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The equilibrium configuration of rapid rotating compact stars and some gravitational effects
Chin.Astron.Astrophys. Act.Astrophys.Sin.
10 (1986) 278-286 6 (1986) 208-219
Pergamon Journals.
Printed in Great Britain 0275-1062/86$10.00+.00
THE EQUILIBRIUM CONFIGUFATION OF RAPID ROTATING COWACT AND SOME GRAVITATIONAL
DONG Jun
Nanjing
STARS
EFFECTS
Aeronautical
Engineering
Institute
SUN Jing-li and ZHU Pei-then Nanjiny University Keywords:
Received
Pulsars
- Hydrodynamics
1985 February
- Relativistic
Effects
26
In this paper, the equilibrium configurations of rapid ABSTRACT rotating compact stars and some gravitational effects are studied within the general relativity by use of the Harrison-Wheeler equation of state and by the self-consistent field method. Numerical calculations show that the equilibrium configuration of a rotating star is a spheroid. For large the eccentricity and mass increase spin velocities, say, w > 3.0 x 102sec-l, very rapidly as the angular velocity increases, for the critical angular velocity of the rotating star, the eccentricity is about 0.7, the increase The difference of the gravitational redshifts at in mass is about lo-35%. the surface of the star caused bv rotation, and the difference of the of rotation or in the light bending when the beam moves in the direction . opposite direction are obvious.
1.
INTRODUCTION
Since the discovery of pulsars in 1967 [l], and their identification with rotating neutron stars [Z], and especially since the discovery of the millisecond pulsar PRS 1937 +214, [3], there has been an upsurge in interest in the investigation of the structure and equilibrium configuration of rapidly rotating compact stars by means of the general relativity. High density and rapid rotation are the two outstanding features of this type of objects. Under the present circumstances when our understanding of the equation of state of superdense matter is still rather we shall adopt the equation of incomplete, state given by Harrison and Wheeler [4], for an approximate description of the state of the “cold matter” after the completion of nuclear burning during the late stage of evolution of the star. Apart from the uncertainty in regard to the equation of state, the main difficulty in the setting up of rapid compact rotators is the mathematical complexity arising from Hartle [S] the introduction of rotation. and Hartle and Thorne [61 have used a perturbation method in a model of a slow, neutron star; this model is rigidly rotating, not suitable for millisecond pulsars.
Butterworth and Ipser [7] have used the selfconsistent field method to study the equilibrium configuration of a rigidly rotating, incompressible, homogeneous fluid and obtained the relation between the general relativity Maclaurin ellipsoids and rotation. In this paper, we shall use the selfconsistent field method, and numerically solve the Einstein field equation and the general relativistic hydrostatic equilibrium equations in an attempt to find the equilibrium configuration of rapid, compact rotators.
2.
BASIC EQUATIONS AND BOUNDARYCONDITIONS
The gravitational metric line element with rotational symmetry can be written in the form d.S’- PL-')(dpl+ dz')+ p'E'c-"(dp - odt)'- Pdr' where the metric functions functions only of p and Z, Einstein equation, R,.-
- 8x T..(
X, v, B and w are and satisfy the
f g,.T >
o is the angular velocity of the local tial system relative to an observer at
(2) iner-
Fast
Compact
infinity, and we have taken G = c = 1. For convenience, we adopt the ellipsoidal surfaces are: confocal
rotating
ellipsoids
confocal
rotating
hyperboloids
planes
$=const.,
where K is by:
the p-
For a chosen axes
279
coordinates
r’+ y’ ----K1 1 - 1’
22 BJa
(c,n,$).
The three
coordinate
--lGrl$l
Os$i2*
semi K(I
Rotator
focal
+ {‘)+(l
K, the
(3) distance.
These
-q’)’
surface
2 of
a-b-KJl+&
coordinates
Kh
are
related
to
the
(p,z,e)
coordinates
pl-‘p
a spheroid
is
uniquely
given
by 5 =Ss,
with
long
c-K.&
(4) and short
(5)
and eccentricity eIn these
(1 + s:)-“’
(6)
coordinates,
the
Einstein
equation
can be written
as
16ne”‘-“K’(C’ (1 + 6’) g,
+ [2E + (1 + 6’) $ Z] ---e ’
2
-“K’B’
$
+ (1 - 11’) 2
(1 + [‘)1(1 - q’) (2)’ 1
+ 4rc”‘-“K’([’
+ 1’)
(l+C’)g
+ [(l - 1’) $ g
+ (1 + r’)(l
(P + E)(l + I”) 1 - V’ 1
+ $) BP
1
+ 2P
(7) -
24
2
- q’)‘($y]
(81
+[45.+(1+5J)(~$4~)]~
+ (1 -
q’) $$
-I- [(l -
q’) (;
g
-
4 Z)
-
4q] $
(9)
(10)
where Y, -
w + [:I( 1 -
+ 2‘9(1
+ 6’) f
11’) -
?‘(I + {‘)I
G -f 2
280
DongpSun et
r, -
2[5’(1 -1’)
f,
e(:* + n’) 1 + r1[‘1’(1 + 5’) -
-
1-v p, _
V(1 f
5’)l
S((’
+ ‘1’) -
~(5’
+
1’)
f
(1 + E’)[:‘(l
-
-
25’1( 1 +
1 s
“i’) + ?$(I
+ [‘)I
1 Ll
I$) $ ;t $ g
+ 6’) - F’(1 - $)I $ _g + 1-
+ 5’)(1
I’)]
c’)( 1 - q’) + -;;
$(l
Jfq(l
‘5x1 -
$) - ‘i’(1 + :‘)I
d, - ( I - $)E$( 1 f $9 - :‘( C) -
-
5’ + ‘1’
259( 1 + :I)( 1 -
- (1 - $)I$(1 Cl -
??(I
f’ + ‘t’
+ (1 + f’)[$‘(l b, -
5’(1 + :‘) -
B
5(5’ + 11’) 1 + 51??1 -‘i”) I-et’ B
11,-
al,
-
II’) -
-
5’)l $)I
7f)
1‘ -
+ f( 1 + P>fl
-
n’)(S$
-
P) + 5&l
f
.$a)’ -
5’( 1 - ‘)‘)‘L 2B
5’ + ‘1’
_ &I + 5’)(1
- $)(S,c
-
n’) + I’?(1 5’ + ‘1)
The equation
of
hydrostatic
equilibrium
- q9
- $0
-t 5Y.L :iZB
is
where v = pBe -2v(i2-w) is the linear velocity of the fluid element in the local inertial system, P and E are the pressure and energy density in the star’s interior, and their relationship is determined by the Harrison-Wheeler equation of state 141. In these coordinates, the total mass is
The rest
mass is (14)
and the
total
angular
momentum is
Fast Compact Rotator
281
The boundary of the object is determined by P=E=O. On the boundary, the metric functions and their derivatives are continuu+2J/r3. ous. At infinity, B+l, v-+-M/r, In our calculations,these boundary conditions are imposed at r= (+3)5,. For A, we take local flatness on the axis of symmetry as its boundary condition, [8,9], that is, h=lnB when n=l. -4 -
THE NUMERICAL CALCULATION
3.
