Galaxy [3]. It follows from evolutionary calculations of the structure of a neutron star (see, e.g., [4-8]) that the superstrong magnetic field (B --- 1011-1013 G [3, 4, ...
Astrophysics. Vol. 40. No. 1, 1997
NATURAL MHD OSCILLATIONS OF A NEUTRON STAR
S. I. Bastrukov, l) I. V. Molodtsova, l) V. V. Papoyan, 2) and D. V. Podgainyi3)
UDC 524.354.6
Natural, low-frequency, hydromagnetic oscillations of an isolated, nonrotating neutron star, which are localized in the peripheral crust, the structure of which is determined by the electron-nuclear plasma (the Ae phase), are studied. The plasma medium of the outer crust is treated as a homogeneous, infinitely conducting, incompressible contitmmn, the motions of which are determined by the equations of magnetohydrodynamics. In the approximation of a constant magnetic field inside the crust (the magnetic fieM outside the star is assumed to have a dipole structure), the spectrum of normal poloidal and toroidal hydromagnetic oscillations, due to presumed residual fluctuations of flow and their associated fluctuations in magnetic field strength, is calculated. Numerical estimates given for the periods of MHD oscillations fall in the range of periods of radio pulsar emission, indicating a close connection between the residual hydromagnetic oscillations and the electromagnetic activity of neutron stars.
1. Introduction One of the most important physical consequences of the principle of conservation of magnetic flux in the process of evolution and collapse of massive stars [1, 2] is that neutron stars are the strongest accumulators of magnetic energy in the Galaxy [3]. It follows from evolutionary calculations of the structure of a neutron star (see, e.g., [4-8]) that the superstrong magnetic field (B --- 1011-1013 G [3, 4, 8]) of these compact objects must be localized mainly in the star's outer envelope (crust) (in which the average density of ionized matter is p ---- 1011 g/cm3). The structure of the crust, the depth of which is variously estimated to be 0.3-1 km, is determined by the perfectly conducting, electron-nuclear plasma (Ae phase) medium, which admits the presence of a magnetic field in the volume occupied by it. The physical properties of the inner core (the average density of which is p --- 1015 g/cm 3) are determined mainly by the highly degenerate Fermi continuum of neutrons. Hence it follows that the physical state of the crust is consistent with the excitation in it of hydromagnetic, long-wave oscillations or the propagation of Alf'v6n waves. In this connection, it is appropriate to note that not long before the discovery of pulsars, Hoyle, Narlikar, and Wheeler [9] suggested that the magnetic energy of a neutron star, stored during the contraction period following their birth in a supernova explosion, may be liberated by the transformation of the energy of residual hydromagnetic oscillations into the energy of electromagnetic radiation. An attempt was recently mad e [10] to estimate the absolute frequencies of the long-wavelength hydromagnetic oscillations based on a homogeneous model of a neutron star. The main result of the analysis in [10] is that the periods of MHD oscillations turn out to be comparable to the periods of pulsar radio emission. The latter fact, in our opinion, is evidence for the hypothesis of hydromagnetic oscillations as the source of the electromagnetic activity of neutron stars. Since the mass distribution in the interior of a neutron star has a stratified character, however, estimates obtained for a homogeneous model cannot be taken as satisfactory, and in this connection it seems appropriate to recalculate the frequencies of MHD oscillations in a more realistic model with allowance for the layered structure of a neutron star. In the present work we give a variational calculation and numerical estimates of the frequencies of natural MHD oscillations localized in the outer crust of a neutron star, i.e., in the region of the electron-nuclear plasma. In the investigated model, a neutron star consists
I)Joint Institute for Nuclear Research, Dubna, Russia; 2)Erevan State University, Armenia; 3)Saratov State University, Russia. Translated from Astrofizika, Vol. 40, No. 1, pp. 77-86, January-March, 1997. Original article submitted 11 September 1996; sent to press 2 October 1996. 46
0571-7256/97/4001-0046518.00
9
Plenum Publishing Corporation
