DC CIRCUIT POWERED BY ORBITAL MOTION: MAGNETIC INTERACTIONS IN COMPACT OBJECT BINARIES AND EXOPLANETARY SYSTEMS

In a seminal paper, Goldreich & Lynden-Bell (1969) developed a DC circuit model for the magnetic interaction between Jupiter and its satellite Io (see also Piddington & Drake 1968). In this model, the orbital motion of Io in the rotating magnetosphere of Jupiter generates electromotive force (EMF), driving a DC current between Io and Jupiter, with the closed magnetic field lines serving as wires. In essence, the Jupiter–Io system operates as a unipolar inductor. This model helped explain Jupiter's decametric radio emissions which are correlated with the motion of Io and the hot spot in Jupiter's atmosphere that is linked magnetically to Io (e.g., Clarke et al. 1996).

In recent years, the DC circuit model of Goldreich & Lynden-Bell has been adapted and applied to other types of astrophysical binary systems, including (1) planets around magnetic white dwarfs (WD; Li et al. 1998); (2) ultracompact WD binaries with periods ranging from a few minutes to an hour (Wu et al. 2002; Dall'Osso et al. 2006, 2007); (3) coalescing neutron star (NS)–black hole (BH) binaries and NS–NS binaries prior to the final merger (McWilliams & Levin 2011; Piro 2012; see also Vietri 1996, Hansen & Lyutikov 2001, and Lyutikov 2011 for more general discussion of unipolar induction in such binaries); and (4) exoplanetary systems containing short-period (∼ days) planets orbiting magnetic MS stars (Laine & Lin 2012; Zarka 2007).

In the DC circuit model, the magnetic interaction torque and the related energy dissipation rate are inversely proportional to the total resistance ${\cal R}_{\rm tot}$ of the circuit, including contributions from the two binary components and the magnetosphere. Thus, it seems that extreme power may be produced for highly conducting binary systems. However, we show in this paper that when ${\cal R}_{\rm tot}$ is smaller than a critical value (∼480v_rel/c Ohms, where v_rel is the orbital velocity as seen in the rotating frame of the magnetic star), the large current flowing in the circuit will break the magnetic connection between the binary components. Thus, there exists an upper limit to the magnetic torque and the energy dissipation rate. In Sections 2–4, we examine the general aspect of the DC circuit model, the upper limit, and its connection with "open circuit" effect ("Alfvén radiation"). We then discuss applications of the model to various astrophysical binaries in Sections 5–8 and conclude in Section 9.

Consider a binary system consisting of a magnetic star (the "primary," with mass M_⋆, radius R_⋆, spin Ω_s, and magnetic dipole moment μ) and a non-magnetic companion (mass M_c, radius R_c). The orbital separation is a and the orbital angular frequency is Ω. The magnetic field strength at the surface of the primary is B_⋆ = μ/R³_⋆. The whole binary system is embedded in a tenuous plasma (magnetosphere). For simplicity, we assume Ω, Ω_s, and ${\mbox{\boldmath $\mu $}}$ are all aligned.

The motion of the non-magnetic companion relative to the magnetic field of the primary produces an EMF ${\cal E}\simeq 2R_c |E|$, where E = v_rel × B/c, with ${\bf v}_{\rm rel}=(\Omega -\Omega _s)a\,{\hat{{\mbox{\boldmath $\phi $}}}}$ and ${\bf B}=(-\mu /a^3){\hat{\bf z}}$. This gives

$$\begin{equation} {\cal E}\simeq {2\mu R_c\over ca^2}\Delta \Omega, \end{equation} \tag{ 1 }$$

where ΔΩ = Ω − Ω_s. The EMF drives a current along the magnetic field lines in the magnetosphere, connecting the primary and the companion through two flux tubes (Figure 1). Note that such a closed DC circuit may not always be established (see Section 4). Assuming that it is, the current in the circuit is given by

$$\begin{equation} {\cal I}={{\cal E}\over {\cal R}_{\rm tot}}, \end{equation} \tag{ 2 }$$

where the total resistance of the circuit is

$$\begin{equation} {\cal R}_{\rm tot}={\cal R}_\star +{\cal R}_c+2{\cal R}_{\rm mag}, \end{equation} \tag{ 3 }$$

with ${\cal R}_\star,\,{\cal R}_c,\,{\cal R}_{\rm mag}$ the resistances of the magnetic star, the companion, and the magnetosphere, respectively. These resistances depend on the properties of the binary components and the magnetosphere, and can vary widely for different types of systems. The energy dissipation rate of the system is then

$$\begin{equation} \dot{E}_{\rm diss}=2{\cal I}^2R_{\rm tot}={2{\cal E}^2\over {\cal R}_{\rm tot}}, \end{equation} \tag{ 4 }$$

where the factor of 2 accounts for both the upper and lower sides of the circuit.

