Preamble

A Simple Model of Coherent Emission Confronting a Number of Puzzles on Pulsars, Magnetars, and Fast Radio Bursts
Bi-Ping Gong∗

Five decades after the discovery of radio pulsars, mainstream theories based on the polar cap still suffer from difficulties in both self-consistency and confrontation with observations \cite{}. Increasing observations of individual subpulses of pulsars—i.e., high brightness temperature, highly polarized, and narrowband nanoshots of the Crab pulsar \cite{2,3}—indicate that they relate to basic emission elements in a pulse window. Moreover, a high degree of circular polarization and rapid orthogonal jump in the position angle of linear polarization are observed in micropulses of both pulsars and Fast Radio Bursts (FRBs) \cite{3-6}, which further requires that such a small element of emission has a cone-core configuration varying rapidly from one subpulse to another. These are both unprecedented challenges and opportunities to understand the origin of pulsar coherent emission. This paper for the first time directly confronts the critical question: the origin of a microstructure with cone-core pattern, by an accumulated magnetic reservoir and tiny dissipation site. The resultant coherent emission expects a position angle jump of 90 degrees occurring simultaneously with sign change of circular polarization and a minimum linear polarization, which is supported by a large number of polarization observations on both pulsars and FRBs and in both microstructures and single pulses.

Such a unified model not only interprets a number of other puzzling questions, but also provides an efficient radiation site for the disk-dome configuration \cite{7,8}, which sheds new light on the mechanism of coherent emission of pulsars and FRBs.

Introduction

Although no consensus has been achieved, investigation on pulsar radiation over five decades forms widely accepted ingredients in the emission mechanism of pulsars \cite{9}. Firstly, the very high brightness temperature requires some form of coherent emission; secondly, the emission involves relativistic electrons and positrons \cite{10} generated in pair cascades \cite{11}; and finally, the emission stems from bipolar regions above the polar cap \cite{9,12}.

Coherent curvature emission (CCE) was the first suggested emission mechanism \cite{10,13}. As pointed out by Melrose \cite{1}, CCE suffers from difficulties of bunch formation and maintenance.

The relativistic plasma emission (RPE) \cite{14} involves a streaming instability generating waves analogous to Langmuir waves, and nonlinear processes in the plasma are required in order to partially convert these waves into escaping radiation. Various suggested instabilities in the first stage are too slow to be effective; the difficulty with the second stage has been referred to as a "bottleneck" \cite{15}.

Linear acceleration emission (LAE) and free-electron maser emission (FEM) \cite{16,17} may be regarded as the second stage of a RPE mechanism \cite{9}. However, radiation from solitons and soliton-like waves undergoing wave collapse is simulated by more generic 2D cases, which casts doubts on the correctness of the soliton radio emission models of neutron stars \cite{18}.

Maybe it is incorrect to assume that the radio source arises from the polar-cap region populated by relativistically outflowing pair plasma \cite{1,9,19}. In other words, the polar-cap gap may be unfavorable for invoking rapid and stable instabilities required in coherent emission.

Reconnection involving Alfvén wave (AW) tends to reproduce coherent emission easier than that in the rotation-driven mechanism, which has been suggested to be responsible for radio pulsars \cite{20}, magnetars \cite{21}, and FRBs \cite{22,23}. Whereas, these reconnection-driven radio emissions have been restricted to the Crab pulsar rather than normal pulsars \cite{24-27}. Because in such Force-free Inside, Dissipative Outside (FIDO) models, the magnetic strength imposed on the current sheet of a few light cylinder radii is approximately the light cylinder magnetic field, which is too weak to account for normal pulsars.

On the other hand, in these reconnection-based models, the reconnections occur either in the whole magnetosphere of the Crab pulsar \cite{17}, or with wave and radiation perpendicular to the magnetic field lines of a long equatorial current sheet \cite{27}, which do not expect a conal-core pattern on pulsar's subpulses and pulse profiles \cite{3,4}.

Apparent similarities show in pulse morphology between radio pulses from PSR B0950+08, a repeating FRB source, and a magnetar \cite{28}, indicating that the same emission mechanism is at play in them. On the other hand, a correlation relating subpulses in pulsed emission of radio-emitting neutron stars to their rotational period (τ ∼ 10⁻³P) is not only seen in magnetars but in members of all classes of radio-emitting rotating neutron stars, regardless of their evolutionary history, power source, or inferred magnetic field strength \cite{29}.

Even shorter subpulses with time scales of 10⁻⁶P have been found, i.e., nanoshots at a time scale of 2 ns in the Crab pulsar, which has a brightness temperature comparable to that of a typical FRB. Therefore, a rapid energy relaxation with an extremely high energy density and length scale of sub-meter is required \cite{2}. Strikingly, FRB 20121102A, the first known repeating FRB source, shows isolated microsecond-duration bursts (also 10⁻⁶P) \cite{30}, as does FRB 20220912A with short-duration structures (∼16 µs), or 'microshots' \cite{31}.

Besides, correlation of coherent radio emission with in-phase and offset high-energy counterparts are shown both in pulsars, magnetars, and FRBs \cite{32-34}, the origin of which is not well understood.

Previous reconnection models have difficulty not only in extending to normal pulsars, giving rise to cone-core beams of subpulses, accounting for rapid Position angle (PA) jumps \cite{3-6}, but also in understanding the huge luminosity discrepancy between pulsars and FRBs \cite{35,36}, as do polar-cap gap models.

This paper directly confronts these long-standing and recent problems through a simple scenario. Firstly, it allows a solar dynamo-like building up of magnetic helicity at the last closed field line region near the light cylinder, playing the role of an energy reservoir. Secondly, plasma condensations eject from such a reservoir carrying field lines out of the light cylinder, which suffers from extremely high differential rotation triggering a forced reconnection at the tip of the last closed field region.

The high energy density at the tiny reconnection site is sufficient to produce electron-positron pairs and accelerate them to relativistic speed. Moreover, along with AW generated by the reconnection, those relativistic pairs undergo efficient particle-wave interaction, which reproduces coherent bunches efficiently.

