Dearborn Observatory
Northwestern University, Evanston, Il 60208
Abstract. While the thermonuclear flash theory well accounts for the general features of Type 1 X-ray bursts, observational details such as irregular time intervals between bursts and quasi-periodic oscillations suggest that other physical factors also significantly contribute to the process. Preliminary multi-dimensional modeling suggests thermal and hydrodynamic mechanisms arising from a localized temperature perturbation can not only lengthen burst intervals but also move enriched nuclear fuel around a neutron star in time to involve the entire star in large-scale bursts. A more detailed study based on these preliminary results promises to offer significant insight into the possible dynamics of irregular X-ray bursters.
Thermonuclear Flash Model of Type I Bursts
X-ray bursts have been observed exclusively from neutron stars in low mass X-ray binaries (LMXBs). For a particular LMXB, the companion is a relatively low mass star. The surface of the companion overspills its Roche lobe, and gas becomes gravitationally bound to the neutron star to form a hot, inspiralling accretion disk. The relatively low level, persistent X-ray flux detected from such a system originates from either the accretion disk, which is hot enough to emit X-rays, or the neutron star's surface, when the accreted gas impacts onto it, releasing gravitational energy as heat and radiation. Sudden and substantial increases in X-ray flux above persistent levels signal the start of an X-ray burst. Estimates of radii based on the assumption that the emitting object is a black-body indicate bright bursts must involve the entire neutron star surface.
While bursts can be classified into two major types, Type I and Type II, most are Type I. Henceforth, reference to X-ray bursts and bursters will mean the Type I variety. Type I burst profiles vary greatly, but characteristic features include a sudden, rapid (1 to 10 seconds) increase in X-ray flux, reaching 5 to 20 times quiescent values, followed by a more extended period of decay (10 seconds to several minutes), during which the energy of the X-rays diminishes from peak values. The thermonuclear flash model (Lewin, van Paradijs, & Taam 1993) successfully explains these general features as resulting from explosive nuclear ignition on the surface of neutron stars. In short, after gas from the accretion disk reaches the surface of the neutron star, it spreads out evenly over it and proceeds to burn in the extremely high pressure environment. At a critical point, degenerate hydrogen and/or helium burning ignites explosively, suddenly heating up the entire surface to around 3 x 107 K, enough to emit strong X-rays. This flash is observed as the start of an X-ray burst. After the explosion, the surface cools, a process observed as the decay of the burst profile. Moreover, as more nuclear fuel continues to accrete onto the surface, the process may repeat. Thus, the model accounts for the general observational features of burst properties such as the energies involved (~1038 to 1039 ergs), their rise time (seconds), duration (~10 to 100 seconds), spectral softening, and recurrence intervals (several hours).
The flash model enjoys widespread consensus. However, some observations are difficult to interpret in its framework. For instance, the bursting behavior of 1735-444 (van Paradijs et al. 1988) can range from intervals of less than an hour, where observations are consistent with explosive helium ignition, to very long intervals of no bursts, during which the hydrogen and helium burning appears to be stable. This raises questions concerning what kinds of conditions favor one type of burning over another, and what causes the sudden change between the two.
Another example is the relationship between the time interval between two bursts and the total burst energy of the later burst. According to the flash model, a longer time interval allows more fuel to accrete, which should make the subsequent burst more powerful. While extensive EXOSAT observations of several bursters such as 1735-444 (Lewin et al. 1980) support this prediction, others do not. For instance, sources as 0748-673 (Gottwald et al. 1986) and 0836-429 (Aoki et al. 1992) have very short burst intervals, around 10 minutes. This is not enough time to accrete enough nuclear fuel to power a second burst. One possible explanation is that a significant amount of fuel may be left over from one burst to be involved in a subsequent one (Hanawa & Fujimoto 1984).
