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Astrophysics, Vol. 57, No. 1, March , 2014
0571-7256/14/5701-0090 ©2014 Springer Science+Business Media New York
Original article submitted December 4, 2013. Translated from Astrofizika, Vol. 57, No. 1, pp. 103-116(February 2014).
RELAXATION OF THE ANGULAR VELOCITY OF PULSARS AFTER GLITCHES
D. M. Sedrakian, M. V. Hayrapetyan, and D. Baghdasaryan
The rotational dynamics of superfluid neutron stars is examined in order to study the relaxation of the
angular velocity of pulsars after glitches. The motion of the neutron-proton vortex system is investigated
taking the sphericity of the superfluid core and vortex pinning and depinning into account. A relaxation
solution is obtained for the angular rotation velocity of pulsars after glitches. In order to compare this
solution with observational data for the Vela pulsar, the inverse problem of finding the initial distribution
of vortices immediately after a glitch is solved.
Keywords: pulsars: angular velocity: relaxation
1. Introduction
Pulsars are a manifestation of neutron stars which is observed as electromagnetic emission from a source in
periodic radio frequency pulses. The pulsed character of this emission has made it possible to determine the rotation
periods of pulsars, since it is assumed that the emission source rotates in synchrony with the pulsar [1]. The rotation
periods P of pulsars range from a few milliseconds to on the order of 1 s [2,3]. Observations show that the rotation
period of pulsars increase constantly because of rotational energy losses. The so-called secular variation in the period
of pulsars is on the order of 1218 1010 −− −~PP� s-1.
Some pulsars have a unique activity in which the angular rotation velocity Ω and its derivative Ω� increase
Erevan State University, Armenia; e-mail: [email protected] [email protected]
91
suddenly, after which they mostly relax to their value before this jump (glitch) [4,5]. Glitches in angular velocity have
been observed in about 100 pulsars [3]. Based on an analysis of glitch activity and the relative magnitudes ΩΔΩ
of the glitches, it is possible to distinguish two groups of pulsars. The first group includes the pulsars Vela PSR
B0833-45 and Crab PSR J0531-21 and others like them for which the magnitude of the glitches is on the order of
69 1010 −− −ΩΔΩ ~ and the time between glitches is on the order of 2-3 years, while there is no correlation between
these quantities. The second group of pulsars includes the pulsars PSR J0537-6910 and PSR B1338-62 [3], which
have periodic glitches, while the magnitude ΩΔΩ of the glitches correlates with the time tg between glitches; that
is, for large tg the glitches are large.
The generally accepted model for pulsars is a magnetized neutron star with mass on the order of �
M~M and
radius on the order of R ~ 10 km. As it rotates, the magnetized neutron star loses kinetic energy of rotation as a result
of magnetic dipole radiation; the rather wide range of values of PP� can be explained in terms of magnetic fields on
the order of 139 1010 −~B G on the stars’ surfaces. Studies of the properties of matter in the interiors of neutron stars
at densities on the order of 1410~ρ g/cm3 indicate the presence of superfluid neutrons and the existence of
superconductivity of the protons in pulsars. The relaxation behavior of the angular velocity of the rotation following
glitches also suggests the existence of a weakly coupled superfluid component. Observations of pulsars show that the
time dependences of the angular rotation velocity ( )t٠and its derivative ( )t� have complicated structures and are
described either by a sum of exponential functions [6] or by a polynomial dependence [7]. The characteristic
relaxation times for the angular velocity after glitches range over a wide interval from on the order of one hour to
several hundred days. The relaxation behavior of the angular velocity of pulsars after glitches can be explained in
terms of a superfluid model for neutron stars. The slowing-down of the rotation of the normal component that takes
place under the influence of an external torque is also accompanied by a continuous transfer of internal angular
momentum from the superfluid component to the normal component. In this case the main theoretical question for
explaining the relaxation of the angular velocity of pulsars is to clarify the mechanism by which the superfluid and
normal components of the neutron star are coupled. It has been shown [8,9] that a system of neutron-proton vortex
clusters parallel to the axis of rotation of a neutron star develops because of “entrainment” of superconducting protons
by the superfluid neutrons in the core of the star. A magnetic field of order 1014 G is generated in the clusters and
the motion of the magnetized neutron vortices during the time the star is slowed down is accompanied by friction
owing to scattering of electrons on the magnetic field of the cluster. This interaction between the two components
of a neutron star makes it possible to introduce a dynamic relaxation time which indicates the characteristic time to
approach steady-state rotation following a sudden change in the angular rotation velocity of one of the components.