-5.
We use
the self-consistentfield method and solve the above equations by iteration. The
.steps
-6
are:
-1
First, for a set of initial values vo, Bo, wo, Ao, we apply the usual method of
numerical integration to the H-W equation of state and the equations (12)-(li) to evaluate P, E, M, MO and J. Assume the K-th and the (K- l)-th iterations differ by the small quantities bv~, ~BK, &K, 6h~. -The equations (7)-(g) are developed into linear partial differential equations in &K, ~BK, hK> in which P, E, v are Taylor-expanded about v&l, BK_~, wK_l, and only first-order terms retained. These equations are then discretized to a set of difference equations over a finite grid in the cn-plane. The latter are then solved by Gauss's method of elimination and the values of &K, ~BK, &K at the grid points found. The equations for h, (10) and (ll), have already been linearized. The iteration is continued until bv is less than a pre-assigned tolerance.
-8 0
I I I I,,,,, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
Fig. 1 Energy density distribution on the equatorial plane for 3 equilibrium configurations: (1) EC= 6(+15)g/cm3, Cl= 1.23(+4)s-’ (2) (3)
EC= l(+15)g/cm3, EC= 2(+13)g/cm3,
a= 5.07(+3)s-1 a= 1.58~~’
4
0.8 (41(3)(2) (I). 0.7I 111
4.
RESULTS OF CALCULATION AND DISCUSSION
We took 10 different values for the central density in the range 2(+13) - 3(+16) g/cm3 and used the static, spherically symmetric solution to initiate the iteration. For each value of the central density, we also took several values of the angular velocity, so as to find the effect of the latter on eccentricity and the equilibrium mass. TABLE 1 gives, for each pair of the central density and the spin velocity, the body’s rest mass, bound total mass, its total total angular momentum, eccentricity, energy, the long and the short axes, and the mean Fig. 1 shows the distribution of the radius. energy density in the equatorial plane n = 0 for three equilibrium configurations. The results of the calculation show that the pressure and density both fall to zero at over the range of showing that, C=Ss, angular velocities used, the equilibrium configuration is indeed a rotating ellipsoid.
a.6 _ I 0.5 -
0.40.30.20.1 0
1
2
3
4
W-N=-‘) Fig. 2 Variation of eccentricity with angular velocity for 4 central densities 1.
The Eccentricity-RotationRelation
Fig. 2 shows the variation of the eccentricity with the rotational velocity for 4 values of the central density, Ed: (1) 6.0(+15), (2) 3.0(+15), (3) 1.0(+15), (4) 3.0(+14) g/cm3. We see that, in all cases, when n< 300 Hz, we
Dong, Sun
282
TABLE
et
al.
The total mass, rest mass, bound enerf?y, angular momentum, long and short ares of the equilibrium eccentricity, radius, configuration for various central densities and angular velocities
1
u
Et
u
c
(set-‘)
(kInI
UL
2.00x10”
1.58
0.283
3.283
2.97~10”
0.681
.22x
.57x10’
.61x10’
2.50~10”
1.58
0.181
0.181
9.69~10”
0.211
.27x10’
.31X10’
.19X10’
1.73
0.191
0.191
1.34X10”
0.714
.59X10’
.30x10*
.41x
1.58
0.209
0.209
1.12X10”
4.49x10+
79.3
79.3
79.2
0.228
0.228
2.76~10”
0.718
86.4
97.5
67.9
0.287
0.287
8.26~10”
7.06x10-’
41.7
41.7
41.7
0.312
0.312
8.68X10.7
0.722
47.2
53.4
37.0
0.411
0.414
0.003
6.04~10”
0.106x10-’
21.2
21.2
21.2
U.19UXlO’
0.412
0.415
0.003
8.09~10”
0.593x10-
21.6
21.6
21.6
U.287~10’
0.414
0.417
0.003
1.27~10”
0.839x10-’
21.8
21.8
21.7
O.R67XlO’
0.420
0.423
0.003
4.23~10”
0.277
22.2
22.5
21.6
0.150x10’
0.423
0.426
0.003
4.92~10”
0.319
22.5
22.9
21.i
0.243~10’
0.503
0.507
0.004
1.61~10”
0.725
23.9
27.1
18.7
I.511
0.558
0.567
0.009
2.92~10”
0.891x10-!
13.9
13.Y
13.9
0.287X10’
0.558
0.567
0.009
7.08~10“
0.448~
Ii.2
14.2
14.2
0.867~10’
0.564
0.574
0.010
2.51~10”
0.128
14.6
14.6
14.5
5.00x10”
0.287X10’ 1.58
1.00x10”
0.867X10’ 3.OOXIO”
1.00x10”
3.00X10”
3.00X10”
6.00X10”
1.58
0.243~10
0.608
0.618
0.010
8.93~10”
0.388
15.3
15.7
14.5
0.403XlO’
0.665
0.677
0.012
1.87~10”
0.559
16.0
17.0
14.1
0.507X10’
0.721
0.735
0.014
3.09x10”
0.712
16.9
19.0
13.3
0.666
0.685
0.019
3.01 x 10”
0.761x10-!
