Observational diversity of magnetized neutron stars

The mass, $M_{\rm ns}$, and radius, $R_{\rm ns}$, of a NS are the targets of extensive theoretical and observational studies, as measurements of both parameters are directly related to the nuclear equation of state (EoS) of superdense matter, and to the nature of the interior composition and state of NSs [460, 461].

A simplistic evaluation of $M_{\rm ns}$ and $R_{\rm ns}$ based on only a few fundamental physical constants is described in some textbooks and reviews (e.g. [119, 490, 555, 719]), which is briefly summarized below. Let us consider that the internal energy of a free Fermi gas composed by ultra-relativistic neutrons $E_{\rm int}\sim N\hbar c/d_{\rm n}$ supports the gravity, $E_{\rm int}>-E_{\rm grav}\sim GM_{\rm ns}^2/R$, where $N\sim M_{\rm ns}/m_{\rm n}$ is the number of neutrons, $\hbar$ is the reduced Planck constant, $d_{\rm n}<\hbar/m_{\rm n}c$ is the separation length of ultra-relativistic neutron, and $m_{\rm n}(\approx m_{\rm p})$ is the neutron ($\approx$ proton) mass. Then, the maximum mass, $M_{\max}$ is estimated to be [460, 461]

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle M_{\max}\sim m_{\rm n} \left(\frac{\hbar c}{Gm_{\rm n}^2}\right)^{3/2} \approx m_p \alpha^{-3/2}_{\rm G} \sim 2 M_{\odot}, \nonumber \end{align} \tag{ 2 }$$

where $\alpha_{\rm G}=Gm_p^2/(\hbar c)=5.9\times 10^{-39}$ is the gravitational fine structure constant. The typical radius of a NS is estimated to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{\rm ns} &\sim N^{1/3}d \nonumber \\ &< \left(\frac{\hbar c}{Gm_{\rm n}^2}\right)^{1/2}\frac{\hbar}{m_{\rm n}c}\approx\alpha_{\rm G}^{1/2}\frac{\hbar}{m_{\rm p}c}\sim3~{\rm km} . \nonumber \end{align} \tag{ 3 }$$

Whereas these give only rough estimates, the orders of which are correct, more accurate theoretical calculations for the realistic nuclear EoS give typical ranges of the maximum mass and the radius of $M_{\max}=$2–3$M_{\odot}$ and $R_{\rm ns}=10$–14 km, respectively (see [460, 461]).

Observationally, NS masses in binary systems have been precisely measured from binary motion, most notably with radio-pulse timing. Figure 2 shows measured masses and orbital periods of known NSs and BHs. One of the most precisely measured NS masses is for the binary pulsar PSR 1913+16, for which the masses of the two NSs were determined with  ∼0.01% accuracy [831]. Figure 3 shows measured $M_{\rm ns}$ values [610], which range from  ∼1$M_{\odot}$ to  ∼$2M_{\odot}$. The fiducial mass, $M_{\rm ns}=1.4M_{\odot}$, is commonly used. In 2010–2013, 2 solar-mass NSs were discovered from three MSPs: PSR J0740+6620 ($M_{\rm ns}=2.17_{-0.10}^{+0.11}M_{\odot}$) [776], PSR J1614−2230 ($M_{\rm ns}=1.908\pm 0.016M_{\odot}$) [49, 192, 253] and PSR J0348+0432 ($M_{\rm ns}=2.01\pm 0.04M_{\odot}$) [34]. In the former two cases, general relativistic Shapiro delay was used to precisely determine $M_{\rm ns}$ (e.g. [261]). A recent candidate for the minimum mass NS is the 4.07 d binary pulsar, PSR J0453+1559 ($M_{\rm ns}= 1.174 \pm 0.004M_{\odot}$) [536]. The ranges of $M_{\rm ns}$ in different NS subclasses, such as binary NSs, pulsars in high-mass x-ray binaries (HMXBs) and low-mass x-ray binaries (LMXBs), are considered to depend on the evolutionary paths of each binary system [610], and are considerably lower than stellar BH masses ($M_{\rm BH}\gtrsim 6M_{\odot}$), as illustrated in figure 3. So far, masses have been typically measured for NSs in binary systems, and it still remains difficult to precisely estimate masses of isolated NSs.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In contrast to measurement of $M_{\rm ns}$, evaluation of $R_{\rm ns}$ (typically $R_{\rm ns}=11$–14 km) is considerably more difficult. One method to determine $R_{\rm ns}$ is using surface radiation, which is usually in the form of x-rays. Observed x-ray flux f  and blackbody temperature T from the surface emission are related as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle 4\pi {\rm d}^2f=L_{\rm th}=4\pi R_{\rm ns}^2\sigma T^4 \quad \rightarrow \quad R_{\rm ns}=d \left(\frac{f}{\sigma T^4}\right)^{1/2}, \nonumber \end{align} \tag{ 4 }$$

where $L_{\rm th}$, d, $\sigma$ are the surface bolometric luminosity, distance to the source, and Stefan–Boltzmann constant, respectively. This method has been applied to, for example, the nearest isolated NS, RX J1856−3754 [106], and x-ray sources in quiescent LMXBs inside globular clusters, for which the distances are known to be measured within 5%–10% accuracy [310].

Several other methods have been developed to constrain the $M_{\rm ns}$–$R_{\rm ns}$ relation (see, for example, [84]). Thermal emission during thermonuclear x-ray bursts from weakly magnetized NSs is a powerful tool for measuring $M_{\rm ns}$ and $R_{\rm ns}$ [179, 608, 748]. With measurement of a gravitational redshift, z, the compactness parameter ($\propto M_{\rm ns}/R_{\rm ns}$; e.g. [96]) can be estimated. Gravitational light-bending affects the observed x-ray pulse profile emitted from a hot spot on the surface of a non-accreting MSP, and thus, precise measurement and theoretical modelling of the x-ray pulse profile provide the compactness parameter (i.e. light-curve modelling method) [98]. A MSP, PSR J0437−4714, with a low-mass helium WD companion is considered to be a promising candidate for the method, for which the mass was independently measured through x-ray and optical observations [51]. Another potential method is to detect gravitationally redshifted absorption lines of highly ionized iron atoms from NS surface radiation (e.g. EXO 0748−676, [167]). This method is expected to be obtained from slowly-rotating and weakly-magnetized NSs, which have an advantage of showing negligible spectral broadening due to Doppler or magnetized effects [209].

Finally, gravitational waves (GWs) from a binary NS merger, named GW170817, were detected by the laser interferometer gravitational-wave observatory (LIGO) along with Virgo [2, 6]. The estimated masses of the two NSs and total mass of the system after the merger are M₁  =  1.36–$1.60M_{\odot}$, M₂  =  1.17–$1.36M_{\odot}$, and $M_{\rm tot}=2.74^{+0.04}_{-0.01}M_{\odot}$, respectively. Although an expected GW signal from the post-merger remnant (NS or BH) was not detected in GW170817 [4, 5], formation of a transiently stable NS is suggested (e.g. [66, 350]) to explain the significant mass ejection, supported from electromagnetic follow-up observations of heated ejecta, so called 'kilonova' (or 'macronova', e.g. [3]). The estimated total mass of GW170817, as well as that of the merged remnant (i.e. a NS or BH), provides a constraint on the maximum mass of cold spherical NSs supported by the EoS [67, 528, 695, 706, 723]. The tidal effects on the GW waveform during its late inspiral phase also limits the NS mass and radius (e.g. [351, 411, 453, 687, 863]). From the observation of GW170817, the radius of a NS with a canonical mass of 1.4 $M_{\odot}$ is constrained to be $R_{\rm ns} \lesssim 13$–14 km [5, 32, 67, 114, 168, 182, 240, 273, 429, 483, 521, 578, 672, 723, 775]. Future observations will give further constraints on the $M_{\rm ns}$–$R_{\rm ns}$ relation with detection of the quasi-periodic oscillations of post-merger remnants (e.g. [65, 349, 795]).

The surface temperature, T, of non-accreting isolated NSs is measurable primarily in soft x-ray and ultra-violet (UV) bands, providing that their magnetospheric non-thermal radiation does not dominate surface radiation. The thermal radiation originates from either or both of the entire surface and one or more small hot spots on the stellar surface. Thermal radiation has been detected from magnetars (section 3.3), XINSs (section 3.5), CCOs (section 3.6), and a few HBPs (section 3.4) and RPPs (section 3.1). After a supernova explosion, a new-born hot isolated NS cools through neutrino emission from the stellar interior at first ($\lesssim$0.01–0.1 Myr), and then, photon cooling from the surface becomes dominant in the later phase [616]. Figure 4 shows a fiducial theoretical cooling curve in [662], which is affected by neutrino emission processes, superfluidity, interior composition, strength and configuration of magnetic fields, chemical composition of stellar atmosphere, and masses of NSs [663, 800, 846].

Zoom In Zoom Out Reset image size

After the first confirmed detection of surface emission by the ROSAT x-ray satellite from four NSs, PSR J0835−4510 (Vela), PSR B0656+14, PSR B0630+18 (Geminga), and PSR B1055−52 [251, 317, 595, 596, 800], the surface temperatures of an increasing number of NSs have been measured (figure 4). In the cooling curve, the observed pulsar ages are estimated from various parameters, including pulsar spin-down timescale (see section 1.3), kinematic age from the proper motion of the targets, energetics of the associated pulsar wind nebulae (PWNe) [760], and ages of the associated supernova remnants (SNRs). Surface radiation can be measured from isolated NSs with ages $\lesssim 10^6$ yr and a typical temperature of $T\sim 10^6$ K, ranging from 10⁵ to 10^6.5 K (0.01–0.3 keV).

In addition to the entire surface emission, the hotter thermal component in soft x-rays (typically  ∼1–$3\times10^6$ K) is the emission from the area of polar caps, which is considered to be heated by bombardment of relativistic particles flowing back to the surface from the pulsar magnetosphere (e.g. [317, 861]). This thermal component may pulsate because the area of the polar caps relative to the bulk NS surface is much smaller than unity, ∼$7\times10^{-4}(P/1~{\rm s}){}^{-1}$ in the dipole case. The energy of the inwardly flowing relativistic particles appears to be powered by the rotation of the NS. In fact, the luminosity from a pair of heated polar caps, $L_{\rm pc}$ is approximately proportional to the spin-down luminosity (see section 1.3), $L_{\rm pc}\sim10^{-3}L_{\rm sd}$ [68, 70]. Consequently, being different from the bulk thermal emission which is significantly reduced at the age of $\gtrsim10^6$ yr (see figure 4), the emission from the polar caps can be observed even in old pulsars ($\tau_{\rm c}\gtrsim10^6$ yr) and MSPs.

The hot surface temperature of magnetars further expanded our knowledge of the above-mentioned classical cooling theories (sections 2.4 and 3.3). Discovered magnetars are systematically hotter than ordinary isolated NSs, as shown in figure 4, the NS-class-divided scatter plot of the age and luminosity. The observations suggest that dissipation of magnetic energy inside a NS becomes an additional energy source for surface x-ray radiation. A simple model assumes a balance between dissipation rate of the magnetic energy and surface radiative-cooling rate (e.g. [644, 646]),

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tr}{{\rm tr}} \displaystyle S\triangle R \frac{{\rm d}u_m}{{\rm d}t} = -S\sigma T^4, \label{eq:thermal_balance} \nonumber \end{align} \tag{ 5 }$$

where S is a size of the hot spot on the surface of a NS, $\newcommand{\tr}{{\rm tr}} \triangle R\sim1$ km is a thickness of the crust where the magnetic energy is dissipated, and $u_m=B^2/8\pi$ is a magnetic energy density in the crust. When the total magnetic field strength in the crust, B, is assumed to be $B=bB_{\rm d}$, where the dipole field, $B_{\rm d}$, is estimated from P and $\dot{P}$ (see section 1.4) and is assumed to follow a simple exponential decay on a timescale, $\tau_{\rm d}$ (i.e. ${\rm d}B/{\rm d}t=-B/\tau_{\rm d}$), equation (5) reduces to

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tr}{{\rm tr}} \displaystyle \triangle R b^{2} B_{\rm d}^2 = 4\pi \tau_{\rm d}\sigma T^4 \quad \rightarrow T\propto B_{\rm d}^{1/2}. \nonumber \end{align} \tag{ 6 }$$

This roughly explains the observed results. In recent years, more dedicated magneto-thermal cooling models have been developed [14, 820] (section 4.4).

The stellar rotation (pulsation) period, P, is a fundamental observable of a NS. The emission from a NS is strongly affected by the presence of a magnetic field, which can control the motion of plasma and can determine the emission geometry. Then, periodic signals with the spin frequency P are observed. Typically, $P\sim 1$ s; however, it ranges from  ∼2 ms for a fast spinning MSP to  ∼5 hours for an accretion-powered x-ray pulsar (see figures 1 and 5).

Zoom In Zoom Out Reset image size

Using the spin angular velocity $\Omega=2\pi/P$, the rotation energy of a NS is given by,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Esd} E_{\rm rot}=\frac{1}{2}I\Omega^2, \nonumber \end{align} \tag{ 7 }$$

where I is the moment of inertia of a NS. The loss rate of this rotational energy, namely the 'spin-down' luminosity, is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Lsd} L_{\rm sd}=-\frac{{\rm d}E_{\rm rot}}{{\rm d}t}=\frac{4\pi^2 I\dot{P}}{P^3}. \nonumber \end{align} \tag{ 8 }$$

One of the energy loss mechanism is the magnetic dipole radiation (e.g. [611]). For a rotating sphere with dipole magnetic field $B_{\rm d}$, the luminosity of magnetic dipole radiation is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Ldip} L_{\rm md}=\frac{B_{\rm d}^2\Omega^4R_{\rm ns}^6}{6c^3}\sin^2\chi, \nonumber \end{align} \tag{ 9 }$$

where $\chi$ is the inclination angle of the dipole magnetic axis relative to the rotation axis. Using a coefficient k, the general spin-down is given by (e.g. [523]),

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \dot{\Omega} = -k \Omega^n. \label{eq:spin-down-equation} \nonumber \end{align} \tag{ 10 }$$

Here the braking index n determines the direction of movement of a pulsar in the $P{{\rm \mbox{--}}}\dot{P}$ plane where its slope is 2  −  n. For a magnetic dipole radiation case ($L_{\rm sd}=L_{\rm md}$), the braking index is n  =  3 derived from equations (8) and (9). For some of other loss processes, n  =  1 for a particle wind braking, and n  =  5 for a GW braking ($\newcommand{\e}{{\rm e}} \propto\epsilon^2 \Omega^6$), where $\newcommand{\e}{{\rm e}} \epsilon=(I_1 - I_2)/I_3$ is equatorial ellipticity, I_m(m  =  1,2,3) are the principal moments of inertia, and I₃ is assumed to be aligned with the spin axis [719]. Observed braking indices ($n\lesssim 3$) are summarized in [232] and plotted in the $P{{\rm \mbox{--}}}\dot{P}$ diagram in figure 6. Isolated NSs spin down gradually, and the solution of equation (10) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:P(t)} P(t)=\left\{\begin{array}{@{}ll@{}} P_{\rm i}\left[1-\frac{(n-1)t}{2\tau_{\rm c}}\right]^{-\frac{1}{n-1}} & (n\neq1) \nonumber \\ P_{\rm i}\exp\left[\frac{t}{2\tau_{\rm c}}\right] & (n=1), \nonumber \\ \end{array} \right. \nonumber \end{align} \tag{ 11 }$$

where $P_{\rm i}$ is an initial period (i.e. $P_{\rm i}=P(t=0)$).

