The possibility that gamma-ray bursts originate from quakes deep in the solid crust of a neutron star is investigated. Seismic waves are radiated if shear stress is relieved by brittle fracture. However they cannot propagate directly to the surface but are temporarily trapped below a reflecting layer. The shaking of the stellar surface couples the seismic waves to Alfven waves which propagate out into the magnetosphere. The crust-magnetosphere transmission coefficient strongly increases with wave frequency and magnetic field strength. Alfven wave luminosities sufficient to power galactic gamma-ray bursts are possible if magnetic fields greater than 100 billion G cover at least part of the stellar surface. As the Alfven waves propagate out into the low density magnetosphere, they become increasingly charge starved, thereby accelerating particles to relativistic energies

Blaes, O.

Blandford, R. D.

Goldreich, P.

Madau, P.

English

NASA Technical Reports Server

N89-24947
247
NEUTRON STARQUAKES AND THE NATURE OF GAMMA-RAY BURSTS
P Madau, O Blaes, R Blandford & P Goldreich
Theoretical Physics, Calif. Inst. of Technology, Pasadena, CA, USA
ABSTRACT
We investigate the possibility that gamma-ray bursts orig-
inate from quakes deep in the solid crust of a neutron star.
Seismic waves are radiated if shear stress is relieved by
brittle fracture. However they cannot propagate directly
to the surface but are temporarily trapped below a re-
flecting layer. The shaking of the stellar surface couples
the seismic waves to Alfv6n waves which propagate out
into the magnetosphere. The crust-magnetosphere trans-
mission coefficient strongly increases with wave frequency
and magnetic field strength. Alfv6n wave luminosities suf-
ficient to power galactic gamma-ray bursts are possible if
magnetic fields _> 1011G cover at least part of the stellar
surface. As the Alfvtn waves propagate out into the low
density magnetosphere, they become increasingly charge
starved, thereby accelerating particles to relativistic ener-
gies.
Keywords: gamma-ray bursts, neutron stars, wave motions
1. INTRODUCTION
discussion of this model can be found in Ref. 8.
2. QUAKE ENERGETICS
The energy released in a typical gamma-ray burst, as-
suming isotropic emission, is
Eb _ -- ergs,
77 10-eer_ cm -2
(i)
where y? specifies the fraction of energy released which is
converted into gamma-rays, F is the observed fluence, and
D is the distance.
We model the crust as plane parallel, chemically homo-
geneous, and subject to a constant Newtouian gravita-
tional acceleration, g = -g_, where _ is a unit vector
in the upward direction. In numerical expressions we take
g = 1014 cms -2. Above neutron drip, the pressure is due
to degenerate electrons. Integrating the equation of hydro-
static equilibrium, we obtain the density profile
Fifteen years after their discovery (Ref. 1), gamma-
ray bursts remain a major unsolved problem of high energy
astrophysics. The bursts lack counterparts at other wave-
lengths, so the source objects are still unidentified. Neu-
tron stars have long been viewed as being the most plau-
sible sources. Individual bursts usually last for a few sec-
onds, with rise times as short as several milliseconds (Ref.
2). The relative lack of low-energy X-rays, L_/L_ __ 0.02,
is an important due to understanding the nature of the
gamma-ray emission region. This X-ray paucity constraint
(Ref. 3) is unlikely to be satisfied in regions of high density
or near the neutron star surface, where substantial ther-
mal reprocessing of gamma-rays into X-rays would occur.
It suggests emission from a region of low-density plasma
well out in the magnetosphere.
The basic idea of starquake models for gamma-ray
bursts is quite simple (Refs. 4-7). Elastic energy released
in a crustquake excites oscillations of the magnetic field
frozen in the surface, and the induced electric field accel-
erates high-energy particles which in turn radiate gamma-
rays. This work is devoted to assessing the merits of star-
quakes as the root cause of gamma-ray bursts. A detailed
(/_,m_)5/2 (gZ_,m,,z 2 + \ s/2
p - 3_r2hS \ c-----------5_ 2gm, lzl) , (2)
where mu is the atomic mass unit, u¢ is the mean molecular
weight per electron, and all other symbols have their usual
meanings. We adopt an intermediate value of/z, = 2.5
when presenting numerical results.
The ions in the solid crust are arranged in a Coulomb
lattice whose shear modulus is given by
= 0.295Z2e2n_/3, (3)
where nl is the ion number density and Z is the atomic
