Photospheric Radius Expansion Research Articles

Constraints on the symmetry energy using the mass-radius relation of neutron stars

The nuclear symmetry energy is intimately connected with nuclear astrophysics. This contribution focuses on the estimation of the symmetry energy from experiment and how it is related to the structure of neutron stars. The most important connection is between the radii of neutron stars and the pressure of neutron star matter in the vicinity of the nuclear saturation density ns. This pressure is essentially controlled by the nuclear symmetry energy parameters Sv and L , the first two coefficients of a Taylor expansion of the symmetry energy around ns. We discuss constraints on these parameters that can be found from nuclear experiments. We demonstrate that these constraints are largely model-independent by deriving them qualitatively from a simple nuclear model. We also summarize how recent theoretical studies of pure neutron matter can reinforce these constraints. To date, several different astrophysical measurements of neutron star radii have been attempted. Attention is focused on photospheric radius expansion bursts and on thermal emissions from quiescent low-mass X-ray binaries. While none of these observations can, at the present time, determine individual neutron star radii to better than 20% accuracy, the body of observations can be used with Bayesian techniques to effectively constrain them to higher precision. These techniques invert the structure equations and obtain estimates of the pressure-density relation of neutron star matter, not only near ns, but up to the highest densities found in neutron star interiors. The estimates we derive for neutron star radii are in concordance with predictions from nuclear experiment and theory.

Double-peaked thermonuclear bursts at the soft–hard state transition in the Rapid Burster

Long suspected to be due to unstable accretion events, the type II bursts exhibited by the Rapid Burster (RB, or MXB 1730-335) still lack an exhaustive explanation. Apart from type II bursts, the transient RB also shows the better-understood thermonuclear shell flashes known as type I bursts. In search of links between these two phenomena, we carried out a comprehensive analysis of all $\textit{Rossi X-ray Timing Explorer}$ observations of the RB and found six atypical type I bursts, featuring a double-peaked profile that is not due to photospheric radius expansion. The bursts appear in a phase of the outburst decay close to the onset of the type II bursts, when the source also switches from the high/soft to the low/hard state. We also report the discovery of a simultaneous low-frequency quasi-periodic oscillation present in the persistent emission as well as in the burst decaying emission. We discuss several scenarios to understand the nature of the peculiar bursts and of the accompanying oscillation, as well as their possible relation with each other and with the type II burst phenomenon. We favour a model consisting of two accretion channels, one polar and one equatorial, in a source viewed at low inclination. We also, for the first time, clearly identify the atoll nature of the RB.

THE HARD X-RAY BEHAVIOR OF AQL X-1 DURING TYPE-I BURSTS

We report the discovery of an anti-correlation between the soft and the hard X-ray lightcurves of the X-ray binary Aql X-1 when bursting. This behavior may indicate that the corona is cooled by the soft X-ray shower fed by the type-I X-ray bursts, and that this process happens within a few seconds. Stacking the Aql X-1 lightcurves of type-I bursts, we find a shortage in the 40--50 keV band, delayed by 4.5$\pm$1.4 s with respect to the soft X-rays. The photospheric radius expansion (PRE) bursts are different in that neither a shortage nor an excess shows up in the hard X-ray lightcurve.

Coherent oscillations and the evolution of the apparent emission area in the decaying phase of radius-expansion bursts from 4U 1636–53

We analysed all archival data of the low-mass X-ray binary 4U 1636--53 with the Rossi X-ray Timing Explorer (1490 observations). We found a total of 336 type-I X-ray bursts from this source. In the time-resolved spectra of 69 of these bursts, close to the peak of the burst, the best-fitting blackbody radius shows the sharp increase and decrease that is typical of photospheric radius-expansion (PRE) bursts. We found that in 17 of these 69 PRE bursts, after the touchdown point, the blackbody radius increases again quickly after about 1 second, and from then on the radius decreases slightly or it remains more or less constant. In the other 52 PRE bursts, after touchdown, the radius of the blackbody stays more or less constant for $\sim 2 - 8$ seconds, and after that it increases slowly. Interestingly, those PRE bursts in which the blackbody radius remains more or less constant for $\simmore 2$ seconds show coherent oscillations in the tail of the burst, whereas those PRE bursts in which the blackbody radius changes rapidly after touchdown show no coherent oscillations in the tail of the burst. We found that the distribution of durations of the post touchdown phase between these two groups of PRE bursts is significantly different; the Kolmogorov-Smirnov probability that the two groups of PRE bursts come from the same parent populations is only $3.5 \times 10^{-7}$. This is the first time that the presence of burst oscillations in the tail of X-ray bursts is associated with a systematic behaviour of the spectral parameters in that phase of the bursts. This result is consistent with predictions of models that associate the oscillations in the tail of X-ray bursts with the propagation of a cooling wake in the material on the neutron-star surface during the decay of the bursts.