Zoom In Zoom Out Reset image size

Pulsar ages are typically estimated using the spin-down 'characteristic-age', $\tau_{\rm c}$, given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \tau_{\rm c}\equiv \frac{P}{2\dot{P}}, \label{eq:characteristic_age} \nonumber \end{align} \tag{ 12 }$$

where $P_{\rm i}\ll P$ and n  =  3 are assumed. The estimated ages are overlaid in figure 1. The historical records of the eight established galactic SNe, each of which is either a Ia-type SN or CCSN, are tabulated in table 1 (see [149, 741] for reviews on East Asian historical documents). Among CCSNe, the Crab pulsar, which was born in 1054, provides a comparison between $\tau_{\rm c}\sim 1250$ yr and true age t  =  965 yr ($\tau_{\rm c}/t\sim 1.3$) at the time of writing in 2019.

The spin-down $\dot{P}$ of RPPs is typically stable and predictable. However, a sudden jump of P occasionally happens, which is called a 'glitch' [670, 690] (see section 4.3). Another timing irregularity is timing noise, which is the difference in the pulse arrival time from the expected spin evolution (e.g. [346]). The noise shows fairly continuous erratic behaviours. External sources such as interstellar turbulence [159] and intrinsic sources such as magnetospheric state change [499] have been discussed for the potential cause of the timing noise. Among RPPs, $\dot{P}$ of MSPs is stable with small timing noise and rare glitches. These potentially good clocks in space will be used to search for GWs at low frequency (pulsar timing array; 10⁻⁹–10⁻⁶ Hz [256, 816]) and for celestial navigation in future [154].

For accretion-powered pulsars in binary systems, the rotation period P can be also changed by accretion due to angular momentum transfer through torque of accreting matter. The derivative $\dot{P}$ depends on the ratio of the co-rotation velocity and rotation velocity of the accreting matter at the innermost radius of the accretion disk (e.g. [283]). For example, very short periods of millisecond pulsars are results of spin-up by accretion from a companion (recycled scenario; [27, 180, 239, 671, 729]). In fact, accreting NSs with spin periods of milliseconds have been discovered in LMXB systems (e.g. [837]; the bottom panel of figure 7). Recently-discovered transitional MSPs also support a close link between radio MSPs and LMXBs [39, 64, 619, 628]. These MSPs are usually weakly magnetized and are not covered in this review.

Zoom In Zoom Out Reset image size

The characteristic magnetic field strength of magnetized NSs is $B\sim 10^{12}$ G (see, 1 T  =10⁴ G), observationally ranging from 10⁸ G to 10¹⁵ G. The broad dispersion of the field strength, structure, and evolution of B is thought to govern the diversity of NSs. The surface dipole magnetic-field strength, $B_{\rm d}$, of isolated NSs is usually estimated by measurement of P and $\dot{P}$ for isolated pulsars, while the magnetic filed of accretion-powered pulsars is estimated with x-ray spectral features and sometimes with their spin periods and x-ray luminosity. Figure 5 shows the P–B diagram converted from the $P{{\rm \mbox{--}}}\dot{P}$ diagram (figure 1), with the data points superposed of accretion-powered pulsars of which B is measured via the spectral feature. This figure shows various NS classes similar to the $P{{\rm \mbox{--}}}\dot{P}$ diagram.

Canonical pulsar models interpret a pulsar as a rotating magnetic dipole and postulate that rotational energy (equation (7)) is converted into electromagnetic energy [606, 611, 612]. Assuming that the Poynting flux from a rotating dipole magnet in vacuum (equation (9)) is equal to spin-down luminosity (equation (8)), we can obtain the dipole magnetic field strength, $B_{\rm d}$, on the NS surface from the relation [288],

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B_{\rm d} &=\sqrt{\frac{3c^3I}{2\pi^2R_{\rm ns}^6}P\dot{P}} \nonumber \\ &= 1.0\times 10^{14}~{\rm G} \left(\frac{P}{1~{\rm s}}\right) ^{1/2} \left(\frac{\dot{P}}{10^{-11}~{\rm s~s^{-1}}}\right)^{1/2}, \label{eq:spindown-B} \nonumber \end{align} \tag{ 13 }$$

where I is a moment of inertia, and the orthogonal rotator case (inclination angle $\chi=90^\circ$) is assumed. The dipole magnetic field is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\bf B}=\frac{B_{\rm d}R_{\rm ns}^3 }{r^3} \left( \cos\theta_{\rm m}{\bf e}_{\rm r} +\frac{1}{2}\sin\theta_{\rm m}{\bf e}_{\theta} \right), \nonumber \end{align} \tag{ 14 }$$

where ${\bf e}_{\rm r}$ and ${\bf e}_{\theta}$ are the polar unit vectors and $B_{\rm d}$ is the strength of the dipole field at the magnetic pole ($r=R_{\rm ns}$ and $\theta_{\rm m}=0$)⁷. The estimated $B_{\rm d}$ is shown in the $P{{\rm \mbox{--}}}\dot{P}$ diagram in figures 1 and 9. A large fraction of the spin-down energy is considered to be converted into particle flows (pulsar winds), and the remaining fraction is into radiation (see section 3.1 and figure 13).

For accreting x-ray pulsars, the dipole magnetic field affects the flow of the accreting matter [180, 456, 666]. Because of the steep radius-dependence of the magnetic pressure of the dipole component ($B_{\rm d}^2/(8\pi)\propto r^{-6}$), the magnetic pressure is higher than the ram pressure of the accreting matter ($\propto$r^−2.5 for the Keplerian disk) at locations close to the pulsar. The inner boundary of the Keplerian disk (the magnetospheric radius, $r_{\rm m}$) is given by,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r_{\rm m}=\xi r_{\rm A}=\xi\left(\frac{B_{\rm d}^4R_{\rm ns}^{12}}{4GM_{\rm ns}\dot{M}^2}\right)^{1/7}, \nonumber \end{align} \tag{ 15 }$$

where $r_{\rm A}$ is the radius at which the magnetic pressure balances with the ram pressure of the spherically accreting matter [221], and the dimensionless parameter $\xi(\sim0.4{{\rm \mbox{--}}}1)$ depends on the configuration of the accretion flow and micro-physical parameters such as magnetic diffusivity [105, 283, 454, 666]. Note that the mass accretion rate is estimated from the observed luminosity ($\dot{M}\propto L_{\rm x}$; see section 2.3). The accreting matter loses (or gains) an angular momentum until the spin period of the pulsar matches the Kepler period at $r_{\rm m}$. In most accretion-powered x-ray pulsars, the spin-up (or -down) timescale is much shorter than their ages (e.g. [220]). Then, the magnetospheric radius equals to the co-rotation radius $r_{\rm co}$ given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle r_{\rm co}=\left(\frac{GM_{\rm ns}P^2}{4\pi^2}\right)^{1/3}. \nonumber \end{align} \tag{ 16 }$$

Using the approximation $r_{\rm m}\approx r_{\rm co}$, the magnetic field is estimated from the spin period and the luminosity with the equation

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B\sim7\times10^{12}~{\rm G}~\xi^{-7/4}\left(\frac{P}{100~{\rm s}}\right)^{7/6}\left(\frac{\dot{M}}{10^{-11}M_{\odot}{\rm yr}^{-1}}\right)^{1/2}. \label{eq:accretion-B} \nonumber \end{align} \tag{ 17 }$$

The rotation periods of accreting pulsars range from  ∼0.1 s to  ∼10⁴ s for HMXBs, and from  ∼10⁻³ s to  ∼1 s for LMXBs. Figure 7 shows the diagram of the spin period versus orbital period, so-called 'Corbet' diagram.

The magnetic field of accreting x-ray pulsars, B, is also measured directly from a spectral feature, called electron cyclotron resonance scattering feature (CRSF). Accretion matter from a companion star is channelled into NS magnetic poles along the strong dipole magnetic field, forming accretion columns. X-ray pulsation originates from these one or two 'hot' accretion columns. In the strong magnetic field close to the surface of a NS of this type, the movement of electrons is restricted to one direction along the strong field lines, and their transverse energies are quantized to Landau levels,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_n=\sqrt{m_{\rm e}^2c^4\left(1+2n\frac{B}{B_{\rm cr}}\right)+p_{\parallel}^2c^2} \nonumber \end{align} \tag{ 18 }$$

which is a function of the electron rest mass energy $m_{\rm e}c^2$, quantum number n, the electron momentum parallel to the magnetic field $p_{\parallel}$, and the magnetic field strength B normalized to the critical field,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B_{\rm cr}\equiv \frac{m_e^2c^3}{\hbar e}=4.414\times 10^{13}~{\rm G}. \label{eq:ciritical_field} \nonumber \end{align} \tag{ 19 }$$

In the non-relativistic momentum case ($p_{\parallel}\ll m_{\rm e}c$) and $B<B_{\rm cr}$, the energy is approximately⁸

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_n -m_{\rm e}c^2 = m_{\rm e}c^2\frac{B}{B_{\rm cr}}n. \label{eq:CRSF1} \nonumber \end{align} \tag{ 21 }$$

X-ray emission from the accretion column are scattered by electron in the magnetic field, provoking transition of electron energy between these Landau levels. This information has been used to measure the magnetic field of accretion-powered x-ray pulsars ($B\sim 10^{12}$ G), and more recently, that of magnetars ($B\sim 10^{14}$–10¹⁵ G) using signatures of proton cyclotron resonance [101, 102, 700, 788]. 'Electron' CRSFs are typically detected in absorption at an energy $E_{\rm cyc}$ of (converted from equation (21))

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\rm cyc}=11.6\left(\frac{B_{\rm cyc}}{10^{12}~ {\rm G}}\right)~{\rm keV}, \nonumber \end{align} \tag{ 22 }$$

where $B_{\rm cyc}$ is the magnetic field strength at the scattering location [150, 518]. Electron CRSFs have been detected from 35 pulsars at the time of writing, using phase-averaged or phase-resolved x-ray spectroscopy. Table in [740] tabulates the full list (see also figure 8).

Zoom In Zoom Out Reset image size

The dipole magnetic field $B_{\rm d}$ of a few highly-magnetized NSs (e.g. magnetars section 3.3) exceeds the quantum critical field $B_{\rm cr}$ (equation (19), figure 5), at which the Landau level separation becomes equal to the rest mass energy of electrons. Exotic physical processes, e.g. vacuum birefringence and photon splitting [320] are observed above this critical field $B_{\rm cr}$. X-ray spectral analyses potentially enable us to investigate electromagnetic interactions under ultra-strong magnetic fields in strongly magnetized NSs.

Figure 9 shows the distribution of the B-fields of various NS classes, compared with those of isolated and binary magnetic WDs [247]. The magnetic fields of isolated WDs are observationally in a range of 10³–10⁹ G, while those of binary magnetic WDs are in a range of $7\times 10^{6}$–$2\times 10^8$ G, determined using Zeeman and cyclotron spectroscopies.

Zoom In Zoom Out Reset image size

A rotation-powered NS dissipates their rotational energy by emitting pulsar wind and electromagnetic radiation and spin down at a steady rate. They are primarily detected in the radio frequency, and some of them are also detected in optical, x-ray, and/or gamma-ray bands. They collectively account for a majority of the observed NSs (figure 1). Most radio pulsars and MSPs are classified as rotation-powered pulsars (RPPs).

Let us consider the radiation originates from rotational energy of a NS. The moment of inertia I of a NS is very high,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle I\sim1.61\times 10^{45}~{\rm g~cm^2}~ \left(\frac{M_{\rm ns}}{1.4M_\odot}\right) \left(\frac{R_{\rm ns}}{12~{\rm km}}\right)^2. \label{eq:I} \nonumber \end{align} \tag{ 23 }$$

With the high moment of inertia and rapid rotation (the typical period of P  =  0.01–1 s), the rotational energy (equation (7)) is also high, as given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\rm rot} \sim3.18\times 10^{46}~{\rm erg}~P_0^{-2}. \nonumber \end{align} \tag{ 24 }$$

Hereafter, the dependence on $M_{\rm ns}$ and $R_{\rm ns}$ is omitted for simplicity, assuming the fiducial values of $M_{\rm ns}=1.4M_{\odot}$ and $R_{\rm ns}=12$ km in equation (23). The loss rate of this rotational energy (equation (8)) is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{\rm sd} \sim6.35\times 10^{35}~{\rm erg~s^{-1}} P_0^{-3}\dot{P}_{-11}. \nonumber \end{align} \tag{ 25 }$$

In the case of magnetic dipole radiation (equation (9)), the loss rate is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{\rm md} \sim 2.88\times10^{35}~{\rm erg~s^{-1}} B_{\rm d,14}^2 P_0^4 \sin^2\chi. \nonumber \end{align} \tag{ 26 }$$

The parameters P and $B_{\rm d}$ are the primary fundamental physical parameters that lead to diversity in the five basic parameters discussed in section 1.

The latent internal heat of NSs makes them thermal x-ray sources (see NS-cooling curves in figure 4). Assuming isotropic radiation from the surface of a NS at temperature T, the surface luminosity is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:Lth} L_{\rm th}=4\pi R_{\rm ns}^2\sigma T^4 \nonumber \end{align} \tag{ 27 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sim\!1.5\times 10^{35}~{\rm erg~s^{-1}} \left(\frac{kT}{0.3~{\rm keV}}\right)^4, \nonumber \end{align} \tag{ 28 }$$

where k is Boltzmann constant. This equation (27) gives measurable parameters of T and $R_{\rm ns}$ for this class, as a very rough estimate. In reality, the radius $R_{\rm ns}$ depends on the assumed models for the surface emission of the star, atmosphere, condensed phase, magnetic field, and also the surface temperature distribution (e.g. [663]).

Several types of x-ray binaries with various types of companion stars are classified into this class (figure 7). Accretion-powered NSs release the gravitational potential energy of mass accretion flow from a companion star and radiate mainly x-rays. In this class, P and $\dot{M}$ are observationally important parameters.

