## Final Activity Report Summary - SOLIDGR (Neutron stars with solid components in general relativity)

Neutron stars provide a meeting point for much extreme physics. In order to model their dynamics, one must account for matter at densities beyond those that can be reached in terrestrial laboratories, strong magnetic fields, an elastic nuclear lattice and various superfluid components.

This project mainly focussed on the modelling of the elastic crust that forms the outer kilometre of a star, which is approximately ten kilometres or so. We developed a versatile mathematical framework that could be used to describe a nuclear lattice through which superfluid neutrons might flow. The description was fully general relativistic, meaning that it took into account effects due to the curved space-time description of gravity.

We analysed the mathematical equations in more detail in order to understand issues concerning shear wave propagation and coupling between the superfluid and crust ions, however this work was not finished by the time of the project completion. In order to investigate some of the astrophysical manifestations of these systems, we calculated the oscillation modes of a neutron star with a crust. These calculations, which were the first of their kind to be carried out in general relativity, could be compared to observational data for quasiperiodic oscillations seen in the tails of magnetar flares.

In addition, we outlined how a comparison with observations could be used to constrain our theoretical neutron star models. As part of the project we also developed a new method for calculating neutron star oscillation modes, including the damping rate because of gravitational wave emission. Furthermore, we considered the magnetar problem within the Newtonian theory of gravity. This allowed us to include magnetic field effects in the analysis and led to a simple toy model that explained the general properties of the observations.

Another aspect of neutron stars with elastic components is that they can sustain shear stresses, and may not be completely symmetric. We developed, again in Newtonian gravity, a new framework for modelling such neutron star 'mountains'. The key issue concerned how large they might be, and we showed that crustal mountains were limited to a height of a fraction of a centimetre on a ten kilometre star. This result should be compared to the upper limit provided by current gravitational wave observations. Our theoretical predictions suggested that observations were not yet able to test our models, but we anticipated that this would change in the not too distance future.

As an extension of this work, we investigated the deformations induced by the neutron star magnetic field. Again, the results indicated that observations were at present not able to test theoretical models. Nevertheless, the results we obtained were important in that they provided an insight into how sensitive the experiments should become before we could expect to glean any real insights into the physics of neutron stars.

This project mainly focussed on the modelling of the elastic crust that forms the outer kilometre of a star, which is approximately ten kilometres or so. We developed a versatile mathematical framework that could be used to describe a nuclear lattice through which superfluid neutrons might flow. The description was fully general relativistic, meaning that it took into account effects due to the curved space-time description of gravity.

We analysed the mathematical equations in more detail in order to understand issues concerning shear wave propagation and coupling between the superfluid and crust ions, however this work was not finished by the time of the project completion. In order to investigate some of the astrophysical manifestations of these systems, we calculated the oscillation modes of a neutron star with a crust. These calculations, which were the first of their kind to be carried out in general relativity, could be compared to observational data for quasiperiodic oscillations seen in the tails of magnetar flares.

In addition, we outlined how a comparison with observations could be used to constrain our theoretical neutron star models. As part of the project we also developed a new method for calculating neutron star oscillation modes, including the damping rate because of gravitational wave emission. Furthermore, we considered the magnetar problem within the Newtonian theory of gravity. This allowed us to include magnetic field effects in the analysis and led to a simple toy model that explained the general properties of the observations.

Another aspect of neutron stars with elastic components is that they can sustain shear stresses, and may not be completely symmetric. We developed, again in Newtonian gravity, a new framework for modelling such neutron star 'mountains'. The key issue concerned how large they might be, and we showed that crustal mountains were limited to a height of a fraction of a centimetre on a ten kilometre star. This result should be compared to the upper limit provided by current gravitational wave observations. Our theoretical predictions suggested that observations were not yet able to test our models, but we anticipated that this would change in the not too distance future.

As an extension of this work, we investigated the deformations induced by the neutron star magnetic field. Again, the results indicated that observations were at present not able to test theoretical models. Nevertheless, the results we obtained were important in that they provided an insight into how sensitive the experiments should become before we could expect to glean any real insights into the physics of neutron stars.