since WD-BD pairs are far easier to observe.

2.1. Photoionisation by Lyman-continuum irradiation

In an external radiation field originating from an O or a B star, a white dwarf, or the ISRF, the radiation energy E > 13.6 eV first dissociates the atmospheric H₂ , which leaves the gas in the upper atmosphere to be dominated by atomic hydrogen. This is supported by Rimmer & Helling (2016; their Fig. 9 ) , who showed that H₂ is transformed into H in the uppermost atmospheric layers of an irradiated planet. Lystrup et al. (2008) pointed out that the upper atmosphere of Jupiter is dominated by H₂, that ${\text{H}}_3^{+}$ has been consistently predicted by models to be a major component of the ionosphere of Jupiter, but that H⁺ predominates at higher altitudes and on the night side. ${\text{H}}_3^{+}$ has a short lifetime and therefore quickly decays into atomic hydrogen, but H⁺ is long-lived as it requires three-body electron recombinations or a radiative recombination reaction. Rimmer et al. (2014) showed that the dominating ionic molecule resulting from cosmic-ray triggered ion-neutral chemistry is ${\text{NH}}_4^{+}$ , which also leads to H₂ dissociation. More work is required to solidify this argument, also to clarify whether super-thermal electrons might be emerging. However, we note that only a very small part (∼0.1%) of the atmosphere will be ionised, so that the bulk (∼99.9%) remains H₂-dominated. We applied a Monte Carlo radiative transfer (MCRT) to investigate the ionisation of atomic hydrogen by Lyman-continuum photons. The abundance of H⁺, and therefore the number of electrons originating from Lyman-continuum irradiation, was derived by applying a Monte Carlo photoionisation code (Wood & Loeb 2000).

The code balances photoionisations with radiative recombinations by the ionisation equilibrium equation in each cell to achieve a new ionisation structure,$$\begin{array}{c}\hfill n(H){\int }_{{\nu }_o}^{\infty }\frac{4\pi J_{\nu }}{h\nu }{a}_{\nu }(H)\text{d}\nu =n_{\mathrm{e}}{n}_{\mathrm{p}}\alpha (H,T),\end{array}$$(1)

where n(H) is the neutral hydrogen atom (HI) density (cm⁻³), J _ν is the mean intensity (erg cm⁻² s⁻¹ Hz⁻¹), a_ν(H)=6.3 × 10⁻¹⁸ cm² is the ionisation cross section for the hydrogen with E > hν_o (E > 13.6 eV), α(H, T) is the recombination coefficient (cm³ s⁻¹), and n_e and n_p (cm⁻³) are the electron and proton densities, respectively. The opacity was updated according to the neutral fraction of hydrogen, and the next radiation transfer iteration was carried out. The Monte Carlo radiation transfer code tracks photon paths and computes the ionisation structure of hydrogen. We used a atmospheric density grid with input from DRIFT-PHOENIX models. It was homogeneously extended onto the 3D MC grid. We therefore used a 3D slab structure with an atmospheric density gradient in order to determine whether a principle effect of LyC irradiation on ultra-cool atmospheres occurred. We are interested in the depth to which external radiation may potentially ionise the atmosphere in addition to cosmic rays, as demonstrated in Rimmer & Helling (2013).

Input parameters. The Monte Carlo code follows the random walks within the atmosphere grid for both direct ionising photons from the source(s) and diffuse photons produced by radiative recombinations direct to the ground-state of hydrogen. We adopted the same approach as Wood & Loeb (2000) and assigned all direct stellar photons an energy of 18eV and diffuse photons an energy of 13.6 eV. The hydrogen ionisation cross sections at these energies are 6.3 × 10⁻¹⁸ cm² and 2.7 × 10⁻¹⁸ cm², respectively. Our simulations only tracked the ionisation of hydrogen, so this two-energy approximation for the direct and diffuse photons is sufficient to determine the depth-dependent ionisation structure. Had we wished to know the detailed depth-dependent ionisation structure of an element with multiple ionisation stages, then we would require an input spectrum as described in the Monte Carlo photoionisation code of Wood et al. (2004).