The gravitational energy for accretion is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tr}{{\rm tr}} \displaystyle \triangle E_{\rm grav}=\frac{GM_{\rm ns}\triangle M}{R_{\rm ns}} \nonumber \end{align} \tag{ 29 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tr}{{\rm tr}} \displaystyle \sim\! 3.1\times 10^{49}~{\rm erg}~ \left(\frac{\triangle M}{10^{-4} M_{\odot}}\right), \nonumber \end{align} \tag{ 30 }$$

where $\newcommand{\tr}{{\rm tr}} \triangle M$ is the mass accreted onto the NS. The corresponding accretion luminosity is

$$\begin{align} \newcommand{\e}{{\rm e}} \newcommand{\tr}{{\rm tr}} \displaystyle L_{\rm acc}=\frac{\triangle E_{\rm grav}}{\triangle t} \nonumber \end{align} \tag{ 31 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sim\!0.98\times 10^{35}~{\rm erg~s}^{-1} \left(\frac{\dot{M}}{10^{-11} M_{\odot}/{\rm yr}}\right)\left(\frac{\Delta t}{10^7~{\rm yr}}\right)^{-1}, \nonumber \end{align} \tag{ 32 }$$

where the accretion rate is normalized by the typical accretion lifetime ($\newcommand{\tr}{{\rm tr}} \Delta t=\triangle M/\dot{M}$) and the dependency on $M_{\rm ns}$ and $R_{\rm ns}$ is omitted. In a spherically symmetric case, the accretion luminosity has an upper limit, called the Eddington luminosity, which is calculated from a balance between the gravitational force and radiation pressure as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{\rm Edd} = \frac{4\pi c GM_{\rm ns}m_p\mu}{\sigma_T} \nonumber \end{align} \tag{ 33 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sim\! 2.2\times 10^{38}~{\rm erg~s^{-1}} \left(\frac{\mu}{1.2}\right) , \label{eq:EddingtonLuminosity} \nonumber \end{align} \tag{ 34 }$$

where $\mu$ is the ratio of the number of nucleons to that of electrons ($\mu=1$ for hydrogen, $\mu=2$ for helium, and  ∼1.2 for the cosmic-abundance elements) and $\sigma_T$ is the Thomson scattering cross-section.

Most of bright x-ray sources in binaries are accretion-powered objects. A subclass of them, called x-ray bursters, hosting weakly magnetized NSs, sometimes exhibit burst activities, and they are powered by another energy source, that is, thermonuclear reactions. These weakly magnetized NSs with thermonuclear x-ray bursts are not in the scope of this review (see [271, 272, 472, 746]). We note that the rotational energy loss is suggested to contribute sometimes to some kind of activity in the accretion-powered class [622–624], most notably radio jets of Cir X-1 [742] and Sco X-1 [278]. However, we do not discuss it in detail in this review.

Magnetars are considered to be primarily powered by the magnetic energy stored in a stellar interior and magnetosphere ([20, 264, 287, 422, 423, 425, 636, 637] for recent numerical works). Figure 10 shows a simplified schematic picture. Their primary observational parameters are P, $B_{\rm d}$, and T. Magnetars are thought to harbour toroidal and/or higher multipole fields, which would be stronger than the dipole field $B_{\rm d}$ at near and interior of a NS (see section 4). Using the volume-averaged strength of magnetic field, $\langle B\rangle$, the available magnetic energy budget is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\rm mag}=\left(\frac{\langle B\rangle^2}{8\pi}\right) \left(\frac{4\pi}{3}R_{\rm ns}^3\right) \nonumber \end{align} \tag{ 35 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sim\! 2.9\times 10^{47}~{\rm erg} \ \langle B\rangle_{15}^2. \nonumber \end{align} \tag{ 36 }$$

Zoom In Zoom Out Reset image size

Dividing this by a typical age of the magnetar class, $t_{\rm mag}\sim 10$ kyr, gives a magnetically-powered luminosity

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{\rm mag}=\frac{E_{\rm mag}}{t_{\rm mag}} \nonumber \end{align} \tag{ 37 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \sim\! 9.2\times 10^{35}~{\rm erg~s^{-1}}~ \langle B\rangle_{15}^2 \left(\frac{t_{\rm mag}}{10~{\rm kyr}} \right)^{-1}. \label{eq:Lmag} \nonumber \end{align} \tag{ 38 }$$

The luminosity in equation (38) is the upper limit on the electromagnetic radiation, given that neutrino emission carries away a significant fraction of the dissipated energy in reality. The photon luminosity highly depends on the depth of the dissipation point in a NS [73, 380].

X-rays are one of the key observational probe for surface and magnetospheric radiations from young and highly-magnetized NSs. Figure 12 shows the observed x-ray luminosity $L_{\rm x}$ of various isolated NS classes, compared with their spin-down luminosity $L_{\rm sd}$. The intrinsic x-ray luminosity at its distance d is estimated to be, from the absorption corrected observed flux $F_{\rm x}$,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle L_{\rm x}=4\pi d^2F_{\rm x} \nonumber \end{align} \tag{ 39 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle =1.2\times 10^{32}~{\rm erg~s^{-1}}~ \left(\frac{d}{1~{\rm kpc}}\right)^2 F_{\rm x,-12}, \nonumber \end{align} \tag{ 40 }$$

where $F_{\rm x,-12}=F_{\rm x}/10^{-12}~{\rm erg~s^{-1}~cm^{-2}}$. As demonstrated in previous subsections, any of the luminosity of the four energy sources can be normalized to units of 10³⁵ erg s⁻¹ despite their different spectral shape if typical observational values of P, $B_{\rm d}$ (or B), and T are assumed (figure 11). Bright x-ray sources with observed fluxes of $\gtrsim$1 mCrab¹⁰ include most of the NS classes within a distance of 10 kpc, demonstrating diversity of NSs.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The conversion efficiency $L_{\rm x}/L_{\rm sd}$ is the capacity of a pulsar to convert its rotational energy to x-ray radiation. The x-ray and spin-down luminosity range for four orders of magnitudes (figure 12). Yet, a relation $L_{\rm x}\propto L_{\rm sd}$ has been established to be statistically significant, though the scatter is large [68, 70, 387, 415, 482, 661, 713]. One of the possible origin of this scatter is thought to be magnetic activities of NSs [724].

A majority of RPPs are detected in the radio band (figure 1). Their primary energy source is the rotational energy provided by the magnetic field braking (see section 2.1). The rotational periods P of RPPs range from  ∼  a few 10 ms to  ∼10 s. The dipole magnetic field of RPPs is in a range of 10¹⁰–10¹³ G except for the MSPs. The nominal value is $B_{\rm d}\sim 10^{12}$ G.

The observed energy spectrum of RPPs is basically composed of thermal blackbody and non-thermal power-law components. The thermal emission is usually seen in optical to soft x-rays, and its origins are considered to be the latent heat inside the NS and the heated polar-cap when it is hit by returning particles from the magnetosphere (see also section 1.2). Non-uniformity of the heated regions results in pulsations in the radiation for the thermal component, of which the pulse profiles usually show a sinusoidal shape. While the thermal emission from the entire stellar surface gives an estimation of the NS radius for a given distance to the source (section 1.1), its temperature provides information of its internal structure and the EOS if compared with theoretical NS-cooling curves (e.g. [846]).

The non-thermal emission is roughly divided into two components: an incoherent broad-band emission from infrared to gamma-rays and a coherent component in the radio band, of which flux density spectra are known to be steep (averaged spectral index of  ∼−1.6 [371, 530]). Particle acceleration and pair-cascade processes create the non-thermal emission. Figure 13 plots its observed luminosity. The numbers of RPPs detected in gamma-rays, x-rays, optical, and radio wavelength are of the orders of  ∼100, ∼100, ∼10, and  ∼1000, respectively (figure 14). The rotationally-induced electric field along the magnetic field lines accelerates charged particles to relativistic speeds. Curvature and/or inverse-Compton-scattered photons from the accelerated particles are further converted to electron-positron pairs through the pair-cascade process [705, 747]. It is considered that the observed $\gamma$-ray photons are emitted by accelerated particles and that the observed optical to hard x-ray emission is emitted by secondary and higher generation pairs [321, 415, 701, 756]. The coherent radio emission would be related to the pair cascade process, although the emission mechanism remains highly uncertain (e.g. [546]). The luminosity in gamma-ray, x-ray, optical and radio bands relative to the spin-down luminosity $L_{\rm sd}$ are 10⁻²–1, 10⁻⁵–10⁻³, 10⁻⁸–10⁻⁵, and 10⁻⁷–0.1¹¹, respectively (figure 13). The light curve of the non-thermal components shows sharp peak(s) because of the formation of caustics [144, 575, 701] and sometimes of a bridge emission. Most gamma-ray pulse profiles display a double-peaked structure [131] and are in general different from those observed in the other energy bands (e.g. [526]).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

RPPs evolve in the $P{{\rm \mbox{--}}}\dot{P}$ diagram (equations (11) and (46) for the cases of steady and decaying magnetic dipole field, respectively), and after crossing the 'death-line', their electromagnetic radiation becomes undetectable. The 'death-line' in the $P{{\rm \mbox{--}}}\dot{P}$ diagram corresponds to the boundary condition where electron-positron pairs can or cannot be produced in the RPP magnetosphere. A rotationaly-induced electric potential difference across the open magnetic field lines sets a limit on the maximum particle energy as $\gamma m_{\rm e}c^2\leqslant e\Delta\phi$, where $\gamma$ is a Lorentz factor of a particle. This potential difference is determined with the magnetic field $B_{\rm lc}$ at the light cylinder $R_{\rm lc}$ at which radius the co-rotation velocity reaches the speed of light, i.e.

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle R_{\rm lc}=\frac{c}{\Omega}=\frac{cP}{2\pi}. \nonumber \end{align} \tag{ 41 }$$

The potential difference is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \Delta \phi&=\frac{1}{2}B_{\rm lc}R_{\rm lc} \nonumber \\ \nonumber &=\frac{1}{2}\frac{B_{\rm d}R_{\rm ns}^3}{R_{\rm lc}^2}\propto B_{\rm d}P^{-2}. \nonumber \end{align} \tag{ 42 }$$

Let us consider the curvature radiation from the maximally accelerated particles as parent photons, of which the characteristic energy is $E_{\rm cur}\propto\gamma^3\propto(B_{\rm d}P^{-2}){}^3\propto L_{\rm sd}^{3/2}$. The pair production criterion is given by

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle E_{\rm cur}>\frac{2m_{\rm e}c^2}{\sin\theta}, \nonumber \end{align} \tag{ 43 }$$

where $\theta$ is the angle of the momentum of the photon to the magnetic field for magnetic pair creation or the collision angle for two-photon pair creation. In the case of $\sin\theta\sim1$, the condition of the pair production ('death line') is given as $L_{\rm sd}=$ const. (see also [615]). This death line is almost consistent with those derived by [140, 705] at $B_{\rm d}>10^{11}$ G, and is observationally consistent with the region where RPPs lose power to radiate radio emission.

Among RPPs, the Crab pulsar is the most extensively studied source, for which a supernova (SN) explosion was recorded in 1054 AD (see table 1). From the rotation period of $P\sim33$ ms and its derivative $\dot{P}\sim4.2\times10^{-13}$ s s⁻¹, the Crab pulsar is one of the most powerful known RPPs ($L_{\rm sd}\sim4.6\times10^{38}$ erg s⁻¹). The pulse was detected in broad energy bands, from radio to even very high-energy gamma-ray bands [23, 24, 33, 817]. The pulse profile shows two main peaks plus some additional minor components in radio bands (e.g. precursor, high- and low-frequency components [318]). Being different from most of other RPPs, the two peaks appear at almost the same rotation phase in all energy bands from radio to gamma-rays [10, 446, 535, 571, 605, 703]. This indicates that both the emission sites for the coherent radio and the incoherent high-energy components lie near the light cylinder [217], although a variation in optical and hard x-ray polarization properties across a rotational phase implies a sign of different emission sites between these energy bands [138, 728, 810]. The Crab pulsar is surrounded by a PWN (section 5.1), the well-known Crab Nebula. The broad-band spectral energy distribution (SED) of the Crab Nebula (figure 15) has been thoroughly studied and established for a remarkably broad band from low-frequency radio to very high energy gamma-rays [337]. Although the Crab Nebula is by now known to show a luminosity fluctuation in x-rays within a level of $\sim5$% [433, 838] and gamma-ray flares (>100 MeV) with short timescales (from hours to days) [8, 109, 110, 762], the Crab Nebula has been and still is used as a standard calibration target in the broad energy range.

Zoom In Zoom Out Reset image size

Other well-known examples of RPPs include three nearby middle-age pulsars, nicknamed 'the Three Musketeers': PSR B0656+14, PSR B1055−52, and Geminga. They were first detected by the Einstein Observatory [88, 141, 164]. Their rotation periods are 200–400 ms and the derived surface magnetic fields are (1–5)$\times 10^{12}$ G. Their x-ray spectra are composed of a cool blackbody from the entire surface, a hot blackbody from smaller hot spots, and a power-law component from the magnetosphere [185].

Some RPPs exhibit several types of irregular timing behaviours in the radio band. One of them is 'nulling', and it is the phenomenon whereby radio pulses cease abruptly and then eventually return to their normal pulsating state [57]. The nulling phenomenon is known to be fairly common in radio pulsars [86], where the null fraction (the fractional duration for which the pulsar is in a null state) ranges from  ∼0.01 to more than 50% [829]. Another somewhat similar one is 'mode changing', and it is a discontinuous change in the pulse profile between two or sometimes more quasi-stable states (or strictly speaking, those that appear to be states). PSR B1237+25 is one of the most famous [56].

The pulsars that show nulling or mode changing occupy the almost same region in the $P{{\rm \mbox{--}}}\dot{P}$ diagram (upper left panel of figure 16). Potentially, nulling and mode-changing may simply be different manifestations of the same basic physics, given that the observational classification between the two phenomena is limited by signal-to-noise ratio of the radio observations (e.g. [855]). Nulling and mode-changing are observed in broad radio-frequency range. However, it has not been known whether these phenomena exist in the higher energy band because of the difficulty of single-pulse detection. Recently, the synchronous radio and x-ray switching between two modes was reported in PSR B0943+10 and PSR B0823+26 [332, 550, 551, 554]. The concurrent multi-wavelength study may help to understand the origin of these phenomena.

Zoom In Zoom Out Reset image size

Some pulsars are known to exhibit a correlation between the discontinuous change in radio emission and that in braking properties [124, 436, 493, 505], and they are called 'intermittent pulsars'. The region of these pulsars in the $P{{\rm \mbox{--}}}\dot{P}$ diagram overlaps those of the nulling and mode changing pulsars (upper right panel of figure 16). The long nulling cycle of days to years could determine the spin-down rate in both the emission 'on' and 'off' states. In the emission 'on' state, the spin-down torque is larger than that in the 'off' state. For example, the pulse frequency derivative changes with switching of $\dot{\Omega}_{\rm on}/\dot{\Omega}_{\rm of\,\!f}=1.77\pm 0.03$ for PSR J1832+0029 [434, 493, 500] and $\dot{\Omega}_{\rm on}/\dot{\Omega}_{\rm of\,\!f}=2.5\pm 0.1$ for PSR J1841−0500 [124]. The fact that a correlation exists suggests that the phenomenon originates in some change in the magnetospheric state [474].