Furthermore, the guide field at the reconnection site distributes the outward jet into triple beams analogous to reconnection occurring in solar wind and near-Earth magnetotail \cite{37-39}, which well accounts for the conal-core pattern exhibited in, e.g., nanoshots of the Crab pulsar \cite{3} and subpulses of pulsars B0950+08 and B1642-03 \cite{4}, and FRBs \cite{5,6,40}. Finally, the coherent emission occurring at the very end of the last closed field line region ensures that the emission propagates in a 'density cavity' without being thermalized by dense plasma \cite{41}.

This paper is structured as follows. Section II: Magnetic helicity building up and relaxation in the form of pulsar wind. Section III: Forced reconnection reproducing coherent bunch, responsible for narrow radio band and giant pulses. Section IV: Origin of rapid PA jump and its correlation with circular polarization and minimum linear polarization. Section V: States of energy accumulation and relaxation on various sources, absence of spin & gamma-ray radiation. Section VI: Association with early magnetosphere models in particular the disk-dome model and predictions.

II. Magnetic Helicity Building Up, Relaxation and Pulsar Wind

In the FIDO models, reconnection occurs by exerting a strength of light cylinder magnetic field upon a long current sheet which is too weak to invoke radiation responsible for normal pulsars. In fact, magnetic energy relaxation through reconnection in the current sheet can change the force-free condition in the magnetosphere, which allows magnetic helicity to build up near the light cylinder. That is, reconnection at the edge of the reservoir (the tip of last closed field line) provides a non-ideal magnetohydrodynamic (MHD) effect resisting strong magnetic tension, which allows magnetic helicity buildup. If all rotating neutron stars involve more or less magnetic helicity buildup, then it is easy to understand that coherent emission can occur not only in normal pulsars and magnetars, but also in pulsars located beyond the death line.

The helicity built up can be investigated by taking into account the centrifugal effect, described by means of MHD equations for a fluid with infinite conductivity \cite{42-47}. Under such perfect MHD conditions, self-similar magnetosphere solutions are sought for flow from a disk. Therefore, with a magnetic cloud of velocity v = u e_r + v_φ e_φ and magnetic field B = B_r e_r + B_φ e_φ in the spherical coordinate system (e_r, e_θ, e_φ), a simple azimuthal equation of motion can be obtained, in which the sum of the angular-momentum term r v_φ and the magnetic-torque term -B_r B_φ/(μρu) is exactly equal to L₀:

r v_φ - r B_φ B_r = const. = L₀

where μ is the magnetic permeability and ρ is the mass density. In such a case, any increase in plasma angular momentum is exactly balanced by a decrease in torque.

Equation (1) depicting conservation of total angular momentum corresponds to a conservation of total energy flux, S = ρ u r² (where Ω is the angular velocity of the roots of lines. The first term is the kinetic energy flux associated with azimuthal velocity. The second term represents energy transported out by the magnetic field, the Poynting energy flux \cite{45}.

As shown in the second term of Equation (1) and Equation (2), disk formation in magnetized cloud cores is hindered by magnetic braking. The stronger the field, the harder to form disks frozen into which is the so-called magnetic tower \cite{48} responsible for launching magnetocentrifugal outflows \cite{44}.

Therefore, effects that weaken the coupling of the magnetic field to matter are proposed, i.e., enhanced resistivity \cite{43}, such that disk formation and thus buildup of magnetic helicity is enabled as poloidal field lines are frozen into the spinning disk.

With rapid rotation and a light cylinder, a pulsar drags its magnetosphere to co-rotate; the local charge density required for co-rotation is the 'Goldreich-Julian' density \cite{49}, which can be presented in terms of pulsar rotation parameters, spin period and the first derivative of spin period:

n_GJ = -ε₀ (Ω · B) = 7 × 10¹⁶ (P/10⁻¹⁵ s)⁻¹/² \cite{50}

The plasma-filled magnetosphere of GJ density, n_GJ, corresponds to a plasma density of ρ = m_e n_e ≈ m_e n_GJ.

Together with the magnetic field at light cylinder, B_lc = B_s [(2πR)/(cP)]³ (where B_s denotes the surface magnetic field), the radial Alfvénic Mach number—the ratio of the momentum term and the magnetic-force term corresponding to Equation (1) and Equation (2)—can be cataloged into two modes:

(ρμ)¹/² u ≪ 0.1, ∼ 0.1, ≫ 0.1

where u ≈ c is assumed.

For pulsars with M_A ≪ 0.1, i.e., the Crab pulsar, magnetic tension overwhelms the momentum term, which hinders helicity buildup. While reconnection at the tip of the last closed field line provides an effect breaking the force-free condition. Such resistance to magnetic tension allows persistent relaxation via reconnection-triggered radiation at the level of pulsar spindown power. In contrast, in the case of M_A ≳ 0.1, magnetic helicity can be built up (forming an energy reservoir) at the cost of suppressed radiation, which is weaker or much weaker than spindown power. Once the reservoir reaches a critical point, either an abrupt relaxation occurs—responsible for, e.g., magnetars—or normal radiation for, e.g., normal pulsars and even old pulsars.

The energy reservoir near the light cylinder can either be double clouds as shown in Fig. 1a, or multi-cloud as shown at the top of Fig. 1b. The magnetic helicity built up gives rise to a current, ∮ B · dl = μI, as shown in Fig. 1a, which can be further simplified as:

(2πR) B_w = μI = μ ρ_e v π (R₂² - R₁²)

where ρ_e = e n_GJ is charge density, and R₁ and R₂ are inner and outer radii of the hollowed thin cylinder respectively, corresponding to a relative thickness of σ = (R₂² - R₁²)¹/² / R_lc as shown at the top of Fig. 1b.

The current I flows around the hollowed cylinder, giving rise to relativistic wind along the open field line with speed v ∼ c. The strength of magnetic field at the base of the hollowed cone, B_w, of Equation (4) can be rewritten as:

B_w = μ e c n_e σ² R_lc

where plasma density is defined as n_e = χ n_GJ, with parameter χ denoting the deviation from the GJ density.

The correlation relating subpulses of radio-emitting neutron stars to their rotational period, τ ∼ 10⁻³P \cite{29}, can be approximated by their light cylinder radii by:

τ ≈ 10⁻³P ∼ 10⁻² R_lc / c

which gives:
- 10⁻⁵ s for P ∼ 10⁻² s (Crab)
- 10⁻⁴ s for P ∼ 10⁻¹ s (normal pulsars)
- 10⁻³ s for P ∼ 100 s (FRBs, magnetars)

Such a time interval τ mimics the travel time of backward electron-positron pairs produced at the tiny reconnection site through the reservoir, around which a current I' is enhanced as shown in Fig. 1a. As a result, it causes fluctuation in the energy reservoir and thus in energy relaxation.

