PULSARS AND COMPACT X-RAY SOURCES : COSMIC LABORATORIES FOR THE STUDY OF NEUTRON STARS AND HADRON MATTER

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PULSARS AND COMPACT X-RAY SOURCES : COSMIC LABORATORIES FOR THE STUDY OF

NEUTRON STARS AND HADRON MATTER

David Pines

To cite this version:

David Pines. PULSARS AND COMPACT X-RAY SOURCES : COSMIC LABORATORIES FOR

THE STUDY OF NEUTRON STARS AND HADRON MATTER. Journal de Physique Colloques,

1980, 41 (C2), pp.C2-111-C2-124. �10.1051/jphyscol:1980219�. �jpa-00219811�

PULSARS AND COMPACT X-RAY SOURCES : COSMIC LABORATORIES FOR THE STUDY OF NEUTRON STARS AND HADRON MATTER,

David Pines

Los Alamos Scientific Laboratory , Los Alamos, N.M. 87545 U.S.A. and Depth of Physics, University of Illinois at Urbana-Champaign, II. 61801, USA +

Abstract.- This paper reviews the dependence of the properties of neutron stars — their maximum mass, mass-radius relation, crustal extent and dynamic response — on models for hadron-hadron interaction and the likely existence of hadron super- fluidity. The extent to which it both has proved possible and might prove possible to determine these and other properties of neutron stars from observations on pul- sars and compact X-ray sources, and especially the pulsating X-ray sources, is summarized. The conclusion is reached that observations are capable of providing, within the next decade, definitive information on both the existence of hadron su- perfluidity and the viability of some of the current models for neutron-neutron interaction inside neutron stars.

I. Introductory remarks.- Some seven years ago, Ruderman, Shaham, and I surveyed the extent to wich it was possible to obtain in- formation concerning neutron star structure from astronomical observations. At that time, attention was focused on the origin of the glitches which had been observed in the Vela and Crab pulsars, and on their post-glitch behavior, which provides eviden- ce for hadron superfluidity in the liquid interior of these neutron stars. Durxng 2 the past seven years, the range of observa- tional possibilities for determining neutron star properties has expanded considerably.

Pulsating x-ray stars have been discovered and identified as accreting rotating magne- tic neutron stars in close binary systems. 3 X-ray observations of the Doppler shift of the period of these stars, when combined with optical observations of the companion

star, yield information on neutron star mas- 4

ses. Their spectra can provide informa- tion about the strength of the magnetic

5

field near the stellar surface while secu- lar changes in their period provide infor- mation on the neutron star magnetic moment, mass, and radius. Further valuable infor-

mation concerning neutron star structure may come as well from the irregular short- term period variations which have been ob- served in all those pulsating x-ray stars whose period variations have been studied in some detail.

To cite another example, the bursts cha- racteristic of many of the x-ray bursters are in all likelihood caused by thermonu- clear runaway processes in. matter freshly

P

accreted on the surface of neutron stars, while a minimum radius for neutron stars may be deduced from the shape of the x-ray spectrum in the tail of these bursts.

Observations which place upper limits on the surface temperature of neutron stars of known age are beginning to provide information about the states of matter and physical processes in the stellar interior which control the cooling of stars follo- wing their initial formation.

It should further be noted that additional glitches have been observed in the Crab and Vela pulsars, while a third, compa- ratively slow pulsar (PSR 1641-45) has exhibited a "giant" glitch only an order 12 of magnitude smaller than the four "super- Permanent address

JOURNAL D E PHYSIQUE Colloque Cl, supplément au n° 3, Tome 41, mars 1980, page C2-111

Résumé.- Pulsars et sources X compactes : laboratoires cosmiques pour l'étude des étoiles à neutrons et de la matière hadronique.

Cet article passe en revue les propriétés des étoiles à neutrons-masse maximum, relation masse-rayon, épaisseur de la croûte et réponse dynamique en fonction des modèles d'interaction hadron-hadron et de la probable superfluiditë hadronique.

On résume ce qui s'est avéré possible (ou pourrait l'être bientôt) pour connaître ces propriétés (ainsi que d'autres) à partir de l'observation des pulsars, des sources X compactes, et surtout des sources X puisantes. En conclusion il sembla que les observations pourront fournir, avant dix ans,.des renseignements détermi- nants à la fois sur la superfluiditë hadronique et sur les quelques modèles cou- rants d'interaction neutron-neutron dans les étoiles à neutrons.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980219

c2-112 JOURNAL DE PHYSIQUE

glitches" observed ifi the Vela pulsar. 11

from the so-called "soft" equations of sta- The maximum mass, mass-radius relation, te (derived from models for which the ave- crustal extent, and dynamic response of a rage system interaction energy is attracti- neutron star are sensitive to hadron inter- ve at nuclear densities) to "stiff" equa- action and hadron superfluidity in the den- tions of state (derived from models for sity region, lor4 g cm-3 lo1' cm-3, so which the average system interaction ener- that to the extent these properties can be gy becomes repulsive at sub-nuclear densi- deduced from observation, it is possible to ties). Representative examples of the cor- view pulsars and compact x-ray sources as responding models are the phenomenological cosmic laboratories for the study of hadron Reid (R) potential for neutron-neutron in- matter under conditions which are not acces- teraction,16 and the tensor interaction sible in the terrestrial laboratory. In (TI) model which assumes that the attracti- this talk I shall describe briefly the ran- ve part of the neutron-neutron interaction ge of theoretical possibilities for these comes from higher order pion exchange. 17 important properties of neutron stars, and The resulting interaction energies (as a then summarize the current output of these function of density) and mass vs. central

