Instability of stellar structures intermediate between dwarfs and neutron star

Cameron, A. G. W.

Cohen, J. M.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19690001924 2020-03-23T22:03:45+00:00Z
THE INSTABILITY OF STELLAR STRUCTURES INTERIMEDIATE
BETWEEN WdITE DWARFS AND NEUTRON STARS.
A.G.W. Cameron
Belfer Graduate School of Science
Yeshiva University, and Goddard
Institute for Space Studies, NASA,
New York, New York
and
Jeffrey M. Cohen
Goddard Institute nor S,--ce Studies
NASA, titer.. York, _\ev, York
69-  (TFIRU)(ACCESSION NU ER)
('^	 33(CODE)
(	 CR R TMX OR AD UMRER) (CATEGORY))
a
*1 _V1
AN
ABSTRACT
Drake has suggested that apparent multimillisecond
periedieities in pulsar signals may correspond to vibrations
of neutron stars. We have reexamined stellar models in the
appropriate density range with an improved equation of state,
and we conlude that suitable stable structures are unlikely.
4
At a recent seminar, Drake (Drake 1968) has reported a
superposition of 10 and 15 millisecond periods on the pulses
received from the puleare AP 2015 and CP 1919. He is reported
to have suggested that these periods represent the intrinsic
vibrational period of a neutron star. However, neutron star
periods much in . excess of one millisecond would be very unlikely
in a neutron star (Tsuruta, Wright, and Cameron 1965). A period
of order 10 milliseconds would be roughly characteristic of a
body with a mean density near 10 13 gm/cm3 , two orders of magnitude
smaller than the mean density of a typical neutron star. The
composite white dwarf-neutron star model curve of Tsuruta and
Cameron (1966) does show a subsidiary mass peak between 1013
and 10 14 gm/cm3 . However, because the treatment of the physics
in this density range was extremely crude, no significance was
claimed for this peak by the above authors. However, the
suggestion by Drake raises an important question as to the
possible existence of stable structures in this density range.
In this note we report on an investigation of this question with
an improved but still crude treatment of the relevant physics.
The following physical phenomena play a key -role in the
equation of state. In the density :range up to about 3 x 10 11 gm/cm3
only ions and electrons are presert under cold degenerate conditions:
In this range the electron Fermi level approaches 30 MeV, and
f	 electron capture reduces the charge to mass ratio of the nuclei.
3
1.
2.
The neutron binding energy of the nuclei approaches zero; the
point where it becomes zero has been denoted the "neutron drip
line" by J.A. Wheeler. At still higher densities and electron
Fermi energies, electron capture continues, and neutrons leave
the nuclei. Gradually a degenerate neutron gas is established
in the volume between the nuclei, and a significant neutron
Fermi energy is established. At this stage the nuclei will
contain unbound neutrons, since neutrons -can only be removed from
the nuclei with energies in excess of the neutron Fermi level.
Of course, neutrons will be readily exchanged between the free
neutron sea and the nuclear interior, but the density of unbound
neutrons in the nuclear interior will considerably exceed that
outside, since the Fermi level of the neutrons inside the nucleus
will be much greater than that outside, and thus the number of
momentum ,itetes per unit energy interval near the Fermi surface
will be much greater inside than outside.
We have treated the physics in the following very approximate
way. We write X71 = M-A for the mass excess of a nucleus. We use
the simplest possible form of the mass equation:
m(A, Z) =61VZZ+ ,(A-Z) +ccA+a (A-2 Z)2 /A+yA 2/3 +6z 2 /A 1/3	 (1)
Here c"" and"-;" are the mass excesses of the proton and neutron,
and the last four terms on the right hand side represent respectively
the volume binding energy, the volume symmetry energy, the surface
energy, and the coulomb energy. This equation is applicable to
3.
nuclei with sharp edges. In what follows we shall make some
approximations valid for large mass numbers A.
The beta decay energy is
Q - n(Z,A) - OJ ( Z+1,A)	 (2)
This leads to the expression
Z ti -Q-^^1,1L-`11k,) + 40 - 40/A - 6/A1/3	 (3)
A M	 80 + 26A
we use values of the coefficients given by Green ( 1954):
a - -15.756 MeV,	 = 23.694 MeV, Y = 17.794 MeV, and
6 = 0.7103 MeV. It may be seen that the last two terms of the
numerator of (3) are small compared to the preceeding ones,
and that the second term in the denominator is fairly small
compared to the iirst. Hence for a given value of the beta
decay energy the ratio Z/A is not strongly dependent on mass
number. Taking as typical A v100, we have
Z ^, 94-Q
A T2 —0
The neutron binding energy is
B = rrO +'^fi(Z,A-1) -%j(Z,A)
This leads to the expression
2 3 B+a+ + 26/A4/3Z
A2	 120 + 6A2/3
It may be seen that the last terms in the denominator and
(4)
(5)
i
(6)
4.
g
numerator of (6) are small compared to the preceeding terms, and
hence for a given neutron binding energy the ratio Z/A is not
strongly dependent on mass number. Again taking as typical
A ^100, we have
	Z2	 B+7.9524	 (7)
	
