The influence of nondipolar magnetic field and neutron star precession on braking indexes of radiopulsars

Some of radiopulsars have anomalous braking index values $n = \Omega \ddot{\Omega} / \dot{\Omega}^2 \sim \pm (10^3 \div 10^4) $. It is shown that such values may be related with nondipolar magnetic field. The precession of neutron star lead to rotation (in reference frame related with neutron star) of vector of angular velocity $\vec{\Omega}$ along direction of neutron star magnetic dipole moment $\vec{m}$ with angular velocity $\vec{\Omega}_{p}$. This process may cause the altering of electric current flow through inner gap and consequently the current losses with the same time scale as precession period $T_{p} = 2\pi / \Omega_{p}$. It occurs because of electric current in inner gaps is determined by Goldreich-Julian charge density $\rho_{GJ} = -\frac{\vec{\Omega} \cdot \vec{B}}{2\pi c}$, which are depend on angle between direction of small scale magnetic field and angular velocity $\vec{\Omega}$. It is essential that pulsar tubes nearby neutron star surface are curved. In current paper it is considered the only inner gaps with steady, electron charge limited flow regime.


INTRODUCTION
Radio pulsars have been discovered more than 40 years ago Hewish (1968) and at present many thousand papers are devoted to these objects. Despite of large progress in the understanding of processes in pulsar magnetospheres still many important questions are unclear. One of them is the value of pulsar braking indices. If a pulsar were just a simple magnet with dipolar magnetic momentum m rotating with angular velocity Ω, then the braking index n =ΩΩ/Ω 2 would be equal to 3. The taking into account of evolution of angle χ between vectors m and Ω yields n = 3 + 2cotχ Davis (1970); Beskin (2006a). Observations of some pulsars are well within theoretical predictions. For example, the braking index of pulsar Crab is equal to 2.5 and pulsar Vela has n ≈ 1.4 Beskin (2006a). So it seems that more sophisticated magnetospheric models will be able to remove this discrepancy, see for example Melatos (1997); Timokhin (2005Timokhin ( , 2006; Contopoulos (2006; Timokhin (2007a,b). However, many isolated pulsars have very large positive or, sometimes, negative braking indices upto |n| ∼ 10 4 . Sometimes such large values may be due to unobserved glitches or timing noise Alpar (2006). For example, in Alpar (2006) it is shown that all negative braking indices may be associated with unresolved glitches which has occurred between intervals of observations of the pulsars. But in some cases refined measurements of braking indices show that at least ⋆ E-mail:tsygan@astro.ioffe.ru some pulsars have large and sometimes negative braking indices n ∼ ±(10 − 10 2 ) Johnston (1999). For example, the pulsars B0656+14 and B1915+13 have braking indices n ≈ 14.1 ± 1.4 and n ≈ 36.08 ± 0.48, correspondingly, while pulsar B2000+32 and B1719-37 have very large negative braking indices n ≈ −226 ± 4.5 and n ≈ −183 ± 10, correspondingly Johnston (1999). There are some explanations of such braking index values. For example, the values like n ∼ ±10 may be related to nonstandard mechanisms of pulsar braking, like neutron star slowing down due to neutrino emission Peng (1982) or due to interaction with circumpulsar disk Menou (2001);Malov (2004); Chen (2006), see Malov (2001) for review of possible mechanisms and its comparison with observations. The large braking indices may be explained by the rapid changing of magnetic field. Such changes may be caused by Hall-drift instabilities ; ; Pons (2007). Also the positive braking indices of old pulsars may be explained by the relaxation of angular velocities of neutron star crust and superfluid between two glitches Alpar (2006). The large braking indices of some pulsars may be related to Tkachenko waves Popov (2008). In this paper we present a some model based on works Beskin (2006b) and , see also Biryukov (2007); Urama (2006); Pons (2007), where it has been shown that large braking indices may be explained by the existence of some internal cyclic process in a pulsar.