Substituting τ ∼ (10⁻² R_lc)/c, the volume of the cylindrical reservoir V ∼ 2π σ² R_lc³ and B_w of Equation (5) into the radiation power, the energy relaxation of the pulsar wind can be estimated to be comparable to the spin-down power:

Ė = B_w² V / (μ τ) = 4 × 10²⁴ (P/10⁻¹⁵ s)⁻³ (Ṗ/10⁻¹⁵) \cite{50}

Taking into account Equation (5), an extremely simple relation between the relative radius and the spin period is obtained:

σ = 1.5 × 10⁻⁴ P⁻¹ χ⁻¹/³

Substituting Equation (8) obtained by equating the Poynting flux from the hollowed cylinder to the spindown power into Equation (5), we get the strength of magnetic field of the reservoir consistent both with the spindown power and the Poynting flux:

B_w = 5 × 10¹⁸ χ σ² (P/10⁻¹⁵ s)⁻¹/²

The strength of B-field complying with the pulsar spindown is stronger or much stronger than that at the light cylinder, i.e., B_w/B_lc ∼ 50 for the Crab pulsar, and B_w/B_lc ∼ 5 × 10⁴ for J0726-2612, as shown in Table 1. In other words, the reservoir magnetic energy of the former is much closer to the light cylinder one, B_lc²/μ, than that of the latter. This enables the former to be dissipated and restored instantaneously; whereas the latter takes a much longer waiting time to do so.

The volume V ∼ 2π σ² R_lc³ of the magnetic energy reservoir approximated in Equation (7) can be represented by both a thin cylinder of thickness and height ∼ σ R_lc and radius R_lc, as shown at the top of Fig. 1b; or a flux tube-like volume of radius ∼ σ R_lc and height 2 R_lc, as shown in the middle of Fig. 1a. The latter corresponds to an area of A ∼ σ² R_lc² ∼ km², as shown in Table 1, which provides a clue to the area of thermal emission observed in X-ray \cite{51}.

The relaxation of energy in the reservoir of the hollowed cylinder, -J · E, depends on the rotation energy of the star, -I_s Ω Ω̇, where I_s is the moment of inertia of the star, and the net Poynting flux flowing out of the hollowed cylinder, ∇ · S, where S is given by Equation (2), which are related by:

-I_s Ω Ω̇ + ∇ · S = -J · E = -J_⊥ E_⊥ - J_∥ E_∥

Notice that the dissipation term is composed of parallel and perpendicular components corresponding to the Poynting flux along the open field lines of the polar region, J_∥ E_∥, and coherent emission in the equatorial plane, J_⊥ E_⊥, respectively. The former contributes to the incoherent wind, and the latter gives rise to coherent pulsed radiation in multi-frequency from gamma-ray to radio, i.e., the Crab pulsar. The orthogonal phase between coherent and incoherent emission as shown in Fig. 1a well accounts for the correlation of coherent radio and incoherent high-energy emission exhibited in some pulsars and FRBs to be discussed later.

III. Forced Reconnection, Coherent and Incoherent Superposition, Narrow Radio Band, and Giant Pulse

The tip of the last closed field line carrying a plasma cloud ejects out of the enhanced Y-point light, which leads to a forced reconnection triggered by strong differential rotation near the light cylinder. The resultant coherent radiation of a microstructure automatically carries a conal-core pattern responsible for the radio emission of pulsars and FRBs, which differs radically from both the coherent bunch proposed decades ago and recent FIDO models.

Once an episodic ejection from the reservoir occurs near the light cylinder, such a plasma cloud is carried to a height Δh ≪ σ R_lc (where σ ≪ 1 and σ R_lc corresponds to the size of reservoir) beyond the light cylinder radius, which is equivalent to pulling a bundle of frozen last closed field lines (with frozen plasma condensation) at time t₀ (at phase φ₀) to a height Δh.

If such a cloud is carried to phase φ₁ at a later time t₁ through co-rotation with the magnetosphere of the pulsar, as shown in Fig. 1a, it would result in cloud speed exceeding the speed of light: (R_lc + Δh) Ω > c.

This can be avoided provided that only the footpoint of the last closed field line is carried from φ₀ to φ₁, for a distance Δl ≈ (φ₁ - φ₀) R_lc. Whereas, the tip of the last closed field line is required to stay near its original phase φ₀ (or in an angle range φ₀ < φ < φ₁) at time t₁. To achieve this, the tip of the last closed field line must be stretched to a length Δl ≫ Δh, which is equivalent to ejecting a cloud with speed v_am ≈ ω' R_lc ≈ c, countering the tangent velocity at the light cylinder, ω' R_lc ≈ -ω R_lc, in the comoving frame of the pulsar magnetosphere.

The stretching force against magnetic tension corresponds to an inflow of Poynting flux into the tip of the last closed field line region, which is equivalent to work done per unit time against magnetic tension at a tiny reconnection site:

S_in = F · v_in = -v · (J × B) ≈ - (B²/μ) v_in

where v_in is the velocity of charge flow compressing and stretching the last closed field line bundle into narrower and narrower shape. Consequently, a reconnection is triggered at a critical ratio of width to length, i.e., Δh/Δl ∼ 1/100 of current sheet \cite{24}.

Once such forced reconnection is triggered, the inflow Poynting flux perpendicular to the magnetic field lines at the reconnection site turns around at the center of the X-line of the reconnection site, towards the outflow direction as illustrated in the middle of Fig. 1b (directions perpendicular to the inflow Poynting flux), so that ∇·S_rec = 0 is held, which corresponds to Hall reconnection \cite{52}. This differs from models of pulsar emission with a diffusion region extending to a few light cylinder radii \cite{24-27}, which corresponds to Sweet-Parker reconnection with ∇·S_rec < 0. The resultant outward Poynting flux converts to three components responsible for pair production, accelerating them to relativistic speed, and generating AW respectively:

S_out = J_rec · E_rec + v_rec · J_rec × B_rec + ...