Direct ionising photons that are absorbed by neutral hydrogen in the Monte Carlo simulation are converted into diffuse photons with a probability α₁/α_A, where α₁ is the hydrogen recombination coefficient direct to the ground-state and α_A is the recombination coefficient to all levels. We adopted a radiative recombination rate α _A = 4 × 10⁻¹³ cm³ s⁻¹ and the ratio α₁/α_A = 0.38, which is appropriate for photoionised gas at 10⁴ K (Osterbrock & Ferland 2006). A key input parameter for the Monte Carlo photoionisation simulations is the flux of LyC photons at 18 eV (688 Å) reaching the top of the atmospheres from the different sources (Table 1). It is determined by the adopted ionising luminosities and source-atmosphere distances.

We conducted test calculations to determine how different input photon energies (corresponding to different cross-section values) may affect our resulting values for the ionisation penetration depth. We find that for the high densities in the brown dwarf atmospheres we study that the total ionising luminosity is the most important parameter for determining the depth of photoionisation and that the energy values adopted for the direct and diffuse photons have a negligible effect.

2.2. Free-free emission

The ionised hydrogen gas in the upper atmospheres of irradiated brown dwarfs may emit free-free radiation (thermal Bremsstrahlung) if the temperature is high enough. If the gas is optically thin, the resulting luminosity is calculated by$$\begin{array}{c}\hfill L_{\mathrm{tot}}^{\mathrm{ff}}=4\pi {\int }_R^{z_{max}}{(R+(z_{max}-z))}^2{ϵ}_{\nu }^{\mathrm{ff}}(z)\text{d}z,\end{array}$$(2)

with R (cm) the radius of the irradiated object, and z_max (cm) the maximum value of the vertical geometrical extension of the atmosphere. The volume emissivity or the total free-free emission power that a gas can emit per unit of volume, per solid angle, and per unit of frequency, ${ϵ}_v^{\text{ff}}\hspace{0.17em}(z)\hspace{0.17em}(\text{erg}\hspace{0.17em}{\text{cm}}^{-3}\hspace{0.17em}{\text{s}}^{-1})$ , of an optically thin gas integrated over a range of energy (Osterbrock & Ferland 2006) is given by the equation$$\begin{array}{c}\hfill {ϵ}^{\mathrm{ff}}=5.44\times {10}^{-39}{T}_{\mathrm{em}}^{1/2}{n}_{\mathrm{e}}{n}_{\mathrm{i}}{\overline{\text{g}}}_B\frac{1}{h}{\int }_{h{\nu }_1}^{h{\nu }_2}{\mathrm{e}}^{-\frac{h\nu }{k_{\mathrm{B}}T}}\text{d}(h\nu )·\end{array}$$(3)

The electron density, n_e (cm⁻³) results from the Monte Carlo calculation. T_em [K] is the temperature of the emitting gas, k_B (eV K⁻¹) is the Boltzmann constant, h (eV s) is the Planck constant, and ḡ_B = 1.2 is the Gaunt factor for hydrogen plasma (Osterbrock & Ferland 2006). The integration boundaries hν₁ = 0.5 KeV and hν₂ = 8 KeV correspond to the X-ray range of the electromagnetic spectra.