THE TORQUE
There are many mechanisms, that may be responsible for pulsar braking Beskin (2006a);Malov (2001). Let us shortly describe two of them. The first of them was proposed before pulsar discovery by Pacini (1967). According to this mechanism the rotation energy E and the rotation momentum M of a neutron star are carried away by magnetic dipole radiation, that, of course, leads to a torque acting on the star. Because of the presence of currents and charges in pulsar magnetosphere, calculation of the strength of this torque is a highly complicated and, at present, uncompleted task, see for example Timokhin (2006Timokhin ( , 2007aSpitkovsky (2006); Beskin (2006a). However, the torque can be estimated within a pure vacuum model of pulsar magnetosphere, where magnetospheric currents and charges are absent Deutsch (1955). In this case, the rotation torque K dip acting on a neutron star, can be written as Davis (1970); Melatos (2000): where Ω = Ω eΩ is the angular velocity of the star, m = m em is the dipolar magnetic momentum of the star, eΩ and em are two unit vectors, directed, correspondingly, along Ω and m, χ is the angle between eΩ and em. In the case of a small rotation speed Ωa/c ≪ 1, where a ≈ 10km is the radius of the star, K0 and R dip can be written as Davis (1970) In Davis (1970); Goldreich (1970) the coefficient ξ is taken to be equal to 1. We guess that the value ξ = 3/5 adopted in Melatos (2000) may be closer to reality, because it corresponds to absence of electric current sheet on the neutron star surface. Thus, we will further consider ξ = 3/5. Expression (1) is valid at any values of Ωa/c, although coefficients K0 and R dip slightly distinguish from their nonrelativistic values, defined by (2) and (3), Melatos (2000).
The second mechanism (current losses) is due to the electric current j, that flows along open field lines, see for example Beskin (1984Beskin ( , 2006aBeskin ( , 1993Beskin ( , 2006c. This current flows from light cylinder, crosses pulsar polar cap diode, intersects the star surface, and goes into deep crustal layers. After this it begins to move to surface and then transforms to the backward current flowing along the separatrix between open and closed field lines. It is worth to note that when the electric current travels inside the crust it sometimes flows across the magnetic field, that leads to torque Kcur acting on the star polar cap. The strength of this torque is calculated in Jones (1976) as: where parameter α(Ω, χ, φ) characterizes the electric current value Beskin (2006c,a). In the case of a circular pulsar tube cross section the parameter can be estimated as Beskin where jN is the density of the electric current, that flows in the northern pulsar tube, and jS is the density of electric current that flows inside the southern pulsar tube, j 0 GJ cos χ = ΩB 0 2π cos χ is the Goldreich-Julian current, B0 = 2m/a 3 is the magnetic field strength at the magnetic pole, SN (η) and SS(η) are the areas of the northern and southern pulsar tubes, S0(η) = πa 2 (Ωa/c) η 3 , η = r/a, r is the distance from the center of the star.
Following Jones (1976); Xu (2001); Wu (2003), we assume that the star can slow down by both mechanisms simultaneously and that the resulting torque K can be described just as a sum of partial torques K dip and Kcur: Thus, the equation of rotation momentum loss can be written as

SPHERICAL SYMMETRY CASE
At first case we assume that a neutron star is an absolutely rigid sphere. Particularly, we neglect any star deformations and any viscosity and dissipation inside the star. Consequently, we can suppose that the star rotation momentum M = I Ω, where I is the momentum of inertia of the star. Under this assumption equation (7) can be rewritten as Now let us to introduce three orthogonal unit vectors ex, ey and ez which rotate together with the star. As vector ez we use em, direction of vectors ex and ey may be arbitrary. Consequently, we have and at any time t the following relations are valid: ( ez, ex) = ( ez, ey) = ( ex, ey) = 0 and e 2 z = e 2 x = e 2 y = 1.(10) Hence, at any time t we can treat these vectors as an orthogonal space basis. So it is possible to write, for example Ω = Ω (sin χ cos φ ex + sin χ sin φ ey + cos χ em) .
Using the last expression, the equation (8) may be rewritten in the form where τ = 3 2 Ic 3 m 2 Ω 2 ≈ 1.5 · 10 8 year P 1s where B12 = B0/10 12 G. It is worth to note that equations (12-14) describe neutron star rotation in an inertial reference frame of "rigid stars" . But all vectors are represented by the basis ( ex, ey, ez) altering in time.