10.1
10.1
10.1
0.287~10’
0.666
0.685
0.019
6.27~10“
O.245xlO-1
10.3
10.3
10.;
0.867~10’
0.668
0.687
0.019
2.13~10”
0.734x10-’
10.6
10.6
10.6
0.243~10
0.686
0.705
0.019
7.16~10”
0.221
11.1
11.2
10.9
0.507x10’
0.764
0.785
0.021
1.71x10”
0.454
11.5
11.9
10.6
0.910x10
0.879
0.904
0.025
3.68~10”
0.720
11.9
13.4
9.30
1.58
0.690
0.712
0.022
2.25~10”
0.698x10-!
8.42
8.42
8.42
0.691
0.713
o-022
1.91X10”
0.661x10-
9.14
9.15
9.13
1.58
0.867~10
1.00x
10”
3.00X10”
IO-’
10’
0.243~10
0.69Y
0.721
0.022
5.77X10”
0.161
9.30
Y.34
9.22
0.507x10
0.790
0.813
0.023
1.31~10”
0.328
9.54
9.72
9.18
0.910x10
0.892
0.918
0.026
2.53~10”
0.578
9.63
10.3
8.41
0.123~10’
0.921
0.951
0.030
3.67~10”
0.728
9.79
11.1
7.61
10’
0.673
0.692
0.019
2.47~10”
0.401x10-
i.74
7.74
7.74
0.867X10’
0.673
0.692
0.019
1.39X10”
0.473x10-
7.78
7.78
7.77
0.243~10’
0.678
0.698
0.020
4.58~10”
0.931x10-
9.20
8.21
8.17
0.507X10’
0.747
0.767
0.020
l.lOXlU”
0.307
6.57
8.71
8.29
0.910x10
0.844
0.866
0.022
2.08~10”
0.494
8.iO
9.12
7.93
0.145x10
0.896
0.922
0.026
3.67~10”
0.732
11.91
10.2
6.95
1.58
0.581
0.584
0.003
1.37X10”
0.347x10-
6.04
6.04
6.04
0.867~10
0.581
0.584
0.003
7.79X10’”
0.534x10-
6.08
6.08
6.07
U.24
0.584
0.587
0.003
2.31~10”
0.731x10-
6.16
6.17
6.15
U.507XlO
0.642
0.646
0.004
5.33X104’
0.246
6.34
6.41
6.21
0.910x10
0.726
0.730
0.004
1.11x10”
0.398
6.62
6.til
6.25
0.190x10
0.752
0.758
0.006
2.84~10”
0.725
7.01
7.97
5.49
1.58
Fast
Compact
283
Rotator
density (labelled the same way as in Fig. 2). Also shown dotted is the curve given by Butterworth and Ipser [7] for a fast-rotating fluid for ya= 0.3, incom ressible MOE1(Y.=0.077. We note there is a considerable difference between our results and the It shows that the Butterworth-Ipser model. equilibrium configuration of the body depends greatly on its internal structure. 2.
MOO Fig. 3 The e-MoR relation for two configurations compared with Butterworth and Ipser’s relation [7] and e varying very slowly always have e< 0.1, with Q, but when Sl > 300 Hz, e varies rapidly with a, reaching a value around 0.7 If we apply these at the critical rotation. results to the known pulsars, then we shall find that, for most pulsars with periods between 0.2 and 2.0 set, their eccentricities and they will be nearly will be very small, spherical, but for three short period pulsars 0531+21, 0833-45 and 1937+214, their eccentricity will be 0.06, 0.02 and 0.5, respectively. In Fig. 3, the eccentricity is plotted against MOQ for 2 values of the central
The Equilibrium Relation
Mass-Rotational
Velocity
Fig. 4 shows the variation of the equilibrium mass with the central density for a) critical n = nmax, (solid curve), angular rotation, b) the static case, 9 = 0 (chained) and c) the curve calculated by Hartle and Thorne by the perturbation method (dashed). We note that, whether or not rotation is included, and whichever method is used for the calculation, the M-E, curve has a minimum at Ed= 2.5(+13) g/cm3 and a maximum at 6.0(+15)g/cm3. Between The these two values, the body is stable. value for the maximum stable mass depends on the model used: our model gave 0.95&, and the H-T model, 0.81EIe.
(6) m 0.8 -
OI 13
I4
I
I
IS
16
17
logE&.cm-‘1 Fig. 5 Variation of the equilibrium mass with the central density for 6 values of the rotation.
0.1
-
0.2
-
aL 13
I
’
14
’
’
IS
1
1
16
I
logE,(g.cm-3) Fig. 4 Variation of mass with the central cases of rotation
the equilibrium density in 3
0.2 0
1 I
I 2
, 3
I 4
kXoWeJz’~ Variation of the equilibrium Fig. 6 mass with the rotational velocity for 4 values of the central density.
Dong,Sun et al.
284
I
D
I
I
I
I
,
,
,
,
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.61.8
...I .~__~..._ _ 0.2 04 0.6 0.8 1.0 1.2 1.4 1.6 1.8
UE, B 1.18
0.18
I__-
_~ .- -
Fig.
0.16
1.16 -
A
. .._
. 10
1
0.14 012
1.10.-
0 10
I.Oa-
0.08
1.06-
0.06
l.o4L:-y-~,
004
--... 1.02------_;~:\~ 1.00. 0
0.02
--_,.‘=_-_ -_ _-_-L====-_-_ L. I t ‘r-- t ___ 0.2 0.4 0.6 0.8 I.0 1.2 1.4 1.6 1.8 1
1
0
0.2 0.4 0.6 0 8
I0
1.2 I.4 I.6 I L1 2.0
e/t,
The metric functions u, w, B, h, shown as functions of Figs. 7- 10 the first coordinate < for three values of the second coordinate n specified in Fig.7, and for two equilibrium configurations. The solid curves are for ~e=6.0 (+15) g/cm3S R=1.23 (+4) /s; the dashed curves are for EC= 3.0 (+14) g/cm , il= 2.43 (+3) /s.