Yet another difficulty is trying to correlate the bursting interval with the persistent X-ray flux. No clear relationships are evident. For instance, in 1820-303, the persistent flux increased five-fold as the burst intervals decreased by half, eventually stopping altogether (Clark et al. 1977). Cessation of bursting when the persistent flux increased was similarly found in other systems such as 1658-298 (Lewin et al. 1978) and GX3+1 (Makishima et al. 1983). However, other systems exhibit bursting when the persistent flux is very high. Examples include Cyg X-2 (Smale 1998), GX17+1 (Tawara et al. 1984), and MXB 1730-335 (the Rapid Burster) in its strong persistent emission phase (Guerriero et al. 1998). Interestingly, bright sources such as these sometimes also exhibit very irregular bursting intervals, where the time between bursts vary greatly. Very bright neutron stars have high accretion rates approaching Eddington values. (Eddington luminosity and accretion rates represent natural limits, above which the radiation pressure exceeds gravitational attraction and minimizes further accretion.) Under these conditions and in a hydrogen deficient environment, helium may not burn over the neutron star symmetrically, but in slowly moving fronts. Instead of the entire surface of the neutron star contributing to a large scale burst, the bursting occurs in limited regions, producing smaller luminosity fluctuations of minute-long duration. Bildsten (1993, 1995) suggests this may account for the very low frequency noise detected from very bright sources. However, this does not explain very bright bursts.
Assuming the flash model validly describes the underlying physical cause of bursts, examples such as these suggest other significant factors exist which affect the nuclear burning. For instance, the current thermal state of the star, which in turn depends on its history of previous bursts, was found to affect nuclear burning (Taam 1980). A star's compositional state also depends on its history and may influence burning (Woosley & Weaver 1985). A model which takes into account the thermal and compositional histories of a burster may give results very different from a model which does not.
New Observations Test an Old Model
Recent observations from more advanced instruments such as RXTE and BeppoSAX have provided new details which both support and challenge the flash model. For example, RXTE observed bursts from a 2.5 ms pulsar, XTE J1808-3658 (SAX J1808.4-3658), and thus convincingly showed that a neutron star can be a burst source (Marshall 1998; Wijnands & van der Klis 1998). Previously, Type I bursts had never been observed from pulsars, suggesting that bursters had low magnetic fields. (A Type II burst had been previously observed from a pulsar, GRO J1744-28, but these kinds of bursts are believed to arise from changes in the properties of the accretion disk.) The discovery of a pulsing Type I burster not only confirms a neutron star can exhibit bursting behavior but also yields an important physical parameter, the magnetic field, previously indeterminable from bursters. The magnetic field of XTE J1808-3658 is estimated to be around 108 to 109 Gauss (Psaltis & Chakrabarty 1999). The existence of a substantial magnetic field on a burster introduces the possibility of channeling matter to the magnetic poles, thus forming hotter regions there.
Recent observations have also revealed rapid fluctuations in both persistent and burst spectra of several X-ray sources. These quasi-periodic oscillations (QPOs) have kilohertz frequencies, and numerous models have been advocated to explain their origin, though none are completely satisfactory. For instance, the sonic-point model (Miller, Lamb, & Psaltis 1998) suggests they are due to interference between a hotter region on the neutron star surface spinning at the star's rotation frequency and concentrations of matter circulating in the accretion disk at Keplarian frequencies. In this picture, the QPOs are beat frequencies between the two more fundamental frequencies. QPOs in persistent flux characteristically occur in pairs 250-350 Hz apart, and while the actual frequencies may drift, the difference between the two is nearly constant. In this model, the difference may be interpreted as the spin frequency of the neutron star.
QPOs have also been observed in bursts. This supports the idea they originate from some surface phenomenon. In one interpretation, the magnetic field of the star funnels gas from the accretion disk onto the magnetic poles, pooling more fuel there. Energy generation due to nuclear burning would naturally be higher in this patch, increasing the temperature above its surroundings. As the burning patches grow in size to cover the rest of the star when a burst develops, this might be observed as strong QPOs at the start of the X-ray burst signal (Strohmayer 2001; Strohmayer, Zhang, & Swank 1997). Other observational details complicate the picture. For instance, oscillations often occur in the tail of the burst, after the nuclear fuel should have been entirely burned. Moreover, the oscillation frequency in the burst tail is often greater than those at the burst onset by 1 to 2 Hz. Another puzzle is why QPOs are not always observed during bursts. In fact, bursts from a given burster may sometimes exhibit QPOs and other times not, while some bursters do not exhibit QPOs at all.