A Giant radio Pulse (GP) is a single pulsed emission in the radio band that is much stronger than average pulses (see [418] for review). One of the distinct features of GPs is their intensity distribution, which follows a power law (e.g. [46, 559]), in contrast to a log-normal function for that of normal pulses (e.g. [117]). GPs are a good probe to study how the magnetosphere changes and what roles it plays during a single rotation (e.g. [160]). About 10 pulsars are so far known to exhibit GPs, e.g. the Crab pulsar and 1.56 ms period PSR B1937+21 [153, 739] (bottom right panel of figure 16), and a couple of more candidates have been suggested (e.g. PSR B0031-07, PSR B1112+50, and PSR B1133+16 [228, 390, 435, 450]). Also notably, two phenomena that are not exactly classified as, but reminiscent to, GPs have been detected from the Vela pulsar (named 'giant micropulses') [374] and from PSR B0656+14 ('spiky emission') [835]. The classification of the two phenomena and GPs is still under debate.

The emission mechanism of GPs has not yet been established, although some theoretical ideas have been proposed [217]. Interestingly, GPs occur at the same pulse phase as peaks of x-rays emissions for pulsars which magnetic fields at the light cylinder is high ($B_{\rm lc}=10^5$–10⁶ G) [174, 418]. This indicates that GPs and high-energy emission may originate in the same region for these pulsars. It is important to study the high-energy emission accompanied by GPs for our understanding of GP generation mechanism. In the Crab pulsar, the flux of the optical pulses coincident with the GPs increases by 3%–11% [720, 745]; therefore, its GPs must have some direct connection to its high-energy emission. Simultaneous observations with other wavelengths (e.g. x-rays and gamma rays) have been performed, although the statistically significant enhancements have not been reported [25, 90, 91, 340, 494, 567, 568].

Rotating RAdio transients (RRATs) are a new and emerging sub-population of RPPs [407, 542] that exhibits sporadically detectable emission of a single radio pulse. RRATs are confirmed to be indeed periodic rotators from timing analyses. The intervals between detected pulses range from minutes to hours (i.e. nulling fractions of  >$95\%$). Although more than 100 RRATs have been detected (see 'The RRATalog' in table A1 in appendix), pulsar timing solutions (P and $\dot{P}$; bottom left panel of figure 16) have been derived from only a fraction (∼25%) of them due to the small number of detected pulses. RRATs have typically longer periods and higher magnetic fields than normal RPPs [115, 172, 197, 386, 405, 407, 541, 542]. It is yet unclear whether the observed $P{{\rm \mbox{--}}}\dot{P}$ distribution of RRATs is genuinely related to their pulsar-intrinsic properties or is rather an observational selection effect whereby longer-periods RPPs are detected with higher signal-to-noise ratios in single-pulse searches [540].

Among RRATs, PSR J1819−1458 is one of the most studied objects. Its magnetic field strength inferred from the measured period and its derivative is $B_{\rm d}=5\times10^{13}$ G [542], which is at a similar level to those of low-B magnetars (section 3.3), high-B pulsars (section 3.4), and XINS (section 3.5) and is close to the upper end of the scale of the magnetic field $B_{\rm d}$ for the RPPs. PSR J1819−1458 exhibit anomalous post-glitch $|\dot{\nu}|$ evolution [503], behaviours similar to which have ever been observed in only two high-B pulsars PSR J1119−6127 and PSR J1846−0258 with bursting activities (see section 3.4). Interestingly, PSR J1119−6127 showed a RRAT-like sporadic emission, in addition to the regular radio emission, at a certain period of time [38, 832]. PSR J1819−1458 is the only RRAT that shows detectable x-ray pulsation [543]. The thermal component of its spectrum has a blackbody temperature of $kT \sim 0.14$ keV and its pulse profile is sinusoidal, of which the peak almost coincides with that in the radio profile peak. These facts are consistent with the model that the thermal x-ray emission comes from the heated polar cap [569]. It was reported that the x-ray photon and radio pulse detections may be correlated on timescales of less than 10 rotation periods (3.4 sigma level), which supports the hypothesis that the (above-mentioned) two emission mechanisms are physically related to each other [569]. In the x-ray spectrum, an absorption line (or lines) is detected at  ∼1 keV [120, 268, 543, 569, 684, 693]. If the line is due to proton resonant cyclotron scattering, then the magnetic field strength is  ∼10¹⁴ G. The x-ray observations also revealed a PWN surrounding PSR J1819−1458 [120, 684]. Since the x-ray luminosity of the PWN relative to the spin-down luminosity of PSR J1819−1458 (i.e. the x-ray efficiency), $L_{\rm pwn}/L_{\rm sd} \sim 0.2$, is much higher than that of normal PWNe (figure 26), this PWN is likely to be powered by the magnetic field [684]. Then, the properties of PSR J1819−1458 characteristic to RRATs are also likely to be related with the strong magnetic field. Although the observational facts are much less clear for the other RRAT sources, this physical relation might hold in the RRAT in general.

Some theoretical ideas for the mechanism of the RRAT have been proposed, e.g. disruption by transit accretion of SN-fallback matter [479], release of plasma trapped in radiation belts [495], and sporadic plasma accretion from a circumstellar asteroid belt [162]. However, if the RRAT is a physically distinct class of RPPs, the birth rate of Galactic NSs would be higher than the Galactic CCSN rate [404, 406].

Alternatively, RRATs may be a mode of normal RPPs, or just a tail of the intensity distribution in terms of pulse-to-pulse variability. Although the single-pulse amplitude of RRATs generally follows a log-normal distribution as in the case for most RPPs, some RRATs exhibit power-law tails as seen for GPs (see section 3.1). Hence, there must be an overlap in population between the RRAT and the RPPs with weak normal pulses and GPs [172]. A few RRATs show oscillations between pulsar-like and RRAT-like modes [115, 230, 733, 855]. In follow-up observations, some RRATs show a weak pulse component which is in itself too weak to be detected as a single pulse [116, 172, 405, 406]. In fact, normal RPPs that have significant pulse-to-pulse intensity modulation, e.g. PSR B0656+14, would be detected as a RRAT if they were located at a larger distance from us [834], i.e. if their flux was considerably weaker. It has been reported that some RRATs discovered at  ∼1 GHz appear as normal pulsars at a lower frequency [196]. Considering such observational diversity, the exact definition of the RRAT is still under debate [407].

Magnetically-powered NSs, commonly called 'magnetars', have been identified through two enigmatic now-historical classes, i.e. SGRs and AXPs [782, 783]. In the $P{{\rm \mbox{--}}}\dot{P}$ diagram in figure 17, most SGRs and AXPs are located above the quantum critical field, $B_{\rm cr}$, exhibiting a narrow pulsation range of P  =  2–12 s and a relatively large derivative range of $\dot{P}=10^{-12}$–10⁻¹⁰ s s⁻¹. The SGR family was historically discovered from its repeated soft gamma-ray bursts [431, 538], whereas the AXP class has been identified as bright soft x-ray pulsars, for which the x-ray luminosity exceeds the spin-down power ($L_{\rm x}>L_{\rm sd}$) [553]. Even though these classes used to be considered as different classes, there is increasing evidence suggesting that they are intrinsically the same class. For example, most of SGRs were shown to have spectra and rotation periods similar to those of AXPs [431], whereas some AXPs showed short bursts similar to those of SGRs [277, 397]. Thus, SGRs and AXPs are nowadays considered to collectively form the magnetar class. As of 2019, 11 SGRs and 12 AXPs have been confirmed, primarily in the x-ray band, and several new candidates have been proposed, as listed in the up-to-date McGill Magnetar Catalog ([600], table A1 in appendix).

Zoom In Zoom Out Reset image size

Common magnetar characteristics are observationally different from those of conventional RPPs as listed below. For comprehensive reviews, see [394, 396, 547, 548, 804, 841].

• (i)  
  A narrow range of slow spin periods of P  =  2–12 s and high spin-down rates of $\dot{P}=10^{-12}$–10⁻¹⁰ s s⁻¹, implying a high surface magnetic field of $B_{\rm d}\sim 10^{14}$–10¹⁵ G, assuming conventional magnetic field braking (equation (13)).
• (ii)  
  Young characteristic ages of $\tau_{\rm c}=$ 1–100 kyr, the spatial distribution concentrated around the Galactic plane (figure 18), and occasional associations with SNRs, all indicating a young population.
• (iii)  
  Persistent x-ray luminosity of $L_{\rm x}$  =  10³⁴–10³⁵ erg s⁻¹ exceeds their spin-down luminosity of $L_{\rm sd}$  =  10³²–10³⁴ erg s⁻¹ and hence excludes rotation-powered interpretation.
• (iv)  
  No Doppler modulation of their x-ray pulses, indicating lack of binary companions, the fact of which excludes accretion-powered interpretation.
• (v)  
  Sporadic powerful short bursts or GFs with super-Eddington luminosity ($L \gg 10^{38}$ erg s⁻¹), indicating recurrent magnetic activities.
• (vi)  
  Blackbody component ($kT\sim 0.3$–0.5 keV) in soft x-rays is hotter than typical RPPs, and hence requires some energy source in addition to the latent heat of the conventional cooling curve.
• (vii)  
  Several magnetars exhibit a hard x-ray power-law component, which is dominant above 10 keV and extends at least up to  ∼100 keV or possibly higher.

Zoom In Zoom Out Reset image size

The rotation periods of SGRs and AXPs are too slow to consider a possibility of them being rotation-powered (point (i) above). None of them exhibit any evidence for mass accretion from a companion star, and hence the accretion-powered scenario is excluded (point (iv) above). Consequently, the 'magnetar hypothesis' whereby they are ultra-strongly magnetized NSs has come to light and is now recognized as the most successful and most widely accepted model [213, 782, 783]. According to the model, a strong magnetic field would be generated by dynamo action shortly after SN explosion, and the radiation is powered by a release of magnetic energies stored in the stellar interior (see also section 5). Numerous theoretical studies have been conducted on the magnetar model (for the comprehensive theoretical review, see [804]), and in recent years magnetars are proposed to be left after some of SNe [391, 513, 843] and GRBs [213, 779, 808] (see section 4.5). It should be noted that alternative interpretations for the radiation model of these classes have been also proposed, e.g. accretion from a fossil disk [26, 136, 531], a quark star model [135, 607, 845], and fast-rotating massive WD [520, 576, 613, 809].

Recent discoveries of transient magnetars have doubled the total number of confirmed magnetars from  ∼11 in 2005 to  ∼23 (for review of the transients, see [679]). Typically, new transient magnetars are discovered from bright short bursts detected with Swift/BAT and/or Fermi/GBM. Once the source location has been approximately determined from a short burst, prompt follow-up observations identify the location of a persistent x-ray pulsar, for which the magnetic field strength is then estimated with measurements of P and $\dot{P}$. The transient magnetars are subsequently monitored for a few months to years.

Figure 19 shows a multi-wavelength SED of a prototypical persistently bright AXP, 4U 0142+61. Magnetars are typically bright in soft x-rays, where the soft x-ray spectrum is quasi-thermal with its blackbody temperature of kT  =  0.2–0.5 keV, the component of which is likely to originate from the stellar surface or its vicinity. An additional power-law (or thermal) component is required in $\lesssim$10 keV to fit the observed spectrum. This component is considered to originate from resonant Compton up-scattering of thermal photons by charges flowing along twisted and closed field lines [784]. The twisted fields are a result of non-potential distortion at the stellar surface and are supported by strong current flowing in the magnetosphere. When the strength of the toroidal field is comparable with that of the poloidal field near the surface, the current density is required to be typically 10³–10⁴ times larger than the averaged current density flowing the open-field region of the normal RPP. Then, the higher charge density in the twisted magnetosphere causes resonant up-scattering for thermal photons from a magnetar [592, 784].

Zoom In Zoom Out Reset image size

In addition, hard x-ray observations revealed a new distinctive component above 10 keV [194, 195, 224, 227, 297, 444, 445], which has a power-law shape with a hard photon index, $\Gamma_{\rm h}\sim 1$, and extends up to at least  ∼100 keV. This persistent hard x-ray component is also detected from transient sources, e.g. an x-ray outburst of 1E 1547.0  −  5408 in 2009 [224]. This hard x-ray emission of magnetars is expected to have a cutoff at a few hundred keV so that it would be consistent with the upper limit at $\gtrsim$1 MeV given by CGRO/COMPTEL [193, 444]. Some theoretical models have been proposed for this hard x-ray component above 10 keV; e.g. optically thin bremsstrahlung [74, 780], synchrotron radiation [339, 780], resonant scattering [60, 61, 71, 592, 825], and down-cascade due to photon splitting [225] from a magnetar magnetosphere. Observationally, a two-component spectral feature of magnetars seems to be common in the magnetar class [225]. The photon index of the hard x-ray component above 10 keV and its relative intensity to the soft thermal emission below 10 keV are proposed to be correlated with the dipole magnetic field strength [227]. This would indicate a spectral evolution of the magnetar class as the magnetic field decays.

In contrast to the absence of bright persistent emission in the MeV band, GFs (e.g. SGR 1806−20 [99, 258]) and short bursts (e.g. 1E 1547.0−5408 [850]) showed radiation in the MeV range, being consistent with a continuous power-law component in SGR 1806−20 and a power-law component with an exponential cut-off in 1E 1547.0−5408. So far, no GeV gamma-ray detections from magnetars have been reported [7, 473]. In lower energy bands, ∼10 optical/IR counterparts and candidates have been detected [600]. Pulsations have been also detected from three of them [204–206, 410]. The profiles of light curves are similar to those in soft x-rays. Given that their observed ratios of the optical to spin-down luminosity are much higher than those of RPPs, the magnetic energy would support the optical emission. In the radio band, pulsations have been detected from 4 magnetars [126, 127, 468, 718]. The radio pulsed-emissions are associated with x-ray outbursts ([125, 710]; see section 4.1).

HBPs are an intermediate class of RPPs with a dipole magnetic-field strength of $B_{\rm d}= 10^{13}$–10¹⁴ G¹², which is higher than those of conventional RPPs (10¹¹–10¹² G) but slightly lower than (but somewhat overlaps with) those of typical magnetars (10¹⁴–10¹⁵ G) (e.g. [588]). Despite the similar dipole field strength of HBPs to those of low-B magnetars, none of magnetar-like activities (short bursts, outbursts, high x-ray luminosity $L_{\rm x} \gtrsim L_{\rm sd}$, etc) have ever been observed from most of HBPs except for two sources (see below in detail), and therefore they are conventionally not classified into magnetars [123, 544]. For example, PSR J1847−0130 ($B=9.4\times10^{13}$ G; [544]) has a dipole magnetic field significantly higher than the critical magnetic field $B_{\rm cr}$, but has shown neither detectable x-rays nor magnetar outbursts so far. No clear definition of HBPs has been yet laid out, majorly because of difficulty in accommodating two exceptional sources, PSR J1846−0258 and PSR J1119−6127 (as described below in detail).