A 2	100
The neutron drip line (B=O) occurs at Z/A = 0.282,
according to (7). From (4), the corresponding beta decay energy
(or, equivalently, electron capture threshold) is Q = 31.96 MeV.
An equation of state can now be constructed, with the pressure
and density parametrically depending on the electron Fermi energy
Ee. The latter quantity specifies n  and P e , the electron number
density and pressure. The total number of protons (inside
nuclei) per cm  is equal to n e , for charge equality. We rewrite
(4) as
Z = 94 - EP	 (8)
	
A	 220
This specifies the number of neutrons in nuclei per cm 3 , and
hence the mass density of the nuclei. The ionic pressure can be
neglected. If E  < 31.96 MeV, the equation of state is thus
e
completely specified. if E Fe > 31.96 MeV, then free neutrons
are also present. We calculate the neutron Fermi energy En
from a modification of (7) :
Z2	-En + 7.9524	 (9)
	
s	
100
5.
From En we also calculate the number density of free neutrons in
the internuclear region and the neutron pressure P n , which will
also apply to the region occupied by nuclei. We assign the
nuclei a density 4 X 1014 gm/cm3 , calculate the fractional
volume occupied by the nuclei, and hence find the mass of the
neutrons in the internuclear region. The total pressure P -
Pe + Pn. The equation of stata in thus completely specified.
It may be noted from equation (8) that Z/A - 0 as Ee - 94
MeV. This indicates, since the total number of protons per
unit volume is fixed by the value of Ee, that ever increasing
numbers of neutrons are absorbed into the nuclei. If individual
nuclear charges become too small, such nuclei will fuse by
pycnonuclear reactions. Thus the fractional volume occupied
by the nuclei approaches unity az the total density approaches
4 X 1014 gm/cm3 and as En 7.9524 MeV.
This equation of state is obviously not a good approximation
near a density of 4 X 10 14 gm/cm3 and certainly cannot be used
beyond this density. However, this limit is quite sufficient
for the present investigation, which is intended only to bring
out the essential physical behavior of the equation of state.
A more refined investigation should take into account higher terms
A.	 in the nuclear symmetry energy, the rounded edge of the nucleus,
corrections to the surface energy when a neutron sea is present,
corrections to the coulomb energy due to the electrons, formation
i
6.
of a crystalline lattice, and nucelar potential energies of the
free neutrons.
With this equation of state the equations of hydrostatic
equilibrium were integrated to obtain stellar models. Figure 1
shows the relation between stellar mass and central density,
showing the well known dwarf mass peak and the region up to
a central density of 4 X 10 14 gm/cm3 . It may be seen
that there is no second mass peak in the -higher density region,
although the mass starts rising slowly above log p=13.5.
In the Tsuruta-Cameron procedure the neutronization of
nuclei was assumed to create new nonrelativistic particles
as the matter was compressed, thus assuring a large value of
the adiabatic index ri . Their secondary peak resulted from
these assumptions. In this improved procedure, the neut-
ronization process initially creates many nonrelativistic
neutrons, but these are progressively absorbed on nuclei as
compression occurs at the higher densities. Hence I
1
1 remains
lcs The values of I 1 obtained for the equation of state de-
scribed here are shown in Figure 2. It may be seen that
attains a maximum value of 1.24 in the neutronization regio..i at
log p- 12.8. To create a secondary meass peak, major portions
of a star would require 1 1 > 4/3, which is not possible here.
g
3
5
f
I
F
The work of J.M.C. was supported by an NAS-,N C post-
doctoral research associateship sponsored by the National
Aeronautics and Space Administration. The work also has been
supported in part by grants from the U.S. Atomic Energy
Commission, the National Science Foundation, and the National
Aeronautics and Space Administration.
REFERENCES
Cohen, J.M., Lapidus, A., and Cameron, A.G , W. 1969, "Pulsating
White Dwarfs Including General Relativistic Effects,"
to be published.
Drake, F.D. 1968, reported in New York Times, October 2, 1968.
Green, A.E.S. 1954, Phys. Rev., 95, 1006.
Harrison, B.K., Thorne, K.S., Wakano, M., and Wheeler, J.A.
1965, Gravitation Theory and Gravitational Collapse,
The University of Chicago Press, Chicago.
Tsuruta, S., and Cameron, A.G.W. 1966, Can. Jr . Prays., 44, 11-195.
Tsuruta, S., Wright, J.P., and Cameron, A.G.W. 1965, Nature,
206, 1137.
FIGURE CAPTIONS
Figure 1. Mass of a degenerate star plotted as a function of
central density, in the range below nuclear density.
Figure 2. Adiabatic Index r1 in the density range up to
nuclear density corresponding to the equation of
stale developed in this paper.
f
fi
F
i
i
i
FI.I
1.0
0.9
0.8
0.7
J 0.6O
Uy
0.5
0.4
0.3
7	 8	 9.	 10	 II	 12	 13	 14	 15
LOG DENSITY (GIVUCM 3 )
0k
1.5
	 ADIABATIC INDEX Ti
i
0.5
14	 158	 9	 10	 II	 12	 13
LOG DENSITY (GM./CM? )
1.0


The instability of stellar structures intermediate between white dwarfs and neutron stars

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