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Astrophysics, Vol. 45, No. 4, 2002

0571-7256/02/4503-0470$27.00 ©2002 Plenum Publishing Corporation

INVERSE PROBLEM OF THE THEORY OF RELAXATION OF THE VELA PULSARSANGULAR VELOCITY AFTER GLITCHES

D. M. Sedrakian and M. V. Hairapetian UDC: 524.354.4

A theory of the relaxation of pulsar angular velocity is compared with observational data for the first eightglitches of the Vela pulsar. The inverse problem of the theory of relaxation is considered and solutions of thisproblem in the regions of exponential and linear relaxation are found. General features in the distribution ofneutron vortices in these regions immediately after a glitch are determined. It is shown that these propertiesmay be related to the size of the glitch in pulsar angular velocity.

Key words: (stars:) pulsars: individual: Vela

1. Introduction

Radio pulsars, which are known to be compact neutron stars, are sources of periodic radio pulses. The time of

the periodic arrival of a given detail in a sequence of pulses determines the period, and with it the pulsars angular rotation

rate. A pulsars angular rotation rate undergoes secular variation on the order of -11513 sec1010 −− ÷≈ΩΩ& It is also well

known that irregular behavior is inherent to pulsars: the angular velocity undergoes glitches and microglitches on the

order of 96 1010 −− ÷≈Ω∆Ω . After a glitch, the pulsars angular rotation rate relaxes to its preglitch value with

characteristic times of from several days to several hundred days. As observations show, the relaxation curve for ( )tΩ∆ & ,

the departure of ( )tΩ& from the preglitch value, has a clearly defined structure, consisting of brief exponential dependences

and one linear dependence [1, 2].

The irregular behavior of pulsar angular velocity with further relaxation indicates the presence of a weakly

coupled, superfluid component within the neutron star. Neutrons in the Aen and npe phases, as well as protons in

the npe phase (core) of the neutron star, change to the superfluid state at a temperature of 109 K. With the stars rotation,

an array of quantized vortex filaments develops in the neutron superfluid. In [3 5] the transient irregularities in pulsar

angular velocity are associated with the dynamics of the motion of neutron vortices in the npe phase of the neutron

star. The effect of entrainment of superconducting protons by the neutron superfluid results in the formation of a neutron

proton vortex cluster with a magnetic field on the order of 1014 G. The interaction between the superfluid component

and the normal component relativistic electrons is accomplished by scattering of those electrons on the magnetic

field of the cluster. As a result, after a glitch there is a continuous transfer of angular momentum from the superfluid to

Translated from Astrofizika, Vol. 45, No. 4, pp. 575-586, October-December, 2002. Original article submitted June5, 2002.

Yerevan State University, Armenia, e-mail: dsedrak@physdep.r.am.

471

the normal component. And this results in the observed relaxation behavior of the neutron stars crust, strongly coupled

with the normal component in the core. In the works cited above, equations were derived for the nonsteady dynamics

of the rotation of a two-component neutron star, on the basis of which a theory of glitches and of relaxation of the angular

rotation rate of the Vela pulsar was developed. The relaxation theory was subsequently generalized within the framework

of the general theory of relativity (GTR) in the Ω approximation [6 8]. Such an analysis made it possible to find

characteristic relaxation times and solve the equations of rotational dynamics of a neutron star with allowance for GTR

corrections.

In the period from 1968 to 2001 the Vela pulsar displayed great activity and underwent 15 small and large glitches.

The theory of relaxation of the Vela pulsars angular velocity was compared with observational data for six glitches [1]

in [5, 9]. This made it possible to find the relative moments of inertia of the regions responsible for the relaxation in

an approximation quadratic in angular velocity. The locations of these regions in the core of the model neutron star were

also found. As a result, it can be concluded that the theory of the relaxation of pulsar angular velocity based on the

dynamics of the motion of the superfluid system in the npe phase of the neutron star is in good agreement with

observations for standard models of neutron stars.

The relaxation theory was compared with observations for the first eight glitches in the Vela pulsars angular

velocity [2] in [10]. The comparison method proposed here differs fundamentally from the method used earlier in [5, 9].

