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Neutron stars are extremely dense objects that harbour extreme physical environments. They consist of a core with a radius of ~10 kilometers and a one-kilometer thick crust. Neutron stars in X-ray binaries accrete matter from a companion star. In transient systems accretion is not continuous, instead accretion outbursts are observed. During an outburst, material accumulates on the surface of the neutron star. This compresses the deeper layers of the crust and induces processes that release heat. Consequently, the neutron star crust heats up and becomes hotter than the core. After the outburst, the crust cools down to restore thermal equilibrium with the core. In this thesis, the thermal evolution of transiently accreting neutron stars is studied. We model the thermal evolution using the code NSCool and compare the results with observations to unravel properties of neutron stars. We improve our modelling methods by taking into account accretion rate variability (Chapter 2). We model the neutron star in KS 1731-260 and find that the constrained properties are strongly affected by variations in accretion rate. In Chapter 3 we model the 20-year outburst history of Aql X-1 and find that the neutron star does not have enough time between its frequent outbursts to restore crust-core equilibrium. Modelling Terzan 5 X-2 reveals that the neutron star in this system might have unusual crust properties (Chapter 4). In Chapter 5, we model how a neutron star reaches an equilibrium state over many outburst cycles and investigate how this equilibrium is affected by various parameters.

We focus on statistical estimation of neutron star parameters, conditional on X-ray data acquired by contemporary and future space telescopes. In particular, we discuss Bayesian inference of: (i) parameters such as masses and radii, which define the spacetime exterior to neutron stars; and (ii) parameters defined by a model of the equation of state of interior dense matter. We tailor to X-ray spectral-timing (also termed pulse-profile or light-curve modelling) with telescopes such as NICER (NASA’s Neutron Star Interior Composition Explorer, installed on the International Space Station). We develop a prototype open-source software package, X-PSI, for neutron star X-ray Pulsation Simulation and Inference. This package is designed to interface native likelihood functionality with open-source posterior sampling software, for use on CPU-nodes of high-performance computing systems. We apply the X-PSI modelling framework to NICER observations of the rotation-powered millisecond pulsar J0030+0451; we jointly constrain its mass and radius, together with the spatial structure of hot surface regions whose X-ray emission signatures are rotationally modulated (or pulsed). The analysis is a proof of concept and acts as a pathfinder for ongoing and future analyses of NICER data. The rotation-powered pulsars targeted by NICER are identified as a logical baseline source-class for community-supported development of an open-source package for neutron star X-ray spectral-timing: specifically, a package that enables the definition and estimation of fundamental parameters pertaining to relativistic stellar structure.

Neutron stars where the output emission is dominated by the decay of an ultra-strong magnetic field, typically in excess of 1014 Gauss, are characterized as magnetars. This decay may occur both gradually and abruptly, which in the latter case results in bright recurrent soft gamma-ray bursts and occasionally giant flares. Here we investigate and develop theoretical models of the trigger and emission mechanisms of these bursts, which involve extreme and fundamental (astro)physical processes. By comparing the model predictions with observed data, we aim to constrain distinct magnetar burst mechanisms. Assuming that magnetar giant flares are triggered by the spontaneous development of a tearing instability in a magnetospheric current sheet, we obtain estimates for the dimension of the corresponding reconnection region. Moreover, we assess inverse Compton and synchrotron emission models for high-energy emission that was observed following the impulsive phase of the giant flare from the magnetar SGR 1806-20 on 2004 December 27. We conclude that the high-energy emission may constitute synchrotron radiation from shock-accelerated particles, whereby the shock is generated between the evaporative outflow from the star and a downward directed reconnection exhaust. We furthermore study the characteristics of the recurrent bursts as manifestations of systems of self-organised critically, such as flares and crust-quakes, and develop a model where we investigate the detectability of a phase-dependence of magnetar burst properties.

A magnetar is a type of neutron star with an extremely strong magnetic field, a hundred million times stronger than the strongest fields made in laboratories. In turn, a neutron star is one of the end products of stellar evolution, and is formed when a massive star has used up all its fuel for nuclear fusion. A neutron star has a mass around twice that of the sun, but a diameter of only twenty km, making it extremely dense. Studying neutron stars, and magnetars in particular, is the only way to study what happens to matter at these extreme densities and magnetic field strengths. Many magnetars are seen through their bursts, which are short bright flashes of radiation on the edge between X-ray and gamma-ray radiation. When radiation interacts with matter, it exerts a small force on that matter. In magnetar bursts, the intensity of the emitted radiation can let this force become as strong as gravity. In Chapters 2 and 3 of this thesis I have studied what happens when this force does overcome gravity, by making numerical models of magnetar atmospheres during a burst. I conclude that during these bursts magnetars will develop highly irregular outflows of matter. In Chapter 4 I have developed a numerical model to test how the atmosphere of a magnetar alters the properties of its emission, in particular to model the observed beaming of magnetar emission, an effect whereby radiation is emitted in a preferred direction. I find that a simple model for the atmosphere of a magnetar can recreate the observed amount of beaming. This makes my model a promising way to use observations of beamed magnetar bursts to learn more about the objects emitting these bursts.

