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Neutron Starquake Mode} for Gamma-Ray Bursts
R. D. BLANDFORD
Han~ard-Smithsonian Center for Astrophysics
and
California Institute of Technology
ABSTRACT
A neutron starquake model for gamma-ray bursts is presented and
critically analyzed. It is suggested that a slowly accreting neutron star may
develop density inversions deep in its crust. These unstable layers, may
be subject to elastic Rayleigh-~ylor instability which can liberate sufficient
gravitational, and perhaps also nuclear, energy to account for individual
bursts. Energy can be transported to the surface by shear waves and slowly
transmitted into the magnetosphere as relativistic Alfven waves. Particle
acceleration and Gray emission from the outer magnetosphere should
ensue. Some observational implications are mentioned.
INTRODUCTION
This is a report on an interpretation of Gray bursts, a phenomenon
studied intensively by space scientists in the United States and the Soviet
Union, as neutron starquakes. The work that I shall report on is collabo-
rative with Omer Blaes, Peter Goldreich, Steve Kooky and Piero Madau.
The basic model was, of course one of the first suggested (e.g., Pacini and
Ruderman 1974; 1§ygan 1975; Fabian, Icke et al. 1976; Muslimov and lLy-
gan 1985; Epstein 1988) after the discovery of Gray bursts by Klebesdal et
al. (1973). Our approach has been to investigate the different components
of the starquake model independently. Only an outline of the model, which
is still only partially complete, can be presented here; [tiller accounts are
given in Blaes e! al. (1989a, 1989b in preparation, l989c in preparation).
In addition, space limitations preclude adequate reference to the extensive
28
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HIGH-ENER~ ASTROPHYSICS
29
literature on this topic. The articles by Bisnovatyi-Kogan and Chechetkm
(l979), Epstein (1988), Lamb (1988), and the book edited by Liang and
Petrosian (1986) are good stardug points.
GAMMA RAY BURSTS FROM OLD NEUTRON STARS
As explained in much greater detail by Hurley in these proceedings, ~y-
ray bursts are observed about every four days producing emission extending
well above lMeV for roughly 1-10s. On average, the total fluences are
~ 10-6 _ 10-4 erg cm-2. Excepting the "soft repeaters," there is a
deficit of X-rays relative to Grays; typically, only a few percent of the
energy is emitted below 10keV. This imposes a serious constraint on the
models. Much circumstantial evidence has been adduced in favor of a local
neutron star origin ("rotational" modulation, "cyclotron" lines, "electron-
positron annih;Hation" lines, "association" of GB790305 with a supernova
remnant and millisecond temporal structure). None of this is compelling;
the interpolation of the first three items can be questioned and the fourth
could arise under more exotic conditions than neutron starquakes. The
observed isotropy of the sources and the source counts (Schmidt, these
proceedings) points to either a local or a cosmological origin.
Using the expected distribution of old pulsars, we find that, in round
numbers, the typical burster would have to be a ~ 10~° yr old neutron
star some ~ 300pc distant producing a ~ 1037 erg burst of Grays every
lOOpyr. The integrated ,-ray energy radiated over the neutron star lifetime
is therefore ~ -1044 erg.
STRUCTURE OF YOUNG NEUTRON STAR CRUSTS
Traditionally, it has been supposed that the structure of the neutron
star crust has the composition computed in a classic paper by Baym et al.
(1971) (or a close variant thereof if we use an improved semi-empincal mass
formula for the nuclear boding energies). In this work, it was assumed
that the composition of the crust would comprise the lowest energy state at
the imposed pressure taking into account the electronic, nuclear and lattice
energies and ignoring any thermal contnbutions. The crust was found to be
made of layers of (mostly) magic nuclei, that became increasingly neutron
rich with depth. It was therefore implicitly assumed that the nuclei would
be able to exchange nucleons freely in order to attain this lowest energy
state. Under normal conditions, nucleon exchange amongst high Z nuclei
requires thermonuclear reactions at a temperature of > 4 x 109K (e.g.,
BisnovaWi-Kogan and Chechetkin 1979, 1986; Thielemann 1989) when the
thermal energy will almost surely result in a mix of nuclei. At lower
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30
AMERICAN AND SOVIET PERSPECTIVES
temperatures, the nuclear composition is frozen, rather like the helium
composition after the epoch of nucleosynthesis in the early universe.