Fig. 5 shows the variation of the equilibrium mass with the central density for 6 values of the rotational speed, R= 0, 1.58, 867, 2430, 5070 and 9100 s-l. Fig. 6 shows its variation with the rotational speed for the same 4 values of the central density as in Fig. 2. These figures show that the increase in the mass is very slow for low rotational speeds, and becomes appreciable only when the rotation is greater than 1000 hz. Also, the effect of the central density on the equilibrium mass is limited by the rotation and for bodies spining at the
critical speed, the effect of the central density is relatively small. A comparison between our result and the H-T result may be noted. The maximum increase in mass we found is 35%, as compared to 10%. This is because of the neglect of higher-order terms in the H-T calculation. 5.
EFFECTS OF ROTATION ON THE GRAVITATIONAL FIELD
Curves of the metric functions of v, W, B and X are shown in Figs. 7-10 for two sets of
Fast
Light
TABLR 2
rotating
body
EC
by
Various
deflection
i.-
Rotating
angle
At
A_
0.15SXIO
0.113
0.115
0.267~10’
0.111
0.117
0.867~10’
0.108
0.120
0.150X10’
0.102
0.126
0.243~10’
0.0984
0.133
0.158~10
0.465
0.472
0.867~10’
0.461
0.460
0.243~10’
0.459
0.482
0.507X10’
0.453
0.489
0.123~10’
0.450
0.493
(XC’)
3.00X10”
Bodies
(radian) Ap=+
.-
0
6.00~10”
0.1148
0
TABLE 3
JL
a. 4795
Gravitational
rotating
Redshift
3.OOXl0’4
1.00x10”
?=O
Surface
of
Various
Rotatinp:
Bodies
q = 0.5385
‘I = 0.9062
0.158X10
1.031
1.030
1.029
1.033
1.032
1.031
0.287~10’
1.035
1.034
1.033
0.867x10’
1.040
1.038
1.036
0.150x104
1.048
1.046
1.042
0.243~10
1.066
1.063
1.060
0.158x10
1.064
1.063
1.063
0.287~10’
1.066
1.065
1.065
O.@67xlO’
1.070
1.069
1.068
0.243x10’
1.082
1.080
1.079
1.106
1.104
1.100
0.507X10’
1.149
1.132
1.127
0.158X10
1.115
1.113
1.112
0.2LixlO
1.126
1.124
1.122
1.128
1.126
1.124
0.243~104
1.137
1.134
1.133
0.507X10’
1.172
1.166
1.158
0.910x10’
1.261
1.246
1.224
0.158X10
1.150
1.148
1.147
0.667~10
1.157
1.154
1.153
0.243~10’
1.162
1.159
1.158
0.507x10’
1.187
1.183
1.177
0.910X10’
1.267
1.250
1.230
0.123~10’
1.344
1.312
1.292
10’
0.867~
10”
the
0.190x10’
0.403x 10”
at
r
body
0 (Id’)
(g*cm-‘)
6.00x
285
Rotator
Reflection
Q
(g-cm-‘)
3.00x
Compact
10’
Dong,
286
density and the values of the central fi = O.l23(+5)/sec ~c= 6(+15)g/cm3, rotation, n = 0.243(+4)/set, and and EC= 3 (+l4) g/cm”, 3 values of n, 0, 0.5385, 0.9062. Based on these data, a discussion on some possible graviational effects now follows. 1.
Deflection
of
2.
al.
The Gravitational
Redshift
the gravitational By definition, z=Ar/At1. In the case under this can be written as 2 -
e-.(1
-
c-y(l
-
VI)-t -
equation of from infinity
where
U’ -
is a constant, a is the radius of the body, and suffix 0 refers to values at p =a. The positive or negative sign is taken according as the light ray is along or against the direction of rotation. The angle by which the light ray is deflected after passing by the object, A’P-~](P(Q)-v~~,]-depends on the direction of the ray. The results of calculation are given in TABLE 2, where A+ is the angle of deflection when the ray is along the direction of rotation, and A- is when against. For M= 0.68 M, and K= 8.41 km, the two deflections may differ by as much as 2.49’. This effect is in agreement with the dragging of local inertial frames predicted by Mach’s Principle.
* _ I/o - 1
V’)_’
The Phenomenon Differential
of
Temperature
In a rotationally symmetric field, the difference in ~a gives rise to the phenomenon of temperature differential on the surface According to Ref. [lo], after of the star. including the relativistic effects, the condition for thermal equilibrium for the body is T = U” x constant Using the results of TABLE 3, we can calculate the difference in temperature between the equator and the poles. If, for E,=6(+15)g/cm, we take a pole temperature of l.O0(+7)K, according to Ref. [ll], then the equator temperature will be l.O4(+7)K, higher by 40 OOOK. We can imagine that this will lead to thermal and material flows and will affect the cooling process of the neutron star.
REFERENCES [ 1I I2 I I3 1 [4I I 5I
Hcwirh, A.. et aL Nutwe. 217(1968). 799. Gold, T., Nature, 218(1968), 731. Backer,D. C. et aI, Ntiure, 300 (1982), 1395. Haoiron. B. K., et al, “Graviution l’ha~ and Grnvitntiod c0B.p~” (1%~). Hude. L. B.. AP. J.. 150(1%7), 1005. C 6 I flnrtle. J. 8.. Thernc. K. S., Ap. J., 153(1968), 807. C 7 I Butt-o& E. M. Ipur, J. R.. Ap. J., 204 (1976), 200. t 8 J Syngc. J. L.. “Rclativiry.the GeneralTheory” (1960). III1
is
TABLE 3 gives the calculated values of the gravitational redshift at the surface of the star in the several case. We see that, in the value is greatest at the each case, equator and least at the poles, the difference becomes more pronounced as the rotational speed increases; the largest difference is some 18%. 3.
[9 I II01
redshift discussion,
where
Light
In the equatorial plane, the the photon trajectory coming at $- is
Sun et
Morgan, L. M., Mqan. T.. Pl~yr. Rev., D2 (1970). 2756. I.andnu. L. D.. Lifshitz, E. M.. “Ststiatical Physics” (1980) Smith t F. C. 3 “Pulsars” (1977).