Features of the motion of the vortex system as the star is slowed down, such as the pinning and depinning of neutron
vortices, have been discussed in Refs. 10 and 11. With these phenomena taken into account, relaxation solutions were
obtained for the angular rotation velocity and its derivative in neutron stars following glitches. These solutions were
compared with observational data on ( )t٠and ( )t� , so it was possible to obtain information on the initial distribution
of the vortices immediately after a glitch and to clarify the role of pinning and depinning in establishing this distribution.
In these studies of the dynamics of the motion of the vortex system in the core of a neutron star, it has been assumed
92
that the density of matter and the coefficient of friction along a rectilinear vortex remain constant, i.e., these studies
have assumed cylindrical symmetry. It is clear that these quantities depend on the spherical radius of the star and vary
along a neutron vortex.
This paper is a study of the motion of neutron-proton vortex clusters in the superfluid spherical core of a
neutron star. New information is obtained about the influence of the distribution of the vortices before and after
glitches on the parameters of the observed relaxation of the angular velocity of pulsars.
2. Basic equations for the rotational dynamics of a two-component neutron star
A lattice of vortex clusters parallel to the rotational axis of the star develops in the superfluid core of a rotating
neutron star. Around each vortex the circulation of the superfluid component is quantized. This fact is described by
the equation
( ) ( ), , ,rot 0 trntrs����
ϑ=� (1)
where
[ ] , , rss���
Ω=� (2)
s�
is the angular rotation velocity of the superfluid liquid, ( )trn ,�
is the vortex density, 0ϑ�
is a vector directed along
a vortex with a modulus equal to the quantum of circulation nm�π=ϑ0 , and nm is the neutron mass. Because of
the axial symmetry of the rotation in a cylindrical coordinate system, from Eqs. (1) and (2) we obtain a first equation
for the rotational dynamics of the superfluid component:
( ). 0
2
nrr
rs ϑ=∂Ω∂
(3)
As the rotation of the neutron star slows down, the vortex density should decrease, i.e., the vortices move
toward the boundary between the core and crust of the neutron star. The motion of the vortices obeys the continuity
equation
( ) , 0div =+∂∂
Lnt
n�
�(4)
where L��
is the local vortex velocity. In cylindrical coordinates Eq. (4) takes the form
93
( ), 1rnr
rrt
n�
∂∂−=
∂∂
(5)
Here it has also been noted that the velocity L��
of the vortices depends only on the cylindrical radius r. Equation
(5) is the second equation for the rotational dynamics of the superfluid component of the neutron star.
As they move the vortices are acted on by forces from both the superfluid component and the normal component.
Since superfluid liquid flows around a vortex, it is acted on by a Magnus force proportional to the difference in the
velocities of the vortex and the superfluid liquid. In addition, a frictional force owing to the interaction of the normal
core of the vortex with the normal component and proportional to the difference in the velocities of the vortex and
the normal component acts on the moving vortex. Finally, the equation of motion of the vortex is given by [8]
( )[ ] ( )( ) ( )[ ] , 0 , , 00 =ϑ−β−−η−ϑ−ρ����� ������
eLeLsLs rrr ������ (6)
where sρ is the density of the superfluid liquid, e��
is the velocity of the normal component, and η and β are,
respectively, the coefficients of transverse and longitudinal friction. In writing Eq. (6) it was assumed previously
[8,10,11] that sρ , η , and β depend on the cylindrical radius r. In the meantime, it is clear that in the spherical core
of a slowly rotating neutron star these quantities depend on the spherical radius R. Thus, along a rectilinear vortex,
different points of which lie at different distances from the star’s center, sρ , η , and β are not constant. In this case
Eq. (6) can be averaged by integrating it with respect to the coordinate z (i.e., along the length of the vortex):
( )[ ] ( )( ) ( )[ ] . 0 , ,2
2-0
2
2-
2
2-0 =ϑ−β−−ηϑ−ρ ∫∫∫ dzrdzrdzR
l
leL
l
leL
l
lsLs
����� ������������ (7)
Since the velocities of the vortex L��
and of the superfluid and normal components s��
and e��
are independent of z,
Eq. (7) can be written in the form
( )[ ] ( )( ) ( )[ ] , 0 , , 00 =ϑ−β−−η−ϑ−ρ����� ������
eLeLsLs rrr ������ (8)
where sρ , η , and β are the average values of sρ , η , and β and now depend only on the cylindrical radius r.