"cosmic laboratories". density curves calculated for these two mo- dels are shown in Fig.1, while cross-sec- Neutron Star Maximum Mass and Internal

Structure tions of corresponding stars of 1.4 M@ are

Extensive theoretical studies have shown shown in Fig.2.

that neutron stars with a mass " 1.4 Ma ha- As may be seen in Fig. la, for a stiff ve radii of the order of 10-16 km, with a equation of state, the interaction energy solid outer crust,

-

^{1-5 km} thick, contai- becomes repulsive at densities somewhat ning increasingly neutron-rich nuclei in a less than that of nuclear matter, and hen- periodic array, free electrons, and free ce acts to assist the neutron kinetic ener- neutrons, beneath which is a liquid inte- gy in opposing the attractive gravitational rior which begins at densities somewhat forces which act to collapse the star. The less than the density of nuclear matter, result, seen in Fig, Lb, is that the maxi- Po= 2.8 x 10 14 9 cm-3, and contains large- mum Pass of stars based on a stiff equation ly superfluid neutrons. The behavior of neu- ^State is greater than that Of stars ba- tron star matter in the crustal region is sed On a soft equation of state while, as comparatively well the remai- may be seen in Fig. 2 , stars calculated ning and key ingredient in determining the with a stiff equation of state have a lo- maximum mass and internal structure of neu- wer central density, a larger radius, and tron stars is the calculation of a reliable a considerably larger crustal volume, than equation

if

state for the neutron liquid do stars of the same mass calculated with phase. It is a difficult calculation, in a soft equation of state. Hence to the ex- part because the basic interaction between tent that such as the

neutrons is still not perfectly known, in stellar radius, and the ratio of the moment part because, for a given interaction model, Of inertia Of the ^Outer 'cr calculation of the ground state energy for

a system at nuclear densities and beyond is a far-from-trivial many-body problem. 14 Pandharipande

,

Pines, and smith15 have shown that one can construct a variety of models for the neutron interaction which are consistent with terrestrial constraints

(the free nucleon scattering data and the experimentally known energy, equilibrium density, and symmetry energy of nuclear mat- ter). The theoretical possibilities for the resulting equation of state span a range

Y I

-

-

z

^-20

-

-

-30

- -

_0.1 ^{I 1}

^.

_0-2 ^{I I} _0.3 ^{1 ,}

^-

n

( 2 I

Fiq. Ia : Interaction energy of neutron mat- Fig. Ib : Neutron star mass as a function ter as calculated with two models of central density for two models of the N-N interaction. of the N-N interaction. The ar-

rows indicate the maximum mass and central density.

Superflutd neutrons

Fig. 2 : Cross section of 1.4 M "Reid" and "TI" stars. For the Reid star, the presence of a possible pion con8ensate, at p 2 2p0, is illustrated; for the TI star, the boundaries between the outer and inner crust, and between the crust and liquid

core, are shown.

c2-114 JOURNAL DE PHYSIQUE

to the total inertial moment, I, can be di- rectly determined from observation, it is possible to reach conclusions concerning a matter of such fundamental importance in phy- sics as the basic interaction between ha- drons on the basis of astronomical observa- tion alone.

If the equation of state is comparatively stiff, the central density of a star as mas- sive as 1,4 Mg is less than twice nuclear matter density, while even the most massive stars possess central densities " 10" g

~ m - ~ . It is unlikely that one will find in the cores of such neutron stars some of the exotic forms of matter which have been con- jectured to exist there, such as quark mat- ter or abnormal matter.18 Indeed, if some- thing close to TI turns out to be the cor- rect interaction model, then neither pion condensation nor the formation of a neutron solid core, both of which have been sugges- ted to begin at densities of the order of twice nuclear density, 18' are likely to occur in stars of

-

^1.4 Mg, and may not oc- cur in nature at all if the process of neu- tron star formation tends to favor stars of this mass. On the otherf'hand, if the equa- tion of state is a good deal less stiff than say, the TI model, pion condensation may oc- cur and act to further soften the equation of state. Two possibilities of this osrt are considered by Baym and PethickI2O who find the resulting maximum stellar mass is

-

^1.5

Mg, corresponding to a central density

-

^{6 x}

1015 3 crnm3. It should further be noted that all present microscopic calculations of the equation of state of neutron matter lead to

<

stars whose maximum mass is

-

2 Me, well be- low the

-

⁵ limit obtained by Hartle 2 1 using only general continuity and causality arguments.