The production of electron-positron pairs depends on the strength of magnetic field at the reconnection site, B_T, which determines the energy density at the reconnection site, ∼ n_e γ² m_e c², where the Lorentz factor of pairs and their number density are γ and n_e respectively. The energy density required, i.e., in the nanoshot of 1,000 Jy during 2 ns of the Crab pulsar, is about 2 × 10¹³ J m⁻³, corresponding to a magnetic field strength B_T ∼ 5 × 10³ T. Such a value of B_T is also consistent with the spindown power of Equation (9), as shown in Table 1.

Such magnetic field strength, and thus energy density at the reconnection site of a small volume ∼ 1 m³, allows not only an energy level well above the criterion required for pair production (1.02 MeV), but also accelerates them to relativistic speed.

On the other hand, the reconnection-induced AW interacting with relativistic pairs leads to a resonance wave that can be depicted by exp(-ω_i t) exp(i ω_r t) \cite{19}, where the real wave frequency is ω_r ≈ k v_A ∼ 1 GHz, with wave number k compatible with radio emission of pulsars and FRBs; and the imaginary one, ω_i = k v_A/(2 R_m), is responsible for damping of the AW. The narrowband radiation exhibited in radio emission of some pulsars and FRBs can be achieved by:

Δω/ω = (ω_r² - ω_i²)¹/²/ω_r = [1 - (1/(2R_m))²]¹/² ≪ 1

which simply requires a Reynolds number of 1 > R_m > 1/2. This in turn corresponds to a dissipation time scale at the reconnection site of τ_rec ∼ (R_m c)⁻¹ ∼ ns, which explains the nanoshots occurring in the Crab pulsar \cite{2}. Moreover, it provides a simple and easy mechanism of wave instability and maintenance required in coherent radio emission of pulsars.

The reconnection-reproduced electron-positron pairs and AW result in particle-wave resonance leading to coherent bunch with enhancement factor F(N_e) ∼ N_e², where N_e is the number of particles in such a bunch. Subsequent interaction of such coherent bunches with the flux tube formed by open field lines around the reconnection site gives rise to coherent emission responsible for radiation of Crab-like young pulsars and regular pulsars \cite{2,4}.

The reconnection occurring at the apex of the last closed field lines produces nanoshots as shown in Fig. 1a, repeating rapidly at a time interval τ_rec ∼ ns. Therefore, the swing of such nanoshots through a pulse window gives rise to a single pulse of a pulsar with enhancement factor:

F_ω(N_e, N) ∼ Σ_N N_e²

in the case of incoherent superposition, where N_e and N are the number of particles in a nanoburst and the number of nanoshots in a single pulse respectively.

A coherent bunch moving along field lines with curvature radius ρ in the vicinity of the light cylinder gives rise to curvature radiation of effective frequency:

ω ∼ (c/ρ) γ³ ∼ 1 GHz

which corresponds to a power of CCE of:

P_cv = (e² c γ⁴)/ρ² ∼ 10⁻³⁰ (γ/10²³)² (ρ/10² m)⁻² ∼ 10¹⁵ W

where the number of plasma in a coherent bunch is N_e = n_e V, with n_e ∼ χ n_GJ ∼ 10 n_GJ ∼ 10²⁰ m⁻³ being the number density, and V ∼ 10³ m³ being the volume of emission site.

The radio power of a nanoburst P_cv ∼ 10¹⁵ W corresponds to a total radio emission energy loss of Ė_na ∼ 10²⁰ W. Considering radio emission energy loss is only a small fraction (typically 10⁻⁶ to 10⁻⁵) of spindown power \cite{50}, the radiation of a nanoshot relates to the spindown power of the Crab pulsar by:

Ė_sd ∼ (10²⁰ W)/(10⁶) (1/f_b) ∼ 10³¹ W

where the beaming factor is f_b = max(ΔΩ/4π, 1/4γ²) ∼ 10⁻⁵ with Lorentz factor γ ∼ 300-1000. In other words, approximately N ∼ 10⁶ nanoshots per second can account for the spindown power of the Crab pulsar of 10³¹ W.

The origin of giant pulses of the Crab pulsar can be interpreted by coherent superposition of 10¹ to 10² short bursts of nanosecond time scale, which gives rise to enhancement factor:

F_ω(N_e, N_c) ∼ Σ_{N_c} N_e²

where N_c is the number of nanobursts (10¹ or 10²) in a subpulse, resulting in an enhancement factor a few orders of magnitude higher than that of incoherent superposition, as shown in Equation (14). In other words, the spindown power can be achieved by coherent superposition of N_c nanobursts as shown in Equation (18), which is much less than the number N corresponding to incoherent superposition as shown in Equation (17).

The critical frequency of the Crab pulsar can be achieved by interaction of a pair of beams (coherent bunches) with the flux tube at an initial pitch angle α_p ∼ 10⁻¹:

ω_c = (e B_T)/(m_e) sin α_p ∼ (5 × 10³ T)(sin α_p/10⁻¹) ∼ 10¹⁹ Hz

which is in the hard X-ray band. The occurrence of coherent superposition as shown in Equation (18) is equivalent to an increase of magnetic field strength B_T by one to two orders of magnitude, so that the critical frequency can reach gamma-ray energies. Such pulsed emission of the Crab pulsar can be extended to 25 GeV via inverse Compton scattering \cite{53}.

Equation (19) is equivalent to pumping the distribution function to ∂f/∂p_⊥ > 0, where p_⊥ is momentum perpendicular to the field line. Such a maser-like process quickly radiates away the perpendicular component of energy, so that the pitch angle reduces substantially at a short time scale:

τ_cool ∼ (γ m_e c²)/(2 σ_T c² γ² U_B sin² α_p) ∼ 10⁻⁴ s

where σ_T is the Thomson cross section and U_B is the magnetic energy density. This time scale can account for the observed time delay between gamma-ray and radio of the Crab pulsar of 280 µs \cite{54}.

Cooled-down electrons or positrons can undergo either cyclotron radiation with mild-relativistic γ ∼ 100 and pitch angle α_p ∼ 10⁻¹, or weak synchrotron radiation with substantially reduced pitch angle α_p ∼ 10⁻³ and significantly weakened magnetic field strength by moving towards the central region where opposite polarity of B-field of flux tube cancels out, as shown in the middle of Fig. 1b. The critical frequency of both can reduce to GHz by Equation (19).