number. An average value of Z = 32 is adopted here. The
elastic energy released in a quake may be expressed as
EQ 2 3~ _(zQ),,,,,_d, (4)
where el/ield is the yield strain at which the crust cracks,
d 2 is the area of the fault plan% and zQ is the depth.
Equations (3) and (4) together require:
Proceedings of the Joint Varenna-Abastumani International School & Workshop on Plasma Astrophysics, held in Varenna, Italy,
24 Aug.-3 Sept. 1988 (ESA SP-285, Vol. 1, Dec. 1988).
https://ntrs.nasa.gov/search.jsp?R=19890015576 2020-03-20T01:38:58+00:00Z
248 P MADAU &AL
d _ 3 × 104 \10SSergs] \ 10_2 ] cm, (5)
where, for convenience, the quake is assumed to take place
at neutron drip.
The characteristic frequency, v0, of elastic waves emit-
ted by the quake is simply the speed at which the frac-
ture propagates divided by d. Since cracks typically prop-
agate at a significant fraction of the shear wave speed,
v, = (/_/p)1/2, we find
~ 10' ( EQ10SSerg s ] \ 1--6-_ / Hz. (6)
The seismic energy density is partitioned into shear
and compressional waves in proportion to the inverse sixth
power of the ratio of their propagation speeds. Most of
the energy is emitted in shear waves since their speed is
typically a few times smaller than that of compressionai
waves.
3. MAGNETOELASTODYNAMICS
The magnetic energy density dominates the rigidity
in the outer layers of the crust of magnetic neutron stars
and exceeds the rest mass energy density in the magneto-
sphere. To include the effects of the magnetic field, B, on
wave propagation, we add the Maxwell stress to the equa-
tions of elastodynamics. For simplicity, we consider the
unperturbed magnetic field to be uniform and we treat
the surface layers as solid because, with our scalings, the
magnetic field completely dominates the stress in the ocean
provided the surface temperature is <_ 3 × 10SK. This is
probably true for old neutron stars.
Linearising the equation of motion and the continuity
equation about a static equilibrium, we obtain
p_- : v.6_ + -1'ic × B + ,pg - v_p, (7)
6p : -v.(p_), (8)
respectively. Here, 6 denotes an Eulerian perturbation,
is the displacement of an element of material from its equi-
librium position, and _j is the perturbed current density.
The pressure due to the degenerate electrons, p, has been
written separately from the elastic stress associated with
the deformation of the Coulomb lattice, <r. The compo-
nents of the perturbed elastic stress tensor are related to
the components of the gradient of _ by
6o-,j = ,_- &jv._+_, \oxi + Oz_] ' (9)
where ,_ is the bulk modulus. In writing equation (9), we
have neglected the extra terms which arise if there is a
static elastic stress associated with the equilibrium state.
It is easy to show that these terms are of order %i,_d << 1
relative to those retained.
The crust is effectively a perfect electrical conductor
at seismic frequencies, so the perturbed electric field is
given by
1 O[
bE = -_-_×B. (10)
Equation (10) combined with Maxwell's equations,
1 O_B
V×rE .... (11)
c 0t
4_r_. 1 OrEv×rB = -- j + --- (12)
C C _ '
may be used to relate 6j to _.
The transmission coefficient for the energy flux between
the crust and the magnetosphere should not be signifi-
cantly affected by the polarisation and propagation angle
of the incident shear wave. We consider then the simplest
possible case, a vertically propagating shear wave polar-
ized such that _ o¢ z × B. For this special case V._,
6p, and 5p all vanish. Next, we take a harmonic time de-
pendence, exp(-i_vt), for the perturbation variables and
combine equations (7)- (12) to derive the wave equation,
_zz /2 + _5w2_ : 0, (13)
where 12 and _ are the effective shear modulus and density:
/2 _ (p + (B cos_rra)2 ]'_ (14)
_-- (p+4_c22). (15)
Here cosa - B_/B. Equations (2) and (3) are used to
obtain the dependence of _ and p on depth. The magneto-
elastic wave speed, _., is equal to (/2/_)_/2. In the magne-
tosphere, z > O,/2 = (Bcosa)2/(4_r), _-- BZ/(4_rcZ), and
the wave equation reduces to that for relativistic Alfv_n
waves,
d_( w 2
+ sec=_ = 0. (16)
dz----_ C"
The transmission coefficient, T, is defined as the ratio
of the transmitted to the incident energy flux. For a ho-
mogeneous crust, it is easy to show that the transmission
coefficient for a vertically propagating shear wave is given
by the familiar formula
T - 4ZcZM (17)
(gc + ZM) 2'
where Zc and ZM are the impedances of the crust and
magnetosphere. Each impedance is the product of the rel-