Recent observations gradually accumulate implications that HBPs exhibit x-ray characteristics that fall somewhere between those of magnetars and RPPs. In the $P{{\rm \mbox{--}}}\dot{P}$ diagram (figure 1), we plot the x-ray detected HBPs among others. Their surface blackbody temperatures are suggested to be higher (kT  =  0.1–0.3 keV) than those of conventional same-aged RPPs and are close to those of magnetars [601, 603, 646, 867] (see also the $\tau_{\rm c}$–kT plot in figure 4). In figure 12, the distribution of most HBPs seem to be contained in that of the ordinary RPPs, and the radiation power of them is typically explained by loss of rotational energy. However, two HBPs have exhibited magnetar-like transient x-ray activities—PSR J1846−0258 in 2006 and PSR J1119−6127 in 2016. Their x-ray luminosities relative to the spin-down power during their transient activities became similar to those of magnetars during outbursts (see section 4.1). As a result, an increasing number of astronomers consider the HBP population to be dormant magnetars, rather than a physically independent class.

We should note that whereas some HBPs show thermal properties with a higher temperature than other RPPs, the others are consistent with the standard cooling curves [603]. Thus, in addition to the lateral heat and the dipole magnetic field, some additional parameters would affect the surface temperature of HBPs. A possible cause for the different behaviour of NSs with similar dipole fields, as in the case of HBPs compared with magnetars and RPPs, is the presence of a toroidal component in the internal field, which is progressively transferred to the external one. The initial magnetic field topology has been pointed out to be a key for the thermal properties and frequency of bursting activity of these NS populations [632, 820].

XINSs (also called x-ray dim isolated NSs, XDINSs) are radio-quiet x-ray point sources that exhibit quasi-blackbody soft x-ray spectra without any association to a SNR; see [312, 548, 803] for reviews. Since the discovery of the prototype object, RX J1856.5−3754, with ROSAT in the 1990s [826, 827], seven XINSs have been discovered by the time of writing (see 'Isolated Neutron Stars in quiescence', table A1 in appendix), and they are referred to as 'The Magnificent Seven' (M7).

All of the M7 (except for RX J1605.3+3249 [641]) have been confirmed to exhibit x-ray pulsations at slow spin periods (P  =  3–12 s, pulse fraction 1%–18%; [638]), indicating inhomogeneous surface temperature distribution. Their period derivatives of $\dot{P}= 10^{-14}$–10⁻¹³ s s⁻¹ are larger than those of RPPs with the similar spin periods, and so are the inferred surface dipole magnetic fields of $B_{\rm d}= 10^{13}$–10¹⁴ G. Their characteristic age $\tau_{\rm c}$ of a few Myr is slightly older than the kinematically estimated age of 0.5–1 Myr [773]. In the $P{{\rm \mbox{--}}}\dot{P}$ diagram (figure 1), their location is close to (but with slightly lower $\dot{P}$ than) that of magnetars. Parallax and proper motion measurements suggest that XINSs are nearby sources (<500 pc). This is supported by low interstellar absorption in their x-ray spectra ($N_{\rm H}<$ a few 10²⁰ cm⁻²). Thus, XINSs are proposed to be nearby middle-aged NSs, presumably the descendants of magnetars or other highly magnetized NSs [383, 384, 813].

Soft x-rays are dominant in the radiation from XINSs ($L_{\rm x}= 10^{30}$–10³² erg s⁻¹) with measured surface temperature of kT  =  0.04–0.1 keV, which is close to the cooler end of magnetars and HBPs. Their x-ray luminosity and temperature from surface radiation are consistent with the cooling curve of similar-aged RPPs and are lower than those of magnetars (figure 4). Faint optical counterparts are discovered from most of them with large x-ray to optical flux ratios ($F_{\rm x}/F_{\rm opt}= 10^4$–10⁵) although the optical fluxes of any of the seven XINSs are higher than the extrapolations of their x-ray spectra [382]. Near-IR band observations give upper limits only except for RX J0806.4−4123, for which a very faint and low-significance counterpart has been identified (H-band 2.5$\sigma$, J-band 1.4$\sigma$ [655, 656]). The $160~\mu$m emission was detected with $>5\sigma$ from two XINSs although the origin is uncertain [659]. None of them exhibit detectable radio emission [372, 385, 427], non-thermal magnetospheric x-ray contribution (see also [851, 852] for an x-ray excess at  ∼1 keV), or magnetar-like burst activity.

The soft thermal spectra of XINSs are mostly well described with a blackbody continuum and a broad absorption feature at 0.2–0.8 keV, except for RX J1856.5−5754, which lacks of the absorption feature [383]. The absorption features in the x-ray spectra of RX J0720.4−3125 and RX J1308.6+2127 show a phase-dependent structure [101, 102]. They are interpreted as proton cyclotron-resonance features or atomic transitions in a strong magnetic field, with required B  =  10¹³–10¹⁴ G in both the interpretations. This strong magnetic field is comparable with or higher than the dipole field inferred from P and $\dot{P}$. Optical linear polarization was recently detected from RX J1856.5−5754 at a polarization degree of $PD\sim16\%$, which supports the existence of QED vacuum birefringence around a magnetized NS [564]. XINSs are similar to CCOs (see the next subsection) in terms of their x-ray spectral properties (section 3.6); however, XINSs are not associated with SNRs like CCOs and have a higher inferred dipole field of $B_{\rm d}=$(1–3)$\times 10^{13}$ G than those of three CCOs, for which the field strength are known, estimated from their P and $\dot{P}$ measurements.

CCOs are soft x-ray point sources that are found at the centres of SNRs and do not fit well in any of the above-mentioned classes. About 10 of them are known so far (see table A1 in appendix). They are typically stable in x-rays and quiet in radio [183, 548]. Despite the fact that they appear to be young systems associated with SNRs, neither associated PWNe nor any counterparts at the other wavelengths have been detected. Arguably, CCOs are a rather ad hoc classification, and some of them can potentially be classified as one of the above-mentioned categories. However, a few CCOs do exhibit clear distinctive properties, such as x-ray pulsations at rotational periods of P  =  100–300 ms, which are faster than those of XINSs and magnetars. The $\dot{P}$ values have been measured for at least three CCOs (Kes79, Pup-A, G296.5), and all of them imply weak surface magnetic fields of $B_{\rm d}\sim 10^{10}$–10¹¹ G, calculated with equation (13). Accordingly, they are referred to as 'anti-magnetars' [293] (see also [97]).

A well-known example of the CCO class is the central NS of SNR Cas A, of which the pulsation has not been detected. Notably, a direct cooling signature was reported from this source [326], although a recent analysis concludes that the signature is not significant [219, 657, 660]. If it exists, this source will be all the more interesting to study the internal structure of the NS.

Another prototypical source is an isolated NS 1E 1207.4−5209, close to the centre of SNR G296.5+10.0. The x-ray spectrum of this source shows absorption at 0.7, 1,4, and 2.1 keV [87, 187, 549, 708], which are considered to be atomic transitions in a strong magnetic field [314, 574], although electron cyclotron resonances in a weak field of  ∼$8\times10^{10}$ G has also been discussed for their origin [87, 187, 292, 316]. Other notable findings of CCOs are a large pulsed fraction of 64% from PSR J1852+0040 in Kes 79 (the two characteristics are suggested to be related to each other) [714] and two antipodal hot spots of different temperatures and areas in RX J0822−4300 in Puppis A [295]. These are interpreted as an existence of a stronger magnetic field than the dipole field.

A leading theory for the origin of a CCO is that a typical magnetic field (10¹²–10¹³ G) is buried by prompt fall-back of supernova material [341]. After 10³–10⁴ yr, the magnetic field re-emerges, and then the CCO becomes a RPP [341, 819]. There is a possible candidate of a CCO descendant, 1RXS J141256.0+792204, dubbed 'Calvera' [707, 721] (and possibly 2XMM J104608.7−594306 [639, 640]); it shows an apparently thermal x-ray emission and neither radio nor $\gamma$-ray emission like the other CCOs [315, 336, 860], although its location in the $P{{\rm \mbox{--}}}\dot{P}$ diagram (P  =  59 ms, $L_{\rm sd}=6\times10^{35}$ erg s⁻¹, and $\tau_{\rm c}=3\times10^5$ yr [315, 860]) is near ordinary young pulsars.

Magnetar-like activity was observed from a mysterious CCO, 1E 161348−5055, at the centre of SNR RCW103 (section 4.1). This suggests a new possibility that the CCO class can exhibit magnetic activity similar to other highly-magnetized NSs, such as magnetars.

Historically, there are two classical x-ray binary systems hosting NSs, i.e. HMXBs and LMXBs. HMXBs are younger systems (typical age of $\lesssim 10^8$ yr), where the mass donors are young O- or B-type giants with $M_{\rm c}\gtrsim 10M_{\odot}$. LMXBs are older binaries ($\gtrsim$10⁸ yr) hosting older K-type optical counterparts with mass $M_{\rm c}\lesssim 1M_{\odot}$. Depending on their system ages, NSs in HMXBs and LMXBs typically have magnetic fields of $B\sim 10^{12}$ G and $B\lesssim 10^{10}$ G, respectively. The stronger magnetic field of HMXB-NSs has been confirmed through detection of CRSFs (section 1.4). Nearly 40 sources exhibit CRSFs at magnetic fields corresponding to clustering around B  =  10¹²–10¹³ G. Weakly-magnetized NSs in LMXBs are related to accretion-induced field reduction (e.g. [81, 93, 579, 725]).

Another subclass in the accreting system is the symbiotic x-ray binary (SyXB), which consists of an x-ray bright NS and an M-type giant primary star. Originally, SyXBs were classified into LMXBs; however, in recent years, SyXBs are recognized as a different class from conventional NSs in LMXBs. The hosting NSs have relatively long spin periods (110–18300 s). There are a few x-ray spectral and timing studies of SyXBs ([226] and references therein).

There is a new peculiar subclass of x-ray binaries, called gamma-ray binaries (e.g. LS I+61 303 and LS 5039), which can potentially host a NS or BH as a compact object [211]. One gamma-ray binary, PSR B1259−63, is known to host a 48 ms RPP in a 3.4 yr eccentric orbit around a massive B2e companion star. This pulsar behaves as an ordinary radio pulsar at the apastron, whereas it becomes an unpulsed x-ray and GeV gamma-ray source at the periastron passage [9, 16]. Recently, another Be-star binary system hosting a radio-emitting pulsar, PSR J2032+4127/MT91 213, was discovered [506]. The enhanced x-ray and TeV gamma-ray emissions were detected as the system approached the periastron [12, 151, 344, 475, 476, 635]; accordingly it is classified as a gamma-ray binary. In addition, a few of the other gamma-ray binaries are expected to host NSs. Because of shorter orbital timescale of other gamma-ray binaries, the scattering in the stellar wind would prevent detection of the expected radio pulse emission [545].

No magnetars have been convincingly confirmed to be in binary systems. Equation (17) suggests that NSs with stronger magnetic field are slower rotators. The most detailed studies to date investigated the following four slowly rotating x-ray pulsars in HMXBs and SyXBs in search for magnetars or magnetar descendants in binary systems: IGR J16358−4726 of $P\sim 1.6$ h [627], 4U 2206+54 of $P\sim 1.6$ h [691], 4U 0114+65 of $P\sim 2.7$ h [481], and 4U 1954+319 of $P\sim 5.4$ h [158]. Nevertheless, no direct evidence, such as clear cyclotron absorption features, has been obtained. Therefore, it is still undetermined whether magnetars are, or can be, hosted in binary systems. The x-ray spectra of slowly rotating NSs are suggested to be harder than others; if so, it is indirect evidence for stronger magnetic fields. Alternatively, this hard x-ray spectrum can be explained by assuming an accretion models, such as the quasi-spherical accretion model [715], in which a canonical magnetic field of  ∼10¹² G is assumed.

Ultraluminous x-ray sources (ULXs) are point-like sources discovered in off-nucleus regions of nearby galaxies, with their x-ray isotropic luminosities exceeding the canonical Eddington limit ($L_{\rm x}\gtrsim 10^{39}$ erg s⁻¹) of a stellar-mass (∼$10M_{\odot}$) BH (see equation (34)). Popular models of ULXs include accretion onto BHs of stellar mass ($M<80{{\rm \mbox{--}}}100M_{\odot}$) at a super-Eddington accretion rate and intermediate mass BHs ($M\sim 10^{2{{\rm \mbox{--}}}4}M_{\odot}$) in a sub-Eddington accretion regime [242, 378]. However, these canonical hypotheses are challenged by following discoveries of four slowly rotating ($P\sim 1$ s) NSs from ULXs in nearby galaxies (4–17 Mpc). Here is a brief summary of them.

The first confirmed ULX NS is ULX M82 X-2 in the nearby galaxy M82 at a distance of 3.6 Mpc; an x-ray coherent pulsation with a period of 1.37 s was discovered [55]. The total x-ray luminosity was $L_{\rm x}\sim 1.8\times 10^{40}$ erg s⁻¹. Three subsequent discoveries of ULX pulsars exhibited similar properties: ULX-1 NGC5907, NGC7793 P13, and NGC 300 ULX1. ULX-1 NGC5907, which is at a distance of 17.1 Mpc, hosts a NS with a pulsation period evolving from 1.43 s in 2003 to 1.13 s in 2014 [364]. A 5.3 day orbital period and variable x-ray luminosity of $L_{\rm x}=$(1.5–10)$\times 10^{40}$ erg s⁻¹ were reported [364]. NGC7793 P13 is located at a distance of 3.6 Mpc, and shows a pulsation period of 0.42 s with an x-ray luminosity of $(2.1-5)\times 10^{39}$ erg s⁻¹ in the 0.4–10 keV band [265, 365]. NGC 300 ULX1 reached the ULX luminosity ($L_{\rm X}\sim3\times10^{39}$ erg s⁻¹) in December 2016 [132]. A spin-up rate of NGC 300 ULX1, $|\dot{P}|\sim5.7\times10^{-7}$ s s⁻¹, is the highest ever observed from an accreting NS. A potential CRSF at  ∼13 keV in the pulsed spectrum was also reported [828]. In addition, although pulsations have not been detected, the possible detection of a CRSF at $E_{\rm cyc}\sim4.5$ keV was reported in the spectrum of an ULX in M51, which would be another candidate of an ULX NS [108, 560]. There are more pulsed ULXs only in bright outburst phases [210, 412, 796–798, 801]. These discoveries lead to the question of whether some or most of ULXs host a NS or stellar mass BH.

Recently, two accreting pulsars (3XMM J004232.1+ 411314, 3XMM J004301.4+413017) with sub-Eddington x-ray luminosities 10³⁷–10³⁸ erg s⁻¹ were discovered in M31, the closest major galaxy to our Galaxy [235, 699, 868]. These new findings have broadened our knowledge of the NS family, adding pulsars to the extragalactic binary-system population.