Hadron.Superf1uidity

The likelihood of hadron superfluidity in nu- clei was first considerated by Bohr, Mottel- son, and who applied the (then) new- ly developed microscopic theory of supercon- ductivity of Bardeen, Cooper, and Schrief- fer23 to nuclear matter and finite nuclei.

Because the fundamental interaction between hadrons is attractive at long dist'ances, the BCS pairing mechanism leads to the formation of a macroscopically occupied condensate in

hadron matter, with energy gaps (and tran- sition temperatures) which may be as large as several MeV. Hence at the comparatively low te~peratures

( 2

keV) expected for all but newlv formed neutron stars. one exoects conditions to be favorable for hadron su- perf luidity

.

At least three distinct hadron superfluids are expected inside neutron stars : (i) In the inner part of the crust (correspondinq to densities 4.3 x I0 11 g ~ r n - ~

2

p 5 2 x lo1 g cm-3 )

,

the free neutrons, which coe- xist there with the neutron-rich nuclei, likely form a

'so

paired superfluid. Becau- se the star is rotating, the neutron super- fluid will not be spatially uniform; rather it will contain an array of vortices, pa- rallel to the axis of rotation of the star, each having quantized circulation,gS'y

.

^dl

24 "

= h/2m, where m is the neutron mass.

The cores of the vortices, where the con- densate wave function decreases to zero, may be either pinned to the crustal nuclei or may thread the spaces between them. F7he- ther or not such pinning occurs in some portions of the crust depends on whether.it is energetically favorable for the "normal"

core region in the neutron superfluid to coincide with the crustal ions; where the coh4rence length in the neutron superfluid is domparable to the size of the nuclei, conditions are favorable for pinning. 2 5

~ e c i n t calculations26 suggest that pinning will take place throughout much of the in- ner crust. (ii) In the quantum liquid re- gime ( p ? 2 x

o1

g cmm3 )

,

where the crus- tal nucleihave dissolved into free neu-

trons and protons, the neutron suverfluid is likelv in a 3 ~ 2 paired state," con- taininq a vortex array.

(iii) The protons in the quantum liauid interior are expected to be superconduc- tins. asain in conseauence of the strons attractive hadron-hadron interaction.

Thev co-rotate with the electrons present there (since anv differential rotation would produce extremelv strona maanetic

fields and hence cost a areat deal of energy), and both protons and electrons may be expected to co-rotate with the nu- clei and electrons in the crust, since the strong magnetic field inside the star

C2-115 is tied to the charged particles in both the within which the accreting plasma co-rota- crust and the interior. ^{2 8} tes with the star. A qualitative picture Hadron superfluidity has a number of possi- of disk accretion Onto a magnetic neutron ble observational cons6quences for pulsars star is shown in Fig. 3. with neutron star and pulsating x-ray sources. If both the

neutrons and protons in the liquid interior are superfluid, the coupling between the normal outer crust and the interior super- fluid neutron liquid is so weak that macros- copic times (-. days to years) may be requi- red for the two parts of the star to come to

equilibr~. Such a dynamic response has F i g - 3 ^I s i d e v i e r o f d i s k a s c r e -

been observed following sudden spinups of t i o n by a r o t a t i n g m a g n e t i c neu- t r o n s t a r ( a f t e x G h o s h and ~ a m b d ) . the Crab and Vela pulsars and provides evi- _{Beyond r} , the stellar magnetic field is dence for hadron superfluidity in these pul- compl6tefy screened, and disk flow is

unperturbed by the magnetosphere, In sars. It may also be observable in the the transition region between r, and rA, power spectrum of frequency variationsin@- the disk flow changes into magnetosphe-

ric flow. In the outer transition zone, sating x-ray sources a question to which I from rs to ro, viscous stresses dominate shall return shortly. Identification of glit- magnetic stresses; in the boundary layer,

6, magnetic stresses dominate, ches or noise processes originating in the

magnetic moments in the expected range of vorticity jumps of the pinned neutron super- 3

lo2'- gauss cm

,

and characteristic

fluid would provide evidence both for the _-1

x-ray luminosities

-

1 0 ~ ~ - 1 0 ~ ~ ergs s

,

pinning mechanism and the presence of neutror.

the Alfv6n surface typically is found at a superfluid in the inner crust. Another pos- 8

distance, RA,

-

¹⁰ cm, large compared to sibility is the direct observation of a Tka-

the neutron star radius, R. As a result, chenkoZ9 mode, a collective shear mode of the

matter which is threaded onto stellar vortex lattice in the interior neutron super-

field near SA will, in the case of a dipo- fluid. The period of such a mode with a wa-

velength comparable to R is

-

^{2R(km) (P (s) ) l f i} lar field, be strongly channeled by the field toward the magnetic poles of the months, so that Tkachenko modes with a period

star, where it forms hot spots of area of the order of months can, in principle, be 2

