The Pale Green Dot: A Method to Characterize Proxima Centauri b Using Exo-Aurorae

M dwarf mass-loss rates, and therefore stellar winds, are not well constrained due to observational sparsity and difficulty (e.g., Wood et al. 2004). To model the M dwarf winds for Proxima Centauri, we adopt the predictions from the modeling efforts of Cohen et al. (2014), who generated an MHD stellar wind model for the M3.5 star EV Lacertae based on available observations. There are two primary differences between EV Lac and Proxima Centauri that we should take into account when considering the stellar wind at our planet's location of interest: (1) the relative mass-loss rates and (2) the difference in rotation rates.

The first of these factors has been estimated by Wood et al. (2005), who find that the mass-loss per unit surface area for Proxima Centauri and EV Lacertae are quite similar. This suggests comparable wind conditions at equal distances in units of their respective stellar radii.

The second factor, the rotation rate, affects the morphology of the stellar wind magnetic field by changing the Alfvén radius. The Alfvén radius, R_A, is defined as the point where the Alfvén Mach number is equal to unity—i.e., ${M}_{A}\,\equiv$ u_sw/v_A = 1, where u_sw is the stellar wind speed and v_A is the Alfvén speed. Interior to R_A (the sub-Alfvénic wind), the magnetic field of the star is mostly radial, and corotates at the angular rate of the star; exterior to R_A (the super-Alfvénic wind) the field begins to lag behind corotation as the magnetic tension is overcome by the flow of the wind. In the super-Alfvénic regime, the interplanetary magnetic field (IMF) exhibits the well-known Parker-spiral (Parker 1958). The Alfvén point is an important boundary that modifies the energy transfer between the stellar wind and the planetary magnetosphere.

To correctly estimate the interactions, it is important to consider Proxima Cen b's orbital distance from its host star, for both the dynamic parameters (mass density, velocity) and the magnetic structure—i.e., we must consider where Proxima Cen b orbits relative to its Alfvén radius, R_A. We note that the rotational period of Proxima Centauri (82.6 days; Collins et al. 2016) is ∼19 times lower than EV Lacertae (4.376 days; Testa et al. 2004). For our purposes, we estimate an average R_A for a simple stellar dipole moment:

$$\begin{eqnarray}&&{R}_{A}={\left(\displaystyle \frac{4\pi {{ \mathcal M }}_{\star }^{2}}{{\dot{M}}_{\star }{\omega }_{\star }{\mu }_{0}}\right)}^{\tfrac{1}{5}},\end{eqnarray} \tag{ 1 }$$

where ${{ \mathcal M }}_{\star }$ is the magnetic dipole moment for the star, ${\dot{M}}_{\star }$ is the mass-loss rate, ${\omega }_{\star }$ is the angular frequency of stellar rotation, and ${\mu }_{0}$ is the vacuum permeability. For EV Lacertae and Proxima Centauri, R_A are ∼65.4 ${R}_{\star }$ (0.075 au) and 115 ${R}_{\star }$ (0.192 au), respectively. This is the average value for a simple dipole moment because we are not including magnetic topology, but nonetheless the value obtained for EV Lac agrees well with the approximate average for the more complicated magnetic treatment simulated in Cohen et al. (2014). The relative orbit for Proxima Cen b is therefore ∼0.76 R_A. Coincidentally, this corresponds well with the simulated Planet B at EV Lac in Cohen et al. (2014), which orbits at ∼0.79 R_A.

Recently, Garraffo et al. (2016) applied an MHD model of stellar winds based on the Zeeman–Doppler Imaging (ZDI) of GJ51, and scaled the magnitude of the surface field to match the anticipated value of 600 G for Proxima Centauri. Their results from the assumed magnetic environment are in line with the values we adopt from Table 2, and our value calculated for the magnetopause distance using Equation (3) below is within the range of their calculations for magnetopause distance for Proxima Cen b. However, the structure of the magnetic topology in the simulation of Garraffo et al. (2016) places Proxima Cen b primarily in the super-Alfvénic wind, contrary to both the simple method above and the bulk of the structure found by Cohen et al. (2014).

Our estimate of R_A does not take into account the complicated magnetic topology of a realistic stellar magnetic field, which could indicate the planet likely orbits primarily through sub-Alfvénic conditions (e.g., Figure 1 of Cohen et al. 2014) or through primarily super-Alfvénic conditions (Figure 2 from Garraffo et al. 2016). Therefore, we consider both super- and sub-Alfvénic conditions for the steady-state stellar wind, using the reported parameters at Planet B from Cohen et al. (2014); see Table 2.

Tidal locking is likely for the expected orbital parameters of Proxima Cen b and the age of the system. We therefore expect a rotational period equal to the orbital period, 11.186 days, or 8.94% of the Earth's rotational frequency. Following the magnetic moment scaling of Stevenson (1983) and Mizutani et al. (1992), we assume the upper limit of the rotationally driven planetary dynamo as ${ \mathcal M }\propto {\omega }^{1/2}{r}_{c}^{3}$, where ${ \mathcal M }$ is the magnetic moment, ω is the rotation rate of the planet, and r_c is the core radius (which we assume to be proportional to the planetary radius). This suggests a magnetic moment for an Earth-radius Proxima Cen b of $\sim 0.3\,{{ \mathcal M }}_{\oplus }$. Taking the upper limit of the expected radius of Proxima Cen b, 1.4 ${R}_{\oplus }$, this gives a magnetic moment of $\sim 0.8\,{{ \mathcal M }}_{\oplus }$, which agrees with the upper limit of Zuluaga & Bustamante (2016). However, Driscoll & Barnes (2015) showed that for an Earth-like terrestrial planet orbiting a star of 0.1 ${M}_{\odot }$ with high initial eccentricity ($e\,\geqslant$ 0.1) within 0.07 au, the planet will circularize before 10 Gyr. On this timescale, the orbital energy dissipated as tidal heating is sufficient to drive a strong convective flow in the planetary interior that could generate a magnetic moment in the range of $\sim 0.8\mbox{--}2.0\,{{ \mathcal M }}_{\oplus }$ during the process of circularization. Given the above, we consider the situation of an Earth magnitude magnetic field for Proxima Cen b, but discuss how each of the methods below can be scaled to various magnetic dipole moments.