of a two-component system, in full analogy with the well-known Baym-Pethick-Pines-Ruderman model [11] (also see [4, 5] and the references therein), which explains pulsar glitches by seismological shear oscillations of the outer crust relative to the inert core. Here we largely rely on the results of [12-14], in which it was shown that the frequencies of nonradial gravitational oscillations [12] turn out to be three or four orders of magnitude higher than the frequencies of hydromagnetic oscillations [10]. This gives reason to treat hydromagnetic oscillations independently from gravitational oscillations.*
2. Variational Method of Calculating MIID Oscillations In the investigated model, the plasma medium of the outer crust is treated as a homogeneous, incompressible continuum (of int'mite conductivity), the motions of which obey the equations of magnetohydrodynamics (see, e.g., [17, 18]), dV = -VW+ P dt
B2
BV)B,
tV = p + _ _ , 87t
div B = 0, aBlat = curl[V x B],
(2.1)
(2.2)
where p and V are the density and velocity of the medium, respectively; B is the magnetic field strength (the magnetic permeability is/~ = 1) and W is the magnetohydrostatic pressure (d/dt is a substantial derivative). Following Chandrasekhar [17], we give the linearized equations of magnetohydrodynamics in a form that explicitly contains the solution describing the propagation of an Alfv(m wave in an incompressible fluid,
0 5 ~ =0,
(2.3)
Ox~
e v, Bk
p -0t
08BI Ot
n,
4n axk
BkOSVt=O, Oxk
0,
(2.4)
08Bt=o, Oxk
(2.5)
where 6Vi and ~Bk are small departures of the velocity and magnetic field strength from their equilibrium values. This form of the MHD equations corresponds to the case in which no gravitational (longitudinal) oscillations are excited and hydromagnetic (transverse) oscillations remain the only activity of the neutron star. The eigenfrequeneies of the hydromagnetic modes corresponding to long-wavelength MHD oscillations can be calculated on the basis of the variational energy principle by the following scheme. Scalar multiplication of (2.4) by ~SV/and integration over the volume of the star (we take 6B I r = R = 0 at the stellar surface) lead to the energy balance equation
Lf o5v2d _• 8V,Bk05B, Ot v
2
4rt v
(2.6)
Oxt
*In the early papers [15, 16] the frequencies of natural pulsations of a neutron star were estimated on the basis of a hydrodynamic model and the assumption that gravitational oscillations have a radial character. Recent investigations [12], however, have revealed the fact that the stellar medium of a degenerate compact object should be treated as a spherical mass of an elastic-like incompressible continuum (the motions of which obey the equations of elastodynamics), in which gravitational oscillations most likely have an essentially nonradial character. 47
To find the normal modes, it is convenient to represent the variations in the flow velocity and the magnetic field strength in the form 6 V i = ai(r)(x(t),
(2.7)
8 Bi = b,(r)~t).
Substituting (2.7) into (2.5), we find Oa l
be = Bt o~x t
(2.8)
M~+K~=O,
(2.9)
The substitution of (2.7) into (2.6) yields
where M is the inertia and K is the stiffness of the hydromagnetic oscillations, which have the form
X=
=
Ib, 4n r
v
b,d .
(2.10)
In the calculations below, we assume, just as in [10], that within the star's outer crust the uniform magnetic field B is directed along the z axis:
B.=,B. Bo=-(l-,2)'/2n.B,--0. ~ - - ~
(2.11)
Note that the magnetic flattening effect is negligible for neutron stars [19], so all the calculations can be made for a spherical configuration. It follows from the above equations that to determine the frequency co2 = K / M of natural MIID oscillations, we need to calculate only the velocity field of the excited flow. 2,1. Poloidal MHD Mode, We use the incompressibility equation to determine the velocities of the flow accompanying nonradial poloidal oscillations. In our case, this equation must be supplemented by the additional condition of impenetrability of the core, the radius of which is designated as R c, reflecting its inertness:
=o, At the surface of the star, of radius R, we impose the standard
for h e =0.
(2.12)
boulKlary condition (2.13)
where R(t) = R[1 + OtL(t)PL(#)], L being the multipole order of the spheroidal distortions of the surface. From (2.3) it follows that aL(r) satisfies the equation div a L = 0,
(2.14)
the solution of which is sought in the form of a poloidal vector field, a L = curlcurl ~L, XL = [Alfi + A~ r - L - IIPL(#).
