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J. D. Nichols; Magnetosphere–ionosphere coupling at Jupiter-like exoplanets with internal plasma sources: implications for detectability of auroral radio emissions, Monthly Notices of the Royal Astronomical Society, Volume 414, Issue 3, 1 July 2011, Pages 2125–2138, https://doi.org/10.1111/j.1365-2966.2011.18528.x
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Abstract
In this paper we provide the first consideration of magnetosphere–ionosphere coupling at Jupiter-like exoplanets with internal plasma sources such as volcanic moons. We estimate the radio power emitted by such systems under the condition of near-rigid corotation throughout the closed magnetosphere, in order to examine the behaviour of the best candidates for detection with next generation radio telescopes. We thus estimate for different stellar X-ray–UV (XUV) luminosity cases the orbital distances within which the ionospheric Pedersen conductance would be high enough to maintain near-rigid corotation, and we then consider the magnitudes of the large-scale magnetosphere-ionosphere currents flowing within the systems, and the resulting radio powers, at such distances. We also examine the effects of two key system parameters, i.e. the planetary angular velocity and the plasma mass outflow rate from sources internal to the magnetosphere. In all XUV luminosity cases studied, a significant number of parameter combinations within an order of magnitude of the jovian values are capable of producing emissions observable beyond 1 pc, in most cases requiring exoplanets orbiting at distances between ∼1 and 50 au, and for the higher XUV luminosity cases these observable distances can reach beyond ∼50 pc for massive, rapidly rotating planets. The implication of these results is that the best candidates for detection of such internally generated radio emissions are rapidly rotating Jupiter-like exoplanets orbiting stars with high XUV luminosity at orbital distances beyond ∼1 au, and searching for such emissions may offer a new method of detection of more distant-orbiting exoplanets.
1 INTRODUCTION
In recent years hundreds of exoplanets have been discovered, many of which (∼21 per cent) have mass greater than or equal to that of Jupiter and orbital semimajor axes of <0.1 au (where 1 au ≈ 1.5 × 1011m), although a significant fraction (∼61 per cent) of planets with Jupiter’s mass or greater have been observed with semimajor axes ≥1 au (see, e.g., the catalogue at exoplanet.eu). The possibility of detection of the auroral radio emissions of ‘hot Jupiter’-like exoplanets close to their parent star has been considered by a number of authors (e.g. Farrell, Desch & Zarka 1999; Zarka et al. 2001, 2007; Farrell et al. 2004; Griebmeier et al. 2004, 2005; Lazio et al. 2004; Stevens 2005; Griebmeier, Zarka & Spreeuw 2007; Jardine & Cameron 2008; Fares et al. 2010; Reiners & Christensen 2010). This interest has been sparked in part by the imminent commencement of high-sensitivity radio observations by next generation radio telescopes such as the Low-Frequency Array (LOFAR), which will have a detection threshold of 1 mJy (where 1 Jansky = 10−26 W m−2 Hz−1) (Farrell et al. 2004; Grießmeier et al. 2007). Such previous consideration of hot Jupiter-like exoplanets has assumed that the auroral radio emission would be caused by a star–planet interaction reminiscent of either the solar wind–Earth interaction or the Io–Jupiter interaction. The former is mediated primarily via reconnection of the planetary and interplanetary magnetic fields at the dayside magnetopause (Dungey 1961). This drives plasma flows within the magnetosphere that generate electric currents flowing between the magnetosphere and the resistive ionosphere, the upward magnetic field-aligned component of which, associated with downward-precipitating electrons, produces auroral and radio emissions. The latter interaction is thought to be mainly associated with the generation of Alfvén waves in the vicinity of the moon Io, caused by its motion through the rapidly rotating planetary magnetic field and plasma (Goertz 1980; Neubauer 1980; Crary & Bagenal 1997). Consideration of such processes has led to the extrapolation of a ‘Radiometric Bode’s Law’ relating incident magnetic power to output radio power for the case of hot Jupiter-like exoplanets orbiting extremely close (at typically ∼10 stellar radii) of their parent stars. It has been concluded that such interaction may generate emissions at or above the LOFAR detection threshold (Farrell et al. 2004; Grießmeier et al. 2007).
However, despite the importance placed by previous authors on stellar wind planet and Io-Jupiter type interactions, significant components of Jupiter’s radio emissions, i.e. the b-KOM, HOM and non-Io-DAM emissions (Zarka 1998), are thought to be generated by the large-scale magnetosphere–ionosphere (M–I) coupling current system associated with the breakdown of corotation of iogenic plasma in Jupiter’s middle magnetosphere, illustrated by Fig. 1 (Hill 1979, 2001; Pontius 1997; Cowley & Bunce 2001; Nichols & Cowley 2003, 2004, 2005). This process generates intense field-aligned electron beams which drive the brightest and most significant of Jupiter’s ultraviolet (UV) auroral emission, i.e. the main auroral oval (Grodent et al. 2003; Clarke et al. 2004; Nichols et al. 2009b), and, coupled with particle mirroring and the absorption of particles in the loss cone, excite the cyclotron maser instability in the high-latitude low-β plasma, which gives rise to the above radio emissions. Observationally, the UV aurora and radio emissions of Jupiter and Saturn have been shown by a number of studies to be closely associated with one another (e.g. Gurnett et al. 2002; Kurth et al. 2005; Pryor et al. 2005; Clarke et al. 2009; Lamy et al. 2009; Nichols et al. 2010a; Nichols, Cowley & Lamy 2010b). Io orbits deep within Jupiter’s magnetosphere at ∼5.9RJ (where RJ represents the equatorial radius of Jupiter equal to 71 373 km), and its volcanoes liberate sulphur and oxygen atoms into a torus surrounding the moon’s orbit at the rate of ∼1000 kg s−1 (e.g. Hill, Dessler & Goertz 1983; Vasyliūnas 1983; Khurana & Kivelson 1993; Bagenal 1997; Dols, Delamere & Bagenal 2008). These atoms are ionized by electron impact ionization and thus become sensitive to the rotating planetary magnetic field, such that the newly created plasma is picked up to corotation velocity. The picked-up plasma is centrifugally unstable and diffuses radially away from the planet, probably via flux-tube interchange motions (Siscoe & Summers 1981; Pontius & Hill 1982; Kivelson et al. 1997; Thorne et al. 1997; Bespalov et al. 2006), before being lost down the dusk flank of the magnetotail via the pinching off of plasmoids (e.g. Vasyliūnas 1983; Woch, Krupp & Lagg 2002; Vogt et al. 2010).
