1. Pavel Bakala Gabriel Török, Zdeněk Stuchlík, Eva Šrámková Institute of Physics Faculty of Philosophy and Science Silesian University in Opava Czech Republic Modelling X-ray power spectra of hot spots on neutron star surface and in thin accretion disc

2. Outline  Introduction and motivation  LSD (Lensing Simulations Device) software code architecture  Preliminary results  Hot spots on the neutron star surface: visualisation, light curves and power spectra  Orbiting hot sport in thin accretion discs with epicyclic frequencies, light curves and power spectra  Conclusions

3. Introduction and motivation The described fully-relativistic LSD software represents a complex, widely configurable tool for modelling of gravitational lensing. The software enables us to define various sources of radiation with different time variability and can process the effects of light propagation along null geodesics including both gravitational and Doppler effects and intensity (de)amplification. The results can be obtained for different classes of static or free-falling observers. There are several motivations behind this study. One of our main objectives lie in timing analysis of the X-ray radiation of binary systems containing a black hole or neutron star within the context of research of the quasi-periodic oscillations. Of significant interest is also visualisation of the general relativistic effects. The obtained results can be of a good use when comparing the models' predictions to the observational data and the visualisation images and movies could be as well used for teaching or popularisation needs. There is a new generation of satellites that has been under preparation in ESA which should soon enable us to directly observe the light curves of the discussed hot spots. Apparently, comparing such observed light curves to those modeled here could represent another tool for exploring the properties of compact objects.

4. LSD software code modular architecture (Lensing Simulations Device) Input: Radiating objects Lensing: Relativistic raytracing Outputs: light curves, power spectra, visualisations Spacetime metric

5. Input module Description of radiating objects C++ classes of objects –Coordinate grid on the surface –Coordinates of surface pixels as functions of time (moving objects, rotation, oscillations of surfaces …) –RGB components of visible radiation –Time variability of intensity –Parent class for user-defined radiating object Predefined objects –Sphere –Thin disc –Radiating spot as a subclass of sphere and disc classes –Stellar background in infinity

6. Lensing modules for relativistic raytracing General features and functions –Solution of emitor-observer problem (raytracing) in given spacetime (identification of constant of motion for proper ray) –Gravitational and doppler frequency shift –Intensity (de)amplification Schwarschild-de Sitter raytracing engine –Spherically symmetric spacetime with repulsive cosmological constant –Direct support of static and radially free-falling observers –Acceleration of modeling using spherical symmetry of the problem –Separate processing for rays of different order ( contribution of flux generated by the first direct rays, first undirect rays, second direct rays) –Planned upgrade for spherically symetric charged spacetime Reisner–Nordström type (coming soon) Kerr-de Sitter raytracing engine –In preparation (coming soon)

7. Output modules Timing processing module Light curves –Processing of time delay – resampling of delayed fluxes into equidistant time intervals –Separate light curves for different rays with and without time delay –Final entire light curve Power spectra –Fourier decomposition (FFT) of final light curves –Filtering of unphysical low frequencies (numerical effects) –Separate spectra for rays of different order Visualisation module Static pictures –Static snapshot for observer local coordinate system (tetrad) –Export into PNM or BMP images Movies –Sequence of static pictures in a given time –Processing of time delay effect (coming soon)

8. Preliminary results – hot spots on neutron star surface Compact star with two hot spots Parameters of the system –Spots angular size –Inclination of the spots –Star spin frequency –Star mass and radius –Inclination of the observer Lensing in Schwarzschild spacetime Distant observer in infinity

9. Preliminary results – hot spots on neutron star surface Critical radius of the star R c ≈ 3.445 M The maximal vieving angle for distant observer is π The whole surface of such star is observable How the shapes of light curves depend on the radius of the star Inclination of hot spots : 1.5 rad Inclination of distant observer : 1.4 rad Spin frequency : 150 Hz Mass of the star : 1.5 M 

10. R star =3.445M R star =15.000M R star =2.700M Radiation of single spot – separate rays Contributions of higher order rays if R star <R c Spot is permanently observable if R star <=Rc Blind time if R star >Rc

11. R star =3.445M R star =2.700M Final light curve –contributions of both spots Final curve as a sum of spots curves Asymmetry of the curve grows with R due to Doppler shift Some numerical effects, smooth curves need higher time resolution R star =15.000M

12. R star =2.700M Final power spectra Dominating double spin frequency Significant occurrence of odd harmonics if R star <=Rc No odd harmonics and only small contributions of even harmonics for R star >>Rc R star =3.445M

13. Visualisation of the rotating star with hot spots Star with R star <Rc Whole surface is visible Red color of the surface White hot spots ON/OFF switchable effects ‒ Rays with particular order ‒ Doppler shift ‒ Gravitational frequency shift

14. Preliminary results –hot spots in thin accretion disc with radial perturbation Geodesic motion of test particle on a stable circular orbit with epicyclic modulation Radial epicyclic frequency Parameters of the system –Radius of the orbit –Radial perturbation amplitude –Star mass and radius –Inclination of the observer Lensing and motion in the Schwarzschild spacetime Distant observer in infinity

15. Light curve Significant modulation by radial perturbation Small contribution of first undirect rays harmonics for

16. Power spectrum Dominating Keplerian orbital frequency Contribution of radial epicyclic frequency and its combination with Keplerian one Spectrum is not so satisfactory for hot spot interpretation of orbital preccesion model (Stella & Vietri model)

17. Conclusion The software is in testing period. Some modules (Kerr-de Sitter lensing, visualisation with time delay proccesing ) are in the development. We are preparing physically more relevant simulations (disc oscilation).