On the spreading layer emission in luminous accreting neutron stars

Emission of the neutron star surface potentially contains information about its size and thus of vital importance for high energy astrophysics. In spite of the wealth of data on the emission of luminous accreting neutron stars, the emission of their surfaces is hard to disentangle from their time averaged spectra. A recent X-ray transient source XTE J1701-462 has provided a unique dataset covering the largest ever observed luminosity range for a single source. In this paper, we extract the spectrum of the boundary layer between the inner part of the accretion disc and the neutron star surface with the help of maximally spectral model-independent method. We show compelling evidences that the energy spectrum of the boundary layer stays virtually the same over factor of 20 variations of the source luminosity. It is rather wide and cannot be described by a single temperature blackbody spectrum, probably because of the inhomogeneity of the boundary layer and a spread in the colour temperature. The observed maximum colour temperature of the boundary/spreading layer emission of kT~2.4-2.6 keV is very close to the maximum observed colour temperature in the photospheric radius expansion X-ray bursts, which is set by the limiting Eddington flux at the neutron star surface. Observed stability of the boundary layer spectrum and its maximum colour temperature strongly supports theoretical models of the boundary/spreading layers on surfaces of luminous accreting neutron stars, which assume the presence of a region emitting at the local Eddington limit. Variations in the luminosity in that case lead to changes in the size of this region, but affect less the spectral shape. Elaboration of this model will provide solid theoretical grounds for measurements of the neutron star sizes using the emission of the boundary/spreading layers of luminous accreting neutron stars.

Models of hydrostatic magnetar atmospheres at high luminosities

We investigate the possibility of Photospheric Radius Expansion (PRE) during magnetar bursts. Identification of PRE would enable a determination of the magnetic Eddington limit (which depends on field strength and neutron star mass and radius), and shed light on the burst mechanism. To do this we model hydrostatic atmospheres in a strong radial magnetic field, determining both their maximum extent and photospheric temperatures. We find that spatially-extended atmospheres cannot exist in such a field configuration: typical maximum extent for magnetar-strength fields is ~10 m (as compared to 200 km in the non-magnetic case). Achieving balance of gravitational and radiative forces over a large range of radii, which is critical to the existence of extended atmospheres, is rendered impossible in strong fields due to the dependence of opacities on temperature and field strength. We conclude that high luminosity bursts in magnetars do not lead to expansion and cooling of the photosphere, as in the non-magnetic case. We also find the maximum luminosity that can propagate through a hydrostatic magnetar atmosphere to be lower than previous estimates. The proximity and small extent of the photospheres associated with the two different polarization modes also calls into question the interpretation of two blackbody fits to magnetar burst spectra as being due to extended photospheres.

EVIDENCE FOR ACCRETION RATE CHANGE DURING TYPE I X-RAY BURSTS

The standard approach for time-resolved X-ray spectral analysis of thermonuclear bursts involves subtraction of the pre-burst emission as background. This approach implicitly assumes that the persistent flux remains constant throughout the burst. We reanalyzed 332 photospheric radius expansion bursts observed from 40 sources by the Rossi X-Ray Timing Explorer, introducing a multiplicative factor fa to the persistent emission contribution in our spectral fits. We found that for the majority of spectra the best-fit value of fa is significantly greater than 1, suggesting that the persistent emission typically increases during a burst. Elevated fa values were not found solely during the radius expansion interval of the burst, but were also measured in the cooling tail. The modified model results in a lower average value of the χ2 fit statistic, indicating superior spectral fits, but not yet to the level of formal statistical consistency for all the spectra. We interpret the elevated fa values as an increase of the mass accretion rate onto the neutron star during the burst, likely arising from the effects of Poynting–Robertson drag on the disk material. We measured an inverse correlation of fa with the persistent flux, consistent with theoretical models of the disk response. We suggest that this modified approach may provide more accurate burst spectral parameters, as well as offering a probe of the accretion disk structure.