10 (1986) 278-286 6 (1986) 208-219
Pergamon Journals.
Printed in Great Britain 0275-1062/86$10.00+.00
THE EQUILIBRIUM CONFIGUFATION OF RAPID ROTATING COWACT AND SOME GRAVITATIONAL
DONG Jun
Nanjing
STARS
EFFECTS
Aeronautical
Engineering
Institute
SUN Jing-li and ZHU Pei-then Nanjiny University Keywords:
Received
Pulsars
- Hydrodynamics
1985 February
- Relativistic
Effects
26
In this paper, the equilibrium configurations of rapid ABSTRACT rotating compact stars and some gravitational effects are studied within the general relativity by use of the Harrison-Wheeler equation of state and by the self-consistent field method. Numerical calculations show that the equilibrium configuration of a rotating star is a spheroid. For large the eccentricity and mass increase spin velocities, say, w > 3.0 x 102sec-l, very rapidly as the angular velocity increases, for the critical angular velocity of the rotating star, the eccentricity is about 0.7, the increase The difference of the gravitational redshifts at in mass is about lo-35%. the surface of the star caused bv rotation, and the difference of the of rotation or in the light bending when the beam moves in the direction . opposite direction are obvious.
1.
INTRODUCTION
Since the discovery of pulsars in 1967 [l], and their identification with rotating neutron stars [Z], and especially since the discovery of the millisecond pulsar PRS 1937 +214, [3], there has been an upsurge in interest in the investigation of the structure and equilibrium configuration of rapidly rotating compact stars by means of the general relativity. High density and rapid rotation are the two outstanding features of this type of objects. Under the present circumstances when our understanding of the equation of state of superdense matter is still rather we shall adopt the equation of incomplete, state given by Harrison and Wheeler [4], for an approximate description of the state of the “cold matter” after the completion of nuclear burning during the late stage of evolution of the star. Apart from the uncertainty in regard to the equation of state, the main difficulty in the setting up of rapid compact rotators is the mathematical complexity arising from Hartle [S] the introduction of rotation. and Hartle and Thorne [61 have used a perturbation method in a model of a slow, neutron star; this model is rigidly rotating, not suitable for millisecond pulsars.
Butterworth and Ipser [7] have used the selfconsistent field method to study the equilibrium configuration of a rigidly rotating, incompressible, homogeneous fluid and obtained the relation between the general relativity Maclaurin ellipsoids and rotation. In this paper, we shall use the selfconsistent field method, and numerically solve the Einstein field equation and the general relativistic hydrostatic equilibrium equations in an attempt to find the equilibrium configuration of rapid, compact rotators.
2.
BASIC EQUATIONS AND BOUNDARYCONDITIONS
The gravitational metric line element with rotational symmetry can be written in the form d.S’- PL-')(dpl+ dz')+ p'E'c-"(dp - odt)'- Pdr' where the metric functions functions only of p and Z, Einstein equation, R,.-
- 8x T..(
X, v, B and w are and satisfy the
f g,.T >
o is the angular velocity of the local tial system relative to an observer at
(2) iner-
Fast
Compact
infinity, and we have taken G = c = 1. For convenience, we adopt the ellipsoidal surfaces are: confocal
rotating
ellipsoids
confocal
rotating
hyperboloids
planes
$=const.,
where K is by:
the p-
For a chosen axes
279
coordinates
r’+ y’ ----K1 1 - 1’
22 BJa
(c,n,$).
The three
coordinate
--lGrl$l
Os$i2*
semi K(I
Rotator
focal
+ {‘)+(l
K, the
(3) distance.
These
-q’)’
surface
2 of
a-b-KJl+&
coordinates
Kh
are
related
to
the
(p,z,e)
coordinates
pl-‘p
a spheroid
is
uniquely
given
by 5 =Ss,
with
long
c-K.&
(4) and short
(5)
and eccentricity eIn these
(1 + s:)-“’
(6)
coordinates,
the
Einstein
equation
can be written
as
16ne”‘-“K’(C’ (1 + 6’) g,
+ [2E + (1 + 6’) $ Z] ---e ’
2
-“K’B’
$
+ (1 - 11’) 2
(1 + [‘)1(1 - q’) (2)’ 1
+ 4rc”‘-“K’([’
+ 1’)
(l+C’)g
+ [(l - 1’) $ g
+ (1 + r’)(l
(P + E)(l + I”) 1 - V’ 1
+ $) BP
1
+ 2P
(7) -
24
2
- q’)‘($y]
(81
+[45.+(1+5J)(~$4~)]~
+ (1 -
q’) $$
-I- [(l -
q’) (;
g
-
4 Z)
-
4q] $
(9)
(10)
where Y, -
w + [:I( 1 -
+ 2‘9(1
+ 6’) f
11’) -
?‘(I + {‘)I
G -f 2
280
DongpSun et
r, -
2[5’(1 -1’)
f,
e(:* + n’) 1 + r1[‘1’(1 + 5’) -
-
1-v p, _
V(1 f
5’)l
S((’
+ ‘1’) -
~(5’
+
1’)
f
(1 + E’)[:‘(l
-
-
25’1( 1 +
1 s
“i’) + ?$(I
+ [‘)I
1 Ll
I$) $ ;t $ g
+ 6’) - F’(1 - $)I $ _g + 1-
+ 5’)(1
I’)]
c’)( 1 - q’) + -;;
$(l
Jfq(l
‘5x1 -
$) - ‘i’(1 + :‘)I
d, - ( I - $)E$( 1 f $9 - :‘( C) -
-
5’ + ‘1’
259( 1 + :I)( 1 -
- (1 - $)I$(1 Cl -
??(I
f’ + ‘t’
+ (1 + f’)[$‘(l b, -
5’(1 + :‘) -
B
5(5’ + 11’) 1 + 51??1 -‘i”) I-et’ B
11,-
al,
-
II’) -
-
5’)l $)I
7f)
1‘ -
+ f( 1 + P>fl
-
n’)(S$
-
P) + 5&l
f
.$a)’ -
5’( 1 - ‘)‘)‘L 2B
5’ + ‘1’
_ &I + 5’)(1
- $)(S,c
-
n’) + I’?(1 5’ + ‘1)
The equation
of
hydrostatic
equilibrium
- q9
- $0
-t 5Y.L :iZB
is
where v = pBe -2v(i2-w) is the linear velocity of the fluid element in the local inertial system, P and E are the pressure and energy density in the star’s interior, and their relationship is determined by the Harrison-Wheeler equation of state 141. In these coordinates, the total mass is
The rest
mass is (14)
and the
total
angular
momentum is
Fast Compact Rotator
281
The boundary of the object is determined by P=E=O. On the boundary, the metric functions and their derivatives are continuu+2J/r3. ous. At infinity, B+l, v-+-M/r, In our calculations,these boundary conditions are imposed at r= (+3)5,. For A, we take local flatness on the axis of symmetry as its boundary condition, [8,9], that is, h=lnB when n=l. -4 -
THE NUMERICAL CALCULATION
3.