Desch & Kaiser (1984) suggested a correlation between incident stellar wind power and the power of planetary radio emissions in the solar system, a so-called "radiometric Bode's law." Zarka (2006), Zarka (2007) extended the work to modern solar system measurements as well as potential exoplanetary systems, and further suggested that a similar "auroral UV-magnetic Bode's law" could exist, though the author notes such scaling would be less generally applicable than the radio case across planetary systems due to the complexities of UV auroral generation for differing planetary atmospheres and magnetospheric dynamics. The calculations in this section can be thought of similarly as a "visible-kinetic Bode's law" for the specific case of exoplanets with an Earth-like atmosphere. A similar relation may also be derived for the magnetic stellar wind interaction (e.g., a "visible-magnetic Bode's law"; details below).

The stellar wind kinetic power delivered to the magnetosphere of the planet can be expressed as

$$\begin{eqnarray}&&{P}_{U}=\rho \,{v}^{3}\,\pi \,{R}_{\mathrm{MP}}^{2},\end{eqnarray} \tag{ 2 }$$

where ρ and v are the stellar wind mass density and velocity relative to planetary motion (∼48 km s⁻¹), respectively, and R_MP is the magnetopause distance along the line connecting the star and planet (sub-stellar point). The latter can be estimated through magnetospheric pressure balance with the stellar wind dynamic ram pressure (e.g., Schield 1969):

$$\begin{eqnarray}&&\displaystyle \frac{2\,{{ \mathcal M }}^{2}}{{K}_{\mathrm{SW}}{\mu }_{0}\,{R}_{\mathrm{MP}}^{6}}={p}_{\mathrm{ram}},\end{eqnarray} \tag{ 3 }$$

where ${ \mathcal M }$ represents the magnitude of the magnetic dipole moment, K_SW is related to particle reflection at the magnetopause (herein the interaction is assumed to consist of inelastic collisions, or K_SW = 1), and R_MP is the distance from the planet at which the magnetic pressure of the planet balances the pressure of the stellar wind. The RHS of Equation (3) represents the dynamic ram (${p}_{\mathrm{ram}}=\rho {v}^{2}$) pressure of the stellar wind, calculated from the values in Table 2.

We consider an Earth-strength magnetic dipole moment of ${ \mathcal M }=8.0\times$ 10¹⁵ Tesla m³. Solving for R_MP in Equation (3) and inserting into Equation (2) provides an estimate of the stellar wind power incident on the planetary magnetopause. Externally driven planetary auroral systems are not typically 100% efficient at converting the incident stellar wind power into electromagnetic auroral emission, and range from ∼0.3% at Neptune, ∼1% at Earth, and up to almost 100% at Jupiter (e.g., Cheng 1990; Bhardwaj & Gladstone 2000). For reference, at Earth, this method gives us a reasonable estimate of the total emitted electromagnetic auroral power of ∼30 GW for nominal solar wind conditions (4 cm⁻³, 400 km s⁻¹), which is consistent with the anticipated power of 1–100 GW, depending on solar and magnetospheric activity. While the intensities of various emissions vary widely with activity and atmospheric conditions, we assume an averaged auroral emission. In order to estimate the emitted power of the O i 5577 Å line, we assume it represents 2% of all emitted electromagnetic power (Chamberlain 1961; Kivelson & Russell 1995, p. 586), as calculated by Equation (2). We note that this assumes an Earth-like atmosphere for the planet; we briefly discuss the effect of different atmospheric compositions in Section 5.

Figure 1 shows the predicted emitted power of the 5577 Å line based on Equation (2) and multiplied by the 2% factor mentioned above and by the conversion efficiency of 1%. For the Earth, this method predicts a power of ${\phi }_{\oplus }\sim 0.68$ GW in the 5577 Å line. Assuming a 5° latitudinal width starting at ∼18° co-latitude and extending equatorward, this corresponds to a photon flux of ∼13.7 kR. This is in agreement with moderate auroral activity (IBC II⁶ , 10 kR 5577 Å emission; see Table 2.1 in Chamberlain 1961), and within a factor of 2–5 of observations during moderate geomagnetic disturbance (2.5–6 kR 5577 Å emission, e.g., Steele & McEwen 1990).

Zoom In Zoom Out Reset image size

Power estimates for the 5577 Å line for a 0.05 au orbit around Proxima Centauri are shown in Table 3. The calculated power is ∼75.3 (54.7) times ${\phi }_{\oplus }$, in the sub-(super-) Alfvénic stellar wind. These are the estimates for a steady-state stellar wind, for a terrestrial planet with an Earth-like magnetic dipole moment. Note that by inspection of Equations (2) and (3), one can see that the expected power scales as ${{ \mathcal M }}^{2/3}$, and so can easily be extended to different planetary dipole moments.

