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GYOTO: a new general relativistic ray-tracing code

Frederic H. Vincent, Thibaut Paumard, Eric Gourgoulhon, Guy Perrin

arXiv:1109.4769v1gr-qcastro-ph.HE

TL;DR

Existing ray-tracing tools address relativistic images and spectra, but GYOTO is presented as a freely available and extensible code supporting broader metrics and objects. It integrates null and timelike geodesics, including in numerically computed 3+1 metrics, and demonstrates image and spectrum calculations for targets such as moving stars and ion tori. The code’s main scope is broad relativistic ray tracing, with scattering omitted from its built-in radiative-transfer treatment and numerical-metric tracing more time-consuming than analytical-metric tracing.

  • Problem

    Ray-tracing research uses many specialized codes, but the paper identifies limited public availability and limited support for non-analytic numerical metrics as gaps.

  • Method

    GYOTO integrates null and timelike geodesics and radiative transfer for astronomical targets, with modular C++ plug-ins and support for numerically computed 3+1 metrics.

  • Results

    GYOTO computes simulated images and spectra for targets including a moving star and an ion torus, while supporting Kerr and numerical metrics.

  • Takeaways & Limitations

    GYOTO provides a public, user-oriented, modular foundation for extending relativistic ray tracing to custom metrics and astronomical objects.

  • Takeaways & Limitations

    Scattering is not included in GYOTO’s built-in radiative-transfer treatment, although it can be added through coefficients or a plug-in.

Abstract

from arXiv · show

GYOTO, a general relativistic ray-tracing code, is presented. It aims at computing images of astronomical bodies in the vicinity of compact objects, as well as trajectories of massive bodies in relativistic environments. This code is capable of integrating the null and timelike geodesic equations not only in the Kerr metric, but also in any metric computed numerically within the 3+1 formalism of general relativity. Simulated images and spectra have been computed for a variety of astronomical targets, such as a moving star or a toroidal accretion structure. The underlying code is open source and freely available. It is user-friendly, quickly handled and very modular so that extensions are easy to integrate. Custom analytical metrics and astronomical targets can be implemented in C++ plug-in extensions independent from the main code.

1. Introduction

GYOTO addresses the need for an accessible, extensible ray-tracing code by combining freely available source code with support for numerically computed metrics. It integrates geodesics and radiative transfer to model images and spectra of relativistic astronomical objects.

  • Motivation: Existing ray-tracing codes target spectra, accretion-structure images, stellar trajectories, polarization, and other relativistic phenomena.Earlier work also addressed stars orbiting Kerr black holes, transfer-function spectra, and Schwarzschild accretion-disk images.
  • Accessibility and extensibility: GYOTO is freely available and exposes its source code for users who need to operate or adapt the algorithm.The code is written in C++ and can be extended through independently compiled plug-ins.
  • Accessibility and extensibility: C++ object-oriented plug-ins let users add numerical metrics and astronomical structures independently of the main GYOTO code.The paper presents this modularity as a response to the limited public availability of existing ray-tracing codes.
  • Novelty: Its distinctive capability is integrating geodesics in non-analytic, numerically computed metrics, enabling studies beyond standard Kerr black holes.The paper identifies testing the no-hair theorem with non-analytic metrics as a study made possible by this capability.
  • Core operation: GYOTO integrates null geodesics backward from an observer’s screen and then solves radiative transfer inside the emitting object.The resulting specific intensity reaching the observer is determined along each computed geodesic.

2.1. Integration of geodesics

GYOTO formulates geodesic integration in Kerr spacetime using conserved quantities and a Hamiltonian approach, with backward ray tracing from observer initial conditions. Adaptive RK4 integration, conservation corrections, and explicit stopping conditions support numerical accuracy while retaining higher-order images.

  • Geodesic formulation: Kerr geodesics are characterized by the four constants (µ, E, L, Q), making the problem completely integrable.E and L arise from spacetime symmetries, while Q is the Carter constant.
  • Geodesic formulation: GYOTO uses a Hamiltonian formulation with position and momentum variables to express the geodesic equations of motion.The formulation applies to timelike and null geodesics, with evolution parameterized by proper time or an affine parameter.
  • Ray-tracing procedure: Null-geodesic integration starts from the observer’s position and photon incidence direction, then proceeds backward toward the astrophysical object.Backward integration avoids computing many rays that would miss a target with a large angular size on the observer’s sky.
  • Numerical integration: Adaptive fourth-order Runge-Kutta integration continues until the photon reaches the object, escapes too far, or approaches too closely to the event horizon.The horizon threshold must remain sufficiently close to preserve photons that swirl near the black hole and form higher-order images.
  • Numerical integration: GYOTO continuously checks conserved quantities and modifies the numerical solution to enforce them during integration.The Hamiltonian formulation avoids sign indeterminations associated with square-root forms, at the cost of these solution adjustments.
  • Numerical integration: The null-geodesic constraint is used to adjust the radial derivative, while angular and radial modifications are limited to 1% of their initial values.The sign choices for the corrected derivatives follow the corresponding RK4 output.
  • Accuracy: A convergence test is provided in the appendix to investigate GYOTO’s numerical accuracy.