When electrons or positrons travel in the flux tube formed by open field lines with further damped pitch angle α_p ≪ 10⁻³, it gives rise to CCE radiation, so that the cone component of a cone-core structure, originating in the two outside beams in the triple beams, usually contains a component of CCE of linear polarization, as well as cyclotron radiation and weak synchrotron radiation with circular or elliptic polarization.

IV. Simultaneous OPM Jump, Circular Polarization and Minimum Linear Polarization

The orthogonal polarization modes (OPM) puzzle has been found in integrated pulse profiles \cite{55}, in which the plane of polarization can be in two perpendicular or nearly perpendicular states occurring at a longitude of depolarization of linear polarization. Interestingly, rapid orthogonal jumps also appear in subpulses of pulsars \cite{3,4} and FRBs \cite{5,6}, during which depolarization of linear polarization is also seen. This makes the long-standing puzzle even more confusing. Here we show that reconnection-induced triple beams provide a simple intrinsic effect responsible for the orthogonal jump, which must accompany depolarization of linear polarization and change in sign of circular polarization. In other circumstances, it can also produce elliptic polarization and the smooth Rotating Vector Model (RVM) \cite{56}.

As addressed in the previous section, a reconnection event at the tiny site reproduces triple beams and hence coherent bunches, which interact with the magnetic field of the flux tube surrounding the reconnection site, giving rise to radiation depicted by the electric field of a single moving electron \cite{57}:

E_rad(r, t) = [n̂ × {(n̂ - β⃗) × a⃗}] / [(1 - n̂ · β⃗)³ R]

At the reconnection site, this can be simplified as:

E_rad|_rec = ε⃗ e^{i(ωt - kz)} = (x̂ ε_x + ŷ ε_y) e^{i(ωt - kz)}

where n̂ is the unit vector along wave propagation direction, and β̇ = a⃗/c is acceleration. The CCE emission stemming from the cone component of a microstructure can be exhibited by centrifugal acceleration a_cc, and thus the polarization vectors E_p and E_e for positrons and electrons respectively, as shown in the middle of Fig. 1b.

The central jet from the reconnection site is responsible for core emission in a subpulse, which gives rise to radiation invoked by linear acceleration a_la, as shown in the middle of Fig. 1b. Due to relativistic beaming, most radiation is emitted in a narrow cone along the acceleration vector, as depicted by the emission power per solid angle \cite{57}, which is also displayed in the middle of Fig. 1b:

dP/dΩ = (e² a²)/(8π c³) γ⁸ (γθ)² / (1 + γ²θ²)⁵

Although the central jet is normally dominated by linear acceleration, weak synchrotron radiation can also contribute to it. Because outer electron and positron beams must move toward the central jet to make the corresponding critical frequency satisfy ω_c ∼ GHz, which can be achieved by, e.g., magnetic field strength B ∼ 10⁻³ T and pitch angle α_p ∼ 10⁻³. In such circumstances, the corresponding synchrotron power is P_syn ∼ 10¹⁵ W, comparable to that of CCE as shown in Equation (16). Consequently, the core component of a microstructure can compose both linear and circular polarization, as can the cone component, as addressed at the end of the previous section and shown in the middle of Fig. 1b.

The PA variation owing to the swing of line-of-sight (LOS) through the cone-core structure as shown in the middle of Fig. 1b can be displayed by projecting the electric field vector of the central ejector to the plane perpendicular to LOS. The smooth change of position angle can be depicted by RVM as well, except the inclination angle α (misalignment between spin and magnetic axis) of RVM is redefined as the misalignment angle between spin axis and the center of the cone-core emission pattern.

The resultant emission originating from linear acceleration of the central jet, denoted as arrows in the circle shown at the right side of the middle of Fig. 1b, can point both outwardly or inwardly depending on the sign of linear acceleration of positrons with respect to the observer.

The polarization of each beam of the triple structure as shown in the middle of Fig. 1b can compose both linear and circular (elliptic) polarization, which can be depicted by two orthogonal plane waves:

ε_xk = ε_k cos α_k e^{iβ_xk}
ε_yk = ε_k sin α_k e^{iβ_yk}

where k = 0 denotes the core, and k = 1 and k = 2 represent the double cone respectively; α_k denotes the angle between the global electric field of each wave, and β_xk, β_yk represent the phase for each electric field component.

Usually when α_k = 0 or ±π/2, the wave is linearly polarized; and α_k = ±π/4 corresponds to right- and left-handed circular polarization respectively.

The Stokes parameters of a wave as shown in Equation (23) are \cite{58}:

I_k = ε_k²
Q_k = ε_k² (cos² α_k - sin² α_k)
U_k = ε_k² sin α_k cos α_k cos δβ_k
V_k = ε_k² sin α_k cos α_k sin δβ_k

where δβ_k = β_yk - β_xk is the phase deviation between two orthogonal electric field components. Therefore, circular and linear polarization can be distinguished by δβ_k ≠ 0 or δβ_k = 0 respectively. In other words, linear polarization requires δβ_k = 0, while the angle α_k is arbitrary, each value of which has its own PA.

Let us examine what happens to the PA when a specific angle α_k = ±π/4 is chosen:

tan ψ_k = U_k/Q_k = tan⁻¹[2 sin α_k cos α_k cos δβ_k / (cos² α_k - sin² α_k)] = ± tan⁻¹(cos δβ_k)

In such a case, a swing of LOS from, e.g., the cone (k = 1) to the core (k = 0) results in three different PA changes corresponding to Equation (25):

ψ_{1,0} = ψ_1 - ψ_0 =
- 0 when α_0 = -α_1 = ±π/4
- ±π/2 when α_1 = α_0 = ±π/4
- undefined when α_1 = α_0 = 0

By Equation (26), as long as α_1 = α_0 = α_2 = ±π/4, an RVM-like smooth PA variation is expected. In this situation, the radiation intensity of linear and circular components are peaked either at the same phase or separated phases, while the signs of α_0, α_1, and α_2 remain unchanged as LOS swings from cone 1 through core 0 to cone 2. This is widely exhibited in both subpulses and single pulses and in both pulsars and FRBs \cite{4-6,40,55,59}.