TR (R/RA)

-

1 km near which the gravita- excited by the accretion torque acting on a

tional energy of the infalling matter (pro- pulsating x-ray source.

tons colliding with the stellar surface ar- Pulsating X-Ray Stars

rive with an energy per particle of order If a rotating magnetic neutroh star is ac-

-

100 MeV) is released.30 The resultinq creting matter from a nearby companion star,

radiation emerqes from the neiqhborhood of it may be visible as a pulsating x-ray star.

the stellar surface in a stronqly aniso- Far from the star its magnetic field will

tropic angular pattern which depends on not influence appreciably the flow of accre-

the details of the accretion process; for ting matter, since it is screened out by cur-

an oblique rotator (a star for which the rents induced in the infalling material. If

magnetic field is not aligned with the a- the matter possesses sufficient angular mo-

xis of rotation), one has a natural mecha- mentum, it cannot fall directly toward the

nism for the product on of pulsed radia- star, but rather will tend to form a disk,

tion with a temperat re in excess of 6 made up of particles gradually spiraling to-

keV. 30,3 1 ward the star on Kepler-like orbits. Suffi-

t

Unlike the pulsar, s ch sources will, on ciently close to the star, the stellar magne- average tend to spin-up, as a result of the

i

tic field is strong enough to force the ac-

torque transmitted to the star by the creting matter to co-rotate with the star at

accreting matter. 30,31 its angular velocity, a. The transition bet-

Observational parameters for nine pulsatinq ween the two regions takes place near the

x-ray stars are given in Table I, where Alfv6n surface, SA, defined as the surface

C2-116 JOURNAL DE PHYSIQUE

comparison is made as well with the results of all sources except Vela X-1, which is of a quantitative investigation of disk ac- possibly a wind-fed sources, lie within an cretion by rotating magnetic neutron stars order of magnitude of one another.

which has been carried out by Ghosh and

showed that a power spectrum analysis of Lamb,6r32 who use two-dimensional hydrody-

the observed angular velocity variations namical equations to study the interaction

would not only test the validity of the sta- between the stellar magnetic field and the

tistical description, but could provide in- accreting plasma at the magnetospheric boun-

formation as well on the relative inertial dary. The fact that they are able to obtain

moments of the crust and the superfluid neu- quantitative agreement between theory and

tron core, thecrust-.core coupling time, and observation for a "standard" neutron star,

the frequencies of any low-lying collective a 1.3 Me star whose radius and inertial mo-

modes. In the absence of internal modes with ment are taken to be those calculated by

a frequency in the range covered by a power Pandharipande, Pines, and smith1 using a

saectrum analvsies of the fluctuations in

"stiff" (TI) equation of state, and a nar-

the an~ular velocity of the crust, one ex- row range of magnetic moments (0.5

;

^p30

<

pects to find a shoulder in the power spec-

^-

s), for all sources except Vela X-1, stron-

trum a t w % I/T, when T is a characteristic gly suggests that these stars, despite their

relaxation time. At low frequencies (UT <<

widely varying periods and luminosities,

1) the star responds to any torque varia- are all magnetic neutron stars accreting

tion like a rigid body of inertial moment, from Keplerian disks. The large magnetic mo-

I, since the crust shares any change in its ment calculated for Vela X- 1 suggests that

angular momentum with the superfluid neu- it might be a wind-fed source, in which ca-

trons in a time T which is short compared se one would obtain the observed spin-up

6 to the time scale for changes in the torque;

rate with p30 in the above range.

on the other hand, at high frequencies (UT

Valuable information about neutron star

> > 1). the crust has no time to share its ,

.

structure may also come from the irregular

anaular momentum with the core, and will short-term ( - days to months) period varia-

tions which have been observed in the four pulsating x-ray sources (Her X-1, Cen X-3, Vela X- 1, and X Per) whose period varia- tions have been studied in some detail. 3 3 Lamb, Pines, and shaham7 suggested that such variations could arise from fluctua- tions in the externa torque on the stellar crust, associated wi h large or small scale variations in the ac retion flow, or from internal torque vari

1

tions, associated with oscillations of the fluid core or the unpin- ning of vortices in the inner crust of the star (see below). They developed a statis-

ti-cal description of the .torque variations in-terms of noise processes, and

t L37 is the luminosity, in erg s-I, divi- ded by

% For each source, Ghosh and Lamb adjust the stellar magnetic moment, measured in units of gauss cm 3

,

to bbtain the best pos- sible agreement between the observed value of Ts and their theoretical value for a 1.3 Mg star whose radius and inertial moment are taken to be those obtained by Pandhari- pande, Pines, and smith&' using a "stiff"

(TI) equation of state. The magnetic moments

-

instead respond like a rigid body of moment of inertia, Ic, with the consequence that the fluctuation level is reduced by a fac-

tor of (Ic/I) 2

.