(2.15)
From (2.12) and (2.13) we find the explicit form of the arbitrary constants A~ and A2:
AL
- L(-i-+ 1)'
AL
:
L(-i-+ l) c
The components of the field of instantaneous displacements 48
R2L§
aL
,
AL -
~L+3 R2L+ 1- - ~ R2L+t" -"
(2.16)
in the spherical coordinate system are represented in the form
ar
=A L
-At,
r 2L.1_ R2cL*I rL+2 PL(IX) ,
(L+I)r2t*I+ LR 2L§
ao = L(L+ !)
r t*2
--~
(2.17)
e~(.).
a,=O,
(2.18) (2.19)
1 where Pt~(t~) is a first-order associated Legendre polynomial:
(.)
pi(.) __ d. The inertial parameter Mr. calculated with this field is [20]
M, ~ L(2 L+ l)
l+
x 'L'' l -
(2.20)
where X = RclR and X varies in the range 0 < X < 1. We emphasize that here o is the density of the electron-nuclear plasma, which is localized in the star's peripheral crust. Substituting (2.11) and (2.15) into (2.8), we trod that the components of magnetic field strength fluctuations have the form
br=-TUf[(L-ALB' l)r 2L~IPL_I(ta)+(L+ 2) R~L+IPL~1(~)],
(2.21)
r
ALB[r2Ldp!
no = ~-zzy.3t
t ~ n2L+lnl t ~]
L-~.J - ~c
Q,I~P4|,
b#--O.
(2.22)
(2.23)
For the stiffness of poloidal hydromagnetic oscillations we obtain the expression
2L+I X2t~_t L+2 X2(2L,I) ] t~+3)(2 t - i ) - L+-----S 2 J"
~+(2 L2L-t
(2.24)
In the limit X --- 0, corresponding to a homogeneous model, for the frequency of poloidal hydromagnetic oscillations we obtain [101 2
2L+I 2 L- 1'
,,,, -- n~ L(L- 1 ) ~
(2.25)
where
t')~ - V] B2 R2 - 4noR 2 '
(2.26)
is the fundamental frequency of Alfv6n pulsations. 49
i
w
L-1 2.0 POLOI'DAL I.-2
1.0 L-S L-t0 Q.
0 0.3
TOROIDA LL/~ I.:,=2 "
0,2 0.t
0
G4
0.8 aR (kin)
Fig. 1. Periods of poloidal (top) and toroidal (bottom) multipole oscillations as a function of the thickness of the outer ernst, A R = R -- R c. Average density p = 4.3.1011 g/cm 3, magnetic field strength B = 0.5.1013 (3.
2.2. Toroidal M H D Mode. In a system with a fixed polar z axis, the toroidal velocity field has the form 6 V = curl XLetL(t), XL = [A I r L + a 2 r - L -
I]PL(/~).
(2.27)
This field corresponds to differentially rotational (torsional) oscillations of the peripheral crust relative to the stationary core. The excitation of nonradial torsional oscillations in an infinitely conducting fluid (the Ae plasma) is possible due exclusively to the presence of the magnetic field, which imparts to it the dynamical properties of an elastic continuum. This is why we use solutions for the velocity field that we obtained earlier in an analysis of gravitational-elastic oscillations of a neutron star [12, 13] to describe hydromagnetic oscillations excited in the Ae plasma. The arbitrary constants A~ and A~ are fixed by the boundary conditions, which are analogous to those used above in the study of spheroidal oscillations. In differentially rotational oscillations, the distortions of the stellar surface are given by the equation R(t) = R[I + aL(t)P~(t~)], so at r = R we must set 6 V , ,=R = R(t) = RP~(rt)& L(t ).
50
(2.28)
1014 [
........
I
.......
"l
.......
"t
.......
1
]
[ 1
2
",, ",
3
4
, % \ \ \
,
'!,,\,, ",.~ ' %
\ \
"l
*~
\
,o,, 1', POLOIDAL TOP, O I D A L
.... 9
0.0 01
,
. . . . . .
I
. . . . . . . .
0.01
I
2"-
. . . . . . . .
0.1
3
I
"
4
t
. . . . . .
1
10
P (sr
Fig. 2. Period of the lowest (dipole) poloidal MIlD mode and toroidal MHD mode on the density of the electron-nuclear plasma. Depth of the outer crust h = 0.5 km. Lines correspond to calculations for the following densities of the outer crust: 1) p = 10s g/cm3; 2) p = 109 g/era3; 3) p = 10 l~ g/cm3; 4) p --- 10 It g/era 3.