In contrast, orthogonal jumps occur whenever α_1 = -α_0 and |α_1| = |α_0| = π/4 as indicated by Equation (26). Such a sign change of α_1 = -α_0 can be achieved either by changing the sign of dominant charge (electrons or positrons) in the cone beams, which is controlled by the guided field at the reconnection site studied extensively in reconnection occurring in solar wind and near-Earth magnetotail \cite{37,39}, or by changing the sign of linear acceleration of the central jet responsible for core emission.

Moreover, by Equation (24), the sign change of α_1 = -α_0 inevitably leads to a sign change of circular polarization, V_1 = -V_0, which automatically explains OPM occurrence accompanying a sign change of circular polarization as exhibited in observations \cite{3-6,59}. Furthermore, such a sign change of α_1 = -α_0 automatically leads to cancellation of linear polarization intensity at, e.g., the joint of cone 1 and core 0 by incoherent superposition of the Stokes parameters:

L_{1,2} = √[(Q_1 + Q_0)² + (U_1 + U_0)²] ≈ 0

which well explains the depolarization of linear polarization intensity during the PA jump. Such prediction of simultaneous change of three observational values is strongly supported by observations from microstructure to single pulses and integrated pulse profiles, and from pulsars to FRBs \cite{4-6,40,55,59,60}.

PA jumps of less than π/2, i.e., approximately 60°, occur \cite{5}. This can be explained by α_1 ≠ π/4 or α_0 ≠ -π/4 in Equation (25), corresponding to elliptical polarization rather than circular, which can be tested by further observations.

Therefore, the polarization of each beam composes a circular component with δβ_k ≠ 0 and α_k = ±π/4, and a linear component with δβ_k = 0, not necessarily requiring α_k = ±π/4, but in the case of orthogonal jumps it does. One of the triple beams, i.e., the cone, can be 100% linearly polarized when dominated by CCE or linear acceleration with δβ_k = 0.

Polarization can vary rapidly from one burst to another depending on different guided fields at the reconnection site, which can change from one reconnection event to another \cite{5,6}. An incoherent superposition of PA from a large number of such microstructures results in the PA of the integrated pulse profile, which usually appears more complicated than that of individual microstructures. Nevertheless, the imprint of the former is obvious, also varying from one single pulse to another as shown in observations \cite{59}.

V. Offset High Energy Components, States of Radiation, Absence of Spin & Gamma-Ray Radiation

After investigating coherent radio emission and its contemporaneous high-energy radiation, we now move on to their non-contemporaneous high-energy emission, which imposes further constraints on magnetosphere and emission site of pulsars and FRBs.

The superb angular resolution of the Chandra X-Ray Observatory of the Crab pulsar exhibits spectral variations consistent with sinusoidal variation of the gamma-ray spectral index, in which the hardest index appears between the main pulses φ = 0 and interpulses φ = 0.4, and the softest index lies between φ = 0.4 and φ = 1.0 \cite{32}, also shown in the middle of Fig. 1b.

Such measurements challenge models of pair cascade processes in pulsar magnetospheres \cite{32}.

In the new scenario, the two phases φ = 0.0 and φ = 0.4 correspond to two bipolar reconnection sites dissipating with J_⊥ E_⊥, which gives rise to multi-frequency coherent emission as shown in Equations (14)-(20) and Fig. 1a.

As the line of sight swings through the phase between these bipolar coherent radiation regions, the radiation is dominated by incoherent emission composed of both thermal radiation from hot spots originating from bombarding of return plasma to the star surface as shown in Fig. 1b (blue flux tubes), and non-thermal Poynting flux (up and down cone) produced by J_∥ E_∥.

Since Hall reconnection produces inward electrons and positrons streaming separately along the separatrix to the star surface, the inward positrons can be accelerated by gaps existing in vacuum regions and bombard the star surface, reproducing hot spots. In contrast, return electrons cannot reach the star surface at high speed and thus cannot produce observable thermal emission like positrons. Consequently, only one hemisphere of the star radiates observable thermal emission. This explains why the softest spectral index occurs in the phase interval Δφ = 0.4-1.0, and the hardest spectral index of gamma-ray radiation appears in Δφ = 0.0-0.4, providing a simple qualitative interpretation for the correlation of spectral index and multi-frequency coherent emission of the Crab pulsar, as shown at the bottom of Fig. 1a. Similarly, PSR J1119-6127 also exhibits in-phase radio and X-ray emission, while its gamma-ray profile peaks at a different phase φ ≈ 0.2 \cite{61,62}.

On the other hand, the huge discrepancy in luminosity between pulsars and FRBs can be explained simply by taking into account coherence effects. A coherent bunch of particle number N radiates with power N² Ė, where Ė is the emission power of a single particle via, e.g., CCE. Consequently, the ratio of radiation power between the microstructure of a pulsar (e.g., the nanoshot of the Crab pulsar) and that of an FRB is:

(Ė_FRB)/(Ė_Crab) ∼ (N_FRB/N_Crab)² ∼ (100)⁶ (10)⁶ ∼ 10⁻¹²

Therefore, a difference in emission site size of 2 orders of magnitude results in ∼12 orders of magnitude deviation in luminosity, in the case of the Crab pulsar nanoshot and an FRB with approximately equal energy density, curvature radius ρ outside the light cylinder, and Lorentz factor γ, as shown in Equation (27).

The huge deviation in luminosity between a magnetar flare and the spindown luminosity of a pulsar (normal or old) can be attributed to their different energy accumulation in the magnetosphere, which can be understood by rewriting Equation (10) into three states:

1. Ė_sd + Ė_pl = Ė_em (normal, spindown)
2. |Ė_sd| ∼ |Ė_pl| (low, nulling, quiescent)
3. |Ė_sd| ≪ |Ė_em| (high, flare)

The first and second terms on the left denote power injection into the energy reservoir at cost of kinetic energy loss of a pulsar (Ė_sd) and power of infalling magnetic energy via twisting field lines into the reservoir (Ė_pl). The term on the right is power output from the reservoir (Ė_em).

Normal pulsars with M_A ≪ 0.1 correspond to extremely strong magnetic tension against field line twisting. It is reconnection at the tiny end of the reservoir (the tip of last closed field line) that provides a non-ideal MHD effect resisting such strong magnetic tension, which allows electromagnetic radiation releasing the kinetic energy of pulsar rotation at a stable spindown power as shown in Equation (28).