This simple approach allowed us to test first estimates for the X-ray luminosity that an irradiated brown dwarf may emit. No model for the formation of a brown dwarf chromosphere/corona is available to consistently derive the X-ray luminosity. Mullan & MacDonald (2016) suggested that Alfvén waves are an efficient mechanism for heating the corona of M dwarfs. They modelled a mechanical energy flux (Alfvén waves) to explain observed X-ray fluxes from the corona of M dwarfs. To the authors’ knowledge, only Wedemeyer et al. (2013) performed a consistent modelling of a chromosphere for an M dwarf. We therefore followed an approach similar to Feigelson et al. (2003), Fuhrmeister et al. (2010) and Schmidt et al. (2015). Fuhrmeister et al. (2005) constructed a semi-empirical chromosphere model for quiescent mid-M dwarfs following examples from the solar community (see references in their paper). Fuhrmeister et al. (2010) applied this model to the flaring M5.5 star CN Leo. A linear temperature rise in transition region and chromosphere was modelled, and the gradient was adjusted. The top of the chromosphere had a prescribed temperature of 8000 K (quiescent chromosphere) or 8500 K (flaring chromosphere). Higher temperatures were considered in order to represent the heating by flares. Schmidt et al. (2015) followed this approach by replacing the temperature in the outer atmosphere with a chromospheric temperature structure consisting of two components (their Fig. 8). The chromospheric temperature rise, the chromosphere break, and the start of the transition region were treated as free parameters, because little is known about the formation of chromospheres on brown dwarfs and late-M dwarfs. They calculated activity strengths for their set of 11820 late-M and brown dwarfs. Here, the 1D model consists of an underlying (cloud-forming) DRIFT-PHOENIX atmosphere in radiative-convective equilibrium and a temperature inversion region representing a chromospheric temperature increase.

3. Results

In the following, we evaluate the effect of Lyman-continuum (LyC) irradiation at 18 eV on the ionisation in a brown dwarf atmosphere in the three different scenarios (i)–(iii) as introduced in Sect. 2. Case (ii) and (iii) are studied in detail with respect to the effect of LyC irradiation on the electrostatic character and regarding the potential magnetic coupling of the atmosphere. We compare the effect of LyC to our previous reference results for local thermal ionisation (Rodríguez-Barrera et al. 2015). Case (i) is only briefly summarised in Sect. 3.1 and then used as a lower-limit reference for cases (ii) and (iii). Our analysis is presented for each plasma and magnetic parameter separately to allow a comparison between the different irradiating environments (Sects. 3.2–3.4).

3.1. Effect of the interstellar radiation field

The interstellar radiation field (ISRF) present in the ISM is a lower limit for high-energy irradiation of an old brown dwarf without a companion (free-floating objects). It has a flux ∼10⁷ photons s⁻¹ cm⁻² (Reynolds 1984; and Table 1). The interstellar radiation field ionises the uppermost atmosphere layers of an ultracool objects (dashed lines in Fig. 2) considerably more than thermal ionisation (asterisks). The upper atmosphere layers reach a degree of ionisation f_e ∼ 10⁻² (f_e = n_e/(n_e + n_gas), n_gas (cm⁻³) – gas density, n_e (cm⁻³) – local electron number density). The inwards-increasing gas density rapidly decreases the influence of the LyC from the ISRF on the local degree of ionisation.

3.2. Effect of Lyman-continuum irradiation on the atmospheric electron budget

Figure 2 shows that the LyC irradiation reaches a similar atmospheric depth and therefore ionises a similar portion of the atmosphere of a brown dwarf, independent of the incident LyC flux arriving from different radiation sources (Table 1). In the case of a young brown dwarf (log(g) = 3.0), the atmospheric portion ionised by the LyC photons has a geometric extension of up to 5000 km, for an older brown dwarf (log(g) = 5.0) it affects only the upper 50 km as the atmosphere is far more compact. These geometrical extensions reach P_gas = 0.1 … 10 bar in the coolest examples studied here (T_eff = 1000 K), but remain in the low-pressure regime for the hottest atmosphere examples we studied (T_eff = 2800 K; see Fig. A.1 for (P_gas − z) correlation). The degree to which the gas is ionised depends on the number of interacting photons, hence on the environment in which the brown dwarf resides.

Star-forming region (case ii). The electron density from the photoionisation through the irradiation of an O3 star is three orders of magnitude higher than that from the ISRF in brown dwarf atmospheres with log(g) = 3.0 and two orders of magnitude higher on brown dwarf atmospheres with log(g) = 5.0. However, type B0 stars produce one and two orders of magnitude more electrons than the effect of the irradiation from the ISRF for brown dwarfs (log(g) = 3.0, 5.0).