There is a further complication. In the supernova explosion that forms
the neutron star, a significant quantity of mass will be ejected with speed
just less than the escape velocity and it will subsequently fall back onto the
neutron star surface. It may take over a year for the last ~ 10-6Me of
mass to reach the surface (e.g., Chevalier 1989) and, by this time, the star
will have cooled to temperatures well below that at which thermonuclear
reactions can occur. The weight of the infalling material can change the
pressure in the original crust significantly, and even if the original crust
had the composition appropriate to cold catalysed equilibrium, it win, ~
general, be out of equilibrium at the new pressure. The crustal composition
clearly depends subtly on the detailed history of the star.
SLOW ACCRETION AND CRUSTAL LOADING
Isolated, old, cold neutron stars moving through the interstellar med-
ium Secrete interstellar gas at a mean rate M ~ 10~0g s~i (e.g., Ostriker
e' al. 1970~. This can only keep the surface temperature at ~ 3 x 105 K
The surface area of a neutron star is A ~ 10~3 con and the surface granter
is ~ 10~4 cm s-2. The original crust can therefore be compressed over
a lifetime t ~ 10~° yr to a pressure p ~ Mgt/A~ 3 x 1028 dyne cm~2,
or equivalently a density p ~ 3 x 10~°g cm23 and an electron chemical
potential (essentially the Fermi energy) pe ~ 10MeV (e.g., Shapiro and
Teukolsky 1983~.
As the (predominantly) hydrogen gas is compressed it will eventually
be able to undergo cold or pycnonuclear reactions to form helium (e.g.,
Salpeter and Van Horn 196~, Shapiro and Teukolsly 1983~. This helium
can undergo a pycnonuclear "triple ~x" reaction to form carbon which in
turn may undergo fusion to oxygen, neon and magnesium. The outcome
is uncertain, but at some point the Coulomb barrier will inhibit fusion
and the nuclei will be compressed by the crustal loading. This behavior
is quite different from what happens at higher mass accretion rates where
the accreting gas is heated by compression faster than it can be cooled by
electron conduction and the hydrogen and helium burns to helium which
in turn burns to carbon and iron either steadily or in flashes as in X-ray
bursters (e.g., Ayasli and Joss 1982~.
When the electron Fermi energy in the compressed lattice becomes
sufflaently large, electron capture will occur and the neutron fraction wH1
increase. This may have to occur via an excited state and so there may
be some heating associated with this process (e.g., Haensel and Zdunik
1989~. However, electron captures can only occur singly as the time scales
for double electron capture are excessively long. Now, even-even nuclei
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Representative terms from entire chapter:
gamma ray
HIGH-ENERGY ASTROPHYSICS
31
are more tightly bound than odd-odd nuclei and the electron Fermi-energy
must be raised to typically ~ lOMeV before an even-even nucleus can
capture an electron. However, as soon as a single capture occurs, there will
invariably be a second capture to leave the nuclei in an even-even state.
For example, at a pressure of p ~ 7 x 1026 dyne cm~2 and a density p ~ 1.5
x 109 dyne cm~3, 56Fe will undergo electron capture to form 56MU which
will rapidly capture a second electron to form 56Cr, with an overall energy
release of 2.1MeV, of which ~ 1.2MeV will be camed off in neutrinos.
The remaining ~ O.9MeV per nucleus is released too slowly to produce
significant heating of the interior.