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Note that because of the finite resistances ${\cal R}_\star$ and ${\cal R}_c$, the magnetic flux tube in general rotates at a rate Ω_F between Ω_s and Ω. The slippage of the companion relative to the magnetic field gives ${\cal E}_c\simeq (2\mu R_c/ca^2)(\Omega -\Omega _F)$. The slippage of the primary's rotation related to the flux tube generates ${\cal E}_\star \simeq (2\mu R_c/ca^2)(\Omega _F-\Omega _s)$ at the magnetic foot on the surface of the primary. The total EMF $({\cal E}_\star +{\cal E}_c)$ is the same as in Equation (1), independent of Ω_F.

The total magnetic force (in the azimuthal direction) on the companion is $F_\phi \simeq (2R_c) (2{\cal I}B_z/c)$, with B_z = −μ/a³. Thus the torque acting on the binary's orbital angular momentum is

$$\begin{equation} T=\dot{J}_{\rm orb}\simeq {4\over c}a\,R_c{\cal I}B_z\simeq -{4\mu R_c\over ca^2}{{\cal E}\over {\cal R}_{\rm tot}}. \end{equation} \tag{ 5 }$$

The torque on the primary's spin is $I_\star \dot{\Omega }_s=-T$ (where I_⋆ is the moment of inertia). The orbital energy loss rate associated with T is then

$$\begin{equation} \dot{E}_{\rm orb}=T\Omega \simeq -{2{\cal E}^2\over {\cal R}_{\rm tot}}\,{\Omega \over \Delta \Omega }. \end{equation} \tag{ 6 }$$

The total energy dissipation rate of the system is

$$\begin{equation} \dot{E}_{\rm diss}=(-\dot{E}_{\rm orb})- I_\star \Omega _s\dot{\Omega }_s =-T\Delta \Omega ={2{\cal E}^2\over {\cal R}_{\rm tot}}, \end{equation} \tag{ 7 }$$

in agreement with Equation (4).

The equations in the previous section clearly show that the binary interaction torque and energy dissipation associated with the DC circuit increase with decreasing total resistance ${\cal R}_{\rm tot}$. Is there a problem for the DC model when ${\cal R}_{\rm tot}$ is too small? The answer is yes.

The current in the circuit produces a toroidal magnetic field, which has the same magnitude but opposite directions above and below the equatorial plane. The toroidal field just above the companion star (in the upper flux tube) is $B_{\phi +}\simeq -(2\pi /c){\cal J}_r$, where ${\cal J}_r\sim -{\cal I}/R_c$ is the (height-integrated) surface current. A more precise expression for ${\cal J}_r$ can be obtained as follows. The magnetic torque on the companion is

$$\begin{equation} T\simeq -a\big(\pi R_c^2\big){\cal J}_rB_z/c\simeq {1\over 2}aR_c^2B_zB_{\phi +}. \end{equation} \tag{ 8 }$$

Comparing this with Equation (5) yields ${\cal J}_r\simeq -4{\cal I}/(\pi R_c)$. Thus the azimuthal twist of the flux tube is

$$\begin{equation} \zeta _\phi =-{B_{\phi +}\over B_z}={8{\cal E}\over cR_c{\cal R}_{\rm tot}|B_z|} ={16 v_{\rm rel}\over c^2{\cal R}_{\rm tot}}, \end{equation} \tag{ 9 }$$

where v_rel = aΔΩ = a(Ω − Ω_s) is the orbital velocity in the corotating frame of the primary star. Clearly, when ${\cal R}_{\rm tot}$ is less than 16v_rel/c², the flux tube will be highly twisted.