Figure 2 shows that Lyman-continuum irradiation provides an efficient ionisation mechanism for the uppermost atmospheric layers in ultra-cool objects. The threshold f_e > 10⁻⁷ marks the point above which the atmospheric gas becomes partially ionised and where plasma behaviour may emerge (Rodríguez-Barrera et al. 2015). In the case of O stars, a shell of fully ionised gas forms in the uppermost atmosphere. This ionisation effect cannot penetrate below a certain atmospheric depth, and other mechanisms may be needed to ionise the gas. For example, thermal ionisation (asterisks) starts to dominate the ionisation of the atmosphere at greater geometrical depth (i.e. higher pressures).

Figure 2 also depicts the location of the cloud layer (open triangles) that results from our DRIFT-PHOENIX atmosphere simulations. Only the outermost cloud layer would be affected by the LyC, either by the high-energy photon directly interacting with the cloud particles, or by the ionised gas depositing on the cloud particle surface. Neither of these processes are considered here, but they can cause an ionisation of the cloud particles similar to the electrification of the Moon surface dust (Helling et al. 2016a) as long as the cloud particles remain stable against electrostatic disruption (Stark et al. 2015; Helling et al. 2016c).

White dwarf – brown dwarf binary (case iii). Figure 2 (bottom panel) shows the degree of gas ionisation produced by thermal ionisation and resulting from the white dwarf LyC flux that impacts the brown dwarf atmosphere. Figure 2 (bottom panel) suggests that the predominant ionisation process is Lyman-continuum irradiation in the upper 30–50 km in the atmosphere of an old brown dwarf and that it ionises the upper atmosphere completely for T_eff = 2000–1000 K. The uppermost layers of the cloud where the smallest particles reside (haze layer; see e.g. Helling et al. 2016b) are affected by the LyC irradiation here as well.

3.3. Plasma parameters

If ω_pe ≫ ν_ne (ω_pe – plasma frequency, ν_ne – collisional frequency neutrals-electrons; both in [s⁻¹]), then electromagnetic interactions dominate electron-neutral interactions in a gas. The plasma frequency is defined as$$\begin{array}{c}\hfill {\omega }_{\mathrm{pe}}={\left(\frac{n_{\mathrm{e}}{e}^2}{{ϵ}_0{m}_{\mathrm{e}}}\right)}^{1/2},\end{array}$$(4)

with e is the electron charge [C] and m_e is the electron mass (kg), and the collision frequency for neutral particles with electrons is$$\begin{array}{c}\hfill {\nu }_{\mathrm{ne}}={\sigma }_{\mathrm{gas}}{n}_{\mathrm{gas}}{v}_{\mathrm{th},\mathrm{e}},\end{array}$$(5)

where v_th,e is the thermal velocity of electrons (v_th,e = [k_BT_s/m_s]^1/2 [m s⁻¹]), and σ_gas is the scattering cross section of particles (${\sigma }_{\text{gas}}=\pi \times r_{\text{gas}}^2=8.8\times {10}^{-17}\hspace{0.17em}{\text{cm}}^2$ ; r_gas ≈ r_H⁺ = 5.3 × 10⁻⁹ cm). Figure 3 shows where in the atmosphere the gas can be treated as a plasma. This study demonstrates that Lyman-continuum irradiation increases the atmospheric volume where the collective long-range electromagnetic interactions dominate compared to our reference study (Rodríguez-Barrera et al. 2015).

Star-forming region (case ii). Figure 3 (top and middle panels) shows that electromagnetic interactions dominate collisional processes (i.e. ω_pe/ν_ne ≫ 1) for all models considered because of external LyC radiation. This is different to the case when only thermal ionisation (solid and dotted lines) is present. The effect of external irradiation in the form of LyC allows for a considerably larger atmospheric volume where electromagnetic interactions dominate compared to thermal ionisation alone. Figure 3 shows that different T_eff and log(g) result in a difference of at least one order of magnitude (or more) in the effect on the atmosphere. All atmosphere cases appear rather similar in the very top layers with respect to their capability for electromagnetic interactions.