The method above has a weakness in that it completely ignores the incident Poynting flux from the IMF, and potential direct magnetic interactions between the stellar wind and planetary magnetic field, e.g., flux merging or reconnection. These interactions can produce a significant amount of magnetospheric energy input, and so they are important to consider. Similar to Equation (2), a scaling relation between power emitted at the 5577 Å line and incident magnetic flux in the stellar wind, akin to a visible-magnetic Bode's law, can be given as (e.g., Zarka 2006; Grießmeier et al. 2007; Zarka 2007)

$$\begin{eqnarray}&&{P}_{B}=\epsilon \,K\left(\displaystyle \frac{v\,{B}_{\perp }^{2}}{{\mu }_{0}}\right)\pi \,{R}_{\mathrm{MP}}^{2},\end{eqnarray} \tag{ 4 }$$

where is the efficiency of reconnection (typically of the order of 0.1–0.2), K is related to the "openness" of the magnetosphere, and for an Earth-like dipole is $K={\sin }^{4}(\theta /2)$, where θ is the angle between the perpendicular IMF and planetary dipole field, B_⊥ is the perpendicular IMF ($\sqrt{{B}_{Y}^{2}+{B}_{Z}^{2}}$), ${\mu }_{0}$ is the vacuum permeability, and R_MP is the magnetopause sub-stellar point discussed above. We can estimate the magnetic interaction at Proxima Cen b using our stellar wind conditions by taking the ratio of Equations (4) and (2):

$$\begin{eqnarray}&&\displaystyle \frac{{P}_{B}}{{P}_{U}}=\displaystyle \frac{\epsilon \,K\,{B}_{\perp }^{2}}{{\mu }_{0}\,\rho \,{v}^{2}},\end{eqnarray} \tag{ 5 }$$

which is essentially the ratio of the perpendicular IMF magnetic pressure to the ram pressure, modulated by magnetic field orientation and reconnection efficiency. In the best-case scenario, K is equal to 1 (indicating θ = π, driving strong reconnection at the magnetopause), and is of the order of 0.2 or so. Assuming this best case, and inserting the values from Table 2 for the sub- and super-Alfvénic cases, one obtains a ratio of ∼0.019 and 0.0084 for the sub- and super-Alfvénic cases, respectively. For the particular stellar wind parameters we have chosen, the kinetic power dominates the anticipated auroral output for Proxima Cen b. It is worth noting, however, that the magnetic environment (both planet and star) is largely unconstrained, and highly dynamic—particularly near active M dwarfs.

Equations (2) and (4) above are decent first approximations, but involve significant uncertainties concerning the energy dissipation in physical phenomena throughout the magnetosphere (i.e., auroral activity; Perreault & Akasofu 1978; Akasofu 1981). Wang et al. (2014) developed a global, 3D MHD model to obtain a fit for the energy coupling between the solar wind and Earth's magnetosphere to estimate the energy transferred directly from the wind into the magnetosphere and auroral precipitation (see their Equation (13) and below). The simulations were performed over 240 iterations across their solar wind parameter space, and the resulting nonlinear fit for the energy transfer to the terrestrial magnetosphere was found to be

$$\begin{eqnarray}&&{P}_{\mathrm{trans}}={K}_{1}\,{n}_{\mathrm{sw}}^{0.24}\,{v}_{\mathrm{sw}}^{1.47}\,{B}_{T}^{0.86}[{\sin }^{2.7}(\theta /2)+0.25],\end{eqnarray} \tag{ 6 }$$

where ${K}_{1}=3.78\times {10}^{7}$ is a coupling constant, n_sw and v_sw are the stellar wind number density (in cm⁻³) and velocity relative to planetary motion (in km s⁻¹), respectively, B_T is the magnitude of the transverse component of the Sun's IMF (${B}_{T}=\sqrt{{B}_{X}^{2}+{B}_{Y}^{2}}$) in nT, and θ is the so-called IMF clock angle ($\tan \theta ={B}_{Y}/{B}_{Z}$). The coordinate system used is the geocentric solar magnetospheric (GSM) system, with $\hat{X}$ pointing from the planet to the star, $\hat{Z}$ aligned with the magnetic dipole axis of the planet (here assumed to be perpendicular to the ecliptic), and $\hat{Y}$ completing a right-handed coordinate system.

Wang et al. (2014) were focused solely on the Earth's magnetosphere, but one can scale to any dipole moment by noting that Equation (6) scales just as in Section 2.3: the dipole moment term is implicitly included in the coupling constant K₁ above and scales with the planetary magnetic dipole magnitude as ${{ \mathcal M }}_{P}^{2/3}$ (Vasyliunas et al. 1982, also Equations (2) and (3) above).

Equation (6) is the total power delivered by the stellar wind to the magnetosphere, which Wang et al. (2014) estimate is ∼13% of the total incident stellar wind energy. They further estimate that 12% of that energy is dissipated by particle precipitation into the auroral regions, yielding a total solar wind/auroral coupling efficiency of ∼1.56%—very similar to the efficiency value of 1% assumed for Earth and Proxima Cen b in Section 2.3. As simple validation for our purposes, we use this method to predict a maximum coupling of auroral particle precipitation (with IMF clock angle $\theta \,=$ π, driving reconnection and likely substorm activity) at Earth of ∼0.17 TW. This is in agreement with terrestrial plasma observations during periods of geomagnetic disturbance (e.g., Hubert et al. 2002). This method is useful in that it provides a direct relationship between the power delivered as auroral particle precipitation and incident stellar wind conditions.

For Proxima Cen b, subjected to the stellar winds from Table 2, this method predicts a total power of auroral particle precipitation of $\sim 10.7$ (5.8) TW for the sub-(super-)Alfvénic stellar wind. The stellar wind parameters in Table 2, however, are a snapshot and not indicative of the highly variable conditions likely experienced at Proxima Cen b.

