Dynamical evolution of idealised star cluster models

This thesis is concerned with the dynamical evolution of globular star clusters modelled as the classical gravitational N-body problem. The models in this thesis are idealised in order to allow the detailed study of particular dynamical aspects of the cluster evolution. Examples of properties which tend to be omitted are stellar evolution, primordial binaries and the effect of an external tidal gravitational field. The methods used in this thesis are gas models, N-body models and physical arguments. One of the main topics in this thesis is gravothermal oscillations in multicomponent star clusters. The evolution of one-component globular clusters, systems with equal particle masses, is quite well understood. However, the evolution of more realistic globular clusters, with a range of particle masses, is a much more complicated matter. The condition for the on-set of gravothermal oscillations in a one-component system is simply that the number of stars is greater than a certain number ( ≈7000). In a multi-component system the relationship between the number of stars at which the gravothermal oscillations first appear and the stellar mass distribution of a cluster is a complex one. In order to investigate this phenomenon two different types of multi-component systems were studied: two-component systems (the simplest approximation of a mass spectrum, Chapter 2) and ten-component systems (which were realisations of continuous power law IMFs, Chapter 3). In both cases the critical number of stars at which gravothermal oscillations first appear are found empirically for a range of stellar mass distributions. The nature of the oscillations themselves are investigated and it is shown that the oscillations can be understood by focusing on the behaviour of the heavier stars within the cluster. A parameter Nef (de nined Mtot/mmax where Mtot is the total mass and mmax is the maximum stellar mass) acts as an approximate stability boundary for multicomponent systems.The stability boundary was found to be at Nef ~- 12000. In this Chapter 4, globular star clusters which contain a sub-system of stellar-mass black holes (BH) are investigated. This is done by considering two-component models, as these are the simplest approximation of more realistic multi-mass systems, where one component represents the BH population and the other represents all the other stars. These systems are found to undergo a long phase of evolution where the centre of the system is dominated by a BH sub-system. After mass segregation has driven most of the BH into a compact sub-system, the evolution of the BH sub-system is found to be in uenced by the cluster in which it is contained. The BH sub-system evolves in such a way as to satisfy the energy demands of the whole cluster, just as the core of a one component system must satisfies the energy demands of the whole cluster. The BH sub-system is found to exist for a significant amount of time. It takes approximately 10trh;i, where trh;i is the initial half-mass relaxation time, from the formation of the compact BH sub-system up until the time when 90% of the sub-system total mass is lost (which is of order 103 times the half-mass relaxation time of the BH sub-system at its time of formation). Based on theoretical arguments the rate of mass loss from the BH sub-system (M2) is predicted to be (βζM)/(αtrh): where M is the total mass, trh is the half-mass relaxation time, and α, β, ζ are three dimensionless parameters. (see Section 4.3 for details). An interesting consequence of this is that the rate of mass loss from the BH sub-system is approximately independent of the stellar mass ratio (m2/m1) and the total mass ratio (M2/M1) (in the range m2/m1 ≥ 10 and M2/M1 ≈ 10-2, where m1, m2 are the masses of individual low-mass and high-mass particles respectively, and M1, M2 are the corresponding total mass). The theory is found to be in reasonable agreement with most of the results of a series of N-body simulations, and all of the models if the value of ζ is suitable adjusted. Predictions based on theoretical arguments are also made about the structure of BH sub-systems. Other aspects of the evolution are also considered such as the conditions for the onset of gravothermal oscillation. The final chapter (Chapter 5) of the thesis contains some concluding comments as well as a discussion on some possible future projects, for which the results in this thesis would be useful

l=1::Weinberg's weakly damped mode in an N-body model of a spherical stellar system

Spherical stellar systems such as King models, in which the distribution function is a decreasing function of energy and depends on no other invariant, are stable in the sense of collisionless dynamics. But Weinberg showed, by a clever application of the matrix method of linear stability, that they may be nearly unstable, in the sense of possessing {\sl weakly} damped modes of oscillation. He also demonstrated the presence of such a mode in an $N$-body model by endowing it with initial conditions generated from his perturbative solution. In the present paper we provide evidence for the presence of this same mode in $N$-body simulations of the King $W_0 = 5$ model, in which the initial conditions are generated by the usual Monte Carlo sampling of the King distribution function. It is shown that the oscillation of the density centre correlates with variations in the structure of the system out to a radius of about 1 virial radius, but anticorrelates with variations beyond that radius. Though the oscillations appear to be continually reexcited (presumably by the motions of the particles) we show by calculation of power spectra that Weinberg's estimate of the period (strictly, $2\pi$ divided by the real part of the eigenfrequency) lies within the range where the power is largest. In addition, however, the power spectrum displays another very prominent feature at shorter periods, around 5 crossing times.Comment: 9 pages, 6 figures, footnote added in Sec.3.2, no other change to the version published in MNRA

The Kinematic Richness of Star Clusters - II. Stability of Spherical Anisotropic Models with Rotation

International audienc

Cross-sectional study of the provision of interventional oncology services in the UK

Objective: To map out the current provision of interventional oncology (IO) services in the UK. Design: Cross-sectional multicentre study. Setting: All National Health Service (NHS) trusts in England and Scottish, Welsh and Northern Ireland health boards. Participants: Interventional radiology (IR) departments in all NHS trusts/health boards in the UK. Results: A total of 179 NHS trusts/health boards were contacted. We received a 100% response rate. Only 19 (11%) institutions had an IO lead. 144 trusts (80%) provided IO services or had a formal pathway of referral in place for patients to a recipient trust. 21 trusts (12%) had plans to provide an IO service or formal referral pathway in the next 12 months only. 14 trusts (8%) did not have a pathway of referral and no plans to implement one. 70 trusts (39%) offered supportive and disease-modifying procedures. One trust had a formal referral pathway for supportive procedures. 73 trusts (41%) provided only supportive procedures (diagnostic or therapeutic). Of these, 43 (59%) had a referral pathway for disease-modifying IO procedures, either from a regional cancer network or through IR networks and 30 trusts (41%) did not have a referral pathway for disease-modifying procedures. Conclusion: The provision of IO services in the UK is promising; however, collaborative networks are necessary to ensure disease-modifying IO procedures are made accessible to all patients and to facilitate larger registry data for research with commissioning of new services

Black box probabilistic numerics

Peer reviewe

Black box probabilistic numerics

Peer reviewe