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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
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
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
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<ES When a heavy fluid rests on top of a lighter fluid in a gravitational field, it is Rayleigh-%ylor unstable (e.g. Chandrasekhar 1961~. Small per- turbations of the interface (with wavelength long enough to overcome the effects of surface tension) will grow through a linear phase in a time ~ (p/p)-~/2(gk)-~12, where k is the horizontal wavenumber, to the "bubble and spike" non-linear phase. However, a neutron star cnlst can withstand shear stress and may be stabilized to Rayleigh-lkylor instability (e.g., Lan- dau and Lifshitz 1986~. The imposition of the no-slip boundary condition at the interface, as opposed to the free movement allowed in a traditional fluid treatment, inhibits the growth of unstable modes. We have performed a linear analysis of the elastic Rayleigh-lkylor instability and find that for a given, sufficiently large density jump at the interface, there wH1 be a range of unstable horizontal wavenumbers centered on roughly twice the recipe cat of the depth of the interface below He surface. The minimum density jump necessary for a single unstable mode depends upon the ratio of the
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~. REFERENCES Ayasli, S., and P.C Joss 1982- Astrophys. J. 256: 637. Baym, G., CJ. Pethick, and P. Sutherland. 1971. Matter at High Densities. Astrophys. J. 170: 299. Baym, G. and D. Pined 1971. Ann. Phys. 66: 816. Bisnovatyi-Kogan, G.S., and V.M. Chechetldn. 1979. Sov. Phys. Uspekhi. 22 89. Bisnovatyi-Kogan, G.S., and V.M. Chechetkin. 1986. In: Helfand, DJ. (ed.~. Origin and Evolution of Neutron Stars. Prow LOU Symp. Kluwer, Dordrecht.
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