Temperature profiles and spectra of accretion disks around rapidly rotating neutron stars

Abstract

In this chapter, we have computed the observed radiation spectrum from accretion disks around rapidly rotating neutron stars using fully general relativistic disk models. This is the first time such a calculation has been made in an exact way, without making any approximations in the treatment of either rotation or general relativity. In computing the observed spectrum from the disk, we explicitly include the effects of Doppler shift, gravitational redshift, and light-bending for an appropriate metric describing space-time around rapidly rotating neutron stars. We find that the effect of light-bending is most important in the high-energy (> 3 keV) part of the observed spectrum. Photons at these high energies originate close to the central star, and hence their trajectories are most affected by the light-bending effect. Depending on the viewing angle, this can enhance the observed flux at ~ 10 keV by as much as 250% compared to that expected if light-bending effects are neglected.

It is to be noted that in this work we have neglected the effect of irradiation of the disk. Miller & Lamb (1996) have discussed such effects on a test particle moving towards a slowly rotating neutron star. A strongly irradiated disk may not remain thin, and the radiation force may relocate the position of the inner edge of the disk. In addition to that, fractions of angular momentum and energy of the accreted matter may be transferred to the irradiating photons, resulting in a redistribution of emitted flux in the disk. These effects will change our calculated spectrum to some extent. Therefore, we aim to modify our calculation in the direction of the work of Miller & Lamb (1996). However, it is to be noted that for rapidly rotating neutron stars, boundary layer emission is small, and hence the effect of irradiation may not be important.

The calculations presented here deal only with the multicolor blackbody disk. In reality, there will be additional contributions to the observed spectrum from the boundary layer as well as a possible accretion disk corona, both of which are likely to add a power-law component at high energies (Popham & Sunyaev 2000, Dove et al. 1997). On the other hand, the spectra presented in Figs 5.2, 5.3, and 5.5 should remain essentially unaffected by boundary layer contribution, as these are for neutron stars rotating near the mass-shed limit for which the boundary layer luminosity will be negligible. For slowly rotating neutron stars, the disk component of the spectrum can be obtained by fitting and removing the contribution of the boundary layer, provided a good model for the boundary layer spectrum is available. Popham & Sunyaev (2000) have made an attempt to compute the boundary layer spectrum in the Newtonian approximation. General relativistic modifications need to be included in these calculations to get a realistic estimate of the spectrum of the boundary layer. We plan to address this issue in a future work. In the slow rotation case, the spectrum of the disk itself may be somewhat modified by the presence of a boundary layer if it extends beyond the disk inner radius assumed in our computations here, thus curtailing the inner edge of the disk.

In addition to the contribution of the boundary layer, the possible contribution of an accretion disk corona to the emergent spectrum could also be significant. To be able to constrain the EOS models of Neutron Stars using the observed spectrum, this contribution must also be accurately estimated. We have not attempted to estimate this in the present work, where we restrict ourselves to thin blackbody and non-magnetic accretion disks in order to understand the effect of the EOS models describing neutron stars on the spectrum of the accretion disk alone. We view this as the first step in accurately modeling the spectra of accreting neutron stars, including the effects of general relativity and rotation. We may mention that the radiation originating in the accretion disk corona would also be modified by the gravitational redshift and light-bending effects, and the technique presented by us here will be useful also in that context.

The comparison of the non-rotating limit of our results with those of the fitting routine GRAD in the X-ray spectral reduction package XSPEC (Ebisawa et al. 1991), shows that the latter model overpredicts the high-energy component of the flux by a large factor. With the help of K. Ebisawa & T. Hanawa, we have been able to trace this disagreement to certain simplifying approximations made in the GRAD code, as well as a couple of incorrect expressions being used there. Conclusions based on the use of the GRAD routine may therefore need to be revised in the light of the new calculations presented here.

The computation of the complete spectrum in the manner presented here is rather time-consuming and therefore not quite suited to routine use. Therefore, in order to make our results available for routine spectral fitting work, we need to present a series of approximate parametric fits to our computed spectra. We do it to some extent in the next chapter.

The spectra presented here will find use in constraining the combined parameter set of the mass, the rotation speed, and, possibly, the EOS, particularly of weakly magnetized, rapidly rotating neutron stars. The relevant signatures are most prominent in hard X-rays, above ~ 10 keV. Sensitive observations of hard X-ray spectra of LMXBs, therefore, are needed to fully utilize the potential of these results.