Under these conditions, it is only possible to achieve restricted nuclear
equilibrium in the crust. This leads to the possibility of an important
instability. Suppose that there is an interface between two layers in which
the nuclei have different atomic weights. For example suppose that a layer
of 56Fe rests on 62Ni. Now suppose that the crust is loaded and compressed
essentially isothermally so that only electron capture can occur. When the
pressure reaches p ~ 7 x 1026 dyne cm~2, the atomic number of the 56Fe
will decrease by two to form 56Cr, increasing the density discontinuous by
~ per cent. However, the 62Ni must be compressed to p ~ 2.1 x 1027 dyne
c~-2 before it can be converted to 62Fe. Mere ~ therefore be a the
when 56Fe is separated from the 62Ni by a thick layer of 56Cr. The 56Cr is
5 per cent denser than the 62Ni. There is therefore a source of mechanical
free energy available if the 56Cr and 62Ni layers can be interchanged. The
development of few per cent density inversions when layers of Me crust are
compressed at constant atomic weight is quite general
ELASTIC RAYLEIGH-TAYLOR INSTABILITY AND STARQUA1
32
AMERICAN AND SOVIET PERSPECTIVES
shear modulus to the bunk modulus. In a bee lattice, of iron this ratio is
0.014 (Baym and Pines 1971) and a density jump of about six per cent is
necessary for linear instability. Some additional complications change these
estimates slightly, but the general conclusion is that, if electron capture-
induced density inversions are formed, then they are only marginally stable,
and we expect that relatively small perturbations from isostapy, such as will
surely be present in a real crust, are sufficient to induce instability. The
growth time for the instability is ~ (p/g2bp)~/2 ~ lops. This is much shorter
than any time scale observed in gamma ray bursts.
The gravitational energr released in a local overturn of about a scale
height of crust at a density of ~ 109 - 10~°g cm~2 is simply computed from
the formula
/`E =-~ J pdp
mp9
where ~ is the total chemical potential. This works out to be about SkeV
per baryon or ~ 1038 erg, adequate to account for a single gamma ray burst
provided that Me efficiency of conversion to gamma rays is high It must
be emphasized that in this model only a of order a cubic scale height of the
crust can overturn in each starquake. In this respect, the starquake model
is similar to earthquakes and dissimilar to the highly successful model of
X-ray bursts In which it is necessary that a burning front cover most of the
stellar surface.
An energy release of ~ SkeV per baIyon, is equivalent to a tem-
perature of several billion degrees, well above the melting temperature.
This introduces the possibility that the crust will become hot enough to
allow thermonuclear reactions to occur and to release additional energy as
nuclear rearrangements are catalyzed by free protons, neutrons and alpha
particles. Up to ~ 30keV per baryon can be released in this manner. It is
not yet clear what conditions are necessary for this to occur.
The proposed model for a starquake releases a lot of energy in the
form of heat. However, unlike in an X-ray burst, the heat is released deep
below the surface, at a depth z > 300m and should take a substantial
time, typically several hours to reach the surface and to be radiated as soft
X-rays.
SEISMIC WAVES
There is however, a much faster way for the energy released to reach
the surface and this is seismically. There are two types of high frequency
waves that can propagate through the crust, pressure waves and shear
waves. As the shear modulus is roughly one percent of the bunk modulus,
the shear speed is about ten per cent of the sound speed. This has the
HIGH-ENERGY ASTROPHYSICS
loo
10 '
EM
A
I_
_4
~4 10-
o
Cal
10<
i 10
o
_.
U3
~ TO'
On
3
10"
10 ~
10~6
33
i
1~
~ /
. . · . ~ . · · . . . . .
104
FREQUENCY (Hz)
los
FIGURE 1 Transmission coefficient as a function of frequency for a vertically propagating
shear wave. The solid lines refer to numerical calculations, while the dashed lines show
WKB solutions. Ibe upper and lower pairs of curves are for 10~2G and lOliG magnetic
field.
consequence that far more power (proportional to the Averse fifth power
of the wave speed in the quadrupolar approximation) from a starquake is
channeled into shear modes and it is only necessary to consider these.
If we consider shear waves in the WE approximation, then in the part
of the crust supported by the degeneracy pressure of relativistic electrons,
a vertically propagating wave's wavelength will decrease or Zi/2, wee its
horizontal displacement will increase or Z-7/4. However, the amplitude
cannot increase indefinitely. When the wavelength becomes comparable
with me pressure scale height, ~ z/4, the wave will be reflected with
high efficiency back towards the stellar core. Refraction in the inner
crust will return the wave back towards the surface. It is very difficult to
estimate the damping rate (probably dominated by stress-induced motion
of dislocations), but if terrestrial measurements are a guide (e.g. Minster
1980), quality factors of several thousand are not out of the question. It
should be emphasized that if most of the energy produced in the explosion
34
AMERICAN AND SOVIET PERSPECTIVES
is dissipated in heating the crust rather than is earned to the surface
seismically then the present model is not viable.