Thus, in both cases theory was compared with observations within the framework of a multilayered model of the relaxation

regions. It was assumed, for example, that the relaxation of pulsar angular velocity with a characteristic time τi represents

the response of that layer in which the average relaxation time is τi. The presence of active and passive regions within

each layer, depending on the pinning conditions and the initial distribution of neutron vortices, was also assumed in [5,

9]. Relaxation occurred in active regions after a glitch, with the average relaxation time of an active region having to

correspond to the observed relaxation time of the pulsars angular velocity. The condition es Ω=Ω && must be satisfied

in passive regions, so that these regions could not take part in the relaxation. For this a distribution of vortices in the

passive regions must be created over the time between two successive pulsar glitches that would correspond to the same

rate of slowing of the angular velocities of the superfluid and normal components. Such a separation of the regions in

the stellar core is very arbitrary, and it would be correct to allow for the influence of the entire layer on the relaxation

process. For this purpose, a new method was proposed in [10] for comparing the theory of relaxation of pulsar angular

velocity with observations. In that paper the inverse problem was solved: from observational data for ( )teΩ& the

distribution of vortices was found and the role of the effects of pinning and depinning of neutron vortices within each

layer was clarified.

The purpose of the present paper is a complete and refined analysis of observational data on the Vela pulsar after

eight glitches and an examination of the solution of the inverse problem in the relaxation theory. Such an analysis enables

us to clarify the general features in the vortex distribution in the stellar core immediately after a glitch. These features

may be related to the observed characteristics of the relaxation of the pulsars angular velocity.

The dynamics of the motion of the two-component superfluid is analyzed in a Newtonian approximation, although

the relaxation time and the moments of inertia of the relaxation regions are calculated in an approximation quadratic in

angular velocity within the framework of the GTR. The stars rotation is axisymmetric, while the behavior of the superfluid

is analyzed in a hydrodynamic approximation.

2. Equations of Motion

The equations of motion of a two-component neutron star with allowance for pinning and depinning of neutron

vortices have the form [11]

472

, ∫ −=Ω+Ω

extsse

e KdIdt

d

dt

dI (1)

[ ] ( ), 0 esps knn

tΩ−Ω−ν−=

∂Ω∂

(2)

( ) , 02 nrr

r s ν=Ω∂∂

(3)

, d

p

p

pp nnn

t

n

τ−

τ−

=∂

∂(4)

where Ωe, I

e, Ω

s, and I

s are the angular velocities and moments of inertia of the normal and superfluid components,

respectively, Kext

is the external braking moment of the forces acting on the star, n(r) and np(r) are the densities of vortices

and pinned vortices at the point r, τp and τ

d are the characteristic times of pinning and depinning of neutron vortices,

ν0 = h/2m

n, where m

n is the mass of a neutron, and k is defined in accordance with [3].

As was shown in [4, 5], the stars relaxation time τ is a rapidly increasing function of mass density in the superfluid

region. In accordance with this, two essentially different regions in the stellar core can be distinguished: we call them

the active and passive regions. In the active region the relaxation time τ is less than or on the order of the observed

relaxation time (τ ≤ 1000 days for the Vela pulsar). We assume that ∆Ω varies only in this region, which is responsible

for a glitch and the post-glitch relaxation of pulsar angular velocity, i.e., in this region we have ∂∆Ω/∂τ ≠ 0. The passive

region, in which the relaxation time is longer than the characteristic times of the observed relaxation, can be divided into

two subregions. In the first, in which the condition τ ≤ τ0 is satisfied, where τ

0 is the pulsars lifetime (τ

0≈ 104 yr for

the Vela pulsar), during the pulsars lifetime a vortex distribution should be created such that the angular velocities Ωs

and Ωe of the superfluid and normal components have the same rate of slowing under the action of the external braking

torque. Consequently, in this subregion we have es Ω=Ω && and ∂∆Ω/∂τ = 0. In the second subregion, in which τ ≥ τ0,

the vortex distribution does not change during the pulsars lifetime, i.e., Ωs = const, and hence ∂∆Ω/∂τ = ∂Ω

e/∂τ in this

subregion. With allowance for the foregoing, Eq. (1) can be reduced to the form [8]

, 1 2

00

0 γ−=∂∆Ω∂

λ++

Ω∫µ

dytp

p

dt

d e(5)

where ∆Ω = Ωs Ω

e, p

0 = I

s/I

e is the relative moment of inertia of the superfluid region, µp

0 is the relative moment of

inertia of the active region, and λp0 is the relative moment of inertia of the first passive subregion. In deriving (5) we

took µ << λ, since the observed relaxation times after a glitch are much shorter than the pulsars lifetime. The quantity

γ2 in (5) is defined as

( )0

012

1

1

p

p

λ++γ

=γ

and

( ) . 1 0

1pI

K

e

ext

+=γ

From (2) and (3) we can obtain one more equation connecting Ωe and ∆Ω, which has the form