We assume that the inner boundary remains at rest: = 0,
for Rc = 0.
(2.29)
As a result, we obtain
1
AL = At.,
2
A p2L,l
AL =-,~L~,c
RL , AL - R2L+I_Rc2L+I.
(2.30)
Using the separable representation (2.7) for the velocity field of torsional oscillations (2.28), we f'md that the components of instantaneous torsional displacements have the form
ar = 0 ,
ao =0,
a, = A L r s
D2L§ 1 L+I P~(I~).I
(2.31)
Substituting (2.11) and (2.31) into (2.8), for the magnetic field strength fluctuations we obtain br=0,
(2.32)
bo=0,
b,: ALB (L+ l)rZ-iP~_i(p.)+
p2L+I
1
LL~T-P~.i(Iz)/, r j
(2.33)
A calculation of the coefficients of inertia and stiffness of toroidal MHD oscillations leads to the following analytical expressions:
51
4npL(L+l)
[
M~ = A~ (2 L+ IX2 L+ 3) R2t+3 1 - ( 2 L+ 3)
X2t.+t+ (2L+I)2x2L,3 (2L+3) 2L-I
2L-1
]
X 212L~ll
(2.34) '
and
/cL =
a 2 L2-( L ++II ) [RaL' 2~1
3 "~ (2 L - IX2 L+ 3)
As one would expect, as
+
X2z. , L(L+2) X~2L.o] 2L+3
"
(2.35)
X(= RclR) ~ 0 we arrive at the result for a homogeneous model [10], %2 = f22(L2_ 1) 2L+3 2L-I
(2.36)
where the fundamental (Alfv6n) frequency fla is determined by Eq. (2.26).
3, Numerical Analysis of the Model and Conclusions The main purpose of the present work is to obtain limits (upper and lower) for the frequencies of natural MHD oscillations of a neutron star. Figure 1 demonstrates the dependence of the frequency of poloidal hydromagnetic oscillations on the depth of the outer crust, AR = R -- Rc = R(1 -- X), which is 0.3-0.8 km according to the data of evolutionary calculations. As seen from this figure, the frequency of poloidal MHD oscillations depends little on AR, in contrast to the frequency of toroidal modes. In our calculations we took the average density of the crust to be p = 4.3.10 tl g/cm 3 and the magnetic field strength to be B = 0.5.1013 G.
Figure 2 gives an idea of the dependence of the periods of dipole (the lowest in frequency) poloidal MHD oscillations and toroidal MHD oscillations on the density of the Ae phase in the outer crust, the depth of which is taken to be 0.5 kin. The mass density in the crust was in the range of 108 g/cm 3 < p < 1011 g/cm 3 [8]. The dashed rectangle outlines the region of typical pulsar parameters, given in the analogous Fig. 5 of the review [3]. It is noteworthy that the homogeneous model predicts that the lowest Alfv6n mode has a multipole order L = 2. For X ~ 0, however, the multipole order of the lowest stable hydromagnetic oscillation corresponds to the dipole mode. The latter statement is illustrated by Fig. 1. A similar situation occurs for gravitational oscillations [20], which are represented as polarization oscillations of the outer crust relative to the massive core and do not involve the position of the star's center of mass. The frequency of the lowest gravitational mode, by the most conservative estimates, proves to be on the order of 103104 sec -l , which corresponds to periods P = 0.01-0.1 msec. The two-component model under consideration thus leads to the conclusion that if the residual effect of a supernova explosion in the birth of a neutron star is manifested in MHD oscillations localized in the peripheral crust, then the periods of those oscillations lie in the range of 10 msec-10 sec, i.e., in the range of the periods of pulsar radio emission. Although our analysis does not involve the mechanism of emission by a neutron star, such close agreement between the calculated periods of MHD oscillations and the periods of the observed activity of radio pulsars indicates that the source of a neutron star's radio emission may be not only rotation (the lighthouse model) but also low-frequency hydromagnetic oscillations. The authors are grateful to Prof. F. Weber for a discussion of a number of questions treated in the present paper.
52
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