GL already speculated in 1969 that the DC circuit would break down when the twist is too large. (For the Jupiter–Io system parameters adopted by GL, the twist |ζ_ϕ| ≪ 1.) Since then, numerous works have confirmed that this is indeed the case. Theoretical studies and numerical simulations, usually carried out in the contexts of solar flares and accretion disks, have shown that as a flux tube is twisted beyond ζ_ϕ ≳ 1, the magnetic pressure associated with B_ϕ makes the flux tube expand outward and the magnetic fields open up, allowing the system to reach a lower energy state (e.g., Aly 1985; Aly & Kuijpers 1990; van Ballegooijen 1994; Lynden-Bell & Boily 1994; Lovelace et al. 1995; Uzdensky et al. 2002). Thus, a DC circuit with ζ_ϕ ≳ 1 cannot be realized: The flux tube will break up, disconnecting the linkage between the two binary components.

A binary system with ${\cal R}_{\rm tot} \lesssim 16v_{\rm rel}/c^2$ cannot establish a steady-state DC circuit. The electrodynamics is likely rather complex. Based on the studies of magnetic star–disk systems, where a similar situation occurs (e.g., Aly & Kuijpers 1990; Uzdensky et al. 2002), we suggest that a quasi-cyclic circuit may be possible, involving several steps: (1) the magnetic field from the primary penetrates part of the companion, establishing magnetic linkage between the two stars; (2) the linked fields are twisted by differential rotation, generating toroidal field from the linked poloidal field; (3) as the toroidal magnetic field becomes comparable to the poloidal field, the fields inflate and the flux tube breaks, disrupting the magnetic linkage; (4) reconnection between the inflated field lines relaxes the shear and restores the linkage. The whole cycle repeats.

In any case, we can use the dimensionless azimuthal twist ζ_ϕ to parameterize the magnetic torque and energy dissipation rate:

$$\begin{eqnarray} && T= {1\over 2}aR_c^2B_zB_{\phi +} = -\zeta _\phi {\mu ^2 R_c^2\over 2a^5}, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} && \dot{E}_{\rm diss} = -T \Delta \Omega =\zeta _\phi \Delta \Omega {\mu ^2 R_c^2\over 2a^5}. \end{eqnarray} \tag{ 11 }$$

The maximum torque and dissipation are obtained by setting ζ_ϕ ∼ 1. If the quasi-cyclic circuit discussed in the last paragraph is established, we would expect ζ_ϕ to vary between 0 and ∼1. Note that in the above, T is negative since we are assuming Ω > Ω_s. A reasonable extension would let ζ_ϕ = ζ(ΔΩ)/Ω, with ζ > 0.

As discussed in GL, the validity of the DC circuit model requires that the slippage of the flux tube relative to the companion during the round-trip Alfvén travel time (t_A) along the flux tube be much less than R_c, i.e., (Ω − Ω_F)at_A ≪ R_c. When this condition is not satisfied or when the poloidal field opens up due to twist (Section 3), the disturbance generated by the companion's orbital motion v_rel will propagate along the field line as Alfvén waves and radiate away (Drell et al. 1965). The Alfvén radiation power associated with the "open circuit" is

$$\begin{equation} \dot{E}_{\rm Alf}\sim {1\over 4\pi }\left(B_z v_{\rm rel}/v_A\right)^2v_A \big(2\pi R_c^2\big) ={1\over 2}B_z^2R_c^2\,{v_{\rm rel}^2\over v_A}, \end{equation} \tag{ 12 }$$

where v_A is the Alfvén speed in the magnetosphere (assuming v_rel ≲ v_A). The associated drag force is simply $F_\phi =-\dot{E}_{\rm Alf}/v_{\rm rel}$, and the torque on the orbit is

$$\begin{equation} T_{\rm Alf}\sim -{1\over 2}B_z^2R_c^2a\,{v_{\rm rel}\over v_A}. \end{equation} \tag{ 13 }$$

Comparing $\dot{E}_{\rm Alf}$ with Equation (11), we find

$$\begin{equation} {\dot{E}_{\rm Alf}\over \dot{E}_{\rm diss}}\sim {v_{\rm rel}\over \zeta _\phi v_A}. \end{equation} \tag{ 14 }$$

Clearly, $\dot{E}_{\rm Alf}$ is always smaller than the maximum $\dot{E}_{\rm diss}$ of a DC circuit. Thus, Equation (11) (with ζ_ϕ ∼ 1) represents the maximum magnetic dissipation rate of the binary system, regardless of the details of the electrodynamics (e.g., close or open circuit).

