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Mathematical model of the rapidly rotating gravitating or
superdense neutron configurations
E.Bespalko, S.Miheev, V.Tsvetkov, I.Tsvetkov
Tver State University, Russia
170000 Russia, Tver, Zhelabova Street, 33, Email:tsvet@tversu.ru
Abstract
In the present work the mathematical model, describing distribution of the density of the super­
dense dense nuclear matter's of the rapidly rotating gravitating magnetic configuration (pulsar) has
been built.
The task under consideration is actual in connection with studying the nature of pulsars, elucidation
of the question about the equation of their superdense nuclear matter's condition and an influence of the
rapid rotation on the distribution of density.
A special interest excites the question about the possible existence of bifurcation points at the pa­
rameter # = # 2
4#G#0
(# ­ angular speed of rotation, G ­ gravitational constant, # 0 ­ density in the center of
configuration), in which there is a branch of nonaxisymmetrical solutions for distribution of density about
the rotating axis. This asymmetry of density's distribution about the rotating axis stipulates for gravi­
tational emission of pulsar the search of which is currently under way [1]. The equation of hydrostatic
equilibrium for gravitational configuration can take the following form for convenience [2­12]:
#+#- # 2
2 (x 2 + y 2 ) = # (PN) +#m (1)
#(#) =
P
# 0
dP (#)
#
(1a), # = -G # #(r # )dV #
|r - r #
| , (1b)
where #­gravitational potential of the configuration, P (#)­pressure, # (PN) ­the contribution of the
post­Newton corrections, #m ­magnetic stresses. The specific kind of the function P (#) depends on
the equation of nuclear matter's condition of the configuration. Further we will use numeral data for
equation of nuclear substance's condition Bete­Jonson (BJ),Oppenheimer­Volko# (OV), Raid (R) for our
calculations [8].
As it will be showed in the following work the parameter of the configuration depends on the choice
of the equation of the condition and subsequently they are the criteria for their choice. In particular at
the meaning # = # k a sharp increase of intensity of the configuration's gravitational emission occurs at
which frequency's registrations # k is defined and arguments appear in favour of the choice of the specific
equation of condition.
The value of post­Newton's corrections | # (PN) | will be equal # # # #(0)
c 2 # # # = 1, 26 · 10 -3 at # 0 = 4 · 10 14
g/cm 3 . That's why we can take in (1) equal to 0.
The function #(#) can be presented as a series on degrees # #
#0 - 1 #
# = # P 0
# 0
# # # 0 + # 1 # #
# 0
- 1 # + # 2 # #
# 0
- 1 # 2
+ ... # (2)
The coe#cient # 0 , # 1 , # 2 for di#erent equation of conditions are chosen from the condition of
consent (2) and (1a) and they are presented in Table 1.
Table 1
OV BJ R
# 0 2.487978 2.415387 2.378167
# 1 0.791516 1.253114 1.177573
# 2 ­1.696462 ­1.162273 ­1.200594
317

In all the cases the mistake of approximation is less than 1% that is quit enough for the equation of
nuclear mutter's condition contains many uncertainties.
The surface of configuration can be found using the condition
#(x, y, z) = 0 (3)
that is it can be known after the equation is solved (1). But the gravitational potential # in (1)
depends on the form of the boundary (3). That's why we'll substitute the exact configuration of the
surface (3) for a pseudo surface #D, the form of which depends on the unknown parameters Z ijk . These
parameters are found out of the condition of the most proximity of #D and (3). The following conditions?
which must be solved together with (1).
According to the works [12, 13, 14] we choose #D in the form of the perturbed ellipsoidal surface
#D : x 2
1 + x 2
2 + x 2
3 +
L
#
i,j,k
Z i,j,k x i
1 x j
2 x k
3 = 1 (4)
where x 1 = x
a1 , x 2 = xy
a1 , x 3 = z
a3 , a 1 , a 3 ­semi­axes of the ellipsoid of rotation, which together with
Z ijk parameterize #D. For smooth surfaces by choice of semi­axis a 1 a 3 we can make coe#cients Z ijk
small in order that use a method of decomposition of configuration's surface depending on the form into
the series of Burman­Lagrange [13] on the degrees of small coe#cients/ Z ijk . In our case |Z ijk | # 10 -4 .
The method of Burman­Lagrange's series with the use of symbolic calculations on a computer becomes
a powerful method of solving very di#cult sums, which couldn't be done using the old classical methods.