( )[ ], 1

12

pe r

rrktdt

d∆Ω−∆Ω

∂∂∆Ω−

∂∆Ω∂−=

Ω(6)

473

where DWp = W

p W

e, while W

p is determined from the density of pinned vortices as follows [11]:

( ) ( ) . , ,0

20 rdrtrn

rtr

r

pp ′′′ν=Ω ∫ (7)

Transforming Eqs. (5) and (6), we obtain an equation for ∆Ω,

( )[ ] , 01

111

0

02

2

0

0 =

γ−∆Ω−∆Ω∂∂∆Ω+

∂∆Ω∂

λ+

µ−∫

µ

dprrrktp

pp

p (8)

which must be satisfied for any k. The latter is expressed in terms of microscopic parameters, such as the electron and

neutron densities and the coefficient of friction between the normal and superfluid components [3, 4]. It is clear that the

above quantities depend on the model of the neutron star. The integrand in (8) is therefore identically equal to zero.

If we also take µ << λ, then from (8) we obtain

( ) ( )[ ] . 002

12

2 =γ−∆Ω−∆Ω∂∂

Ωτ∆Ω+

∂∆Ω∂

pe

rrrt (9)

where 1/τ = 2kΩe(0), τ is the stars relaxation time, and Ω

e(0) is the pulsars angular velocity immediately after a glitch.

Later on, to determine ∆Ω from Eq. (9) we must solve it together with (4). For relatively small glitches, i.e., for

n ≈ n0, where n

0 = 2Ω

e(0)/ν

0, the solution of Eq. (9) has the form [11]

( ) ( ) ( )( ), 100

20tx

ttxtx etdee −′− −∆Ω−′γ=∆Ω−∆Ω ∫ (10)

where

( ) ( )( )ατα+−−α+

ττ

α+τα+

= tp

ettx 12

2

111

1(11)

and α = τd/τ

p, while ∆Ω

0 is the initial value of ∆Ω immediately after the glitch. A two-zone model of the active region

was adopted in [11] to explain both a glitch and the post-glitch relaxation of pulsar angular velocity. It was assumed

that in the zone of preparation for a glitch, acts of pinning must predominate over acts of depinning of neutron vortices,

i.e., the condition α >> 1 must be satisfied. But α << 1 was taken in the relaxation zone, i.e., the vortices are almost

free. Such behavior of a is not obligatory to explain the relaxation of pulsar angular velocity after a glitch. It can be

assumed that the condition α >> 1 is satisfied in the entire region, i.e., τd >> τ

p. If we also take τ

d << τ

g, where τ

g is the

time between two successive glitches, then for x(t) we obtain

( ) . 1

τα= t

tx (12)

Then for ∆Ω we obtain the expression

[ ]( ). 1020τ−−∆Ω−ταγ=∆Ω−∆Ω te (13)

Substituting this expression into (5), we obtain an expression for ( )teΩ& after a glitch:

( ) [ ]( ) . 11 2

002

0

0 γ−−∆Ω−ταγλ+

−=Ω ∫µ

τ− dyep

pt t

e&

(14)

Let us proceed to find the initial condition ∆Ω0. Over the time t

g between glitches the quantity ∆Ω will have the value

474

determined from (13) for t = tg >> τ:

( ) , , 12ταγ=∆Ω gtr (15)

where α1(r) is the value of α(r) before the glitch in angular velocity. The change in ∆Ω during a glitch is

( ) ( ) ( ) ( ) ( ) , 00 , , 0 esgeesgsg ttrtr ∆Ω+∆Ω−=

Ω−Ω+

Ω−Ω=∆Ω−∆Ω (16)

where ∆Ωs and ∆Ω

e are the sizes of the glitches in the superfluid and normal components, respectively. Then for the initial

condition ∆Ω0 we obtain

. 120 es ∆Ω−∆Ω+ταγ=∆Ω (17)

If we introduce the designations ∆α = α1 α

2 and

, 2 α∆τγ=Ω′∆ (18)

then for the departure ( )teΩ∆ & from its steady-state value we obtain

( ) ( ) . 1

00

0 dye

p

pt

t

see ∫µ τ−

τ∆Ω−Ω′∆−∆Ω

λ+−=Ω∆ &

(19)

If we allow for only the pinning of vortices, then ∆Ω′ in (19) is determined from the formula [12]

. 1ln4 2

2

τγ

Ω′∆+=ττp

gt(20)