The cooling phase of Type I X-ray bursts observed with RXTE in 4U 1820−30 does not follow the canonical F ∝ T4 relation

We analysed the complete set of bursts from the neutron-star low-mass X-ray binary 4U 1820-30 detected with the Rossi X-ray Timing Explorer (RXTE). We found that all are photospheric radius expansion bursts, and have similar duration, peak flux and fluence. From the analysis of time-resolved spectra during the cooling phase of the bursts, we found that the relation between the bolometric flux and the temperature is very different from the canonical F - T^4 relation that is expected if the apparent emitting area on the surface of the neutron star remains constant. The flux-temperature relation can be fitted using a broken power law, with indices 2.0$\pm$0.3 and 5.72$\pm$0.06. The departure from the F - T^4 relation during the cooling phase of the X-ray bursts in 4U 1820-30 could be due to changes in the emitting area of the neutron star while the atmosphere cools-down, variations in the colour-correction factor due to chemical evolution, or the presence of a source of heat, e.g. residual hydrogen nuclear burning, playing an important role when the burst emission ceases.

PHOTOSPHERIC RADIUS EXPANSION IN SUPERBURST PRECURSORS FROM NEUTRON STARS

Thermonuclear runaway burning of carbon is in rare cases observed from accreting neutron stars as day-long X-ray flares called superbursts. In the few cases where the onset is observed, superbursts exhibit a short precursor burst at the start. In each instance, however, the data was of insufficient quality for spectral analysis of the precursor. Using data from the propane anti-coincidence detector of the PCA instrument on RXTE, we perform the first detailed time resolved spectroscopy of precursors. For a superburst from 4U 1820-30 we demonstrate the presence of photospheric radius expansion. We find the precursor to be 1.4-2 times more energetic than other short bursts from this source, indicating that the burning of accreted helium is insufficient to explain the full precursor. Shock heating would be able to account for the lacking energy. We argue that this precursor is a strong indication that the superburst starts as a detonation, and that a shock induces the precursor. Furthermore, we employ our technique to study the superexpansion phase of the same superburst in greater detail.

Constraining neutron star EoS from cooling stages of X-ray bursts

AbstractThermal emission during X-ray bursts is a powerful tool to determine neutron star masses and radii, if the Eddington flux and the apparent radius in the cooling tail can be measured accurately, and distances to the sources are known. We propose here an improved method of determining the basic stellar parameters using the data from the cooling phase of long, photospheric radius expansion bursts covering a large range of luminosities. For this purpose, we computed a large set of atmosphere models for burst luminosities varying by two orders of magnitude and for various chemical compositions and surface gravities. We show that the variation of the inverse square root of the apparent blackbody radius with the flux, observed during the photospheric radius expansion bursts from a number of sources at low accretion rate is entirely consistent with the theoretical expectations of the color-correction factor evolution. However, for bursts happening at higher accretion rates the observed evolution is inconsistent with theory, implying that accretion strongly disturbs the neutron star atmosphere. These findings have profound implications for the recent claims on determination of the neutron star radii and masses from such bursts. Our method allows us to determine both the Eddington flux and the ratio of the stellar apparent radius to the distance much more reliably. For 4U 1724-307, we find a lower limit on the neutron star radius of 13 km, independently of the chemical composition. These results suggest that the matter inside neutron stars is characterized by a stiff equation of state.

CONSTRAINTS ON THE MASS AND RADIUS OF THE ACCRETING NEUTRON STAR IN THE RAPID BURSTER

The Rapid Burster (MXB 1730-335) is a unique object, showing both type I and type II X-ray bursts. A type I burst of the Rapid Burster was observed with Swift/XRT on 2009 March 5, showing photospheric radius expansion for the first time in this source. We report here on the mass and radius determination from this photospheric radius expansion burst using a Bayesian approach. After marginalization over the likely distance of the system (5.8-10 kpc) we obtain M=1.1+/-0.3 M_sun and R=9.6+/-1.5 km (1-sigma uncertainties) for the compact object, ruling out the stiffest equations of state for the neutron star. We study the sensitivity of the results to the distance, the color correction factor, and the hydrogen mass fraction in the envelope. We find that only the distance plays a crucial role.