The conditions of proximity (3),(4) can be formed entering the function #:
# = 1
4## 0
#
#D
# 2
d# # 2
; # = e 1
3 # 1 + 1 - e 2
e 2
cos 2 # # 1
2
, e = a 3
a 1
(5)
Weighting multiplier 1
# 2 in (5) has been chosen for the conveniency of calculations in order the integrals
in # and # to have the same structure. Evidently the parameter # #D = # 1
2 will present a measure of
mistake in our equations substituting the exact surface of the configuration for #D. In the approximation
we use it increases with decrease e from 0 to 4, 88 · 10 -2 . Minimum # leads to the equations:
# i,j,k = ##
#Z i,j,k
= 0, # 1 = a 1
#
#a 1
#(Z i,j,k = 0) = 0 (6)
# 2 = a 3
#
#a 3
#(Z i,j,k = 0) = 0
Let's present the configuration's density # in the form of polynomial of degree P
# =
P
#
a,b,c
# a,b,c x a
1 x b
2 x c
3 (7)
If we chose P quite large, the with a high degree will approximate the density of the real configuration.
Calculation # abc for the meaning P = 4, 6, 8 and L = 2, 4, 6 is impossible without using the method
of symbolic calculations on a computer.
To find # abc in the system of symbolic calculations MAPLE, we have made a program.
As the programme has turned out to be large and complex the necessity to test it. We have done
two tests. A limited transition e # 1 to the symmetrical case was done. It is easily calculated using an
independent programme. Besides ## = 4#G# and operating on # by Laplac's operator we get density
multiplied by 4#G#. Both the tests have been done successfully for di#erent meaning L P . The analogous
programme in the system MAPLE has been done for the functional #.
Will present # and # (m) also in the form of polynomial of degree P on coordinates x a
1 , x b
2 , x c
3
# =
P
#
a,b,c
# a,b,c x a
1 x b
2 x c
3 ; # (m) = km
P
#
a,b,c
# (m)a,b,c x a
1 x b
2 x c
3 , km = B 2
0
8#G# 0 a 2
1
(8)
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In this case the system of equations (1),(6) can be presented as the system of algebraic equations
relatively to # abc and Z ijk :
# a,b,c +K 0 # a,b,c - #(# 2a - # 2b )# c0 = # (m)a,b,c (9)
where
# i,j,k = 0, # i = 0, K 0 = P 0
2#G# 2
0 a 2
1
, # = # 2
4#G# 0
.
We shall divide the coe#cients defining the structure of configuration # abc and Z ijk into symmetric
and antisymmetric parts about the axis of rotation
# a,b,c = # a+b
2 # !
# a
2 # ! # b
2 # ! # a+b,c + # 1(ab)c X 2 + # [ab]c X (10)
Z i,j,k = # i+j
2 # !
# i
2 # ! # j
2 # ! Z i+j,k + # 1(ij)k X 2 + Z [ij]k X
.
Here and hence a, b, c and i, j, k are even. And again entered values satisfy the correlation of symmetry:
# 1(ab)c = # 1(ba)c , # [ab]c = -# [ba]c (11)
Z 1(ij)k = Z 1(ji)k , Z [ij]k = -Z [ji]k
Symmetrizing and anti­symmetrizing at the first two indices the system of equations (9) we get a new
system of equations for defining # (ab)c , # [ab]c , Z (ab)c , Z [ab]c :
# (ab)c +K 0 # (ab)c - #(# 2a - # 2b )# c0 = 0, # (ij)k = 0 (11a)
# [ab]c +K 0 # [ab]c = km# (m)[ab]c , # [ij]k = 0 (11b)
At the output (11) we have omitted the small member # (m),(ab)c , which leads only to the small
correction km .
The coe#cients in equations (11), (11b) are integrals J ac , which we'll calculate numerically. The
calculation of these integrals and all other calculations we'll make with accuracy 10 -30 .
From physical point of view # is a free parameter, and e is a calculated one. However, for conveniency
of calculations it is more simple to consider e as a free parameter, # ­ as the calculated parameter. The
result is that all the parameters of the configuration will depend on e, # 0 , P 0 .
Substituting (10) into (9) system of equations which is a polynomial in the series of the small parameter
X . That's why it's possible to use the method of decomposition in the small parameter X . In the first
approximation we shall put X = 0 and find the meaning # ac and Z ik corresponding the figure of rotation.
In this approximation we'll put K 0 = K 00 , # = # 0 .
Two­dimensional arrays unknowns in the system (11) Z ik , # ac , K 0 = K 00 , # = # 0 identify as
ym (m = 1, 2...N 1 ).
It is easy to find a formula for defining N 1
N 1 = 1
8 (P + 2)(P + 4) + 1
8 (L + 2)(L + 4) (12)
At P = 6 and L = 2 we have N 1 = 13. Connection between the variables will take the form:
y 1 = # 02 , y 2 = # 04 , y 3 = # 06 , y 4 = # 20 , y 5 = # 22 , y 6 = # 24 , y 7 = # 40 ,
y 8 = # 42 , y 9 = # 60 , y 10 = Z 02 , y 11 = Z 20 , y 12 = K 0 , y 13 = #
.