evant propagation speed and effective density. Thus,
B 2 cos a
Zc = (#/2)_/_ and Zm- 4_rc (18)
To determine the reflection and transmission amplitudes
for an inhomogeneous crust, we perform two numerical in-
tegrations of equation (13) from z << 0 to z > 0 starting
NEUTRON STARQUAKES 249
from different lower boundary conditions. Then we choose
the appropriate linear combination of the two boundary
conditions such that the solution satisfies the radiation
condition,. _' = iw seca _/c, for z > 0. The curve in figure
1 displays T as a function of frequency for various vertical
magnetic field strengths. Only a small fraction T of the
incident wave flux leaks into the magnetosphere.
10 o
10-1
/
i0-2
Q
L)
zlO'_
0
_/t10-5
or_
_i0-6
10-7
i i i i i i L II i i i i i i i i
10-8103 10 4 10 5
FREQUENCY (Hz)
Figure 1: The transmission coefficient as a function of fre-
quency for a vertically propagating wave. The upper and
lower pairs of curves are for 1012 and 1011G fields respec-
tively.
The fate of the reflected energy is clearly of interest.
The angular momentum barrier (or equivalently, the 1/r
dependence of the horizontal wave number in a spherical
star) and an increase in wave speed with depth both refract
waves upward. Shear waves will not propagate in the fluid
interior below the crust. Those which reach this region
may be reflected if the rigidity drops sufficiently abruptly
to zero at the inner boundary of the crust. Otherwise, as
the waves slow down their radial wave vectors and am-
plitudes will increase, and their energy will ultimately be
dissipated as heat.
The waves which do return will bounce many times off
the surface and will spread throughout the crust. The
characteristic storage time for wave energy in the crust,
T(v), is twice the time it takes the waves to cross the crust,
2to, divided by T(v). For vertically propagating waves,
f0 "Q s. (xg)
2 dz 5 × 10-elzql 1/2
r(v) T(v) v-_-_ T(v)
With the usual scalings, r _ ls at v0 : 104Hz.
Our analysis of the propagation of shear waves has
neglected damping. Theoretical calculations of the degen-
erate electron viscosity in a solid neutron star crust (Ref.
9) included electron scattering by phonons, impurity ions,
other electrons, and free neutrons. We neglect chemical
impurities because of the large uncertainty in their con-
centration and type. At neutron drip, the damping time
is of order 4 × 10% for 104I-Iz waves, the dominant scat-
tering mechanism being due to phonons. This is probably
an underestimate because of the likely presence of other
sources of electron scattering, especially lattice imperfec-
tions. Even so, it is still much longer than the typical
timescale for a gamma-ray burst.
Up to this point we have considered how a vertically
propagating shear wave polarized orthogonal to the ambi-
ent magnetic field couples to an Alfv6n wave. Since the
magnetosphere can also support fast magnetosonic waves,
it is natural to inquire whether more general seismic dis-
turbances couple to them. Now, the dispersion relation for
fast magnetosonic waves is isotropic and reads
o_2 = (ck)L (20)
which differs from the anisotropic dispersion relation for
Alfv6n waves,
_,_= (okcos¢) _. (2_)
Here, k is the magnitude of the wave vector, and ¢ is the
angle between k and B. For fast magnetosonic waves and
Alfv6n waves, Shell's law reads
sin O, c
-- (22)
sin _i V, '
sin 0_ c cos ¢
- (23)
sin Oi '_,
Here Oi and 0_ denote the angles of incidence and transmis-
sion. At the reflecting layer, _, << c, and thus only waves
having very small angles of incidence can couple to fast
magnetosonic waves. For Alfv6n waves the only restriction
is that the transmitted wave vector be almost orthogonal
to B.
The displacement amplitude of a propagating shear
wave increases with height in the crust as a consequence of
the conservation of energy flux. Because the strain amph-
tude, e, is proportional to the gradient of the displacement,
and the magnitude of the wave vector, k = w/6,, also in-
creases with height, e reaches a maximum near the surface.
The following argument relates the maximum strain am-
plitude to the Alfv6n wave luminosity.
We assume that the entire neutron star surface radi-
ates Alfv6n waves. Then, the fractional perturbation of
the surface magnetic field associated with luminosity L is
-g- ~ 10SSe_sss-1 _ (24)
In terms of the surface displacement amplitude, equations
(10) and (11) imply
B c '
250 PMADAU&AL
Thus, we estimate the totalsurfacestrainto be:
L _1/2 B -i
(26)