-5.
We use
the self-consistentfield method and solve the above equations by iteration. The
.steps
-6
are:
-1
First, for a set of initial values vo, Bo, wo, Ao, we apply the usual method of
numerical integration to the H-W equation of state and the equations (12)-(li) to evaluate P, E, M, MO and J. Assume the K-th and the (K- l)-th iterations differ by the small quantities bv~, ~BK, &K, 6h~. -The equations (7)-(g) are developed into linear partial differential equations in &K, ~BK, hK> in which P, E, v are Taylor-expanded about v&l, BK_~, wK_l, and only first-order terms retained. These equations are then discretized to a set of difference equations over a finite grid in the cn-plane. The latter are then solved by Gauss's method of elimination and the values of &K, ~BK, &K at the grid points found. The equations for h, (10) and (ll), have already been linearized. The iteration is continued until bv is less than a pre-assigned tolerance.
-8 0
I I I I,,,,, 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I.0
Fig. 1 Energy density distribution on the equatorial plane for 3 equilibrium configurations: (1) EC= 6(+15)g/cm3, Cl= 1.23(+4)s-’ (2) (3)
EC= l(+15)g/cm3, EC= 2(+13)g/cm3,
a= 5.07(+3)s-1 a= 1.58~~’
4
0.8 (41(3)(2) (I). 0.7I 111
4.
RESULTS OF CALCULATION AND DISCUSSION
We took 10 different values for the central density in the range 2(+13) - 3(+16) g/cm3 and used the static, spherically symmetric solution to initiate the iteration. For each value of the central density, we also took several values of the angular velocity, so as to find the effect of the latter on eccentricity and the equilibrium mass. TABLE 1 gives, for each pair of the central density and the spin velocity, the body’s rest mass, bound total mass, its total total angular momentum, eccentricity, energy, the long and the short axes, and the mean Fig. 1 shows the distribution of the radius. energy density in the equatorial plane n = 0 for three equilibrium configurations. The results of the calculation show that the pressure and density both fall to zero at over the range of showing that, C=Ss, angular velocities used, the equilibrium configuration is indeed a rotating ellipsoid.
a.6 _ I 0.5 -
0.40.30.20.1 0
1
2
3
4
W-N=-‘) Fig. 2 Variation of eccentricity with angular velocity for 4 central densities 1.
The Eccentricity-RotationRelation
Fig. 2 shows the variation of the eccentricity with the rotational velocity for 4 values of the central density, Ed: (1) 6.0(+15), (2) 3.0(+15), (3) 1.0(+15), (4) 3.0(+14) g/cm3. We see that, in all cases, when n< 300 Hz, we
Dong, Sun
282
TABLE
et
al.
The total mass, rest mass, bound enerf?y, angular momentum, long and short ares of the equilibrium eccentricity, radius, configuration for various central densities and angular velocities
1
u
Et
u
c
(set-‘)
(kInI
UL
2.00x10”
1.58
0.283
3.283
2.97~10”
0.681
.22x
.57x10’
.61x10’
2.50~10”
1.58
0.181
0.181
9.69~10”
0.211
.27x10’
.31X10’
.19X10’
1.73
0.191
0.191
1.34X10”
0.714
.59X10’
.30x10*
.41x
1.58
0.209
0.209
1.12X10”
4.49x10+
79.3
79.3
79.2
0.228
0.228
2.76~10”
0.718
86.4
97.5
67.9
0.287
0.287
8.26~10”
7.06x10-’
41.7
41.7
41.7
0.312
0.312
8.68X10.7
0.722
47.2
53.4
37.0
0.411
0.414
0.003
6.04~10”
0.106x10-’
21.2
21.2
21.2
U.19UXlO’
0.412
0.415
0.003
8.09~10”
0.593x10-
21.6
21.6
21.6
U.287~10’
0.414
0.417
0.003
1.27~10”
0.839x10-’
21.8
21.8
21.7
O.R67XlO’
0.420
0.423
0.003
4.23~10”
0.277
22.2
22.5
21.6
0.150x10’
0.423
0.426
0.003
4.92~10”
0.319
22.5
22.9
21.i
0.243~10’
0.503
0.507
0.004
1.61~10”
0.725
23.9
27.1
18.7
I.511
0.558
0.567
0.009
2.92~10”
0.891x10-!
13.9
13.Y
13.9
0.287X10’
0.558
0.567
0.009
7.08~10“
0.448~
Ii.2
14.2
14.2
0.867~10’
0.564
0.574
0.010
2.51~10”
0.128
14.6
14.6
14.5
5.00x10”
0.287X10’ 1.58
1.00x10”
0.867X10’ 3.OOXIO”
1.00x10”
3.00X10”
3.00X10”
6.00X10”
1.58
0.243~10
0.608
0.618
0.010
8.93~10”
0.388
15.3
15.7
14.5
0.403XlO’
0.665
0.677
0.012
1.87~10”
0.559
16.0
17.0
14.1
0.507X10’
0.721
0.735
0.014
3.09x10”
0.712
16.9
19.0
13.3
0.666
0.685
0.019
3.01 x 10”
0.761x10-!
