The space velocities of radio pulsars

Known models proposed to explain the high space velocities of pulsars based on asymmetry of the transport coefficients of different sorts of neutrinos or electromagnetic radiation can be efficient only in the presence of high magnetic fields (to 10¹⁶ G) or short rotation periods for the neutron stars (of the order of 1 ms). This current study shows that the observed velocities are not correlated with either the pulsar periods or their surface magnetic fields. The initial rotation periods are estimated in a model for the magnetedipolar deceleration of their spin, aßsuming that the pulsar ages are equal to their kinematic ages. The initial period distribution is bimodal, with peaks at 5 ms and 0.5 s, and similar to the current distribution of periods. It is shown that asymmetry of the pulsar electromagnetic radiation is insufficient to give rise to additional acceleration of pulsars during their evolution after the supernova explosion that gave birth to them. The observations testify to deceleration of the motion, most likely due to the influence of the interstellar medium and interactions with nearby objects. The time scale for the exponential decrease in the magnetic field τ_D and in the angle between the rotation axis and magnetic moment τ_ß are estimated, yielding τ_β = 1.4 million years. The derived dependence of the transverse velocity of a pulsar on the angle between the line of sight and the rotation axis of the neutron star corresponds to the expected dependence for acceleration mechanisms associated with asymmetry of the radiation emitted by the two poles of the star.     

The radio emission of anomalous pulsars

Previously developed methods for estimating the angle β between the spin axis of a neutron star and its magnetic moment together with observational data for anomalous X-ray pulsars (AXPs) indicate that these objects are nearly aligned rotators, and that the drift model can be applied to them. The magnetospheres of aligned rotators are appreciably more extended than in pulsars with large values of β. With such extents for the magnetosphere, the conditions for the generation of transverse waves via the cyclotron instability are satisfied. The expected spectrum of the resulting radiation is very steep (its spectral index is α > 3), consistent with the observed radio spectra of known AXPs (α > 2). A large magnetosphere also favors the appearance of appreciable pitch angles for relativistic electrons, and therefore the generation of synchrotron emission. The maximum of this emission falls in the microwave range. This mechanism provides appreciable fluxes at frequencies of tens of gigahertz and can explain the observed enhanced AXP radiation in this range.     

Peculiarities of radio pulsars with emission outside the radio range

Parameters of 100 radio pulsars detected outside the radio range (he pulsars) are compared with those of pulsars radiating only in the radio (n pulsars). The periods of he pulsars are, on average, appreciably shorter than those of n pulsars: 〈P〉 = 0.10 and 0.56 s, respectively. The distribution of the magnetic field at the light cylinder is shifted toward higher magnetic fields for the pulsars with high-energy radiation, compared to the distribution for pulsars radiating only in the radio. The magnetic fields at the light cylinder are 〈B _lc〉 = 9×10³ G for he radio pulsars, and 〈B_lc〉 = 56 G formost purely radio pulsars. This suggests the generation of high-energy nonthermal radiation in radio pulsars at the peripheries of their magnetospheres. The distribution of the spin-energy loss rate dE/dt is uniform for he pulsars, and is characterized by a higher average value \(\left( {\left\langle {\log \frac{{dE}} {{dt}}} \right\rangle = 35.53} \right) \) , compared to n pulsars, \(\left( {\left\langle {\log \frac{{dE}} {{dt}}} \right\rangle = 32.60} \right) \) . The spatial distribution of he pulsars is nonuniform: they form two well separated clouds.     

The spectra of hard radiation from radio pulsars

The kinetic equation for the distribution function of relativistic electrons is solved taking into account quasi-linear interactions with waves and radiative processes. Mean values of the pitch angles ψ are calculated. If the particles of the primary beam with Lorentz factors γ_b～10⁶ are resonant, then the condition γ_bψ_b?1 is satisfied, the particle distribution is described by the function f _‖(γ) ∝ γ^?4, and the synchrotron radiation spectrum is characterized by the spectral index α=3/2. On the other hand, if a cyclotron resonance is associated with particles of the high-energy tail of the secondary plasma (γ_t～10⁵), then γ_tψ_t?1, and the distribution function has two parts—f _‖(γ) ∝ γ and f _‖(γ) ∝ γ^?2—which correspond to the spectral indices α₁=+1 and α₂=?0.5. This behavior is similar to that observed for the pulsar B0656+14. The predicted frequency of the maximum ν_m=7.5×10¹⁶ Hz coincides with the peak frequency for this pulsar. The model estimate for the total synchrotron luminosity of a typical radio pulsar with hard radiation L _s =3×10³³ erg/s is in agreement with observed values.     