In this case the equations of the system (11) can take the form of algebraic equations:
f l (y m , e) = 0 (13)
319

For numerical solution (13) we'll use regularized analogue of Newton's method with the parameter of
regularization # = 10 -6 [16]. Omitting index the component ym , f l we have the next iterative scheme:
y (y+1)(e) = y n (e) - # n [#f 2 (y (n) (e), e) +
#
f # (y (n) (e), e)f # (y (n) (e), e)] -1 â
â #
f # (y (n) (e), e)f(y (n) (e), e) (14)
where # n ­the step of iteration (# 0 # # n # 1), f # (y (n) (e), e) ­the matrix of Jacobi,
#
f # (y (n) (e), e) ­
transposed matrix of Jacobi.
The variable # f 2 (y (n) (e), e) = # n (e) is discrepancy and defines precision of the system of equations
(13), which in our case is 10 -30 .
Scheme's realization programme (14) in the same packet MAPLE as well as the above mentioned
symbolic calculations # abc # ijk turned out to be e#ective for the number of iterations for every parameter
e was not more that ten. It is very important taking into consideration a slow speed of numerical
calculations in the system MAPLE.
Precision of approximation of the found configuration of perturbed ellipsoidal surface #D for the case
P = 6, L = 2 increases with decrease e from 0 to 4, 8 · 10 -2 .
We put the received numerical data ym (e) into (11b). Then (11b) will contain only unknown data
# [ab]c , Z [ij]k . Just as in the previous case we'll convert these three ­dimensional arrays into one­
dimensional xm (m = 1, 2, ...N 2 ), and we'll also put x 1 = # [20]0 = 1. In case P = 6 L = 8 N 2 = 8.
We'll consider an influence of magnetic strains in the simplest model when two coe#cients # (m)[20]0 =
-# (m)[02]0 = k
2
, k­ the index of magnetic field's speed's degrease at the distance from the magnetic axis
are di#erent from 0.
We'll put #m = - 1
2 kkm . As the system (11b) is antisymmetric at the first two indices it will contain
only add on X members. At first we'll solve this system in the linear X approach. It is well afar from
critical point at the parameter e k and respectively # k , where the definer at the unknown x i is essentially
di#erent from 0. In the proximity of e k it is necessary to take into consideration (11b) member of order
X 3 . In the linear X approach (1b) take the form:
# N2
# p=1
A p
l
(y(e), e)x p
# X = #m # l1 (15)
The variables x p , X are solved analytically as function #m at the set numerals y(e) and e with the
help of MAPLE.
The meanings e k are found out of the condition X(e # e k ) # #. The calculations are presented in
table 2.
Table 2
OV BJ R
e k 0.58673 0.59207 0.59135
# k 0.09675 0.09684 0.09695
It is necessary to note the proximity of the meanings e k and # k for the three equations of condition.
(OV, BJ ,R).
The linear approach can be applied in the region |e - e k | # #
2
3
m . At |e - e k | # #
2
3
m the system (15) is
substituted for cubical about X system
(A p
l
)X + (B p+r
l x p x t x r )X 3 = #m # l1 ; l, p, t, r = 1, 2...N 2 (16)
Here the summing at the repeated indices is done.
Taking l #= 1 into (15) we'll find solutions x p = x p (e), which preserve their meanings and at e = e k .
At |e - e k | # #
2
3
m taking l = 1 into (12) we'll find X = X(e, #m ), besides X # #m .
At |e - e k | # #
2
3
m we'll put X = X(e, #m ) into (16) and then we have a cubical equation in this field
for X .
#(e - e k )X + #X 3 = #m (17)
320

where # and # are constant coe#cients, which meaning are presented in table 3.
Table 3
OV BJ R
# 0.62596 0.41016 0.42876
# 10.8375 2.3442 2.7674
In the critical point e = e k , X k = X(e k ) = #
1
3
m/#, is asymmetry of density's distribution is much
more than the linear approach. Note that for pulsar #m # 10 -9 - 10 -12 .
If #m = 0, then at e > e k (# < # k ) (17) has one decision X = 0, that corresponds to axial symmetrical
configuration. But at e < e k (# > # k ) there will be 2 roots X = 0 (symmetric solution) and X =
# #
#
(e k - e) ( asymmetric solution). A well­know dynamic violation of axial symmetry for nonlinear
equations in configuration's density distribution occurs.
Dependence #X = X(e) are presented in table 4.
Table 4
e X/#m (BJ) e X/#m (OV)
1 5.297 1 3.38238
0.95 6.08079 0.95 3.87927
0.9 7.13011 0.9 4.54211
0.85 8.60169 0.85 5.46700
0.8 10.80368 0.8 6.84062
0.75 14.4357 0.75 9.07916
0.7 21.49619 0.7 13.33520
0.65 40.88923 0.65 24.40889
0.6 306.19759 0.6 119.48844
0.59208 296239.23811 0.587 5983.47999
0.5920717750 483161352.66966 0.58677 43234.55271
0.59207177 52902983654.17490 0.5867334 4567205.28503
0.59207176997 151567305446.40869 0.5867330528 617880443.66757
0.59207176996 400620145034.52280 0.5867330502 ­110332016807.96199
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