It is easy to see that equation (12) describes the losses of neutron star rotation energy and equation (13) describes the evolution of the inclination angle χ. In the case of α = 1 the equation (12) may be rewritten as So in this case parameter τ is equal to two characteristic pulsar ages τ = P/(2Ṗ ). For any other values of parameter α there is no so simple interpretation of the time τ . But it is possible to treat the parameter τ just as some characteristic time over which large changes of the angular velocity Ω and the inclination angle χ occur. The equation (14) describes the precession of the neutron star and Tp may be interpreted as precession period. It is worth to note that the period Tp is 3-4 orders of magnitude smaller than the characteristic time τ . So it seems that the neglecting of the altering of values Ω and χ over times compared with precession period Tp may be a good approximation. Consequently, if we assume α = α(Ω, χ, φ) it will be possible to write In this case, in equation (12) only parameter α is able to change significantly over precession period and hence Thus, the braking index n =ΩΩ/Ω 2 can be estimated as Good (1985) n If we take into account the equation (14), expression (20) may be rewritten as If we assume that α ∼ 1, cos χ ∼ sin χ ∼ 1, then the braking index n may be estimated as If we take into account the equation (3) then the braking index estimation (21) takes the form It is necessary to note that within the current paper we assume that the magnetic dipole moment m does not change significantly over the precession period Tp. Particularly, the expression (23) is valid only if the time corresponding to large changes of the magnetic dipole momentum m is large enough compared with the period Tp.
Now following Goldreich (1970); Tsygan (2009) we define the value of function f (Ω, χ, φ) averaged over precession time as Then, averaged equations (12) and (13) may be written as It is easy to see that case < α >= 1 corresponds to "stationary" or, better to say, equilibrium value of the inclination angle, when < dχ dt >= 0 Goldreich (1970); Gurevich (2007); Istomin (2007). In this case the average speed of angular velocity decrease also does not depend on the inclination angle: The last equation may correspond to the results of Beskin (2006b); Biryukov (2007) where it is shown that an average braking index of many pulsars is close to n ≈ 5. It is worth to note that if the parameter < α > decreases with increasing angle χ, this equilibrium state is stable, see Tsygan (2009) for details.

NONDIPOLAR MAGNETIC FIELD
It is widely accepted that besides large-scale dipolar magnetic field, small scale nondipolar magnetic field may exist near neutron star surface. This field has a spatial scale about ℓ ∼ (0.3 − 3)km and rapidly falls to zero when the distance from the neutron star surface increases to become fully negligible at the light cylinder. This allows to suppose that this component does not exert any direct influence on magnetic dipole braking, that appears to occur near the light cylinder. It seems that the same reasons are applicable to the direct influence of the small scale component on current losses, see for example Tsygan (2009). However, the existence of the small scale component may lead to drastic changes of the electric current that flows through the inner gaps of the pulsar Shibata (1991). Thus, the indirect influence of the nondipolar field on current losses and torque Kcur is supposed to play a decisive role in pulsar braking.
The influence of small scale magnetic field upon electric currents was considered in many papers, c.f. Arons (1979); Gil (2001Gil ( , 2006a. In order to estimate its influence on current losses we will use a simple model, proposed in Palshin (1998), see also Kantor (2003); Tsygan (2009). According to this model the inner gaps fully occupy the pulsar tubes cross sections and are situated close to the neutron star surface. It is assumed that magnetic field strength on the star surface is not large enough to prevent free emission of electrons, so when pulsar diode is placed onto star surface it will be operate in electron charge limited steady flow regime. In the neighbourhood of the inner gap small scale magnetic field will be modeled by the field of a small magnetic dipole m1, which is embedded into neutron star crust under magnetic (dipolar) pole at the depth ∆ · a (see fig. 1 and fig. 2). We assume that ∆ ≈ 1/10 and that m1 is perpendicular to the main dipole momentum m. So the vector m1 may be written in the form where ν = B1/B0 is ratio of small scale magnetic field strength B1 = m1/(∆ · a) 3 to the strength B0 = 2m/a 3 of the large scale dipolar magnetic field at the magnetic (dipolar) pole. We also assume that the changes of values ν, ∆ and γ, as well as, the value of magnetic momentum m, are negligible, at least, over a period Tp of neutron star precession. A similar model has been proposed in Gil (2002a), where a small dipole m1 is placed not strictly under the magnetic pole and may have arbitrary direction. The case of pure axisymmetrical small scale magnetic field has been investigated in Gil (2002a,b); Asseo (2002); Tsygan (2000). It is worth to note that the field of dipole m1 is able to describe small scale magnetic field only in vicinity of the pulsar diode. We does not any intention to suppose that this dipole is able to describe small scale magnetic component over whole neutron star surface. We rather suppose that the small scale field is close to a sum of 10 2 − 10 3 such small dipoles and that we use only one of them, which is the closest to pulsar diode.