Magnetars sometimes exhibit sporadic enhancement of electromagnetic radiation, called 'outburst'. The observed magnetar outbursts in x-rays have shown the following properties in general (see [236, 679] for reviews):

• Soft (∼1–10 keV) x-ray flux increases by a factor of 10–10³ from the quiescent level ($\lesssim$10³³ erg s⁻¹), reaching the maximum x-ray luminosity $L_{\rm x}$ of 10³⁴–10³⁵ erg s⁻¹.
• Flux enhancement over the pre-outburst quiescent level remains for a few months with gradual decay.
• Soft x-ray spectrum below 10 keV exhibits clear softening (temperature decrease and/or photon index increase) during outbursts as the x-ray flux decreases. In a few cases, hard x-rays above 10 keV have been detected during outbursts.
• During the early phase of outbursts, the magnetar shows greatly shorter-time scale bursting activities (e.g. short bursts).
• Outbursts are often accompanied by change of pulse profiles in shape and pulsed fraction, by glitches, and by enhancement of $\dot{P}$. The x-ray profile usually becomes more complex (e.g. increased number of peaks) than before the outburst.

The first observed transient magnetar was the 5.5 s period x-ray pulsar XTE J1810−197, from which an outburst was observed in 2002–2003 [361]. Since then, at least 18 transient outbursts from 13 magnetars have been detected by the time of writing (see Magnetar Outbursts Online Catalog [600]). The distribution of transient magnetars in the $P{{\rm \mbox{--}}}\dot{P}$ diagram is shown in figure 20. Figure 21 shows, as an example, long- and short-term light-curves and x-ray spectrum during an outburst (in 2009) of a fast spinning AXP, 1E 1547.0−5408, which is situated at the centre of a small SNR, G327.24−0.13 [279] and was discovered in 1980 with the Einstein x-ray satellite. Two separate outbursts have been so far recorded from this pulsar in 2008 and 2009 (figure 21(b)). During the outburst in 2009, the position of this pulsar in the $L_{\rm sd}$–$L_{\rm x}$ diagram was moving from the region typical for magnetars towards that for RPPs (figure 12). Several of the magnetars that have shown transient activities seem to be in a quiescent state most of the time and remain undetected until they become active. An outburst of magnetars is considered to be triggered by a sudden energy release of the internal (e.g. [458, 782]) or external magnetic fields (e.g. [621]; see also [396, 804] for recent reviews). The schematic illustration for transient activity is shown in figure 10. The increasing number of detected transient magnetars raises questions about the birth rate of magnetars and their total number in our Galaxy. In addition, this hidden population is potentially important for studying the rate of supernova explosions in our Galaxy.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Magnetars used to be thought to be radio quiet [113, 170, 463]. Theoretically, the radio quietness is considered to be suppression of the electron-positron pair-production cascade due to photon splitting, i.e. a QED process at $B\gtrsim B_{\rm cr}$ in which a single photon splits into two low-energy photons [59]. However, a significant fraction of transient magnetars are now known to exhibit radio emission. At the time of writing, 4 magnetars (XTE J1810−197 [127], 1E 1547.0−5408 [126], PSR J1622−4950 [468], and SGR 1745−2900 [718]) and a magnetar-like HBP PSR J1119−6127 are known to emit pulsed radio-emission.

The radio pulsations are associated with x-ray outbursts. In fact, the first radio pulsation from a magnetar was detected from XTE J1810−197 during an x-ray outburst [127]. Radio emission from it had never been detected before the x-ray outburst in 2003. Interestingly, the radio emission suddenly disappeared in 2008 [125]. In 2018, the x-ray outbursts and the reactivation of coherent radio emission were reported [177, 294, 470]. Another notable example is PSR J1622−4950; the radio emission from it was also re-detected during an x-ray reactivated state after a long period of radio-inactivity [130]. Other radio magnetars also show similar transient properties [30, 126, 710]. The observed relation between the radio flux and the spin-down evolution [122, 130, 710] may indicate that the rotation energy supports the radio pulsation.

It was proposed that radio-emitting magnetars satisfy $L_{\rm sd}>L_{\rm x}$ in quiescence [685]; however, two sources (SGR 1627−41 and PSR J1846−0258) do not follow the relation. Non-detection of radio pulses from most of magnetars would possibly be because of beaming effects or effects of dense medium in the surrounding SNRs.

Similar to radio RPPs, the magnetar radio pulses are highly linear polarized [129, 214, 438, 468, 718] and show sharper pulse profiles than those in the other wavelengths [122]. Since the position angle of the linear polarization appears to be related to the direction of the magnetic field, the change of the position angle (swing) gives the geometry of stellar rotation and magnetic configuration (e.g. inclination angle) [669]. Judging from the swings of the polarization position angle for 1E 1547.0−5408 and PSR J1622−4950, they have nearly aligned rotators [129, 469]. By contrast, in XTE J1810−197, the position angle swing deviates from the dipole magnetic field model [438, 712]. On the basis of the radio pulse-profiles, XTE J1810−197 would be probably a nearly orthogonal rotator [512]. It should be noted that the polarization angle swings evolve in a timescale of several days [462, 469, 712].

The peak flux and pulse-profile of XTE J1810−197 are highly variable in a timescale of a day or shorter, and accordingly interstellar scintillation cannot account for their variability. As for the pulse-energy distributions, that of PSR J1622−4950 follows a log-normal distribution [469], which is similar to those of normal RPPs (e.g. [117]), whereas an additional high-energy excess component is apparent in those of XTE J1810−197 and SGR J1745−2900 [496, 712].

The radio energy spectrum of radio-pulsating magnetars is relatively flat (spectral index  ∼0, [128, 129, 469, 791, 792], but see also [631, 718]), compared with that of RPPs (−1.6, e.g. [371, 530]). Note that these values of spectral indices may have a considerable uncertainty from systematic errors due to the flux variation. These flat spectra allowed us to observe the source up to very high frequencies.

Hard radio spectra of these magnetars are useful to investigate the dense region of interstellar matter. In the dense matter region, the scattering timescale of the radio pulse is much longer than the rotation period in lower frequency, so that the lower frequency pulses would not be detected as a pulsed emission. For example, SGR 1745−2900 is situated near the galactic centre (projected separation  ∼0.1 pc) [408, 573], and has the largest value of dispersion measure of 1778 pc cm⁻³ among the known pulsars [214, 680, 718]. Angular and temporal broadenings of radio pulses and variation of the derived rotation measure provide information of the magnetic field structures and ionized interstellar matter near the Galactic centre [104, 200, 202, 214, 736, 836].

In recent years, magnetar outbursts have been detected from low-B magnetars, which exhibit typical magnetar transient behaviours, yet has a surface dipole field of the same order as that of canonical RPPs. Three sources of this class have been discovered by the time of writing: SGR 0418+5729 [681, 683], Swift J1822.3−1606 [682], and 3XMM J185246.6+003317 [686]. The first source, SGR 0418+572, was discovered on 2009 June 5 with two bursts [812]. Its period derivative was determined to be $\dot{P}=4\times 10^{-15}$ s s⁻¹, which is considerably smaller than those of any of the known magnetars. The dipole magnetic field strength is accordingly calculated to be $B_{\rm d}\sim 6\times 10^{12}$ G [683]. The other two sources showed similar characteristics. The three sources are located below the QED critical field in the $P{{\rm \mbox{--}}}\dot{P}$ diagram. They are currently considered to be either magnetars with much stronger non-dipolar field than normal magnetars [820] or aged-magnetars whose dipole magnetic field has decayed over their life time [424].

There is some observational indication to support the former scenario, i.e. low-B magnetars have strong non-dipolar fields. A variable absorption feature was detected from SGR 0418+5729 at 1–5 keV [788]. Similar signatures were also reported from Swift J1822.3−1606 at  ∼2 and 5–12 keV [700]. Assuming that these features are proton cyclotron resonance feature, the corresponding magnetic field is estimated to be 10¹⁴–10¹⁵ G, which is larger than the dipole magnetic field evaluated from P and $\dot{P}$. The observation implies that these magnetars harbour small-scale magnetic fields (or higher multipoles) near the stellar surface that are much stronger than their dipole fields. Bursting activities (section 4.2) are likely to be related to such fields. The small-scale magnetic fields near the surface are thought to be transferred from the strong internal magnetic fields through some conceivable mechanisms, e.g. Hall drift (see section 4.4).

Magnetar-like outbursts or bursting activities have been detected also from two HBPs, PSR J1846−0258 and PSR J1119−6127. PSR J1846−0258 is a HBP with $P\sim0.326$ s and $B_{\rm d}\sim4.9\times10^{13}$ G, located in SNR Kes 75 [296]. Observations by RXTE discovered several short bursts from J1846−0258 on 2006 May 31 and July 27 [276]. The fluxes in the 0.5–2 keV and 2–10 keV bands increased by a factor of  ∼17 and  ∼5.5, respectively [276]. J1846−0258 exhibited a large spin-up glitch, $\newcommand{\tr}{{\rm tr}} \triangle \nu/\nu\sim 3\times 10^{-6}$, and following a net spin-down, 'over-recovery' (see section 4.3), reported in some magnetars [487, 489]. PSR J1119−6127 is a radio-emitting HBP with a (radio) period of 0.41 s, located at the centre of SNR G292.2−0.5. Combining with $\dot{P}=4\times 10^{-12}$ s s⁻¹, the spin-down magnetic field of this pulsar is $B_{\rm d}=4.1\times 10^{13}$ G. Short bursts were detected from this pulsar on 2016 July 27 and 28 with Swift, and subsequent x-ray outbursts were monitored [44]. The x-ray flux in the 0.5–10 keV band increased by a factor of  >160 from the preceding quiescent state. Strong x-ray pulsation was observed for the first time above 2.5 keV, with phase-averaged emission detectable up to 25 keV. In addition, a spin-up glitch, $\newcommand{\tr}{{\rm tr}} \triangle \nu/\nu\sim 5.8\times 10^{-6}$, and following glitch 'over-recovery' was detected [45], similar to J1846−0258. In the $L_{\rm sd}$–$L_{\rm x}$ diagram, PSR J1119−6127, and PSR J1846−0258 are located at the boundary between magnetars and RPPs (figure 12).

Different from magnetars and PSR J1846−0258, PSR J1119−6127 shows the regular radio pulsed emission. The radio properties in the transition phase from quiescence to an outburst possibly provide new insights to study the connection between HBPs and magnetars. The radio characteristics changed at the outbursts. After the outburst, the radio emission was quenched [111], and reappeared two weeks later [112]. The reactivated emission display a magnetar-like radio spectral flattening [630]. The pulse profile after the reactivation changed from a single Gaussian peak to a two-peaked profile, and showed significant profile evolution over several months [176, 514]. Moreover, radio and x-ray simultaneous observations showed that the coherent radio emission shuts off multiple times, in coincident with the occurrence of x-ray bursts [40]. The pair plasma produced in bursts is considered to cease and/or absorb the radio emission [40, 847].

In 2016, a magnetar-like outburst was detected from a CCO, 1E 161348−5055 [802], at the centre of RCW 103 (G332.4−0.4) with a SNR age of  ∼2 kyr [175, 678]. Monitoring observations show that the total radiation energy, the flux decay pattern, and the spectrum evolution are all consistent with general characteristics of the observed magnetar outbursts [100, 234]. This peculiar system exhibits an extremely long periodicity of 6.67 h [186] with variable x-ray flux (n.b., these features are very different from any other CCOs). Deep infrared observations during x-ray quiescence gave a significant limit on the companion mass, and it rules out an accreting binary model for the x-ray luminosity [79, 188, 234, 642, 769]. In order to explain such a long spin period, a model of fall-back accretion that was not ejected by the supernova has been proposed [186, 238, 342, 363, 480, 649, 790]. In an accreting system, there is an equilibrium period, $P_{\rm eq}\propto B_{\rm d}^{6/7}\dot{M}^{-3/7}$ from equation (17), where the magnetospheric radius reaches the co-rotation radius, $r_{\rm m}\simeq r_{\rm co}$. Stronger the magnetic field of the NS is, larger the magnetospheric radius is, and hence smaller the Keplerian velocity at the magnetospheric radius is and longer the equilibrium period $P_{\rm eq}$ is. In order for a NS to reach the period of P  =  6.67 h within 2 kyr (the SNR age) as of 1E 161348−5055, the dipole field is required to satisfy $B_{\rm d}\gtrsim10^{15}$ G [342], which is similar to other magnetars.

Transient magnetars usually exhibit repeated and sporadic bursting activities. They are typically associated with the early phase of outbursts (a few weeks after an onset). In most cases, the isotropic x-ray luminosity of a burst exceeds the Eddington luminosity (equation (34)). The strong magnetic field suppresses electron motion perpendicular to the field; as a result, the opacity of photons to excite this motion (X mode) decreases by several orders of magnitude compared with the Thomson scattering. Then, the Eddington luminosity is modified as $B>B_{\rm cr}$ [335, 570, 614, 782, 806, 815].

Magnetar bursts are phenomenologically classified into the following three types: (1) rare but extremely intense GFs with $L_{\rm x}\gtrsim 10^{45}$ erg s⁻¹ with a duration of approximately a few hundred seconds [246, 358, 538], (2) intermediate flares with $L_{\rm x} \sim 10^{41}$–10⁴³ erg s⁻¹ with a duration of a few seconds [366, 432, 604], (3) considerably more frequently occurring short bursts with $L_{\rm x} \sim 10^{38}$–10⁴¹ erg s⁻¹ with a duration of  ∼0.1 s [366, 583]. It is noted that the observed radiation energy roughly follows a single power-law distribution from short bursts to GFs.

'GFs' are the most spectacular manifestations of the magnetar class. Three historical 'GFs' have been observed from SGR 0526−66 (Large Magellanic Cloud), SGR 1900+14 (12.5 kpc in our Galaxy), and SGR 1806−20 (8.7 kpc in our Galaxy) in 1979, 1998, and 2004, respectively, in the last  ∼40 years with total radiated energies of 10⁴⁴–10⁴⁶ erg for each GF [358, 359, 538, 617] (see [547] for review). Figure 22 shows two examples of GFs from SGR 1900+14 and SGR 1806−20.

Zoom In Zoom Out Reset image size

GFs are one of the most luminous transient events in our Galaxy, except SNe, releasing as much energy as the Sun radiates in a quarter of a million years. The initial spike in the light curve of a GF involves a very rapid (<1 ms) rise to the peak, lasting a few hundred milliseconds, with a hard energy-spectrum extending up to the MeV range. The energy released in the initial spike is  ∼$4\times 10^{46}$ erg [771]. This initial spike is followed by a long pulsating tail modulated at the spin period of the NS, which lasts for a few hundred seconds. The x-ray pulse profile changes before and after a GF, indicating a re-arrangement of the magnetic field structure during the event. The rate of GFs with energy above  ∼10⁴⁶ erg is estimated to be  ∼10⁻⁴ yr⁻¹ per a magnetar [751].

Since GFs are extremely bright, extragalactic magnetar GFs occurring in nearby galaxies are detectable at least during bright initial spikes. Their spectra and durations are expected to be similar to those of classical short (<2 s) GRBs [212, 216, 357, 360]. The estimated fraction of potential extragalactic GFs among the observed short GRBs ranges from a few to  ∼15%, according to recent searches of the GRB catalogues [357, 464, 524, 585, 597, 617, 652, 751, 789]. There are a few candidates for extragalactic GFs. For example, GRB 051103 in (the line of sight of) M81 is one of them, whose location, light curve, and energy spectrum are found to be consistent with the interpretation that it is a GF in M81 at 3.6 Mpc [259, 360]. GRB 070701 close to M31 is another [598].