Gravitational wave (GW) emission drives the orbital decay of the NS binary, with timescale

$$\begin{equation} t_{\rm GW}={a\over |\dot{a}|}={5c^5a^4\over 64G^3M_\star ^3 q(1+q)} \simeq 0.012\left(\!{a\over 30\,{\rm km}}\!\right)^{\!4}{\rm s}, \end{equation} \tag{ 15 }$$

where in the last equality we have adopted M_⋆ = 1.4 M_☉ and mass ratio q = M_c/M_⋆ = 1. The magnetic torque tends to spin up the primary when Ω > Ω_s. Spin–orbit synchronization is possible only if the synchronization time t_syn = I_⋆Ω/|T| is less than t_GW at some orbital radii. With I_⋆ = κM_⋆R²_⋆ and the torque (10), we find

$$\begin{eqnarray} t_{\rm syn}&=&{2\kappa (1+q)\over \zeta _\phi \Omega } \left(\!{GM_\star ^2\over B_\star ^2R_\star ^4}\!\right) \!\left(\!{a\over R_c}\!\right)^2\nonumber \\ &\simeq& 2\times 10^7\zeta _\phi ^{-1}\!\left(\!{B_\star \over 10^{13}\,{\rm G}} \!\right)^{\!-2}\!\left({a\over 30\,{\rm km}}\right)^{7/2}{\rm s}, \end{eqnarray} \tag{ 16 }$$

where in the second line we have adopted κ = 0.4 and R_⋆ = R_c = 10 km. Clearly, even with magnetar-like field strength (B_⋆ ∼ 10¹⁵ G) and maximum efficiency (ζ_ϕ ∼ 1), spin–orbit synchronization cannot be achieved by magnetic torque. (It was already known that tidal torque, due to both equilibrium tide (Kochanek 1992; Bildsten & Cutler 1992) and resonant tide (Lai 1994; Ho & Lai 1999), cannot synchronize the NS spin during binary inspiral.)

For the same reason, the effect of magnetic torque on the number of GW cycles during binary inspiral, N, is small. We find

$$\begin{equation} {dN\over d\ln f}={1\over 1+\alpha }\left({dN\over d\ln f}\right)_0, \end{equation} \tag{ 17 }$$

where f = Ω/π is the GW frequency and

$$\begin{equation} \left(dN\over d\ln f\right)_0={5c^5(1+q)^{1/3}\over 96\pi q M_\star ^{5/3} (\pi Gf)^{5/3}} \end{equation} \tag{ 18 }$$

is the usual leading-order point-mass GW cycles. The correction factor due to the magnetic torque is

$$\begin{equation} \alpha = {2t_{\rm GW}\over J_{\rm orb}/|T|}= {2I_\star \over \mu _m a^2}\left({t_{\rm GW}\over t_{\rm syn}}\right), \end{equation} \tag{ 19 }$$

where μ_m = M_⋆q/(1 + q) is the reduced mass of the binary. With Equations (15) and (16), we see that GW phase error α(dN/dln f)₀ is much less than unity even for B_⋆ ∼ 10¹⁵ G and maximum ζ_ϕ ∼ 1.

The energy dissipation rate is

$$\begin{eqnarray} \dot{E}_{\rm diss}&=&\zeta _\phi \left(\!{v_{\rm rel}\over c}\!\right){B_\star ^2R_\star ^6 R_c^2c\over 2a^6}\nonumber \\ &=& 7.4\times 10^{44}\zeta _\phi \left(\!{B_\star \over 10^{13}\,{\rm G}}\!\right)^{\!2}\!\left(\!{a\over 30\,{\rm km}}\!\right)^{\!\!-13/2} \!{\rm erg\,s}^{-1},\qquad \end{eqnarray} \tag{ 20 }$$

where in the second line we have used v_rel ≃ aΩ (for Ω_s ≪ Ω) and adopted canonical parameters (M_⋆ = M_c = 1.4 M_☉, R_⋆ = R_c = 10 km). The total energy dissipation per ln a is

$$\begin{eqnarray} {dE_{\rm diss}\over d\ln a}&=&\dot{E}_{\rm diss}t_{\rm GW}\nonumber \\ &\simeq& 8.9\times 10^{42}\zeta _\phi \!\left(\!{B_\star \over 10^{13}\,{\rm G}}\!\right)^{\!2}\!\left({a\over 30\,{\rm km}}\right)^{\!\!-5/2} \!{\rm erg}.\qquad \end{eqnarray} \tag{ 21 }$$

Some fraction of this dissipation will emerge as the electromagnetic radiation counterpart of binary inspiral. Whether it is detectable at extragalactic distance depends on the microphysics in the magnetosphere, including particle acceleration and radiation mechanism (e.g., Vietri 1996; Hansen & Lyutikov 2001).