In the case of a thin pulsar tube Rt ≪ ∆ · a, where Rt is its radius, and at small altitudes η = r/a ≪ κ c/Ωa ∼ 10 2 the Goldreich-Julian density (inside the pulsar tube) may be written as At ν 1/3 the function f (η) may be approximated as Palshin (1998) where the coefficient κ ≈ 0.15 describes general relativistic frame dragging Muslimov (1992), B -magnetic field strength, λ = ν (∆η) 3 / (η − 1 + ∆) 3 . In the case of k = 0 the function f (η) is equal to the cosine of the angle between the direction of magnetic field B and angular velocity Ω.
In the case of a thin long inner gap, when the radius Rt of pulsar tube is small compared with the gap height zca, see fig. 1, the density j of electric current flowing through the inner gap may be written as Tsygan (2009) where η0 is the altitude where the inner gap begins and j 0 GJ = ΩB 2π . Following Tsygan (2009) we assume that the electric potential Φ monotonically increases inside the pulsar diode and the diode is placed as close to the surface as possible. Hence using expression (30) one may write at cos(φ − γ) < 0 and in the case of cos(φ − γ) > 0 see Tsygan (2009) for details.
Suppose that the small scale magnetic field near the northern inner gap may be described by a small dipole with parameters ν = νN and γ = γN , and the small scale magnetic field in a neighbourhood of southern inner gap (and, of course, inside it) may be described by a small dipole with ν = νS and γ = γS. Then, the parameter α may be written as where SN and SS is cross section areas of northern and southern pulsar tube. Let us firstly consider the most simple case when νN = νS = ν and SN = SS = S0. The values of braking index n for angle χ = 30 • are shown on fig. 3 and 4. They have been calculated by the substitution of expressions (30) and (34) into equation (21). It is easy to see that the values ν ≈ 0.1 mostly correspond to n ∼ 10−10 2 and for ν ∼ 1 the braking index may be as large as ∼ 10 3 .
On fig. 7 and 8 it is shown the braking indices when angle χ is not arbitrary but correspond to equilibrium value χeq, which is defined as < α > (χ = χeq) = 1. It is easy to see that in this case at ν = 0.1 the braking index n ∼ ±(10 − 20) and at ν = 0.7 the braking index may achieve values as large as n ∼ 1.5 · 10 3 .
In the case of weak small scale magnetic field ν ≪ 1 and ν tan χ < 1 the electric current through the inner gap may be approximated as j ≈ j The value of parameter α averaged over precession period Tp may be written as Thus, the equilibrium angle χ is equal to At this angle the expression (36) may be rewritten as and in the case of νN = νS and γN = γS = 0 parameter α may be estimated as This expression shows that at the equilibrium value of angle χ parameter α is changing from ≈ 3 4 to ≈ 3 2 or, in other words, is changing in two times over the precession period. Consequently, the braking index n may be a crudely estimated as n ∼ 2π τ /Tp ∼ 10 3 . Now let us discuss a more sophisticated case when the pulsar tube cross section St depends on angle χ. We will use the dependence that was derived in Biggs (1990): where and µ = 1 + 1 2 cos 2 χ − | cos χ| 2 + 1 4 cos 2 χ. When χ increases from 0 to π 2 the value µ changes from 0 to 1/3. At χ = 0 • the coefficient g(χ) is equal to 1 and at χ = 90 • it is equal to g π 2 = 4 27 1 4 ≈ 0.620, so the area of pulsar tube cross section decreases with increasing angle χ. Some other dependencies of pulsar tube areas on the angle χ is calculated, for example, in Dyks (2004); Muslimov (2009);Beskin (2006a). Such a dependence may be either decreasing or increasing, for example, in Beskin (2006a) it is shown that the area of a pulsar tube increases with χ from 1.59S0 at χ = 0 • upto 1.96S0 at χ = 90 • .
In the case of νN = νS = ν and SN = SS = S0g 2 (χ) the resulting braking indices n for angle χ = 30 • are shown on fig. 5 and 6. Braking indices at equilibrium values of the angle χ are shown on fig. 9 and 10.

AN AXISYMMETRIC CASE
In this section we will suppose that a neutron star is an absolutely rigid axisymmetric body and that its symmetry axis coincides with dipolar magnetic momentum m. The rotation momentum of the star may be written as where Ix = Iy = I ⊥ and Iz = I || = I ⊥ − ∆I are momenta of inertia of the star. Hence, the equation (7) of momentum loss takes the form: −π − 3π 4 − π 2 − π 4 0 π 4 π 2 3π 4 π φ ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.7 Figure 3. The dependence of braking index n on precession phase φ. The changing of phase φ from −π to π corresponds to one precession period and takes Tp seconds. It is assumed that S N = S S = S 0 , ν N = ν S = ν, γ N = γ S = 0, χ = 30 • , P = 1s and the neutron star is spherical.
where K ef f is an effective rotation torque acting on the star Here we neglect the small term ∆I em em ·˙ Ω = (∆I/Iz) em em · K and introduce the coefficient where ∆I33 = ∆I/10 33 g cm 2 .