Recent efforts to evolve a neutron star atmosphere using a rotating, one-dimensional, hydrostatic model support the theory that QPOs are due to rotating hot spots on the star surface and suggest possible explanations for other QPO phenomenology (Cumming & Bildsten 2000). For instance, angular momentum conservation as a burning shell expands and relaxes during bursts may account for the 1 to 2 Hz QPO frequency change in burst tails. Moreover, the rotational behavior was found to strongly depend upon the composition of the burning layers. This would be consistent with studies which correlate burst duration with QPO occurrence. Bursts of long duration (indicative of hydrogen burning) in GS 1826-24 do not exhibit QPOs (Kong et al. 2000), while those of short duration (indicative of pure helium burning) in KS 1731-26 do exhibit QPOs (Muno et al. 2000). However, additional observational studies are needed to strengthen the case.
Clearly, additional observational and theoretical work is needed to understand old and new aspects of bursts such as these in the framework of the flash model. In terms of computational modeling, particularly lacking are multidimensional simulations involving hydrodynamics which take into account thermal and compositional histories of the star. Only recently have computational speeds and methodology improved to make these complex calculations feasible. Such work forms the basis of this proposal, and preliminary results encourage further efforts.
Irregular Bursting Behavior from Bright X-Ray Sources
Proceeding with the idea that localized patches of burning may occur on the surface of a neutron star, a recent investigation by Taam, Bayliss, and Sandquist (2000) offer insights into a possible reason why bursting intervals can be irregular in some bright LMXBs. In this study, a localized temperature perturbation was introduced in the initial conditions to simulate a region on the neutron star's surface which is hotter than its immediate surroundings. The thermal evolution was followed, and the temperature perturbation was found to be quickly smoothed out by thermal diffusion. As additional mass accreted near Eddington values onto the surface, a burst finally resulted. However, the time it took for the burst to occur was much longer than in the control case in which no temperature perturbation was introduced. (The control case produced regular bursting intervals.) Thus it was demonstrated that localized temperature disturbances due to localized accretion of fuel can significantly affect X-ray burst duration. Irregularities in burst duration, then, may be due to varying degrees of such localization.
If fuel is localized as portrayed in the above scenario, it must be able to spread over the entire neutron star in a relatively short amount of time, since in order to form a bright burst with a rapid increase in luminosity, the explosion must involve most of the neutron star nearly simultaneously. Given that the radius of a neutron star is about 106 cm, matter would need to move relatively quickly (greater than 105 - 106 cm/s) to account for the observed burst properties. To discover what might cause such motion, I (Lin) recently initiated an investigation which forms the basis of the present proposal. The preliminary results strongly suggest that hydrodynamical mechanisms exist which may move matter and energy rapidly away from a localized region where energy generation occurs at a higher rate than its immediate surroundings.
Using a multidimensional, hydrodynamical code, I constructed and evolved one and two-dimensional, non-rotating models of the outer surface of a neutron star. First, 1D calculations were performed to see whether significant hydrodynamical waves would develop as a result of introducing energy at a constant rate in a limited region of the domain. Both a vertical model and a lateral model were constructed, which represent simple cross-sectional slices of a neutron star surface. Energy generation was simulated by adding a constant energy generation rate of 1017 erg g-1 s-1 to a small region on the domain. Sound waves developed from the region, but the velocities induced were insignificant (~50 cm/s), far too small to carry matter and energy very far in the timescale of interest. However, in both vertical and lateral models, the site where energy was added did exhibit small, persistent differences in pressure and density from its surroundings, of the order of 10-4 in terms of fractional difference. Moreover, after the initial wave passes, the regions in the wake of the wave showed a fractional difference in pressure and density of 10-7. Since differences of these magnitudes suggested the possibility that hydrodynamic circulation may develop in a 2D domain, the results encouraged me to proceed to a 2D model.
The 2D calculations yielded a much more dynamic picture (Figure 1). At first, a circulation pattern develops around the energy generation region. Later, much larger velocity outflows develop, set up by an initial pressure wave that propagates through the computational domain at the sound speed (~108 cm/s). The magnitude of the velocity outflows was found to be related to the size of the region. For example, in the case when the region is 667 cm long, the magnitude of the outflowing velocities are on the order of 105 cm/s. If the magnitudes of these velocity fields persist on the timescale of enhanced nuclear energy generation (around 10 seconds), they will propel fuel around the star in time to involve most of it in a large-scale burst.