3. Comparison with Observations

As we said above, the most detailed observational data are available for the Vela pulsar. An analysis of these data

[2] shows that the behavior of eΩ∆ & after a glitch can be described by the interpolation formula

( ) , 3

1

AAteat~

j

t

jej −+−=Ω∆ ∑

=

τ−

& (21)

[in Eq. (17) of [10], B = A was not taken into account]. As shown in [2], for eight glitches the characteristic times of

exponential relaxation are τ1 = 10 h, τ

2 = 3.2 days, and τ

3 = 32.7 days. The values of the coefficients a

1, a

2, a

3, and A

for the eight glitches are given in Table 1. To compare Eq. (19) that we derived with Eq. (21), we assume that each term

in (21) represents the response of four layers of the active region of the neutron stars core: in three of them the average

relaxation time equals the observed time τj, while in the fourth layer the relaxation time is on the order of the times

between glitches.

In determining the relaxation regions we use one of the standard models of neutron stars with a mass M = 1.4 MΘ,

a radius R = 10.13 km, and a total moment of inertia I = 1.156⋅1045 g⋅cm2 [13, 14]. The relaxation time was calculated

in [9] with allowance for GTR corrections. As shown in [10], a comparison of Eqs. (19) and (21) leads to the integral

equations

( ) , 3 ,2 ,1 ,1

==τ

τ−τ−

∫−

ieadre

rfc i

i

i

ti

R

R

t

(22)

475

( ) , 3

AAtdre

rfctR

R

+−=τ

τ−

∫ (23)

for determining the unknown quantity f(r) = ∆Ωc ∆Ω′ ∆Ω

s. Here we have c = 3.1⋅10-6 cm1. For the first relaxation

region, with an average relaxation time τ1 = 10 h, we have R

0 = 9.61 km and R

1 = 9.57 km, for the second region, with

τ2 = 3.2 days, R

2 = 9.533 km, and for the third, with τ

3 = 32.7 days, R

3 = 9.47 km. The region of linear relaxation was

confined to a radius R = 9.36 km, it being assumed that the region with a radius R ≤ 9.36 km does not take part in the

relaxation, since the corresponding relaxation times are unobservable.

To solve Eqs. (22) and (23) we use the method of regularization of the solution of integral equations [15]. In Figs.

1 8 we show the quantity ∆Ωc f = ∆Ω′ + ∆Ω

s as a function of stellar radius r, found for the eight glitches in the Vela

pulsars angular velocity. As seen from the figures, ∆Ω′ + ∆Ωs has fundamentally different behavior in the first two sections

of exponential relaxation, i.e., for 9.533 km ≤ R ≤ 9.61 km and in the region of exponential relaxation with τ = 32.7

days and of linear relaxation, i.e., for 9.36 km ≤ R ≤ 9.533 km.

We first consider the first two parts of the region of exponential relaxation, i.e., for 9.533 km ≤ R ≤ 9.61 km. Here

∆Ω′ + ∆Ωs is on the order of the size of the glitch and is positive. The positive value of ∆Ω′ + ∆Ω

s may be due to a

change ∆Ωs in the angular velocity of the superfluid component, i.e., an increase in the distribution density of vortices

in this region. But this region borders on the region of a glitch with a characteristic time of less than a few minutes

[12]. In the glitch region the pinning of vortices is important, being necessary for the accumulation of a sufficient

number of vortices for its sudden release to result in the observed glitch. A redistribution of neutron vortices in this region

is therefore unlikely, i.e., ∆Ωσ ≈ 0. Then the condition ∆Ω′ > 0 must be satisfied, which is consistent with Eq. (20),

obtained with allowance only for the pinning of neutron vortices. Small irregular changes in ∆Ω′ + ∆Ωs in this region

may be due to variation of ∆Ωs, which may be random.

TABLE 1. Characteristics ai and A of the Exponential and Linear Sections of Relaxation

Following the First Eight Glitches in the Vela Pulsars Angular Velocity

∆Ωc) size of the glitch; t

g) time between glitches.