It is easy to see that the only difference between equations (8) and (44)  I by its component I ⊥ and the replacing of the coefficient R dip by the coefficient R ef f . Consequently, upto these two exchanges all the formulas of previous section remain applicable to the axisymmetrical neutron star. Particularly, the braking index n may be estimated as Again, supposing that α ∼ 1, cos χ ∼ sin χ ∼ 1, one can crudely estimate the braking index as ∼ τ 250year where τ = P/(2Ṗ ) is characteristic pulsar age and I ≈ I ⊥ is moment of inertia of a undeformed (spherical) neutron star, I45 = I/10 45 g cm 2 . Some examples of dependence of braking index n on the precession phase φ are shown on fig. 11-13. This dependence may be found by calculating the braking index n spher for a spherically symmetrical neutron star and then multiplying it by R ef f /R dip : The axisymmetrical deformations of the neutron stars may be caused by internal magnetic field that resides inside neutron star crust or inside its core Goldreich (1970). The deformation caused by internal magnetic field may be estimated as −π − 3π 4 − π 2 − π 4 0 π 4 π 2 3π 4 π φ ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.7 Figure 11. The dependence of braking index n on precession phase φ. The changing of phase φ from −π to π corresponds to one precession period and takes Tp seconds. It is assumed that where Bin is the strength of the internal magnetic field and M is the mass of the star. The coefficient ζ depends on magnetic field profile. In the case of dipolar magnetic field ζ may be estimated as ζ = 25/8 Ferraro (1954), the same value is used in Goldreich (1970). For other configurations it may be, for example, as small as ζ = 1/18 Haskell (2008). If we assume that Bin = B0 = 2m/a 3 then the coefficient R ef f can be estimated as where rg = 2GM/c 2 is gravitational radius of the neutron star. Firstly it is easy to see that in the case of ζ ∼ 1/18 the −π − 3π 4 − π 2 − π 4 0 π 4 π 2 3π 4 π φ ν = 0.1 ν = 0.3 ν = 0.5 ν = 0.7 Figure 13. The same as fig. 11, but ∆I = −1 · 10 −11 I.
braking indices does change significantly. Secondly if ζ 1 then the braking indices will be ∼ 20 · ζ times more than in the spherically-symmetrical case and, particularly, at value ζ = 25/8 used by Goldreich (1970) the braking index could become as large as n ∼ 10 4 − 10 5 .
In some cases the neutron star interiors may be superconductive. If the neutron star matter is a superconductor of the second type, the internal magnetic field is able to form magnetic flux tubes. In this case the coefficient ζ increase substantially ζ ∼ B f l /Bin ∼ 10 3 , where B f l ∼ 10 15 G is the strength of magnetic field inside magnetic flux tubes and Bin is average strength of internal magnetic field Akgun (1987). This leads to B f l /Bin ∼ 10 3 time larger deformation of the star and, consequently, to very short precession period like Pp ∼ 50 cos χ P 1s years .
Hence, the coefficient R ef f increases significantly and leads to the increasing of braking index, that can reach n ∼ 10 7 − 10 8 . As the absolute majority of normal isolated radiopulsars do not have such large braking indices it may be possible to conclude that there is no superconductivity of the second type in neutron star interiors.
The dependence of value  The formula (48) allows to estimate the lower limit of neutron star deformation ∆I/I. The corresponding values are shown on fig. 16. It shows that braking indices of majority of pulsars may be explained by the existence of deformation ∆I/I ∼ 10 −13 − 10 −11 that agrees with estimation of ∆I received by Goldreich (1970).

CONCLUSION
In this paper we present a some explanation of large values of braking indices of pulsars. The proposed model is based on four main assumptions: The strength of this field must be enough large to curve pulsar tube but enough small to allow free emission of electrons from star surface.
(ii) There are inner gaps in pulsar tubes and a electric current flowing across inner gaps depends on angle between small scale magnetic field and angular velocity Ω of the star.
(iii) Braking torque depends on current that flows across inner gaps.