XMM-NEWTONFINDS THAT SAX J1750.8–2900 MAY HARBOR THE HOTTEST, MOST LUMINOUS KNOWN NEUTRON STAR

We have performed the first sensitive X-ray observation of the low-mass X-ray binary SAX J1750.8-2900 in quiescence with XMM-Newton. The spectrum was fit to both a classical black body model, and a non-magnetized, pure hydrogen neutron star atmosphere model. A power law component was added to these models, but we found that it was not required by the fits. The distance to SAX J1750.8-2900 is known to be D = 6.79 kpc from a previous analysis of photospheric radius expansion bursts. This distance implies a bolometric luminosity (as given by the NS atmosphere model) of (1.05 +/- 0.12) x 10^34 (D/6.79 kpc)^2 erg s^-1, which is the highest known luminosity for a NS LMXB in quiescence. One simple explanation for this surprising result could be that the crust and core of the NS were not in thermal equilibrium during the observation. We argue that this was likely not the case, and that the core temperature of the NS in SAX J1750.8-2900 is unusually high.

Thermonuclear X-ray bursts from the 401-Hz accreting pulsar IGR J17498−2921: indication of burning in confined regions

We use the 2011 Rossi X-ray Timing Explorer (RXTE) proportional counter array (PCA) data of the 401 Hz accreting pulsar and burster IGR J17498-2921 to perform timing analysis and time-resolved spectroscopy of 12 thermonuclear X-ray bursts. We confirm previously reported burst oscillations from this source with a much higher significance (8.8\sigma). We notice that the bursts can be divided into three groups: big photospheric radius expansion (PRE) bursts are about ten times more luminous than medium bursts, while the latter are about ten times more luminous than small bursts. The PCA field-of-view of these observations contains several known bursters, and hence some of the observed bursts might not be from IGR J17498-2921. The oscillations during big bursts at the known pulsar frequency show that these bursts were definitely from IGR J17498-2921. We find that at least several of the other bursts were also likely originated from IGR J17498-2921. Spectral analysis reveals that the luminosity differences among various bursts are primarily due to differences in normalizations, and not temperatures, even when we consider the effects of colour factor. This shows burning on a fraction of the stellar surface for those small and medium bursts, which originated from IGR J17498-2921. The low values of the upper limits of burst oscillation amplitude for these bursts suggest a small angle between the spin axis and the magnetic axis. We find indications of the PRE nature of a medium burst, which likely originated from IGR J17498-2921. If true, then, to the best of our knowledge, this is the first time that two PRE bursts with a peak count rate ratio of as high as {\approx} 12 have been detected from the same source.

SYSTEMATIC UNCERTAINTIES IN THE SPECTROSCOPIC MEASUREMENTS OF NEUTRON-STAR MASSES AND RADII FROM THERMONUCLEAR X-RAY BURSTS. II. EDDINGTON LIMIT

Time resolved X-ray spectroscopy of thermonuclear bursts observed from low mass X-ray binaries offer a unique tool to measure neutron star masses and radii. In this paper, we continue our systematic analysis of all the X-ray bursts observed with RXTE from X-ray binaries. We determine the events which show clear evidence for photospheric radius expansion and measure the Eddington limits for these accreting neutron stars using the bolometric fluxes attained at the touchdown moments of each X-ray burst. We employ a Bayesian technique to investigate the degree to which the Eddington limit for each source remains constant between bursts. We find that for sources with a large number of radius expansion bursts, systematic uncertainties are at a 5-10% level. Moreover, in six sources with only pairs of Eddington-limited bursts, the distribution of fluxes is consistent with a ~10% fractional dispersion. This indicates that the spectroscopic measurements of neutron star masses and radii using thermonuclear X-ray bursts can reach the level of accuracy required to distinguish between different neutron star equations of state, provided that uncertainties related to the overall flux calibration of X-ray detectors are of comparable magnitude.

