Table Of ContentRef. p. 91 56.1 Neutron stars 1
5 Special types of stars
5.1...5.5 in Subvolume b; see table of contents.
5.6 Compact objects
5.6.1 Neutron stars
5.6.1.1 General properties
The concept of neutron stars as theoretically possible stable structures was introduced by Landau [l]. Baade
and Zwicky [2] suggested that such stars might be formed in supernova explosions, and Oppenheimer and Volkoff
[3] calculated the first neutron star model using a Fermi gas of non-interacting neutrons.
Neutron star models are computed by integrating the general-relativistic equation of hydrostatic equilibrium
which was derived by Tolbert, Oppenheimer, and Volkoff (T.O.V. equation)
dP ~G(@+P/cz)(~+4~r3P/cz)
--
(1)
dr r’(1 - 2GYJ$k2) .
Here e(r) is the rest mass plus potential and kinetic energy density (Q= mn + E/C’ with m = rest mass of neutron,
IZ= number density of neutrons), and G the constant of gravity. %X(r)i s the mass within radius r, and P(r) denotes
the pressure. The radius R of the star is defined as usual by P(R) = 0.
A schematic illustration of the structure of neutron stars computed from (1) is shown in Fig. 1. Typical radii are
10.. .20 km, masses z 1 W,, and central densities exceed the density of nuclear matter, e. = 2.8. 1014 g cmW3.
Outer crust (nuclei.electronsl
Fig. 1. Cross-section through a representative neutron star
with a mass of 1.3 9X,.
r = radius, Q= density, Q,= central density.
Grewing
2 5.6.1 Neutron stars [Ref. p. 9
In the outermost regions. the structure of a neutron star is expected to be the same as in the interior of white dwarf
stars: 56Fe nuclei surrounded by a sea of degenerate electrons and a density of about 104g cmw3 at the surface.
The iron nuclei. because of their mutual electrostatic repulsion, are expected to form a crystalline lattice (as soon
as the temperature has dropped below about 10” K), thereby giving the neutron star a solid outer crust.
Beneath this crust the density increases.c ausing an increased electron Fermi energy which in turn results in electron
capture by nuclei. leading to neutron-rich nuclei. These form the inner crust, another crystalline lattice, and are
embedded in a sea of degenerate electrons. Still further inward, the further density increase causes the number
of free neutrons to rise rapidly until. at about 2. lOI g cm- 3, the nuclei completely dissolve into a neutron sea.
The neutrons are likely to be superfluid. At densities higher than 2. lOI g cm- 3 the properties of matter are still
not completely clear but it is expected that a sea of electrons, superfluid neutrons and superconducting protons
exists. If the core density is even higher, as some models suggest, new particles such as muons and hyperons will
bc created.
Whereas this coarse description of the structure of neutron stars applies to most model calculations. there are
important differences in detail bctwecn the various computations. These result from the differences in the equations
of state used which relate the pressure P(r) to the density e(r) in the different density regimes. In the low density
regime the equation of state is well understood, and the principal progress has been a better description of the
propcrtics of matter near the surface in the strong magnetic fields that prevail (see below). At densities near go
recent work has brought out the sensitivity of the stellar radius and crust thickness to microscopic details of the
equation of state in this region. At the highest densities. the recent discussion has concentrated e.g. on the pion
condensation and quark matter (seee .g. [4]).
The properties of matter at high densities and their implication for the structure of neutron stars have recentl)
been reviewed. e.g. by Canuto [S], and Baym and Pethick [6,4]. These authors have also compiled extensive
lists of references to the original literature.
5.6.1.2 Results from model calculations
5.6.1.2.1 Masses, radii and moments of inertia
Theoretically determined masses. radii and moments of inertia depend on the choice of the equation of state.
As some uncertainty still remains about the correct form of the equation of state at the very high densities that
occur in neutron stars, we give here the results of Arnett and Bowers [7] who calculated the structure of slowI)
rotating neutron stars for a variety of equations of state listed in Table 1, p. 3.