The quiescent state denotes that spindown power is transferred to helicity buildup, with B_w ≫ B_lc, rather than radiation. The relaxation of such piled-up energy occurs in two ways: firstly, release of accumulated energy at spindown power level, i.e., in RRATs (Rotating RAdio Transients), a subclass of neutron stars emitting sporadic, short-lived radio bursts with intervals of silence lasting minutes to hours, like RRAT J1819-1458 as shown in Table 1; secondly, energy dissipation of piled-up energy for, e.g., hundreds of days for SGR J1935+2154, corresponding to luminosity much larger than spindown power \cite{35,36}.

The magnetar SGR J1935+2154 is not only one of the most active magnetars detected so far, but also the unique confirmed source of FRBs. FRB 20200428 bursts are distributed in a wide phase range, i.e., the Westerbork bursts and the CHIME 8 October bursts (vertical dashed) differ considerably from that of FAST, appearing to be anti-aligned with the persistent pulse profile in X-ray detected by NICER or XMM-Newton (vertical solid black), as shown at the bottom of Fig. 1a.

Interestingly, five months after its X-ray outburst associated with FRB 20200428, a radio pulsar phase emerged associated with X-ray spectral hardening \cite{34}. Furthermore, the double peaks of such radio pulsar phase are also anti-aligned with the persistent X-ray pulse profile detected by NICER or XMM-Newton \cite{33,34}, resembling the correlation of radio emission and offset gamma-ray emission exhibited in the Crab pulsar.

By Equation (3), the radial Alfvénic Mach number of magnetar SGR J1935+2154 is M_A ≈ 0.1, which is much larger than that of the Crab pulsar as shown in Table 1. This allows helicity buildup in SGR J1935+2154 approximately 3 orders of magnitude stronger than in the Crab pulsar as shown in Table 1. Therefore, the X-ray outburst and FRB state of SGR J1935+2154 correspond to a high dissipation state, while the radio pulsar phase corresponds to a lower radiation state when the reservoir is sufficient only for spindown-level radiation.

The absence of spin exhibited in FRB 20200428, the high state of SGR J1935+2154, may stem from multi-reservoir buildup in the magnetosphere and hence multi-reconnection sites which contaminate the Crab pulsar-like oblique radiation. Five months after such high-state energy relaxation via X-ray outburst associated with FRB 20200428, it enters a low state dominated by oblique radio emission appearing as a radio pulsar.

No significant gamma-ray emission from any SGR or AXP has been detected by Fermi LAT yet \cite{65}. This can be well interpreted by Equation (19). The magnetic field of the reconnection region of the Crab pulsar, B_w ∼ 5 × 10³ T, corresponds to a critical frequency of 10¹⁹ Hz. In comparison, the corresponding magnetic field is B_w ∼ 10¹ T for typical magnetars, e.g., SGR J1935+2154, which expects a frequency in the X-ray band, ω_c ∼ 10¹⁷ Hz, by Equation (19).

Consequently, an enhanced magnetic field at the Y-point plus a tiny reconnection near the light cylinder provides a unified scenario responsible for ordinary pulsars, magnetars, RRATs, FRBs, and even pulsars located beyond the death valley.

VI. Association with Early Magnetosphere Models and Predictions

The main differences between the new model and FIDO models are: (a) helicity buildup in case of non-ideal force-free condition near the Y-point or energy reservoir; and (b) forced reconnection at the tip of the last closed field lines in the vicinity of the light cylinder (LC).

These two items themselves provide a new formation at the Y-point, by accumulated magnetic reservoir connecting with a tiny dissipation site undergoing rapid Hall reconnection. The superposition of a large number of such reconnection events automatically builds up a Y-shaped configuration.

Furthermore, it provides a possible 'physical solution' avoiding the singularity of the pulsar equation at the LC \cite{66}, because Hall reconnection occurring near the LC allows continuity in current and energy flux across the LC in the case of plasma speed not exceeding the speed of light.

In microphysics, such Hall reconnection gives rise to coherent radio emission, by which the problem of the disk-dome configuration believed unable to produce appreciable radiation \cite{7} can be fixed.

With assumptions of GJ-like force-free condition, i.e., free escape of particles (both electrons and ions) from the neutron star surface and no pair creation \cite{7}, researchers have identified a ghost behind the elegant picture by discovering a completely different magnetospheric solution, in which electric field pulls electrons outward and the equatorial disk contains trapped ions, separated by large accelerating gaps existing in vacuum regions between the domes and the disk. These features are the consequence of the charge-separated nature of the solution \cite{8}.

The disk-dome configuration is believed to be the state of every neutron star below the death line (a 'dead' pulsar), which produces no appreciable radiation \cite{7,8}.

As one possible 'solution' to the tension between force-free magnetosphere and breakdown dissipation, the model of this paper deviates from previous ones by piled-up Y-point and a tiny dissipation site, which can be treated as adding a tiny dissipation site to the last closed field region of the disk-dome configuration. The resultant forced reconnection (in fact Hall reconnection) occurring near the LC leads to radiation enabling the disk-dome configuration to produce appreciable radiation for normal pulsars and even old pulsars.

Moreover, Hall reconnection produces: (i) inward electrons and positrons streaming separately along the separatrix to the star surface; and (ii) outward electron and positron beams interacting with surrounding open field lines responsible for cone emission of a microstructure.

Furthermore, trapped positrons piled up at the Y-point are also carried out by Hall reconnection as the central jet, which is responsible for the core component of the triple beams in the cone-core configuration. In other words, electrons pulling out from open field line region are balanced by positrons piled up at the Y-point across the LC by a large number of reconnection events N, so that the current (by positrons) can be continuous across the LC: I_in ∼ Σ_N I_out.

It ends up with electrons drifting along open field lines and positrons escaping from closed field line region across the LC via a large number of reconnections, so that circuit closure is automatically achieved.

Furthermore, reconnection occurring near the LC also corresponds to continuous energy flux across the LC as shown in Equation (17): ε_in ∼ Σ_N ε_out.

Such energy relaxation from the piled-up Y-point is achieved by 'persistent' emission of triple beams triggered by Hall reconnection. On the other hand, such reconnection of tiny length scale in the vicinity of the LC automatically avoids plasma velocity exceeding the speed of light, v > c.