Magnetospheric substorms, related to transient populations of energized particles driven by magnetic reconnection in the magnetotail, can drive strong increases in auroral particle precipitation. Though not a one-to-one indicator, substorm activity can be associated with periods of strong reconnection at the magnetopause—correlated with a significant negative B_Z component in the IMF. In the present work, we assume θ = π, or B_Y = 0, to obtain an upper limit to substorm influence under our model. Although this is not a strict definition, Wang et al. (2014) calibrated the model used here to include periods of substorm activity and high hemispheric energy input. Assuming with this strong negative B_Z that a substorm is driven at Proxima Cen b, we predict an energy input of ∼28.3 (15.9) TW for the sub-(super-)Alfvénic wind. To compare directly to the 5577 Å line auroral power output such as that calculated in Section 2.3, we must link these values to the aurora by including the efficiency of precipitating charged particles in the production of auroral emission for the 5577 Å line, which will be done below.

To calculate the auroral 5577 Å photon flux, we use the precipitating auroral particle powers above obtained from Equation (6), and combine with the anticipated size of the auroral oval and an observed conversion efficiency for electron precipitation to 5577 Å emission. This gives the photon flux in kR, ${\phi }_{5577}$:

$$\begin{eqnarray}&&{\phi }_{5577}={P}_{\mathrm{in}}\,{A}_{\mathrm{mag}}^{-1}\,{\epsilon }_{e},\end{eqnarray} \tag{ 7 }$$

where P_in is 12% (discussed above) of P_trans from Equation (6), A_mag is the summed area of both the northern and southern auroral ovals (we assume N–S symmetry), and ${\epsilon }_{e}$ is the efficiency with which magnetospheric electrons are converted to auroral emission of the 5577 Å oxygen line. We use the reported values from Steele & McEwen (1990; noted below), who used ground-based observations of auroral line intensities and the related satellite observations of energetic electron flux to draw a relation between electron precipitation and auroral photon flux. We then integrate the resulting flux over a nominal $5^\circ$ auroral oval (for each hemisphere), the co-latitude of which is dependent on the sub-stellar magnetopause distance (discussed below).

Steele & McEwen (1990) reported the conversion efficiency for the 5577 Å O i line as 1.73 ± 0.51 (1.23 ± 0.44) kR/(erg cm⁻² s⁻¹) for a magnetospheric Maxwellian electron population of characteristic temperature 1.8 (3.1) keV. In the present work, we take the average values for these populations, $\sim 1.48$ kR/(erg cm⁻² s⁻¹). We assume the fraction of total hemispheric power (P_in) delivered by electrons to be 0.8 (Hubert et al. 2002), so this factor is included in the P_in factor.

The magnetopause distance we calculate via Equation (3) for the Earth-like magnetic dipole moment is ∼4.2 (3.3) R_P for the (sub-)super-Alfvénic conditions. From these values, we can provide a simple estimate of the total auroral oval coverage. The magnetic co-latitude of the boundary between open and closed flux for our assumed ideal planetary dipole geometry (i.e., the co-latitude where the field structure no longer intersects the planetary surface) is sin⁻¹(1/$\sqrt{{R}_{\mathrm{MP}}}$) (Kivelson & Russell 1995). Discrete auroral activity occurs primarily due to energized plasma originating from closed field structure stretched out behind the planet in the stellar wind, i.e., the magnetotail. This field structure intersects the planet equatorward of the open/closed boundary co-latitude. If we assume a nominal $5^\circ$ auroral oval width beginning at the co-latitude obtained, and extending equatorward, we calculate a single-hemisphere coverage of ∼1.17 × 10¹⁷ cm² for the auroral oval under sub-Alfvénic conditions, and ∼1.30 × 10¹⁷ cm² under super-Alfvénic conditions.

Following the above, we obtain a photon flux value of ${\phi }_{5577}$ ∼2.26 (1.16) MR for the sub-(super-)Alfvénic wind conditions. This corresponds to our predicted emission power in Table 3, method 2 (quiet) of ∼0.090 (0.049) TW under steady-state sub-(super-)Alfvénic conditions. For the maximum emission during a magnetospheric substorm, we obtain values of ∼0.24 (0.14) TW for sub-(super-)Alfvénic winds.

There is another case of interactions that we should consider that involves stellar activity—flaring and coronal mass ejections (CME). During these events, stellar wind densities could increase by a factor of ∼10, velocities by a factor of ∼3, and IMF magnitude by a factor of ∼10–20 (Khodachenko et al. 2007; Gopalswamy et al. 2009). Inserting such ratios in the 3D MHD-fit predicted power in from Equation (6), we predict transient maximum 5577 Å emissions of ∼8.10 (4.40) TW for the sub-(super-)Alfvénic CME conditions. For the maximum emission during a magnetospheric substorm under CME conditions, we obtain values of ∼21.42 (12.10) TW for sub-(super-)Alfvénic winds. These transient CME conditions can have timescales of $\sim 10\mbox{--}{10}^{3}$ minutes per event, with multiple, consecutive events possible. Given that Davenport et al. (2016) report such high stellar activity for Proxima Centauri, Proxima Cen b could experience CME impacts for a large percentage of its orbital phase (e.g., Khodachenko et al. 2007).

The above results all assume a large, Earth-like planetary magnetic dipole moment for Proxima Cen b. If, in fact, the planet does not sustain a global dynamo, it will only be protected by a relatively thin spherical shell (∼1000 km) of plasma in the upper atmosphere—similar to Earth's ionosphere.