White dwarf – brown dwarf binary (case iii). Figure 3 (bottom panel) shows that the collective long-range electromagnetic interactions dominate the collisions between electron and neutral particles at p_gas ≤ 10⁻⁴ bar for the atmosphere with T_eff = 2000 K and at p_gas ≤ 10⁻² bar for the coldest atmosphere with T_eff = 1000 K. The comparison of these results with the effect of the thermal ionisation shows that Lyman-continuum irradiation increases the atmospheric volume where the collective long-range electromagnetic interactions dominate by six orders of magnitude in pressure, which means that the affected region increases substantially. The LyC irradiation from a white dwarf causes a larger volume of the brown dwarf atmosphere to form a plasma, and it causes a stronger electromagnetic interaction compared to LyC from O and B stars in star-forming regions (case ii) for a given atmosphere structure (compare panels in Fig. 3).

3.4. Magnetic parameters

We have quantified the degree of ionisation and the plasma frequency and how they are affected by the Lyman-continuum irradiation in comparison to thermal ionisation in a substellar atmosphere for different Galactic environments. In this section, we show if and how the magnetic coupling changes as a result of external LyC photoionisation. The atmospheres are analysed with respect to a critical magnetic flux and the classical Reynolds number.

A gas is magnetised if ω_c,s ≫ ν_n,s (ω_c,s – cyclotron frequency, ν_n,s – collisional frequency, both [s⁻¹]). A critical magnetic flux, B_s, can be derived that is required to ensure ω_c,s/ν_n,s ≫ 1 (Sect. 5.1 in Rodríguez-Barrera et al. 2015):$$\begin{array}{c}\hfill B_{\mathrm{s}}=\frac{m_{\mathrm{s}}}{e}{\sigma }_{\mathrm{gas},\mathrm{e}}{n}_{\mathrm{gas}}{\left(\frac{k_{\mathrm{B}}{T}_{\mathrm{s}}}{m_{\mathrm{s}}}\right)}^{1/2}[T],\end{array}$$(6)

where ${\sigma }_{\text{gas}}=\pi \times r_{\text{gas}}^2$ the collision, or scattering, (m²) is the collision, or scattering, cross section, m_s is the mass of the species s in (kg), and T_s is the temperature of the species in [K]. T _s = T_gas is assumed. Hence, if a local magnetic field B ≫ B_s, then magnetic coupling will be possible.

The magnetic Reynolds number, R_m, provides a measurement of the diffusivity of the magnetic field in a given atmospheric gas. The magnetic Reynolds number can be calculated by R_m = vL/η, where L (m) is the pressure scale length of the considered plasma and η (m² s⁻¹) is the diffusion coefficient. The pressure length scale is assumed to be L = 10³ m, which is a representative scale height value for a brown dwarf with log(g)=5 (Helling et al. 2011). The diffusion coefficient, η, can be approximated by η ≈ η_d (Rodríguez-Barrera et al. 2015), where η_d is the decoupled diffusion coefficient ${\eta }_{\text{d}}=c^2\hspace{0.17em}{v}_{\text{ne}}/{\omega }_{\text{pe}}^2$ . The plasma has reached the ideal MHD regime if R_m ≫ 1 and the coupling between the plasma and the background magnetic field is complete.

Star-forming region (case ii). Figure 5 shows where electrons resulting from LyC irradiation from an O star can be magnetised in an M dwarf/browndwarf/planet atmosphere compared to thermal ionisation, that is, where B ≫ B_s. We furthermore observe that

• for atmospheres with T_eff = 2800 K (M dwarfs and young brown dwarfs), a typical background magnetic field flux of B = 10³ G magnetises the ambient electrons at p_gas < 5 bar,

• for atmospheres with T_eff = 1000 K (planets), a typical background magnetic field flux of B = 10 G magnetises the electrons at p_gas < 5 × 10⁻² bar.