Sketch of a meridian cross-section through a Jupiter-like exoplanet’s inner and middle magnetosphere, showing the principal physical features involved. The arrowed solid lines indicate magnetic field lines, the arrowed dashed lines the magnetosphere–ionosphere coupling current system, and the dotted region the rotating disc of outflowing plasma. After Cowley & Bunce (2001).
Sketch of a meridian cross-section through a Jupiter-like exoplanet’s inner and middle magnetosphere, showing the principal physical features involved. The arrowed solid lines indicate magnetic field lines, the arrowed dashed lines the magnetosphere–ionosphere coupling current system, and the dotted region the rotating disc of outflowing plasma. After Cowley & Bunce (2001).
As the plasma diffuses radially outward, its angular velocity drops (inversely with the square of the distance if no torques act) due to conservation of angular momentum, such that a radial gradient of angular velocity is set up in the equatorial plasma. This angular velocity gradient, when mapped along the magnetic field to the ionosphere, causes an equatorward-directed (for Jupiter’s magnetic field polarity) ionospheric Pedersen current to flow, the J×B force of which opposes the drag of the neutral atmosphere on the subrotating plasma. Angular momentum is transferred between the ionosphere and the equatorial plasma by the sweep-back of magnetic field lines into a lagging configuration, such that the ionospheric Pedersen current is balanced in the equatorial plane by an outward-directed (again, for Jupiter’s magnetic field polarity) radial current, the J×B force of which tends to return the equatorial plasma back to corotation with the planet. Current continuity between these two field-perpendicular currents is maintained via field-aligned (Birkeland) currents, the inner upward component of which is thought to generate Jupiter’s main auroral oval emission (Cowley & Bunce 2001; Hill 2001; Southwood & Kivelson 2001). The current system was studied in detail theoretically by Nichols & Cowley (2003), who considered the effect of two poorly constrained but important system parameters, the effective ionospheric Pedersen conductance Σ*P, and the plasma mass outflow rate
, and they derived analytic approximations appropriate for small and large radial distances, the former of which will be instrumental in the present work. Nichols & Cowley (2004) went on to examine the effect on the current system of modulation of the ionospheric Pedersen conductance due to auroral electron precipitation, and Nichols & Cowley (2005) and Ray et al. (2010) have studied the effect of field-aligned voltages. Cowley, Nichols & Andrews (2007) considered the effect on the current system of solar wind-induced expansions and contractions of Jupiter’s magnetosphere, and recently the modulation of the current system by diurnal variation of the ionospheric Pedersen conductance caused by solar illumination has been considered by Tao, Fujiwara & Kasaba (2010).
Both giant planets in our Solar system which have been extensively studied by in-situ spacecraft, i.e. Jupiter and Saturn, possess moons which actively outgas into the near-planetary space. At Jupiter this is Io, which emits material at the rate of ∼1000 kg s−1 as discussed above, whilst at Saturn this is Enceladus, whose cryo-volcanoes emit water group ions at estimated rates ranging from a few to a few hundred kg s−1 (e.g. Pontius & Hill 2006; Khurana et al. 2007). It is thus reasonable to assume that such active moons may be relatively prevalent amongst Jupiter-like exoplanets, and the resulting implications for detectability of auroral radio emissions is therefore considered here. We begin from the assumption that Jupiter-like exoplanets are strongly illuminated by their parent star, such that the ionospheric Pedersen conductance is high enough to maintain near-rigid corotation throughout the magnetosphere, a condition which, as reviewed below, maximizes the field-aligned current density for a given magnetosphere. We then compute the currents, and thus the resulting radio power output, for varying configurations of Jupiter-like exoplanets. Note that such strong illumination could be associated with close ‘hot Jupiters’, but equally applies to more distant planets orbiting active stars. The parameters examined are the plasma mass outflow rate, planetary orbital distance and rotation rate. We show that, for planets with host stars more active than the Sun, only relatively modest modifications from the jovian system parameters are required to produce potentially detectable configurations, and by doing so we open up the catalogue of potential candidates for detection by radio telescopes such as LOFAR to a class of planet previously overlooked.
2 THEORETICAL BACKGROUND
In this section we discuss the equations which describe the system, along with the approximations appropriate to the case of Jupiter-like exoplanets. The analysis follows that which has been described previously by Hill (1979), Pontius (1997), Cowley & Bunce (2001) and Nichols & Cowley (2003), such that here we provide only a brief outline.
2.1 ‘Baseline’ magnetic field model
is the incomplete gamma function. This flux function is shown by the solid red line in Fig. 2(b), and is typically factors of ∼5 greater than the corresponding dipole flux function shown by the long-dashed line.Plot showing (a) the magnitude of the north–south magnetic field strength threading the equatorial plane |Bze | in nT, (b) the associated flux function Fe in nT R2p and (c) the ionospheric colatitude θi to which the field lines threading the equatorial plane at radial distance ρe map. The red solid lines show the ‘baseline’ model values given by equations (4) and (5), while the black long-dashed lines show the planetary dipole values for comparison. The other solid lines show the values modified by using equations (8) and (9) for assumed magnetopause stand-off distances (Rmp/Rp) ≤ 85, and by simply extending equations (4) and (5) for (Rmp/Rp) > 85. The colours blue, green, red and black correspond to (Rmp/Rp) values of 20, 50, 85 and ≥150, respectively. The horizontal dotted lines in panels (b) and (c) show the value of F∞≈ 2.841 × 104 nT R2p, and the corresponding limiting ionospheric colatitude of the outer boundary of the model at ∼15°.