Neutron Star Properties from Astronomical Observations

The extent to which it has proved possible to determine properties of neutron stars by combining theory with observations on pulsars and compact X-ray sources is sumna- rized in Tables 1-3, Some brief comments follow.

(i) Mass determinations

The best determined neutron star mass from X-ray observations is that of Her X-1, for 'which Middleditch and Nelson obtain (1.3

0.1) Mg from an analysis of optical pulsa- tions.%/~he masses of the other four stars ofor which mass determinations have been ma- de are not known as well (MCen X-3 = 1.9

t

1.2 MB; MSMC X - 1 = 1.1 _{0.6 1%;} MVela X-1

= 1.7 f 0.3 Mg; .E14U1538-52 = 1.9 2 ^{1.1 M@),} /primarily because a 10% uncertainty in the difficult-to-measure orbital velocity

(1) 6 / From Ghosh and Lamb

-

S o u r c e SMC X- 1 Her X-1 4 U OII5+63 Cen X-3 A 0535+26 GX 1+4 Vela X-1 GX 301-2 X-Per

Table 2, Observing neutron stars : Properties which are sensitive to the hadron equation of state Table 1 . Observational and theoretic41 parameters of nine pulsating x-ray sources

P (s) 0.71

1.24 3.6 4.84 104 121 283 700 836

+

Calculated property for a 1.4 Mo star

td See Pandharipande, Pines, and Smieh,E1 for further details Property

Mmax'M@

R (km)

Ic/I

Reference oblateness,

E

,

and critical strain angle, Oc/u or minimum angular velocity at time of crust formation, nmin. Pf

0

45 2 I = (1/10 )g cm )

45

Interval, At between starquakes due to pulsar spindown

**

5 0 1 20.9

5 6 4 0.1 0.3 4x10-~

Observation

T (yes-rs) s

(1.3f0.4) x lo3 (3.320.6) X 1 0

%3.1 lo4 (3.6+0.6) x 1 0 3 29 f 8

5 0 f 13

(1.0+0.4) lo4 120 f 6 0

(5.9+1,.5) x 1 0 3

Interaction Model

R TI

Current Status

(a1-/aK

^(ro)

0.11 0.35 0.30 0.29 9.7 3.4 0.35

6.7 x

LO-^

0. LO

zz

'30 0.50 0.47 1.4 4.5 3.3 0.93 8 6 0.3 4.8

1.4

2

^Mmax/Mo

5

3 Minimum radius, R . % 9 f 2

mLn

(Ic/I) > 112 if Id S T

<

l o 2 ^d

Origin of 3Sd clock not yet determined

0.2

2

^{I~~ 9 . 2}

*!it

(At)Crab % SY X-ray Doppler shift +

optical observations Black body fit to burster spectrum No structure in Her X-1 Q(T) noise

Stellar wobble as origir:

of 3 5 d periodicity of Her X-l

ro ⁽¹⁰ 8 c") 0.36

1.1 2.0 2.4 1.9 I. 1 4 0 1.2 3 8

1.4 5 0.2 1.8 2 ^0.2

9.8* 16*

0.04" 3.56%

E~ ,= - 0 5 E " ^-001

0 .

a :

n

= 4600

fir-

³²⁰

or or

oC/p % .05 oC/p %

1

Crab pulsar luminosity, 0 . 9 ~ 2*

38 3 9

L ~ , (10 S L ~ S I O )erg s-' Successive macroglitches in fast pulsar P(t)

(~t):~~%10 ² y (At)Erab%5y

JOURNAL DE PHYSIQUE

T&le 3. Observing neutron stars : Other properties

*

In arrivina at a maximum surface temperature, I have assumed black bodv radiation, taken R c 15 km, and neglectedanv aeneral relativistic corrections

r Property

B ~ u r f ace

Magnetic moment, p

T ~ u r f ace

**

Results obtained by R. Harnden and collaborators, using the Einstein observatory, as reported by D o Q. L 45/ e

curves can lead to an uncertainty in the 1.4 Me star (a result which is roughly con- neutron star mass of order 1 Mg. There e- sistent with the calculations of ~ o s s y f o r xists the possibility that all these neu- a helium-burning flash in the surface la- tron stars have approximately the same mass yers of an accreting neutron star) and that

(c 1.4 M ~ ) ; in this connection it is worth X-ray bursters are located symmetrically noting that Taylor et a ~ . ~ ' have shown that about the galactic center it a distance of if the binary pulsar consists of two neutron r~ 9 kpc, he finds the radius of the emit- stars (and definitive evidence is Still lac- tinq reaion to be 8.5

+

^{1.5 kg.'/} This is king on this point), then both have masses likely an underestimate of the radius for