Magnetic fields of radio pulsars

The mechanism of magnetodipole braking of radio pulsars is used to calculate new values of the surface magnetic fields of neutron stars. The angles β between the spin axes and magnetic moments of the neutron stars were estimated for 376 radio pulsars using three different methods. It is shown that small inclinations of magnetic axes dominate. The equatorial magnetic fields for the considered sample of pulsars are calculated using the β values obtained. As a rule, these magnetic fields are a factor of a few higher than the corresponding values in known catalogs.     

The distribution of space velocities of radio pulsars

The distribution of the directions of the space velocities of 67 radio pulsars is shown to be strongly anisotropic. This anisotropy cannot be explained by the structure of our Galaxy or by various types of solar motions. Pulsars with stronger surface magnetic fields B have higher velocities V. The mean value of V for B < 10¹⁰ G is 108 km/s, while 〈V〉 = 340 km/s for B > 10¹⁰ G. These results must be taken into account when identifying a mechanism to explain the observed pulsar velocities and their anisotropy.     

Integrated radio luminosities of pulsars

The integrated radio luminosities of 311 long-period (P > 0.1 s) and 27 short-period (P < 0.1 s) pulsars have been calculated using a new compilation of radio spectra. The luminosities are in the range 10²⁷ ? 10³⁰ erg/s for 88% of the long-period pulsars and 10²⁸ ? 10³¹ erg/s for 88% of the short-period pulsars. We find a high correlation between the luminosity L and the estimate L ₁ = S ₄₀₀ d ² from the catalog of Taylor et al. The factor η for the transformation of the rotational energy of the neutron star into radio emission increases-decreases with increasing period for long-period and short-period pulsars. The mean value of η is ?3.73 for the long-period and ?4.85 for short-period pulsars. No dependence was found between L and the pulsar’s kinematic age t _k = |z|/〈v _z〉, where |z| and 〈v _z〉 = 300 km/s are the pulsars’ height above the plane of the Galaxy and mean velocity. A dependence of L on the rate of rotational energy losses ? was found for both groups of pulsars. It is shown that L ∝ ? ^1/3 for the entire sample. The pulsar luminosity function is constructed, and the total number and birth rate of pulsars in the Galaxy are calculated.     

The electric fields of radio pulsars with asymmetric nondipolar magnetic fields

The effect of the curvature of open magnetic field lines on the generation of electric fields in radio pulsars is considered in the framework of a Goldreich-Julian model, for both a regime with a free outflow of electrons from the neutron-star surface and the case of a small thermoemission current. An expression for the electron thermoemission current in a strong magnetic field is derived. The electric field associated with the curvature of the magnetic flux tubes is comparable to the field generated by the relativistic dragging of the inertial frames.     

Pulsed optical emission by radio pulsars

The luminosity L of radio pulsars due to synchrotron radiation by the primary beam at the magnetosphere periphery is derived. There is a strong correlation between the observed optical luminosities of radio pulsars and the parameter $\dot P/P^4$ (where P is the pulsar period). This correlation predicts appreciable optical emission from several dozen pulsars, in particular, from all those with P<0.1 s. Agreement with optical observations can be achieved for Lorentz factors of the secondary plasma γ_p=2–13. Plasma with such energies can be produced only when the magnetic-field structure near the neutron-star surface deviates substantially from a dipolar field. The peak frequency of the synchrotron spectrum should shift toward higher values as the pulsar period P decreases; this is, in agreement with observational data for 27 radio pulsars for which emission has been detected outside the radio band.     

The formation of millisecond radio pulsars in close binary systems

An analysis of the basic parameters of a sample of radio and X-ray pulsars that are members of close binary systems is used to separate them into several families according to the nature of the pulsar companions and the previous evolution of the systems. To quantitatively describe the main parameters of close binaries containing neutron stars, we have performed numerical modeling of their evolution. The main driving forces of the evolution of these systems are the nuclear evolution of the donor, the magnetically coupled and radiation-induced stellar winds of the donor, and gravitational-wave radiation. We have considered donors that are low-mass stars in various stages of their evolution, nondegenerate helium stars, and degenerate stars. The systems studied are either the products of the normal evolution of close binaries with large initial component-mass ratios or result from inelastic collisions of old neutron stars with single and binary low-mass, main-sequence stars in the dense cores of globular clusters. The formation of single millisecond pulsars requires either the dynamical disruption of a low-mass (?0.1M_⊙) donor or its complete evaporation under the action of the X-ray radiation of the millisecond pulsar. The observed properties of binary radio pulsars with eccentric orbits combined with the bimodal spatial-velocity distribution of single radio pulsars suggest that it may be possible to explain the observed rotational and spatial motions of all radio pulsars as a result of their formation in close binaries. In this case, neutron stars formed from massive single stars or the components of massive wide binaries probably cannot acquire the high spatial velocities or rapid rotation rates that are required for the birth of a radio pulsar.     