Parameters 1 2 3 4 5 6 7 8

a1

0.001 0.0002 0.0 0.0004 0.48 0.26 0.89 2.11

(1013 rad⋅sec2)

a2

1.98 6.13 1.64 4.77 3.92 5.8 4.71 6.8

(1013 rad⋅sec2)

a3

2.85 3.03 2.02 7.17 0.74 6.17 2.8 4.52

(1013 rad⋅sec2)

A 49.62 53.34 78.75 54.55 115.89 45.78 75.78 37.45

(1022 rad⋅sec3)

c∆Ω 1.66 1.45 1.4 2.16 0.82 1.45 0.92 1.28

(104 rad⋅sec1)

tg

912 1491 1009 1227 272 1067 1261 907

(days)

476

r, km

0.002

9.4

-0.002

-0.004

-0.006

-0.008

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

r, km9.4 9.45 9.5 9.55 9.6

0.002

-0.002

-0.004

-0.006

-0.008

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 1

Fig. 2

0.002

-0.002

-0.004

-0.005

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 3

0.001

-0.001

-0.003

r, km9.4 9.45 9.5 9.55 9.6

477

r, km

0.002

9.4

-0.002

-0.004

-0.006

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 4

r, km

0.002

-0.001

-0.002

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 5

9.4

0.001

r, km

0.002

-0.004

-0.006

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

Fig.6

9.4

-0.008

-0.002

478

Starting with R ≤ 9.533 km, the quantity ∆Ω′ + ∆Ωs is of alternating sign, and its absolute value may far exceed

the size of the glitch. One can try to explain this behavior of ∆Ω′ + ∆Ωs only by a change ∆Ω

s in the angular rotation

rate of the superfluid component, taking ∆Ω′ = 0. This means that during a glitch there is only a transfer of some number

of neutron vortices from one part of the star to another. Then the change in vortex density must be determined from the

condition of local conservation of the number of neutron vortices. As seen from Figs. 1-8, the negative value of

∆Ω′ + ∆Ωs can exceed the positive value severalfold in absolute value. This means that it is impossible to explain the

behavior of ∆Ω′ + ∆Ωs by a redistribution of vortices alone, especially in the region of linear relaxation, in which its

changes are the most pronounced. It is therefore impossible to explain the calculated behavior of ∆Ω′ + ∆Ωs without

allowance for the depinning phenomenon. We therefore assume the existence of rare acts of depinning for this region,

so that we have α >> 1. If we also assume that a increases after a glitch and relaxation of the pulsars angular velocity,

and we note that the relaxation time t is a rapidly increasing function of density, then ∆Ω′ = γ2τ∆α is a negative quantity,

the absolute value of which increases with increasing density. In such an analysis we can explain the asymmetric behavior

of ∆Ωσ + ∆Ω′ in the region of exponential relaxation with a characteristic relaxation time τ ≈ 32 days and in the region

r, km

0.002

-0.001

-0.003

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 7

9.4

0.001

-0.002

r, km

0.002

-0.002

-0.004

9.45 9.5 9.55 9.6

∆Ω′ + ∆Ωs, rad⋅sec1

Fig. 8

9.4

0.001

-0.003

-0.005

-0.001

Figs. 1-8. Dependence of ∆Ω′ + ∆Ωs on r found from the solution of the inverse

problem for the first eight glitches in the Vela pulsars angular velocity.

479

of linear relaxation. The deep negative minima in the region of linear relaxation, in particular, can be explained by an

increase in τ∆α with increasing mass density.

The large glitches in the Vela pulsars angular velocity are the same in general features. Thus, the relative change

in angular velocity is on the order of ∆Ω/Ω ≈ 106, while the time between successive glitches is about two or three years.

Using our results in solving the inverse problem, however, we can explain some details in the difference between various

glitches and the post-glitch relaxation. As seen from a comparison of Figs. 1-8 with Table 1, there is some correlation

between the size of a glitch in pulsar angular velocity and the behavior of ∆Ωs + ∆Ω′ immediately after a glitch. Thus,

for glitches with a size ∆Ωe

≥ 1.3 rad⋅sec1 (glitches 1, 2, 3, 4, 6, and 8) the minimum of ∆Ωs + ∆Ω′ is larger in absolute

value and lies deeper than for relatively small glitches, i.e., for ∆Ωe

≤ 1.3 rad⋅sec1 (glitches 5 and 7). This means that

during large glitches the neutron vortices are redistributed from the relatively deep part of the region of linear relaxation,

which does not take part in relaxation during small glitches. If there were observational data available enabling us to

analyze all the subsequent glitches of the Vela pulsar (15 large and small glitches have now been detected), it would be

possible to say more about the role of the size of the glitch in the distribution of neutron vortices after the next glitch.

The authors wish to thank the FAR/ANSEF fund for financial support under grant No. PS51-01 and CRDF grant

N12006/NFSAT PH N067-02.

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