1 - 3 9 (0.15)M0. neutron stars of this mass, since strong

ii) Neutron star radii from X-ray bursters ma9necic fieldst if Present, might cons- T~~ shape of the X-ray spectrum in the train the emission region to be less than tail of many observed X-ray bursts can be the entire surface, while the recent calcu- fit to

first

approximation by a blackbody lations of Swank et al.2' suggest that the spectrum

with

^a decreasing temperature, a influence of electron scattering on the ra- result which suggests that one is observing diative opacity of the outer surface layers radiation from a cobling surface of cons- of the star could lead to an underestimate tant

size,w

Van Paradijs

2/

has analyzed of the radius of the emitting region by as SAS-3 observations of X-ray bursts from .ten much as a two-

sources, and finds a striking uniformity in

their properties. He concludes that if the iii) Crustal extent from the interval bet- peak luminosity in each burst is a standard ween successive macroglitches in the candle, the size of the cooling surface is Crab pulsar, assuming these originate approximately the same for these burst sour- in pulsar spindown.

ces; with the further assumption~~that the Information on stellar structure mav peak luminosity corresponds to the Eddington also be provided bv the crustquake explana- limit, LE = 1 . 2 5 ~ 1 0 ~ ~ (M/M@) erg s-'t for a tion for the macroqlitches observed in the

Observation Spectral features in pulsed fraction of X radiation

Spin-up for stars accreting from Keplerian disks

Pulsar luminosities

Upper limit on flux of non-pulsed soft X-rays from radio pulsars*

Current Status 4x1012 5 X-1 2 6x10'~ G B ~ O 1 15-63 % 2x1012 G

0.1

2

p 3o

<

^{10 2}

Best fit for 1.3 M TI star

0

is

u30

c 0.5 0.026 5

u30

2 4.5

<

TCrab 2.5~10~ O x

< H E *

TVela

-

^{1.2~10~ OK}

Crab pulsar. According to starquake theory 37r38! as the pulsar slows down, strain energy builds up in the crust until the cri- tical strain angle, oc/p, is reached, at which point one has a crustquake, whidh

acts to relieve strain by reducing I by an amount A1 (hence a sudden spinup, or macro- glitch). The time to the next crustquake is then simply the time for pulsar spindown to replenish the strain released in the pre-

3 8/

vious quake, and is- 2 2 t (AE) = T(w /Q ) (AE)

q 9

where A E = (AI/I)

,

T(5 Q/Q) is the slowing- down time, and

w2 _q = ( ~ A ~ / B I )

.

The latter parameter is a rapidly varying function of stellar mass and the hadron e- quation of state through its dependence on B, the elastic energy content of the star

(which is itself proportional to the crus- tal extent), and A, the gravitational ener- gy stored in the star. Assuming the Crab

X-1, assuming its mass 1.3 Me, to obtain the reference oblateness of Her X-1 and, from that, to infer either the critical strain angle or a minimum value, Q p , of the angular velocity of the star at crust formation. To the extent, therefore, that comparatively large amplitude stellar wob-

4 1/

ble may exist in pulsating X-ray sources- and any observed long term periodicities may be identified as stellar wobble (an unambiguous identification has not as yet

proved possible for Her X-l), valuable in- formation about the structure and/or the evolutionary history of a neutron star may be obtained.

V) Neutron star structure and dynamics from an analysis of period variations

Boynton and ~ e e t e r G ~ h a v e used the Uhuru data on Her X-1 to compute the power spec- trum of angular velocity variations. They were able to establish that the period wan- dering observed for this source was due to torque fluctuations, as suggested by Lamb, Pines, and shahad/, and that these fluctu- pulsar be a 0 star leads to a ations are describable as red Q-' frequency prediction of a 5 year interval between the noise over a wide raJsge of frequencies. On observed 1969 and 1975 spinups, in accord the hand, Boynton and Deeter did not with observation, while if it were a 1.3 Ma ,iaent shoulder in the power spec-

that have been loo trum, from which they conclude that either the

Y.- 15/ crust-superfluid coupling time in Her X-1

iv) Stellar wobble

Because neutron stars are oblate and possess a solid outer crust, they may exe- cute a free precession mode analogous to the Chandler wobble of the earth. The wobble frequency, QW, of a neutron star reflects both its structure and its evolutionary his- tory; it is related to the rotation frequen- cy by

3 9 '

B and A are the same coefficients which ap- pear in Eq. (2), while cot the reference oblateness, is determined by the angular velocity,

no,

at which the star solidifies and its subsequent seismic history. For li- quid core neutron stars, B << A, while for solid core neutron stars, B % A. As Pandha- ripande, Pines, and smith%/ have emphasi- zed,

if

stellar wobble is the origin of the observed 35 periodicity of Her X-1,S1one d can use calculated values of A and B for Her

is shorter than ld {a likely possibility on theoretical grounds), or is longer than lood, or that Is, the moment of inertia of- the superfluid neutrons in the liquid inte- rior, is less than or comparable to Ic, the crustal moment of inertia (an equally like- ly possibility, since Is % Ic for a 1.4 Mo TI star).

vi) Is the hadron equation of state soft or stiff ?