Piro (2012) recently applied the DC circuit model to NS binaries. In the case when the magnetic NS primary dominates the total resistance, he found that the magnetic torque synchronizes the NS spin prior to merger and significantly affects the gravitational wave cycles, even for modest (∼10¹² G) NS magnetic fields. Using the resistance computed by Piro, we find from Equation (9) that the corresponding azimuthal twist ζ_ϕ is much larger (by a factor ∼10⁸ at Ω = 100 s⁻¹) than unity, violating the upper limit discussed in Section 3.

If one assumes that the magnetosphere resistance is given by the impedance of free space, ${\cal R}_{\rm mag}=4\pi /c$, then the corresponding twist is ζ_ϕ = 2v_rel/(πc), which satisfies our upper limit. The energy dissipation rate is then

$$\begin{eqnarray} \dot{E}_{\rm diss}&=&\left(\!{v_{\rm rel}\over c}\!\right)^2\! {B_\star ^2R_\star ^6 R_c^2c\over \pi a^6}\nonumber \\ &=& 1.7\times 10^{44}\left(\!{B_\star \over 10^{13}\,{\rm G}}\!\right)^{\!2} \!\left({a\over 30\,{\rm km}}\right)^{\!\!-7}{\rm erg\ s^{-1}}. \end{eqnarray} \tag{ 22 }$$

Not surprisingly, this is approximately the same as the Alfvén power $\dot{E}_{\rm Alf}$ (see Equation (14)) with v_A = c and is in agreement with the estimate of Lyutikov (2011).

The situation is similar to the case of NS–NS binaries. In the membrane paradigm (Thorne et al. 1986), a BH of mass M_H resembles a sphere of radius R_c = R_H = 2GM_H/c² (neglecting BH spin) and impedance ${\cal R}_{\rm H}=4\pi /c$. Neglecting the resistances of the magnetosphere and the NS, the azimuthal twist of the flux tube in the DC circuit is

$$\begin{equation} \zeta _\phi ={4v_{\rm rel}\over \pi c}, \end{equation} \tag{ 23 }$$

which satisfies our upper limit (Section 3). The energy dissipation rate is (cf. Lyutikov 2011; McWilliams & Levin 2011)

$$\begin{eqnarray} \dot{E}_{\rm diss}&=&\left(\!{v_{\rm rel}\over c}\!\right)^2\! {2B_\star ^2R_\star ^6 R_{\rm H}^2c\over \pi a^6}\nonumber \\ &\simeq& 5.7\!\times \! 10^{42}\!\left(\!{B_\star \over 10^{13}\,{\rm G}}\!\right)^{\!2} \!\!\left(\!{M_{\rm H}\over 10\ M_\odot }\!\right)^{\!\!\!-4} \!\!\!\left(\!{a\over 3R_{\rm H}}\!\right)^{\!\!-7}\!\!\!{\rm erg\,s}^{-1},\nonumber\\ \end{eqnarray} \tag{ 24 }$$

where we have assumed M_BH/M_⋆ ≫ 1.

Wu et al. (2002) and Dall'Osso et al. (2006, 2007; see also Wu 2009) developed the DC circuit model for ultracompact double WD binaries, particularly for the systems RX J1914+24 (period P = 9.5 minutes) and RX J0806+15 (P = 5.4 minutes). Usual mass transfer models appear to have difficulties explaining some of the properties of these systems (e.g., the observed orbital decay). The DC circuit model seeks to account for the observed X-ray luminosity (10³⁵–10³⁶ erg s⁻¹ for RX J1914+24 assuming a distance of 100 pc) without mass accretion, while allowing for orbital decay driven by gravitational radiation.