The X-ray binary systems consist of two stars, rotating around each other. One of them (primary) is a compact star (neutron star, strange star, or black hole) and the other one (secondary companion) is a main-sequence star or an evolved star (red sub-giant, blue super-giant, or white dwarf). When the companion star fills its Roche-lobe, matter from its surface flows towards the compact star. Due to initial angular momentum, this matter cannot fall radially; rather it follows a spiral path and forms a disk. Such a disk is called an accretion disk. Due to viscous dissipation, energy is radiated from the disk. As the temperature of the inner portion of the disk is very high (~ 10^7 K), X-rays are generated in this region. If the compact star has a hard surface (i.e., if it is not a black hole), the inflowing matter hits this surface and another component of X-ray is produced in a thin layer, called the boundary layer. However, it is to be remembered that disk accretion is not the only mechanism for the accretion process. Such a process can also happen from the wind of the companion star.

There are two classes of X-ray binary systems: HMXBs and LMXBs. The secondary companion in an HMXB is a high-mass star (generally, O or B type). As mentioned in Chapter 1, the age of such a system is rather low (~ 10^7 years). Most of the energy coming out of such systems is in the visible range. On the other hand, an LMXB consists of a low-mass (< 1 M?) companion star, with the age typically ~ 10^9 years, and most of its radiated energy is in X-rays.

For our work, we have chosen LMXB systems with neutron stars or strange stars as the central accretors. These systems offer several advantages over the HMXBs in understanding the properties of compact stars. For example, most of the energy (in X-rays) from such systems comes from the accretion disks. The motion of matter in these regions is expected to be influenced by the mass-radius relation and the total angular momentum of the compact star. Therefore, the analysis of X-ray spectra from these systems may shed light on the properties of the compact stars. Moreover, the accretion disks of such systems may extend very close to the stellar surface, as the magnetic fields of the primary stars in LMXBs are expected to be decayed to lower values (~ 10^8 G; see Bhattacharya & Datta 1996 and Bhattacharya & van den Heuvel 1991). This ensures that the observed spectra can actually reflect some properties of the compact stars. Besides, accretion via wind is negligible in LMXBs, which may make the spectral calculation for such systems simpler than that for HMXBs.

LMXBs exhibit many complex spectral and temporal behaviors. It is a challenge to explain these phenomena using theoretical models. However, most of the existing models for spectral fitting are Newtonian. But near the surface of a compact star, the accretion flow is expected to be governed by the laws of general relativity due to the presence of strong gravity. Therefore, general relativistic models should be used for the purpose of fitting to get the correct best-fit values of the parameters. Besides, the principal motivation behind the study of the spectral and temporal behaviors of compact star LMXBs is to understand the properties of very high (~ 10^15 g cm^-3) density matter at the compact star core (van der Klis 2000). This is a fundamental problem of physics, which cannot be addressed by any kind of laboratory experiment. The only way to answer this question is to assume an equation of state (EOS) model for the compact star core, to calculate the structure parameters of the compact star, and then to calculate an appropriate spectral model. By fitting such models (for different chosen EOSs) to the observational data, one can hope to constrain the existing EOS models and hence to understand the properties of high-density matter. However, general relativistic calculation is essential to calculate the structure parameters of a compact star and therefore to constrain the EOS models.

It is expected that the compact stars in LMXBs are rapidly rotating due to accretion-induced angular momentum transfer. LMXBs are thought to be the progenitors of millisecond (ms) radio pulsars (Bhattacharya & van den Heuvel 1991) like PSR 1937+21 with P ~ 1.56 ms (Backer et al. 1982). The recent discovery of ms (P ~ 2.49 ms) X-ray pulsations in XTE J1808-369 (Wijnands & van der Klis 1998) has strengthened this hypothesis. Therefore, it is necessary to calculate the structure of a rotating compact star considering the full effect of general relativity. This was done by Cook et al. (1994) and the same procedure was used by Thampan & Datta (1998) to calculate the luminosities of the disk and the boundary layer.