MAGNETO SPHERIC ALFVEN WAVES
If the neutron star is endowed with a substantial surface magnetic field,
then it is possible for there to be an appreciable transmission of energy into
the magnetosphere. The magnetosphere will be magnetically-dominated
and will support two types of hydromagnetic waves, fast magnetosonic
modes, which are similar to vacuum electromagnetic waves, and relativistic
Aliven modes (e.g. Melrose 1980) in which field parallel conduction currents
compete with displacement current. The phase velocity of an Alfven mode
propagating at an angle c' to the field is c cos car, which can be much less
than the speed of the light and can be comparable with the shear speed
in the crust for a wave propagating nearly perpendicular to the field. If
we treat the surface layers as a plane discontinuity, then standard notions
of impedance matching suggest that there will be a significant transmission
coefficient. More careful calculations, that take into account the magnetic
contn~utions to the shear stress in the crust, verily this and, for example,
give transmission coefficients of a few per cent for 10kEIz waves and a 10~2G
field. This has the attractive consequence that energy can be stored as waves
propagating around the crust for several hundred vertical propagation time
scales, typically of order a second, and comparable with the length of the
bursts.
Relativistic Alden waves in the magnetosphere will also change as
they propagate away from the star. In particular, the magnetic amplitude
B/B will increase or B-~/2. They are likely to become non-linear at ~ 5 -
10 stellar radii and may create large parallel electric fields. If this is the
location of Me ~y-ray emission then only a small fraction of the ,-rays would
be re-radiated from the stellar surface as X-rays as me observations seem
to require.
EMISSION OF GAMMA RAYS
We now turn to the component of the whole problem where there
is the greatest prospect for confronting present and future observations,
and where I have least to suggest! ~ have identified a region some 5-10
stellar radii from the surface as Me emission site and the location of large,
parallel electric fields, capable of accelerating stray electrons and positrons
to radiation reaction-limited energies ~ 10TeV. The magnetosphere will
be relatively starved of charge-carIying particles, but there is the strong
possibility of creating fresh electron-positron pairs, primarily through two
photon production. These pairs can themselves be re-accelerated and
HIGH-ENERGY ASTROPHYSICS
.'
.2
Z -. 6
-.8
-1.0
_4
-1.2
-1.4 _
-1.6
-1.8
-2.0
35
1 1 ' ' ' '
-
(a)
. . . .
-20 -18 -16 -14 -12 -10 ~
HEIGHT (x 103 cm)
-2 ~2 ~
FIGURE 2 The displacement amplitude as a function of height above the surface for a
vertically propagating lOkHz shear wave in a lOl1G magnetic field. The evanescent zone is
identified by the pair of vertical lines.
produce further generations of particles until the electric fields are locally
shorted out. The details are unclear.
Two comments can be made, however. Firstly, it is not jUSt necessary
tO create electrons and positrons; ,-ray photons are also necessary (e.g.,
Zdziarski and Lamb 1988~. One way in which they may be multiplied is
through the following cycle. An incident soft photon Is Compton scattered
by a relativistic electron moving along the field in its ground state of gy-
rational motion and Hereby creating a ,-ray. The electron then recoils
Into an excited gyrational state, from which it will quickly de-excite by
radiating Doppler-shifted cyclotron radiation which can then be scattered
and repeat the cycle. Photon yields of Y ~ mec2/hcoG, where WG iS the
gyro frequency are possible and so a runaway growth of soft photons only
requires mere to be a Thomson depth of ~ Y~i, typically ~ 10-4. Unfor-
tunately, preliminary calculations of the inverse Compton and relativistic
electron synchrotron radiation emitted imply a spectrum that is steeper
than reported. More realistic modeling should be pursued.