Plot showing (a) the magnitude of the north–south magnetic field strength threading the equatorial plane |Bze | in nT, (b) the associated flux function Fe in nT R2p and (c) the ionospheric colatitude θi to which the field lines threading the equatorial plane at radial distance ρe map. The red solid lines show the ‘baseline’ model values given by equations (4) and (5), while the black long-dashed lines show the planetary dipole values for comparison. The other solid lines show the values modified by using equations (8) and (9) for assumed magnetopause stand-off distances (Rmp/Rp) ≤ 85, and by simply extending equations (4) and (5) for (Rmp/Rp) > 85. The colours blue, green, red and black correspond to (Rmp/Rp) values of 20, 50, 85 and ≥150, respectively. The horizontal dotted lines in panels (b) and (c) show the value of F∞≈ 2.841 × 104 nT R2p, and the corresponding limiting ionospheric colatitude of the outer boundary of the model at ∼15°.
2.2 Modification of the field structure by internal field strength and subsolar magnetopause stand-off distance
, where n = 1 (Busse 1976; Sano 1993), 3/4 (Mizutani, Yamamoto & Fujimura 1992), or 1/2 (Stevenson 1983; Mizutani et al. 1992). We thus take n = 3/4 as representative of these choices, such that and where ΩJ is Jupiter’s planetary rotation period given by ΩJ = 1.76 × 10−4 rad s−1, and B*ze and F*e are given by equations (4) and (5), respectively.Plot showing (left axis) subsolar magnetopause distance Rmp and (right axis) stellar wind dynamic pressure versus orbital radius Rorb computed using equation (10) and the Huddleston et al. (1998) empirical relation for Jupiter given by equation (13) (solid line) and the vacuum dipole relation given by equation (12) (dashed line), both for jovian values of the planetary magnetic moment. Also shown by the vertical dotted line at 5.2 au is the range of magnetopause distances observed for Jupiter, thus the range over which the Huddleston et al. (1998) relation was defined, i.e. 45–100 RJ, and the horizontal dotted lines show the orbital radii to which this range corresponds in the present study, i.e. ∼1.7–10 au.
Plot showing (left axis) subsolar magnetopause distance Rmp and (right axis) stellar wind dynamic pressure versus orbital radius Rorb computed using equation (10) and the Huddleston et al. (1998) empirical relation for Jupiter given by equation (13) (solid line) and the vacuum dipole relation given by equation (12) (dashed line), both for jovian values of the planetary magnetic moment. Also shown by the vertical dotted line at 5.2 au is the range of magnetopause distances observed for Jupiter, thus the range over which the Huddleston et al. (1998) relation was defined, i.e. 45–100 RJ, and the horizontal dotted lines show the orbital radii to which this range corresponds in the present study, i.e. ∼1.7–10 au.
2.3 Steady-state plasma angular velocity and current system equations
2.4 High-conductance approximation and conductance estimations
, as shown in Fig. 4(a) for roughly jovian values of the ionospheric Pedersen conductance Σ*P and equatorial plasma mass outflow rate
. Specifically, the typical scale over which the plasma falls from rigid corotation is called the ‘Hill distance’ρH after T. W. Hill, who first derived this scale length for a dipole magnetic field (not to be confused with the radius of the Hill sphere within which a body’s gravitational field is dominant). The scale length was modified by Nichols & Cowley (2003) for the case of a magnetic field that varies as a power law with arbitrary exponent m, and for which the field lines map from the equatorial plane to a narrow band in the ionosphere. This is appropriate for the stretched current sheet magnetic field of Jupiter’s middle magnetosphere, as can be appreciated from the ionospheric mapping shown in Fig. 2(c), in which between 20 and 150RJ the current sheet field models indicated by the solid lines map between ∼15° and 17°, whereas the dipole field indicated by the dashed line maps to a much broader region between ∼5° and 13°. The current sheet scale length is given by where F○ is the value of the flux function at the narrow ionospheric band, taken here to be equal to F∞, shown by the dotted horizontal line in Fig. 2. Thus, for a given equatorial distance, the plasma angular velocity increases for increasing values of
. Nichols & Cowley (2003) therefore derived analytic approximations to the solution appropriate to both the inner region, where the plasma near-rigidly corotates, and the outer region, where the plasma can be considered to be either free of ionospheric torque or stagnant. The inner region approximation for the equatorial plasma angular velocity, which Nichols & Cowley (2003) suggested is valid out to a distance ρS lim of is the leading term in the series solution of equation (15), taking
as the formal expansion parameter and (ω/Ωp) = 1 at ρe = 0 (see Nichols & Cowley 2003 for further details), such that as shown by the dashed lines in Fig. 4(a). Substitution of equation (24) into equations (19) and (21) thus yields and for the total radial current and ionospheric field-aligned current density, respectively. These limiting currents, which are dependent only on
and not Σ*P, are those required to maintain near-rigid corotation throughout the magnetosphere, and can thus be viewed alternatively as a high-Σ*P approximation for a given value of
.Plot showing profiles of the current system parameters using the full solution of equation (15) (solid lines) and the high-conductance approximation given by equation (24) (dashed lines). Parameters shown are (a) the equatorial plasma angular velocity, (b) the azimuthally integrated equatorial radial current and (c) the field-aligned current density. Three results are shown for
and 2 × 10−4 mho s kg−1, with
set to 1000 kg s−1 in panels b and c. Note only one dashed line is shown in panels b and c since the high-conductance approximations (given by equations (25) and (26) for the radial and field-aligned currents, respectively) are independent of Σ*P. The tick marks show the limit of validity of the approximation as suggested by Nichols & Cowley (2003).
Plot showing profiles of the current system parameters using the full solution of equation (15) (solid lines) and the high-conductance approximation given by equation (24) (dashed lines). Parameters shown are (a) the equatorial plasma angular velocity, (b) the azimuthally integrated equatorial radial current and (c) the field-aligned current density. Three results are shown for
and 2 × 10−4 mho s kg−1, with
set to 1000 kg s−1 in panels b and c. Note only one dashed line is shown in panels b and c since the high-conductance approximations (given by equations (25) and (26) for the radial and field-aligned currents, respectively) are independent of Σ*P. The tick marks show the limit of validity of the approximation as suggested by Nichols & Cowley (2003).