In our work, we have calculated the structure parameters of a rapidly rotating neutron star and the metric coefficients in and around it. Then we have computed disk temperature profiles and disk spectra for various EOS models and many (M, ?) combinations. We have considered strange stars as well. But recently, the story has been complicated considerably by the finding that ordinary neutron beta-decay may be energetically allowed in nuclear matter (Lattimer et al. 1991), so that the cooling rate may be comparable to that of strange quark matter. There are other possible ways (for example, study of pulsar glitches, oscillation, and maximum rotation rate of the stars) to distinguish between these two kinds of stars (see Madsen 1998 for discussions).

In Chapter 7, we have computed the structure parameters of a strange star and calculated the corresponding disk temperature profiles and luminosities. We have then compared these values with those for a neutron star and tried to distinguish between them.

In sections 8.2 and 8.3, we discuss the conclusions from the calculations of disk temperature profile and spectrum, respectively. We give the summary of the work with strange stars in section 8.4. In section 8.5, we discuss the future prospects, and in section 8.6, we give the final conclusion.

It can be seen (see Fig. 5.1) that the inclusion of light-bending effect enhances the predicted flux from the disk. This is because, due to light-bending, the disk subtends a larger solid angle to the observer than it otherwise would. We also see (from Fig. 5.2) that the spectrum, especially the high-energy part, is very sensitive to the accretion rate. This may be useful for constraining the accretion rate by fitting the observational data with our model.

The inclination angle i is a very important parameter in determining the shape of the spectrum. For lower energies, the observed flux is higher for lower values of i, while this effect is opposite at higher energies due to Doppler blue shift. We can also notice (from Fig. 5.4) that the behavior of the disk spectrum is not monotonic with inclination. This is expected from the non-monotonic behavior of the disk temperature profile. We can also expect to constrain EOS models by spectral fitting, as the disk spectrum is fairly sensitive to the chosen EOSs. However, as the spectrum is a function of a large number of free parameters, it is very difficult to constrain the equations of state in a decisive way. But it may actually be possible with the data of new generation X-ray satellites with very good spectral resolution.

The calculations presented here deal only with the thin Keplerian blackbody disk. In reality, there may be other X-ray emitting components (boundary layer, accretion disk corona, etc.) present in the LMXB source. In addition to that, the disk may not be thin, Keplerian, or a blackbody. Our results will change for such cases. For example, our temperature profile and hence the spectrum will not be valid for a non-Keplerian disk. The effect of such uncertainty in the nature of the source may be more important than the effect of general relativity and rapid rotation. However, there is no competition between these two kinds of effects. General relativistic modifications should be considered to calculate the spectra from all the X-ray emitting components to have the full general relativistic spectrum of a source. However, as this is a first step for this kind of work, we chose the simplest system, i.e., a thin blackbody disk.

As mentioned in section 5.5, our results for non-rotating neutron stars did not match with those of the spectral fitting routine GRAD. With the help of Ebisawa & Hanawa (private communication), we traced this mismatch to certain simplifying approximations, as well as a couple of errors made in the GRAD code.

The computation of our model spectrum is rather time-consuming and therefore not quite suited to routine use. To facilitate direct comparison with observations, we have presented a simple empirical function which describes the numerically computed relativistic spectra well. This empirical function (which has three parameters, including normalization) also describes the Newtonian spectrum adequately. Thus, the function can in principle be used to distinguish between the two. In particular, the best-fit value of one of the parameters (?-parameter) is ~ 0.4 for the Newtonian model, implying the possibility of distinguishing these objects from each other by sensitive observations. Through future X-ray detectors, one can, in principle, fit the observational data by this model spectrum and constrain the equation of state and the parameters of the source effectively. However, computation of such a general relativistic spectrum is time-consuming and rather unsuitable for the fitting procedure. Therefore, it is important to create a series of parametric fits to this spectrum to make it available for routine spectral fitting work.

The main purpose of the study of the properties of an LMXB with a neutron star or a strange star as the central accretor is to understand the properties of very high-density matter and to address the question of the existence of strange quark matter. These can be achieved only by fitting the observed spectral and temporal behavior of such sources with appropriate theoretical models. It is not possible to incorporate all the important factors in such a model in a single work, and one should proceed step by step. In our work, we have included two major factors, namely, the effects of general relativity (essential for constraining EOS) and rapid rotation, which we think is a step forward towards our aim.