Eleven of these models (models A...M) employ a non-rclativisticmany-body formalism in constructing the equation
of state. Two models (N and 0) are based on a relativistic description of the interactions and a consistent (but
simplified) relativistic many-body theory. With two exceptions (models G and M), all of the equations of state
represent normal systems. i.e. Fermi fluids. Models H and L represent a very soft and a very stiff equation ofstate.
respectively. Some of the models can hardly bc considered as realistic, e.g. model H; they have, however, been
included as a reference. At densities below about lOI g cm- 3 the equation of state is believed to be much better
known. and in general for these densities the equations of state as given by Baym, Pethick, and Sutherland [18]
and by Baym. Bethe. and Pcthick [19] have been used. Model F represents a reasonable alternative to this, and
ir has been included to test the sensitivity of models near mass peak on the low-density region. For further details.
see [7].
Grewing
Table 1. Equations of state, considered by Arnett and Bowers [7],
Model Ref. Interactions Many-body theory Density range Composition*)
Q in [g cme3]
A Pandharipande 8 Reid soft core - adapted Variational principle applied to
to nuclear matter correlation function
B Pandharipande 9 Same as A; arbitrary Same as A
reduction for hyperon-
hyperon attraction
C Bethe-Johnson 10 Modified Reid soft core Constrained variational principle 1.71 .10’4...3.23.1016 n, p, A, C ’ so,A ’ so, A+ ’
D Bethe-Johnson 10 Same as C; more Same 1.7.1014...2.26.1016 Same
realistic adaptation to
hyperon matter
E Moszkowski 11 Reid soft core; modified Reaction matrix n,p, Z-,A”,A-
hyperon interactions based
on quark theory
F Arponen 12 Thomas Fermi model Brueckner G-matrix n, P, em, P’
G Canuto-Chitre 13 Modified Reid soft core. T-matrix; includes spin dependence n (solid)
8
4 Localization via non-
5 relativistic harmonic
w
oscillators
H Ideal neutron gas 3 None Fermi statistics Entire range n
I Cameron-Cohen- 14 Levinger-Simmons Hartree-Fock approximation with 1.O.lOi4~~~5.35.10’5 n, P, em, P*
Langer-Rosen velocity-dependent two-body potential
L Pandharipande 15 Nuclear attraction due to Mean-field approximation for scalar; ~>4.386.10” n
and Smith scalar exchange variational method
M Pandharipande 15 Nuclear attraction due to Constrained variational method e>8.428.1013 n
and Smith pion-exchange tensor
interactions
N Walecka 16 Relativistic mean-field scalar Mean-field approximation (relativistic) n
plus vector exchange fitted
to nuclear matter
0 Bowers, Gleeson, 17 Non-perturbative, phenom- Relativistic finite-density Green’s
and Pedigo enological approximation functions
to relativistic meson
exchange
*) n, p: nucleons; e-, u+ : leptons; A, C, E, A: hyperons.
56.1 Neutron stars [Ref. p. 9
Figs. 2;1..c. Rcprcscntativc equations of state for cold new
tron stars. In la. the equation of SI;IIC for a t’rcc gas of neu-
trons is sho\vn by the dashed curw for comparison. The
region contained in the rcctanylar box in 2a is shoun
cnlarpzd in 2b and 2c. In 2b. results arc sho\vn for models
a A...]. in Zc rcwlls for models L...O (from [7]. SW also
Table I). P in [dyn cm-‘]. 0 in [g cm-‘].
5 10 15
loge -
36.5
36.0
I
4
F
35.5
Ref. p. 91 5.6.1 Neutron stars 5
Figs. 2a.. .c illustrate the equations of state used in the calculations by Arnett and Bowers [7]. Figs. 3a, b give the
resulting neutron star masses as a function of the stars’ central densities Q,. The equations of state based on non-
relativistic interactions (A...G) produce maximum masses 1.36<%R< 1.85 mO, and have central densities in the
range 3.0~1015~~~6.3~10’gs cme3. Models L, N, 0, the last two of which are relativistic, lead to neutron stars
with 2.39~!&,,,~2.7O%R,, having central densities in the range 1.4~10’5~~~2~101g5 cmm3. With the exception
of models A, G, and M, there is a marked tendency for the value of W,,, to increase and move to lower central
densities as the equation of state becomes stiffer.