Ordinary short bursts (classification 3 in the previous discussion) are relatively frequent [142, 302, 303]. The short bursts are probably related to some mechanisms involving rapid dissipation of the magnetic field in a localized region (e.g. [284, 508, 510, 620, 621, 755, 857, 858]). The power-law distribution of the logN–logS fluence of short bursts of magnetars implies statistical similarity to solar flares and seismology on the Earth (Gutenberg–Richeter law) [583].

A glitch is a sudden spin-up of a pulsar. Figure 23 plots in the $P{{\rm \mbox{--}}}\dot{P}$ diagram the known pulsars from which a glitch has ever been detected. The relative intensity of the glitches is $\Delta\nu/\nu=10^{-11}$–10⁻⁵, where $\nu$ and $\Delta \nu$ are the rotation frequency (=P⁻¹) and the difference of the frequency before and after the glitch, respectively. In most cases, a glitch is followed by a recovery phase, in which the spin-up rate of $\dot{\nu}(=-\dot{P}P^{-2})$ increases with a relative intensity of $\Delta\dot{\nu}/\dot{\nu}=10^{-4}$–10⁻³ [52, 233, 262, 504, 859]. The intense radiative changes such as flares or outbursts are not associated with the glitches in RPPs. Therefore, the origin of the glitches of RPPs should be related to the internal structure of the NS. Several mechanisms to explain the RPP glitches have been proposed, e.g. a rapid transfer of angular momenta from neutron superfluid in the inner crust to the rest of the star (e.g. [31, 323]).

Zoom In Zoom Out Reset image size

By contrast to RPPs, magnetars sometimes show effectively 'spin-down glitches' (e.g. [208, 398]). For example, the spin after the glitch recovery phase is slower than the pre-glitch ones ('over-recovery', e.g. [275]). Moreover, an apparent 'anti-glitch' has been once reported (1E 2259  +586, [43]). One of efficient extraction mechanisms of the torque of a NS is the variation of field line structure at the outer magnetosphere (∼$R_{\rm lc}$), where the lever-arm is the largest. The magnetar glitches are sometimes accompanied by radiative flares, outbursts, and/or pulse-profile changes (e.g. [207, 396]). Then, a change in the magnetic field structure in both the inner and outer magnetospheres, such as movement of a crust and an injection of magnetic helicity (e.g. [133, 139, 620, 621]), may be a trigger of an outburst and/or a flare(s).

Interestingly, the HBPs that have shown magnetar-like bursting phenomena, PSR J1846−0258 and J1119−6127, have also shown irregular post-glitch evolution [38, 45, 176, 441, 487, 489, 832]. For PSR J1846−0258, the net frequency change after the recovery following the glitch associated with the x-ray outburst in 2006 was negative. The braking index changed from $n=2.65\pm0.01$ to $n=2.16\pm0.13$ after the glitch. The other one, PSR J1119-6127, showed unusual radio pulse-profile changes and short radio bursts following the 2007 glitch. The radio activity was present only after the glitch and lasted no more than  ∼3 months, the fact of which strongly implies that the two phenomena were related to each other [832]. These are similar to the magnetar glitches.

A variety of NS classes exhibit a variety of electromagnetic radiation and timing behaviours. Observationally, in the $P{{\rm \mbox{--}}}\dot{P}$ and P–B phase spaces (figures 5 and 17), the magnetar, HBP, and XINS populations are located in overlapping regions. In recent years, as similar magnetar-like activities have been discovered from non-magnetar classes (see section 4.1), manifestations that used to be distinct have become indistinct. This implies that these populations are likely to be evolutionarily linked to each other, which is hopefully explained coherently with decay of the magnetic field as a primary parameter (see, for example, a sophisticated theoretical model [820]).

At the time of writing, it is unclear how a magnetic field is held inside a NS. The internal magnetic field strength and configuration would be set by the time of crust formation (∼100 s from the core-collapse [749]). During the proto-NS phase (before the crust formation), the magnetic field might be amplified through some combination of convection, differential rotation, and magnetic instabilities, depending on the progenitor core properties, e.g. a rotation profile and a seed field (see [248, 552, 738] for recent reviews). Recently, a new type of instability called the 'chiral plasma instability' was proposed as a possible mechanism to generate the strong toroidal and poloidal magnetic fields of magnetars [599]. If this instability is at work, the resulting helical magnetic field naturally explains not only the gigantic magnetic field of magnetars, but also their linked configuration of the toroidal and poloidal magnetic fields. Some numerical studies for a NS have begun to address this chiral magnetic effect [189, 537]. The initial field configuration at their birth significantly affects the subsequent field evolution [285, 298, 348, 424, 647, 648, 692, 807, 820], and it may explain the observed population varieties of NSs [648].

Studies of the stability of the magnetic field configuration of a non-degenerate star suggest that a star with only a toroidal [763] or poloidal field [529] is unstable; in other words, both magnetic field components are required for stability [764]. It is the same for NSs even though the magnetic field in place is orders of magnitude stronger [21, 252, 264]. In recent years, possible signatures of stellar free precession were reported in the hard x-ray band relative to the soft x-rays from two magnetars, 4U 0142+61 and 1E 1547.0−45408. These signatures are considered to originate from deformation of the stellar shape into a prolate shape due to strong internal toroidal magnetic fields (∼10¹⁶ G) [516, 517, 519].

Here is the standard picture of the evolution of the magnetic fields of a NS. After the proto-NS phase, the internal magnetic field decays (see details [281, 396, 426, 552, 804] for recent reviews). The dissipated magnetic field energy could heat a NS and affect the observables. The velocity differences of each component inside a NS (neutrons, protons, electrons, and, perhaps other 'exotic' particles) affects the magnetic-field evolution. The electron flow with a velocity relative to the ion of $v_{\rm e}=j/n_{\rm e}ec$ induces the Ohmic dissipation and Hall drift, where j  is a current density and $n_{\rm e}$ is a number density of electrons. The Hall-drift timescale of $t_{\rm Hall}\propto v_{\rm e}^{-1}$ is inversely proportional to B through Ampère's law, ${\bf j}=(c/4\pi)\nabla\times{\bf B}$. Then, a stronger magnetic field can generate higher multipole magnetic fields with shorter timescale before it dissipates, and as a result, the Ohmic decay is enhanced (e.g. [173, 282, 289, 347, 375, 582, 717, 811]). The Hall drift is a predominant mechanism of the magnetic-field evolution in the solid crust. The field advection accumulates the magnetic stress in the solid crust, which could cause plastic deformation and result in (magnetar) glitches and outbursts [18, 19, 72, 73, 299–301, 376, 457–459, 471, 477, 478, 509, 632, 645, 696, 697, 785, 830, 839, 840].

In the NS core, ambipolar diffusion caused by the motion of plasma relative to that of neutrons affects the magnetic-field evolution [73, 134, 286, 289, 311, 353, 354, 381, 625, 626, 716, 783]. The Lorentz force ${\bf j}\times{\bf B}/c\propto B^2$ drives charged particles through neutrons at the ambipolar-diffusion velocity. The diffusion timescale is proportional to B⁻². The ambipolar diffusion is opposed by the following two factors. One is the friction caused by nuclear collisions between protons and neutrons, and the other is chemical potential difference between the plasmas and neutrons caused by a local deviation from the chemical $\beta$-equilibrium. For the high temperature core ($T\gtrsim10^9$ K), a high rate of weak interactions, $e + p \leftrightarrow n$, quickly erase the chemical potential difference, and thus the ambipolar diffusion velocity is mainly determined by the force balance between the friction and the Lorentz force. As the core cools down, the pressure gradient due to the chemical potential difference becomes significant, and eventually dominant, to balance with the Lorentz force and to determine the ambipolar diffusion velocity. It should be noted that most of the dissipated energy in the core is extracted as neutrino emission. For the surface temperature of typical magnetars of  ∼$4\times10^6$ K, the core temperature should be  ∼10⁹ K [664]. The required strength of the core magnetic field for this core temperature (∼10⁹ K) during  ∼10 kyr is $B\gtrsim10^{16}$ G [73]. The magnetic-field evolution in the core strongly depends on the superfluidity and superconductivity [107, 218, 286, 304, 625].

A few pieces of phenomenological discussion about the magnetic field decay have been published [156, 178, 584, 750]. A simple analytical decay model was proposed [156], which was basically in agreement with recent numerical calculations. It was further investigated in subsequent studies. Here is a brief summary of the analytical decay model. The basic equation of the model is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{{\rm d}B}{{\rm d}t}=-aB^{1+\alpha_{\rm d}}, \label{eq:B-field-evolution} \nonumber \end{align} \tag{ 44 }$$

where t is the time since the birth of an NS and $\alpha_{\rm d}$ ($\geqslant 0$) and a ($\geqslant 0$) are the constant parameters that govern the magnetic-field decay. The solution of the equation is

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle B(t) = \left\{\begin{array}{@{}ll@{}} B_{\rm i} \left(1+\alpha_{\rm d} \frac{t}{\tau_{\rm d}}\right)^{-1/\alpha_{\rm d}} & {(\alpha_{\rm d} \ne 0)} \nonumber \\ B_{\rm i}\exp\left(-\frac{t}{\tau_{\rm d}}\right) & (\alpha_{\rm d}=0) \end{array} \right. \label{eq:B-time-dependence} \nonumber \end{align} \tag{ 45 }$$

where $B_{\rm i}$ is the initial magnetic field and $\newcommand{\e}{{\rm e}} \tau_{\rm d}\equiv 1/(aB_{\rm i}^{\alpha_{\rm d}})$ is the characteristic time-scale until which the power-law-like magnetic decay dominates. From the phenomenological analysis, the B-dependent decay with an index of $1\lesssim \alpha_{\rm d}\lesssim 2$ is favoured [178].

The magnetic field decay following equation (45) with $0\leqslant\alpha_{\rm d}<2$ leads to a saturation period of $\newcommand{\e}{{\rm e}} P_{\infty}[\equiv P(t\rightarrow\infty)]$, in which the magnetar asymptotically reaches in the $P{{\rm \mbox{--}}}\dot{P}$ diagram the narrow vertical strip along the $\dot{P}$ axis (figure 6). Integrating equation (13) with time and using equation (45) for the dipole field $B_{\rm d}(t)$, we can obtain the rotation period $P(t)$ from

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:period_evolution} P^2(t)=P_{\rm i}^2+\frac{4\pi^2R_{\rm ns}^6B_{\rm i}^2\tau_{\rm d}}{3c^3I(2-\alpha_{\rm d})}\left[1-\left(1+\alpha_{\rm d}\frac{t}{\tau_{\rm d}}\right)^{(\alpha_{\rm d}-2)/\alpha_{\rm d}}\right], \nonumber \end{align} \tag{ 46 }$$

where the notations used in equation (13) are adopted. This equation (46) shows the existence of a saturation period for $0<\alpha_{\rm d}<2$. With time-integration of equation (13) with a range from 0 to $\infty$, the saturation period is calculated to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{eq:saturation-period} P_{\infty}^2=P_{\rm i}^2+\frac{4\pi^2R_{\rm ns}^6B_{\rm i}^2\tau_{\rm d}}{3c^3I(2-\alpha_{\rm d})}. \nonumber \end{align} \tag{ 47 }$$

Substituting equations (13), (45) and (47) to equation (46) yields the evolution track in the $P{{\rm \mbox{--}}}\dot{P}$ diagram:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \dot{P}=\frac{4\pi^2R_{\rm ns}^6B_{\rm i}^2}{3Ic^3P}\left[1-\left(\frac{P^2-P_{\rm i}^2}{P_{\infty}^2-P_{\rm i}^2}\right)\right]^{\frac{2}{2-\alpha_{\rm d}}}~(0\leqslant\alpha_{\rm d}<2). \label{eq:P-evolution} \nonumber \end{align} \tag{ 48 }$$

Evolution tracks with $\alpha_{\rm d}=1.5$, $P_{\infty}=10.8$ s, $B_{\rm i}=10^{15}$ G, 10¹⁴ G, 10¹³ G and 10¹² G are shown in figure 6 as black curves.

The observed $P_{\infty}$ values of magnetars, which are isolated NSs, are approximately  ∼12 s: $P \sim 11.0$ s for 1RXS J170849.0−400910, $P \sim 11.8$ s for 1E 1841−045, and $P\sim 11.6$ s for 3XMM J185246.6+003317. No isolated x-ray pulsars with a period of  >12 s are known so far. This typical period of  ∼11 s is slightly longer than those of RPPs, and is similar to that of the XINS class. This period distribution (P  <  12 s) indicates that spin-down would be turned off in the course of magnetic-field decay.

Figure 24 shows a comparison between the age estimated from SNR x-ray diagnostics and the pulsar characteristic age, $\tau_{\rm c}$ (equation (12)). The pulsar characteristic age is overestimated for older magnetars. This discrepancy can be explained by an appropriate magnetic-field decay model (e.g. [502]).

Zoom In Zoom Out Reset image size

Extragalactic transient phenomena have provided another implication on the magnetar formation. Theories propose that some fraction of GRBs and bright SNe are powered by newborn magnetars at their central regions.

Superluminous SNe (SLSNe) were discovered with peak luminosities of 10–100 times higher than those of normal CCSNe [577, 668]. SLSNe can be classified in two different spectral types; those without narrow hydrogen emission lines (referred to as SLSN-I) and those with (referred to as SLSN-II). The mechanism of the SLSN-II is well explained with shock interaction with surrounding circumstellar material (but see also [198] for counter-argument). However, the SLSN-I does not show evidence for the interaction. The high luminosity of SLSN-I would require several solar masses of ⁵⁶Ni to be synthesized during the explosion, and thus an additional energy input is discussed. A spin-down energy injection from a rapidly rotating magnetized NS has been proposed to explain SLSN-I [308, 391, 556, 590]. Optical light curves of SLSN-I are successfully fitted using a magnetized-NS-powered model, with dipole magnetic fields of 10¹³–10¹⁴ G and initial spin periods $P_{\rm i}$ of 1–8 ms [391]. Note that some alternative models have been also proposed, e.g. pair-instability SNe and additional energy injection from BH accretion [203]. The recent detection of radio emission coincident with a SLSN, PTF10hgi [215], was successfully explained with a model of a central-engine-powered nebula, which gives a consistent result for the age and luminosity of the unresolved radio source at the position of the SLSN-I. Since the event rate of the SLSNe, which is only  ∼$0.01\%$ of the CCSNe rate [665], is much lower than the formation rate of the galactic magnetars (∼$10\%$ of that of NSs [75, 431]), formation of a rapidly rotating new-born magnetar in the extragalactic transients is, if there is any, suggested to be very rare. Combined with their short spin-down timescale of $E_{\rm rot}/L_{\rm sd}\lesssim$ a few years, SLSN-I could provide unique information about the characteristics of new-born magnetars.

