The influence of small-scale magnetic field on the evolution of inclination angle and precession damping in the framework of 3-component model of neutron star
K. Y. Kraav M. V. Vorontsov D. P. Barsukov






^1 Peter the Great St. Petersburg Polytechnic University, Saint Petersburg, Russia  
^2 Ioffe Institute, Saint-Petersburg, Russia 

bars.astro@mail.ioffe.ru
 


The evolution of inclination angle and precession damping of radio pulsars is considered.
It is assumed that the neutron star consists of 
3 "freely"  rotating components: the crust and two core components,
one of which contains pinned superfluid vortices.
We suppose that each component rotates as a rigid body.
Also the influence of the small-scale magnetic field on the star's braking
 process is examined.
Within the framework of this model the star simultaneously
can have glitch-like events combined with 
long-period precession (with periods 10-10^4 years).
It is shown that the case of the small quantity of pinned superfluid vortices
seems to be more consistent with observations.




§ INTRODUCTION

Radio pulsars can be considered as exceptionally stable clocks.
But sometimes besides smooth braking their rotation suffers
some irregularities called glitches 
<cit.>. 
Usually it is related to superfluid pinning. 
Pinned vortices do not allow superfluid 
to participate in pulsar braking.
It causes the sudden unpinning of vortices at some moments and 
the transfer of angular momentum from superfluid to the crust
<cit.>.
That is observed as a glitch.

There are some evidences that isolated neutron stars 
may precess with large periods T_p of precession. 
Pulsar B1821-11 precesses with period 
T_p∼ 500 <cit.>.
Some other pulsars show periodic variations
which may be explained by precession with 
T_p∼ 100 - 500
<cit.>. 
The changing of pulse profile of Crab pulsar 
also may be related to precession with T_p∼ 10^2
<cit.>.
The increase of radio luminosity of Geminga pulsar <cit.>
may be related to precession with T_p > 10 as well.
The non-regular pulse variations known as "red noise"<cit.> 
with timescales T ∼ 1  - 10^4
also can be related to precession
<cit.>.

The problem is that the pinning of vortices leads to precession with periods 
T_p∼ P · (I_tot / L_g ) ∼ (10^2 - 10^6)   P
<cit.>,
where P is  pulsar period, I_tot is moment of inertia of the star,
L_g is the angular momentum of pinned superfluid.
Such precession seems to damped very quickly and it is incompatible with
existence of the long-period precession <cit.>.

In this paper we consider a model of rotating neutron star, proposed in <cit.>.
It allows the coexistence of long-term precession and quasi-glitch events.
We also take into account the influence of the small-scale magnetic
field on pulsar braking.







§ BASIC EQUATIONS

We assume that neutron star consists of 3 components which we will call the crust (c-component), g-component and r-component.


The crust (c-component) .
We suppose that it rotates as a rigid body with angular velocity
Ω⃗_c.
It is the outer component so the angular velocity Ω⃗_c 
is the observed pulsar angular velocity Ω⃗,
Ω = 2 π / P.
We suppose that 

    M⃗_c = I_cΩ⃗_cṀ⃗̇_c = K⃗_ext + N⃗_gc + N⃗_rc,

where M⃗_c is angular momentum of the crust, 
I_c is its moment of inertia, 
K⃗_ext is the external torque of magnetospheric origin
acting on the crust,
N⃗_gc and N⃗_rc are the torques 
acting on the crust due to its interaction with g and r
components correspondingly.





g-component. It is the one of two inner components. 
We assume that it consists of normal matter 
rotating as a rigid body with angular velocity Ω⃗_g
and superfluid matter 
firmly pinned to the normal matter
so that superfluid vortices rotate together with the normal matter
with angular velocity Ω⃗_g:

    M⃗_g = I_gΩ⃗_g + L⃗_gṀ⃗̇_g = N⃗_cg + N⃗_rgL̇⃗̇_g = 
     [ Ω⃗_g×L⃗_g],

where M⃗_g is the total angular momentum of g-component,
I_g is the moment of inertia of its normal matter,
L⃗_g is angular momentum of pinned superfluid,
N⃗_cg and N⃗_rg are the torques 
acting on g-component due to its interaction with 
the crust and  r-component correspondingly. 





r-component . It is the second inner component.
We assume that it rotates as a rigid body
with angular velocity Ω⃗_r:

    M⃗_r = I_rΩ⃗_rṀ⃗̇_r = N⃗_cr + N⃗_gr,

where M⃗_r is the angular momentum of r-component,
I_r is its moment of inertia,
N⃗_cr and N⃗_gr are the torques 
acting on the r-component due to its interaction with 
the crust and g-component correspondingly.  