If the vortex velocity is written in the form [10]
[ ] , , ϕϕ++Ω= eer rreL ���
���(9)
then the solutions of Eq. (8) for the unknown components of the vortex velocity r� and ϕ� relative to the normal
component are given by
94
( ),
1
,
2 r
esr
k
k
rk
��
�
+=
Ω−Ω=
ϕ(10)
where
( ).
12
0
0
⎟⎟⎠
⎞⎜⎜⎝
⎛η
β−ρϑ+
ηβ−ρϑ=
s
sk
Note that in the core of the neutron star the condition ( )00 1 ϑρβ−ϑρ>>η ss holds, and we also have k << 1. We take
these conditions into account in the following.
We note here that Eq. (8) holds for vortices moving relative to the normal and superfluid components of the
plasma. Some fraction of the vortices may become attached to pinning centers and rotate with the normal component.
The local velocity for the pinned vortices is zero, i.e., 0=L��
. If we denote the density of pinned vortices by np,
Eq. (5) takes the form
( )( ), 1rp rnn
rrt
n�−
∂∂−=
∂∂
(11)
and from Eqs. (3) and (11) we can obtain the equation
( ) ( ) . , 02
rps rnnrt
�−ϑ−=Ω∂∂
(12)
for the time behavior of the rotational velocity of the superfluid component of the plasma.
3. Equation for the rotation of neutron stars
It is generally known that a neutron star is slowed down by radiation from the entire electromagnetic spectrum.
According to observations and modern theories, the torque that slows down a star has a power law dependence on the
angular rotation velocity next ~K Ω . A large change in the angular velocity of a star can take place over a time
comparable to the lifetime ΩΩτ �20 ~ , which is on the order of 104 years for the Vela pulsar. Meanwhile, the
characteristic relaxation times for the angular velocity of the Vela pulsar after glitches can be as long as a few hundred
days, so that we can assume that the torque that slows down a star is constant during the time between glitches, i.e.,
Kext
= const.
We now obtain an equation for the normal and superfluid components of a neutron star. The rotation equation
95
for the normal component is
, extinte
e KKdt
dI +=Ω
(13)
where Kint
is the internal torque produced by the superfluid component and Ie is the moment of inertia of the normal
component. The internal torque from friction between the normal and superfluid components is given by
[ ] ( ) , , , dVtrnrFK frint ∫= ���
(14)
where ( )eLfrF ��
���−η= is the frictional force per unit length of a neutron vortex. Using Eqs. (9) and (10) for the
vortex velocity and its components and using the fact that pinned vortices rotate with the normal component of the
star, we obtain the following equation for the internal torque [10]:
∫ ∂Ω∂
−=sI
ss
int dIt
K0
, (15)
where Is is the moment of inertia of the superfluid component. With Eq. (15), Eq. (13) for the rotation of the normal
component takes the final form
∫ =∂Ω∂
+Ω sI
extsse
e KdItdt
dI
0. (16)
The equation for the rotation of the superfluid component of the neutron star can be derived from Eq. (12) by
integrating it over the superfluid region:
( ) . 00
00
=−ϑ+∂Ω∂ ∫∫
ss I
sr
p
I
ss dI
rnndI
t
�
(17)
We shall use Eqs. (16) and (17) to study the relaxation of the angular velocity of pulsars after glitches when the time
dependence of the density np of pinned vortices is known.