= 1000 kg s−1 and Σ*P = 0.2, 1 and 5 mho. It is apparent that the current magnitudes increase with increasing Σ*P, asymptoting towards the limiting currents shown by the dashed lines. Thus, the high-conductance approximation represents the condition under which maximum currents, and therefore maximum output radio power, are generated. The minimum conductance required such that the high-conductance approximation is valid throughout the magnetosphere is found by setting Rmp=ρS lim from equation (23), such that which, upon substitution of equation (22) and rearranging for Σ*P, yields Since we are interested in current systems powerful enough to be detected across interstellar distances, as possibly possessed by Jupiter-like exoplanets strongly irradiated by their parent stars, such a high-conductance is clearly appropriate and will be used as the basis for the results which follow.2.5 Parallel voltage, precipitating electron energy flux and radio power output
Plot showing the field aligned current j∥i normalized to the limiting current j∥i○ produced by field-aligned voltage Φ normalized to the unaccelerated population thermal energy Wth, for the three current–voltage relations given by equations (33) for RB = 16, appropriate for an accelerator height of ∼2.5 RJ (dashed line), 34 (solid line) and 35 (dotted line). ‘Rel.’ means ‘relativistic’, ‘Non-rel.’ means ‘non-relativistic’ and RB is the ratio of the ionospheric magnetic field strength to the field strength at the top of the voltage drop.
Plot showing the field aligned current j∥i normalized to the limiting current j∥i○ produced by field-aligned voltage Φ normalized to the unaccelerated population thermal energy Wth, for the three current–voltage relations given by equations (33) for RB = 16, appropriate for an accelerator height of ∼2.5 RJ (dashed line), 34 (solid line) and 35 (dotted line). ‘Rel.’ means ‘relativistic’, ‘Non-rel.’ means ‘non-relativistic’ and RB is the ratio of the ionospheric magnetic field strength to the field strength at the top of the voltage drop.
3 RESULTS
We now consider the application of the analysis presented in Section 2 to various configurations of exoplanets. We first show in Fig. 7 the effect on the current system of taking different values of Rmp, whilst maintaining jovian values for the planetary angular velocity Ωp=ΩJ (and thus Beq=BJ) and mass plasma outflow rate
, thus simulating transplanting the jovian system to different orbital distances, assuming that the XUV flux is at all distances sufficiently high such that high-conductance approximation holds everywhere. It is apparent from Fig. 7 that for values of Rmp less than the reference value of 85RJ the current amplitudes are, as expected from e.g. equation (25), reduced owing to the elevated magnetic field strength |Bze |, which we recall is increased via equation (8) for a smaller magnetosphere due to flux conservation. For these model parameters, very compressed magnetospheres with Rmp smaller than ∼20RJ, are such that the field-aligned current density is everywhere less than j∥i○, hence no field-aligned voltages are required and there is no bright discrete auroral emission. Hence, even if we neglect tidal locking which would in reality act to prevent the fast rotation of such planets (discussed further below), very close-orbiting (Rorb≲ 0.25 au) ‘hot Jupiter’ systems jovian-like in all respects apart from the orbital distance would exhibit no main auroral oval auroras. On the other hand, similar magnetospheres larger than 85RJ exhibit common current profiles, extending to further distances and thus modestly further towards the pole, depending on the size of the magnetosphere. The monotonically decreasing field strength with increasing equatorial radius results in an associated increasing of the current magnitudes towards the pole, and thus precipitating electron energy flux, towards the outer boundary of the model. Increasing the size of the magnetosphere thus increases the emitted radio power, as shown in Fig. 7(d).
Plot showing the current system parameters versus ionospheric colatitude, for magnetopause stand-off distances of Rmp = 20, 50, 85 and 200Rp (blue, green, red, and black lines, respectively), and with Ωp=ΩJ and
. The labels in panel (a) correspond to the values of Rorb which result in magnetospheres of these sizes, according to equations (10) and (13). Parameters shown are (a) the azimuthally integrated ionospheric Pedersen current IP in A as given by equation (25), (b) the field-aligned current density at the top of the ionosphere j∥i in A m−2 given by equation (26), (c) the minimum field-aligned voltage Φmin in V required to drive the current in panel (b) as given by equation (34) and (d) the precipitating electron energy flux Ef in W m−2 as given by equation (36). The labels in panel (d) show the radio power obtained by integrating the energy fluxes using equations (37) and (38). Note there are no blue lines in panels (c) and (d) since for this case the current density in panel (b) is everywhere below the minimum current j∥i○ for which field-aligned voltages are required [shown by the horizontal dotted line in panel (b)].
Plot showing the current system parameters versus ionospheric colatitude, for magnetopause stand-off distances of Rmp = 20, 50, 85 and 200Rp (blue, green, red, and black lines, respectively), and with Ωp=ΩJ and
. The labels in panel (a) correspond to the values of Rorb which result in magnetospheres of these sizes, according to equations (10) and (13). Parameters shown are (a) the azimuthally integrated ionospheric Pedersen current IP in A as given by equation (25), (b) the field-aligned current density at the top of the ionosphere j∥i in A m−2 given by equation (26), (c) the minimum field-aligned voltage Φmin in V required to drive the current in panel (b) as given by equation (34) and (d) the precipitating electron energy flux Ef in W m−2 as given by equation (36). The labels in panel (d) show the radio power obtained by integrating the energy fluxes using equations (37) and (38). Note there are no blue lines in panels (c) and (d) since for this case the current density in panel (b) is everywhere below the minimum current j∥i○ for which field-aligned voltages are required [shown by the horizontal dotted line in panel (b)].