1
II
2.c
1.E
I
g
k
1.C
0.5
b
c II I II II I II II
'lk 15 16 15 16
log ec - blec-
Figs. 3a, b. The gravitational mass of neutron stars !lJ&
versus their central density Q, for models A through I (3a),
and models L through 0 (3b). For comparison model B is
repeated in both figures (from [7], see also Table 1).
e, in [g cmm3]
In Fig. 4 the gravitational mass-radius relation is shown for a total of nine different equations of state. From
these results it is evident that the range of radii is fairly limited and lies for most models in the range 8...20 km.
Fig. 5 shows the resulting moment of inertia as a function of the central density for all the models that have been
considered by Arnett and Bowers [7]. For central densities below about 1015 g cme3 there is a strong dependence
of the calculated moment of inertia on the central density which in turn depends on the stiffness of the equation
of state. As there is some hope of determining the moment of inertia of neutron stars from observations (see
below) this should reveal the behaviour of matter at very high densities.
Grewing
6 5.6.1 Neutron stars [Ref. p. 9
2.7:
2x
2.2:
2.K
1.75
1.5C
i
&
:- 1.25
6
1.03
0.7:
053
i
0.25 \
I I I I III I pl I I I
0
8 IO 20 LO 60 80 3 14 15.0 15.5 16.0
log R - 4 ec -
Fig. 1. The gravitational mass-radius rclntion for sclccted Fig. 5. Tbc moment of inertia I versns central density Q,
equations of state (from 171. xc also Table I). R in [km] for all models calculated by Arnctt and Bowers ([7]. see
also Table I). I in [g cm’]. Q, in [g cme3]
5.6.1.2.2 Maximum-mass considerations
Due to the uncertainties that still remain in the equation of state that applies at the very high densities that
occur in neutron stars. it is not yet clear what maximum mass stable neutron stars can have. A number ofattempts
have. however. been made to derive such a limit. Rhoades and Rufini [20] applied an extremal principle in
selecting an equation of state that products a maximum critical mass in the framework of Einstein’s theory of
general relativity. le Chatelier’s principle and the principle of causality. They find a value !IJl,,,,,=3.2 9J31,.V ery
similar limits were found in two more recent calculations [21,22]. However, under extreme conditions. these
limits can be pushed to roughly 8 %Jnl,a s shown in [22], or even 11 1111a, s shown in [23].
5.6.1.2.3 Magnetic fields
Magnetic fields are observed on main-sequence stars. The interior of such stars can be safely assumed to remain
in a highly conducting state until the ultimate collapse of the core. Ruderman and Sutherland [24] suggest that
turbulent convection in the core of presupernova stars results in the field building up to an equipartition value
of 2 3. lo9 Gauss. Conservation of flux during the collapse to a neutron star then results in a surface field strength
of ~4.10’~ Gauss. The field probably has significant multipole structure, but in the outer magnetosphere it is
usually assumed to be bipolar. Probably, thcrc is also a huge torodial field interior to the neutron star. The mag-
nitude of this is unknown but is probably at least as strong as the exterior polodial lield [25].
Several scientists have raised the question of magnetic field decay in neutron stars. The only region within such
stars where the field could decay due to ohmic dissipation on a reasonably short time scale is the crust (e.g. [26.27]).
Grewing
Ref. p. 91 5.6.1 Neutron stars 7
5.6.1.2.4 Cooling calculations for neutron stars
The significantly increased sensitivity of recent X-ray experiments and the anticipated sensitivities of future
X-ray experiments have greatly stimulated new calculations of the cooling of young neutron stars. Recent com-
putations are given in [28...31]. References to earlier papers are given by these authors.
The most recent calculation by Nomoto and Tsuruta [31] is based on an exact stellar evolution code, incorporating
the full general relativistic version of the stellar structure equations with the best physical input currently available.
This approach takes into account the effect of the finite time scale of thermal conduction which has been neglected
in most of the earlier computations.
In Fig. 6 both the surface temperature and the photon luminosity are given as a function of age for a neutron
star with YJIo= 1.3 mm, according to calculations by Nomoto and Tsuruta [31].
37
36
6.6
6.4 35
1
6.2 I
34 ',
I--
$ 6.0 z
33
5.8
32
5.6
-3 -2 -1 0 1 2 3 4 5 6 73'
log/-
Fig. 6. The surface temperature T and the photon luminosity L,, from a representative neutron star model calculation
(Wo= 1.3 !JJI,) after Nomoto and Tsuruta [31].