In this paper we also neglect the contribution of outer gaps. If a some part of electric current flows through outer gaps then this part is determined only by outer gap electrodynamics and, consequently, does not depend from small scale magnetic field. It decreases the variation of current losses over the precession period and, consequently, leads to the decreasing of braking index. In the case of young pulsars like Crab and Vela, which have small braking indices, we suppose that current losses is almost fully determined by currents flowing through outer gaps and the contribution of inner gaps current is negligible. In this paper it is also assumed that magnetic dipole braking exists and does not depends on electric current flowing across inner gaps. This assumption is widely used, c.f. Eliseeva (2006); Yue (2007); Gurevich (2007); Istomin (2007), but, as mentioned in Beskin (2006a), it must be treated with caution. It is shown that in the case of force free magnetosphere and absence of electric current flowing along pulsar tube the orthogonal pulsar χ = π/2 does not slow down at all Beskin (1983Beskin ( , 1984; Mestel (1999). It gives the reason to suppose that magnetic dipole braking does not exist or, at least, must be depend on the electric current Beskin (2006a). In such case presented model can provide the only qualitative explanation of the existence of large braking indices. And the quantitative estimations, of course, will strongly depend on relation between magnetic dipole braking and current losses torques.
The presence of a long period precession is the weakest point of the model. At present the precession is discovered only at a few isolated neutron stars. And these pulsars have the precession periods like Tp ∼ 1 − 10 years Link (2007). The presence of pined superfluid in neutron star crust substantially increases precession speed Shaham (1977). The precession periods Tp larger than (10 2 − 10 4 )P may exist only when vortices of superfluid can not been pinned anywhere in star, are able to move freely and do not coexist with magnetic flux tubes Link (2006Link ( , 2007. The small number of isolated pulsars with observed precession force us to exclude the triaxial precession and to assume that neutron star is axisymmetrical and symmetry axis coincide with magnetic dipole moment m. In this case in frame reference related to "rigid stars" vector m rotate with constant angular velocity ω. And because of ω ≈ Ω its trajectory coincides with the case of unprecessing star, see Appendix A. Also, in order to prevent the observation of such precession it is necessary to assume that pulsar tube structure does not precess too. It particularly means that pulsar tube crossection is circular or depends on only vectors Ω and m and does not depend from small scale magnetic field. Also, it means that distributions of energy of primary electrons and pair multiplicity over pulsar tube crossection are close to axisymmetrical.

APPENDIX A: THE PRECESSION OF AXISYMMETRICAL STAR
Consider the neutron star rotation in an extreme case when it is possible to neglect the first and second terms in expression (45) which contain vectors em and eΩ compared with the last term which contains [ eΩ × em]. Particularly, it means that any rotation energy losses are neglected and it is assumed that the angular velocity Ω and inclination angle χ do not change with time. Consequently, this approximation is valid only over time scales that are small compared with the characteristic time τ or pulsar age τ , although this time scales may be comparable with or larger than the precession period Tp. In this case the equation (44) may be written as With this equation it easy to obtain dΩ/dt = 0 and dχ/dt = 0 and, consequently, neither the characteristic time τ nor the coefficient R ef f are changing. Let us introduce the vector eω = cos β eΩ − sin β em and calculate its time derivative In this case the first of equations (9) may be rewritten as It means that em and, consequently, m just rotate with the constant angular velocity ω = eωΩ/ cos β around the constant vector ω, see for example Link (2003). As Ω = ω + em Ω tan(β), Ω would also rotate with the angular velocity ω around the constant direction eω. It is worth to note that the angle β is very small: β = arctan ξ m 2 Ic 2 a + ∆I I cos χ ∼ 10 −12 ξ 4 B 2 12 I45 + ∆I 10 −12 I cos χ, and, consequently, ω is almost undistinguished from the star angular velocity Ω. It seems that such a rotation substantially increases the difficulty of direct observations of pulsar precession. It is easy to see, that in the reference frame Kω, rotating with the angular velocity ω, vectors em and eΩ are constant in time and the star rotates around vector em with the angular velocity (− Ωp) Link (2003) Because of radio radiation may be generated at a some distance from the center of pulsar tube, the precession may be manifested as subpulses drift or variation of pulse profile Asseo (2002); Ruderman (2006). In order to exclude such manifestations it is necessary to assume that average distribution of radio sources in pulsar tube is axisymetrical. Also, it needs to assume that pulsar tube crossection is circular or, at least, its profile depend on vectors Ω and m and does not depend on small scale magnetic field. The model of pulsar tube presented in Biggs (1990) satisfies this criteria. This paper has been typeset from a T E X/ L A T E X file prepared by the author.