and Ωp. There are presently no published theoretical limitations on the maximum plasma mass outflow rate from exoplanetary moons, which presumably would vary with the number of volcanic moons in a given system as well as the orbital and geological parameters of each satellite and planet. The theory considered in this paper is valid for all non-zero values of the plasma mass outflow rate, and it is worth noting that at Jupiter the solar wind and ionosphere also act as plasma sources with rates of <102 and >rsim 20 kg s−1, respectively (Hill et al. 1983), such that in practice the mass outflow rate will probably be non-zero in most systems, even those devoid of active moons. It therefore seems reasonable to compute the radio power emitted for a range of
values within an order of magnitude of the canonical jovian figure of 1000 kg s−1. The rotation rate of a planet is governed by its initial angular velocity and subsequent tidal dissipation. The origin of the former is poorly understood, but candidates include the relative motion of planetesimals during protoplanetary accretion and, for the gas giants specifically, hydrodynamic flows during accretion of hydrogen and helium (Lissauer 1993). A crude estimation of the maximum angular velocity allowed by centrifugal stability of a planet is where G is the gravitational constant equal to ∼6.67 × 10−11 N m2 kg−2, and which for a planet of Jupiter’s mass and radius yields Ωmax≃ 3.3 ΩJ. However, Jupiter lies roughly on the boundary between the regime of bodies supported by Coulomb pressure, for which Mp∝R3p, and degeneracy pressure, for which Mp∝R−3p, such that Jupiter’s radius is near maximal for bodies of solar composition. The net effect is that all bodies from solar composition giant planets through brown dwarfs to the very lowest mass stars are expected to have radii similar to Jupiter, and it is thus possible for planets with mass of, e.g. 10MJ to have angular velocities up to ∼11ΩJ. The time-scale τsyn required to de-spin a planet by tidal dissipation is given by where Q is the planet’s tidal dissipation factor typically given by Q = 105 for jovian planets, Ωorb is the Keplerian angular velocity of the planet, and where we note that factors of order unity have been omitted (Goldreich & Soter 1966; Showman & Guillot 2002). For planets of jovian parameters orbiting solar-type stars at 0.1, 1 and 100 au, this expression yields tidal synchronization time scales of 108, 1014 and 1020 yr, respectively, while for planets more susceptible to tidal locking e.g. those with Mp = 0.1MJ and M★ = 10 M⊙, the synchronisation time scales are modified to 105, 1011 and 1023 yr. Thus, close-orbiting ‘hot Jupiters’ are expected to rapidly become tidally locked, but for planets orbiting beyond 1 au, tidal effects will be negligible over time scales of e.g. the present age of the Solar system of ∼4.6 × 109 yr, such that the planetary angular velocity may not be much reduced from its initial value at formation.With the above considerations in mind, we now examine the system using four spot combinations of the parameters
and ΩJ, i.e. [
, (Ωp/ΩJ) = 1], [
, (Ωp/ΩJ) = 1], [
, (Ωp/ΩJ) = 3] and [
, (Ωp/ΩJ) = 3]. Results are shown in Fig. 8, which is divided into four sections of two panels corresponding to the above pairs of parameters. The solid line in the top panel of each section shows the limiting conductance given by equation (28). The dependence of the limiting conductance on the parameters Rorb,
and Ωp, obtained from equations (4), (5), (10), (14) and (28), is approximately
, such that it increases slightly faster than linearly with distance in each panel, and compared to the values shown Fig. 8(a), those in Figs 8(b), (c) and (d) are thus multiplied by factors of 10, ∼0.41 and ∼4.1, respectively. The estimated Pedersen conductance is also shown by the dashed lines in each panel for stellar XUV luminosities of (LXUV ★/L XUV ⊙) = 110 100 and 1000 (from bottom to top, respectively). Despite these XUV luminosities being orders of magnitude larger than the solar value, since Σ*P∝ (LXUV ★/L XUV ⊙)1/2 the conductance values are multiplied from the solar values by factors of only 1, 3.2, 10 and 32, respectively. However, it is obvious that the increased conductance generated by the higher XUV luminosities greatly increases the maximum orbital distance R*orb at which the high-conductance approximation holds, as shown in each section by the four vertical dotted lines, for example in section (a) at ∼4.4, 5.3, 7.3 and 10.9 au corresponding to conductances of ∼3.6, 4.6, 6.6 and 10.6 mho, respectively. As shown in Figs 8(b), (c) and (d), these values of R*orb are then modified from, e.g., the high XUV luminosity case of ∼10.9 au to ∼3.5, 17.7 and 5.4 au, due to increased
, Ωp and both
and Ωp, respectively.
Plot showing results using four combinations of the input parameters Ωp and
, i.e. (a) [
, (Ωp/ΩJ) = 1], (b) [
, (Ωp/ΩJ) = 1), (c) [
, (Ωp/ΩJ) = 3] and (d) [
, (Ωp/ΩJ) = 3]. Each section consists of two panels. The top shows the limiting conductance given by equation (28) (solid line) and the estimated Pedersen conductance in mho given by equation (30) for four values of the stellar XUV luminosity, i.e. (LXUV ★/L XUV ⊙) = 110 100 and 1000 (dashed lines from bottom to top, respectively). The bottom shows with the left axis the radio power in W emitted, obtained by integrating the precipitating electron energy fluxes using equations (37) and (38), and with the right axis the maximum distance in parsecs at which sources emitting such powers would be detectable, assuming a detection threshold of 1 mJy. All parameters are plotted versus orbital distance Rorb. The vertical dotted lines in each section indicate where the estimated Pedersen conductances equal the limiting conductance, i.e. show the maximum distances R*orb to which the high-conductance approximation is valid, and thus the maximum radio powers for these parameters. The lines in the bottom of each panel are shown dotted beyond the maximum values of R*orb for (LXUV ★/L XUV ⊙) = 1000, indicating that this region corresponds to XUV luminosities above the upper limit for a young solar-type star as indicated by Ribas et al. (2005).