Standard cooling curves are shown for four cases: Case S with superfluid nucleons and case N with non-superfluid (normal)
nucleons both for non-magnetized neutron stars (zero magnetic field), and cases MS (superfluid) and MN (non-superfluid)
for the magnetic field H = 4.4.10” Gauss. For details of the equation of state used, see [31]. t in [a], T in [K], L,, in [erg s-‘1
5.6.1.3 Observational results
5.6.1.3.1 Masses, radii and moments of inertia
Optical and X-ray observations of binary X-ray sources provide the possibility of determining the masses
of neutron stars (see also 6.1.4). For a recent review, see Bahcall[33] who refers extensively to the original literature.
The mass determination involves the derivation of either an optical or an X-ray mass function, or both. This
function is defined by
F(%$, W,, i)=iUlT sin3i/(%R, +YJIJ2, (1)
where %Rm%,,I & are the masses of the two stars in binary orbit and i is the inclination of the plane of the orbit to
the plane of the sky (see also 6.1.3.1). The mass function for either star is determined directly from observations
by measuring either the size of the star’s orbit or its radial velocity. From Newton’s laws for two mass points
moving in elliptical orbits under the influence of their mutual gravitational attraction one finds that
F('YJl,,W ,, i)=(4n/GP2)(a2 sini) (2)
and
F(!JJll, %R2,i) =(P/2nG) (1 -e2)312 (u2 sini) (3)
with a2 =semi-major axis of the orbit of star 2, P=orbital period, e=orbit eccentricity, u2 sini=amplitude of
the orbital velocity curve, and G = constant of gravity.
Grewing
8 56.1 Neutron stars [Ref. p. 9
For pulsating X-ray sources one can detcrminc the X-ray mass function Fx from the ampiitude of the observed
delays of the pulse arrival time. n2 sin i/c, and the measured orbital period. From Eq. (I) we then have
9J1,= F,q( 1 + q)‘/sin’ i , (4)
where 4 is the ratio of the mass of the X-ray star. 9131xto, the mass of the optical star. !lJI,,,. In order to obtain the
individual masses of the components. also the optical mass function must be known. For a careful discussion of
the possible sources of error that enter into such an analysis, set Bahcall [33]:
Table 2. Masses of neutron stars in X-ray binaries [33].
Source !u1, c’9Jlo] Comments
3u 0900-40 l.O~!LIIx~3.4 Successful test of method of analysis.
SMC X-l O.S59J?x=( 1.8 X-ray heating and possible extra light source
complicate system.
Cen X-3 0.15 9J31xs4.4 Optical velocity amplitude predicted to be
15...80 km s-‘.
Her X-l 0.459J31xs2.2 Measurements required during X-ray “off period of
optical velocity amplitude and ellipsoidal light
variations. The optical velocity amplitude predicted
to bc SO...130 km s-‘.
3u 1700-37 0.6 5 9J1, Assumed mass of the optical primary exceeds 10 ‘9JI,.
These results are basically in agrccmcnt with the model predictions discussed in 5.6.1.2.1. However, the range
of uncertainties is still too large to favour any of the particular neutron star models discussed in 5.6.1.2.1.
Observations of X-ray burst sources can in principle bc used to independently constrain the mass-radius relation
for neutron stars as was pointed out by van Paradijs [34].
Information on moments of inertia of neutron stars may be obtained from observations of the secular rates of
change of their spin periods. A comparison of the slow-down rate of the Crab pulsar with the luminosity of the
Crab nebula yields a lower bound on its moment of inertia of > 1.5.1044g cm* [25]. Observed speed-ups of
putsatins X-ray sources. that indicate a time scale of z 102...105 years, combined with model descriptions of
accretion torques, indicate moments of inertia which are consistent with this result as well as with the model
predictions [35...37].
5.6.1.3.2 Magnetic fields
Surface magnetic fields of neutron stars in active radiopulsars and binary X-ray sources are inferred from
the rates of energy loss (set 5.6.2) and from the structure of the radiation and the spin-up rates (see 5.6.3), respcc-
tively. In either case the results are consistent with fields in the order of 10” ...10’3 Gauss; the values depend.
however. on a number of model assumptions which enter into the analysis.
