ЯДЕРНАЯ ФИЗИКА, 2009, том 72, № 9, с. 1607-1610
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
DYNAMIC STABILITY OF NONSPHERICAL BODIES
© 2009 G. S. Bisnovatyi-Kogan1),2),3)*, O. Yu. Tsupko1),2)**
Received September 18, 2008; in final form, February 27, 2009
We derive and solve equations, describing in a simplified way the Newtonian dynamics of a self-gravitating nonspherical nonrotating star after its loss of a linear stability, and investigate nonlinear stages of contraction. We find that only pure spherical models can collapse to singularity, but any kind of nonsphericity leads to a dynamic stabilization of the collapse, and formation of regularly or chaotically oscillating body. Therefore nonspherical star without dissipative processes will never reach a singularity. A real collapse happens after damping of the oscillations due to energy losses, shock-wave formation, or viscosity. Detailed analysis of the nonlinear oscillations is performed using a Poincare map construction.
PACS: 04.40.-b, 05.45.-a, 05.45.Pq, 95.10.Fh, 95.30.Sf, 97.10.Bt, 97.10.Sj, 97.10.Cv
1. INTRODUCTION
Dynamic stability of spherical stars is determined by an average adiabatic power 7 = g'logp I s- For a density distribution p = p0p(m/M), the star in the Newtonian gravity is stable against dynamical
collapse when f0R (7- > 0 [1, 2]. This
approximate criterium becomes exact for adiabatic stars with constant 7. Here, p0 is a central density, M is a stellar mass, m is the mass inside a Lagrangian radius r, so that m = 4n /Qr pr2dr, M = m(R), R is a stellar radius. Collapse of a spherical star may be stopped only by a stiffening of the equation of state, like neutron-star formation at late stages of evolution, or formation of fully ionized stellar core with 7 = 5/3 at collapse of clouds during star formation. Without a stiffening, a spherical star in the Newtonian theory would collapse into a point with an infinite density (singularity).
Here we show that deviation from the spherical symmetry in a nonrotating star with zero angular momentum leads to a dynamic stabilization, and non-spherical star without dissipative processes will never reach a singularity. Therefore collapse to a singularity is connected with a secular type of instability, even without rotation.
We calculate a dynamical behavior of a nonspher-ical, nonrotating star after its loss of a linear stability and investigate nonlinear stages of contraction. We
1)Space Research Institute of Russian Academy of Sciences, Moscow.
2)Moscow Engineering Physics Institute, Russia.
3) Joint Institute for Nuclear Research, Dubna, Russia. E-mail: gkogan@iki.rssi.ru
E-mail: tsupko@iki.rssi.ru
use approximate system of dynamic equations, describing three degrees of freedom of a uniform self-gravitating compressible ellipsoidal body [3, 4]. We obtain that the development of instability leads to the formation of a regularly or chaotically oscillating body, in which dynamical motion prevents the formation of the singularity. We find regions of chaotic and regular pulsations by constructing a Poincare diagram for different values of the initial eccentricity and initial entropy.
The case 7 = 5/3 was considered in [3] and [4], but this case is not interesting for the present work, because isentropic spherical star with 7 = 5/3 always stops contraction, and never suffers collapse to singularity. In the present work we consider mainly spheroidal figures with 7 = 4/3 (see also [5]). A spherical star with 7 = 4/3 collapses to singularity at small enough K, and we show here how deviations from a spherical form prevent formation of any singularity.
2. EQUATIONS OF MOTION AND NUMERICAL RESULTS Let us consider 3-axis ellipsoid with semiaxes a = b = c:
x2 y2 z2
= 1,
a2 b2 c2 and uniform density p. A mass m of the uniform ellipsoid is written as (V is the volume of the ellipsoid)
m = pV = pabc. 3
Let us assume a linear dependence of velocities on coordinates:
a x
Ux =
b y
cz
Uz =
1607
b
a
с
1608
The gravitational energy of the uniform ellipsoid is defined as[6]
BISNOVATYI-KOGAN, TSUPKO
for the oblate spheroid k = c/a < 1, 3
U9 = -
3Gm
du
10 J a/(a2 + u) (b2 + u) (c2 + u)
3Gm b =---—b
du
_5_1 e
2 'J (b2 + u)A 3m b (abc)1/3' 0
a = —
2a2 (k2 - 1) 1
+
k
cosh 1 k
vP^T,
The equation of state P = KpY is considered here, with y = 4/3. For y = 4/3, the thermal energy of the ellipsoid is E\_h ~ F 1/3 ~ (abc) 1/3, and the value
£ = Eih{abcf3 = 3 AT
remains constant in time. A Lagrange function of the ellipsoid is written as
L = Ukin - Upoi i Upot = + Eth i
C/kin = \p J(Vx +Vy+ Vl)dV =
V
= + + Eth =
3
ca =
a2(k2 - 1)
1
a (a2c)1/3' kcosh-1k
1
+ -
for the prolate spheroid k = c/a > 1, and
1- e
c (a2c)1/3
a = — -
(4)
(5)
(abc)11/3'
Equations of motion describing behavior of three semiaxes (a, b, c) are obtained from the Lagrange function in the form
3 Gm f du 5 1 e
a= 2 aJ (a2+u)A + 3ma(a6c)V3'
0
for the sphere, where the equilibrium corresponds to eeq = 1. At e = eeq the spherical star has zero total energy, and it may have an arbitrary radius. For smaller e < eeq the sphere should contract to singularity, and for e > eeq there will be a total disruption of the star with an expansion to infinity. Near the spherical shape we should use expansions around k = 1, what leads to equations of motion valid for both oblate and prolate cases
1 - e e 3 1 - k
,. _ 1 - e /4e 4\ 1 -k 5/
(6)
In nondimensional variables the total energy is written as
a2 c2 3 arccos k
H =--1-----,
5 10 5 a VT^
+
(7)
+
3
5 (a2c)1/3
(oblate),
3Gm c =---—c
du
+ 5 1
(c2 + u)A 3m c (abc)1/3 '
A2 = (a2 + u)(b2 + u)(c2 + u).