4. Pinning and depinning of neutron vortices
As indicated above, the motion of the vortices during slowing-down of a star is accompanied by pinning and
depinning of the vortices. This phenomenon leads to the time dependence for the density of the pinned vortices which
96
shows up in Eq. (17). Characteristic pinning times pτ , during which a moving neutron vortex can become attached
to a pinning center, and depinning times dτ , during which a pinned vortex can be freed, have been introduced [10,11].
In accordance with this, the equation for the time dependence of the density of pinned vortices takes the form
( ) ( ) ( )( )
( )( ) .
, , , ,
r
trn
r
trntrn
t
trn
d
p
p
pp
τ−
τ−
=∂
∂(18)
Since the relative changes in the angular velocity of pulsars during glitches are small, i.e., 96 1010 −− ÷ΩΔΩ ~ ,
we may assume that during relaxation of the angular velocity the density of vortices also remains nearly constant, i.e.,
0nn ≈ . In this case Eq. (18) can be linearized and the time dependence of the density of pinned vortices takes the
form
( ), 10pt
pp ennτ′−−= (19)
where
. , 00 nnp
pp
dp
dpp τ
τ′=
τ+τττ
=τ′ (20)
Here we note that part of the neutron vortex is in the crust of the neutron star, which contains atomic nuclei. Neutron
vortices can become pinned to these nuclei [10], so that when the segment of a vortex contained in the star’s crust
is longer, pinning becomes more probable. Thus, the characteristic pinning time pτ decreases as the star’s radius r
increases. Assuming, as well, that depinning events are random, we can treat the characteristic depinning time dτ as
independent of the star’s radius r. In the steady-state case, i.e., for ∞→t , the density of pinned vortices approaches
the value 0pn given by Eq. (20). For strong vortex pinning, i.e., when dp τ<<τ , Eq. (20) implies that pp τ≈τ′ and
00 nnp ≈ , i.e., in the steady state all the vortices will be pinned to pinning centers. With weak pinning the condition
dp τ>>τ holds; then dp τ≈τ′ and 000 →ττ≈ nn
p
dp , i.e., in the steady state all the vortices will be free.
5. Relaxation equation for the angular velocity of pulsars after glitches
During the secular slowing down of pulsars the neutron vortices move toward the boundary of the star’s core.
The movement of the vortices is accompanied by friction so that the rate of slowing down of the normal and superfluid
components will be the same in the steady state. However, with differential rotation of a two-component neutron star
an instability will develop that is eliminated by a sudden change in the angular rotation velocity of the normal
97
component. After a glitch the angular velocity of the normal component slows down under the influence of the internal
and external torques. We now find the time dependence ( )teΩ for relatively small glitches in the angular velocity
of the pulsar ( 69 1010 −− −ΩΔΩ ~ for the Vela pulsar). We denote the difference in the angular velocities of the
superfluid and normal components by es Ω−Ω=ΔΩ . Then Eq. (16) transforms to
∫ γ−=∂ΔΩ∂
++
Ω 1
01
0
0 , 1
dytp
p
dt
d e(21)
where ns IIp =0 is the ratio of the total moments of inertia of the superfluid and normal components of the star,
( ) ns IrIppy == 0 , ( )rIs is the moment of inertia of the superfluid component in a sphere of radius r, and
( )01 1 pKext +−=γ . Now we transform the equation for the superfluid component of the neutron star (17). Substituting
es Ω+ΔΩ=Ω and the expression (10) for the vortex velocity r� in Eq. (17) and noting that eΩ is independent of
the radius r, we obtain
( ) . 1
00 dt
ddynnk
te
pΩ
−=⎟⎠⎞⎜
⎝⎛ ΔΩ−ϑ+∂ΔΩ∂∫ (22)
We introduce the notation
( ) , 2
, 1
20
000 ϑ
Ω=
τ′=Ω e
e nr
k (23)
where n0 is the vortex density in the star’s core, which we assume to be constant because the glitches in the angular
velocity of pulsars are small. With Eqs. (23), Eq. (22) takes the form
( ).