A totally independent determination can be derived from the observation (Triimper et al. [38]) of an emission
(or absorption) feature in the X-ray spectrum of Her X-l at z 58 keV (~42 keV) if this is interpreted as cyclotron
emission (absorption) by electrons in the strong magnetic field of the neutron star. The field strength turns out
to be ~6.10’~ Gauss (~4.10’~ Gauss), respectively for Her X-l.
5.6.1.3.3 Surface temperatures
In Fig. 7. we have compiled the presently available results for a total of 11 objects.
Grewing
5.6.1.4 References for 5.6.1 9
logf-
Fig. 7. Nine upper limits and two possible measurements
of surface temperatures T of neutron stars (and correspond-
ing photon luminosities Lph) as function of their age. For
comparison the theoretical cooling curves [31] discussed
in 5.6.1.2.4 have been repeated (see Fig. 6).
Identification of the sources:
1) Cas A [39] (5) SW 1006 [42] (9) G 350.0-18 [40]
(2) Kepler [40] (6) RCW 103 [43] (10) G 22.7-0.2 1401
(3) Tycho [40] (7) RCW 86 [40] (11) Vela [44]
(4) Crab [41] (8) W28 [40]
t in [a], T in [K], Lph in [erg s-r]
5.6.1.4 Referencesf or 5.6.1
1 Landau, L.: Phys. Z. Sowjetunion I (1932)2 85.
2 Baade,W ., Zwicky, F.: Proc. Nat. Acad. Sci. 20 (1934)2 54.
3 Oppenheimer,J .R., Volkoff, G.M.: Phys. Rev. 55 (1939)3 74.
4 Baym, G., Pethick, C.J.: Annu. Rev. Astron. Astrophys. 17 (1979)4 15.
5 Canuto, V.: Annu. Rev. Astron. Astrophys. 12 (1974)1 67.
6 Baym, G., Pethick, C.J.: Annu. Rev. Nucl. Sci. 25 (1975)2 7.
7 Arnett, W.D., Bowers,R .L.: Astrophys. J. Suppl. 33 (1977)4 11.
8 Pandharipande,V .: Nucl. Phys. Al74 (1971)6 41.
9 Pandharipande,V .: Nucl. Phys. Al78 (1971)1 23.
10 Bethe,H .A., Johnson,M .: Nucl. Phys. A230 (1974)1 .
11 Moszkowski, S.: Phys. Rev. D9 (1974)1 613.
12 Arponen, J.: Nucl. Phys. Al91 (1972)2 57.
13 Canuto, V., Chitre, S.M.: Phys. Rev. D9 (1974)1 587.
14 Cohen, J.M., Langer, W.D., Rosen,L .C., Cameron,A .G.W.: Astrophys. SpaceS ci. 6 (1970)2 28
15 Pandharipande,V ., Smith, R.A.: Nucl. Phys. Al75 (1975)2 25.
16 Walecka,J .D.: Ann. Phys.8 3 (1974)4 91.
17 Bowers,R .L., Gleeson,A .M., Pedigo,R .D.: Phys.R ev. D12 (1975)3 043.
18 Baym, G., Pethick, C., Sutherland,P .: Astrophys. J. 170 (1972)2 99.
19 Baym, G., Bethe,H .A., Pethick, C.: Nucl. Phys. A175 (1971)2 25.
20 Rhoades,C .E., Ruffini, R.: Phys. Rev. Lett. 32 (1974)3 24.
21 Chitre, D.M., Hartle, J.B.: Astrophys. J. 207 (1976)5 92.
22 Durgapal, M.C., Rawat, P.S.: Mon. Not. R. Astron. Sot. 192 (1980)6 59.
23 Hegyi, D.J.: Astrophys. J. 217 (1977)2 44.
24 Ruderman,M .A., Sutherland,P .G.: Nature Phys. Sci. 246 (1973)9 3.
25 Ruderman,M .A.: Annu. Rev. Astron. Astrophys. 10 (1972) 427.
26 Heintzmann, H., Grewing, M.: Z. Physik 250 (1971)2 54.
27 Ewart, G.M., Guyer, R.A., Greenstein,G .: Astrophys. J. 202 (1975)2 38.
Grewing
10 5.6.2 Radiopulsars [Ref. p. 22
28 Tsuruta. S.: Phys. Rep. 56 (1979) 273.
29 Glen. G.. Sutherland. P.G.: Astrophys. J. 239 (1980) 671.
30 Riper. K.A. van. Lamb. D.Q.: Astrophys. J. (Lett.) 244 (1981) L13.
31 Nomoto. K.. Tsuruta. S.: NASA TM 82145 (1981).
32 Pandharipnnde. V.R.. Pines. D., Smith, R.A.: Astrophys. J. 208 (1976) 550.
33 Bnhcall. J.N.: Annu. Rev. Astron. Astrophys. 16 (1978) 241.