In the crust frame of references
the equations of rotation (<ref>)-(<ref>)
can be rewritten as

    Ω̇⃗̇    =    R⃗_gc + R⃗_rc + S⃗_ext,
    
    μ̇⃗̇_cg    +     [Ω⃗×μ⃗_cg ]
    + [Ω⃗×ω⃗_g]
    + [μ⃗_cg×ω⃗_g ]
    = R⃗_cg + R⃗_rg 
    - R⃗_gc - R⃗_rc - S⃗_ext,
    
    μ̇⃗̇_cr    +     [Ω⃗×μ⃗_cr ]
    = R⃗_cr + R⃗_gr - R⃗_gc - R⃗_rc - S⃗_ext,
    
    ω̇⃗̇_g    =    [ μ⃗_cg×ω⃗_g],

where 
μ⃗_ij = Ω_j - Ω_i, 
N⃗_ij = - N⃗_ji, 
R⃗_ij = N⃗_ij / I_j,
i,j = c,g,r,
S⃗_ext = K⃗_ext / I_c
and ω⃗_g = L⃗_g / I_g.







In the sake of simplicity we suppose that

    N⃗_ij = - I_j (  α_ijμ^||_ije⃗_Ω 
               + β_ijμ⃗^⊥_ij
               + γ_ij [ e⃗_Ω×μ⃗^⊥_ij ]
         ),

where α_ij,β_ij,γ_ij are some constants,
e⃗_Ω = Ω⃗ / Ω 
and we have introduced parallel  component A^|| = A⃗·e⃗_Ω and perpendicular component A⃗^⊥ = A⃗ - A^||e⃗_Ω for any vector A⃗.   

First let us consider the equilibrium state for zeroth
external torque K⃗_ext=0.
In this case,
the whole star rotates as a rigid body (μ⃗_ij = 0)
and, hence, R⃗_ij = 0.
Equations (<ref>)-(<ref>) may be written as

    Ω̇⃗̇ = 0
    ω̇⃗̇_g = 0 
    ω⃗_g = ω_ge⃗_Ω.


Let us further consider a small perturbation to the equilibrium state.
We will treat values μ⃗_ij, ω⃗_g^⊥
and S⃗_ext
as small perturbations and neglect any term quadratic in these values. 

Hence, equations (<ref>)-(<ref>) may be written as

    Ω̇    =     R_gc^|| + R_rc^|| + S_ext^||,
    
    μ̇_cg^||    =    
      R_cg^|| + R_rg^||
    - R_gc^|| - R_rc^|| - S_ext^||,
    
    μ̇_cr^||    =    
    R_cr^|| + R_gr^|| - R_gc^|| - R_rc^|| - S_ext^||,
    
    ω̇_g^||    =     0,
    
    Ωė⃗̇_Ω    =    R⃗_gc^⊥ + R⃗_rc^⊥ + S⃗_ext^⊥,
    
    μ̇⃗̇_cg^⊥    -     
      (ω_g^|| - Ω)   
      [ e⃗_Ω×μ⃗_cg^⊥ ]
    + [Ω⃗×ω⃗_g^⊥]
    = R⃗_cg^⊥ + R⃗_rg^⊥ 
    - R⃗_gc^⊥ - R⃗_rc^⊥ - S⃗_ext^⊥,
    
    μ̇⃗̇_cr^⊥    +     
    [Ω⃗×μ⃗_cr^⊥ ]
    = R⃗_cr^⊥ + R⃗_gr^⊥ 
    - R⃗_rc^⊥ - R⃗_gc^⊥ - S⃗_ext^⊥,
    
    ω̇⃗̇_g^⊥    =     
    - ω_g^||/Ω (
         R⃗_gc^⊥ + R⃗_rc^⊥ + S⃗_ext^⊥ 
       + [ Ω⃗×μ⃗_cg ]
      ).
 