1
0 0
0
dt
ddy
n
nn
tep Ω
−=⎟⎟⎠
⎞⎜⎜⎝
⎛ −⋅
τ′ΔΩ+
∂ΔΩ∂∫ (24)
Equations (21) and (24) yield the following equation for ΔΩ :
( ), 0
0
0 =γ−−
⋅τΔΩ+
∂ΔΩ∂
n
nn
tp
(25)
where
( ). 1; 1 01
0
pp
+γ=γ+τ′=τ (26)
98
Here τ is the dynamic relaxation time of the star. On substituting Eq. (19) for the density of the pinned vortices in
Eq. (25), we obtain the final form of the equation for ΔΩ [11]:
( ). 0
1
0
00 =γ−−−
τΔΩ+
∂ΔΩ∂
τ′−
n
enn
t
ptp
(27)
Therefore, Eqs. (21) and (27) fully describe the relaxation behavior of the angular velocity eΩ of pulsars after glitches.
6. Solution of the relaxation equations and the inverse problem
Before finding the relaxation solution for ( )teΩ from Eqs. (21) and (27), we note that Eq. (27) can be
integrated if we use the notation
pd ττ=α
and reduce Eq. (27) to the form
. 01
1
1
=γ−⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛+
α+αα
τΔΩ+
∂ΔΩ∂ τα
+α−p
t
et (28)
The solution of Eq. (28) has the form
( ) ( ) ( ) ( )( ), 100
0tx
ttxtx etdeet −′− −ΔΩ−′γ=ΔΩ−ΔΩ ∫ (29)
where
( )( )
( ),
11
1 1
2
2ptp e
ttx
ατα+−
τα+
τα+
τα+= (30)
and 0ΔΩ is the initial value of ΔΩ at the time of the glitch. Here, as in Ref. 11, we assume that the neutron vortices
are almost free, i.e., 1<<α . With this assumption the expression for x(t) takes the form
( ) . 1
1
τα+= t
tx (31)
99
Substituting Eq. (31) in Eq. (29), we obtain
( ) ( )[ ]( ), 11 00τ−−ΔΩ−α+γτ=ΔΩ−ΔΩ tet (32)
where we have set 11 ≈α+ in the exponent. Before finding an expression for t∂ΔΩ∂ from Eq. (32) and substituting
it in Eq. (21), we shall comment on the values of t∂ΔΩ∂ in the core of a neutron star. The value of t∂ΔΩ∂ depends
on the values of the dynamic relaxation time of the star, τ , which can be calculated on specifying a model for the
neutron star. Using the same neutron star model as in Refs. 8, 9, and 11, we have calculated τ and plotted the result
as a function of r in Fig. 1. This figure shows that τ increases rapidly from values on the order of a few seconds
at the boundary between the star’s core and crust to on the order of 108 years for a radius of 561 .R ≈ km, where the
proton superconductivity vanishes. We refer to the region of the star where the average dynamic relaxation time
reaches values on the order of the observed relaxation times (1000 days for the Vela pulsar) as the active region. The
active region lies in the range of radii from Ract
= 9.25 km to R0 = 9.59 km where the relaxation time ranges from an
hour to 1000 days. It may be assumed that the active region contributes to the observed relaxation of the angular
velocity of the normal component. This means that Eq. (32) for ΔΩ applies in the active region. We refer to the
rest of the star in the region actRrR ≤≤1 where τ exceeds the observed relaxation times as the passive region. We
can also see from Fig. 1 that in the region of the star with radii actRrR ≤≤2 , where R2 = 7 km, the dynamic relaxation
time τ is shorter than the pulsar lifetime 0τ . In this region, over the lifetime, a steady-state distribution of the vortices
such that the rates of slowing down of the angular velocities of the superfluid and normal components are the same
should develop; that is es Ω=Ω �� and 0=∂ΔΩ∂ t in this region. In the rest of the passive region, i.e., for 21 RrR ≤≤ ,
the condition 0τ≥τ holds. Thus, in this region the angular velocity of the superfluid component does not vary during
the lifetime of the pulsar and remains equal to the value it had when the star underwent a transition into a superfluid
state; that is, 0=Ωs� and dtdt eΩ−=∂ΔΩ∂ in this region. Given the above remarks, on substituting Eq. (32) in
Fig. 1. The dynamic relaxation time τ of a neutronstar as a function of the star’s radius r.