Plot showing results using four combinations of the input parameters Ωp and
, i.e. (a) [
, (Ωp/ΩJ) = 1], (b) [
, (Ωp/ΩJ) = 1), (c) [
, (Ωp/ΩJ) = 3] and (d) [
, (Ωp/ΩJ) = 3]. Each section consists of two panels. The top shows the limiting conductance given by equation (28) (solid line) and the estimated Pedersen conductance in mho given by equation (30) for four values of the stellar XUV luminosity, i.e. (LXUV ★/L XUV ⊙) = 110 100 and 1000 (dashed lines from bottom to top, respectively). The bottom shows with the left axis the radio power in W emitted, obtained by integrating the precipitating electron energy fluxes using equations (37) and (38), and with the right axis the maximum distance in parsecs at which sources emitting such powers would be detectable, assuming a detection threshold of 1 mJy. All parameters are plotted versus orbital distance Rorb. The vertical dotted lines in each section indicate where the estimated Pedersen conductances equal the limiting conductance, i.e. show the maximum distances R*orb to which the high-conductance approximation is valid, and thus the maximum radio powers for these parameters. The lines in the bottom of each panel are shown dotted beyond the maximum values of R*orb for (LXUV ★/L XUV ⊙) = 1000, indicating that this region corresponds to XUV luminosities above the upper limit for a young solar-type star as indicated by Ribas et al. (2005).
The effects of these changes on the maximum radio power are shown in the bottom panels of each section of Fig. 8. The left axes indicate the emitted powers, obtained by integrating the precipitating electron energy fluxes using equations (37) and (38), whilst the right axes show the distance s in parsecs obtained from equations (39) and (40) with these powers and an assumed spectral flux density of 1 mJy. Note that the right axes in Figs 8(c) and (d) are different to those in Figs 8(a) and (b), since the spectral flux density is dependent on Δν, which is proportional to Ω3/4p through Beq. As also shown in Fig. 7, the radio power increases with orbital distance, with a slight discontinuity in the gradient where Rmp = 85 Rp. This discontinuity is simply an artefact introduced by the change in the behaviour of the equatorial magnetic field model at the reference boundary of the ‘baseline’ magnetic field model. For Rmp < 85 Rp, as the radius of the magnetosphere increases with increasing orbital distance Rorb, the equatorial magnetic field strength decreases due to flux conservation, such that the current intensities and power output increase as discussed above (c.f. the blue and green lines in Fig. 7). On the other hand, as the radius of the magnetosphere further increases such that
the current intensity at a given colatitude remains the same, but extends increasingly towards the pole (c.f. the red and black lines in Fig. 7). The latter has less of an effect that the former, such that the power output increases less quickly with Rorb after Rmp passes 85Rp. Thus, for the values of
and Ωp in Fig. 8(a), the maximum radio powers Pr for the four XUV luminosity cases are ∼7.2 × 1010, 1.8 × 1011, 8.2 × 1011 and 2.5 × 1012 W, corresponding to maximum observable distances s of ∼0.4, 0.7, 1.5 and 2.6 pc, respectively. However, again taking the high XUV luminosity case as an example, this power is modified as shown in Figs 8(b), (c) and (d) to ∼2.6 × 1012, 8.6 × 1013 and 3.4 × 1014 W, corresponding to maximum observable distances of ∼2.7, 10 and 20 pc, respectively. Thus, from these four spot combinations of
and Ωp it is apparent that the maximum radio power Pr is relatively insensitive to changes in
, since the higher powers available at a given orbital distance Rorb for increased
are compensated for by a decrease in the maximum distance to which the high-conductance approximation is valid R*orb. On the other hand, whilst changing Ωp does have competing effects on different components of the system, the overall behaviour is one in which increased Ωp results in increased power. We further note that, although, as discussed above, the compressed vacuum dipole relation for the subsolar magnetopause stand-off distance is not strictly appropriate for the rotationally driven, plasma-filled magnetospheres we consider here, the Huddleston et al. (1998) relation was derived using observations of jovian values over the range ∼45–100RJ, and extrapolation beyond this range may lead to uncertainties. We have thus also computed the above values using equation (12) in place of equation (14), and find that the results are qualitatively similar, except that the values of R*orb, Pr and s are modified by factors of ∼2–4, ∼0.1–0.4 and ∼0.3–0.6, respectively. As noted by Cowley et al. (2002) and Nichols & Cowley (2003), the effect of the stretching of the planetary field from a dipolar configuration into a magnetodisc structure is to amplify the upward field-aligned current density associated with the aurora and radio emissions by 1–2 orders of magnitude, such that the precipitating electron energy flux, a quantity proportional to the square of the field-aligned current density through equation (36), is modified by 2–4 orders of magnitude. Using equation (12) in place of equation (14) compresses the equatorial magnetic field to a more dipole-like form, such that the radio power available at a given orbital radius is significantly reduced. However, the effect of a closer boundary is also to reduce the conductance required to maintain near-rigid corotation throughout the magnetosphere, such that the maximum power reduction is partly mitigated by increased orbital radius at which the maximum powers are available. However, we again note that equation (12) does not reproduce results consistent with observations of Jupiter’s magnetosphere, such that in the discussion that follows we employ results obtained using the Huddleston et al. (1998) relation.
We now extend the analysis to cover the full parameter space within the considerations discussed above. We thus examine the behaviour of the system over
and 0.3 < (Ωp/ΩJ) < 10. We should note that angular velocities above ∼3 ΩJ would require planets more massive than Jupiter and may thus have larger dipole moments, although as discussed above for simplicity we do not take this into account here and will be the subject of future study. Results are shown in Fig. 9 for the four different XUV luminosity cases. The colour indicates the maximum radio power Pr available for each pair of parameter values, as shown in Fig. 8 by the powers at the locations of the vertical dotted lines, and the black contours show the maximum observable distances of sources emitting such radio powers, again assuming a spectral flux density threshold of 1 mJy. Note the kinks are an artefact of the model indicating the change of behaviour where Rmp = 85 RJ as discussed above. The white contours indicate the orbital distances R*orb at which these powers are available, which we note increase with decreasing
and, to a lesser extent with increasing Ωp, as previously noted. This plot confirms the overall behaviour apparent in Fig. 8, i.e. the radio power is essentially independent of
, increases with ΩJ, and increases with stellar XUV luminosity. Thus, for stars with solar XUV luminosity, planets with (Ωp/ΩJ) ∼ 5 are required to produce radio emissions detectable from beyond ∼10 pc, but this is reduced to e.g. (Ωp/ΩJ) ≃ 2 for stars with (LXUV ★/L XUV ⊙) = 1000 and
. In all XUV luminosity cases, a significant number of parameter combinations within an order of magnitude of the jovian values are capable of producing emissions observable beyond 1 pc, in most cases requiring exoplanets orbiting at distances between ∼1 and 50 au. For the two higher XUV luminosity cases, parameter combinations within an order of magnitude of Jupiter’s could generate emissions detectable beyond ∼50 pc.
