2.1.1. Magnetospheric Interactions of NS–NS/BH Binaries

It has long been suggested that the magnetospheric interactions in compact star binaries can lead to energy dissipation (e.g., [55]). Following [62,69], three cases are considered in this review as shown in Figure 1: case 0 with *B*<sup>c</sup> < *µ*∗*a* −3 , case 1 with *µ*<sup>∗</sup> ∼ −*µ*c, and case 2 with *µ*<sup>∗</sup> ∼ *µ*c, where the subscripts (∗, c) represent the NS and its companion, respectively. The magnetic dipole moment is *µ* = *BR*<sup>3</sup> , where *B* is the surface magnetic field and *R* is the radius. We consider the binary to be of a separation *a*, a mass ratio *q* = *M*c/*M*∗, and an orbital angular velocity Ω = [*GM*∗(1 + *q*)/*a* 3 ] 1/2. Within this framework, the energy dissipation rate of the NS binary system can be well formulated.

Case 0 can be well understood within the unipolar induction directcurrent (DC) circuit model, i.e., the weakly magnetized NS or non-magnetic BH is moving across the magnetic field lines inside the magnetosphere of the NS. This generates an electromotive force (EMF) E ' 2*Rc*|**E**| on the two poles of the companion, where **E** = **v** × **B**c/*c*, **v** = (**Ω** − **Ω**∗) × *a*, and **Ω**<sup>∗</sup> is the spin of the NS. This EMF can drive currents along the magnetic field lines, which makes a closed DC circuit. Note this DC circuit may not always be stable [56]. The resistance of the magnetosphere is R = 4*π*/*c* [70]. The luminosity can then be estimated as [56],

$$L\_{\rm UI} \approx 1.2 \times 10^{42} M\_\*(1+q) \mu\_{\*,30}^2 (R\_\circ/10 \,\text{km})^2 (a/30 \,\text{km})^{-7} \,\text{erg/s},\tag{2}$$

where the mass *M*<sup>∗</sup> is in units of solar mass *M*. Note that, here and below, we adopt the approximation of Ω Ω∗, this is appropriate, as we are considering the last few seconds before the merger. Simulations of inspiraling NS–NS/BH binaries indicate that the main features are well captured by this model (e.g., [58,59,61,63,71,72]).

**Figure 1.** We show the schematics of three typical magnetic field configurations in inspiraling NS binaries: case 0 with *B*<sup>c</sup> < *µ*∗*a* −3 , case 1 with *µ*<sup>∗</sup> ∼ −*µ*c, and case 2 with *µ*<sup>∗</sup> ∼ *µ*c. The red winding arrows represent the emitted photons.

The other cases are more complicated. During the shrinking of the orbit, the magnetospheres of NSs would interact with each other, dissipating the orbital kinetic energy. The location of the interaction is around *r<sup>i</sup>* = *a*/(1 + *e* 1/3), where *<sup>µ</sup>*∗*<sup>r</sup>* −3 *<sup>i</sup>* = *µ*c(*a* − *ri*) −3

and *e* = *µ*c/*µ*∗. If the magnetic field lines from two stars are anti-parallel with each other (Case 1), magnetic reconnection is expected. After the reconnection, the magnetic field lines would connect the two stars directly, leading to the formation of a DC circuit, such as in Case 0 driven by the EMF with an electric field *E* ≈ *a*Ω*B*c*c* −1 . We find that the energy dissipation rate from the DC circuit is generally larger than that in the magnetic reconnection in the last few seconds before the merger, and it can be formulated as [62],

$$L\_{\rm a,UI} \approx 3.8 \times 10^{44} (R\_{\odot}/10 \,\text{km})^{-3} (\epsilon^{5/3} + \epsilon^2) \mu\_{\*,90}^2 (a/30 \,\text{km})^{-2} \,\text{erg/s} \,\text{.}\tag{3}$$

If the magnetic field lines are parallel with each other (Case 2), the field lines would experience compressing at *r<sup>i</sup>* , and the compression location would rotate around the main star at an angular speed Ω − Ω∗. When the compression location moves away, the compressed field lines will relax. This alternate compression and relaxation would lead to an electric field *E* ∼ *µ*∗*r* −2 *<sup>i</sup>* Ω*c* <sup>−</sup><sup>1</sup> and an energy dissipation rate [62],

$$L\_{\rm P} \approx 1.8 \times 10^{43} (0.19/\eta - 0.08)(1 + \epsilon^{1/3})^3 (1 + \epsilon) \mu\_{\*,30}^2 (a/30 \,\text{km})^{-9/2} \,\text{erg/s} \,\text{s} \,\tag{4}$$

where *ηr<sup>i</sup>* is the width of the compression region.

It can be found that the energy dissipation rate increases non-linearly with time. Comparing with Equation (1), we found that it would be detectable only in the last few milliseconds to seconds depending on the magnetic field and distance to the observer. In general, Case 1 would have the highest energy dissipation rate, while Case 0 would have the lowest. We noticed that, for real cases, the magnetic axis may have an inclination angle with respect to the orbital axis, and in these cases, the energy dissipation rate would lie in between the above scenarios. As the poloidal field is the dominant component, we ignored the contribution of the toroidal magnetic field, which are caused by the revolution of the binary system and are observed in many simulations (e.g., [58,59,61,63,71,72]).

The opening angle of the radiation depends on the actual magnetic configuration and the orbital phase. In all cases, the generated acceleration electric field is not parallel to the *B* field. The radiation process would then be dominated by the synchrotron radiation [62]. Based on the threshold (Equation (1)), the magnetic field should be *B* > 10<sup>13</sup> G. While in such a high magnetic field, the high-energy photon will be absorbed, leading to a synchrotron-pair cascade. Using Monte-Carlo simulations, we find the spectral energy distribution (SED) can be well described by a cutoff-power law, with a photon index around −2/3 and peak energy at <MeV [62,73]. This could be understood as the synchrotron radiation by the mildly relativistic electrons with *γ* . 10, as high-energy photons emitted by higher-energy electrons will be absorbed to produce pairs. Therefore, the radiation cone will be of half opening-angle ∼ 1/*γ* = O(0.1), and this radiation cone is rotating with an angular speed at Ω − Ω∗. Note that the magnetospheric interaction can create more open field lines than the isolated NSs, we would expect the outer gap acceleration to operate at around *r<sup>i</sup>* , and so the curvature radiation may dominate after the electrons/positrons, losing their perpendicular moment. Overall, in these cases, the radiation opening angle will be much larger than that of jetted GRBs [62]. This can also be seen from the Poynting flux direction from magneto-hydrodynamics simulations (e.g., [57,58]).