Using Equation (11) with parameters appropriate for compact WD binaries, we find

$$\begin{eqnarray} &&\dot{E}_{\rm diss}=3.8\times 10^{29}\zeta _\phi \left(\!{\Delta \Omega \over \Omega }\! \right)\left(\!{\mu \over 10^{32}\,{\rm G\,cm^3}}\!\right)^2 \left(\!{R_c\over 10^4\,{\rm km}}\!\right)^{\!2}\nonumber \\ &&\qquad \times \left(\!{M_\star +M_c\over 1\,M_\odot }\!\right)^{\!\!-5/3} \left(\!{P\over 10\,{\rm min\rm utes}}\!\right)^{\!\!-13/3}\!{\rm erg\ s}^{-1}. \end{eqnarray} \tag{ 25 }$$

Note that μ = 10³² G cm³ corresponds to B_⋆ ≃ 0.5 MG (for R_⋆ = 6000 km), approaching the field strengths of intermediate polars. Obviously, even with the maximum asynchronization (ΔΩ/Ω = 1) and maximum efficiency (ζ_ϕ ∼ 1), $\dot{E}_{\rm diss}$ falls far short of the observed X-ray luminosities. Wu et al. (2002) calculated the resistance of the WD and used Equation (4) to obtain a much higher energy dissipation power (see their Figure 3)—evidently, their result (which was also adopted by Dall'Osso et al. 2006) corresponds to ζ_ϕ ≫ 1, violating our upper limit.

Laine & Lin (2012) recently applied the DC circuit model to study the interaction of close-in planets with the magnetosphere of host stars. From their calculation of the resistances of the planet, host star, and magnetosphere, they suggested that magnetic interaction may affect the orbital evolution of close-in super-Earths on a few Myr timescale and produce hot spots on the surface of the host stars (see also Zarka 2007).

Applying Equation (10) to a planetary system (M_c = M_p, R_c = R_p), we find that the magnetic torque induces orbital decay (assuming Ω > Ω_s, as is the case for close-in planets in a few day orbit) at the rate

$$\begin{equation} {\dot{a}\over a}=-{\zeta _\phi \mu ^2 R_p^2\over a^5M_p(GM_\star a)^{\!1/2}}. \end{equation} \tag{ 26 }$$

The timescale is

$$\begin{eqnarray} &&{a\over |\dot{a}|}=5.7\times 10^{15}\zeta _\phi ^{-1} \left(\!{M_{\star }\over 1\,M_\odot }\!\right)^{\!1/2} \!\left({R_{\star }\over 1\,R_\odot }\!\right)^{\!-6} \!\left(\!{B_\star \over 1\,{\rm G}}\!\right)^{\!-2}\nonumber \\ &&\qquad \times \left(\!{R_p\over 1\,R_J}\!\right)\! \left(\!{\bar{\rho }_p\over 1\,{\rm g\ cm}^{-3}}\!\right)\! \left({a\over 0.04\,{\rm AU}}\right)^{\!11/2}{\rm yr}, \end{eqnarray} \tag{ 27 }$$

where $\bar{\rho }_p$ is the mean density of the planet. Thus, it is clear that even at maximum efficiency (ζ_ϕ ∼ 1), a hot Jupiter (R_p ∼ R_J) or super-Earth (R_p ∼ 0.1R_J) in a P ∼ 3 day orbit (a ≃ 0.04 AU) around a solar-type star (with a typical dipole field B_⋆ of a few Gauss) will suffer negligible orbital decay due to the magnetic torque. If super-Earths migrate to their close-in locations in the early T Tauri phase of the star, when the stellar magnetic field is stronger (a few kG), they may experience appreciable orbital evolution due to magnetic interactions.

The energy dissipation rate is (Equation (11))

$$\begin{eqnarray} &&\dot{E}_{\rm diss}=9.5\times 10^{20}\zeta _\phi \left(\!{\Delta \Omega \over \Omega }\!\right)\! \left(\!{M_\star \over 1\,M_\odot }\!\right)^{\!\!1/2}\!\! \left(\!{R_\star \over 1\,R_\odot }\!\right)^{\!6}\! \left(\!{B_\star \over 1\,{\rm G}}\!\right)^2\nonumber \\ &&\qquad \times \left(\!{R_p\over 1\,R_J}\!\right)^{\!2} \left(\!{a\over 0.04\,{\rm AU}}\!\right)^{\!\!-13/2}\!{\rm erg\ s}^{-1}. \end{eqnarray} \tag{ 28 }$$

If this energy is accumulated (e.g., building up twist from ζ_ϕ ∼ 0 to ζ_ϕ ∼ 1) over time Δt ≃ 2π/ΔΩ ∼ 3 days and then released suddenly (∼ hours), the energy release is ${\sim}\dot{E}_{\rm diss} \Delta t/2\sim 10^{26}$ erg (for the canonical parameter values adopted in the above equation). This is much smaller than the energy release of solar flares (10²⁹–10³² erg s) and of superflares from solar-type stars (∼10³³ erg s; see Maehara et al. 2012).