Coloured plots indicating the maximum radio powers Pr available using different pairs of system parameters
and Ωp for, in panels (a)-(d) respectively, the XUV luminosity cases (LXUV ★/LXUV ⊙) = 1, 10, 100 and 1000. Also shown by the black contours are the maximum distances s in pc at which sources of these powers are observable, assuming a detection threshold of 1 mJy, and the white contours show the orbital distances R*orb in au at which these maximum powers are available. Labelled crosses refer to the spot combinations shown in Fig. 8.
Coloured plots indicating the maximum radio powers Pr available using different pairs of system parameters
and Ωp for, in panels (a)-(d) respectively, the XUV luminosity cases (LXUV ★/LXUV ⊙) = 1, 10, 100 and 1000. Also shown by the black contours are the maximum distances s in pc at which sources of these powers are observable, assuming a detection threshold of 1 mJy, and the white contours show the orbital distances R*orb in au at which these maximum powers are available. Labelled crosses refer to the spot combinations shown in Fig. 8.
4 SUMMARY AND DISCUSSION
In this paper we have provided the first consideration of magnetosphere–ionosphere coupling at Jupiter-like exoplanets. We have estimated the radio power emitted by such systems under the condition of near-rigid corotation throughout (a condition which maximizes the field-aligned currents and thus the radio power for a given magnetosphere), in order to examine the behaviour of the best candidates for detecting internally generated radio emission with next generation radio telescopes such as LOFAR. We have thus estimated for different stellar XUV luminosity cases the orbital distances within which the ionospheric Pedersen conductance would be high enough to maintain near-rigid corotation, and we have then considered the magnitudes of the large-scale magnetosphere–ionosphere currents flowing within the systems, and the resulting radio powers, at such distances. We have also examined the effects of two key system parameters, i.e. the planetary angular velocity and the plasma mass outflow rate.
The key results of the study can be summarized as follows.
The radio power emitted increases with increasing system size, and thus increases with orbital distance within the limit of validity of the high-conductance approximation.
The limiting orbital distance, which defines the maximum radio power available for a given set of system parameters, increases with stellar XUV luminosity and planetary rotation rate, and decreases with magnetospheric plasma mass outflow rate.
The overall effect is that the radio power emitted increases with planetary rotation rate, but is essentially independent of plasma mass outflow rate since the higher powers available at a given orbital distance for increased plasma mass outflow rate are compensated for by a decrease in the maximum orbital distance to which the high-conductance approximation is valid.
In all XUV luminosity cases studied, a significant number of parameter combinations within an order of magnitude of the jovian values are capable of producing emissions observable beyond 1 pc, in most cases requiring exoplanets orbiting at distances between ∼1 and 50 au. For the higher XUV luminosity cases the observable distances for jovian mass planets can reach ∼20 pc, and massive, rapidly rotating planets could be detectable beyond ∼50 pc.
However, we should note here the limitations of the simple model used in this study. First, we have not considered how the structure of the magnetic field changes with planetary angular velocity and plasma mass outflow rate. As noted above, Jupiter’s magnetosphere is partly inflated by the centrifugal force of iogenic plasma (Caudal 1986), such that systems with higher angular velocity or plasma loading would be expected to be further inflated, although this would be somewhat mitigated by the associated increase of the planetary field strength in the case of the angular velocity. However, as discussed in Section 3, the effect of the stretching of the planetary field from a dipolar configuration into a magnetodisc structure is to amplify the upward field-aligned current density associated with the aurora and radio emissions by 1–2 orders of magnitude, and therefore the power values derived here may be viewed as lower limits in this regard. Secondly, we note that the field-aligned voltages obtained by the current–voltage relation used in this study may underestimate the true voltages if the location of the accelerator is fixed at a few planetary radii up the the field lines, such that these power values might again be underestimates. In a related point, we note that simply defining the plasma angular velocity as we have done ignores the effects of the significant field-aligned voltages, which act to decouple the equatorial and ionospheric plasma angular velocities (Nichols & Cowley 2005; Ray et al. 2010), an issue which should be studied in further works. Further, we note that we have not considered at all the stellar wind interaction mediated by magnetic reconnection at the dayside magnetopause, which at Jupiter could be associated with the many variable and sometimes extremely bright polar auroras observed (Pallier & Prangé 2001; Waite et al. 2001; Grodent et al. 2003; Bunce, Cowley & Yeoman 2004; Nichols et al. 2009a,b), and would thus sporadically increase the power output. Nor have we considered the effect of changing stellar wind dynamic pressure, which at Jupiter is known to modulate the intensity of the UV and radio emissions by factors of ∼3 (Gurnett et al. 2002; Cowley et al. 2007; Clarke et al. 2009; Nichols et al. 2009b), and again, although the details of how this affects Jupiter’s auroral and radio emissions remain to be fully determined, this effect may also act to increase the output powers from those computed here. However, we also reiterate that our assumption of solar values for the stellar wind velocity and density may represent underestimates for stars more active than the Sun, such as those much younger. Higher stellar wind dynamic pressure values would decrease the size of the magnetosphere for a given orbital distance, thus reducing the emitted radio power. It is clear that the evolution of internally generated radio emissions over the lifetime of a star should therefore be considered in future studies. We should also recall that we have only considered here planetary magnetic fields with the same polarity as those of Jupiter and Saturn, i.e. with magnetic and spin axes co-aligned to first order, since this is the configuration that has been most studied for bodies in our Solar system. However, it is probable that only 50 per cent of Jupiter-like exoplanets exhibit this polarity. The effect of reversed polarity is to reverse the direction of the current system, such that downward currents are replaced with upward currents and vice versa, such that the source population for the field-aligned currents in this case may be very different to that considered here (see e.g. Bunce et al. 2004 for a discussion of field-aligned currents induced at Jupiter’s dayside magnetopause). The emissions from planets with the opposite polarity should be studied in future works.