10.1
10.1
10.1
0.287~10’
0.666
0.685
0.019
6.27~10“
O.245xlO-1
10.3
10.3
10.;
0.867~10’
0.668
0.687
0.019
2.13~10”
0.734x10-’
10.6
10.6
10.6
0.243~10
0.686
0.705
0.019
7.16~10”
0.221
11.1
11.2
10.9
0.507x10’
0.764
0.785
0.021
1.71x10”
0.454
11.5
11.9
10.6
0.910x10
0.879
0.904
0.025
3.68~10”
0.720
11.9
13.4
9.30
1.58
0.690
0.712
0.022
2.25~10”
0.698x10-!
8.42
8.42
8.42
0.691
0.713
o-022
1.91X10”
0.661x10-
9.14
9.15
9.13
1.58
0.867~10
1.00x
10”
3.00X10”
IO-’
10’
0.243~10
0.69Y
0.721
0.022
5.77X10”
0.161
9.30
Y.34
9.22
0.507x10
0.790
0.813
0.023
1.31~10”
0.328
9.54
9.72
9.18
0.910x10
0.892
0.918
0.026
2.53~10”
0.578
9.63
10.3
8.41
0.123~10’
0.921
0.951
0.030
3.67~10”
0.728
9.79
11.1
7.61
10’
0.673
0.692
0.019
2.47~10”
0.401x10-
i.74
7.74
7.74
0.867X10’
0.673
0.692
0.019
1.39X10”
0.473x10-
7.78
7.78
7.77
0.243~10’
0.678
0.698
0.020
4.58~10”
0.931x10-
9.20
8.21
8.17
0.507X10’
0.747
0.767
0.020
l.lOXlU”
0.307
6.57
8.71
8.29
0.910x10
0.844
0.866
0.022
2.08~10”
0.494
8.iO
9.12
7.93
0.145x10
0.896
0.922
0.026
3.67~10”
0.732
11.91
10.2
6.95
1.58
0.581
0.584
0.003
1.37X10”
0.347x10-
6.04
6.04
6.04
0.867~10
0.581
0.584
0.003
7.79X10’”
0.534x10-
6.08
6.08
6.07
U.24
0.584
0.587
0.003
2.31~10”
0.731x10-
6.16
6.17
6.15
U.507XlO
0.642
0.646
0.004
5.33X104’
0.246
6.34
6.41
6.21
0.910x10
0.726
0.730
0.004
1.11x10”
0.398
6.62
6.til
6.25
0.190x10
0.752
0.758
0.006
2.84~10”
0.725
7.01
7.97
5.49
1.58
Fast
Compact
283
Rotator
density (labelled the same way as in Fig. 2). Also shown dotted is the curve given by Butterworth and Ipser [7] for a fast-rotating fluid for ya= 0.3, incom ressible MOE1(Y.=0.077. We note there is a considerable difference between our results and the It shows that the Butterworth-Ipser model. equilibrium configuration of the body depends greatly on its internal structure. 2.
MOO Fig. 3 The e-MoR relation for two configurations compared with Butterworth and Ipser’s relation [7] and e varying very slowly always have e< 0.1, with Q, but when Sl > 300 Hz, e varies rapidly with a, reaching a value around 0.7 If we apply these at the critical rotation. results to the known pulsars, then we shall find that, for most pulsars with periods between 0.2 and 2.0 set, their eccentricities and they will be nearly will be very small, spherical, but for three short period pulsars 0531+21, 0833-45 and 1937+214, their eccentricity will be 0.06, 0.02 and 0.5, respectively. In Fig. 3, the eccentricity is plotted against MOQ for 2 values of the central
The Equilibrium Relation
Mass-Rotational
Velocity
Fig. 4 shows the variation of the equilibrium mass with the central density for a) critical n = nmax, (solid curve), angular rotation, b) the static case, 9 = 0 (chained) and c) the curve calculated by Hartle and Thorne by the perturbation method (dashed). We note that, whether or not rotation is included, and whichever method is used for the calculation, the M-E, curve has a minimum at Ed= 2.5(+13) g/cm3 and a maximum at 6.0(+15)g/cm3. Between The these two values, the body is stable. value for the maximum stable mass depends on the model used: our model gave 0.95&, and the H-T model, 0.81EIe.
(6) m 0.8 -
OI 13
I4
I
I
IS
16
17
logE&.cm-‘1 Fig. 5 Variation of the equilibrium mass with the central density for 6 values of the rotation.
0.1
-
0.2
-
aL 13
I
’
14
’
’
IS
1
1
16
I
logE,(g.cm-3) Fig. 4 Variation of mass with the central cases of rotation
the equilibrium density in 3
0.2 0
1 I
I 2
, 3
I 4
kXoWeJz’~ Variation of the equilibrium Fig. 6 mass with the rotational velocity for 4 values of the central density.
Dong,Sun et al.
284
I
D
I
I
I
I
,
,
,
,
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.61.8
...I .~__~..._ _ 0.2 04 0.6 0.8 1.0 1.2 1.4 1.6 1.8
UE, B 1.18
0.18
I__-
_~ .- -
Fig.
0.16
1.16 -
A
. .._
. 10
1
0.14 012
1.10.-
0 10
I.Oa-
0.08
1.06-
0.06
l.o4L:-y-~,
004
--... 1.02------_;~:\~ 1.00. 0
0.02
--_,.‘=_-_ -_ _-_-L====-_-_ L. I t ‘r-- t ___ 0.2 0.4 0.6 0.8 I.0 1.2 1.4 1.6 1.8 1
1
0
0.2 0.4 0.6 0 8
I0
1.2 I.4 I.6 I L1 2.0
e/t,
The metric functions u, w, B, h, shown as functions of Figs. 7- 10 the first coordinate < for three values of the second coordinate n specified in Fig.7, and for two equilibrium configurations. The solid curves are for ~e=6.0 (+15) g/cm3S R=1.23 (+4) /s; the dashed curves are for EC= 3.0 (+14) g/cm , il= 2.43 (+3) /s.