Figure 4 shows the magnetic Reynolds number as the result of the thermal ionisation plus the photoionisation by Lyman-continuum irradiation from an O star and from the ISRF. The results from Fig. 4 suggest that

• a decrease in T_eff, for a given log(g) and [M/H], decreases the atmospheric volume where R_m≫ 1,

• an ideal MHD be haviour (R_m ≫ 1) can be considered in the upper atmosphere owing to the significant increase in electron density (R_m ∝ n_e) by the effect of the Lyman-continuum irradiation compared to the effect of thermal ionisation alone.

White dwarf – brown dwarf binary (case iii). Figure 5 (bottom) shows that LyC irradiation from white dwarfs would not only magnetise electrons in brown dwarf atmospheres, but also in M dwarf and giant gas planet atmospheres:

• for atmospheres with T_eff = 2000 K (brown dwarfs and M dwarfs) with a background magnetic field flux of B = 10³ G, electrons are magnetised for p_gas < 5 bar,

• for atmospheres with T_eff = 1000 K (giant gas planets), a background magnetic field flux of B = 10 G can magnetise the electrons at p_gas < 5 × 10⁻² bar.

Figure 4 (bottom panel) shows the magnetic Reynolds number for LyC irradiation from a WD and the ISM compared to the effect of thermal ionisation. The results are similar to case (ii), except that the Reynolds number reaches higher values if the brown dwarf were affected by LyC from a white dwarf.

3.5. Free–free emission luminosity from strongly ionised, optically thin atmospheres

When we assume that these uppermost highly ionised atmosphere layers are optically thin, we can estimate the resulting luminosity from free-free emission (Eq. (2)) for the three cases of Galactic environment considered here. The emissivity ${ϵ}^{\text{ff}}\sim T_{\text{em}}^{1/2}{n}_e{n}_i\times f(e^{1/T_{em}})$ (Eq. (3)) requires information about the local densities, n_e, n_i, and the temperature of the emitting gas. The local electron number densities are a direct result of our calculation. For the local electron temperature, T _em [K], we tested two cases: T_em = T_gas with T_gas the gas temperature as result of the DRIFT-PHOENIX model atmosphere. This case assumes that no effective heating of the ionised gas occurs. In the second case, we used T_em = T_chrom with T_chrom = const, a preset temperature of expected chromospheric values, representing potential heating through photoionisation and mechanical waves. We have no information about chromospheric temperatures on brown dwarfs, and suggestions in the literature vary: Burgasser et al. (2013) suggested that the brown dwarf 2MASS J13153094-2649513 (L7) emits electrons with temperature of T_em ≈ 10⁹–10¹⁰ K, Osten et al. (2009) quoted a temperature of T_em ≈ 10⁶ K, and Schmidt et al. (2015) applied T_em = 10⁴ K. In order to provide first estimates based on our ionisation calculations, we followed the examples by Fuhrmeister et al. (2010) and Schmidt et al. (2015) in postulating an isothermal temperature layer with temperatures higher than in the inner atmosphere. Such a layer could also be the result of MHD wave heating as a result of the magnetic coupling shown in Fig. 5. The temperature of the emitting electrons may also decouple from the gas because their mass is significantly lower than that of neutrals and ions. Hence, because the electrons will lose a smaller portion of their energy during a collision, they may not be in thermal equilibrium with their surrounding in regions from which X-ray and radio emission emerge. We tested two values, T_em = 10⁵ K and T_em = 10⁶ K.

Table 2 summarises the free-free emission luminosities, $L_{\text{tot}}^{\text{ff}}$ , that would occur if the LyC ionised gas were optically thin and had the temperature T_em. The highest amount of free-free emission luminosity is to be expected from brown dwarfs orbiting white dwarfs, the lowest from brown dwarfs in the interstellar medium. The strong increase of the values in Table 2 when we increased the local gas temperature to 10⁶ K results from the steepness of the high-energy tail of the Planck function.