A NEUTRON STAR STIFF EQUATION OF STATE DERIVED FROM COOLING PHASES OF THE X-RAY BURSTER 4U 1724–307

Thermal emission during X-ray bursts is a powerful tool to determine neutron star masses and radii, if the Eddington flux and the apparent radius in the cooling tail can be measured accurately, and distances to the sources are known. We propose here an improved method of determining the basic stellar parameters using the data from the cooling phase of photospheric radius expansion bursts covering a large range of luminosities. Because at that phase the blackbody apparent radius depends only on the spectral hardening factor (color-correction), we suggest to fit the theoretical dependences of the color-correction versus flux in Eddington units to the observed variations of the inverse square root of the apparent blackbody radius with the flux. We show that spectral variations observed during a long photospheric radius expansion burst from 4U 1724-307 are entirely consistent with the theoretical expectations for the passively cooling neutron star atmospheres. Our method allows us to determine both the Eddington flux (which is found to be smaller than the touchdown flux by 15%) and the ratio of the stellar apparent radius to the distance much more reliably. We then find a lower limit on the neutron star radius of 14 km for masses below 2.2M_sun, independently of the chemical composition. These results suggest that the matter inside neutron stars is characterized by a stiff equation of state. We finally show that the apparent blackbody emitting area in the cooling tails of the short bursts from 4U 1724-307 is two times smaller than that for the long burst and their evolution does not follow the theory. This makes their usage for determination of the neutron star parameters questionable and casts serious doubts on the results of previous works that used for the analysis similar bursts from other sources. [abridged]

The cooling phase of Type I X-ray bursts in 4U 1636−53

Time-resolved spectra during the cooling phase of thermonuclear X-ray bursts in low-mass X-ray binaries (LMXBs) can be used to measure the radii and masses of neutron stars. We analyzed ~ 300 bursts of the LMXB 4U 1636-53 using data from the Rossi X-ray Timing Explorer. We divided the bursts in three groups, photospheric radius expansion (PRE), hard non-PRE and soft non-PRE bursts, based on the properties of the bursts and the state of the source at the time of the burst. For the three types of bursts, we found that the average relation between the bolometric flux and the temperature during the cooling phase of the bursts is significantly different from the canonical $F \propto T^4$ relation that is expected if the apparent emitting area on the surface of the neutron star remains constant as the flux decreases during the decay of the bursts. We also found that a single power law cannot fit the average flux-temperature relation for any of the three types of bursts, and that the flux-temperature relation for the three types of bursts is significantly different. Finally, for the three types of bursts, the temperature distribution at different flux levels during the decay of the bursts is significantly different. From the above we conclude that hard non-PRE bursts ignite in a hydrogen-rich atmosphere, whereas for soft non-PRE and PRE bursts the fuel is helium-rich. We further conclude that the metal abundance in the neutron star atmosphere decreases as the bursts decay, probably because the heavy elements sink faster in the atmosphere than H and He.

THE EQUATION OF STATE FROM OBSERVED MASSES AND RADII OF NEUTRON STARS

We determine an empirical dense matter equation of state (EOS) from a heterogeneous data set of six neutron stars: three Type-I X-ray bursters with photospheric radius expansion, studied by Özel et al., and three transient low-mass X-ray binaries. We critically assess the mass and radius determinations from the X-ray burst sources and show explicitly how systematic uncertainties, such as the photospheric radius at touchdown, affect the most probable masses and radii. We introduce a parameterized EOS and use a Markov chain Monte Carlo algorithm within a Bayesian framework to determine nuclear parameters such as the incompressibility and the density dependence of the bulk symmetry energy. Using this framework we show, for the first time, that these parameters, predicted solely on the basis of astrophysical observations, all lie in ranges expected from nuclear systematics and laboratory experiments. We find significant constraints on the mass–radius relation for neutron stars, and hence on the pressure–density relation of dense matter. The predicted symmetry energy and the EOS near the saturation density are soft, resulting in relatively small neutron star radii around 11–12 km for M = 1.4 M☉. The predicted EOS stiffens at higher densities, however, and our preferred model for X-ray bursts suggests that the neutron star maximum mass is relatively large, 1.9–2.2 M☉. Our results imply that several commonly used equations of state are inconsistent with observations.

