Pavel Bakala

1
Modelling X-ray power spectra of hot spots on
neutron star surface and in thin accretion disc

• Pavel Bakala
• Gabriel Török, Zdenek Stuchlík, Eva Šrámková
• Institute of Physics
• Faculty of Philosophy and ScienceSilesian
  University in Opava
• Czech Republic

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
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Input moduleDescription 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

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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 ReisnerNordström type (coming soon)
• Kerr-de Sitter raytracing engine
• In preparation (coming soon)

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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

• 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?

• Critical radius of the star
• Rc 3.445 M
• The maximal vieving angle for distant observer is
  p
• The whole surface of such star is observable

10
Radiation of single spot separate rays
Rstar3.445M
Rstar2.700M

• Contributions of higher order rays if RstarltRc
• Spot is permanently observable if RstarltRc
• Blind time if RstargtRc

Rstar15.000M
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Final light curve contributions of both spots
Rstar2.700M
Rstar3.445M

• 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

Rstar15.000M
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Final power spectra
Rstar2.700M
Rstar3.445M

• Dominating double spin frequency
• Significant occurrence of odd harmonics if
  RstarltRc
• No odd harmonics and only small contributions of
  even harmonics for RstargtgtRc

Rstar15.000M
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Visualisation of the rotating star with hot spots

• Star with RstarltRc
• 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).