36
AMERICAN AND SOVIET PERSPECTIVES
The second point is that it is necessary to account for the "cyclotron"
lines in detail. A good start has been made by Loredo et al. (1989,
preprint) who point out that the relative equivalent widths of the two lines
observed in both GBS70303 and GB880205 can be accounted for in terms
of resonance scattering. However, the atmosphere that is postulated is
neither radiatively nor dynamically self-consistent. In addition there is no
clear reason why a narrow cyclotron feature should be produced when the
field strength will vary by a factor ~ 2 over the surface. In the present
model, it might be possible to attribute the X-ray continuum to heating of
the surface consequent to the starquake and localized to the site of the
quake. Again, further study is necessary.
OBSERVATIONAL IMPLICATIONS
In this report, I have outlined some studies of a particular model for
gamma ray bursts involving neutron starquakes. The model, as described,
is fairly fragile especially on energetic and demographic grounds. In order
to account for the observed frequencies and fluences of observed bursts, it
is necessary to tap most of the available nuclear energy from the accreted
interstellar gas and to observe most of the "dead" neutron stars within a
Galactic scale height every thousand years.
There are alternative possible components to the model which may
be substituted. For example, the neutron star cores may contain magnetic
fields of far greater strength than are measured on the surface and these
may be subject to instability. The discussions of seismic and hydromagnetic
waves are not seriously affected by this change. Alternatively, it may be that
the radiative conditions on the surface really do allow the "X-ray paucity
constraint" to be satisfied and that most of the ,-ray emission originates
here. This would still allow the energy to derive from the sort of deep
starquake described above. Another possibility is that a minority of neutron
stars accrete at a substantially greater rate than the interstellar rate and
yet do not produce X-ray bursts; alternatively some pulsars may be able
to store substantial nuclear fuel from their formation and still be able to
detonate small pieces of it some ~ 10~° yr later.
It is therefore quite hard to test the details of the present model ob-
servationally. Nevertheless, it is possible and indeed is far more important
to test the most general features of neutron starquake models. In view
of the reported anisotropy of bursts, only nearby stars can be involved.
They must therefore repeat Existing constraints cannot yet rule out a
neutron star origin (Hartmann 1989 preprint; Pac~ynsH 1989 preprint),
but it is anticipated that either Sequent repetition or anisotropy should be
measurable soon. (If the frequency of bursts reflects current rather than
integrated accretion then a much stronger anisotropy is needed.) Should
HIGH-ENERGY ASTROPHYSICS
37
neither effect become apparent then a cosmological population of exotic
objects or a major revision of our ideas on the early stellar evolution of our
Galaxy would probably be indicated. The detection of more "cyclotron" and
"redshifted electron-positron annihilation" lines would clearly be a boon.
The discovery of rotational modulation of the foyer would be compelling
evidence for neutron stars. The discovery of an ultra-violet afterglow from
the position of a burst, perhaps using Hubble Space Telescope would also
constitute valuable evidence for neutron stars.
This model, in common with several others, does require there to be a
strong surface magnetic field, typically > 10~iG. This raises the perplexing
question of the evolution of neutron star magnetic fields, a subject which
recent observations have made highly confusing. On the one hand, it
appears that radio pulsar torques decay in a few million years (e.g., Lyne
and Manchester 1988) and become very small in old, millisecond pulsars.
In addition models of low mass X-ray binaries generally require them to be
weakly magnetized. On the other hand, if old neutron star really do possess
fields of strength ~ 2 x 10~2G, as the "cyclotron" lines suggest, then
we have to understand the reason for these different evolutions. Perhaps
a resolution can be found in alignment (e.g. Candy and Blair 1986) or
the erasure of low mulitpoles in favor of higher multipoles, (e.g. Flowers
and Ruderman 1976~. A clearer understanding of radio and X-ray pulsar
magnetic fields would have immediate implications for the study of gamma
ray bursts.
We eagerly await the launch and successful deployment of HST, GRO,
GRANAT and SPEKTRUM-y which should provide answers to some of
these pressing questions.
ACKNOWLEDGEMENTS
In addition to my collaborators listed in the introduction, I am par-
ticularly indebted to Fnednch Thielemann for advice on nuclear physics.
I thank the Harvard-Smithsonian Center for Astrophysics for hospitality.
I also gratefully acknowledge financial support of the Guggenheim Foun-
dation, the Smithsonian Institution and the National Science Foundation
(AS1~15325~.
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