r, km
log(
τ),
days
�
��
��
�
�
�
�
������ �� � ��� �
100
Eq. (21) we obtain the final form of the relaxation solution for the deviation from the steady-state value of the
derivative of the pulsar’s angular velocity:
( )( ) , 11 0
00
0 ∫μ τ−
τΔΩ−+αγτ
λ+−=ΩΔ dy
e
p
p t
e� (33)
where 0pμ is the relative moment of inertia of the active and 0pλ is the relative moment of inertia of the region with
radius 21 RrR ≤≤ .
We now proceed to determine the initial condition 0ΔΩ in Eq. (33). Assuming that immediately prior to a
glitch of the pulsar, i.e., over the time tg between two successive glitches, ΔΩ has reached its steady-state value, then
for τ>>gt Eq. (32) yields
( ), 1 1α+γτ=ΔΩ∞ (34)
where 1α is the value of a before the glitch. Then the initial condition 0ΔΩ immediately after the glitch will differ
from ∞ΔΩ , by the amount acquired over the short time of the glitch, i.e.,
( ) , 1 10 eses ΔΩ−ΔΩ+α+γτ=ΔΩ−ΔΩ+ΔΩ=ΔΩ ∞ (35)
where sΔΩ and eΔΩ are the changes in the angular velocities of the superfluid and normal components of the star
over the time of the glitch. If we denote the value of α after the glitch by 2α , then on substituting the initial condition
(35) in Eq. (33) we obtain the following expression for the relaxation solution:
( ) , 1 00
0 ∫μ τ−
τΔΩ−Ω′Δ−ΔΩ
λ+−=ΩΔ dy
e
p
p t
see� (36)
where
( ). 21 α−αγτ=αΔγτ=Ω′Δ (37)
The solution (36) can be compared with the observed value of eΩΔ � for those pulsars for which there are
sufficiently detailed data on glitches and relaxation of the angular velocity. One such pulsar is the Vela pulsar PSR
B0833-45, which has undergone 16 large glitches in angular velocity [3]. An analysis of the observational data for
the first eight glitches of the Vela pulsar shows that these data can be interpolated most precisely using the following
time dependence [12]:
( ) ∑ =
τ− −⋅+⋅α−=ΩΔ3
1,
j g
tje A
t
tAet j�
(38)
101
where 101 =τ h, 232 .=τ d, and 7323 .=τ d. The other parameters for the interpolation are given in Table 1. For
a known dependence ( )teΩΔ � and a glitch magnitude eΔΩ and with a chosen neutron star model, it is possible to find
the unknown value of sΔΩ+Ω′Δ from the integral equation (36). This is an incorrectly stated problem and its
solution requires the use of special methods for inverse problems [13]. Here, as in Ref. 11, Tikhonov’s regularization
technique [13] is used. In Ref. 11, however, for comparing the solution (36) with Eq. (38) the active region of the
star was divided into four parts, within each of which the average relaxation time was equated to a value jτ . This
partition was arbitrary and was made in order to simplify the calculations. Here we have not subdivided the active
region of the star, and the inverse problem was solved for the entire integral equation (36). The calculations were done
using the MATHEMATICA program package.