For the sub-Alfvénic case, the interaction is a unipolar interaction similar to the Jupiter–Io interaction (Zarka 2007). In this case, the power dissipated by the wind is similar to the form of Equation (4), where and K are replaced by a single parameter indicating the fraction of magnetic flux convected onto the "obstacle" (the ionosphere), and R_MP becomes the size of the "obstacle"—for an Earth-like ionosphere, $\sim 1.16$ planetary radii. While there is no dipolar focusing mechanism for the particle precipitation in this case, it is worth considering the energized particles flowing on the flux tube connecting the unmagnetized planet with the star, producing maxima on the unmagnetized body in the plane perpendicular to the IMF (see, e.g., Saur et al. 2000, 2004).

Assuming 100% of incident magnetic energy flux is convected onto the planet and ionosphere, the expected 5577 Å auroral power becomes 5.9 × 10⁻⁴ TW for the sub-Alfvénic stellar wind conditions in Table 2. This interaction is likely insignificant in the context of remote sensing.

For the super-Alfvénic flow, this could be considered as analogous to Venus' situation, which sustains no global magnetic field. In this case, the discrete aurora would obviously not be expected due to a lack of magnetic structure, though induced airglow is still a consideration. Lacking a planetary magnetosphere, the magnetic structure fails to focus precipitating particles into the upper atmosphere of such a planet, though there is still magnetic interaction at the planet. The ionosphere is a spherical, conducting shell, and so interacts with the magnetic flux from the IMF as it drapes over and around the planet. Energized particles in the impacting stellar wind magnetic flux could still dip down into the upper atmosphere, depositing sufficient energy to produce airglow—this is especially true for the strong flows from CME activity, or fast stellar wind flow.

A study of the intensity of 5577 Å oxygen emission at Venus, relative to Earth, for CME/flare events from the Sun was performed by Gray et al. (2014). The results indicated that the airglow was relatively on par with that of Earth's upper atmosphere, varying between 10 and a few hundred Rayleigh, which, if integrated over an entire hemisphere of an Earth-like planet, gives a value less than 1% of the discrete values given in Table 3. For the super-Alfvénic flow in Table 2, the number density is ∼1500 times greater than the average at Venus, and the velocity is a factor of ∼0.5 that at Venus or Earth. Given the power scales as $\rho \,{v}^{3}$, this is a factor of ∼200 greater power delivered to the planet. Assuming that airglow at an unmagnetized Proxima Cen b scales linearly with the incident power, this would give a brightness of ∼2–60 kR, which is at most a factor of ∼5 times the value for Earth using Method 1 and 2 in Table 3, or on the order of 10⁻³ TW. Given our discussion of auroral detectability below (Section 3), we do not expect that this signal could be observed with either current or upcoming missions.

The preceding estimates are mostly conservative. It is possible that all the auroral numbers reported for the sub-Alfvénic cases above could be a factor of four to five (or more) larger. We are assuming a simple dipolar interaction with the stellar wind, which is not specifically the case for a planetary dipole in the sub-Alfvénic stellar wind; these interactions are more akin to the interactions of Ganymede and Io with the corotating magnetosphere of Jupiter, with the formation of Alfvén wings. Modeling efforts by Preusse et al. (2007) showed that for a giant planet with a dipole magnetic moment, field-aligned currents (which are associated with auroral activity) are significantly stronger for planets orbiting inside the Alfvén radius of their stellar host. Our estimates, therefore, could be viewed as lower limits. It is also worth noting that Cohen et al. (2014) suggested that a transition between the sub- and super-Alfvénic conditions would likely produce enhanced magnetospheric activity and therefore could lead to a periodicity in the auroral activity depending on combined planetary orbital and stellar rotational phases.

For planets in the solar system, only Mars and Earth exhibit observed, significant 5577 Å emission for both diffuse airglow and discrete aurora. Mars does not presently have a global magnetic field, but there are crustal regions containing the remnants of previous magnetization that exist and focus particles into the upper atmosphere to produce a relatively weak (inferred ∼30 R at 5577 Å) discrete aurora that is ∼10 times the strength of the nominal airglow (e.g., Acuña et al. 2001; Bertaux et al. 2005; Lilensten et al. 2015). On Earth, the airglow and aurora are typically in the range of 0.01–1 kR and 1–1000 kR, respectively. During transient periods of minimal auroral activity and maximum airglow emission, emissions can be roughly equivalent, but the average ratio of airglow emission to auroral emission is ≤1% (e.g., Chamberlain 1961; Greer et al. 1986). Even for a constant, planet-wide 1 kR airglow on an Earth-sized planet at Proxima Cen b (${R}_{P}\sim 6371$ km), the total signal from the observer-facing hemisphere would be ∼4.54 × 10⁸W, which is ∼1% of the lowest signal from Table 3 and would not be detectable.

Nevertheless, it is important to note that the FUV flux from Proxima Centauri is nearly two orders of magnitude higher than that of the Sun (Meadows et al. 2016). Airglow stemming from recombination of photodissociated O₂ and CO₂ could thus be significantly stronger on Proxima Cen b than on Earth. Barthelemy & Cessateur (2014) stress the importance of stellar UV/FUV emissions on the production of UV and visible aurorae, and note that, e.g., Lyα flux can contribute up to 25% to the production of the O(1D) red-line. However, even if Proxima Cen b had a sustained 100 kR airglow—one hundred times the maximum Earth airglow—its emission would be comparable to the lowest estimate of auroral emission in Table 3, which is still unlikely to be detectable (see Section 3). We therefore ignore this potential contribution in the present work, noting that a detailed photochemical treatment would be required to pin down the expected airglow emission at Proxima Cen b.