A spindown mechanism for short-period radio pulsars

It is shown that a model with accretion in a “quasi-propeller” mode can explain the observed spindown of pulsars with periods P<0.1 s. The mean accretion rate for 39 selected objects is \(\dot M = 5.6 \times 10^{ - 11} M_ \odot /year\). If \(\dot M\) is constant during the pulsar’s lifetime, the neutron star will stop rotating after 10⁷ years. The mean magnetic field at the neutron-star surface calculated in this model, \(\bar H_0 = 6.8 \times 10^8 G\), is consistent to an order of magnitude with the values of H₀ for millisecond pulsars from known catalogs. However, the actual value of H₀ for particular objects can differ from the catalog values by appreciable factors, and these quantities must be recalculated using more adequate models. The accretion disk around the neutron star should not impede the escape of the pulsar’s radiation, since this radiation is generated near the light cylinder in pulsars with P<0.1 s. Pulsars such as PSR 0531+21 and PSR 0833-45 have probably spun down due to the effect of magnetic-dipole radiation. If the difference in the braking indices for these objects from n=3 is due to the effect of accretion, the accretion rate must be of the order of 10¹⁸ g/s.     

The origin of the rotational and spatial velocities of radio pulsars

We analyze possible origins of the observed high rotational and spatial velocities of radio pulsars. In particular, these can be understood if all radio pulsars originate in close binary systems with orbital periods of 0.1–100 days, with the neutron star being formed by a type Ib,c supernova. The high spatial velocities of pulsars (v _p up to 1000 km/s) reflect the high Keplerian velocities of the components of these binaries, while their short periods of rotation (P _p < 4 s) are due to the rapid rotation of the presupernova helium-star components with masses of 2.5–10 M_⊙, which is synchronous with their orbital rotation. Single massive stars or components in wide binaries are likely to produce only slowly rotating (P _p > 4 s) neutron stars or black holes, which cannot be radio pulsars. As a result, the rate of formation of radio pulsars should be a factor of a few lower than the rate of type II and type Ib,c supernovae estimated from observations. This scenario for the formation of radio pulsars is supported by (i) the bimodal spatial velocity distribution of radio pulsars; (ii) the coincidence of the observed spatial velocities of radio pulsars with the orbital velocities of the components of close binaries with nondegenerate helium presupernovae; (iii) the correlation between the orbital and rotational periods for 22 observed radio pulsars in binaries with elliptical orbits; and (iv) the similarity of the observed rate of formation of radio pulsars and the rate of type Ib,c supernovae.     

Classification of radio pulsars using the principle-components method

The principle-components method is used as a basis to analyze the distributions of known radio pulsars in spaces of eigenvectors of correlation matrices for various samples of pulsars and classification parameters (from 4 to 11 parameters characterizing the physical and kinematic properties of the objects). Pulsars with periods P < 0.1 s form a separate cluster, far from the cluster formed by “normal” pulsars with P ～ 1 s, in all the studied spaces. These two groups also differ appreciably in their other parameters (period derivatives, magnetic fields, pulse widths). In particular, the spatial velocities of short-period pulsars (106 km/s) are appreciably lower than those displayed by long-period pulsars (334 km/s). The distributions of the pulsars at southern (Z < 0) and northern (Z > 0) Galactic latitudes do not differ; i.e., there is no anisotropy in the motions in these two directions perpendicular to the Galactic plane, or in the corresponding distributions of the pulsar parameters.     

The death lines of radio pulsars for dipolar and asymmetric magnetic fields

The effect of curvature of open magnetic-field tubes on the death lines of radio pulsars is studied. The solution is obtained in the framework of a Goldreich-Julian model for both dipolar and asymmetric magnetic fields. The tube-axis curvature can shift the death line appreciably toward either longer or shorter periods. If the field is dipolar and gamma rays are generated by the inverse Compton effect, the formation of secondary plasma is more efficient near the death line. In the case of an asymmetric magnetic field, the generation of radio emission beyond the tube of open field lines is possible.     

Differences in the parameters of radio pulsars with short and long periods

A comparative analysis of various parameters of pulsars with short (P < 0.1 s) and long (P > 0.1 s) periods is carried out. There is no correlation between the radio and gamma-ray luminosities of the pulsars and their surfacemagnetic fields, but there is a correlation between the X-ray luminosity and the surfacemagnetic field. A dependence of the X-ray and gamma-ray luminosities on the magnetic field at the light cylinder is also found. This result provides evidence for the formation of hard, non-thermal emission at the periphery of the magnetosphere. An appreciable positive correlation between the luminosity and the rate of rotational energy loss by the neutron star is observed, supporting the idea that all radio pulsars have the same basic source of energy. The efficiency of the transformation of rotational energy into radiation is significantly higher in long-period pulsars. The dependence of the pulse width on the pulsar period is steeper for pulsars with short periods than for those with long periods. The results obtained support earlier assertions that there are differences in the processes generating the emission in pulsars with P < 0.1 s and those with P > 0.1 s.     