7. Discussion of results
Figure 2 shows plots of sΔΩ+Ω′Δ as a function of the star’s radius r for the first eight observed glitches in
the angular velocity of the Vela pulsar. The graphs all have a common oscillatory variation in this quantity that
depends on the radius of the star. However, sΔΩ+Ω′Δ behaves differently near the boundary of the core and crust
of the star, i.e., in the region r > 9.53 km, and far from this boundary, i.e., in the region r < 9.53 km. The difference
is both in the amplitude of the oscillations and in the sign of sΔΩ+Ω′Δ . We begin by examining the region r < 9.53
km where the function sΔΩ+Ω′Δ changes sign. Equation (37), however, shows that Ω′Δ is proportional to the
dynamic relaxation time τ of the star, which increases monotonically with decreasing radius of the star. The function
( )rα also cannot change sign, so that for r < 9.53 km the oscillatory variation is in the quantity sΔΩ . This means
TABLE 1. Relaxation Parameters for the Angular Velocity of the Vela Pulsar after
Eight Glitches
Time of glitch 1969 1971 1975 1978 1981 1982 1985 1988
a1 (10-13
s-1) 0.001 0.0002 0.0 0.0004 0.48 0.26 0.89 2.11
a2 (10-13
s-1) 1.91 5.94 1.57 4.88 3.76 5.89 4.64 6.90
a3 (10-13
s-1) 2.9 3.2 2.18 6.99 0.91 6.05 2.91 4.31
A (10-13 s-1) 49.62 53.34 78.75 54.55 115.89 45.78 75.78 37.45
tg (days) 912 1491 1009 1227 272 1067 1261 907
ΩΔΩ/ (10-6) 2.35 2.05 1.99 3.06 1.14 2.05 1.30 1.81
102
that during a glitch some amount of neutron vortices shift from one part of the star into a neighboring part [11]. These
kinds of changes in the vortex density should be symmetric with respect to increases and decreases, while the graphs
in Fig. 2 are asymmetric with respect to the axis r. Note, however, that Ω′Δ makes an additional contribution to the
sum sΔΩ+Ω′Δ which should be a negative monotonically increasing function with decreasing radius r of the star.
This, in turn, means that α is smaller prior to a glitch in the pulsar’s angular velocity than after the glitch.
Fig. 2. Dependence of sΔΩ+Ω′Δ on the star’s radius for eight glitches in the
angular velocity of the Vela pulsar.
r, km
������
���� ���� ����
sΔΩ+Ω′Δ 0 rad/s
������
�����
����
�����
r, km
�������
���� ���� ����
sΔΩ+Ω′Δ 0 rad/s
�������
�������
������
������
r, km
������
���� ���� ����
sΔΩ+Ω′Δ 0 rad/s
������
�����
����
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103
In the region r > 9.53 km at the boundary of the core and crust of the neutron star, sΔΩ+Ω′Δ is positive and
undergoes small oscillations about its mean. These oscillations can be explained by small variations in the density of
the neutron vortices. It can be assumed that the small variations in sΔΩ compared to the change in this quantity in
the region r < 9.53 km are caused by strong pinning of the neutron vortices, since as the vortices approach the core-
crust boundary the length of a vortex in the crust of the neutron star increases. But in this case a positive value of
sΔΩ+Ω′Δ cannot be explained by the dependence of Ω′Δ on the star’s radius r according to Eq. (37) since the above
assumption implies that Ω′Δ is a monotonic negative function of r.
We have assumed above that the vortices move with weak pinning and have obtained a relaxation solution (36)
under the condition that 1<<α . This condition may not be satisfied in the region r > 9.53 km because of an increase
in the probability of pinning and a reduction in the characteristic pinning time pτ . However, in a study where only
pinning of vortices was considered, it has been shown [10] that the general form of the relaxation solution (36) does
not change in this case. Only then Ω′Δ is given by the expression
. 1ln4
2
⎟⎟⎠
⎞⎜⎜⎝
⎛γτΩ′Δ+=
ττp
gt(39)
This expression implies that Ω′Δ is positive and almost constant because of the small variation in the relaxation time
τ in the region r > 9.53 km.
In sum, a comparison of the relaxation theory with observational data on glitches of the Vela pulsar has made
it possible to clarify the role of the initial conditions for the distribution of the neutron vortices to the relaxation
characteristics. The initial vortex distribution immediately following a glitch is caused by the glitch itself and by the
features of the vortex motion between two successive glitches in the angular velocity.
One of the authors (M.V.H.) thanks the Volkswagen Stiftung for financial support (grant Az:85182).
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