34 Paradijs. J. van: Astrophys. J. 2.34( 1979) 609.
35 Elsner. R.F.. Lamb. F.K.: Nature 262 (1976) 356.
36 Rappnport, S.. Joss. P.: Nature 266 (1977) 123.
37 Ghosh. P.. Lamb. F.K.: Astrophys. J. 232 (1979) 259.
38 Triimpcr. J.. Pictsch. W.. Reppin. C.. Vogcs. W., Staubcrt, R., Kendziorra. E.: Astrophys. J.(Lett.) 219(1978)
L105.
39 Murray. S.S.. Fabbiano. G., Fabian. AC.. Epstein. A., Giacconi. R.: Astrophys. J. (Lett.) 234 (1979) L69.
40 Hclfand. D.J.. Chanan. G.A.. Novick. R.: Nature 283 (1980) 337.
41 Toor. A.. Seward. F.J.: Astrophys. J. 216 (1977) 560.
42 Pyc. J.P.. Pounds. K.A.. Rolf. D.P.. Seward, F.D., Smith. A.. Willingate. R.: Mon. Not. R. Astron. Sot. 194
(1981) 569.
43 Tuohy, 1.. Garmirc. G.: Astrophys. J. (Lett.) 239 (1980) L107.
44 Harndcn. F.R.. jr.. Hertz. P., Gorcnstein. P., Grindlay. J.. Schrcier. E., Seward. F.: Bull. American Astron.
sot. 11 (1979) 424.
5.6.2 Radiopulsars
5.6.2.1 General properties
Discnvcrcd by Hcwish et al. [l]. thcsc objects show the following basic characteristics:
broad-band radio noise in the form of a periodic scquencc of pulses
hi_rhly variable pulse intcnsitics
a pulse duration that is typically 3 % of the pulse period P
individual pulses that often consist of two or more subpulscs with a duration of typically 1...2!% of the period
within the subpulses some pulsars show microstructure
the radio frequency spectrum of pulsars is fairly steep with a power law index of typically - 1.5
only very few pulsars hnvc been detcctcd outside the radio band (e.g. the Crab and the Vela pulsar)
in all cases where accurate observations have been made. the pulsar periods arc found to increase in a regular
way with time. The typical rate of change is lo- ” s s- ’
with only two exceptions all known radiopulsars are single neutron stars without a companion.
For a dctniled description of thcsc and other characteristics and a compilation of further references. see [2.3]
5.6.2.2 Surveys
Following the detection of the first pulsar in 1967 [ 11.a nd successful pulsar searches by many different groups.
the vast majority ofall known pulsars was first dctcctcd in one offivc major surveys: the first and second Molonglo
survey (Large and Vaughan [4] and rcfcrcnccs thcrcin: 36 pulsars in total. 31 new: and Manchester et al. [S]:
27-l total. 155 new): the Jodrcll Bank survey (Davies et al. [6.7]: 51 total. 31 new); the University of Massachusetts
Arccibo survey (Hulsc and Taylor [S. 91: 50 total, 40 new); and the University of Massachusetts NRA0 survq
(Damnshck et al. [lo]: 43 total. 17 new). Except for the Molonglo surveys which cover most of the sky south of
6= + 20.. all the other senrchcs concentrated on regions in the vicinity of the galactic plane. The original Molonglo
survey and the Jodrcll Bank survey had reduced sensitivity for pulsars with short periods (SO.3 s) and!or high
dispersion (n:Iiz 150cmm3 pc, see below). Thcsc limits have been pushed considerably further in the more recent
surveys, the Arccibo survey e.g. maintaining its full sensitivity up to DM= 1280 cmm3 pc. and it was less sensitive
by only about 3 dB for periods as short as 0.06 s.
Chewing