In order to calculate the magnetospheric torque K⃗_ext
acting on the crust we use the model proposed in
<cit.>.
It is assumed that neutron star is braking simultaneously
by both magnitodipolar and current losses. Hence,

    K⃗_ext = - Ĩ_tot/τ_0 ( e⃗_Ω - (1-α) cosχ e⃗_m
           - R_eff [e⃗_Ω×e⃗_m] 
      ),

where m⃗ = m e⃗_m is the dipolar magnetic moment
of neutron star,
χ is the inclination angle 
(the angle between e⃗_Ω and e⃗_m, 
see fig.<ref>),
τ_0 = 3/2c^3/m^2Ω^3Ĩ_tot,
Ĩ_tot = I_c + I_g + I_r,
the coefficient 
R_eff is related to the magnetic field inertia
<cit.>.
In the paper we assume that
R_eff = 9/10c/Ω r_ns∼ 5 · 10^3( P/1s) 
<cit.>,
where r_ns is neutron star radius.

The coefficient α is related to the 
value of the current flowing through the pulsar tubes.
In this paper we assume that 
there are only "inner gaps" 
with free electron emission from neutron star surface
in pulsar tube.
Hence, the magnitude of the current depends on the structure of surface
small-scale magnetic field (see fig. <ref>).


The value of α averaged over precession angle ϕ_Ω (see fig <ref>) 

    <α>(χ) = 1/2π∫_0^2πα(χ,ϕ_Ω) dϕ_Ω


is shown in fig <ref>.
Here, ν = B_sc / B_dip, 
B_sc is the induction of small-scale magnetic field,
B_dip = 2 m / r_ns^3 is the induction of
dipolar field on the magnetic pole of neutron star. 








§ QUASISTATIC APPROXIMATION

Firstly let us take into account that
τ_rel≪τ_0,
where 
τ_rel∼max( 1 /α_ij, 1/β_ij, 1/γ_ij)
∼ (1 - 10^7) s is relaxation time
<cit.>.
Hence, we can consider the rotation of neutron star
under the small slowly varying torque K⃗_ext.
In this case, it is possible to neglect terms
μ̇⃗̇_ij and ω̇⃗̇_g^⊥ 
in (<ref>)-(<ref>). 
Using (<ref>)-(<ref>)
we can immediately obtain that

    Ω̇ =  K_ext^||/Ĩ_tot

so the star brakes as a rigid body with 
moment of inertia equal to Ĩ_tot.

Equations (<ref>) and (<ref>) give us

    μ_cg^||    =     -  S_ext^|| I_c/Ĩ_tot·α_cr + α_rg + α_gr/α_cgα_cr + α_rgα_cr 
             + α_cgα_gr,
    
    μ_cr^||    =     -  S_ext^|| I_c/Ĩ_tot·α_cg + α_rg + α_gr/α_cgα_cr + α_rgα_cr 
             + α_cgα_gr.

Let us  introduce a complex number A^⊥ associated to arbitrary perpendicular vector A⃗^⊥ in the following way:
if e⃗_z = e⃗_Ω then
A⃗^⊥ = A_xe⃗_x + A_ye⃗_y
and A^⊥ = A_x + i A_y.
Hence, equations (<ref>)-(<ref>)
give us

    μ_cg^⊥    =    S_ext^⊥/Δ_⊥·( iΩ + ξ_cr + ξ_gr),
    
    μ_cr^⊥    =    S_ext^⊥/Δ_⊥·( iΩ + ξ_gr),
    
    ω_g^⊥    =    S_ext^⊥/Ω Δ_⊥·( ω_g^|| ( iΩ + ξ_cr + ξ_gr)
           - Ω ξ_cg
           + ξ_cgξ_cr + ξ_cgξ_gr + ξ_rgξ_cr),

where ξ_pq = β_pq + i γ_pq and  

    Δ_⊥ = Ω^2
     - i Ω  ( ξ_cr + ξ_rc + ξ_gr + ξ_gc )
     - ( ξ_gcξ_cr + ξ_gcξ_gr + ξ_rcξ_gr).