Type I X-ray bursts and burst oscillations in the accreting millisecond X-ray pulsar IGR J17511−3057

We report the discovery of burst oscillations at the spin frequency in ten thermonuclear bursts from the accreting millisecond X-ray pulsar (AMXP) IGR J17511-3057. The burst oscillation properties are, like those from the persistent AMXPs SAX J1808.4-3658 and XTE J1814-338, anomalous compared to burst oscillations from intermittent pulsars or non-pulsing LMXBs. Like SAX J1808.4-3658 they show frequency drifts in the rising phase rather than the tail. There is also evidence for harmonic content. Where IGR J17511-3057 is unusual compared to the other two persistent pulsars is that oscillations are not detected throughout all bursts. As accretion rate drops the bursts get brighter and their rise/decay time scales become shorter, while the oscillation amplitude falls below the detection threshold: first in the burst peak and then also in the rise. None of the bursts from IGR J17511-3057 show evidence for photospheric radius expansion (which might be expected to suppress oscillation amplitude) which allow us to set an upper limit to the distance of 6.9 kpc. We discuss the implications of our results for models of the burst oscillation mechanism.

The X-ray spectrum of the newly discovered accreting millisecond pulsar IGR J17511−3057

We report on a 70 ks XMM–Newton Target of Opportunity (ToO) observation of the newly discovered accreting millisecond pulsar, IGR J17511−3057. Pulsations at 244.833 9512(1) Hz are observed throughout the outburst with an rms-pulsed fraction of 14.4(3) per cent. Pulsations have been used to derive a precise solution for the Porb= 12 487.51(2) s binary system. The measured mass function indicates a main-sequence companion star with a mass between 0.15 and 0.44 M⊙. The XMM–Newton 0.5–11 keV spectrum of IGR J17511−3057 can be modelled by at least three components, which we interpret, from the softest to the hardest, as multi-coloured disc emission, thermal emission from the neutron star surface and thermal Comptonization emission. Spectral fit of the XMM–Newton data and of the Rossi X-ray Timing Explorer (RXTE) data, taken in a simultaneous temporal window, well constrain the Comptonization parameters: the electron temperature, kTe= 51+6−4 keV, is rather high, while the optical depth (τ= 1.34+0.03−0.06) is moderate. The energy dependence of the pulsed fraction supports the interpretation of the cooler thermal component as coming from the accretion disc, and indicates that the Comptonizing plasma surrounds the hot spots on the neutron star surface, which in turn provides the seed photons. Signatures of reflection, such as a broadened iron Kα emission line and a Compton hump at ∼30 keV, are also detected. We derive from the smearing of the reflection component an inner disc radius of ≳40 km for a 1.4 M⊙ neutron star, and an inclination between 38° and 68°. XMM–Newton also observed two type I X-ray bursts, whose fluence and recurrence time suggest that the bursts are ignited in a nearly pure helium environment. No photospheric radius expansion is observed, thus leading to an upper limit on the distance to the source of 10 kpc. A lower limit of 6.5 kpc can be also set if it is assumed that emission during the decaying part of the burst involves the whole neutron star surface. Pulsations are observed during the burst decay with an amplitude similar to the persistent emission. They are also compatible with being phase locked to pre-burst pulsations, suggesting that the location on the neutron star surface where they are formed does not change much during bursts.

SUPER-EDDINGTON FLUXES DURING THERMONUCLEAR X-RAY BURSTS

It has been known for nearly three decades that the energy spectra of thermonuclear X-ray bursts are often well-fit by Planck functions with temperatures so high that they imply a super-Eddington radiative flux at the emitting surface, even during portions of bursts when there is no evidence of photospheric radius expansion. This apparent inconsistency is usually set aside by assuming that the flux is actually sub-Eddington and that the fitted temperature is so high because the spectrum has been distorted by the energy-dependent opacity of the atmosphere. Here we show that the spectra predicted by currently available conventional atmosphere models appear incompatible with the highest-precision measurements of burst spectra made using the Rossi X-ray Timing Explorer, such as during the 4U 1820-30 superburst and a long burst from GX 17+2. In contrast, these measurements are well-fit by Bose-Einstein spectra with high temperatures and modest chemical potentials. Such spectra are very similar to Planck spectra. They imply surface radiative fluxes more than a factor of three larger than the Eddington flux. We find that segments of many other bursts from many sources are well-fit by similar Bose-Einstein spectra, suggesting that the radiative flux at the emitting surface also exceeds the Eddington flux during these segments. We suggest that burst spectra can closely approximate Bose-Einstein spectra and have fluxes that exceed the Eddington flux because they are formed by Comptonization in an extended, low-density radiating gas supported by the outward radiation force and confined by a tangled magnetic field.