I conclude from Table 2 that current observational evidence, while far from con- clusive, tends to favor neutron interaction models for which the calculated interaction energy becomes repulsive at sub-nuclear densities, For the TI model the calculated properties of neutron stars lie comfortably within the suggested ".observational" range while stars calculated using a soft hadron equation of state possess properties which are at, or near, one limit of the range ob-

c2- 120 JOURNAL DE PHYSIQUE

tained from an analysis of the current obser- tron star cooling-/ is such that no de- vations. It is to be hoped that through fur- finitive conclusions can yet be drawn from ther analysis of existing data (such as that these observations concerning the states of in progress on the twelve pointed observa- matter (i.e., pion condensates) and physi- tions of Vela X-1 made with HEAO I by Boyn- cal processes in the interior of these stars ton, Lamb, Pravdo, and Wood), better theore-

tical calculations of these and other proper- Hadron Superfluidity ties, and further observations, the question

may be decidable in the near future. Evidence for hadron superfluidity in neu- vii) Measuring neutron star magnetic fields tron stars has come from observations of

The spectra of pulsating X-ray sources the dynamic behavior of the Crab and Vela can provide information about the strength pulsars following sudden spinup,-/ Their of the magnetic field near the stellar sur- post-glitch behavior is rougfily consistent face, whereas measurements of secular period with the two-component model of Baym et changes provide information about the dipole al.z/involving a crust; of moment of inertia component of the surface magnetic field. The Ic, coupled to the superflhid neutrons, of observation by Triimper et al.?' of a feature moment of inertia Is, average angular fre- in the pulsed high energy spectrum of Her X- quency Rs, responding to a torque N(t), with

4 and by Wheaton et

slag/

of a comparable the crust superfluid neutron interaction feature in the pulsed spectrum of 4U0115-63 charaterized by a counling time, T ~ ; has been interpreted as indicating the exis-

tence of surface fields of some 4-6x10 12

IcAc = N(t)

-

^{I c W c}

-

^{Rs)/Tc (4)}

gauss in Her X-1 -(depending on whether the

feature is ascribed to cyclotron emission of IsRs = Ic(Rc

-

^{Rs)/~c (5)}

absorption), and % 2 x 1 0 ~ ~ gauss in 4U0115-63.

These values may be compared with the fit After a sudden initial jump, (AR)o in a, which Ghosh and ~ambg~obtain for the obser- the post-glitch behavior described by (4) ved spin-up torques for these pulsating sour- and (5) is

ces, p30 % 0.5 for Her X-1, and p30 a 1.4 for

400115-63. Taken together, these results rug- R(t) = Do (t)

+

(An) bevt/'+ (1-Q)] (6) gest that there are important non-dipolar ma-

gnetic fields in these neutron stars. where Qo(t) is the extrapolated frequency

~t is also worth remarking that the range of in the absence of the glitch, magnetic moments obtained by Ghosh and Lamb

from the torque on pulsating X-ray stars Q = (Is/I) (~-(AQ

/ ( a h ) -

^{I&, and T =}

(0.1

<

^p&, lo2) is comparable to that obtained

C s

⁰¹

by assuming that the pulsar luminosity is gi- ^TC (Is/I)

.

ven by p2R4/c2;%/ the latter calculation Eight macroglitches [(AQ/Q

2

^{lo-'] have}

yields mabnetic moments which range from p30 thus far been observed in pulsars, In each ,

% .026, for the binary pulsar, to p30 % 4.5, of the six examples for which a detailed for the Crab pulsar. analysis of post-glitch behavior has been viii) Neutron star surface temperatures carried out, it has proved possible to fit

~ h u s far no observations of steady black the data with the post-glitch function, body emission from a neutron star have been Equation (6); the results of such fits are made, although many attempts have been made given in Table 4, As was noted seven years to detect such radiation from the Crab pul- ago,L1 a key test of the simple two-compo- sar and upper limits on this emission have nent model is whether or not all post- recently been obtained for a number of other glitch functions in a given pulsar have candidate neutron stars by investigators u- the same Q and T; an examination of Table sing HEAO 2, the "Einstein1' Observatory (cf. 4 shows this is not quite the case. This

45/ is scarcely surprising, given the diffi- Table 3). As has recently been emphasized,--

our current theoretical understanding of neu- culties of data analysis (frequency noise

introduces ambiguity into the seperation bet ween the post-glitch function and noise fluc tuations in the reduction of the data-{, 47 and the over-simplifications inherent in the two-component model (the crust-core coupling is characterized by a single relaxation time ,and no effects of pinned vorticity are taken into account). A conservative interpretation of the observational data would be that re- sults thus far strongly suggest the existen- ce of at least two distinct components of the star which require macroscopic times to

come to equilibrium with each other, while the time scales involved are not unexpected for crust-superfluid coupling.

Anderson, Shaham, and I have been continuing our study of the effects of pinned vorticity with particular attention to the question of whether this can provide an explanation of both the origin of the Vela and PSR 1641-45

"superglitches" and of the variations in observed in the Vela pos-glitch behavior. We conclude that giant vorticity jumps,

TABLE 4. Observations of Glitches and Post-Glitch Behavior for Three Pulsars {I)

"'(nn/o) is the relative iumo in freouencv at the time of the glitch; 0 and T are the two component parameters introduced in Ea.