In the case of weak viscosity
|ξ_pq| ≪Ω we have 

    μ⃗_cg^⊥≈1/Ω 
           [ e⃗_Ω×S⃗_ext^⊥ ]
    μ⃗_cr^⊥≈1/Ω 
           [ e⃗_Ω×S⃗_ext^⊥ ] 
    ω⃗_g^⊥≈ω_g^||/Ω^2 
           [ e⃗_Ω×S⃗_ext^⊥ ].

In the last equation we also assume that |ξ_pq| ≪ω_g^||.





Taking into account equations (<ref>) - (<ref>)
and (<ref>) one can rewrite
equations (<ref>) and (<ref>)
 as

    Ω̇    =      - Ω/τ_0( sin^2χ + αcos^2χ),
    
    χ̇    =    
     - Ĩ_tot/I_c 1/τ_0 sinχ cosχ·( B   (1-α) - Γ  R_eff),
    
    ϕ̇_Ω    =    
     - Ĩ_tot/I_c 1/τ_0 cosχ·( B   R_eff + Γ  (1-α) ),
 
where coefficients B and Γ are defined as

    B + iΓ = Ω/Δ_⊥·(Ω - i(ξ_cr + ξ_gr) ).

In the case of weak viscosity
|ξ_pq| ≪Ω we have

    B ≈ 1 - γ_gc + γ_rc/ΩΓ≈β_gc + β_rc/Ω.

And consequently equation (<ref>) gives that
as long as inclination angle χ differs 
from 0^∘ and 90^∘
the star will precess
with period

    T_p = τ_0/ R_eff· I_c/Ĩ_tot· 2π/cosχ∼ (10^-5 - 10^-3)  τ_0∼ 10 - 10^3.

Also it is worth to note that as long as
quasistatic approximation is valid
the pinned superfluid L⃗_g does not influence
on star rotation.






 
We suppose that the period of precession
T_p≪( I_c /Ĩ_tot) ·τ_0≪τ_0.
Hence,
we can average equations (<ref>) and (<ref>)
over precession
obtaining equation

    dχ/d P = 
     - 1/P·Ĩ_tot/I_c·sinχcosχ· B   (1-< α >) - Γ  R_eff/sin^2χ + < α > cos^2χ.



















§ QUASI-GLITCH EVENTS

According to equation (<ref>) the direction of L⃗_g
follows  the vector e⃗_Ω
so the pinned superfluid vortices are directed almost
along the crust rotation axis.
Consequently, the direction of pinned vortices precesses together with e⃗_Ω.
However, due to ideal pinning (see the last equation in (<ref>)) 
the magnitude of vector L⃗_g does not change at all.
Hence, due to pulsar braking the difference between
velocities of superfluid and normal matter will grow.
So the glitch must occur in g-component at some moment.

We suppose that the glitch occurs and relaxes at time-scale 
much smaller than the period of precession T_p.
It allows us to consider e⃗_Ω as a constant vector 
in equations (<ref>), (<ref>), (<ref>)-(<ref>) 
and, for simplicity, neglect the acting of external torque S⃗_ext.