In summary, we predict a steady-state auroral emission at 5577 Å from Proxima Cen b that is of the order of 100 times stronger than seen on Earth for a quiet magnetosphere, corresponding to an emitted auroral power for the O i line on the order of $\sim 0.090\,(0.049)$ TW for the sub-(super-)Alfvénic winds using method 2 (Section 2.4). We believe that this method yields more realistic results than the purely kinetic power estimate in method 1 (Section 2.3), due to the inclusion of magnetic interactions in method 2—though the magnitudes are similar to within a factor of two. Assuming Proxima Cen b is an Earth-like terrestrial planet, our maximum transient power estimate for the 5577 Å line for CME conditions that drive a magnetospheric substorm is ∼21.42 TW, or ∼30,000 times stronger than on Earth under nominal solar wind conditions. The actual values for Proxima Cen b will naturally change based on planetary parameters (e.g., magnetic dipole moment, magnetospheric particle energy distributions, substorm onset, atmospheric Joule heating) and stellar activity. By our analysis, a $\sim {10}^{3}$ (or higher) enhancement compared to Earth as suggested by O'Malley-James & Kaltenegger (2016) is only possible due to one or more of the following: transient magnetospheric conditions driven by either CME or substorm activity, a magnetic dipole significantly stronger than Earth's, or higher stellar mass-loss than predicted (Wood et al. 2005; Cohen et al. 2014).

To estimate the signal-to-noise as a function of spectral resolution, we need to estimate the auroral spectral line width. The O i 5577 Å green line has no hyperfine structure and because it is a forbidden line, it has negligible ($\lesssim {10}^{-15}\,{\mathring{\rm{A}}}$) natural width (Hunten et al. 1967). Spectroscopic observations of the O i airglow by Keck/HIRES (Slanger et al. 2001) and by HARPS (Anglada-Escudé et al. 2016, see Section 4) are unresolved, revealing the resolution element width of the instrument used for the observation at the wavelength of the line ($\sim 0.1$ Å Keck/HIRES; $\sim 0.05$ Å HARPS) rather than the full width at half maximum (FWHM) of the line.

To determine the width of the line, we examine several line broadening mechanisms that play a key role in terrestrial atmospheres. The planet's rotation will broaden the O i line, but calculations by Barnes et al. (2016) and Ribas et al. (2016) show that Proxima Cen b is likely tidally locked with a rotation period of 11.2 days, resulting in negligible instantaneous rotational broadening (FWHM = 0.002 Å). Pressure broadening can also be safely neglected since O i auroral emission occurs in terrestrial atmospheres at an elevation of $\sim 100$ km where the atmosphere is thin (Slanger et al. 2001). Similarly, broadening due to atmospheric turbulence can also safely be neglected due to the stratospheric origin of the line. Thermal Doppler broadening should therefore be the dominant line broadening mechanism, resulting in a Gaussian line profile. For the 5577 Å O i line, Doppler broadening gives the following scaling relation:

$$\begin{eqnarray}&&\mathrm{FWHM}=2{\rm{\Delta }}\lambda =0.014{\left(\displaystyle \frac{T}{200{\rm{K}}}\right)}^{1/2}\ {\mathring{\rm{A}}} ,\end{eqnarray} \tag{ 8 }$$

where T is the temperature of the emitting layer, for which we adopt the value of 200 K (see Slanger et al. 2001). An FWHM of 0.014 Å is in good agreement with the Fabry–Perot interferometric line width measurements of Wark (1960).

Given the relatively short period of Proxima Cen b and the long exposure times expected for high-resolution spectroscopy, we must also consider the possibility of broadening due to the orbital motion of the planet over the course of an observation. One could take a series of shorter exposures, but this strategy will introduce significant instrumental noise, which is likely to overwhelm any planetary signals. In Figure 2, we plot the orbital broadening of the 5577 Å line as a function of the exposure time, calculated from the maximum change in the RV of the planet over the course of the observation and assuming an inclination of 90°. The effect is strongest at full and new phases (dotted line), where the time derivative of the RV is highest, and weakest at quadrature (solid line), where the derivative is smallest. Two intermediate phases are also shown. The FWHM given by Equation (8) is indicated by a horizontal red line; orbital broadening becomes significant as the curves approach this line. In general, observations made at quadrature with exposure times of up to ∼6 hr cause negligible broadening. At all other phases, however, broadening becomes significant in a matter of one or a few hours. At full and new phase, the line width doubles after an exposure of only 40 minutes. However, at these phases the RV of the planet relative to the star is zero, and as we argue in Section 4 below, disentangling stellar and planetary emission becomes difficult. In the discussion that follows, we therefore focus on observations made close to quadrature.

Zoom In Zoom Out Reset image size

Figure 3 shows a high-resolution model spectrum of Proxima Cen b at quadrature, illustrating an auroral emission feature that could be expected from the planet. We injected a Gaussian line at 5577 Å with $\mathrm{FWHM}=0.014$ Å, normalized to a steady-state Proxima Cen b O i auroral power of ${L}_{{\rm{O}}{\rm{I}}}=0.1$ TW. This O i auroral power yields an equivalent width of $\sim 3.63({L}_{{\rm{O}}{\rm{I}}}/1\mathrm{TW})$ Å relative to our model of the reflected planetary spectrum at quadrature.

Zoom In Zoom Out Reset image size

To unambiguously detect a narrow emission feature, such as the example shown in Figure 3, the telescope resolving power ($R\equiv \lambda /{\rm{\Delta }}\lambda$) needs to be taken into account. The typical resolving power being considered for future space-based coronagraph mission concepts is $R\approx 100$, which is appropriate for the detection of molecular absorption bands in the optical and NIR given the relatively low planet–star contrast ratio (Robinson et al. 2016). However, at 5577 Å an R = 100 spectrograph has a spectral element width of ${\rm{\Delta }}\lambda \approx 56$Å, over $\sim {10}^{3}$ times broader than the O i green line width. Future space-based high-contrast exoplanet imaging missions would need to fly with higher resolution spectrographs to detect the O i 5577 Å line.