Another exotic event related with magnetars is fast radio bursts (FRBs), which are bright radio flashes (0.02–150 Jy) with a duration of milliseconds [491, 786]. The unknown astrophysical origin of FRBs has received a considerable attention (for review, see [161, 400, 401, 447, 634]). As of July 2019, ∼82 FRBs have been reported (see [633] for the FRB catalogue). FRBs are considered to be of extragalactic origin on the basis of the fact that their large dispersion measure (114–2596 cm⁻³ pc) exceeds the contribution from propagation through our Galaxy.

The identification of FRB 121102 as a repeating source was a breakthrough [737]. Subsequent direct localization identified the host galaxy (a dwarf star-forming galaxy) and determined its redshift ($z = 0.192\,73$, 972 Mpc) [137, 525]. The corresponding FRB luminosity of this repeating source is $L_{\rm FRB}=$(0.03–6)$\times 10^{42}$ erg s⁻¹. Furthermore, a quasi-steady 180 $\mu$Jy radio source was discovered within the error circle ($\lesssim$40 pc) of the FRB direction [766]. Recently, the second repeating FRB source, FRB 180814.J0422+73, was discovered by CHIME [145]. On the basis of the observed properties, young and magnetized NSs are one of the promising candidates of FRBs. Several theoretical evaluations of the radiation mechanism have been proposed, e.g. hypothetical radio emission from magnetar GFs at their initial spikes [507, 650] and GPs from young RPPs [163, 392, 414, 650].

Relativistic particle outflow ('pulsar wind') from a NS interacts with surrounding remnant gas and magnetic fields, creating a PWN behind a termination shock ([409, 689]; see [47, 269, 399, 654, 694] for recent reviews). The broadband SED of a PWN is composed of synchrotron radiation extending from the radio to a few hundred MeV and an inverse-Compton scattering component in the higher energy range [53, 269, 758]. The wind at the origin around the light cylinder is thought to be highly magnetized, i.e. $\sigma \gg 1$, where $\sigma$ is the magnetization parameter defined as the ratio of the electromagetic energy to the plasma kinetic energy. By contrast, the energy in a PWN is particle-dominant, $\sigma \lesssim 0.01$ (for at least TeV-detected ones [759, 760, 794]; see also [534]). It still remains unknown where the electromagnetic energy is converted to the kinetic energy of plasma, more specifically, whether it is ahead of the shock or behind. The plasma behind the shock is not simply thermalized but accelerated to high energies as indicated by a synchrotron spectrum from a non-thermal particle energy distribution, of which the acceleration mechanism (e.g. magnetic reconnection, shock acceleration, or stochastic acceleration) is still under debate ([379, 727] for recent reviews).

The termination shock is formed, at which the pulsar-wind pressure is balanced with the ambient pressure $P_{\rm ext}$. This condition is given by $L_{\rm sd} / (4 \pi c R_s^2) = P_{\rm ext}$, where R_s is the distance between the shock and NS. The size of the shock is thus evaluated to be

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \label{nebula-eq1} R_s &= 1.7 \times 10^{-3} {{\rm pc}}~ L_{\rm sd,33}^{1/2}P_{\rm ext,-10}^{-1/2} \nonumber \\ &\sim 0.34^{\prime \prime} \left(\frac{d}{1~{\rm kpc}}\right)^{-1}L_{\rm sd,33}P_{\rm ext,-10}^{-1/2}, \nonumber \end{align} \tag{ 49 }$$

where d is the distance from the observer to the pulsar. Modern x-ray mirrors with a fine angular resolution have provided capability to resolve some nearby PWNe (most are powered by energetic pulsars with $L_{\rm sd} \gtrsim 10^{35}$ erg s⁻¹), which approximately corresponds to $R_s > 1^{\prime \prime}$. The actual PWN size (high synchrotron emissivity region) observed with Chandra, shown in figure 25 from [388], is typically ten times larger than the shock size evaluated from equation (49) and shows a large scatter by an order of magnitude.

Zoom In Zoom Out Reset image size

The energy sources of the emission from both a PWN and its pulsar are the rotational energy of a central NS, which is directly related to the pulsar spin-down luminosity $L_{\rm sd}$. In section 2.5, we have already learnt a correlation between the spin-down luminosity and the x-ray luminosity for (rotation-powered) pulsars (a $L_{\rm x}$–$L_{\rm sd}$ plot in figure 12; [68, 70, 387, 415, 482, 661, 713, 724]). Given that PWNe too are powered by the rotation energy of their central NSs and the synchrotron cooling time for x-ray emitting electrons is much shorter than the age of NSs, one might expect that there is a similar relation between the x-ray synchrotron luminosity $L_{\rm x}$ and $L_{\rm sd}$ for PWNe. Statistical studies have suggested that indeed there is, as shown in figure 26 [388, 482].

Zoom In Zoom Out Reset image size

The radiation efficiencies to the spin-down luminosity of $\newcommand{\e}{{\rm e}} \eta_{\rm psr} = L_{\rm x}/L_{\rm sd}$ for (rotation-powered) pulsars and $\newcommand{\e}{{\rm e}} \eta_{\rm pwn} = L_{\rm pwn}/L_{\rm sd}$ for PWNe are also found to be correlated to each other (figure 27; [388, 724]). Both of $\newcommand{\e}{{\rm e}} \eta_{\rm psr}$ and $\newcommand{\e}{{\rm e}} \eta_{\rm pwn}$ show a large dispersion, ranging 10⁻⁵–0.1 (figure 27). Although systematic uncertainties of the distances of the sources may contribute to a small part of the dispersion, they alone cannot be attributed to the large dispersion [724]. Another potential cause may be possible emission anisotropy; however, it is also excluded for the case of $\newcommand{\e}{{\rm e}} \eta_{\rm pwn}$ at least, because PWNe should not have a large anisotropy in the radiation, being different from the pulsar magnetospheric emission. These indicate that an additional physical parameter(s) may govern the x-ray luminosity [388, 387, 482], such as pair multiplicity and energy distribution function of wind particles. Interestingly, the fact that there is a correlation between $\newcommand{\e}{{\rm e}} \eta_{\rm pwn}$ and $\newcommand{\e}{{\rm e}} \eta_{\rm psr}$ (figure 27; [388, 724]) suggests that the physical process responsible for the large dispersion of $\newcommand{\e}{{\rm e}} \eta_{\rm pwn}$ has an effect also on the pulsar magnetosphere in $\newcommand{\e}{{\rm e}} \eta_{\rm psr}$. In addition, a marginal positive correlation between $\newcommand{\e}{{\rm e}} \eta_{\rm psr}$ and the photon indices in the x-ray energy spectra $\Gamma$ of PWNe was reported [388], whereby $1 \lesssim \Gamma \lesssim 2$ corresponds to a particle distribution index of $1 \lesssim p \lesssim 3$.

Zoom In Zoom Out Reset image size

Magnetars could have the PWN-like extended emission if their rotation energy converts to particle wind [757]. However, the observational confirmation is difficult, because the spin-down luminosity of most magnetars is $L_{\rm sd}<10^{35}$ erg s⁻¹. Possible spatially extended emissions associated with magnetars have been reported [822, 854]. However, they could be interpreted as dust scattering halo of their past activities [237, 602], rather than PWNe. Recently, the first PWN-like emission around a magnetar, Swift J1834.9−0846, was discovered [853]. The magnetic energy of magnetars as the energy source of the wind have been also proposed [305, 793], and there are some observational hints that the nebula surrounding J1119−6127 at three months after the bursts is brighter than a pre-burst observations in x-ray [95]. The creations of transient radio nebulae after giant flares were also reported [257, 266]. Future multi-wavelength observations will distinguish them.

NS is usually formed from a gravitational CCSN with a Fe or O/Ne/Mg core of a massive star. In an ordinary CCSN, the inner part of the progenitor forms a NS, while the outer matter is ejected by bounced shock and forms an expanding shell, which is referred to as a SNR. Since SNRs are typically bright in x-rays for $\lesssim$10 kyr from its birth, only young NSs are associated with x-ray-illuminating SNRs, as shown in the $P{{\rm \mbox{--}}}\dot{P}$ (figure 1) and distance-size diagrams (figure 28). The size and expanding velocity of a SNR give the estimation of the true age of the associated NS. General descriptions of the SNR evolution are given in, for example, [799] (and a library for simulating SNR evolution in Python is presented in [466]).

Zoom In Zoom Out Reset image size

After decades of theoretical and observational studies of SNe, how the initial physical parameters of a NS are determined, especially the conditions of the magnetar formation, still remains as an unresolved important question. In the conventional magnetar model, a magnetar is supposed to be born through an energetic explosion of a massive star (mass of 'zero-age main sequence', $M_{\rm ZAMS}\gtrsim 30$–$40M_{\odot}$). Some of massive progenitors are expected to evolve into Wolf–Rayet stars and eventually explode as stripped-envelope SNe [325]. However, neither this type of progenitors nor magnetar-forming SNe have yet been observationally identified. x-ray diagnostics of a magnetar-associated SNR are expected to provide the type, progenitors, and birth environment of a SN (for related reviews, see [143, 730, 731, 821]).

Early studies of galactic SNRs hosting magnetars (1E 1841−045 in SNR Kes 73 and 1E 2259+586 in SNR CTB109) estimated the SN explosion energies to be $E_{\rm SN}=10^{50}$–10⁵¹ erg, which are consistent with those of ordinary CCSNe (∼10⁵¹ erg) [824]. Table 2 lists the explosion energies of magnetars and magnetar-like HBPs associated with SNRs that have been estimated from observations by the time of writing, and table 3 lists the known progenitor masses. There is no strong evidence for higher explosion energies of magnetar-related SNe than ordinary SNe, whereas the estimated progenitor masses suggest higher mass progenitors for magnetar-host SNe [824].

Table 2. List of explosion energy evaluated from magnetar-associated SNRs.

Source name	Explosion energy (erg)	Distance (kpc)	Reference
Magnetars
N49 (SGR 0526−66)	$(1.3\pm 0.3)\times 10^{51}$ erg	50	[824]
CTB 109 (1E 2259+586)	$(0.7\pm 0.3)\times 10^{51}$ erg	3	[709]
Kes 73 (1E 1841−045)	$(0.5\pm 0.3)\times 10^{51}$ erg	7.0	[824]
	$(0.30^{+0.28}_{-0.18})\times 10^{51}$ erg	8.5	[449]
G292.2−0.5 (PSR J1119−6127)	$(0.6^{+0.1}_{-0.2})\times 10^{51}$ erg	8.4	[448]
Kes 75 (PSR J1846−0248)	$1.9\times 10^{51}$	6	[465]

Table 3. List of progenitor masses estimated from magnetar-associated SNRs.

Source name	Mass ($M_{\odot}$)	Reference
Magnetars
SGR 1806−20	$48^{+20}_{-8}$	[85]
	∼50	[250]	Young massive star cluster (age  ∼4 Myr)
1E 2259+586	∼40	[584]	Ejecta of SNR CTB109
CXO J1647−455	∼40	[581]	Young massive star cluster (age  ∼4 Myr)
		[698]
1E 1048.1−5937	30–40	[267]	Stellar wind bubble blown by the progenitor
SGR 0526−66	26	[805]	Ejecta of SNR N49 in LMC
1E 1841−045	>20	[449]	Suggested by abundances of SNR Kes 73
SGR 1900+14	17 $\pm$ 2	[181]	Aged star cluster (∼14 Myr)

Another important question about the NS formation is whether a SN can form a low-B NSs rather than strongly magnetized NSs. Observationally, there is no clear evidence to indicate a difference between SNRs hosting CCOs and those hosting other types of highly-magnetized NSs [533]. However, there are a few observational indications for low-B NSs to be formed during SN explosions. One of them is the fact that the magnetic fields of CCOs derived with the $P{{\rm \mbox{--}}}\dot{P}$ method, $B_{\rm d} \sim10^{10}$ G, are lower than those of other magnetized young NSs. An exception is RCW 103, of which the transient activity suggests an existence of magnetic activity, hence a strong magnetic field (see section 4.1). Another interesting implication comes from the peculiar x-ray binary, Circinus X-1 (Cir X-1). In 1984 and 2010, this NS exhibited type-I x-ray bursts [484, 770], which occur under weak magnetic fields, e.g. B  <  10¹⁰ G [89, 263]. Then, after circa 2000, when Cir X-1 entered a historically low luminosity state, the surrounding SNR was discovered with Chandra. Strikingly, the estimated age of the SNR is only  <3 kyr [327, 328]. This fact is a strong indication that the magnetic field of Cir X-1 was weak ($B\lesssim 10^{10}$) when it was born, providing that Cir X-1 was born with the SN that generated the discovered  <3 kyr-old SNR. Given that both CCOs and Cir X-1 are young, (1–10 kyr), these facts suggest that a significant fraction of NSs are born with weaker magnetic fields (see also [404]).

In addition to CCSNe, there are alternative channels of NS formation, e.g. accretion-induced collapse (AIC) of a WD (e.g. [593]) and binary NS mergers (e.g. [349]). The AIC of a WD would potentially produce weakly-magnetized NSs and strongly-magnetized NSs [781]. Note that the formation of a NS through AIC was suggested from an x-ray observation of a SyXB [226].

The measured kick velocities of 293 NSs are plotted in figure 29. See [50, 241, 345] for comparison with various models of velocity distribution. The typical kick velocity of NSs of a few hundred km s⁻¹ is by an order of magnitude faster than the observed velocity of normal stars, which ranges between 10–30 km s⁻¹. The pulsar kick mechanism is important to understand the formation of NSs in SN explosion [369]. Asymmetric explosion of SNe [118, 370] and anisotropic emission of the neutrinos from the NS [92, 842], among others, are proposed as the kick mechanism. An alignment of the kick direction and the rotation axis of a NS is also observationally suggested [455, 589, 594, 673], which may be related to late-time accretion [580]. Though the number of the magnetars with the known kick velocities is still limited to 4 (SGR 1806−20, SGR 1900+14, XTE 1810−197, and PSR J1550-5418; listed in table 4; see also figure 29), their velocity is consistent with those of canonical pulsars and hot young isolated NSs.

Zoom In Zoom Out Reset image size

Table 4. List of known kick velocities of magnetars.

Source name	Linear transverse velocity	Reference
Magnetars
4U 0142+61	102 $\pm$ 26 km s−1	[768]
SGR 1900+14	130 $\pm$ 30 km s−1	[767]
1E 2259+586	157 $\pm$ 17 km s−1	[768]
XTE 1810−197	212 $\pm$ 35 km s−1	[329]
1E 1547.0−5408	280$^{+130}_{-120}$ km s−1	[191]
SGR 1806−20	350 $\pm$ 100 km s−1	[767]

Young bright XINS
4 sources (e.g. RX J1856.5−3754)	350 $\pm$ 180 km s−1	[774]

CCO
RX J0822−4300 (SNR Puppis A)	672 $\pm$ 115 km s−1	[69]