Let us assume that before glitch the neutron star rotates as a rigid body.
At some moment (t=0) a small amount of angular momentum ΔL⃗_g = Δ L_ge⃗_Ω is is transfered from superfluid part to normal part of g-component.
Then the solution of equations
(<ref>)-(<ref>)
gives us (see fig. <ref>)

    Ω(t) = ΔΩ( 1 - e^-p_+t - Q (1-e^-p_-t) ),

where 
ΔΩ = ΔΩ_∞/1-Q,
ΔΩ_∞ = Δ L_g/Ĩ_tot,
the coefficients p_+ and p_- (p_+ > p_-)
are the roots of equation

    p^2    -    ( α_cg + α_rg + α_cr + α_gr 
         + α_gc + α_rc)   p 
    +
    
        +    ( α_gc + α_rg + α_cg) ·( α_cr + α_gr + α_rc)
    + ( α_rc - α_rg) ·( α_gr - α_gc)
    = 0

and 

    Q = Ĩ_totα_cg - I_c p_+/Ĩ_totα_cg - I_c p_-

In the case of 
α_cg≫( 1 + I_c/I_r) α_rc,
 ( 1 + I_g/I_r) α_rg
we have

    p_+≈( 1 + I_g/I_c) α_cg
    p_-≈Ĩ_tot/I_c + I_g ( α_cr + α_gr)
    
    1 - Q ≈I_c+I_g/Ĩ_tot


If one want to relate these quasi-glitch event to
observed glitches then
1 / p_+ and 1 / p_- should be interpreted as the glitch growth and relaxation times respectively.
We obtain that
1 / p_+≤ 1 <cit.> and
1 / p_-∼ 1 - 10^2 <cit.>.
Unfortunately, 1-Q ∼ 10^-2-10^-1 in our model.
That may be not so bad for glitches in some pulsars
like Crab (Q ≥ 0.8 <cit.>)
or J0205+6446 (Q ≈ 0.77 <cit.>), 
but obviously contradicts glitches in most pulsars for which Q ≪ 1 <cit.>.
In particular, our model does not describe glitches in Vela pulsar 
Q ≤ 0.2 <cit.>.
 







§ RESULTS

In present paper we  will use the following component interaction model.
The crust and g-component interact with r-component
like normal matter with superfluid and
there are normal viscosity between the crust and g-component <cit.> :

    β_cr = β_gr = Ωσ/1+σ^2α_cr = α_gr = 2 β_gcγ_cr = γ_gr = - σβ_gcβ_cg = α_cgγ_cg = 0.

The solution of equation (<ref>) for
different initial inclination angles
χ and initial period P = 10
in the case of 
σ = 10^-10, α_cg = 10^-1 s^-1,
I_c / Ĩ_tot = 10^-2,
I_g / Ĩ_tot = 10^-3
is shown in fig. <ref> and <ref>.

We can see that in most cases 
the star forgets initial value of 
inclination angle χ very rapidly
and evolves to equilibrium inclination angle χ_eq,
at which

    sinχcosχ·( B   (1-< α >) - Γ  R_eff) ≈ 0.

The subsequent evolution of angle χ is caused by 
the slow changing of equilibrium angle χ_eq due to
pulsar braking and consequently growing of R_eff.

The solution (<ref>) for
initial inclination angle 
χ = 45^∘, initial period P = 10
and
different values of ν,
in case of α_cg = 10^-1 s^-1,
I_c / Ĩ_tot = 10^-2,
I_g / Ĩ_tot = 10^-3
is shown in fig. <ref>.
In this case 1-Q ≈ 10^-2.


The same but for
I_c / Ĩ_tot = 10^-1,
I_g / Ĩ_tot = 10^-5 
is shown in fig <ref>-<ref>.
In this case Q ≈ 0.9. 
The increase of I_c leads to slower evolution to
equilibrium angle χ_eq.
The equilibrium inclination angle evolves slowly due to both
the decrease of I_g and, hence, lesser dissipation, 
and the increase of I_c and, hence,
large precession period T_p.
And, consequently, evolutionary tracks may pass
through the most pulsars.


The same but for
I_g / Ĩ_tot = 10^-3,
I_r / Ĩ_tot = 10^-3 
is shown in fig. <ref> -  <ref>. 
In this case, I_c≈ I_tot and Q ≈ 10^-3.
The star rotates almost as a rigid body.
Consequently, the inclination angle changes very slowly and
the pulsars usually do not reach the equilibrium angle
during their life.  