Figure 4 shows planet–star contrast ratios in a spectral element centered on the 5577 Å O i auroral line as a function of spectrograph resolving power and auroral power, assuming an FWHM of 0.014 Å (i.e., negligible orbital broadening). The FWHM of the auroral line and equivalent width (${W}_{\lambda }$) as a function of auroral power are represented as "resolving powers," where ${R}_{\mathrm{FWHM}}={\lambda }_{{\rm{O}}{\rm{I}}}/\mathrm{FWHM}$ and ${R}_{W}={\lambda }_{{\rm{O}}{\rm{I}}}/{W}_{\lambda }$, respectively. In Figure 4, the dashed-white line gives the resolving power such that the spectral element width matches the equivalent width of the line at a given auroral power. The dashed-orange line gives the resolving power such that the spectral element width matches the FWHM of the line. That is, a fixed FWHM = 0.014 Å yields ${R}_{\mathrm{FWHM}}=4\times {10}^{5}$. Optimal observations occur when the planet–star contrast (indicated by the contours) is highest. An increase in the contrast of the emission line is only achieved when the width of a spectral element is smaller than the equivalent width of the line. For resolving powers greater than R_FWHM, multiple spectral elements are needed to span the emission line, which may introduce additional unwanted read noise and dark current. Therefore, observations should be made in the wedge between the FWHM resolving power and the equivalent width resolving power. Our predicted steady-state auroral emission ($\sim 0.1$ TW) requires that spectrographs achieve $R\gtrsim {10}^{5}$.

Zoom In Zoom Out Reset image size

Using the auroral power estimates from Section 2, we explore the feasibility of detecting the 5577 Å O i auroral emission line with five different ground-based telescope configurations: the 3.6 m High Accuracy Radial velocity Planet Searcher (HARPS), the 8.2 m Very Large Telescope (VLT) with and without a coronagraph, and a Thirty Meter Telescope (TMT) concept with and without a coronagraph (Johns et al. 2012; Udry et al. 2014; Skidmore et al. 2015). We also model the detection using two future space-based coronagraph concepts: the 16 m Large UV/Optical/IR Surveyor (LUVOIR; Kouveliotou et al. 2014; Dalcanton et al. 2015) and the 6.5 m Habitable Exoplanet Imaging Mission (HabEx; Mennesson et al. 2016).

High spectral resolution coronagraphy with the VLT will require an update to the SPHERE high-contrast imager and a coupling with the ESPRESSO spectrograph, as described in Lovis et al. (2016). Note that given the VLT's 8.2 m diameter, the SPHERE coronagraph must achieve an inner working angle no more than ${\theta }_{\mathrm{IWA}}=2.7\lambda /D$ to observe at wavelengths as long as 5577 Å given the maximal planet–star angular separation of 37 mas for Proxima Cen b. Our HARPS, TMT, LUVOIR, and HabEx telescope models use $R={\rm{115,000}}$, while for VLT we use $R={\rm{120,000}}$. All models assume a total telescope and instrument throughput of 5% and a quantum efficiency of 90%. Coronagraph noise estimates use the model presented in Robinson et al. (2016) with updated parameters from Meadows et al. (2016), and consider noise due to speckles, dark counts, read noise, telescope thermal emission, and zodi and exozodi light. Ground-based coronagraphy assumes a conservative design contrast of 10⁻⁵ (Dou et al. 2010; Guyon et al. 2012), while space-based assumes 10⁻¹⁰ (Meadows et al. 2016) unless stated otherwise. Typically, telescope detectors have a maximum exposure time to mitigate the damaging effect of cosmic ray strikes (see Robinson et al. 2016). Therefore, integration times longer than one hour require multiple readouts, introducing more detector noise. Non-coronagraph telescope calculations assume only stellar noise at the photon limit; their values are therefore lower limits, and may increase significantly due to stellar activity (see Section 4). To prevent significant orbital broadening, we assume that observations are made for one hour at a time at or close to quadrature; longer exposure times are achieved by stacking multiple observations. For exposure times much longer than an hour, stacking will appreciably increase the read noise and dark current for coronagraph observations, where the star is nulled, but not for the non-coronagraph observations, where the stellar photons dominate the noise budget.

Our integration time calculations follow those described in Robinson et al. (2016). For the stellar spectrum, we adopt the steady-state Proxima Centauri spectrum of Meadows et al. (2016) and neglect the impact of flares on the stellar continuum. We assume that the quoted auroral power emitted via the 5577 Å O i line is constant throughout the entire observation.

For observations without a coronagraph, both the stellar flux and reflected stellar flux define the continuum from which we wish to resolve the auroral emission feature. Observations with a coronagraph need only resolve the auroral emission above the coronagraph noise and reflected stellar continuum. With these considerations in mind, we simulate the net planetary emission as a combination of reflected stellar continuum and auroral emission. We compute the flux from the reflected stellar continuum by assuming that the planet is a Lambertian scatterer at quadrature with a planetary geometric albedo of 0.3 and a planetary radius of $1.07{R}_{\oplus }$ following Barnes et al. (2016). We then inject the expected flux from the auroral line at its wavelength. We integrate over all spectral elements that contain the auroral line flux, taking the auroral photon count rate as our signal and all other sources as noise as in Robinson et al. (2016). For the oxygen 5577 Å line width of $\sim 0.014$Å (Section 3.1) and our nominal resolving power, this corresponds to one spectral element.