Parameters of radio pulsars in binary systems and globular clusters

The parameters of radio pulsars in binary systems and globular clusters are investigated. It is shown that such pulsars tend to have short periods (of the order of several milliseconds). Themagnetic fields of most of the pulsars considered are weak (surface fields of the order of 10⁸?10⁹ G). This corresponds to the generally accepted view that short-period neutron stars are spun up by angular momentum associated with the stellar wind from a companion. However, the fields at the light cylinders in these objects are two to three orders of magnitude higher than for the main population of single neutron stars. The dependence of the pulse width on the period does not differ from the corresponding dependences for single pulsars, assuming the emission is generated inside the polar cap, at moderate distances from the surface or near the light cylinder. The radio luminosities of pulsars in binary systems do not show the correlation with the rate of loss of rotational energy that is characteristic for single pulsars, probably due to the influence of accreting matter from a companion. Moreover, accretion apparently decreases the power of the emergent radiation, and can explain the observed systematic excess of the radio luminosity of single pulsars compared to pulsars in binary systems. The distributions and dependences presented in the article support generally accepted concepts concerning the processes occurring in binary systems containing neutron stars.     

Average pulse profiles of radio pulsars at 102 and 111 MHz

Average profiles for 180, mostly faint, pulsars at 102 and 111 MHz are presented. Pulse shapes have been obtained for the first time for most of the faint pulsars (about 50% of the total sample). A comparison with high-frequency data (mostly at 234, 408, 610, 925, and 1408 MHz) demonstrates appreciable changes in the profile width and shape with frequency. For most pulsars, the number of components is preserved, but the intensities of individual components can change. As a rule, the profile width increases with decreasing frequency. The possible generation of emission at the cyclotron frequencies is considered.     

A comparative analysis of the integrated radio luminosities of normal and millisecond pulsars

We present the results of a comparative statistical analysis of the integrated radio luminosities of millisecond and normal pulsars and their dependences on other parameters of the pulsars. The analysis is based on our own measurements of the flux densities, spectra, and integrated radio luminosities of the millisecond pulsars, as well as data from the literature used to determine the integrated radio luminosities for 545 pulsars, 50 of them millisecond pulsars. Despite large differences in their periods P, period derivatives $\dot P$ , magnetic fields B, and ages τ, the integrated radio luminosities of the millisecond and normal pulsars and their dependences on other parameters are approximately the same. The integrated radio luminosity depends on the parameter B/P ², which is proportional to the potential difference in the polar-cap gap; this may indicate that the radio energy of pulsars is determined by the energy of primary particles accelerated in the polar-cap gap. Secular decreases in the radio luminosities of both normal and millisecond pulsars were also detected.     

Evolution of the angle between the magnetic moment and the rotation axis of radio pulsars

It is shown that, when angular-momentum losses of a radio pulsar are represented as a sum of magnetic-dipole and current losses, the angle between the magnetic moment and rotation axis of the radio pulsar tends to some equilibrium value (near 45°). This process takes place on a timescale of the order of the pulsar’s characteristic age. Taking into account the non-dipolarity of the pulsar’s magnetic field changes this equilibrium angle.     

Magnetic fields and the $$\dot P - P$$ diagram for radio pulsars

Various mechanisms for the loss of angular momentum of neutron stars are analyzed. Theoretical predictions about the evolution of the period are compared with the observed distribution of pulsars on the log\(\dot P\)log(P) diagram. Pulsars with short periods (P≤0.1 s) cannot be fit well by any of the models considered. Their braking index is n=?1, which requires the development of a new braking mechanism. The evolution of pulsars with P>1.25 s is described by the law \(\dot P \propto P^2\), probably due to processes internal to the neutron stars. The observational data for pulsars with 0.1<P≤1.25 s can be fit with a hybrid model incorporating internal processes and magnetic-dipole losses. The magnetic fields in pulsar catalogs should be recomputed in accordance with the results obtained. For example, the magnetic fields obtained for two magnetars with P=5.16 s and P=7.47 s are B _s =1.7×10¹³ and 2.9×10¹³ G, which are lower than the critical field B_cr=4.4×10¹³ G. For a substantial fraction of pulsars, their characteristic ages \(\tau = P/2\dot P\) cannot serve as measures of their real ages.