With the chosen scalings, the maximum strain is danger-
ously close to unity. A dynamic yield strain as large as
0.1 is not unusual. Thus, we cannot predict whether shear
waves transfer their energy to Alfv_n waves and/or crum-
ble the crust, generating heat. However, it is clear that
the neutron starquake model is not viable if bursts are sig-
nificantly more distant, and consequently more luminous,
than estimated above.
4. ALFVI_N WAVES IN THE MAGNETOSPHERE
We have already discussed the magnetic field pertur-
bations required for Alfvdn waves to carry the luminosity
of a typical gamma-ray burst. Because Alfvdn waves trans-
port energy without loss along the equilibrium field lines,
their relative amplitudes vary according to
6B B_I/2 (27)
-- O(
B
Equations (24) and (27) indicate that the Alfvdn waves are
likely to become nonlinear not very far from the star.
Up to this point, we have been making the implicit as-
sumption that the plasma density in the magnetosphere is
high enough so that the MHD limit applies to Alfvdn wave
propagation. We shall now show that this is a questionable
assumption.
The corotation charge number density (Ref. 1O),
nc_ _ -- --_7 × 109 cm -s (28)Pce _ _ '
setsa lower limiton the magnetospheric plasma density,
n. Here, P is the rotationalperiod of the neutron star.
Much higher densitiesmay existifaccretionisoccurring,
or ifan energeticprocess expellsplasma from the stellar
atmosphere.
For low plasma densitiesa nonlinearitymay occur
even at relativefieldamplitudes 6B/B << 1. Because the
wave vectorisnearly orthogonal to the magnetic fieldnear
the surface,equation (10) implies that the wave vector is
nearly parallelto _E. Therefore,the displacement current
and V × _B are nearly orthogonal. From equation (12),
we see that a substantialphysical current is necessary to
support the wave. If the plasma density is so low that
the required driftvelocitiesexceed c, the wave is charge
starved. This occurs where
_B 2nec 10_ 4 n (29)
B roB -_s "
Based on the values of _B/B given by equation (24), the
Alfvdn waves are expected to be charge starved close to
the stellar surface unless n >> nc_. The fate of such waves
requires further investigation.
Once the Alfvdn waves go nonlinear, a substantial part
of the wave energy is probably transferred to the electrons.
An upper limit to the Lorentz factors is obtained by bal-
ancing the electrostatic acceleration with curvature radia-
tion reaction. We find
( BIB3'' ( B 3''( R
q'_ 10v \ 10-' ] \IO--_-GG] \_] . (30)
5. CONCLUSIONS
Starquakes have the followingvirtues.They can easily
provide the energy necessaryto power a singleburst. They
releaseenergy in the low entropy form of seismic waves
which are transformed into Alfvdn waves on timescales
characteristicof observed gamma-ray bursts. The Alfvdn
waves may become nonlinearfar from the the stellarsur-
faceand acceleratethe ambient plasma to energiesthat are
limited by radiationreactionlossesdue to the emission of
gamma-rays. Thus, it is possible to imagine an efficient
conversion of elasticenergy to gamma-ray energy.
Of course, the above scenario is repletewith uncer-
tainty.It isquite possiblethat plasticflow,and not brittle
fracture,is the manner in which stressis relievedin neu-
tron starcrusts.The source Of the freeenergy necessaryto
replenishthe crustalstressin old neutron starsisnot iden-
tified.The conversion of seismicwaves into Alfvdn waves,
on the appropriate timescalesand without the production
of excessivecrustal strains,requiresthat magnetic fields
>_ 1011G cover at least part of the stellarsurface. The
propagation of Alfv_n waves in low density,neutron star
magnetospheres ispoorly understood. Their behavior and
the manner in which they accelerateparticlesprobably de-
pends upon the degree to which the neutral plasma density
exceeds the corotation change density.
This research was supported by NASA grant NAGW
1303 and NSF grants AST86-15325 and AST86-1279. O.
B. holds a Chaim Weizmann researchfellowshipat Caltech.
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2. Liang, E. P., and Petrosian, V. 1986. Gamma-Ray
Bursts, AIP Conf. Proc. 141.
3. Imamura, 3. N., and Epstein, R. I.1987. Ap. J. 313,
711.
4. Pacirfi,F.,and Rudcrman, M. 1974. Nature 251,399.
5. Tsygan, A. I.1975. Astr. Ap. 44, 21.
6. Fabian, A.C., Icke, V., and Pringle, :I.E. 1976. Astr.
Sp. Sci., 42, 77.
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Neutron starquakes and the nature of gamma-ray bursts

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