Neutron stars are the most compact known stars; their cores are of higher density than an atomic nucleus. Their rotation rates are generally very predictable, with a slow decrease over time. This spin-down is occasionally interrupted, however, by abrupt 'glitches' when the rotation rate increases slightly. Glitches are attributed to a sudden coupling between a neutron star's crust and an interior superfluid component - and therefore the nature and frequency of glitches encodes important information about physics at densities which cannot be attained in terrestrial laboratories. This thesis contains analysis of both observational and theoretical aspects of the physics of neutron-star glitches. It describes glitch detectability limits and a search algorithm, which together allow for a more systematic analysis of glitch data than was previously possible. These are applied to one of the most closely-observed neutron stars, the Crab pulsar, finding it has an unexpected deficit of small glitches. Next, a multifluid model of the neutron-star interior is used to simulate glitches; it is shown that different kinds of glitch and post-glitch behaviour can be explained within this theoretical approach. One highly-magnetised neutron star with particularly notable glitch properties is PSR J1119-6127. This thesis presents a detailed observational study of its glitches and discusses their implications for theoretical models of the dynamics of the neutron star's crust and exterior magnetic field. Finally, the aftermath of a large glitch from the Vela pulsar is analysed, and the potential to use this data to constrain models of the interior superfluid is discussed.

X-ray bursts originate from accreting neutron stars (NSs) in X-ray binaries (XRBs). They come in two flavours: thermonuclear bursts are due to the sudden runaway burning of the material accreted on the surface; accretion bursts signal a sudden change in the mass accretion rate, leading to enhanced emission in the innermost regions of the accretion flow. While thermonuclear bursts have been observed from 105 NSs as of writing, accretion bursts remain enigmatically confined to only two sources. One of them, the Rapid Burster (RB), is the only one to have shown both types of bursts. My PhD project has dealt with studying this extraordinary source to understand the reason behind its uniqueness and, through that, answer open questions regarding both thermonuclear burning and unstable accretion.

The research in this thesis details ways to characterise and understand variability in short magnetar bursts. Magnetars, neutron stars with exceptionally strong magnetic fields, show variability on many different time scales. Particularly at high X-ray energies, they show bright bursts that are believed to be created by what we call starquakes: a sudden release of energy that triggers global oscillations of the star's solid crust and possibly the interior. These oscillations, observed in the X-ray light curves of the bright giant flares, are of particular interest, because they have to potential to probe the stellar composition and constrain related nuclear physics. Because standard statistical methods fail when applied to magnetar bursts, the research in this thesis aims at developing new methods to search for and characterise oscillations in the light curves of short magnetar bursts, and at un- derstanding variability in these sources as well as the underlying physical processes in general. With this new methodology, several oscillations were found in short bursts from two different magnetars, indicating that similar processes are active in both the short recurrent bursts and the longer, brighter giant flares. Additionally, a new model was developed for the underlying aperiodic variability, which, for the first time, allows a comparison of the X-ray variability in magnetar bursts with toy models of the underlying physics.

De voortplantingssnelheid van de vlam ontstaan door een ‘type 1-explosie’ bij neutronensterren hangt sterk af van de rotatie van de ster en de thermische geleiding van de materie. De materie naast de brandende gebieden wordt door geleiding verhit, maar hydrodynamische effecten versnellen de voortplanting van de vlam terwijl ze het branden stabiel houden. Deze effecten worden veroorzaakt door de zogeheten Corioliskracht die afhangt van de rotatie van de ster. Yuri Cavecchi bestudeerde verschijnselen die plaatsvinden rond en op het oppervlak van neutronensterren: hun ‘type 1-explosie’ (veroorzaakt onder meer een intense flits van röntgenstraling) en de effecten die hun zwaartekracht heeft op het uitgezonden licht. Neutronensterren hebben ongeveer dezelfde massa als onze zon, terwijl hun straal slechts ongeveer 10 km is.