Together with the earlier finding that bursts are delayed due to such perturbations, this mechanism may be one explanation why some bright X-ray sources burst irregularly. That is, a localized region of greater burning of nuclear fuel due to greater accretion on a certain part of the star not only significantly delays the time required to develop a burst due to thermal effects, but also establishes flows due to hydrodynamical effects sufficient to carry the fuel to the rest of the star in time for the entire star to be involved in a burst.
Research Plan
While promising, these results are preliminary. A more complete investigation will involve careful parameter studies of how changing different variables affects the dynamic evolution of the system. One parameter which is currently rather arbitrary is the dimensions of the energy generation region. Previous calculations suggest enhanced nuclear reactions will occur between densities of 2x105 and 2x106 g/cm3, helping to constrain the vertical limits. However, the lateral extent of the region is a parameter which needs to be explored. In reality, the magnetic polar cap is likely to be on the order of a kilometer or longer. While computational limitations may prohibit studies of such extent, it may be possible to adequately scale the velocities which develop from smaller regions of different sizes, to the limits of computational feasibility. One would expect that larger flows would develop from larger regions of energy generation, since more energy is being introduced. In fact, an investigation using the current model supports this physically intuitive idea: the velocity of the developing flows scales roughly linearly with
the lateral extent of the energy generating region. A full parameter study is warranted to explore this effect.
Another improvement to the current calculation will be the inclusion of a nuclear reaction network, which may lead to unexpected dynamics. Energy generation from nuclear reactions are currently introduced into the system by simply adding a constant term to the total energy in the hydrodynamical equations. While the energy generation rate used is consistent with independent calculations involving a nuclear reaction network, it may change when hydrodynamics is included. The preliminary simulation suggests a mechanism by which the rate may actually increase via inflow of matter from deeper in the star into the energy generation region, thereby enriching the nuclear fuel with heavier elements, possibly increasing the energy generation rate. Another possibility is the formation of other burning regions in other parts of the domain, due to dynamic flows which develop. Such potential results would represent additional mechanisms which enhance mass flow around the star, worthy of careful study.
Yet another avenue of questions involve how lateral differences in the initial conditions affect how the system evolves. Baroclinic instability, a hydrodynamical instability which causes horizontal fluid motion by releasing gravitational potential energy in the skewed pressure and compositional states, has been studied for its possible role in transporting angular momentum in the interiors of stars (Spruit & Knobloch 1984), and in accreting compact objects (Fujimoto 1988, 1993). Baroclinic instability may likely occur in systems with strong differential rotation or high accretion rates. Such motion may enhance turbulence and significant mixing of fuel to deeper parts of the star. Fujimoto proposed that flashes which originate in the deeper layers due to such fuel enrichment may explain bursters with very short intervals, too short for surface accretion to explain. However, Cumming and Bildsten (2000) recently found that mixing between burning and ash layers by baroclinic instability does not occur in strongly stratified systems, but that baroclinic instability may operate within a differentially rotating burning layer. Fujimoto also proposed that varying degrees of turbulent mixing may explain irregular bursting behavior, although a study has yet to explore the possibility. Presently, my research is based on a model which has a uniform lateral distribution of matter. In the future, I plan to introduce inhomogeneities in pressure and density as initial conditions and carry out multidimensional investigations to examine how the developing hydrodynamic flows depend on varying degrees of the instability. Other factors such as nuclear burning and thermal diffusion will be included to examine more complex interactions.
Summary of Motivations and Objectives
The effort to better comprehend X-ray bursts has already revealed important aspects of the basic physics of neutron stars where extremely high densities, pressures, and temperatures are prevalent. Many aspects of X-ray bursts such as what determines the burst intervals and what causes quasi-periodic oscillations have yet to be explained, which will likely involve more careful consideration of thermal, compositional, and multidimensional factors. Observational hints that bursters can have significant magnetic fields suggest that such physical interactions may significantly affect accretion dynamics, leading to localized temperature differences. Thermal diffusion studies indicate such differences effectively delay the time intervals between bursts. Taken together with the preliminary multidimensional, hydrodynamical studies, which reveal significant flows developing from such a region, this may be one mechanism which explains the variety of bursting intervals observed. Further investigations involving full parameter studies of physical variables, improved initial models, and the inclusion of baroclinic instabilities promise improved insight into the nature of X-ray bursting behavior. Significant progress is expected within the duration of this proposal.