This paper for the first time directly confronts the most critical question raised by current observations: a microstructure with cone-core pattern, by an enhanced Y-point and tiny dissipation site, as shown in Fig. 2. Firstly, reconnection-triggered triple beams automatically give rise to a cone-core configuration responsible for polarization behavior of pulsars and FRBs. That is, an OPM jump must occur simultaneously with sign change of circular polarization and minimum linear polarization percentage. This is strongly supported by numerous polarization observations on both pulsars and FRBs in both microstructures and integrated pulse profiles.

Secondly, the dissipation term is composed of parallel and perpendicular components corresponding to Poynting flux along open field lines of the polar region, J_∥ E_∥, responsible for pulsar wind; and along the equatorial plane, J_⊥ E_⊥, responsible for coherent pulsed radiation in multi-frequency from high-energy emission respectively. The orthogonal phase between coherent and incoherent emission well accounts for the correlation of coherent radio and incoherent high-energy emission exhibited in some pulsars and FRBs.

The predictions are as follows:

1. The predicted correlation of PA jump with sign change of circular polarization and minimum linear polarization, as well as occurrence of elliptic polarization as shown under Equation (26), is expected to occur in: (a) shorter and shorter microstructures, i.e., nanoshots of the Crab pulsar and microshots of FRBs; and (b) microstructures of pulsars with 360-degree radiation.

2. Hall reconnection triggered by enhanced magnetic field (much greater than that of the light cylinder) leads to radiation propagating in a 'density cavity' along the flux tube formed by open field lines, which should leave imprints in both the rotation measure (RM) of the star and PA behavior in microstructures and single pulses.

3. Narrowband radiation occurs in radio emission of some young pulsars and FRBs with small Reynolds number, 1 > R_m > 1/2, as shown in Equation (13), corresponding to very short microstructure time scales. For pulsars and FRBs with R_m ≫ 1 and much longer microstructure time scales, broadband emission is expected.

4. The energy budget responsible for pulsars, magnetars, RRATs, and FRBs depends on the radial Alfvénic Mach number, denoting efficiency of helicity buildup of a star as shown in Equation (3).

5. Subpulses overlapped with each other along the rotation phase for most pulsars correspond to storms of beams radiating in a pulse profile \cite{67}. In contrast, pulsars located beyond the death line like PSR J0250+5854 \cite{68}, which should be invisible in polar-cap gap models, can radiate in raindrops of weak subpulses as shown in Table 1.

6. The predicted correlation of coherent radio with in-phase and offset incoherent high-energy emission should occur both in oblique rotators and approximately aligned rotators, which apparently deviates from polar-cap models. Further multi-frequency observations of pulsars and FRBs together with their polarization behavior, i.e., OPM and RVM, will reveal unprecedented magnetosphere geometry and emission mechanism of pulsars and FRBs.

FIG. 1 [FIGURE:1]. A schematic plot exhibiting coherent emission induced by forced reconnection, not to scale. Panel (a) top: The neutron star at the center of the light cylinder. The small red cylinders near the light cylinder are energy reservoirs created by twisting magnetic field, which give rise to Poynting flux parallel to open field lines. Magnetic reconnection triggered by outward ejectors across the light cylinder invokes coherent emission, while inward ejectors streaming along the separatrix bombard the star surface, causing thermal radiation along open field lines. Panel (a) bottom: The correlation of radio and high-energy emission of SGR J1935+2154 and the Crab pulsar. Panel (b) top: An overview of energy reservoirs (multi-site), which result in multi-ejectors across the light cylinder giving rise to coherent emission. The sum of dissipation from these ejectors is equivalent to a dissipation region approximated by a thin cylinder near the light cylinder. Panel (b) middle: Dissipation of an outward ejector operates via reconnection at the tip of the last closed field line, which produces triple beams. The two outside beams (responsible for cone emission) interact with magnetic field in the flux tube at pitch angle α_p ∼ 10⁻¹, giving rise to synchrotron radiation at high energy, which cools rapidly to the radio band with power comparable to CCE. The central jet produces emission originating from linear acceleration. The resultant linear polarization is denoted by arrows in the circle shown at the right side of the middle of Fig. 1b. Whether arrows point outwardly or inwardly depends on the sign of linear acceleration of positrons with respect to the observer. Panel (b) bottom: The swing of LOS through such triple beams crosses the joint of the cone-core configuration, leading to OPM jumps responsible for observations \cite{5,6}. The left side at the bottom of Fig. 1b denotes two different distributions of beams in the reconnection site controlled by different guided fields.

FIG. 2 [FIGURE:2]. A schematic plot of the model with an enhanced Y-point and tiny dissipation site. In the plot, T, dt and P denote waiting time between single pulses, the time duration of a microstructure and the spin period respectively. E_B represents the magnetic energy piled up at the Y-point.

TABLE I [TABLE:1]. Observational parameters and derived parameters (left and right of the references respectively) of selected pulsars, magnetars and FRB.

P (s) Ṗ (ss⁻¹) Ė_sd (W) B_s (T) reference B_lc (T) R_lc (m) B_w/B_lc B_w (T) 0.033 4.23×10⁻¹³ 4×10²⁶ 4×10⁸ Crab 2×10²⁹ 4×10⁹ 2×10⁻¹² J1119-6127 0.41 3×10²⁵ 1×10¹⁰ 2×10⁻¹² J1819-1458 4.26 3×10²⁵ 3×10⁹ 3×10⁻¹³ J0726-2612 3.44 J1935+2154 3.24 1.4×10⁻¹¹ 4×10²⁷ 2×10¹⁰ J0250+5854 23.5 2.7×10⁻¹⁴ 8×10²¹ 3×10⁹ reference 1×10² 2×10⁶ 5×10⁻³ 5×10¹ 5×10³ 1×10⁻⁵ 1×10⁻³ 2×10⁸ 4×10⁻⁴ 6×10² 3×10² 2×10⁻³ 1×10⁻³ 2×10⁸ 4×10⁻⁵ 4×10³ 4×10⁰ 6×10⁻⁴ 2×10⁸ 5×10⁻⁵ 5×10⁴ 3×10¹ 7×10⁻⁴ 2×10⁸ 3×10⁻⁵ 3×10⁴ 2×10¹ 2×10⁻⁶ 1×10⁹ 3×10⁻⁶ 5×10⁴ 1×10⁻¹ 6×10¹

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