Finally, we note that the implication of these results is that the best candidates for detection of such internally generated radio emissions are rapidly rotating Jupiter-like exoplanets orbiting stars with high XUV luminosity at orbital distances beyond ∼1 au. This type of exoplanet has not previously been considered as potential detection candidates for next generation radio telescopes, but searching for such emissions may offer a new method of detection of more distant orbiting exoplanets less likely to be detected by those techniques which are biased towards close-orbiting ‘hot Jupiters’. However, dual detections with radio telescopes and conventional methods would best constrain the planetary parameters.
JDN was supported by STFC Grant ST/H002480/1, and wishes to thank S. W. H. Cowley, M. A. Barstow, M. R. Burleigh and G. A. Wynn for constructive discussions during this study, and also thanks the referee for providing helpful comments on the manuscript.















































![Plot showing the current system parameters versus ionospheric colatitude, for magnetopause stand-off distances of Rmp = 20, 50, 85 and 200Rp (blue, green, red, and black lines, respectively), and with Ωp=ΩJ and . The labels in panel (a) correspond to the values of Rorb which result in magnetospheres of these sizes, according to equations (10) and (13). Parameters shown are (a) the azimuthally integrated ionospheric Pedersen current IP in A as given by equation (25), (b) the field-aligned current density at the top of the ionosphere j∥i in A m−2 given by equation (26), (c) the minimum field-aligned voltage Φmin in V required to drive the current in panel (b) as given by equation (34) and (d) the precipitating electron energy flux Ef in W m−2 as given by equation (36). The labels in panel (d) show the radio power obtained by integrating the energy fluxes using equations (37) and (38). Note there are no blue lines in panels (c) and (d) since for this case the current density in panel (b) is everywhere below the minimum current j∥i○ for which field-aligned voltages are required [shown by the horizontal dotted line in panel (b)].](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnras/414/3/10.1111/j.1365-2966.2011.18528.x/2/m_mnras0414-2125-f7.jpeg?Expires=1548413673&Signature=40a8oT~f8h5vmEhsCyyT~N631Df-yY8UIMWj~iO4w0RQ47IQQhP6qAnSfDUJEIUuQyZmAapwMSrv9BEr~GRqUGcReADXhoQkCIuyeOttVDkj0utiZbVklk9A528QIq7RykwBzH2rAppcEFIFyIgaE77u1euCUExjPh2~otDvgqeHvvn~uxF1O8wafUc4Yx5FRGUXT7-j4TR3fXQyRzJEh1SjDXxEjzibOthBZlHWoQ5zDY2Y4G6UTWk6cz1SYFmTo5Gt-jYQonm8QgcM7~NjWhOanOEltNYD0ZrXZaTVgzfFm8N9i~9dPmLnumOn0mrG-womwG6xI-od9kUzFT8-jg__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)


![Plot showing results using four combinations of the input parameters Ωp and , i.e. (a) [, (Ωp/ΩJ) = 1], (b) [, (Ωp/ΩJ) = 1), (c) [, (Ωp/ΩJ) = 3] and (d) [, (Ωp/ΩJ) = 3]. Each section consists of two panels. The top shows the limiting conductance given by equation (28) (solid line) and the estimated Pedersen conductance in mho given by equation (30) for four values of the stellar XUV luminosity, i.e. (LXUV ★/L XUV ⊙) = 110 100 and 1000 (dashed lines from bottom to top, respectively). The bottom shows with the left axis the radio power in W emitted, obtained by integrating the precipitating electron energy fluxes using equations (37) and (38), and with the right axis the maximum distance in parsecs at which sources emitting such powers would be detectable, assuming a detection threshold of 1 mJy. All parameters are plotted versus orbital distance Rorb. The vertical dotted lines in each section indicate where the estimated Pedersen conductances equal the limiting conductance, i.e. show the maximum distances R*orb to which the high-conductance approximation is valid, and thus the maximum radio powers for these parameters. The lines in the bottom of each panel are shown dotted beyond the maximum values of R*orb for (LXUV ★/L XUV ⊙) = 1000, indicating that this region corresponds to XUV luminosities above the upper limit for a young solar-type star as indicated by Ribas et al. (2005).](https://oup.silverchair-cdn.com/oup/backfile/Content_public/Journal/mnras/414/3/10.1111/j.1365-2966.2011.18528.x/2/m_mnras0414-2125-f8.jpeg?Expires=1548413673&Signature=wJsbdEiTj8egeLRZ5Gr2jcl91WAS2kt~XSTm6Rn8vUQYfzT27JweFRHQBuI5SPYG1DSr~-Ag4lKZNF7SY52KINWoduE96D9lRiGVbceuM6pv7TCo3IhwkJeSuLnQMd4P5GwmtqCpvF-cP-WHB3GwCu1~hXk3mdeFrEfU-d~-z2dbd6CwokN0F5Blq8uTw0sKATVkOGzmGRMY~TMUopvUuN7EYAMqHYfMtPHW9IoSFbaN1fz90YymU3d2~XBJdSndSUbiOS6MxNE3w5f~~P-FbxqzFahDkGP7eO7tY3TBnBxHphu7FSHzkZDMdbv6SzUxIVT6gM9tTqIu60dKUJCykA__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA)