Fig. 5 shows the variation of the equilibrium mass with the central density for 6 values of the rotational speed, R= 0, 1.58, 867, 2430, 5070 and 9100 s-l. Fig. 6 shows its variation with the rotational speed for the same 4 values of the central density as in Fig. 2. These figures show that the increase in the mass is very slow for low rotational speeds, and becomes appreciable only when the rotation is greater than 1000 hz. Also, the effect of the central density on the equilibrium mass is limited by the rotation and for bodies spining at the
critical speed, the effect of the central density is relatively small. A comparison between our result and the H-T result may be noted. The maximum increase in mass we found is 35%, as compared to 10%. This is because of the neglect of higher-order terms in the H-T calculation. 5.
EFFECTS OF ROTATION ON THE GRAVITATIONAL FIELD
Curves of the metric functions of v, W, B and X are shown in Figs. 7-10 for two sets of
Fast
Light
TABLR 2
rotating
body
EC
by
Various
deflection
i.-
Rotating
angle
At
A_
0.15SXIO
0.113
0.115
0.267~10’
0.111
0.117
0.867~10’
0.108
0.120
0.150X10’
0.102
0.126
0.243~10’
0.0984
0.133
0.158~10
0.465
0.472
0.867~10’
0.461
0.460
0.243~10’
0.459
0.482
0.507X10’
0.453
0.489
0.123~10’
0.450
0.493
(XC’)
3.00X10”
Bodies
(radian) Ap=+
.-
0
6.00~10”
0.1148
0
TABLE 3
JL
a. 4795
Gravitational
rotating
Redshift
3.OOXl0’4
1.00x10”
?=O
Surface
of
Various
Rotatinp:
Bodies
q = 0.5385
‘I = 0.9062
0.158X10
1.031
1.030
1.029
1.033
1.032
1.031
0.287~10’
1.035
1.034
1.033
0.867x10’
1.040
1.038
1.036
0.150x104
1.048
1.046
1.042
0.243~10
1.066
1.063
1.060
0.158x10
1.064
1.063
1.063
0.287~10’
1.066
1.065
1.065
O.@67xlO’
1.070
1.069
1.068
0.243x10’
1.082
1.080
1.079
1.106
1.104
1.100
0.507X10’
1.149
1.132
1.127
0.158X10
1.115
1.113
1.112
0.2LixlO
1.126
1.124
1.122
1.128
1.126
1.124
0.243~104
1.137
1.134
1.133
0.507X10’
1.172
1.166
1.158
0.910x10’
1.261
1.246
1.224
0.158X10
1.150
1.148
1.147
0.667~10
1.157
1.154
1.153
0.243~10’
1.162
1.159
1.158
0.507x10’
1.187
1.183
1.177
0.910X10’
1.267
1.250
1.230
0.123~10’
1.344
1.312
1.292
10’
0.867~
10”
the
0.190x10’
0.403x 10”
at
r
body
0 (Id’)
(g*cm-‘)
6.00x
285
Rotator
Reflection
Q
(g-cm-‘)
3.00x
Compact
10’
Dong,
286
density and the values of the central fi = O.l23(+5)/sec ~c= 6(+15)g/cm3, rotation, n = 0.243(+4)/set, and and EC= 3 (+l4) g/cm”, 3 values of n, 0, 0.5385, 0.9062. Based on these data, a discussion on some possible graviational effects now follows. 1.
Deflection
of
2.
al.
The Gravitational
Redshift
the gravitational By definition, z=Ar/At1. In the case under this can be written as 2 -
e-.(1
-
c-y(l
-
VI)-t -
equation of from infinity
where
U’ -
is a constant, a is the radius of the body, and suffix 0 refers to values at p =a. The positive or negative sign is taken according as the light ray is along or against the direction of rotation. The angle by which the light ray is deflected after passing by the object, A’P-~](P(Q)-v~~,]-depends on the direction of the ray. The results of calculation are given in TABLE 2, where A+ is the angle of deflection when the ray is along the direction of rotation, and A- is when against. For M= 0.68 M, and K= 8.41 km, the two deflections may differ by as much as 2.49’. This effect is in agreement with the dragging of local inertial frames predicted by Mach’s Principle.
* _ I/o - 1
V’)_’
The Phenomenon Differential
of
Temperature
In a rotationally symmetric field, the difference in ~a gives rise to the phenomenon of temperature differential on the surface According to Ref. [lo], after of the star. including the relativistic effects, the condition for thermal equilibrium for the body is T = U” x constant Using the results of TABLE 3, we can calculate the difference in temperature between the equator and the poles. If, for E,=6(+15)g/cm, we take a pole temperature of l.O0(+7)K, according to Ref. [ll], then the equator temperature will be l.O4(+7)K, higher by 40 OOOK. We can imagine that this will lead to thermal and material flows and will affect the cooling process of the neutron star.
REFERENCES [ 1I I2 I I3 1 [4I I 5I
Hcwirh, A.. et aL Nutwe. 217(1968). 799. Gold, T., Nature, 218(1968), 731. Backer,D. C. et aI, Ntiure, 300 (1982), 1395. Haoiron. B. K., et al, “Graviution l’ha~ and Grnvitntiod c0B.p~” (1%~). Hude. L. B.. AP. J.. 150(1%7), 1005. C 6 I flnrtle. J. 8.. Thernc. K. S., Ap. J., 153(1968), 807. C 7 I Butt-o& E. M. Ipur, J. R.. Ap. J., 204 (1976), 200. t 8 J Syngc. J. L.. “Rclativiry.the GeneralTheory” (1960). III1
is
TABLE 3 gives the calculated values of the gravitational redshift at the surface of the star in the several case. We see that, in the value is greatest at the each case, equator and least at the poles, the difference becomes more pronounced as the rotational speed increases; the largest difference is some 18%. 3.
[9 I II01
redshift discussion,
where
Light
In the equatorial plane, the the photon trajectory coming at $- is
Sun et
Morgan, L. M., Mqan. T.. Pl~yr. Rev., D2 (1970). 2756. I.andnu. L. D.. Lifshitz, E. M.. “Ststiatical Physics” (1980) Smith t F. C. 3 “Pulsars” (1977).