Table 4 shows the integration times required to achieve a signal-to-noise of six on the 5577 Å O i auroral emission line above the stellar and reflected planetary continuum as a function of auroral power. We simulated contrast ratios and integration times for emitted auroral powers at the O i 5577 Å line ranging from ${10}^{-3}\,\mathrm{to}\,{10}^{3}$ TW to bracket all potential auroral fluxes. The 10⁻³ TW lower limit corresponds to a strong 5577 Å emission from Earth (Earth total electromagnetic auroral power is of the order of 10⁻² TW with 5577 Å typically ∼2% of this value). The upper limit of 10³ TW is an extreme case that is an order of magnitude stronger than the largest value predicted in Section 2. Values in between correspond to the different cases considered in Table 3, which depend on the planetary dipole moment, magnetospheric substorm activity, and whether CME conditions are present. For reference, the estimated steady-state Proxima Cen b value calculated in Section 2 is $\sim 0.1$ TW.

The weak 10⁻³ TW aurora is indistinguishable from the purely reflecting planet–star contrast near the 5577 Å O i auroral emission line (Meadows et al. 2016; Turbet et al. 2016) and effectively demonstrates why high-resolution spectroscopy is not typically considered for high-contrast imaging. For an Earth-like planet, the auroral power estimates from Section 2 ($\sim 0.1$ TW) make detecting the O i emission line infeasible with current instruments, even though the contrast ratio at the line is relatively strong ($\sim 8\times {10}^{-7}$). Although unlikely, if the auroral power was much higher ($\sim {10}^{3}$ TW) and sustained over the period of the observation, detection of O i emission could be possible with current instruments in tens of hours. Realistically, however, these estimates suggest that current instruments are likely not capable of detecting an O i aurora on Proxima Cen b.

For a SPHERE-ESPRESSO coupling (Lovis et al. 2016), the integration times required to detect an O i auroral line are slightly more favorable over a wide range of possible auroral powers, but still prohibitively long under most plausible circumstances. If Proxima Cen b has a Neptune-strength magnetic dipole moment, and observations were made during substorm conditions (when the power in the O i 5577 Å line reaches ∼1 TW with a contrast ratio of $7\times {10}^{-6}$), a coronagraph-equipped VLT would have to integrate for $\sim 400$ hr. However, if observations coincided with periods of more vigorous stellar activity such as during a concurrent CME and substorm, the auroral output could reach $\sim 100$ TW and contrast ratios of $7\times {10}^{-4}$. An upgraded SPHERE (denoted by VLT+C in Table 4) may be able to detect this signal in under an hour. Since CMEs and fast stellar wind streams can have timescales of $\sim 10$ hr, and substorms up to several days (Gonzalez et al. 1994, 1999), the high level of transient activity may be observable. Under near constant CME activity, storm conditions could potentially last for weeks or longer, improving the chances of detecting auroral emission (Khodachenko et al. 2007).

Even future observations with TMT, HabEx, and LUVOIR outfitted with instruments optimized for high-resolution, high-contrast coronagraphy will be unable to detect a steady-state 0.1 TW O i aurora on Proxima Cen b. However, a coronagraph-equipped TMT could detect a substorm strength aurora of $\sim 10$ TW in about 10 hr, while LUVOIR could make the predicted substorm auroral observation in about 30 hr. Only auroral powers $\gtrsim 10$ TW would be detectable with HabEx. Auroral powers of the order of 100 TW arising from a concurrent CME and substorm could be observed by the TMT, HabEx, and LUVOIR in well under an hour.

Finally, we consider how improvements in ground-based instrumentation might expand the ability to detect exo-aurorae. In the top panel of Figure 5, we model a coronagraph-equipped TMT concept that achieves a design contrast of $C={10}^{-7}$ and has negligible instrumental noise (e.g., no dark current and read noise). Low-resolution observations with a resolving power smaller than the equivalent width resolving power, $R\lt \lambda /{W}_{\lambda }$, yield longer integration times at fixed auroral power as the auroral signal is diluted by additional stellar continuum photons from larger spectral elements, which increases the noise. For high resolutions that exceed the equivalent width resolving power, $R\gt \lambda /{W}_{\lambda }$, the auroral signal dominates the planetary continuum as the spectral element more tightly bounds the narrow emission feature, yielding little additional improvement in integration times.

Zoom In Zoom Out Reset image size

In the bottom panel of Figure 5, we vary coronagraph design contrasts for observations with and without instrumental noise. We find that a TMT with coronagraphic starlight suppression, negligible instrumental noise, a design contrast of $C={10}^{-7}$ and $R\gt {10}^{5}$ allows for a detection of our predicted steady-state O i auroral emission (auroral power of $\sim 0.1$ TW) over about 40 hr (see also Table 4). The discontinuities that occur at high resolving powers are due to the need for additional spectral elements to span the width of the O i auroral line.

Despite the likely increased strength of aurorae on Proxima Cen b compared to Earth, observing a steady-state 0.1 TW aurora requires sufficiently long integration times, which is not currently feasible, nor will it be feasible with the next generation of instruments, unless ideal instrumental performance is achieved. O i auroral detection may only be possible if observations coincide with magnetospheric substorms or periods of vigorous stellar activity, such as CMEs, which can induce much stronger aurorae ranging from 1–10 TW (and up to ∼100 TW if Proxima Cen b has a stronger magnetic dipole than Earth). These transient events are frequent on Proxima Centauri (Davenport et al. 2016) and may persist on timescales comparable to the integration times needed to detect strong aurorae.