2.2. Radiative transfer and spectra computation

GYOTO computes observed specific intensities and spectra by integrating radiative transfer along geodesics through emitting objects. Its screen model converts pixel intensities into observed fluxes, with examples for optically thick and thin sources.

  • Radiative transfer: Radiative transfer is integrated along the geodesic segment inside an emitter using emission and absorption coefficients evaluated in an observer’s frame.The code computes the specific-intensity increment from these coefficients along the ray.
  • Radiative transfer: GYOTO uses invariant intensity and invariant emission and absorption coefficients to formulate relativistic radiative transfer across reference frames.The invariant quantities do not depend on the frame in which they are evaluated, and the equation is then expressed in the emitter’s comoving frame.
  • Optically thick spectra: For optically thick objects, the code integrates the transfer equation from a boundary where the specific intensity vanishes to the point of interest.The method is illustrated for a geometrically thin, optically thick accretion disk around a Schwarzschild black hole.
  • Examples: Figure 1 presents an emitted spectrum for a Schwarzschild black hole at 45° inclination, with disk radii 6 M to 10 M, power-law parameter p = −3, and only the first-order image.The figure caption identifies the modeled disk and image-order restriction.
  • Screen and flux: The screen represents a point-like spectro-imaging detector whose pixels sample sky directions and whose field of view and sampling define the observed flux calculation.Pixel flux uses the intensity reaching each pixel, its incidence angle, and the pixel’s solid-angle element.
  • Optically thin spectra: For optically thin objects, GYOTO computes specific-intensity maps over a chosen set of observed frequencies and derives the corresponding observed fluxes.The calculation uses the radiative-transfer and flux relations for each frequency in the selected panel.

2.3. Implemented astrophysical objects

GYOTO implements ray-tracing targets including moving stars, thin accretion disks, Rossby-wave-unstable disks, and ion tori, producing images and spectra with different radiative-transfer prescriptions.

  • Star: GYOTO computes images of moving stars orbiting Kerr black holes using a simple finite-radius model centered on the star’s timelike geodesic.The model neglects internal physics and tidal effects.
  • Star: Primary through quaternary stellar images correspond to geodesics that swirl around the black hole zero, one, two, or three times.The example uses a star orbiting the ISCO of a Kerr black hole with spin parameter a = 0.9 M.
  • Thin disks: GYOTO includes a geometrically thin infinite accretion disk with an inner radius at the ISCO and reproduces the Schwarzschild-disk image in agreement with prior results.The example uses an inclination angle of 70◦.
  • Thin disks: A thin disk with a density extremum can model Rossby wave instability, including density waves and nearby vortices, with its observed flux computed from surface density and radius.The density profile is computed using the VAC code.
  • Ion tori: GYOTO models geometrically thick, optically thin ion tori and computes images with trivial or synchrotron radiative transfer and spectra for different black-hole spins.The synchrotron spectrum is produced for the ion torus shown at the Galactic center.

3. Ray-tracing in a numerically computed metric

GYOTO supports ray-tracing in numerically computed spacetimes represented in the 3+1 formalism, extending applications beyond analytic Kerr black holes.

  • 3+1 metric representation: Numerical-relativity spacetimes supplied in 3+1 form allow GYOTO to ray-trace objects beyond Kerr black holes, including rotating and binary neutron stars and stellar core collapse.The 3+1 representation uses spacelike hypersurfaces and fields including the lapse, shift, spatial metric, and extrinsic curvature.
  • 3+1 metric representation: The spacetime metric is reconstructed from the lapse N, shift βi, and spatial metric γij using the 3+1 line element.The reconstruction is given by equation (27).
  • Integration techniques: GYOTO implements two geodesic-integration techniques when the metric is provided in 3+1 form.One technique reconstructs the four-metric before computing four-dimensional Christoffel symbols.

Four-dimensional method:

The four-dimensional method reconstructs the metric’s connection and integrates geodesics as curves parameterized by proper time or an affine parameter.