To obtain a numerical solution of equations we write them in nondimensional variables. We solve here numerically the nondimensional equations of motion for a spheroid with a = b = c, which are written as
3
aa =
2a2(1 - k2)
k
arccos k
1
+ -
3
ca =
a2 (1 - k2) 1
VT^PJ
k arccos k
1
a (a2c)1/3
VT^W \
+
, (1)
(2)
+
c (a2c)1/3
c2
3 cosh-1 k
H = — +--,_
5 10 5a VF^l
+
3 + -
5 (a2c)1/3
(prolate),
3d2
H --
10
a2 c2
— H--
5 10
— — (1 —e) (sphere), 5a
_3_
5a
1 S 262\ 1 + 3+ur) +
3e / 5
+ 5a l + 3 +
(8)
(9) (10)
S = 1 - k, |S| < 1 (around the sphere).
In Eqs. (1)—(10) only nondimensional variables are used.
e
e
e
2
e
0
e
e
e
afl,EPHA^ OH3HKA tom 72 № 9 2009
DYNAMIC STABILITY OF NONSPHERICAL BODIES
1609
da/dt 8
0.2
a
Part of the Poincare map forthree regular and two chaotic trajectories in case of y = 4/3, H = —1/5, e = 2/3. The values of semiaxis and time derivative (a, a) are taken in the minimum of another semiaxis c. Thick solid curve is the bounding curve.
Solution of the system of equations (1)—(6) was performed for initial conditions at t = 0: c0 = 0, different values of initial a0,a0,k0, and different values of the constant parameter e. Evidently, at k0 = = 1, a0 = 0, e < 1 we have the spherical collapse to singularity. The most interesting result was obtained at k0 = 1, and all other cases with deviations from spherical symmetry. It is clear that at e = 0 a weak singularity is reached during formation of a pancake with infinite volume density, and finite gravitational force. At e > 0 no singularity was reached in case of spheroid. At e > 1 the total energy of spheroid is positive, H > 0, determining the full disruption of the body. At H < 0 the oscillatory regime is established at any value of e < 1. The type of oscillatory regime depends on initial conditions, and may be represented either by regular periodic oscillations, or by chaotic behavior.
3. THE POINCARE SECTION
To investigate regular and chaotic dynamics we use the method of Poincare section [7] and obtain the Poincare map for different values of the total energy H. Let us consider a spheroid with semiaxes = = b = c. This system has two degrees of freedom. Therefore in this case the phase space is four-dimensional: a, a, c, C. If we choose a value of the Hamiltonian H0, we fix a three-dimensional energy surface H(a, a, c, C) = H0. During the integration of Eqs. (1)—(6) which preserve the constant H, we fix moments ti, when C = 0. At these moments there are only two independent values (i.e., a and a), because
the value of c is determined uniquely from the relation for the Hamiltonian at constant H. At each moment ti we put a dot on the plane (a, a).
For the same values of H and e we solve equations of motion (1)—(6) at initial c = 0, and different a, a. For each integration we put the points on the plane (a, a) at the moments ti. These points are the intersection points of the trajectories on the three-dimensional energy surface with a two-dimensional plane C = 0, called the Poincare section. For each fixed combination of e, H we get the Poincare map. The regular oscillations are represented by closed lines on the Poincare map, and chaotic behavior fills regions of finite area with dots. Example of the Poincare map is represented in the figure.
Condition c = 0 splits in two cases, of a minimum and of a maximum of c. The Poincare maps are drawn separately, either for the minimum, or for the maximum of c, and both maps lead to identical results [5].
4. DISCUSSION
The main result following from our calculations is the indication of a degenerate nature of formation of a singularity in unstable Newtonian self-gravitating gaseous bodies. Only pure spherical models can collapse to singularity, but any kind of nonsphericity leads to nonlinear stabilization of the collapse by a dynamic motion, and formation of regularly or chaotically oscillating body. Therefore, nonspherical star without dissipative processes will never reach
ftOEPHAfl OH3HKA tom 72 № 9 2009
1610
BISNOVATYI-KOGAN, TSUPKO
a singularity. A real collapse happens after damping of the oscillations due to energy losses, shock-wave formation, or viscosity.
This conclusion is valid for all unstable equations of state, namely, for adiabatic with 7 < 4/3. In addition to the case with 7
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