  
  
  
  
 
  
  
 
 



§ DISCUSSION

We consider a model proposed in <cit.>
that allows simultaneously
the long-period precession and quasi-glitch events
with taking into account the influence of the small-scale
magnetic field on pulsar braking.
For simplicity we consider only the case of axial symmetric precession
and do not take into account that the presence of the small-scale magnetic
field makes precession triaxial
<cit.>. 
The main problem of proposed model is the exact nature of components,
especially, of g-component.
Strictly speaking, we only postulate
the existence of components with some properties.

Let us speculate a little about 
possible nature of these components.


The g-component may consist of tangles of closed fluxoids,
normal matter inside the tangles and pinned superfluid 
(see fig. <ref>).
These tangles "freely"  flow inside the star core.
The collapse of closed fluxoids is prevented
by repulsion of pinned vortices.
Similar configurations are considered in 
<cit.>, where vortices are pinned to fluxoids forming
the regular toroidal magnetic field inside the neutron star core,
and  <cit.>, where the "freely"  flowing of magnetic field tangles is discussed.
We expect that L_g∼ 0.1   I_gΩ.
Hence, if we suppose that L_g∼ 10^-2 I_totΩ <cit.>
then I_g∼ 10^-3 I_tot.
In this case, c-component is exactly the cruct, so 
I_c∼ 10^-2 I_tot <cit.>
and r-component consists of normal and superfluid matter
located outside the tangles, so I_r≈ I_tot.

We suppose that strong interaction between the crust and the tangles
may be related to small number of fluxoids got out the tangles.
It is worth to note that a tangle located deep inside the core
weakly interacts with the crust and may produce
something like "slow glitch"  <cit.>.

The main problem of such configuration is the stability of tangles
and its partial destruction during glitches.

The g-component also may be created by small rigid core which can exist
in central region of the star <cit.> (see fig. <ref>) with superfluid vortices pinned to it.
However, in this case, we must assume that the vortices are extremely rigid,
so a vortex pinned by its central part to the rigid core is fixed outside the rigid core as well.
Consequently, if we assume that the radius of rigid core
r_g∼ (0.1-0.2)   r_ns then
the moment of inertia of "normal"  matter inside the rigid core
I_g∼ 0.1 ( r_g / r_ns )^3∼ (10^-6-10^-5)   I_tot
However,  it controls the vortices movement and, hence, the superfluid flow 
in volume ∼ r_g^2· r_ns, see fig. <ref>.
Consequently, we can estimate the angular momentum of pinned superfluid:
L_g∼( r_g / r_ns)^2 I_totΩ∼ 10^-2 I_totΩ.
In this case, c-component consists of the crust
and normal matter of the core:
I_c∼ 10^-1 I_tot <cit.>,
and r-component is superfluid which is not pinned 
to rigid core I_r≈ I_tot.
We suppose that the strong interaction 
between g-component and the crust
requires that the magnetic field penetrates from crust 
up to boundaries of rigid core.
Consequently, core superconductivity must be absent 
(at least outside the rigid core).


In the paper we assume that the interaction between
c-component and g-component is strong
in order to provide rapid angular momentum transfer
from g-component to the crust.
Hence, if I_g∼ I_tot, precession is damped
very rapidly and the inclination angle quickly evolves
to χ≈ 0^∘ or χ≈ 90^∘.
So in order to save precession during pulsar life time
it is necessary to suppose that I_g≪ I_tot.
We can reduce this restriction if we suppose
that the friction between the crust and g-component
increases during glitch.
For example, we can assume that the crust and g-component
interact due to weak viscous friction between two glitches
but during the the glitch the angular momentum is rapidly transfered 
by Kelvin waves <cit.>,
Alfven or sound waves. 
We can also reduce the restriction if we assume
that g-component is slowly destroyed.
For example, the tangles of fluxoids slightly
destroyed during each glitch. 


Authors thank O.A. Goglichidze and D.M.Sedrakian for
usefull discussion which has led to writting this paper,
A.N.Biryukov for usefull discussion of pulsar precession
and V.S.Beskin, I.F.Malov, E.B.Nikitina, A.N.Kazantsev, V.A. Urpin
and A.I.Chugunov for help and  useful discussions.


  
  
 



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