It is instructive to compare the magnetic torque T_mag = T (Equation (10)) with the tidal torque (due to tide raised on the star by the planet). Parameterizing tidal dissipation by the quality factor Q_⋆, we have

$$\begin{equation} |T_{\rm tide}|=\left({9\over 4Q_\star ^{\prime }}\right){GM_p^2R_\star ^5\over a^6}, \end{equation} \tag{ 29 }$$

where Q'_⋆ = 3Q_⋆/(2k₂) and k₂ is the Love number (Goldreich & Soter 1966). Thus,

$$\begin{eqnarray} {|T_{\rm mag}|\over |T_{\rm tide}|}&=&3\times 10^{-12}\zeta _\phi Q_\star ^{\prime } \left(\!{R_\star \over 1\,R_\odot }\!\right)\! \left(\!{B_\star \over 1\,{\rm G}}\!\right)^{\!2}\nonumber \\ && \times \left(\!{R_p\over 1\,R_J}\!\right)^{\!\!-4}\! \left(\!{\bar{\rho }_p\over 1\,{\rm g\,cm}^{-3}}\!\right)^{\!-2}\! \left({a\over 0.04\,{\rm AU}}\right). \end{eqnarray} \tag{ 30 }$$

With Q'_⋆ ∼ 10⁶–10⁸, the magnetic torque generally cannot compete with the tidal torque.

Although we have shown that the orbital motion of a close-in (non-magnetic) planet cannot generate significant dissipation on the surface of a magnetic host star (see Equation (28)), our result does not exclude other possible forms of star–planet magnetic interactions, such as the intrinsic coronal activity of the star modified by the orbiting planet (e.g., Zarka 2007; Cohen et al. 2011; Lanza 2012). Continued search for the signature of star–planet interaction would be useful (e.g., Shkolnik et al. 2008; Pillitteri et al. 2011; Fares et al. 2012; Miller et al. 2012).

Closed DC circuit connecting a magnetic primary and a non-magnetic companion (Section 2) can be an efficient unipolar engine in a binary system, potentially more efficient than an "open" circuit engine (see Section 4). The power of this DC engine is inversely proportional to the total resistance ${\cal R}_{\rm tot}$ of the circuit. However, we have shown that when ${\cal R}_{\rm tot}$ is less than a critical value, ∼ 16v_rel/c² (in cgs units; see Equation (9)) or 480(v_rel/c) Ohms, the magnetic flux tube connecting the two binary components will be highly distorted and the circuit will break. In this case, a quasi-cyclic unipolar engine may operate in the system (Section 3). Thus, there exists an upper limit to the magnetic interaction torque and the associated energy dissipation rate (Equations (10) and (11)) of any unipolar engine in magnetic binaries. Several previous applications of the DC circuit model to different types of binary systems apparently violated this upper limit. We have shown the following.

• 1.  
  In coalescing double neutron star (NS) or NS–BH binaries. Magnetic interactions cannot synchronize the NS spin with the orbital motion and have a negligible effect on the phase evolution of the gravitational waveform, even for magnetar field strengths. Nevertheless, the energy dissipation associated with the unipolar engine can produce electromagnetic radiation prior to binary merger, although the detectability of this radiation depends on the microphysics of the binary magnetosphere.
• 2.  
  In ultracompact white dwarf binaries. Magnetic energy dissipation induced by orbital motion is too small to account for the observed X-ray luminosities. Thus, the puzzling behaviors of several sources (RX J1914+24 and RX J0806+15) cannot be explained by the unipolar inductor circuit model.
• 3.  
  In close-in exoplanetary systems. Interaction between hot Jupiters or super-Earths and the magnetosphere of their host stars does not lead to appreciable orbital evolution, with the possible exception of the early T Tauri phase, when the stellar dipole magnetic field is higher than 10³ G. Magnetic energy dissipation induced by the orbital motion of the planet is generally negligible compared to the observed energy releases in stellar flares or superflares.

I thank Doug N.C. Lin for many valuable discussions on this topic during the winter of 2010–2011. This work has been supported in part by the grants NSF AST-1008245, NASA NNX12AF85G, and NNX10AP19G.