(6). b ~ h e references to the observations are as follows : ( 1 ) V. Radhakrishnan and R. N.

Manchester, Nature

222,

228 (1969) ; P. E.

Reichley and G. S. Downs, Nature

222,

²²⁹

(1969). (2) P. E. Reichley and G. S. Downs, Nature Phys. Sci.

234,

48 (1971). (3) R. N.

Manchester, P. A. Hamilton, and W. M. Goss, Nature

259,

291 (1976). (4) G. S. Downs, IAU

Tel. Circ. 3274 (1978). (5) P.E. Boynton et al., Ap. J.

175,

217 (1972). (6) E. Lbtsen, Nature

258,

688 (1975). (7) R.N. Manchestel L. N. Newton, W. M. Goss, and P. A. Hamil- ton,

&.

^Not.

^5.

^Astron.

^{e. 184,}

^{35 17}

(1978).

Date 8/69 8/7 1 10/75

7/78 9/69 10/72 2/75

"39/77

resulting from thesudden unpinning o f a substantial concentration of vortices in those regions of the star in which the pinning force is maximum, of-fer a (bQ/Ql0

%2.3x10-~

%2x10-~

s2x104 s3 x 1

o - ~

2 x l 0 - ~ 3.7x10-~

1.9x10-~

Pulsar Vela Vela Vela vela Crab Crab Crab 1641-45

Reference 1 2 3 4 5 6 6 7

Q

~0.15

?

"30.22

?

"30.93 s0.96 '~0.96

?

T

"3450 d

?

"3450 d

?

"34.1 d 15 d 15 d 85 Y

c2- 122 JOURNAL DE PHYSIQUE

promisFng explanation for both the magni- tude and frequency of the superglitches;

a progress report on our work will be given by Shaham at this meeting.

The quite different values of Q ( = IS/I, in the simple two-component model) observed for the Crab and Vela pulsars suggest at first sight that these neutron stars have rather different masses, and that neither pulsar has the structure anticipated for a 1.4 Ma TI star (Is/I s 1/2). An alternative, and perhaps more plausible possibility, is that all pulsars have comparatively similar va- lues of Is/I, and that the variations in Q inferred from post-glitch behavior reflect different glitch origins and/or a more com- plex pdst-glitch behavior. A further indi- cation that this might be the case comes from the recent analysis by Boynton and Dee- ter of the Princeton timing data on the Crab pulsars'. They find that the behavior of the frequency noise is similar to that of Her X-1; the spectrum is "red" Cl-2noise over a considerable frequency range, and does not exhibit a prominent s oulder. They conclude that either the crust superfluid coupling time is shorter than

1

d or longer than 100 d

(a result in apparent/ contradiction with the 4-15 d relaxation times inferred from the post-glitch behavior of the Crab pulsar) or that Is 2 I, (a result compatible with a 1.4 M TI star, but likewise in contradiction

0

with the simplified two-component model fit --

to the post-glitch behavior). The results of Boynton and Deeter show the coupling between the crust and the neutron superfluid is considerably more complex than is descri- bed by the two-component model; it deserves more careful study theoretically, while observations of the noisy behavior of the pulsating X-ray sources offer a particularly promising possibility for further investiga-

4 8 ting superfluid behavior.- Conclud4ng Remarks

We have learned a great deal about neutron stars from the discovery and.study of pulsa- ting and bursting X-ray sources in the 1970

Is, and from observations of further pulsar macroglitches. It is possible now to design investigations and develop theory with thq goal of achieving a fundamental understanding

~f neutron stars. Further detailed timing

only offer considerable promise of provi- ding valuable information about the proper- ties and internal structure of neutron stars, but also the fascinating possibility of reaching a definitive conclusion concer- ning both the existence of hadron superflui- dity and the validity of some of the cur- rent models for hadron-hadron interaction in neutron stars. Thus the use of compact X-ray sources as a cosmic laboratory for the study of hadron matter under extreme conditions is a very real possibility for the 1980's; let us hope that the flight of a satellite devoted to temporal and broad- band spectrosco~ic studies of the X-ray

starsbefore the middle of the next decade will permit the scientific community to

turn that possibility into a reality.

Acknowledqments

This paper is a revised and expanded ver- sion of an invited paper presented at a Workshop on Compact Galactic X-Ray Sources, held in Washington, LC., in April, 1979.

(D. Pines, in Compact Galactic X-Ray Sour-

=,

^F. K. Lamb and D. Pines, e d ~ , p. 126 (Physics Dept, UIUC, Urbana, IL., 1979). I thank my colleagues there for many useful conversations, the National Science Foun- dation for its support, and the Aspen Cen- ter for physics and the Los Alamos Scien- tific Laboratory for their hospitality du- ring the preparation of this manuscript.

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