Star-forming region (case ii). The free-free emission luminosities in the X-ray energy interval 0.5–8 KeV in Table 2 reach values that are similar to observations from non-accreting brown dwarfs at X-ray wavelengths (Sicilia-Aguilar et al. 2008; Kashyap et al. 2008 and Feigelson et al. 2003) if the upper LyC ionised atmosphere can be heated substantially. The emissivity is very low if the emitting part of the atmosphere would emit with the local LTE gas temperature (Fig. 1). Schmidt et al. (2015) presented the H _α emission of ultra-cool (M7-L8) brown dwarfs and suggested that these observations required the presence of a chromosphere. Forbrich & Preibisch (2007) suggested that the candidate brown dwarf B185839.6-365823, with log L_x = 28.43 (L_x ∼ 2.7 × 10²⁸ erg s⁻¹), required a high plasma temperature in order to fit the observed spectrum. Another example for a very late dwarf, the M8.5 brown dwarf B185831.1-370456, was observed with a luminosity from free-free emission of log L_x = 26.9 (L_x ∼ 8 × 10²⁶ erg s⁻¹). Neuhäuser et al. (1999) reported a ∼1 Myr old brown dwarf (Cha H_α 1) with a quiescent emission at X-ray wavelengths of log L_x = 28 [erg s⁻¹]. They suggested that magnetic activity must be present to explain these observations. Feigelson et al. (2003) detected 525 objects in a massive cluster (Orion nebula) at energies of 0.5–8 KeV (X-rays); 144 of these objects are catalogued as brown dwarfs (logM ≤ −0.2 M_⊙) (their Table 1) with an average of the X-ray emitted luminosity of log L_x ≈ 28.84 erg s⁻¹. Our estimates of the free–free luminosity that is emitted from brown dwarfs that form a shell of highly ionised but optically thin gas in star-forming regions are of the same order of magnitude. Our estimates, which are summarised in Table 2, show that the resulting luminosity from free-free emission in the energy interval 0.5–8 KeV increases if

• the object forms a chromosphere with a substantially increased temperature compared to LTE values.

• the T_eff decreases.

An increase in surface gravity, hence a decrease in atmospheric scale heights, does lead to an increased luminosity in most but not all cases.

White dwarf – brown dwarf binary (case iii). Table 2 also presents the resulting luminosity from free-free emission (thermal Bremsstrahlung irradiation) of a brown dwarf that is ionised by Lyman-continuum irradiation from a WD star. The results show that $L_{\text{tot}}^{\text{ff}}$ is higher in a white dwarf – brown dwarf system than in a star-forming region or through the ISRF because of the close orbital separation between the white dwarf and the brown dwarf. Recent optical and near-infrared observations of the close white dwarf – brown dwarf non-interacting binary system WD0137-349 by Longstaff et al. (2017) showed that He, Na, Mg, Si, K, Ca, Ti, and Fe lines were emitted from the brown dwarf companion. This is a strong indication for a temperature inversion in the upper atmosphere, and in combination with the detected Hα emission, it is a strong indication that a high-temperature plasma is present in this cool companion to a white dwarf. This conclusions for such systems is supported by Casewell et al. (2018), who detected Mg I and Ca II emission lines in addition to Hα lines in the (shortest-period) non-interacting, white dwarf – brown dwarf post-common-envelope binary known as EPIC 21223532.

4. Discussion

Unexpectedly powerful emissions at radio, X-ray, and H _α wavelengths from very low-mass objects (ultra-cool dwarfs) have been observed by different groups (e.g. Williams et al. 2014; Burgasser et al. 2013; Berger 2002; Route & Wolszczan 2012 and see Pineda et al. 2017 for a recent survey). Sorahana et al. (2014) suggested that weakened H₂O (2.7 μm), CH₄ (3.3 μm), and CO (4.6 μm) absorption in combination with moderate Hα emission could be linked to chromospheric activity and its effect on the underlaying atmosphere structure. Observations by Schmidt et al. (2016, 2015) support this interpretation. They postulated the presence of a chromosphere to reproduce extensive Hα-activity survey data for brown dwarfs. Schmidt et al. (2016) reported an old L0 dwarf emitting powerful emissions at H _α and near-infrared wavelengths. Such powerful emissions suggest that magnetic activity must be present even in old-type brown dwarfs, and it implies the presence of a magnetised atmospheric plasma even in such ultra-cool objects. However, no consistent MHD simulations for brown dwarfs have been carried out so far. Wedemeyer et al. (2013) studied the formation of a chromosphere on fully ionised M dwarfs. Non-ideal (i.e. partially ionised) MHD simulations by Tanaka et al. (2015) suggested the formation of a temperature inversion by magneto-convection processes and Alfvén-wave heating (see also Brady & Arber 2016; Mullan & MacDonald 2016 and Reep & Russell 2016) in giant gas planets. According to our work, this might also work for brown dwarfs and lead to the formation of a chromosphere.