  • Four-dimensional method: The method integrates the standard four-dimensional geodesic equation after obtaining the Christoffel symbols of the reconstructed metric.The metric is expressed in coordinates xα before integration.
  • Four-dimensional method: The parameter λ is proper time for timelike geodesics and an affine parameter for null geodesics.The geodesic is represented as xα = Xα(λ).

Three-dimensional method:

GYOTO integrates geodesics in 3+1 numerical spacetimes by evolving coordinate-time equations using numerical metric fields and spectral representations. This enables trajectories in rotating-star metrics, while spectral evaluation provides high accuracy at increased computational cost.

  • Three-dimensional method:: The 3+1 geodesic system parametrizes the geodesic by coordinate time and uses the lapse, shift, spatial metric, extrinsic curvature, and their spatial derivatives.The auxiliary velocity V^i is the particle’s linear momentum divided by its energy as measured by the Eulerian observer.
  • Three-dimensional method:: GYOTO integrates the geodesic equations with a fourth-order Runge-Kutta algorithm, requiring 3+1 field values at arbitrary points.Spectral methods provide fields directly, whereas finite-difference spacetimes require interpolation of their data.
  • Three-dimensional method:: Spectral numerical metrics represent fields with Nr × Nθ × Nϕ coefficients across spherical-coordinate computational domains and polynomial bases.The basis combines Chebyshev polynomials with trigonometric and parity-adapted functions for the radial, angular, and azimuthal coordinates.
  • Three-dimensional method:: Spectral field evaluation has exponentially decreasing error with increasing degrees of freedom, while its arithmetic cost scales as Nr × Nθ × Nϕ.Around 20 degrees of freedom are generally sufficient to reach approximately 10^-14 relative accuracy in double precision, but numerical-metric ray tracing is more time-consuming than analytical-metric ray tracing.

4. GYOTO code in a nutshell

GYOTO provides command-line, interactive, graphical, and library interfaces for ray tracing, with modular plug-ins for adding metrics and astrophysical objects. Its current code is not itself parallel, although image computation can be split across jobs.

  • 4. GYOTO code in a nutshell: The gyoto command-line utility ray-traces a single frame from an XML scenery description and stores the output in FITS format.
  • 4. GYOTO code in a nutshell: The ygyoto Yorick extension supports interactive use and complex scripts, while the gyotoy graphical interface interactively traces timelike Kerr geodesics.The graphical interface is illustrated with an equatorial orbit in a quasi-extremal Kerr metric.
  • 4. GYOTO code in a nutshell: GYOTO’s core functionality is packaged in the libgyoto C++ shared library for access from other code.
  • 4. GYOTO code in a nutshell: A plug-in system lets users add metrics and astrophysical objects without modifying or recompiling GYOTO.The standard objects and metrics are themselves implemented as plug-ins, providing ready examples.
  • 4. GYOTO code in a nutshell: GYOTO is not yet parallel internally, but its image computation can be divided into regions and run through job-submission systems such as Torque.
  • 4. GYOTO code in a nutshell: The code is available online with documentation covering its architecture and software prerequisites, and it has been tested on Linux and Mac OS X.

5. Conclusion

The paper presents GYOTO as a public, user-oriented C++ ray-tracing code designed for extensibility. Its distinctive capability is handling numerical metrics from 3+1 numerical-relativity computations, while future work targets broader object and metric coverage and GPU parallelization.

  • 5. Conclusion: GYOTO is a public code designed for non-specialist users and integrates null and timelike geodesics in the Kerr metric.Its C++ object-oriented design supports development for specific user needs.
  • 5. Conclusion: GYOTO handles numerical metrics produced by 3+1 numerical-relativity computations, extending ray tracing beyond analytic metrics.The paper identifies this capability as a specificity of GYOTO compared with other ray-tracing codes.
  • 5. Conclusion: Future development will add astrophysical objects, diversify supported numerical metrics, and investigate GPU-based parallelization.

Appendix: Convergence test

The convergence test evaluates flux error against a 2000 × 2000-pixel reference screen. GYOTO reaches small errors rapidly, while a 1000 × 1000 image required about 15 minutes for the reported laptop computation.

  • Appendix: Convergence test: The convergence test compares fluxes computed with different screen resolutions against a reference flux from Npix = 2000 × 2000 pixels.
  • Appendix: Convergence test: The error is below 1 % at Npix = 100 × 100 and below 0.05 % at Npix = 1000 × 1000.The 1000 × 1000 resolution was used for the figures in the article.
  • Appendix: Convergence test: About 15 min were required to compute figure 3 at Npix = 1000 × 1000 on one core of a 2.4 GHz Intel Core 2 Duo laptop processor.The observer was located at r = 100 M from the black hole.
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