The radio emission from brown dwarfs has also been examined by Nichols et al. (2012), for instance, who studied the properties of radio emissions in ultra-cool dwarfs assuming the occurrence of auroral regions. They suggested that the coupling between the atmospheric plasma and the magnetic field causes a (postulated) high-latitude ionosphere and generates auroral processes. A loss of particles to the ionosphere was suggested. Speirs et al. (2014) presented a theoretical approach for cyclotron radio emission from Earth’s auroral region, showing that the radio emission results from a backward-wave cyclotron-maser emission process.

Pineda et al. (2017) have provided a concise overview of these different approaches, suggesting that a transition from Sun-like chromospheric processes to Jupiter-like large-scale magnetospheric current systems occurs at the low-mass end of the main sequence. Our paper has added to this discussion by addressing the question of the origin of the pool of electrons that allows a chromosphere to form or a magnetospheric current system to develop, assuming that a magnetic field is present. Although we do not present a consistent simulation to demonstrate the formation of a chromospheric region on brown dwarfs, first estimates of the X-ray luminosity of a hot, optically thin gas that is ionised by Lyman-continuum radiation suggest values that are similar to observations of non-accreting brown dwarfs in star-forming regions.

The required high electron density could affect the cloud layer in brown dwarf atmospheres. Helling et al. (2016c) suggested that cloud particles might be destroyed if the electron temperature is > 10⁵ K. This would occur if a chromosphere-like temperature increase were to coincide with some part of the cloud layer where a strong gas ionisation occurs (only the uppermost atmosphere, Fig. 2).

5. Conclusions

Lyman-continuum irradiation causes a considerable increase in ionisation in the upper and outermost atmospheric regions of ultra-cool objects, forming a shell of substantial local ionisation. We showed that these atmospheric regions exhibit a far stronger plasma than can be achieved with thermal ionisation alone, and it would therefore be reasonable to call this part of the atmosphere an ionosphere. If a sufficient global magnetic field is present, magnetic coupling of these highly ionised atmospheric layers occurs, and we may therefore expect the formation of a chromosphere if the object is sufficiently convective such that MHD heating mechanisms similar to that on M dwarfs can occur (Wedemeyer et al. 2013).

Different Galactic environments were investigated (ISRF, star-forming region, white dwarf - brown dwarf binary). Lyman-continuum radiation from the interstellar radiation field has the smallest effect on the degree of ionisation. The outer atmosphere of a brown dwarf binary as companion of a white dwarf, however, can be expected to be fully ionised. More rigorous follow-up simulations are therefore warranted in order to study effects such as magnetically driven mass loss or even mass transfer in WD-BD systems.

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Acknowledgments

M. I. R.-B. and Ch. H. acknowledge financial support of the European Community under the FP7 by the ERC starting grant 257431. Ch. H. thanks for the hospitality of the Kaptyn Astronomical Insititut at the University of Groningen, and travel support from NWO and LKBF. We thank A. Sicilia-Aguilar and P. Rimmer for the insightful and valuable discussions of the manuscript. Most literature search was performed using the ADS. We gratefully acknowledge our local computer support.

References

Appendix A: Supplementary details

This appendix contains supplementary information about the relation between the local gas pressure, P_gas, and the geometrical extension of the atmospheres, z, in Fig. A.1.

All Tables

All Figures